Subgroups — C₆₇×#61578 #30740829

Order 17152.

2114 conjugacy classes of subgroups.

The subgroup lattice has 890 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3506	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#548×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3321	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3313	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3271	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3269	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3213	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3194	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3190	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3113	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3112	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#3004	C₂	1	2	yes	maximal, complemented
14	C₆₇×#3002	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2973	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
19	C₂×#135×C₆₇	C₂²	1	4	yes	complemented
20	#559×C₆₇	—	2	2	no	—
21	#556×C₆₇	C₂²	1	2	yes	complemented
22	#555×C₆₇	C₂²	1	2	yes	complemented
23	#553×C₆₇	—	2	2	no	—
24	#550×C₆₇	C₂²	1	2	yes	complemented
25	#548×C₆₇	—	2	2	no	—
26	#548×C₆₇	—	2	2	no	—
27	#548×C₆₇	C₂²	1	4	yes	—
28	#548×C₆₇	C₂²	1	4	yes	—
29	#547×C₆₇	C₂²	1	2	yes	complemented
30	#538×C₆₇	—	2	1	no	—
31	#538×C₆₇	—	2	1	no	—
32	#535×C₆₇	—	2	1	no	—
33	#533×C₆₇	C₂²	1	1	yes	characteristic
34	#532×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#124×C₆₇	—	2	1	no	—
36	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
37	#525×C₆₇	C₂²	1	2	yes	—
38	#525×C₆₇	C₂²	1	2	yes	—
39	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
40	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#121×C₆₇	—	2	1	no	—
42	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
43	#520×C₆₇	—	2	1	no	—
44	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄×D₄×C₆₇	—	2	1	no	—
46	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
47	#497×C₆₇	C₂²	1	2	yes	complemented
48	#496×C₆₇	C₂²	1	2	yes	complemented
49	#489×C₆₇	C₂²	1	2	yes	complemented
50	#489×C₆₇	C₂²	1	2	yes	complemented
51	#485×C₆₇	C₂²	1	4	yes	complemented
52	#483×C₆₇	C₂²	1	2	yes	—
53	#482×C₆₇	C₂²	1	4	yes	complemented
54	#480×C₆₇	C₂²	1	2	yes	—
55	#477×C₆₇	C₂²	1	2	yes	complemented
56	#475×C₆₇	C₂²	1	2	yes	complemented
57	#469×C₆₇	C₂²	1	4	yes	complemented
58	#468×C₆₇	C₂²	1	4	yes	complemented
59	#467×C₆₇	C₂²	1	4	yes	complemented
60	#465×C₆₇	C₂²	1	4	yes	complemented
61	#464×C₆₇	C₂²	1	4	yes	complemented
62	#463×C₆₇	C₂²	1	4	yes	complemented
63	#462×C₆₇	C₂²	1	2	yes	complemented
64	#460×C₆₇	C₂²	1	2	yes	complemented
65	#452×C₆₇	C₂²	1	4	yes	complemented
66	#452×C₆₇	C₂²	1	4	yes	complemented
67	#451×C₆₇	C₂²	1	2	yes	—
68	#451×C₆₇	C₂²	1	2	yes	—
69	#451×C₆₇	C₂²	1	4	yes	complemented
70	#450×C₆₇	C₂²	1	2	yes	—
71	#449×C₆₇	C₂²	1	4	yes	complemented
72	#449×C₆₇	C₂²	1	4	yes	complemented
73	#448×C₆₇	C₂²	1	2	yes	—
74	#448×C₆₇	C₂²	1	2	yes	—
75	#448×C₆₇	C₂²	1	4	yes	complemented
