Subgroups — C₆₇×#61094 #30740345

Order 17152.

2738 conjugacy classes of subgroups.

The subgroup lattice has 930 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#577×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3496	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3283	C₂	1	2	yes	maximal, complemented
5	D₄×SD₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3276	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3272	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3267	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3266	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3152	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3063	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3035	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#3013	C₂	1	2	yes	maximal, complemented
15	C₆₇×#3009	C₂	1	2	yes	maximal, complemented
16	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
17	C₆₇×#2935	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	complemented
19	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	—
20	#577×C₆₇	—	2	16	no	—
21	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
22	C₂×#137×C₆₇	C₂²	1	2	yes	complemented
23	C₂×#137×C₆₇	C₂²	1	2	yes	complemented
24	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	complemented
26	C₂×#135×C₆₇	C₂²	1	4	yes	complemented
27	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
28	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
29	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
30	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
31	D₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
32	D₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
33	#547×C₆₇	—	2	1	no	—
34	#547×C₆₇	—	2	2	no	—
35	#547×C₆₇	—	2	2	no	—
36	#547×C₆₇	C₂²	1	2	yes	—
37	#547×C₆₇	C₂²	1	2	yes	complemented
38	#547×C₆₇	C₂²	1	2	yes	complemented
39	#546×C₆₇	—	2	2	no	—
40	#546×C₆₇	C₂²	1	4	yes	—
41	D₄²×C₆₇	—	2	1	no	—
42	D₄²×C₆₇	C₂²	1	2	yes	—
43	#537×C₆₇	C₂²	1	2	yes	—
44	#536×C₆₇	—	2	2	no	—
45	#536×C₆₇	C₂²	1	2	yes	complemented
46	#534×C₆₇	—	2	2	no	—
47	#532×C₆₇	—	2	1	no	—
48	#532×C₆₇	C₂²	1	1	yes	characteristic
49	#532×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
51	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
52	C₂×#121×C₆₇	—	2	2	no	—
53	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic, complemented
54	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
55	#497×C₆₇	C₂²	1	4	yes	complemented
56	#494×C₆₇	C₂²	1	2	yes	complemented
57	#492×C₆₇	C₂²	1	2	yes	complemented
58	#470×C₆₇	C₂²	1	4	yes	complemented
59	#468×C₆₇	C₂²	1	4	yes	complemented
60	#467×C₆₇	C₂²	1	4	yes	complemented
61	#465×C₆₇	C₂²	1	4	yes	complemented
62	#464×C₆₇	C₂²	1	2	yes	complemented
63	#463×C₆₇	C₂²	1	2	yes	complemented
64	#462×C₆₇	C₂²	1	1	yes	characteristic, complemented
65	#461×C₆₇	C₂²	1	4	yes	complemented
66	#461×C₆₇	C₂²	1	2	yes	complemented
67	#461×C₆₇	C₂²	1	2	yes	complemented
68	#461×C₆₇	C₂²	1	1	yes	characteristic, complemented
69	#460×C₆₇	C₂²	1	1	yes	characteristic, complemented
70	#459×C₆₇	C₂²	1	1	yes	characteristic, complemented
71	#452×C₆₇	C₂²	1	4	yes	complemented
72	#452×C₆₇	C₂²	1	4	yes	complemented
73	#452×C₆₇	C₂²	1	4	yes	complemented
74	#451×C₆₇	C₂²	1	4	yes	complemented
75	#450×C₆₇	C₂²	1	4	yes	complemented
76	#450×C₆₇	C₂²	1	4	yes	complemented
77	#449×C₆₇	C₂²	1	4	yes	complemented
78	#448×C₆₇	C₂²	1	4	yes	complemented
79	#442×C₆₇	C₂²	1	2	yes	complemented
80	#440×C₆₇	C₂²	1	2	yes	complemented
81	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
82	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
83	#436×C₆₇	C₂²	1	2	yes	complemented
84	#435×C₆₇	C₂²	1	4	yes	complemented
85	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
86	#428×C₆₇	C₂²	1	1	yes	characteristic, complemented
87	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
88	#418×C₆₇	C₂²	1	1	yes	characteristic
89	#417×C₆₇	C₂²	1	1	yes	characteristic, complemented
90	#416×C₆₇	C₂²	1	2	yes	complemented
91	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
92	#408×C₆₇	C₂²	1	2	yes	complemented
93	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
94	2₋¹⁺⁴×C₆₇	—	2	8	no	—
95	2₋¹⁺⁴×C₆₇	—	4	4	no	—
96	H(5,𝔽₂)×C₆₇	—	2	8	no	—
97	H(5,𝔽₂)×C₆₇	—	4	4	no	—
98	C₂×#41×C₆₇	—	2	2	no	—
99	C₂×#41×C₆₇	—	2	2	no	—
100	C₂×#41×C₆₇	—	2	4	no	—
101	C₂×#41×C₆₇	C₂³	1	2	yes	—
102	C₂×#41×C₆₇	C₂³	1	4	yes	—
103	C₂×#41×C₆₇	C₂³	1	4	yes	—
104	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×#41×C₆₇	C₂³	1	2	yes	—
106	C₂²×Q₈×C₆₇	—	2	2	no	—
107	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
108	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
109	C₂²×D₄×C₆₇	—	2	4	no	—
110	C₂²×D₄×C₆₇	—	2	2	no	—
