Subgroups — C₆₇×#61381 #30740632

Order 17152.

1794 conjugacy classes of subgroups.

The subgroup lattice has 792 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3501	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3389	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3386	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3385	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3384	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3381	C₂	1	2	yes	maximal, complemented
8	Q₈×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3165	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3164	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3083	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3081	C₂	1	2	yes	maximal, complemented
13	C₆₇×#3076	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2962	C₂	1	1	yes	maximal, characteristic, complemented
16	C₆₇×#2946	C₂	1	2	yes	maximal
17	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
18	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
19	Q₈²×C₆₇	C₂²	1	2	yes	complemented
20	#557×C₆₇	C₂²	1	2	yes	complemented
21	#554×C₆₇	—	2	1	no	—
22	#554×C₆₇	—	2	2	no	—
23	#554×C₆₇	—	2	2	no	—
24	#554×C₆₇	C₂²	1	2	yes	—
25	#554×C₆₇	C₂²	1	4	yes	—
26	#554×C₆₇	C₂²	1	2	yes	complemented
27	D₄×Q₈×C₆₇	—	2	1	no	—
28	D₄×Q₈×C₆₇	—	2	2	no	—
29	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
30	D₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
31	#544×C₆₇	—	2	2	no	—
32	#544×C₆₇	C₂²	1	2	yes	—
33	#544×C₆₇	C₂²	1	2	yes	—
34	#543×C₆₇	—	2	2	no	—
35	#533×C₆₇	—	2	1	no	—
36	#533×C₆₇	C₂²	1	1	yes	characteristic
37	#533×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
39	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
41	#520×C₆₇	C₂²	1	2	yes	—
42	#518×C₆₇	C₂²	1	2	yes	—
43	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
45	#501×C₆₇	C₂²	1	4	yes	complemented
46	#501×C₆₇	C₂²	1	4	yes	complemented
47	#501×C₆₇	C₂²	1	4	yes	complemented
48	#500×C₆₇	C₂²	1	4	yes	complemented
49	#500×C₆₇	C₂²	1	2	yes	complemented
50	#499×C₆₇	C₂²	1	4	yes	complemented
51	#498×C₆₇	C₂²	1	2	yes	complemented
52	#479×C₆₇	C₂²	1	2	yes	complemented
53	#479×C₆₇	C₂²	1	4	yes	complemented
54	#478×C₆₇	C₂²	1	4	yes	—
55	#478×C₆₇	C₂²	1	4	yes	—
56	#478×C₆₇	C₂²	1	2	yes	complemented
57	#478×C₆₇	C₂²	1	4	yes	complemented
58	#477×C₆₇	C₂²	1	1	yes	characteristic, complemented
59	#477×C₆₇	C₂²	1	4	yes	complemented
60	#476×C₆₇	C₂²	1	4	yes	—
61	#476×C₆₇	C₂²	1	1	yes	characteristic, complemented
62	#475×C₆₇	C₂²	1	1	yes	characteristic, complemented
63	#474×C₆₇	C₂²	1	2	yes	—
64	#474×C₆₇	C₂²	1	2	yes	—
65	#474×C₆₇	C₂²	1	4	yes	—
66	#474×C₆₇	C₂²	1	4	yes	—
67	#474×C₆₇	C₂²	1	1	yes	characteristic, complemented
68	#474×C₆₇	C₂²	1	4	yes	complemented
69	#446×C₆₇	C₂²	1	2	yes	complemented
70	#446×C₆₇	C₂²	1	4	yes	complemented
71	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
72	#442×C₆₇	C₂²	1	4	yes	—
73	#442×C₆₇	C₂²	1	2	yes	—
74	#440×C₆₇	C₂²	1	2	yes	—
75	#440×C₆₇	C₂²	1	4	yes	—
76	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
77	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
78	C₄×D₈×C₆₇	C₂²	1	2	yes	—
79	C₄×D₈×C₆₇	C₂²	1	4	yes	—
80	#428×C₆₇	C₂²	1	2	yes	—
81	#428×C₆₇	C₂²	1	1	yes	characteristic
82	#428×C₆₇	C₂²	1	2	yes	—
83	#427×C₆₇	C₂²	1	2	yes	—
84	C₂×#107×C₆₇	C₂²	1	2	yes	—
85	#424×C₆₇	C₂²	1	2	yes	—
86	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
87	#418×C₆₇	C₂²	1	1	yes	characteristic
88	#417×C₆₇	C₂²	1	1	yes	characteristic
89	#416×C₆₇	C₂²	1	2	yes	—
90	C₂×#102×C₆₇	C₂²	1	2	yes	—
91	#405×C₆₇	C₂²	1	2	yes	—
92	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
93	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²×Q₈×C₆₇	—	2	2	no	—
95	C₂²×Q₈×C₆₇	—	2	2	no	—
96	C₂²×Q₈×C₆₇	—	2	1	no	—
97	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₂³×C₄×C₆₇	—	2	1	no	—