76	#447×C₆₇	C₂²	1	2	yes	—
77	#441×C₆₇	C₂²	1	4	yes	complemented
78	#440×C₆₇	C₂²	1	4	yes	complemented
79	#435×C₆₇	C₂²	1	4	yes	complemented
80	#435×C₆₇	C₂²	1	4	yes	complemented
81	#432×C₆₇	C₂²	1	1	yes	characteristic
82	#432×C₆₇	C₂²	1	1	yes	characteristic
83	#419×C₆₇	C₂²	1	4	yes	complemented
84	#416×C₆₇	C₂²	1	4	yes	complemented
85	#407×C₆₇	C₂²	1	2	yes	—
86	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
87	C₂×#41×C₆₇	—	2	1	no	—
88	C₂×#41×C₆₇	—	2	2	no	—
89	C₂×#41×C₆₇	—	4	4	no	—
90	C₂×#41×C₆₇	C₂³	1	4	yes	—
91	C₂×#41×C₆₇	C₂³	1	2	yes	—
92	C₂×#41×C₆₇	C₂³	1	2	yes	—
93	C₂²×Q₈×C₆₇	—	2	1	no	—
94	C₂²×Q₈×C₆₇	—	2	2	no	—
95	C₂²×Q₈×C₆₇	—	4	4	no	—
96	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
98	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂²×D₄×C₆₇	—	2	1	no	—
100	C₂²×D₄×C₆₇	—	2	1	no	—
101	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
102	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
103	C₂³×C₄×C₆₇	—	2	1	no	—
104	C₂³×C₄×C₆₇	—	2	1	no	—
105	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
107	#137×C₆₇	—	4	8	no	—
108	#135×C₆₇	—	4	8	no	—
109	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
110	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
111	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
112	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
113	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
114	C₂×D₈×C₆₇	C₂³	1	4	yes	—
115	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
116	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
117	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
118	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
119	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
120	#127×C₆₇	—	2	2	no	—
121	#127×C₆₇	C₂³	1	1	yes	characteristic
122	#126×C₆₇	—	2	2	no	—
123	#126×C₆₇	—	2	2	no	—
124	#125×C₆₇	—	2	1	no	—
125	#124×C₆₇	—	2	2	no	—
126	#124×C₆₇	—	2	4	no	—
127	#124×C₆₇	—	2	4	no	—
128	#124×C₆₇	—	2	2	no	—
129	#124×C₆₇	—	2	2	no	—
130	#124×C₆₇	—	2	1	no	—
131	#124×C₆₇	—	2	1	no	—
132	#124×C₆₇	—	2	1	no	—
133	#124×C₆₇	C₂³	1	2	yes	—
134	#124×C₆₇	C₂³	1	2	yes	—
135	#123×C₆₇	—	2	2	no	—
136	#123×C₆₇	—	2	2	no	—
137	#123×C₆₇	—	2	2	no	—
138	#123×C₆₇	—	2	2	no	—
139	C₂²⋊Q₈×C₆₇	—	2	2	no	—
140	C₂²⋊Q₈×C₆₇	—	2	1	no	—
141	C₂²⋊Q₈×C₆₇	—	2	4	no	—
142	C₂²⋊Q₈×C₆₇	—	2	4	no	—
143	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
144	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
145	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
146	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
147	#121×C₆₇	—	2	4	no	—
148	#121×C₆₇	—	2	4	no	—
149	#121×C₆₇	—	2	2	no	—
150	#121×C₆₇	—	2	2	no	—
151	#121×C₆₇	—	2	2	no	—
152	#121×C₆₇	—	2	1	no	—
153	#121×C₆₇	—	2	2	no	—
154	#121×C₆₇	—	2	2	no	—
155	#121×C₆₇	C₂³	1	2	yes	—
156	#121×C₆₇	C₂³	1	2	yes	—
157	#121×C₆₇	C₂³	1	2	yes	—
158	#121×C₆₇	C₂³	1	4	yes	—
159	#120×C₆₇	—	2	2	no	—
160	#120×C₆₇	—	2	2	no	—
161	C₄×Q₈×C₆₇	—	2	2	no	—
162	C₄×Q₈×C₆₇	—	2	1	no	—
163	C₄×Q₈×C₆₇	—	2	1	no	—
164	C₄×Q₈×C₆₇	—	2	1	no	—