111	C₂²×D₄×C₆₇	—	2	2	no	—
112	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
113	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
114	C₂³×C₄×C₆₇	—	2	1	no	—
115	#137×C₆₇	—	2	16	no	—
116	#137×C₆₇	—	2	8	no	—
117	#137×C₆₇	—	4	4	no	—
118	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
119	AGL(1,ℤ/8)×C₆₇	D₄	1	8	yes	complemented
120	#135×C₆₇	—	2	8	no	—
121	#135×C₆₇	—	2	16	no	—
122	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
123	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
124	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
125	C₂×SD₁₆×C₆₇	—	2	8	no	—
126	C₂×SD₁₆×C₆₇	—	2	16	no	—
127	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
128	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
129	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
130	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
131	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
132	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
133	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
134	C₂×D₈×C₆₇	C₂³	1	2	yes	—
135	C₂×D₈×C₆₇	C₂³	1	2	yes	—
136	#131×C₆₇	—	2	8	no	—
137	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
138	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
140	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
141	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
142	C₄⋊Q₈×C₆₇	—	2	2	no	—
143	C₄⋊Q₈×C₆₇	—	2	2	no	—
144	C₄⋊Q₈×C₆₇	—	2	2	no	—
145	C₄⋊Q₈×C₆₇	—	2	1	no	—
146	C₄⋊Q₈×C₆₇	—	2	1	no	—
147	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
148	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
149	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
150	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
151	#127×C₆₇	D₄	1	1	yes	characteristic
152	#127×C₆₇	C₂³	1	1	yes	characteristic
153	#124×C₆₇	—	2	2	no	—
154	#124×C₆₇	D₄	1	2	yes	—
155	#124×C₆₇	C₂³	1	2	yes	—
156	#123×C₆₇	—	2	4	no	—
157	#123×C₆₇	—	2	4	no	—
158	#123×C₆₇	—	2	4	no	—
159	#123×C₆₇	—	2	4	no	—
160	#123×C₆₇	—	2	4	no	—
161	C₂²⋊Q₈×C₆₇	—	2	2	no	—
162	C₂²⋊Q₈×C₆₇	—	2	4	no	—
163	C₂²⋊Q₈×C₆₇	—	2	4	no	—
164	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
165	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
166	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
167	#121×C₆₇	—	2	4	no	—
168	#121×C₆₇	—	2	2	no	—
169	#121×C₆₇	—	2	2	no	—
170	#121×C₆₇	—	2	2	no	—
171	#121×C₆₇	—	2	2	no	—
172	#121×C₆₇	—	2	1	no	—
173	#121×C₆₇	—	2	1	no	—
174	#121×C₆₇	—	2	2	no	—
175	#121×C₆₇	—	2	4	no	—
176	#121×C₆₇	D₄	1	2	yes	—
177	#121×C₆₇	D₄	1	2	yes	—
178	#121×C₆₇	C₂³	1	4	yes	—
179	#120×C₆₇	—	2	4	no	—
180	#120×C₆₇	—	2	4	no	—
181	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
182	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
183	C₄×D₄×C₆₇	—	2	2	no	—
184	C₄×D₄×C₆₇	—	2	2	no	—
185	C₄×D₄×C₆₇	—	2	1	no	—
186	C₄×D₄×C₆₇	—	2	1	no	—
187	C₄×D₄×C₆₇	—	2	1	no	—
188	C₄×D₄×C₆₇	—	2	1	no	—
189	C₄×D₄×C₆₇	—	2	1	no	—
190	C₄×D₄×C₆₇	—	2	1	no	—
191	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
192	C₄×D₄×C₆₇	C₂³	1	2	yes	—
193	C₄×D₄×C₆₇	C₂³	1	2	yes	—
194	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
195	C₄×D₄×C₆₇	C₂³	1	2	yes	—
196	C₄×D₄×C₆₇	C₂³	1	2	yes	—
197	#117×C₆₇	—	2	1	no	—
198	#117×C₆₇	C₂³	1	1	yes	characteristic
199	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
200	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
201	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
202	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
203	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
204	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
205	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
206	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
207	C₂×C₄²×C₆₇	—	2	1	no	—
208	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
209	#107×C₆₇	C₂³	1	2	yes	—
210	#106×C₆₇	C₂³	1	2	yes	—
211	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
212	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
213	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
214	#103×C₆₇	C₂³	1	2	yes	—
215	#103×C₆₇	C₂³	1	4	yes	—
216	#103×C₆₇	C₂³	1	4	yes	—
217	#103×C₆₇	C₂³	1	2	yes	—
218	#103×C₆₇	C₂³	1	2	yes	—
219	#102×C₆₇	C₂³	1	2	yes	—
220	#102×C₆₇	C₂³	1	2	yes	—
221	#102×C₆₇	C₂³	1	2	yes	—
222	#102×C₆₇	C₂³	1	2	yes	—
223	#102×C₆₇	C₂³	1	2	yes	—
224	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
225	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
226	#97×C₆₇	C₂³	1	2	yes	—
227	C₄×C₈×C₆₇	C₂³	1	2	yes	—
228	C₂⁴×C₆₇	—	2	1	no	—
229	C₂⁴×C₆₇	—	2	4	no	—
230	#41×C₆₇	—	2	4	no	—
231	#41×C₆₇	—	2	4	no	—
232	#41×C₆₇	—	2	2	no	—
233	#41×C₆₇	—	2	8	no	—
234	#41×C₆₇	—	2	8	no	—
235	#41×C₆₇	—	2	4	no	—
236	#41×C₆₇	—	2	4	no	—
237	#41×C₆₇	—	4	4	no	—