99	AGL(1,ℤ/8)×C₆₇	Q₈	1	8	yes	complemented
100	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
101	C₂×D₈×C₆₇	C₂³	1	2	yes	—
102	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
105	C₄⋊Q₈×C₆₇	—	2	2	no	—
106	C₄⋊Q₈×C₆₇	—	2	2	no	—
107	C₄⋊Q₈×C₆₇	—	2	2	no	—
108	C₄⋊Q₈×C₆₇	—	2	4	no	—
109	C₄⋊Q₈×C₆₇	—	2	2	no	—
110	C₄⋊Q₈×C₆₇	—	2	1	no	—
111	C₄⋊Q₈×C₆₇	—	2	1	no	—
112	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
113	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
114	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
115	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
116	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
117	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
118	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
119	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
120	#125×C₆₇	—	2	2	no	—
121	#125×C₆₇	—	2	4	no	—
122	#125×C₆₇	—	2	2	no	—
123	#125×C₆₇	D₄	1	2	yes	—
124	#125×C₆₇	C₂³	1	4	yes	—
125	#125×C₆₇	C₂³	1	4	yes	—
126	#125×C₆₇	C₂³	1	2	yes	—
127	C₂²⋊Q₈×C₆₇	—	2	2	no	—
128	C₂²⋊Q₈×C₆₇	—	2	2	no	—
129	C₂²⋊Q₈×C₆₇	—	2	4	no	—
130	C₂²⋊Q₈×C₆₇	—	2	1	no	—
131	C₂²⋊Q₈×C₆₇	—	2	1	no	—
132	C₂²⋊Q₈×C₆₇	—	2	2	no	—
133	C₂²⋊Q₈×C₆₇	—	2	4	no	—
134	C₂²⋊Q₈×C₆₇	—	2	2	no	—
135	C₂²⋊Q₈×C₆₇	—	2	4	no	—
136	C₂²⋊Q₈×C₆₇	—	2	4	no	—
137	C₂²⋊Q₈×C₆₇	—	2	4	no	—
138	C₂²⋊Q₈×C₆₇	—	2	4	no	—
139	C₂²⋊Q₈×C₆₇	—	2	4	no	—
140	C₂²⋊Q₈×C₆₇	—	2	4	no	—
141	C₂²⋊Q₈×C₆₇	—	2	4	no	—
142	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
143	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
144	C₄×Q₈×C₆₇	—	2	2	no	—
145	C₄×Q₈×C₆₇	—	2	2	no	—
146	C₄×Q₈×C₆₇	—	2	1	no	—
147	C₄×Q₈×C₆₇	—	2	1	no	—
148	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
149	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
150	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
151	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
152	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
153	C₄×D₄×C₆₇	—	2	1	no	—
154	C₄×D₄×C₆₇	—	2	1	no	—
155	C₄×D₄×C₆₇	—	2	2	no	—
156	C₄×D₄×C₆₇	—	2	2	no	—
157	C₄×D₄×C₆₇	—	2	4	no	—
158	C₄×D₄×C₆₇	—	2	1	no	—
159	C₄×D₄×C₆₇	—	2	1	no	—
160	C₄×D₄×C₆₇	C₂³	1	2	yes	—
161	C₄×D₄×C₆₇	C₂³	1	2	yes	—
162	C₄×D₄×C₆₇	C₂³	1	4	yes	—
163	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
164	C₄×D₄×C₆₇	C₂³	1	4	yes	—
165	C₄×D₄×C₆₇	C₂³	1	4	yes	—
166	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
167	#117×C₆₇	—	2	2	no	—
168	#117×C₆₇	—	2	1	no	—
169	#117×C₆₇	C₂³	1	1	yes	characteristic
170	#117×C₆₇	C₂³	1	2	yes	—
171	#117×C₆₇	C₂³	1	2	yes	—
172	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
173	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
174	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
175	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
178	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
179	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
180	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
181	C₂×C₄²×C₆₇	—	2	1	no	—
182	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
183	#107×C₆₇	C₂³	1	4	yes	—
184	#107×C₆₇	C₂³	1	2	yes	—
185	#107×C₆₇	C₂³	1	4	yes	—
186	#107×C₆₇	C₂³	1	4	yes	—
187	#107×C₆₇	C₂³	1	2	yes	—
188	#107×C₆₇	C₂³	1	4	yes	—
189	#107×C₆₇	C₂³	1	2	yes	—
190	#106×C₆₇	C₂³	1	2	yes	—
191	#106×C₆₇	C₂³	1	2	yes	—
192	#106×C₆₇	C₂³	1	4	yes	—
193	#106×C₆₇	C₂³	1	4	yes	—
194	#106×C₆₇	C₂³	1	2	yes	—
195	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
196	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
197	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
198	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
199	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
200	#103×C₆₇	C₂³	1	2	yes	—
201	#103×C₆₇	C₂³	1	2	yes	—
202	#103×C₆₇	C₂³	1	2	yes	—
203	#102×C₆₇	C₂³	1	4	yes	—
204	#102×C₆₇	C₂³	1	2	yes	—
205	#102×C₆₇	C₂³	1	4	yes	—
206	#102×C₆₇	C₂³	1	2	yes	—