165	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
166	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
167	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
168	C₄×D₄×C₆₇	—	2	2	no	—
169	C₄×D₄×C₆₇	—	2	2	no	—
170	C₄×D₄×C₆₇	—	2	2	no	—
171	C₄×D₄×C₆₇	—	2	2	no	—
172	C₄×D₄×C₆₇	—	2	2	no	—
173	C₄×D₄×C₆₇	—	2	4	no	—
174	C₄×D₄×C₆₇	—	2	4	no	—
175	C₄×D₄×C₆₇	—	2	1	no	—
176	C₄×D₄×C₆₇	D₄	1	2	yes	—
177	C₄×D₄×C₆₇	D₄	1	2	yes	—
178	C₄×D₄×C₆₇	C₂³	1	2	yes	—
179	C₄×D₄×C₆₇	C₂³	1	2	yes	—
180	C₄×D₄×C₆₇	C₂³	1	2	yes	—
181	#117×C₆₇	—	2	1	no	—
182	#117×C₆₇	—	2	2	no	—
183	#117×C₆₇	C₂³	1	2	yes	—
184	#117×C₆₇	C₂³	1	2	yes	—
185	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
189	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
190	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
191	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
192	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
193	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
194	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
195	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
196	C₂×C₄²×C₆₇	—	2	1	no	—
197	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
198	#107×C₆₇	C₂³	1	4	yes	—
199	#106×C₆₇	C₂³	1	4	yes	—
200	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
201	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
202	#103×C₆₇	C₂³	1	4	yes	—
203	#103×C₆₇	C₂³	1	4	yes	—
204	#103×C₆₇	C₂³	1	4	yes	—
205	#103×C₆₇	C₂³	1	4	yes	—
206	#103×C₆₇	C₂³	1	4	yes	—
207	#102×C₆₇	C₂³	1	4	yes	—
208	#102×C₆₇	C₂³	1	4	yes	—
209	#102×C₆₇	C₂³	1	4	yes	—
210	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
211	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
212	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
213	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
214	#97×C₆₇	C₂³	1	2	yes	—
215	#97×C₆₇	C₂³	1	2	yes	—
216	C₂⁴×C₆₇	—	2	1	no	—
217	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
218	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
219	#41×C₆₇	—	2	8	no	—
220	#41×C₆₇	—	4	4	no	—
221	#41×C₆₇	—	4	4	no	—
222	#41×C₆₇	—	4	4	no	—
223	#41×C₆₇	—	4	4	no	—
224	#41×C₆₇	—	4	8	no	—
225	C₂×Q₈×C₆₇	—	2	4	no	—
226	C₂×Q₈×C₆₇	—	2	1	no	—
227	C₂×Q₈×C₆₇	—	2	1	no	—
228	C₂×Q₈×C₆₇	—	2	4	no	—
229	C₂×Q₈×C₆₇	—	2	1	no	—
230	C₂×Q₈×C₆₇	—	2	1	no	—
231	C₂×Q₈×C₆₇	—	2	8	no	—
232	C₂×Q₈×C₆₇	—	4	4	no	—
233	C₂×Q₈×C₆₇	—	4	2	no	—
234	C₂×Q₈×C₆₇	—	4	4	no	—
235	C₂×Q₈×C₆₇	—	4	8	no	—
236	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
237	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
238	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
239	C₂×D₄×C₆₇	—	2	2	no	—
240	C₂×D₄×C₆₇	—	2	4	no	—
241	C₂×D₄×C₆₇	—	2	4	no	—
242	C₂×D₄×C₆₇	—	2	2	no	—
243	C₂×D₄×C₆₇	—	2	2	no	—
244	C₂×D₄×C₆₇	—	2	2	no	—
245	C₂×D₄×C₆₇	—	2	2	no	—
246	C₂×D₄×C₆₇	—	2	1	no	—
247	C₂×D₄×C₆₇	—	2	1	no	—
248	C₂×D₄×C₆₇	—	2	1	no	—
249	C₂×D₄×C₆₇	—	4	2	no	—
250	C₂×D₄×C₆₇	—	4	2	no	—
251	C₂×D₄×C₆₇	—	4	4	no	—
252	C₂×D₄×C₆₇	—	4	4	no	—