238	#41×C₆₇	—	4	4	no	—
239	#41×C₆₇	—	4	8	no	—
240	#41×C₆₇	—	4	16	no	—
241	#41×C₆₇	—	4	8	no	—
242	#41×C₆₇	C₂×D₄	1	4	yes	complemented
243	C₂×Q₈×C₆₇	—	2	4	no	—
244	C₂×Q₈×C₆₇	—	2	2	no	—
245	C₂×Q₈×C₆₇	—	2	8	no	—
246	C₂×Q₈×C₆₇	—	2	4	no	—
247	C₂×Q₈×C₆₇	—	2	4	no	—
248	C₂×Q₈×C₆₇	—	4	4	no	—
249	C₂×Q₈×C₆₇	—	4	8	no	—
250	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
251	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
252	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
253	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
254	C₂×D₄×C₆₇	—	2	1	no	—
255	C₂×D₄×C₆₇	—	2	8	no	—
256	C₂×D₄×C₆₇	—	2	4	no	—
257	C₂×D₄×C₆₇	—	2	2	no	—
258	C₂×D₄×C₆₇	—	2	4	no	—
259	C₂×D₄×C₆₇	—	2	4	no	—
260	C₂×D₄×C₆₇	—	2	1	no	—
261	C₂×D₄×C₆₇	—	2	1	no	—
262	C₂×D₄×C₆₇	—	2	4	no	—
263	C₂×D₄×C₆₇	—	2	1	no	—
264	C₂×D₄×C₆₇	—	2	2	no	—
265	C₂×D₄×C₆₇	—	4	4	no	—
266	C₂×D₄×C₆₇	—	4	8	no	—
267	C₂×D₄×C₆₇	—	4	16	no	—
268	C₂×D₄×C₆₇	—	8	4	no	—
269	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
270	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
271	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
272	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
273	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
274	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
275	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
276	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
277	C₂²×C₄×C₆₇	—	2	1	no	—
278	C₂²×C₄×C₆₇	—	2	2	no	—
279	C₂²×C₄×C₆₇	—	2	4	no	—
280	C₂²×C₄×C₆₇	—	2	1	no	—
281	C₂²×C₄×C₆₇	—	2	1	no	—
282	C₂²×C₄×C₆₇	—	2	1	no	—
283	C₂²×C₄×C₆₇	—	2	4	no	—
284	C₂²×C₄×C₆₇	—	2	4	no	—
285	C₂²×C₄×C₆₇	—	2	2	no	—
286	C₂²×C₄×C₆₇	—	2	2	no	—
287	C₂²×C₄×C₆₇	—	2	2	no	—
288	C₂²×C₄×C₆₇	—	8	2	no	—
289	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
290	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
291	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
292	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
293	Q₁₆×C₆₇	—	2	4	no	—
294	Q₁₆×C₆₇	—	2	8	no	—
295	Q₁₆×C₆₇	—	4	4	no	—
296	SD₁₆×C₆₇	—	2	4	no	—
297	SD₁₆×C₆₇	—	2	8	no	—
298	SD₁₆×C₆₇	—	2	8	no	—
299	SD₁₆×C₆₇	—	2	4	no	—
300	SD₁₆×C₆₇	—	2	8	no	—
301	SD₁₆×C₆₇	—	4	4	no	—
302	SD₁₆×C₆₇	C₂×D₄	1	8	yes	complemented
303	D₈×C₆₇	—	2	4	no	—
304	D₈×C₆₇	C₂×D₄	1	8	yes	complemented
305	M₁₆×C₆₇	—	2	2	no	—
306	M₁₆×C₆₇	—	2	8	no	—
307	M₁₆×C₆₇	C₂×D₄	1	4	yes	complemented
308	C₂×C₈×C₆₇	—	2	4	no	—
309	C₂×C₈×C₆₇	—	2	8	no	—
310	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
311	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
312	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
313	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
314	C₄⋊C₄×C₆₇	—	2	1	no	—
315	C₄⋊C₄×C₆₇	—	2	2	no	—
316	C₄⋊C₄×C₆₇	—	2	2	no	—
317	C₄⋊C₄×C₆₇	—	2	4	no	—
318	C₄⋊C₄×C₆₇	—	2	2	no	—
319	C₄⋊C₄×C₆₇	—	2	2	no	—
320	C₄⋊C₄×C₆₇	—	2	4	no	—
321	C₄⋊C₄×C₆₇	—	2	4	no	—
322	C₄⋊C₄×C₆₇	—	2	1	no	—
323	C₄⋊C₄×C₆₇	—	2	1	no	—
324	C₄⋊C₄×C₆₇	—	2	1	no	—
325	C₄⋊C₄×C₆₇	—	2	1	no	—
326	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
327	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
328	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
329	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
330	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
331	C₂²⋊C₄×C₆₇	—	2	4	no	—
332	C₂²⋊C₄×C₆₇	—	2	1	no	—
333	C₂²⋊C₄×C₆₇	—	2	4	no	—
334	C₂²⋊C₄×C₆₇	—	2	4	no	—
335	C₂²⋊C₄×C₆₇	—	2	2	no	—
336	C₂²⋊C₄×C₆₇	—	2	2	no	—
337	C₂²⋊C₄×C₆₇	—	2	2	no	—
338	C₂²⋊C₄×C₆₇	—	2	4	no	—
339	C₂²⋊C₄×C₆₇	—	2	1	no	—
340	C₂²⋊C₄×C₆₇	—	2	1	no	—
341	C₂²⋊C₄×C₆₇	—	2	2	no	—
342	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
343	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
344	C₄²×C₆₇	—	2	1	no	—
345	C₄²×C₆₇	—	2	1	no	—
346	C₄²×C₆₇	—	2	1	no	—
347	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
348	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
349	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
350	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
351	C₂³×C₆₇	—	2	1	no	—
352	C₂³×C₆₇	—	2	2	no	—
353	C₂³×C₆₇	—	2	4	no	—
354	C₂³×C₆₇	—	2	2	no	—
355	C₂³×C₆₇	—	4	8	no	—
356	C₂³×C₆₇	—	8	1	no	—
357	C₂³×C₆₇	—	8	1	no	—
358	C₂³×C₆₇	—	8	4	no	—
359	C₂³×C₆₇	—	8	4	no	—
360	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
361	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
362	Q₈×C₆₇	—	2	4	no	—
363	Q₈×C₆₇	—	2	1	no	—
364	Q₈×C₆₇	—	2	4	no	—
365	Q₈×C₆₇	—	2	2	no	—
366	Q₈×C₆₇	—	2	4	no	—
367	Q₈×C₆₇	—	2	2	no	—
368	Q₈×C₆₇	—	4	8	no	—
369	Q₈×C₆₇	—	4	4	no	—
370	Q₈×C₆₇	C₂²×D₄	1	2	yes	—
371	D₄×C₆₇	—	2	4	no	—
372	D₄×C₆₇	—	2	8	no	—
373	D₄×C₆₇	—	2	4	no	—
374	D₄×C₆₇	—	2	4	no	—
375	D₄×C₆₇	—	2	4	no	—
376	D₄×C₆₇	—	2	2	no	—
377	D₄×C₆₇	—	2	2	no	—
378	D₄×C₆₇	—	2	1	no	—