207	#102×C₆₇	C₂³	1	2	yes	—
208	#102×C₆₇	C₂³	1	2	yes	—
209	#102×C₆₇	C₂³	1	2	yes	—
210	#97×C₆₇	C₂³	1	2	yes	—
211	#97×C₆₇	C₂³	1	4	yes	—
212	C₄×C₈×C₆₇	C₂³	1	2	yes	—
213	C₄×C₈×C₆₇	C₂³	1	4	yes	—
214	C₂⁴×C₆₇	—	2	1	no	—
215	#41×C₆₇	—	2	2	no	—
216	#41×C₆₇	C₂×Q₈	1	4	yes	complemented
217	C₂×Q₈×C₆₇	—	2	1	no	—
218	C₂×Q₈×C₆₇	—	2	2	no	—
219	C₂×Q₈×C₆₇	—	2	1	no	—
220	C₂×Q₈×C₆₇	—	2	4	no	—
221	C₂×Q₈×C₆₇	—	2	4	no	—
222	C₂×Q₈×C₆₇	—	2	2	no	—
223	C₂×Q₈×C₆₇	—	8	2	no	—
224	C₂×Q₈×C₆₇	—	8	4	no	—
225	C₂×Q₈×C₆₇	—	8	4	no	—
226	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
227	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
228	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
229	C₂×D₄×C₆₇	—	2	1	no	—
230	C₂×D₄×C₆₇	—	2	1	no	—
231	C₂×D₄×C₆₇	—	2	2	no	—
232	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	complemented
233	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
234	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
235	C₂²×C₄×C₆₇	—	2	1	no	—
236	C₂²×C₄×C₆₇	—	2	1	no	—
237	C₂²×C₄×C₆₇	—	2	2	no	—
238	C₂²×C₄×C₆₇	—	2	1	no	—
239	C₂²×C₄×C₆₇	—	2	1	no	—
240	C₂²×C₄×C₆₇	—	2	2	no	—
241	C₂²×C₄×C₆₇	—	2	4	no	—
242	C₂²×C₄×C₆₇	—	2	4	no	—
243	C₂²×C₄×C₆₇	—	2	2	no	—
244	C₂²×C₄×C₆₇	—	2	4	no	—
245	C₂²×C₄×C₆₇	—	8	2	no	—
246	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
247	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
248	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
249	SD₁₆×C₆₇	C₂×Q₈	1	8	yes	complemented
250	D₈×C₆₇	C₂×Q₈	1	8	yes	complemented
251	M₁₆×C₆₇	—	2	2	no	—
252	M₁₆×C₆₇	C₂×Q₈	1	4	yes	complemented
253	C₂×C₈×C₆₇	—	2	4	no	—
254	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
255	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
256	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
257	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
258	C₄⋊C₄×C₆₇	—	2	2	no	—
259	C₄⋊C₄×C₆₇	—	2	1	no	—
260	C₄⋊C₄×C₆₇	—	2	2	no	—
261	C₄⋊C₄×C₆₇	—	2	2	no	—
262	C₄⋊C₄×C₆₇	—	2	4	no	—
263	C₄⋊C₄×C₆₇	—	2	4	no	—
264	C₄⋊C₄×C₆₇	—	2	4	no	—
265	C₄⋊C₄×C₆₇	—	2	4	no	—
266	C₄⋊C₄×C₆₇	—	2	2	no	—
267	C₄⋊C₄×C₆₇	—	2	4	no	—
268	C₄⋊C₄×C₆₇	—	2	1	no	—
269	C₄⋊C₄×C₆₇	—	2	1	no	—
270	C₄⋊C₄×C₆₇	—	2	1	no	—
271	C₄⋊C₄×C₆₇	—	2	2	no	—
272	C₄⋊C₄×C₆₇	—	2	1	no	—
273	C₄⋊C₄×C₆₇	—	2	4	no	—
274	C₄⋊C₄×C₆₇	—	2	4	no	—
275	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
276	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
277	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
278	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
279	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
280	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
281	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
282	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
283	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
284	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
285	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
286	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
287	C₂²⋊C₄×C₆₇	—	2	2	no	—
288	C₂²⋊C₄×C₆₇	—	2	1	no	—
289	C₂²⋊C₄×C₆₇	—	2	4	no	—
290	C₂²⋊C₄×C₆₇	—	2	4	no	—
291	C₂²⋊C₄×C₆₇	—	2	2	no	—
292	C₂²⋊C₄×C₆₇	—	2	2	no	—
293	C₂²⋊C₄×C₆₇	—	2	1	no	—
294	C₂²⋊C₄×C₆₇	—	2	1	no	—
295	C₂²⋊C₄×C₆₇	—	2	4	no	—
296	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
297	C₄²×C₆₇	—	2	1	no	—
298	C₄²×C₆₇	—	2	1	no	—
299	C₄²×C₆₇	—	2	1	no	—
300	C₄²×C₆₇	—	2	4	no	—
301	C₄²×C₆₇	—	2	2	no	—
302	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
303	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
304	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
305	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
306	C₄²×C₆₇	C₂⁴	1	4	yes	—
307	C₂³×C₆₇	—	2	1	no	—
308	C₂³×C₆₇	—	2	2	no	—
309	C₂³×C₆₇	—	2	4	no	—
310	C₂³×C₆₇	—	8	1	no	—
311	C₂³×C₆₇	—	8	1	no	—
312	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