253	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
254	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
255	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
256	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
257	C₂²×C₄×C₆₇	—	2	1	no	—
258	C₂²×C₄×C₆₇	—	2	1	no	—
259	C₂²×C₄×C₆₇	—	2	1	no	—
260	C₂²×C₄×C₆₇	—	2	4	no	—
261	C₂²×C₄×C₆₇	—	2	2	no	—
262	C₂²×C₄×C₆₇	—	2	4	no	—
263	C₂²×C₄×C₆₇	—	2	2	no	—
264	C₂²×C₄×C₆₇	—	2	2	no	—
265	C₂²×C₄×C₆₇	—	4	2	no	—
266	C₂²×C₄×C₆₇	—	4	2	no	—
267	C₂²×C₄×C₆₇	—	4	4	no	—
268	C₂²×C₄×C₆₇	—	4	4	no	—
269	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
270	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
271	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
272	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
273	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
274	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
275	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
276	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
277	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
278	Q₁₆×C₆₇	—	2	8	no	—
279	Q₁₆×C₆₇	—	4	4	no	—
280	SD₁₆×C₆₇	—	2	8	no	—
281	SD₁₆×C₆₇	—	4	4	no	—
282	SD₁₆×C₆₇	—	4	4	no	—
283	D₈×C₆₇	—	4	4	no	—
284	M₁₆×C₆₇	—	4	4	no	—
285	C₂×C₈×C₆₇	—	4	4	no	—
286	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
287	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
288	C₄⋊C₄×C₆₇	—	2	1	no	—
289	C₄⋊C₄×C₆₇	—	2	4	no	—
290	C₄⋊C₄×C₆₇	—	2	2	no	—
291	C₄⋊C₄×C₆₇	—	2	2	no	—
292	C₄⋊C₄×C₆₇	—	2	2	no	—
293	C₄⋊C₄×C₆₇	—	2	2	no	—
294	C₄⋊C₄×C₆₇	—	2	1	no	—
295	C₄⋊C₄×C₆₇	—	2	1	no	—
296	C₄⋊C₄×C₆₇	—	2	2	no	—
297	C₄⋊C₄×C₆₇	—	2	1	no	—
298	C₄⋊C₄×C₆₇	—	2	1	no	—
299	C₄⋊C₄×C₆₇	—	2	2	no	—
300	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
301	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
302	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
303	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
304	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
305	C₂²⋊C₄×C₆₇	—	2	4	no	—
306	C₂²⋊C₄×C₆₇	—	2	2	no	—
307	C₂²⋊C₄×C₆₇	—	2	2	no	—
308	C₂²⋊C₄×C₆₇	—	2	2	no	—
309	C₂²⋊C₄×C₆₇	—	2	2	no	—
310	C₂²⋊C₄×C₆₇	—	2	2	no	—
311	C₂²⋊C₄×C₆₇	—	2	2	no	—
312	C₂²⋊C₄×C₆₇	—	2	2	no	—
313	C₂²⋊C₄×C₆₇	—	2	2	no	—
314	C₂²⋊C₄×C₆₇	—	2	2	no	—
315	C₂²⋊C₄×C₆₇	—	2	4	no	—
316	C₂²⋊C₄×C₆₇	—	2	2	no	—
317	C₂²⋊C₄×C₆₇	—	2	2	no	—
318	C₂²⋊C₄×C₆₇	—	2	2	no	—
319	C₂²⋊C₄×C₆₇	—	2	2	no	—
320	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
321	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
322	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
323	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
324	C₄²×C₆₇	—	2	2	no	—
325	C₄²×C₆₇	—	2	2	no	—
326	C₄²×C₆₇	—	2	2	no	—
327	C₄²×C₆₇	—	2	2	no	—
328	C₄²×C₆₇	—	2	2	no	—
329	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
330	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
331	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
332	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
333	C₂³×C₆₇	—	2	2	no	—