379	D₄×C₆₇	—	2	2	no	—
380	D₄×C₆₇	—	4	2	no	—
381	D₄×C₆₇	—	4	8	no	—
382	D₄×C₆₇	—	4	8	no	—
383	D₄×C₆₇	—	4	8	no	—
384	D₄×C₆₇	—	4	4	no	—
385	D₄×C₆₇	—	4	2	no	—
386	D₄×C₆₇	—	8	1	no	—
387	D₄×C₆₇	—	8	4	no	—
388	D₄×C₆₇	C₂²×D₄	1	2	yes	—
389	D₄×C₆₇	C₂²×D₄	1	4	yes	—
390	C₂×C₄×C₆₇	—	2	1	no	—
391	C₂×C₄×C₆₇	—	2	4	no	—
392	C₂×C₄×C₆₇	—	2	2	no	—
393	C₂×C₄×C₆₇	—	2	4	no	—
394	C₂×C₄×C₆₇	—	2	4	no	—
395	C₂×C₄×C₆₇	—	2	1	no	—
396	C₂×C₄×C₆₇	—	2	1	no	—
397	C₂×C₄×C₆₇	—	2	2	no	—
398	C₂×C₄×C₆₇	—	2	2	no	—
399	C₂×C₄×C₆₇	—	2	4	no	—
400	C₂×C₄×C₆₇	—	4	1	no	—
401	C₂×C₄×C₆₇	—	4	8	no	—
402	C₂×C₄×C₆₇	—	4	8	no	—
403	C₂×C₄×C₆₇	—	4	8	no	—
404	C₂×C₄×C₆₇	—	4	2	no	—
405	C₂×C₄×C₆₇	—	4	4	no	—
406	C₂×C₄×C₆₇	—	4	4	no	—
407	C₂×C₄×C₆₇	—	4	2	no	—
408	C₂×C₄×C₆₇	—	4	4	no	—
409	C₂×C₄×C₆₇	—	8	2	no	—
410	C₂×C₄×C₆₇	—	8	1	no	—
411	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
412	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
413	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
414	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
415	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
416	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
417	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
418	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
419	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
420	C₈×C₆₇	—	2	2	no	—
421	C₈×C₆₇	—	2	2	no	—
422	C₈×C₆₇	—	2	2	no	—
423	C₈×C₆₇	C₂²×D₄	1	4	yes	—
424	C₂²×C₆₇	—	2	1	no	—
425	C₂²×C₆₇	—	2	1	no	—
426	C₂²×C₆₇	—	2	4	no	—
427	C₂²×C₆₇	—	2	2	no	—
428	C₂²×C₆₇	—	2	4	no	—
429	C₂²×C₆₇	—	2	4	no	—
430	C₂²×C₆₇	—	2	4	no	—
431	C₂²×C₆₇	—	4	1	no	—
432	C₂²×C₆₇	—	4	2	no	—
433	C₂²×C₆₇	—	4	2	no	—
434	C₂²×C₆₇	—	4	1	no	—
435	C₂²×C₆₇	—	4	2	no	—
436	C₂²×C₆₇	—	4	2	no	—
437	C₂²×C₆₇	—	4	2	no	—
438	C₂²×C₆₇	—	8	4	no	—
439	C₂²×C₆₇	—	16	8	no	—
440	C₂²×C₆₇	D₄²	1	2	yes	—
441	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
442	C₄×C₆₇	—	2	1	no	—
443	C₄×C₆₇	—	2	1	no	—
444	C₄×C₆₇	—	2	2	no	—
445	C₄×C₆₇	—	2	4	no	—
446	C₄×C₆₇	—	2	1	no	—
447	C₄×C₆₇	—	4	1	no	—
448	C₄×C₆₇	—	4	2	no	—
449	C₄×C₆₇	—	4	4	no	—
450	C₄×C₆₇	—	4	1	no	—
451	C₄×C₆₇	—	4	2	no	—
452	C₄×C₆₇	D₄²	1	2	yes	—
453	C₄×C₆₇	#577	1	2	yes	complemented
454	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
455	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
456	#61094	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
457	C₂×C₆₇	—	2	2	no	—
458	C₂×C₆₇	—	4	4	no	—
459	C₂×C₆₇	—	4	2	no	—
460	C₂×C₆₇	—	4	4	no	—
461	C₂×C₆₇	—	8	2	no	—
462	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
463	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
464	C₂×C₆₇	#3586	1	1	yes	characteristic
465	C₂×#577	C₂×C₆₇	1	2	yes	complemented
466	#3499	C₂×C₆₇	1	1	yes	characteristic, complemented
467	#3496	C₂×C₆₇	1	1	yes	characteristic, complemented
468	#3283	C₂×C₆₇	1	2	yes	complemented
469	D₄×SD₁₆	C₂×C₆₇	1	2	yes	complemented
470	#3276	C₂×C₆₇	1	2	yes	complemented
471	#3272	C₂×C₆₇	1	2	yes	complemented
472	#3267	C₂×C₆₇	1	4	yes	complemented
473	#3266	C₂×C₆₇	1	4	yes	complemented
474	#3152	C₂×C₆₇	1	1	yes	characteristic, complemented
475	#3063	C₂×C₆₇	1	2	yes	complemented
476	#3035	C₂×C₆₇	1	1	yes	characteristic, complemented
477	C₂×#461	C₂×C₆₇	1	1	yes	characteristic, complemented
478	#3013	C₂×C₆₇	1	2	yes	complemented
479	#3009	C₂×C₆₇	1	2	yes	complemented
480	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic, complemented
481	#2935	C₂×C₆₇	1	1	yes	characteristic, complemented
482	C₆₇	#61094	1	1	yes	characteristic, direct product factor, Sylow
483	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	complemented
484	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	—
485	#577	—	2	16	no	—
486	C₂×#137	C₂²×C₆₇	1	2	yes	complemented
487	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
488	C₂×#137	C₂²×C₆₇	1	2	yes	complemented
489	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
490	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
491	C₂×#135	C₂²×C₆₇	1	4	yes	complemented
492	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
493	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
494	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
495	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
496	D₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
497	D₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
498	#547	—	2	1	no	—
499	#547	—	2	2	no	—
500	#547	—	2	2	no	—
501	#547	C₂²×C₆₇	1	2	yes	—
502	#547	C₂²×C₆₇	1	2	yes	complemented
503	#547	C₂²×C₆₇	1	2	yes	complemented
504	#546	—	2	2	no	—
505	#546	C₂²×C₆₇	1	4	yes	—
506	D₄²	—	2	1	no	—
507	D₄²	C₂²×C₆₇	1	2	yes	—
508	#537	C₂²×C₆₇	1	2	yes	—
509	#536	—	2	2	no	—
510	#536	C₂²×C₆₇	1	2	yes	complemented
511	#534	—	2	2	no	—
512	#532	—	2	1	no	—