313	Q₈×C₆₇	—	2	4	no	—
314	Q₈×C₆₇	—	2	1	no	—
315	Q₈×C₆₇	—	4	2	no	—
316	Q₈×C₆₇	—	4	2	no	—
317	Q₈×C₆₇	—	4	4	no	—
318	Q₈×C₆₇	—	8	1	no	—
319	Q₈×C₆₇	—	8	2	no	—
320	Q₈×C₆₇	—	8	4	no	—
321	Q₈×C₆₇	—	8	4	no	—
322	Q₈×C₆₇	C₂²×Q₈	1	2	yes	—
323	D₄×C₆₇	—	2	4	no	—
324	D₄×C₆₇	—	2	4	no	—
325	D₄×C₆₇	—	2	2	no	—
326	D₄×C₆₇	—	2	1	no	—
327	D₄×C₆₇	C₂²×Q₈	1	2	yes	—
328	D₄×C₆₇	C₂²×Q₈	1	4	yes	—
329	C₂×C₄×C₆₇	—	2	1	no	—
330	C₂×C₄×C₆₇	—	2	4	no	—
331	C₂×C₄×C₆₇	—	2	2	no	—
332	C₂×C₄×C₆₇	—	2	1	no	—
333	C₂×C₄×C₆₇	—	2	1	no	—
334	C₂×C₄×C₆₇	—	2	2	no	—
335	C₂×C₄×C₆₇	—	2	2	no	—
336	C₂×C₄×C₆₇	—	2	4	no	—
337	C₂×C₄×C₆₇	—	2	2	no	—
338	C₂×C₄×C₆₇	—	4	1	no	—
339	C₂×C₄×C₆₇	—	4	2	no	—
340	C₂×C₄×C₆₇	—	4	4	no	—
341	C₂×C₄×C₆₇	—	4	2	no	—
342	C₂×C₄×C₆₇	—	8	4	no	—
343	C₂×C₄×C₆₇	—	8	2	no	—
344	C₂×C₄×C₆₇	—	8	4	no	—
345	C₂×C₄×C₆₇	—	8	4	no	—
346	C₂×C₄×C₆₇	—	8	1	no	—
347	C₂×C₄×C₆₇	—	8	2	no	—
348	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
349	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
350	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
351	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
352	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
353	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
354	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
355	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
356	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
357	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
358	C₈×C₆₇	—	2	2	no	—
359	C₈×C₆₇	—	2	2	no	—
360	C₈×C₆₇	—	2	2	no	—
361	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
362	C₂²×C₆₇	—	2	1	no	—
363	C₂²×C₆₇	—	2	1	no	—
364	C₂²×C₆₇	—	2	2	no	—
365	C₂²×C₆₇	—	2	4	no	—
366	C₂²×C₆₇	—	4	1	no	—
367	C₂²×C₆₇	—	4	2	no	—
368	C₂²×C₆₇	—	4	1	no	—
369	C₂²×C₆₇	—	4	2	no	—
370	C₂²×C₆₇	—	8	4	no	—
371	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
372	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
373	C₄×C₆₇	—	2	1	no	—
374	C₄×C₆₇	—	2	1	no	—
375	C₄×C₆₇	—	2	4	no	—
376	C₄×C₆₇	—	2	2	no	—
377	C₄×C₆₇	—	2	1	no	—
378	C₄×C₆₇	—	2	4	no	—
379	C₄×C₆₇	—	4	1	no	—
380	C₄×C₆₇	—	4	2	no	—
381	C₄×C₆₇	—	4	2	no	—
382	C₄×C₆₇	—	4	1	no	—
383	C₄×C₆₇	—	4	2	no	—
384	C₄×C₆₇	—	4	4	no	—
385	C₄×C₆₇	D₄×Q₈	1	2	yes	—
386	C₄×C₆₇	#576	1	2	yes	—
387	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
388	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
389	#61381	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
390	C₂×C₆₇	—	2	2	no	—
391	C₂×C₆₇	—	4	2	no	—
392	C₂×C₆₇	—	4	4	no	—
393	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
394	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
395	C₂×C₆₇	#3586	1	1	yes	characteristic
396	#3503	C₂×C₆₇	1	1	yes	characteristic, complemented
397	#3501	C₂×C₆₇	1	1	yes	characteristic, complemented
398	#3389	C₂×C₆₇	1	4	yes	complemented
399	#3386	C₂×C₆₇	1	2	yes	complemented
400	#3385	C₂×C₆₇	1	4	yes	complemented
401	#3384	C₂×C₆₇	1	2	yes	complemented
402	#3381	C₂×C₆₇	1	2	yes	complemented
403	Q₈×D₈	C₂×C₆₇	1	2	yes	complemented
404	#3165	C₂×C₆₇	1	1	yes	characteristic, complemented
405	#3164	C₂×C₆₇	1	2	yes	complemented
406	#3083	C₂×C₆₇	1	2	yes	complemented
407	#3081	C₂×C₆₇	1	2	yes	complemented
408	#3076	C₂×C₆₇	1	1	yes	characteristic, complemented
409	C₂×#474	C₂×C₆₇	1	1	yes	characteristic, complemented
410	#2962	C₂×C₆₇	1	1	yes	characteristic, complemented
411	#2946	C₂×C₆₇	1	2	yes	—
412	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
413	C₆₇	#61381	1	1	yes	characteristic, direct product factor, Sylow
414	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
415	Q₈²	C₂²×C₆₇	1	2	yes	complemented
416	#557	C₂²×C₆₇	1	2	yes	complemented
417	#554	—	2	1	no	—
418	#554	—	2	2	no	—
419	#554	—	2	2	no	—
420	#554	C₂²×C₆₇	1	2	yes	—
421	#554	C₂²×C₆₇	1	4	yes	—
422	#554	C₂²×C₆₇	1	2	yes	complemented
423	D₄×Q₈	—	2	1	no	—
424	D₄×Q₈	—	2	2	no	—