334	C₂³×C₆₇	—	2	2	no	—
335	C₂³×C₆₇	—	2	2	no	—
336	C₂³×C₆₇	—	2	2	no	—
337	C₂³×C₆₇	—	4	1	no	—
338	C₂³×C₆₇	—	4	1	no	—
339	C₂³×C₆₇	—	4	1	no	—
340	C₂³×C₆₇	—	4	1	no	—
341	C₂³×C₆₇	—	4	2	no	—
342	C₂³×C₆₇	—	4	2	no	—
343	C₂³×C₆₇	—	4	2	no	—
344	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
345	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
346	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
347	Q₈×C₆₇	—	2	4	no	—
348	Q₈×C₆₇	—	2	2	no	—
349	Q₈×C₆₇	—	2	4	no	—
350	Q₈×C₆₇	—	2	4	no	—
351	Q₈×C₆₇	—	4	8	no	—
352	Q₈×C₆₇	—	4	2	no	—
353	Q₈×C₆₇	—	4	2	no	—
354	Q₈×C₆₇	—	8	1	no	—
355	Q₈×C₆₇	H(5,𝔽₂)	1	4	yes	—
356	D₄×C₆₇	—	2	4	no	—
357	D₄×C₆₇	—	2	4	no	—
358	D₄×C₆₇	—	4	2	no	—
359	D₄×C₆₇	—	4	2	no	—
360	D₄×C₆₇	—	4	8	no	—
361	D₄×C₆₇	—	4	2	no	—
362	D₄×C₆₇	—	4	2	no	—
363	D₄×C₆₇	—	4	8	no	—
364	D₄×C₆₇	—	4	4	no	—
365	D₄×C₆₇	—	8	1	no	—
366	D₄×C₆₇	—	8	2	no	—
367	D₄×C₆₇	—	8	2	no	—
368	D₄×C₆₇	—	8	2	no	—
369	D₄×C₆₇	—	8	2	no	—
370	C₂×C₄×C₆₇	—	2	4	no	—
371	C₂×C₄×C₆₇	—	2	2	no	—
372	C₂×C₄×C₆₇	—	2	2	no	—
373	C₂×C₄×C₆₇	—	2	4	no	—
374	C₂×C₄×C₆₇	—	2	4	no	—
375	C₂×C₄×C₆₇	—	2	2	no	—
376	C₂×C₄×C₆₇	—	2	2	no	—
377	C₂×C₄×C₆₇	—	2	2	no	—
378	C₂×C₄×C₆₇	—	2	2	no	—
379	C₂×C₄×C₆₇	—	2	2	no	—
380	C₂×C₄×C₆₇	—	2	2	no	—
381	C₂×C₄×C₆₇	—	4	1	no	—
382	C₂×C₄×C₆₇	—	4	2	no	—
383	C₂×C₄×C₆₇	—	4	1	no	—
384	C₂×C₄×C₆₇	—	4	2	no	—
385	C₂×C₄×C₆₇	—	4	1	no	—
386	C₂×C₄×C₆₇	—	4	8	no	—
387	C₂×C₄×C₆₇	—	4	8	no	—
388	C₂×C₄×C₆₇	—	4	4	no	—
389	C₂×C₄×C₆₇	—	4	2	no	—
390	C₂×C₄×C₆₇	—	4	2	no	—
391	C₂×C₄×C₆₇	—	8	2	no	—
392	C₂×C₄×C₆₇	—	8	2	no	—
393	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
394	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
395	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
396	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
397	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
398	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
399	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
400	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
401	C₈×C₆₇	—	2	4	no	—
402	C₈×C₆₇	—	2	4	no	—
403	C₂²×C₆₇	—	2	1	no	—
404	C₂²×C₆₇	—	2	1	no	—
405	C₂²×C₆₇	—	2	2	no	—
406	C₂²×C₆₇	—	2	2	no	—
407	C₂²×C₆₇	—	2	2	no	—
408	C₂²×C₆₇	—	2	2	no	—
409	C₂²×C₆₇	—	2	2	no	—
410	C₂²×C₆₇	—	2	2	no	—
411	C₂²×C₆₇	—	4	4	no	—
412	C₂²×C₆₇	—	4	2	no	—
413	C₂²×C₆₇	—	4	2	no	—
414	C₂²×C₆₇	—	4	2	no	—
415	C₂²×C₆₇	—	4	2	no	—
416	C₂²×C₆₇	—	4	2	no	—
417	C₂²×C₆₇	—	4	2	no	—
418	C₂²×C₆₇	—	8	2	no	—
419	C₂²×C₆₇	—	8	4	no	—
420	C₂²×C₆₇	#575	1	2	yes	—
421	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
422	C₄×C₆₇	—	2	1	no	—
423	C₄×C₆₇	—	2	1	no	—
424	C₄×C₆₇	—	2	4	no	—
425	C₄×C₆₇	—	2	2	no	—
426	C₄×C₆₇	—	2	1	no	—
427	C₄×C₆₇	—	2	2	no	—
428	C₄×C₆₇	—	2	1	no	—
429	C₄×C₆₇	—	2	1	no	—
430	C₄×C₆₇	—	4	2	no	—
431	C₄×C₆₇	—	4	2	no	—
432	C₄×C₆₇	—	4	2	no	—
433	C₄×C₆₇	—	4	2	no	—
434	C₄×C₆₇	—	4	2	no	—
435	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
436	#61578	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
437	C₂×C₆₇	—	2	2	no	—
438	C₂×C₆₇	—	4	2	no	—
439	C₂×C₆₇	—	4	2	no	—