513	#532	C₂²×C₆₇	1	1	yes	characteristic
514	#532	C₂²×C₆₇	1	1	yes	characteristic
515	C₂×C₄⋊Q₈	—	2	1	no	—
516	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
517	C₂×#121	—	2	2	no	—
518	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic, complemented
519	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
520	#497	C₂²×C₆₇	1	4	yes	complemented
521	#494	C₂²×C₆₇	1	2	yes	complemented
522	#492	C₂²×C₆₇	1	2	yes	complemented
523	#470	C₂²×C₆₇	1	4	yes	complemented
524	#468	C₂²×C₆₇	1	4	yes	complemented
525	#467	C₂²×C₆₇	1	4	yes	complemented
526	#465	C₂²×C₆₇	1	4	yes	complemented
527	#464	C₂²×C₆₇	1	2	yes	complemented
528	#463	C₂²×C₆₇	1	2	yes	complemented
529	#462	C₂²×C₆₇	1	1	yes	characteristic, complemented
530	#461	C₂²×C₆₇	1	4	yes	complemented
531	#461	C₂²×C₆₇	1	2	yes	complemented
532	#461	C₂²×C₆₇	1	2	yes	complemented
533	#461	C₂²×C₆₇	1	1	yes	characteristic, complemented
534	#460	C₂²×C₆₇	1	1	yes	characteristic, complemented
535	#459	C₂²×C₆₇	1	1	yes	characteristic, complemented
536	#452	C₂²×C₆₇	1	4	yes	complemented
537	#452	C₂²×C₆₇	1	4	yes	complemented
538	#452	C₂²×C₆₇	1	4	yes	complemented
539	#451	C₂²×C₆₇	1	4	yes	complemented
540	#450	C₂²×C₆₇	1	4	yes	complemented
541	#450	C₂²×C₆₇	1	4	yes	complemented
542	#449	C₂²×C₆₇	1	4	yes	complemented
543	#448	C₂²×C₆₇	1	4	yes	complemented
544	#442	C₂²×C₆₇	1	2	yes	complemented
545	#440	C₂²×C₆₇	1	2	yes	complemented
546	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
547	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
548	#436	C₂²×C₆₇	1	2	yes	complemented
549	#435	C₂²×C₆₇	1	4	yes	complemented
550	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
551	#428	C₂²×C₆₇	1	1	yes	characteristic, complemented
552	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
553	#418	C₂²×C₆₇	1	1	yes	characteristic
554	#417	C₂²×C₆₇	1	1	yes	characteristic, complemented
555	#416	C₂²×C₆₇	1	2	yes	complemented
556	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
557	#408	C₂²×C₆₇	1	2	yes	complemented
558	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
559	2₋¹⁺⁴	—	2	8	no	—
560	2₋¹⁺⁴	—	4	4	no	—
561	H(5,𝔽₂)	—	2	8	no	—
562	H(5,𝔽₂)	—	4	4	no	—
563	C₂×#41	—	2	2	no	—
564	C₂×#41	—	2	2	no	—
565	C₂×#41	—	2	4	no	—
566	C₂×#41	C₂³×C₆₇	1	2	yes	—
567	C₂×#41	C₂³×C₆₇	1	4	yes	—
568	C₂×#41	C₂³×C₆₇	1	4	yes	—
569	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
570	C₂×#41	C₂³×C₆₇	1	2	yes	—
571	C₂²×Q₈	—	2	2	no	—
572	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
573	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
574	C₂²×D₄	—	2	4	no	—
575	C₂²×D₄	—	2	2	no	—
576	C₂²×D₄	—	2	2	no	—
577	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
578	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
579	C₂³×C₄	—	2	1	no	—
580	#137	—	2	16	no	—
581	#137	—	2	8	no	—
582	#137	—	4	4	no	—
583	AGL(1,ℤ/8)	—	2	16	no	—
584	AGL(1,ℤ/8)	D₄×C₆₇	1	8	yes	complemented
585	#135	—	2	8	no	—
586	#135	—	2	16	no	—
587	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
588	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
589	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
590	C₂×SD₁₆	—	2	8	no	—
591	C₂×SD₁₆	—	2	16	no	—
592	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
593	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
594	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
595	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
596	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
597	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
598	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
599	C₂×D₈	C₂³×C₆₇	1	2	yes	—
600	C₂×D₈	C₂³×C₆₇	1	2	yes	—
601	#131	—	2	8	no	—
602	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
603	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
604	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
605	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
606	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
607	C₄⋊Q₈	—	2	2	no	—
608	C₄⋊Q₈	—	2	2	no	—
609	C₄⋊Q₈	—	2	2	no	—
610	C₄⋊Q₈	—	2	1	no	—
611	C₄⋊Q₈	—	2	1	no	—
612	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
613	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
614	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
615	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
616	#127	D₄×C₆₇	1	1	yes	characteristic
617	#127	C₂³×C₆₇	1	1	yes	characteristic
618	#124	—	2	2	no	—
619	#124	D₄×C₆₇	1	2	yes	—
620	#124	C₂³×C₆₇	1	2	yes	—
621	#123	—	2	4	no	—
622	#123	—	2	4	no	—
623	#123	—	2	4	no	—
624	#123	—	2	4	no	—
625	#123	—	2	4	no	—
626	C₂²⋊Q₈	—	2	2	no	—
627	C₂²⋊Q₈	—	2	4	no	—
628	C₂²⋊Q₈	—	2	4	no	—
629	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
630	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