425	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
426	D₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
427	#544	—	2	2	no	—
428	#544	C₂²×C₆₇	1	2	yes	—
429	#544	C₂²×C₆₇	1	2	yes	—
430	#543	—	2	2	no	—
431	#533	—	2	1	no	—
432	#533	C₂²×C₆₇	1	1	yes	characteristic
433	#533	C₂²×C₆₇	1	1	yes	characteristic
434	C₂×C₄⋊Q₈	—	2	1	no	—
435	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
436	C₂×C₂²⋊Q₈	—	2	2	no	—
437	#520	C₂²×C₆₇	1	2	yes	—
438	#518	C₂²×C₆₇	1	2	yes	—
439	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
440	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
441	#501	C₂²×C₆₇	1	4	yes	complemented
442	#501	C₂²×C₆₇	1	4	yes	complemented
443	#501	C₂²×C₆₇	1	4	yes	complemented
444	#500	C₂²×C₆₇	1	4	yes	complemented
445	#500	C₂²×C₆₇	1	2	yes	complemented
446	#499	C₂²×C₆₇	1	4	yes	complemented
447	#498	C₂²×C₆₇	1	2	yes	complemented
448	#479	C₂²×C₆₇	1	2	yes	complemented
449	#479	C₂²×C₆₇	1	4	yes	complemented
450	#478	C₂²×C₆₇	1	4	yes	—
451	#478	C₂²×C₆₇	1	4	yes	—
452	#478	C₂²×C₆₇	1	2	yes	complemented
453	#478	C₂²×C₆₇	1	4	yes	complemented
454	#477	C₂²×C₆₇	1	1	yes	characteristic, complemented
455	#477	C₂²×C₆₇	1	4	yes	complemented
456	#476	C₂²×C₆₇	1	4	yes	—
457	#476	C₂²×C₆₇	1	1	yes	characteristic, complemented
458	#475	C₂²×C₆₇	1	1	yes	characteristic, complemented
459	#474	C₂²×C₆₇	1	4	yes	—
460	#474	C₂²×C₆₇	1	2	yes	—
461	#474	C₂²×C₆₇	1	2	yes	—
462	#474	C₂²×C₆₇	1	4	yes	—
463	#474	C₂²×C₆₇	1	1	yes	characteristic, complemented
464	#474	C₂²×C₆₇	1	4	yes	complemented
465	#446	C₂²×C₆₇	1	2	yes	complemented
466	#446	C₂²×C₆₇	1	4	yes	complemented
467	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
468	#442	C₂²×C₆₇	1	4	yes	—
469	#442	C₂²×C₆₇	1	2	yes	—
470	#440	C₂²×C₆₇	1	2	yes	—
471	#440	C₂²×C₆₇	1	4	yes	—
472	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
473	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
474	C₄×D₈	C₂²×C₆₇	1	2	yes	—
475	C₄×D₈	C₂²×C₆₇	1	4	yes	—
476	#428	C₂²×C₆₇	1	2	yes	—
477	#428	C₂²×C₆₇	1	1	yes	characteristic
478	#428	C₂²×C₆₇	1	2	yes	—
479	#427	C₂²×C₆₇	1	2	yes	—
480	C₂×#107	C₂²×C₆₇	1	2	yes	—
481	#424	C₂²×C₆₇	1	2	yes	—
482	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
483	#418	C₂²×C₆₇	1	1	yes	characteristic
484	#417	C₂²×C₆₇	1	1	yes	characteristic
485	#416	C₂²×C₆₇	1	2	yes	—
486	C₂×#102	C₂²×C₆₇	1	2	yes	—
487	#405	C₂²×C₆₇	1	2	yes	—
488	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
489	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
490	C₂²×Q₈	—	2	2	no	—
491	C₂²×Q₈	—	2	2	no	—
492	C₂²×Q₈	—	2	1	no	—
493	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
494	C₂³×C₄	—	2	1	no	—
495	AGL(1,ℤ/8)	Q₈×C₆₇	1	8	yes	complemented
496	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
497	C₂×D₈	C₂³×C₆₇	1	2	yes	—
498	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
499	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
500	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
501	C₄⋊Q₈	—	2	2	no	—
502	C₄⋊Q₈	—	2	2	no	—
503	C₄⋊Q₈	—	2	2	no	—
504	C₄⋊Q₈	—	2	4	no	—
505	C₄⋊Q₈	—	2	2	no	—
506	C₄⋊Q₈	—	2	1	no	—
507	C₄⋊Q₈	—	2	1	no	—
508	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
509	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
510	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
511	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
512	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
513	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
514	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
515	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
516	#125	—	2	2	no	—
517	#125	—	2	4	no	—
518	#125	—	2	2	no	—
519	#125	D₄×C₆₇	1	2	yes	—
520	#125	C₂³×C₆₇	1	2	yes	—
521	#125	C₂³×C₆₇	1	4	yes	—
522	#125	C₂³×C₆₇	1	4	yes	—
523	C₂²⋊Q₈	—	2	2	no	—
524	C₂²⋊Q₈	—	2	2	no	—
525	C₂²⋊Q₈	—	2	4	no	—
526	C₂²⋊Q₈	—	2	1	no	—
527	C₂²⋊Q₈	—	2	1	no	—