440	C₂×C₆₇	—	8	2	no	—
441	C₂×C₆₇	—	8	2	no	—
442	C₂×C₆₇	#3484	1	1	yes	characteristic
443	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
444	C₂×C₆₇	#3586	1	1	yes	characteristic
445	#3506	C₂×C₆₇	1	1	yes	characteristic, complemented
446	C₂×#548	C₂×C₆₇	1	1	yes	characteristic, complemented
447	#3321	C₂×C₆₇	1	4	yes	complemented
448	#3313	C₂×C₆₇	1	4	yes	complemented
449	#3271	C₂×C₆₇	1	4	yes	complemented
450	#3269	C₂×C₆₇	1	4	yes	complemented
451	#3215	C₂×C₆₇	1	1	yes	characteristic, complemented
452	#3213	C₂×C₆₇	1	1	yes	characteristic, complemented
453	#3194	C₂×C₆₇	1	2	yes	complemented
454	#3190	C₂×C₆₇	1	2	yes	complemented
455	#3113	C₂×C₆₇	1	1	yes	characteristic, complemented
456	#3112	C₂×C₆₇	1	1	yes	characteristic, complemented
457	#3004	C₂×C₆₇	1	2	yes	complemented
458	#3002	C₂×C₆₇	1	2	yes	complemented
459	#2973	C₂×C₆₇	1	1	yes	characteristic, complemented
460	C₆₇	#61578	1	1	yes	characteristic, direct product factor, Sylow
461	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
462	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
463	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
464	C₂×#135	C₂²×C₆₇	1	4	yes	complemented
465	#559	—	2	2	no	—
466	#556	C₂²×C₆₇	1	2	yes	complemented
467	#555	C₂²×C₆₇	1	2	yes	complemented
468	#553	—	2	2	no	—
469	#550	C₂²×C₆₇	1	2	yes	complemented
470	#548	—	2	2	no	—
471	#548	—	2	2	no	—
472	#548	C₂²×C₆₇	1	4	yes	—
473	#548	C₂²×C₆₇	1	4	yes	—
474	#547	C₂²×C₆₇	1	2	yes	complemented
475	#538	—	2	1	no	—
476	#538	—	2	1	no	—
477	#535	—	2	1	no	—
478	#533	C₂²×C₆₇	1	1	yes	characteristic
479	#532	C₂²×C₆₇	1	1	yes	characteristic
480	C₂×#124	—	2	1	no	—
481	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
482	#525	C₂²×C₆₇	1	2	yes	—
483	#525	C₂²×C₆₇	1	2	yes	—
484	C₂×C₂²⋊Q₈	—	2	1	no	—
485	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
486	C₂×#121	—	2	1	no	—
487	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
488	#520	—	2	1	no	—
489	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
490	C₂×C₄×D₄	—	2	1	no	—
491	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
492	#497	C₂²×C₆₇	1	2	yes	complemented
493	#496	C₂²×C₆₇	1	2	yes	complemented
494	#489	C₂²×C₆₇	1	2	yes	complemented
495	#489	C₂²×C₆₇	1	2	yes	complemented
496	#485	C₂²×C₆₇	1	4	yes	complemented
497	#483	C₂²×C₆₇	1	2	yes	—
498	#482	C₂²×C₆₇	1	4	yes	complemented
499	#480	C₂²×C₆₇	1	2	yes	—
500	#477	C₂²×C₆₇	1	2	yes	complemented
501	#475	C₂²×C₆₇	1	2	yes	complemented
502	#469	C₂²×C₆₇	1	4	yes	complemented
503	#468	C₂²×C₆₇	1	4	yes	complemented
504	#467	C₂²×C₆₇	1	4	yes	complemented
505	#465	C₂²×C₆₇	1	4	yes	complemented
506	#464	C₂²×C₆₇	1	4	yes	complemented
507	#463	C₂²×C₆₇	1	4	yes	complemented
508	#462	C₂²×C₆₇	1	2	yes	complemented
509	#460	C₂²×C₆₇	1	2	yes	complemented
510	#452	C₂²×C₆₇	1	4	yes	complemented
511	#452	C₂²×C₆₇	1	4	yes	complemented
512	#451	C₂²×C₆₇	1	2	yes	—
513	#451	C₂²×C₆₇	1	2	yes	—
514	#451	C₂²×C₆₇	1	4	yes	complemented
515	#450	C₂²×C₆₇	1	2	yes	—
516	#449	C₂²×C₆₇	1	4	yes	complemented