631	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
632	#121	—	2	4	no	—
633	#121	—	2	2	no	—
634	#121	—	2	2	no	—
635	#121	—	2	2	no	—
636	#121	—	2	2	no	—
637	#121	—	2	1	no	—
638	#121	—	2	1	no	—
639	#121	—	2	2	no	—
640	#121	—	2	4	no	—
641	#121	D₄×C₆₇	1	2	yes	—
642	#121	D₄×C₆₇	1	2	yes	—
643	#121	C₂³×C₆₇	1	4	yes	—
644	#120	—	2	4	no	—
645	#120	—	2	4	no	—
646	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
647	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
648	C₄×D₄	—	2	2	no	—
649	C₄×D₄	—	2	2	no	—
650	C₄×D₄	—	2	1	no	—
651	C₄×D₄	—	2	1	no	—
652	C₄×D₄	—	2	1	no	—
653	C₄×D₄	—	2	1	no	—
654	C₄×D₄	—	2	1	no	—
655	C₄×D₄	—	2	1	no	—
656	C₄×D₄	C₂³×C₆₇	1	2	yes	—
657	C₄×D₄	C₂³×C₆₇	1	2	yes	—
658	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
659	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
660	C₄×D₄	C₂³×C₆₇	1	2	yes	—
661	C₄×D₄	C₂³×C₆₇	1	2	yes	—
662	#117	—	2	1	no	—
663	#117	C₂³×C₆₇	1	1	yes	characteristic
664	C₂×C₄⋊C₄	—	2	1	no	—
665	C₂×C₄⋊C₄	—	2	1	no	—
666	C₂×C₄⋊C₄	—	2	2	no	—
667	C₂×C₄⋊C₄	—	2	2	no	—
668	C₂×C₄⋊C₄	—	2	2	no	—
669	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
670	C₂×C₂²⋊C₄	—	2	1	no	—
671	C₂×C₂²⋊C₄	—	2	4	no	—
672	C₂×C₄²	—	2	1	no	—
673	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
674	#107	C₂³×C₆₇	1	2	yes	—
675	#106	C₂³×C₆₇	1	2	yes	—
676	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
677	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
678	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
679	#103	C₂³×C₆₇	1	4	yes	—
680	#103	C₂³×C₆₇	1	4	yes	—
681	#103	C₂³×C₆₇	1	2	yes	—
682	#103	C₂³×C₆₇	1	2	yes	—
683	#103	C₂³×C₆₇	1	2	yes	—
684	#102	C₂³×C₆₇	1	2	yes	—
685	#102	C₂³×C₆₇	1	2	yes	—
686	#102	C₂³×C₆₇	1	2	yes	—
687	#102	C₂³×C₆₇	1	2	yes	—
688	#102	C₂³×C₆₇	1	2	yes	—
689	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
690	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
691	#97	C₂³×C₆₇	1	2	yes	—
692	C₄×C₈	C₂³×C₆₇	1	2	yes	—
693	C₂⁴	—	2	1	no	—
694	C₂⁴	—	2	4	no	—
695	#41	—	2	4	no	—
696	#41	—	2	4	no	—
697	#41	—	2	2	no	—
698	#41	—	2	8	no	—
699	#41	—	2	8	no	—
700	#41	—	2	4	no	—
701	#41	—	2	4	no	—
702	#41	—	4	4	no	—
703	#41	—	4	4	no	—
704	#41	—	4	8	no	—
705	#41	—	4	16	no	—
706	#41	—	4	8	no	—
707	#41	C₂×D₄×C₆₇	1	4	yes	complemented
708	C₂×Q₈	—	2	4	no	—
709	C₂×Q₈	—	2	2	no	—
710	C₂×Q₈	—	2	8	no	—
711	C₂×Q₈	—	2	4	no	—
712	C₂×Q₈	—	2	4	no	—
713	C₂×Q₈	—	4	4	no	—
714	C₂×Q₈	—	4	8	no	—
715	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
716	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
717	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
718	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
719	C₂×D₄	—	2	1	no	—
720	C₂×D₄	—	2	8	no	—
721	C₂×D₄	—	2	4	no	—
722	C₂×D₄	—	2	2	no	—
723	C₂×D₄	—	2	4	no	—
724	C₂×D₄	—	2	4	no	—
725	C₂×D₄	—	2	1	no	—
726	C₂×D₄	—	2	1	no	—
727	C₂×D₄	—	2	4	no	—
728	C₂×D₄	—	2	1	no	—
729	C₂×D₄	—	2	2	no	—
730	C₂×D₄	—	4	4	no	—
731	C₂×D₄	—	4	8	no	—
732	C₂×D₄	—	4	16	no	—
733	C₂×D₄	—	8	4	no	—
734	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
735	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
736	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
737	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
738	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
739	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
740	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
741	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
742	C₂²×C₄	—	2	1	no	—
743	C₂²×C₄	—	2	2	no	—
744	C₂²×C₄	—	2	4	no	—
745	C₂²×C₄	—	2	1	no	—
746	C₂²×C₄	—	2	1	no	—
747	C₂²×C₄	—	2	1	no	—
748	C₂²×C₄	—	2	4	no	—
749	C₂²×C₄	—	2	4	no	—
750	C₂²×C₄	—	2	2	no	—
751	C₂²×C₄	—	2	2	no	—
752	C₂²×C₄	—	2	2	no	—
753	C₂²×C₄	—	8	2	no	—
754	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
755	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
756	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
757	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
758	Q₁₆	—	2	4	no	—
759	Q₁₆	—	2	8	no	—
760	Q₁₆	—	4	4	no	—
761	SD₁₆	—	2	4	no	—
762	SD₁₆	—	2	8	no	—
763	SD₁₆	—	2	8	no	—
764	SD₁₆	—	2	4	no	—
765	SD₁₆	—	2	8	no	—
766	SD₁₆	—	4	4	no	—
767	SD₁₆	C₂×D₄×C₆₇	1	8	yes	complemented
768	D₈	—	2	4	no	—
769	D₈	C₂×D₄×C₆₇	1	8	yes	complemented
770	M₁₆	—	2	2	no	—
771	M₁₆	—	2	8	no	—
772	M₁₆	C₂×D₄×C₆₇	1	4	yes	complemented
773	C₂×C₈	—	2	4	no	—
774	C₂×C₈	—	2	8	no	—
775	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
776	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
777	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
778	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
779	C₄⋊C₄	—	2	1	no	—
780	C₄⋊C₄	—	2	2	no	—
781	C₄⋊C₄	—	2	2	no	—
782	C₄⋊C₄	—	2	4	no	—
783	C₄⋊C₄	—	2	2	no	—