528	C₂²⋊Q₈	—	2	2	no	—
529	C₂²⋊Q₈	—	2	4	no	—
530	C₂²⋊Q₈	—	2	2	no	—
531	C₂²⋊Q₈	—	2	4	no	—
532	C₂²⋊Q₈	—	2	4	no	—
533	C₂²⋊Q₈	—	2	4	no	—
534	C₂²⋊Q₈	—	2	4	no	—
535	C₂²⋊Q₈	—	2	4	no	—
536	C₂²⋊Q₈	—	2	4	no	—
537	C₂²⋊Q₈	—	2	4	no	—
538	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
539	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
540	C₄×Q₈	—	2	2	no	—
541	C₄×Q₈	—	2	2	no	—
542	C₄×Q₈	—	2	1	no	—
543	C₄×Q₈	—	2	1	no	—
544	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
545	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
546	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
547	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
548	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
549	C₄×D₄	—	2	1	no	—
550	C₄×D₄	—	2	1	no	—
551	C₄×D₄	—	2	2	no	—
552	C₄×D₄	—	2	2	no	—
553	C₄×D₄	—	2	4	no	—
554	C₄×D₄	—	2	1	no	—
555	C₄×D₄	—	2	1	no	—
556	C₄×D₄	C₂³×C₆₇	1	2	yes	—
557	C₄×D₄	C₂³×C₆₇	1	2	yes	—
558	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
559	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
560	C₄×D₄	C₂³×C₆₇	1	4	yes	—
561	C₄×D₄	C₂³×C₆₇	1	4	yes	—
562	C₄×D₄	C₂³×C₆₇	1	4	yes	—
563	#117	—	2	2	no	—
564	#117	—	2	1	no	—
565	#117	C₂³×C₆₇	1	1	yes	characteristic
566	#117	C₂³×C₆₇	1	2	yes	—
567	#117	C₂³×C₆₇	1	2	yes	—
568	C₂×C₄⋊C₄	—	2	1	no	—
569	C₂×C₄⋊C₄	—	2	2	no	—
570	C₂×C₄⋊C₄	—	2	4	no	—
571	C₂×C₄⋊C₄	—	2	2	no	—
572	C₂×C₄⋊C₄	—	2	2	no	—
573	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
574	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
575	C₂×C₂²⋊C₄	—	2	1	no	—
576	C₂×C₂²⋊C₄	—	2	4	no	—
577	C₂×C₄²	—	2	1	no	—
578	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
579	#107	C₂³×C₆₇	1	4	yes	—
580	#107	C₂³×C₆₇	1	4	yes	—
581	#107	C₂³×C₆₇	1	2	yes	—
582	#107	C₂³×C₆₇	1	2	yes	—
583	#107	C₂³×C₆₇	1	4	yes	—
584	#107	C₂³×C₆₇	1	2	yes	—
585	#107	C₂³×C₆₇	1	4	yes	—
586	#106	C₂³×C₆₇	1	2	yes	—
587	#106	C₂³×C₆₇	1	2	yes	—
588	#106	C₂³×C₆₇	1	4	yes	—
589	#106	C₂³×C₆₇	1	2	yes	—
590	#106	C₂³×C₆₇	1	4	yes	—
591	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
592	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
593	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
594	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
595	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
596	#103	C₂³×C₆₇	1	2	yes	—
597	#103	C₂³×C₆₇	1	2	yes	—
598	#103	C₂³×C₆₇	1	2	yes	—
599	#102	C₂³×C₆₇	1	4	yes	—
600	#102	C₂³×C₆₇	1	4	yes	—
601	#102	C₂³×C₆₇	1	2	yes	—
602	#102	C₂³×C₆₇	1	2	yes	—
603	#102	C₂³×C₆₇	1	2	yes	—
604	#102	C₂³×C₆₇	1	2	yes	—
605	#102	C₂³×C₆₇	1	2	yes	—
606	#97	C₂³×C₆₇	1	2	yes	—
607	#97	C₂³×C₆₇	1	4	yes	—
608	C₄×C₈	C₂³×C₆₇	1	2	yes	—
609	C₄×C₈	C₂³×C₆₇	1	4	yes	—
610	C₂⁴	—	2	1	no	—
611	#41	—	2	2	no	—
612	#41	C₂×Q₈×C₆₇	1	4	yes	complemented
613	C₂×Q₈	—	2	1	no	—
614	C₂×Q₈	—	2	2	no	—
615	C₂×Q₈	—	2	1	no	—
616	C₂×Q₈	—	2	4	no	—
617	C₂×Q₈	—	2	4	no	—
618	C₂×Q₈	—	2	2	no	—
619	C₂×Q₈	—	8	2	no	—
620	C₂×Q₈	—	8	4	no	—
621	C₂×Q₈	—	8	4	no	—
622	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
623	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
624	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
625	C₂×D₄	—	2	1	no	—
626	C₂×D₄	—	2	1	no	—
627	C₂×D₄	—	2	2	no	—
628	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	complemented
629	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
630	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
631	C₂²×C₄	—	2	1	no	—
632	C₂²×C₄	—	2	1	no	—
633	C₂²×C₄	—	2	2	no	—
634	C₂²×C₄	—	2	1	no	—
635	C₂²×C₄	—	2	1	no	—
636	C₂²×C₄	—	2	2	no	—
637	C₂²×C₄	—	2	4	no	—
638	C₂²×C₄	—	2	4	no	—
639	C₂²×C₄	—	2	2	no	—
640	C₂²×C₄	—	2	4	no	—
641	C₂²×C₄	—	8	2	no	—
642	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
643	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
644	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
645	SD₁₆	C₂×Q₈×C₆₇	1	8	yes	complemented