517	#449	C₂²×C₆₇	1	4	yes	complemented
518	#448	C₂²×C₆₇	1	2	yes	—
519	#448	C₂²×C₆₇	1	2	yes	—
520	#448	C₂²×C₆₇	1	4	yes	complemented
521	#447	C₂²×C₆₇	1	2	yes	—
522	#441	C₂²×C₆₇	1	4	yes	complemented
523	#440	C₂²×C₆₇	1	4	yes	complemented
524	#435	C₂²×C₆₇	1	4	yes	complemented
525	#435	C₂²×C₆₇	1	4	yes	complemented
526	#432	C₂²×C₆₇	1	1	yes	characteristic
527	#432	C₂²×C₆₇	1	1	yes	characteristic
528	#419	C₂²×C₆₇	1	4	yes	complemented
529	#416	C₂²×C₆₇	1	4	yes	complemented
530	#407	C₂²×C₆₇	1	2	yes	—
531	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
532	C₂×#41	—	2	1	no	—
533	C₂×#41	—	2	2	no	—
534	C₂×#41	—	4	4	no	—
535	C₂×#41	C₂³×C₆₇	1	4	yes	—
536	C₂×#41	C₂³×C₆₇	1	2	yes	—
537	C₂×#41	C₂³×C₆₇	1	2	yes	—
538	C₂²×Q₈	—	2	1	no	—
539	C₂²×Q₈	—	2	2	no	—
540	C₂²×Q₈	—	4	4	no	—
541	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
542	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
543	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
544	C₂²×D₄	—	2	1	no	—
545	C₂²×D₄	—	2	1	no	—
546	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
547	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
548	C₂³×C₄	—	2	1	no	—
549	C₂³×C₄	—	2	1	no	—
550	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
551	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
552	#137	—	4	8	no	—
553	#135	—	4	8	no	—
554	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
555	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
556	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
557	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
558	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
559	C₂×D₈	C₂³×C₆₇	1	4	yes	—
560	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
561	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
562	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
563	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
564	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
565	#127	—	2	2	no	—
566	#127	C₂³×C₆₇	1	1	yes	characteristic
567	#126	—	2	2	no	—
568	#126	—	2	2	no	—
569	#125	—	2	1	no	—
570	#124	—	2	2	no	—
571	#124	—	2	4	no	—
572	#124	—	2	4	no	—
573	#124	—	2	2	no	—
574	#124	—	2	2	no	—
575	#124	—	2	1	no	—
576	#124	—	2	1	no	—
577	#124	—	2	1	no	—
578	#124	C₂³×C₆₇	1	2	yes	—
579	#124	C₂³×C₆₇	1	2	yes	—
580	#123	—	2	2	no	—
581	#123	—	2	2	no	—
582	#123	—	2	2	no	—
583	#123	—	2	2	no	—
584	C₂²⋊Q₈	—	2	2	no	—
585	C₂²⋊Q₈	—	2	1	no	—
586	C₂²⋊Q₈	—	2	4	no	—
587	C₂²⋊Q₈	—	2	4	no	—
588	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
589	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
590	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
591	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
592	#121	—	2	4	no	—
593	#121	—	2	4	no	—
594	#121	—	2	2	no	—
595	#121	—	2	2	no	—
596	#121	—	2	2	no	—
597	#121	—	2	1	no	—
598	#121	—	2	2	no	—
599	#121	—	2	2	no	—
600	#121	C₂³×C₆₇	1	4	yes	—
601	#121	C₂³×C₆₇	1	2	yes	—
602	#121	C₂³×C₆₇	1	2	yes	—
603	#121	C₂³×C₆₇	1	2	yes	—
604	#120	—	2	2	no	—
605	#120	—	2	2	no	—