784	C₄⋊C₄	—	2	2	no	—
785	C₄⋊C₄	—	2	4	no	—
786	C₄⋊C₄	—	2	4	no	—
787	C₄⋊C₄	—	2	1	no	—
788	C₄⋊C₄	—	2	1	no	—
789	C₄⋊C₄	—	2	1	no	—
790	C₄⋊C₄	—	2	1	no	—
791	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
792	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
793	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
794	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
795	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
796	C₂²⋊C₄	—	2	4	no	—
797	C₂²⋊C₄	—	2	1	no	—
798	C₂²⋊C₄	—	2	4	no	—
799	C₂²⋊C₄	—	2	4	no	—
800	C₂²⋊C₄	—	2	2	no	—
801	C₂²⋊C₄	—	2	2	no	—
802	C₂²⋊C₄	—	2	2	no	—
803	C₂²⋊C₄	—	2	4	no	—
804	C₂²⋊C₄	—	2	1	no	—
805	C₂²⋊C₄	—	2	1	no	—
806	C₂²⋊C₄	—	2	2	no	—
807	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
808	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
809	C₄²	—	2	1	no	—
810	C₄²	—	2	1	no	—
811	C₄²	—	2	1	no	—
812	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
813	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
814	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
815	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
816	C₂³	—	2	1	no	—
817	C₂³	—	2	2	no	—
818	C₂³	—	2	4	no	—
819	C₂³	—	2	2	no	—
820	C₂³	—	4	8	no	—
821	C₂³	—	8	1	no	—
822	C₂³	—	8	1	no	—
823	C₂³	—	8	4	no	—
824	C₂³	—	8	4	no	—
825	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
826	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
827	Q₈	—	2	4	no	—
828	Q₈	—	2	1	no	—
829	Q₈	—	2	4	no	—
830	Q₈	—	2	2	no	—
831	Q₈	—	2	4	no	—
832	Q₈	—	2	2	no	—
833	Q₈	—	4	8	no	—
834	Q₈	—	4	4	no	—
835	Q₈	C₂²×D₄×C₆₇	1	2	yes	—
836	D₄	—	2	4	no	—
837	D₄	—	2	8	no	—
838	D₄	—	2	4	no	—
839	D₄	—	2	4	no	—
840	D₄	—	2	4	no	—
841	D₄	—	2	2	no	—
842	D₄	—	2	2	no	—
843	D₄	—	2	1	no	—
844	D₄	—	2	2	no	—
845	D₄	—	4	2	no	—
846	D₄	—	4	8	no	—
847	D₄	—	4	8	no	—
848	D₄	—	4	8	no	—
849	D₄	—	4	4	no	—
850	D₄	—	4	2	no	—
851	D₄	—	8	1	no	—
852	D₄	—	8	4	no	—
853	D₄	C₂²×D₄×C₆₇	1	2	yes	—
854	D₄	C₂²×D₄×C₆₇	1	4	yes	—
855	C₂×C₄	—	2	1	no	—
856	C₂×C₄	—	2	4	no	—
857	C₂×C₄	—	2	2	no	—
858	C₂×C₄	—	2	4	no	—
859	C₂×C₄	—	2	4	no	—
860	C₂×C₄	—	2	1	no	—
861	C₂×C₄	—	2	1	no	—
862	C₂×C₄	—	2	2	no	—
863	C₂×C₄	—	2	2	no	—
864	C₂×C₄	—	2	4	no	—
865	C₂×C₄	—	4	1	no	—
866	C₂×C₄	—	4	8	no	—
867	C₂×C₄	—	4	8	no	—
868	C₂×C₄	—	4	8	no	—
869	C₂×C₄	—	4	2	no	—
870	C₂×C₄	—	4	4	no	—
871	C₂×C₄	—	4	4	no	—
872	C₂×C₄	—	4	4	no	—
873	C₂×C₄	—	4	2	no	—
874	C₂×C₄	—	8	2	no	—
875	C₂×C₄	—	8	1	no	—
876	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
877	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
878	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
879	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
880	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
881	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
882	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
883	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
884	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
885	C₈	—	2	2	no	—
886	C₈	—	2	2	no	—
887	C₈	—	2	2	no	—
888	C₈	C₂²×D₄×C₆₇	1	4	yes	—
889	C₂²	—	2	1	no	—
890	C₂²	—	2	1	no	—
891	C₂²	—	2	4	no	—
892	C₂²	—	2	4	no	—
893	C₂²	—	2	2	no	—
894	C₂²	—	2	4	no	—
895	C₂²	—	2	4	no	—
896	C₂²	—	4	2	no	—
897	C₂²	—	4	1	no	—
898	C₂²	—	4	1	no	—
899	C₂²	—	4	2	no	—
900	C₂²	—	4	2	no	—
901	C₂²	—	4	2	no	—
902	C₂²	—	4	2	no	—
903	C₂²	—	8	4	no	—
904	C₂²	—	16	8	no	—
905	C₂²	D₄²×C₆₇	1	2	yes	—
906	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
907	C₄	—	2	1	no	—
908	C₄	—	2	1	no	—
909	C₄	—	2	1	no	—
910	C₄	—	2	2	no	—
911	C₄	—	2	4	no	—
912	C₄	—	4	2	no	—
913	C₄	—	4	1	no	—
914	C₄	—	4	1	no	—
915	C₄	—	4	2	no	—
916	C₄	—	4	4	no	—
917	C₄	D₄²×C₆₇	1	2	yes	—
918	C₄	#577×C₆₇	1	2	yes	complemented
919	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
920	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
921	C₂	—	2	2	no	—
922	C₂	—	4	4	no	—
923	C₂	—	4	4	no	—
924	C₂	—	4	2	no	—
925	C₂	—	8	2	no	—
926	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
927	C₂	C₂×#577×C₆₇	1	1	yes	characteristic
928	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #532×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#577×C₆₇	C₂	semidirect
2. C₆₇×#3499	C₂	semidirect
3. C₆₇×#3496	C₂	semidirect
4. C₆₇×#3283	C₂	semidirect
5. D₄×SD₁₆×C₆₇	C₂	semidirect
6. C₆₇×#3276	C₂	semidirect
7. C₆₇×#3272	C₂	semidirect
8. C₆₇×#3267	C₂	semidirect
9. C₆₇×#3266	C₂	semidirect
10. C₆₇×#3152	C₂	semidirect
11. C₆₇×#3063	C₂	semidirect
12. C₆₇×#3035	C₂	semidirect
13. C₂×#461×C₆₇	C₂	semidirect
14. C₆₇×#3013	C₂	semidirect
15. C₆₇×#3009	C₂	semidirect
16. C₄×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
17. C₆₇×#2935	C₂	semidirect
18. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
21. C₂×#137×C₆₇	C₂²	semidirect