646	D₈	C₂×Q₈×C₆₇	1	8	yes	complemented
647	M₁₆	—	2	2	no	—
648	M₁₆	C₂×Q₈×C₆₇	1	4	yes	complemented
649	C₂×C₈	—	2	4	no	—
650	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
651	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
652	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
653	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
654	C₄⋊C₄	—	2	2	no	—
655	C₄⋊C₄	—	2	1	no	—
656	C₄⋊C₄	—	2	2	no	—
657	C₄⋊C₄	—	2	2	no	—
658	C₄⋊C₄	—	2	4	no	—
659	C₄⋊C₄	—	2	4	no	—
660	C₄⋊C₄	—	2	4	no	—
661	C₄⋊C₄	—	2	4	no	—
662	C₄⋊C₄	—	2	2	no	—
663	C₄⋊C₄	—	2	4	no	—
664	C₄⋊C₄	—	2	1	no	—
665	C₄⋊C₄	—	2	1	no	—
666	C₄⋊C₄	—	2	1	no	—
667	C₄⋊C₄	—	2	2	no	—
668	C₄⋊C₄	—	2	1	no	—
669	C₄⋊C₄	—	2	4	no	—
670	C₄⋊C₄	—	2	4	no	—
671	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
672	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
673	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
674	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
675	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
676	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
677	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
678	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
679	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
680	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
681	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
682	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
683	C₂²⋊C₄	—	2	2	no	—
684	C₂²⋊C₄	—	2	1	no	—
685	C₂²⋊C₄	—	2	4	no	—
686	C₂²⋊C₄	—	2	4	no	—
687	C₂²⋊C₄	—	2	2	no	—
688	C₂²⋊C₄	—	2	2	no	—
689	C₂²⋊C₄	—	2	1	no	—
690	C₂²⋊C₄	—	2	1	no	—
691	C₂²⋊C₄	—	2	4	no	—
692	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
693	C₄²	—	2	1	no	—
694	C₄²	—	2	1	no	—
695	C₄²	—	2	1	no	—
696	C₄²	—	2	4	no	—
697	C₄²	—	2	2	no	—
698	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
699	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
700	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
701	C₄²	C₂⁴×C₆₇	1	4	yes	—
702	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
703	C₂³	—	2	1	no	—
704	C₂³	—	2	2	no	—
705	C₂³	—	2	4	no	—
706	C₂³	—	8	1	no	—
707	C₂³	—	8	1	no	—
708	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
709	Q₈	—	2	4	no	—
710	Q₈	—	2	1	no	—
711	Q₈	—	4	2	no	—
712	Q₈	—	4	2	no	—
713	Q₈	—	4	4	no	—
714	Q₈	—	8	2	no	—
715	Q₈	—	8	4	no	—
716	Q₈	—	8	4	no	—
717	Q₈	—	8	1	no	—
718	Q₈	C₂²×Q₈×C₆₇	1	2	yes	—
719	D₄	—	2	4	no	—
720	D₄	—	2	4	no	—
721	D₄	—	2	2	no	—
722	D₄	—	2	1	no	—
723	D₄	C₂²×Q₈×C₆₇	1	2	yes	—
724	D₄	C₂²×Q₈×C₆₇	1	4	yes	—
725	C₂×C₄	—	2	1	no	—
726	C₂×C₄	—	2	4	no	—
727	C₂×C₄	—	2	2	no	—
728	C₂×C₄	—	2	1	no	—
729	C₂×C₄	—	2	1	no	—
730	C₂×C₄	—	2	2	no	—
731	C₂×C₄	—	2	2	no	—
732	C₂×C₄	—	2	4	no	—
733	C₂×C₄	—	2	2	no	—
734	C₂×C₄	—	4	1	no	—
735	C₂×C₄	—	4	2	no	—
736	C₂×C₄	—	4	4	no	—
737	C₂×C₄	—	4	2	no	—
738	C₂×C₄	—	8	2	no	—
739	C₂×C₄	—	8	4	no	—
740	C₂×C₄	—	8	4	no	—
741	C₂×C₄	—	8	4	no	—
742	C₂×C₄	—	8	1	no	—
743	C₂×C₄	—	8	2	no	—
744	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
745	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
746	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
747	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
748	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
749	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
750	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
751	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
752	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
753	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
754	C₈	—	2	2	no	—
755	C₈	—	2	2	no	—
756	C₈	—	2	2	no	—
757	C₈	C₂²×Q₈×C₆₇	1	4	yes	—
758	C₂²	—	2	1	no	—
759	C₂²	—	2	1	no	—
760	C₂²	—	2	2	no	—
761	C₂²	—	2	4	no	—
762	C₂²	—	4	1	no	—
763	C₂²	—	4	2	no	—
764	C₂²	—	4	1	no	—
765	C₂²	—	4	2	no	—
766	C₂²	—	8	4	no	—
767	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