22. C₂×#137×C₆₇	C₂²	semidirect
23. C₂×#137×C₆₇	C₂²	semidirect
25. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
26. C₂×#135×C₆₇	C₂²	semidirect
27. C₂×#135×C₆₇	C₂²	semidirect
29. C₂²×SD₁₆×C₆₇	C₂²	semidirect
30. C₂×#131×C₆₇	C₂²	semidirect
31. D₄×Q₈×C₆₇	C₂²	semidirect
32. D₄×Q₈×C₆₇	C₂²	semidirect
37. #547×C₆₇	C₂²	semidirect
38. #547×C₆₇	C₂²	semidirect
45. #536×C₆₇	C₂²	semidirect
51. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
53. C₄×#41×C₆₇	C₂²	semidirect
55. #497×C₆₇	C₂²	semidirect
56. #494×C₆₇	C₂²	semidirect
57. #492×C₆₇	C₂²	semidirect
58. #470×C₆₇	C₂²	semidirect
59. #468×C₆₇	C₂²	semidirect
60. #467×C₆₇	C₂²	semidirect
61. #465×C₆₇	C₂²	semidirect
62. #464×C₆₇	C₂²	semidirect
63. #463×C₆₇	C₂²	semidirect
64. #462×C₆₇	C₂²	semidirect
65. #461×C₆₇	C₂²	semidirect
66. #461×C₆₇	C₂²	semidirect
67. #461×C₆₇	C₂²	semidirect
68. #461×C₆₇	C₂²	semidirect
69. #460×C₆₇	C₂²	semidirect
70. #459×C₆₇	C₂²	semidirect
71. #452×C₆₇	C₂²	semidirect
72. #452×C₆₇	C₂²	semidirect
73. #452×C₆₇	C₂²	semidirect
74. #451×C₆₇	C₂²	semidirect
75. #450×C₆₇	C₂²	semidirect
76. #450×C₆₇	C₂²	semidirect
77. #449×C₆₇	C₂²	semidirect
78. #448×C₆₇	C₂²	semidirect
79. #442×C₆₇	C₂²	semidirect
80. #440×C₆₇	C₂²	semidirect
81. C₄×SD₁₆×C₆₇	C₂²	semidirect
82. C₄×D₈×C₆₇	C₂²	semidirect
83. #436×C₆₇	C₂²	semidirect
84. #435×C₆₇	C₂²	semidirect
85. C₈×D₄×C₆₇	C₂²	semidirect
86. #428×C₆₇	C₂²	semidirect
87. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
89. #417×C₆₇	C₂²	semidirect
90. #416×C₆₇	C₂²	semidirect
91. C₂×#103×C₆₇	C₂²	semidirect
92. #408×C₆₇	C₂²	semidirect
93. C₄×M₁₆×C₆₇	C₂²	semidirect
119. AGL(1,ℤ/8)×C₆₇	D₄	semidirect
242. #41×C₆₇	C₂×D₄	semidirect
302. SD₁₆×C₆₇	C₂×D₄	semidirect
304. D₈×C₆₇	C₂×D₄	semidirect
307. M₁₆×C₆₇	C₂×D₄	semidirect
453. C₄×C₆₇	#577	semidirect
456. #61094	C₆₇	direct (N × H)
465. C₂×#577	C₂×C₆₇	semidirect
466. #3499	C₂×C₆₇	semidirect
467. #3496	C₂×C₆₇	semidirect
468. #3283	C₂×C₆₇	semidirect
469. D₄×SD₁₆	C₂×C₆₇	semidirect
470. #3276	C₂×C₆₇	semidirect
471. #3272	C₂×C₆₇	semidirect
472. #3267	C₂×C₆₇	semidirect
473. #3266	C₂×C₆₇	semidirect
474. #3152	C₂×C₆₇	semidirect
475. #3063	C₂×C₆₇	semidirect
476. #3035	C₂×C₆₇	semidirect
477. C₂×#461	C₂×C₆₇	semidirect
478. #3013	C₂×C₆₇	semidirect
479. #3009	C₂×C₆₇	semidirect
480. C₄×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
481. #2935	C₂×C₆₇	semidirect
482. C₆₇	#61094	direct (N × H)
483. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
486. C₂×#137	C₂²×C₆₇	semidirect
487. C₂×#137	C₂²×C₆₇	semidirect
488. C₂×#137	C₂²×C₆₇	semidirect
490. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
491. C₂×#135	C₂²×C₆₇	semidirect
492. C₂×#135	C₂²×C₆₇	semidirect
494. C₂²×SD₁₆	C₂²×C₆₇	semidirect
495. C₂×#131	C₂²×C₆₇	semidirect
496. D₄×Q₈	C₂²×C₆₇	semidirect
497. D₄×Q₈	C₂²×C₆₇	semidirect
502. #547	C₂²×C₆₇	semidirect
503. #547	C₂²×C₆₇	semidirect
510. #536	C₂²×C₆₇	semidirect
516. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
518. C₄×#41	C₂²×C₆₇	semidirect
520. #497	C₂²×C₆₇	semidirect
521. #494	C₂²×C₆₇	semidirect
522. #492	C₂²×C₆₇	semidirect
523. #470	C₂²×C₆₇	semidirect
524. #468	C₂²×C₆₇	semidirect
525. #467	C₂²×C₆₇	semidirect
526. #465	C₂²×C₆₇	semidirect
527. #464	C₂²×C₆₇	semidirect
528. #463	C₂²×C₆₇	semidirect
529. #462	C₂²×C₆₇	semidirect
530. #461	C₂²×C₆₇	semidirect
531. #461	C₂²×C₆₇	semidirect
532. #461	C₂²×C₆₇	semidirect
533. #461	C₂²×C₆₇	semidirect
534. #460	C₂²×C₆₇	semidirect
535. #459	C₂²×C₆₇	semidirect
536. #452	C₂²×C₆₇	semidirect
537. #452	C₂²×C₆₇	semidirect
538. #452	C₂²×C₆₇	semidirect
539. #451	C₂²×C₆₇	semidirect
540. #450	C₂²×C₆₇	semidirect
541. #450	C₂²×C₆₇	semidirect
542. #449	C₂²×C₆₇	semidirect
543. #448	C₂²×C₆₇	semidirect
544. #442	C₂²×C₆₇	semidirect
545. #440	C₂²×C₆₇	semidirect
546. C₄×SD₁₆	C₂²×C₆₇	semidirect
547. C₄×D₈	C₂²×C₆₇	semidirect
548. #436	C₂²×C₆₇	semidirect
549. #435	C₂²×C₆₇	semidirect
550. C₈×D₄	C₂²×C₆₇	semidirect
551. #428	C₂²×C₆₇	semidirect
552. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
554. #417	C₂²×C₆₇	semidirect
555. #416	C₂²×C₆₇	semidirect
556. C₂×#103	C₂²×C₆₇	semidirect
557. #408	C₂²×C₆₇	semidirect
558. C₄×M₁₆	C₂²×C₆₇	semidirect
584. AGL(1,ℤ/8)	D₄×C₆₇	semidirect
707. #41	C₂×D₄×C₆₇	semidirect
767. SD₁₆	C₂×D₄×C₆₇	semidirect
769. D₈	C₂×D₄×C₆₇	semidirect
772. M₁₆	C₂×D₄×C₆₇	semidirect
918. C₄	#577×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#577×C₆₇	2
C₆₇×#3499	2
C₆₇×#3496	2
C₆₇×#3283	2
D₄×SD₁₆×C₆₇	2
C₆₇×#3276	2
C₆₇×#3272	2
C₆₇×#3267	2
C₆₇×#3266	2
C₆₇×#3152	2
C₆₇×#3063	2
C₆₇×#3035	2
C₂×#461×C₆₇	2
C₆₇×#3013	2
C₆₇×#3009	2
C₄×AGL(1,ℤ/8)×C₆₇	2
C₆₇×#2935	2
#61094	67

Subgroups — C67×#61094 #30740345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61094 #30740345