768	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
769	C₄	—	2	1	no	—
770	C₄	—	2	4	no	—
771	C₄	—	2	1	no	—
772	C₄	—	2	1	no	—
773	C₄	—	2	4	no	—
774	C₄	—	2	2	no	—
775	C₄	—	4	2	no	—
776	C₄	—	4	1	no	—
777	C₄	—	4	2	no	—
778	C₄	—	4	4	no	—
779	C₄	—	4	1	no	—
780	C₄	—	4	2	no	—
781	C₄	D₄×Q₈×C₆₇	1	2	yes	—
782	C₄	#576×C₆₇	1	2	yes	—
783	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
784	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
785	C₂	—	2	2	no	—
786	C₂	—	4	4	no	—
787	C₂	—	4	2	no	—
788	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
789	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
790	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #533×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3503	C₂	semidirect
2. C₆₇×#3501	C₂	semidirect
3. C₆₇×#3389	C₂	semidirect
4. C₆₇×#3386	C₂	semidirect
5. C₆₇×#3385	C₂	semidirect
6. C₆₇×#3384	C₂	semidirect
7. C₆₇×#3381	C₂	semidirect
8. Q₈×D₈×C₆₇	C₂	semidirect
9. C₆₇×#3165	C₂	semidirect
10. C₆₇×#3164	C₂	semidirect
11. C₆₇×#3083	C₂	semidirect
12. C₆₇×#3081	C₂	semidirect
13. C₆₇×#3076	C₂	semidirect
14. C₂×#474×C₆₇	C₂	semidirect
15. C₆₇×#2962	C₂	semidirect
19. Q₈²×C₆₇	C₂²	semidirect
20. #557×C₆₇	C₂²	semidirect
26. #554×C₆₇	C₂²	semidirect
30. D₄×Q₈×C₆₇	C₂²	semidirect
45. #501×C₆₇	C₂²	semidirect
46. #501×C₆₇	C₂²	semidirect
47. #501×C₆₇	C₂²	semidirect
48. #500×C₆₇	C₂²	semidirect
49. #500×C₆₇	C₂²	semidirect
50. #499×C₆₇	C₂²	semidirect
51. #498×C₆₇	C₂²	semidirect
52. #479×C₆₇	C₂²	semidirect
53. #479×C₆₇	C₂²	semidirect
56. #478×C₆₇	C₂²	semidirect
57. #478×C₆₇	C₂²	semidirect
58. #477×C₆₇	C₂²	semidirect
59. #477×C₆₇	C₂²	semidirect
61. #476×C₆₇	C₂²	semidirect
62. #475×C₆₇	C₂²	semidirect
67. #474×C₆₇	C₂²	semidirect
68. #474×C₆₇	C₂²	semidirect
69. #446×C₆₇	C₂²	semidirect
70. #446×C₆₇	C₂²	semidirect
71. C₈×Q₈×C₆₇	C₂²	semidirect
99. AGL(1,ℤ/8)×C₆₇	Q₈	semidirect
216. #41×C₆₇	C₂×Q₈	semidirect
232. C₂×D₄×C₆₇	C₂×Q₈	semidirect
249. SD₁₆×C₆₇	C₂×Q₈	semidirect
250. D₈×C₆₇	C₂×Q₈	semidirect
252. M₁₆×C₆₇	C₂×Q₈	semidirect
389. #61381	C₆₇	direct (N × H)
396. #3503	C₂×C₆₇	semidirect
397. #3501	C₂×C₆₇	semidirect
398. #3389	C₂×C₆₇	semidirect
399. #3386	C₂×C₆₇	semidirect
400. #3385	C₂×C₆₇	semidirect
401. #3384	C₂×C₆₇	semidirect
402. #3381	C₂×C₆₇	semidirect
403. Q₈×D₈	C₂×C₆₇	semidirect
404. #3165	C₂×C₆₇	semidirect
405. #3164	C₂×C₆₇	semidirect
406. #3083	C₂×C₆₇	semidirect
407. #3081	C₂×C₆₇	semidirect
408. #3076	C₂×C₆₇	semidirect
409. C₂×#474	C₂×C₆₇	semidirect
410. #2962	C₂×C₆₇	semidirect
413. C₆₇	#61381	direct (N × H)
415. Q₈²	C₂²×C₆₇	semidirect
416. #557	C₂²×C₆₇	semidirect
422. #554	C₂²×C₆₇	semidirect
426. D₄×Q₈	C₂²×C₆₇	semidirect
441. #501	C₂²×C₆₇	semidirect
442. #501	C₂²×C₆₇	semidirect
443. #501	C₂²×C₆₇	semidirect
444. #500	C₂²×C₆₇	semidirect
445. #500	C₂²×C₆₇	semidirect
446. #499	C₂²×C₆₇	semidirect
447. #498	C₂²×C₆₇	semidirect
448. #479	C₂²×C₆₇	semidirect
449. #479	C₂²×C₆₇	semidirect
452. #478	C₂²×C₆₇	semidirect
453. #478	C₂²×C₆₇	semidirect
454. #477	C₂²×C₆₇	semidirect
455. #477	C₂²×C₆₇	semidirect
457. #476	C₂²×C₆₇	semidirect
458. #475	C₂²×C₆₇	semidirect
463. #474	C₂²×C₆₇	semidirect
464. #474	C₂²×C₆₇	semidirect
465. #446	C₂²×C₆₇	semidirect
466. #446	C₂²×C₆₇	semidirect
467. C₈×Q₈	C₂²×C₆₇	semidirect
495. AGL(1,ℤ/8)	Q₈×C₆₇	semidirect
612. #41	C₂×Q₈×C₆₇	semidirect
628. C₂×D₄	C₂×Q₈×C₆₇	semidirect
645. SD₁₆	C₂×Q₈×C₆₇	semidirect
646. D₈	C₂×Q₈×C₆₇	semidirect
648. M₁₆	C₂×Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3503	2
C₆₇×#3501	2
C₆₇×#3389	2
C₆₇×#3386	2
C₆₇×#3385	2
C₆₇×#3384	2
C₆₇×#3381	2
Q₈×D₈×C₆₇	2
C₆₇×#3165	2
C₆₇×#3164	2
C₆₇×#3083	2
C₆₇×#3081	2
C₆₇×#3076	2
C₂×#474×C₆₇	2
C₆₇×#2962	2
C₆₇×#2946	2
C₄×AGL(1,ℤ/8)×C₆₇	2
#61381	67

Subgroups — C67×#61381 #30740632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61381 #30740632