606	C₄×Q₈	—	2	2	no	—
607	C₄×Q₈	—	2	1	no	—
608	C₄×Q₈	—	2	1	no	—
609	C₄×Q₈	—	2	1	no	—
610	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
611	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
612	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
613	C₄×D₄	—	2	2	no	—
614	C₄×D₄	—	2	2	no	—
615	C₄×D₄	—	2	2	no	—
616	C₄×D₄	—	2	2	no	—
617	C₄×D₄	—	2	2	no	—
618	C₄×D₄	—	2	4	no	—
619	C₄×D₄	—	2	4	no	—
620	C₄×D₄	—	2	1	no	—
621	C₄×D₄	D₄×C₆₇	1	2	yes	—
622	C₄×D₄	D₄×C₆₇	1	2	yes	—
623	C₄×D₄	C₂³×C₆₇	1	2	yes	—
624	C₄×D₄	C₂³×C₆₇	1	2	yes	—
625	C₄×D₄	C₂³×C₆₇	1	2	yes	—
626	#117	—	2	1	no	—
627	#117	—	2	2	no	—
628	#117	C₂³×C₆₇	1	2	yes	—
629	#117	C₂³×C₆₇	1	2	yes	—
630	C₂×C₄⋊C₄	—	2	1	no	—
631	C₂×C₄⋊C₄	—	2	1	no	—
632	C₂×C₄⋊C₄	—	2	2	no	—
633	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
634	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
635	C₂×C₂²⋊C₄	—	2	1	no	—
636	C₂×C₂²⋊C₄	—	2	1	no	—
637	C₂×C₂²⋊C₄	—	2	1	no	—
638	C₂×C₂²⋊C₄	—	2	1	no	—
639	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
640	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
641	C₂×C₄²	—	2	1	no	—
642	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
643	#107	C₂³×C₆₇	1	4	yes	—
644	#106	C₂³×C₆₇	1	4	yes	—
645	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
646	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
647	#103	C₂³×C₆₇	1	4	yes	—
648	#103	C₂³×C₆₇	1	4	yes	—
649	#103	C₂³×C₆₇	1	4	yes	—
650	#103	C₂³×C₆₇	1	4	yes	—
651	#103	C₂³×C₆₇	1	4	yes	—
652	#102	C₂³×C₆₇	1	4	yes	—
653	#102	C₂³×C₆₇	1	4	yes	—
654	#102	C₂³×C₆₇	1	4	yes	—
655	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
656	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
657	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
658	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
659	#97	C₂³×C₆₇	1	2	yes	—
660	#97	C₂³×C₆₇	1	2	yes	—
661	C₂⁴	—	2	1	no	—
662	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
663	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
664	#41	—	2	8	no	—
665	#41	—	4	4	no	—
666	#41	—	4	4	no	—
667	#41	—	4	4	no	—
668	#41	—	4	4	no	—
669	#41	—	4	8	no	—
670	C₂×Q₈	—	2	4	no	—
671	C₂×Q₈	—	2	1	no	—
672	C₂×Q₈	—	2	1	no	—
673	C₂×Q₈	—	2	4	no	—
674	C₂×Q₈	—	2	1	no	—
675	C₂×Q₈	—	2	1	no	—
676	C₂×Q₈	—	2	8	no	—
677	C₂×Q₈	—	4	4	no	—
678	C₂×Q₈	—	4	2	no	—
679	C₂×Q₈	—	4	4	no	—
680	C₂×Q₈	—	4	8	no	—
681	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
682	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
683	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
684	C₂×D₄	—	2	2	no	—
685	C₂×D₄	—	2	4	no	—
686	C₂×D₄	—	2	4	no	—
687	C₂×D₄	—	2	2	no	—
688	C₂×D₄	—	2	2	no	—
689	C₂×D₄	—	2	2	no	—
690	C₂×D₄	—	2	2	no	—
691	C₂×D₄	—	2	1	no	—
692	C₂×D₄	—	2	1	no	—
693	C₂×D₄	—	2	1	no	—
694	C₂×D₄	—	4	2	no	—
695	C₂×D₄	—	4	2	no	—
696	C₂×D₄	—	4	4	no	—
697	C₂×D₄	—	4	4	no	—
698	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
699	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
700	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
701	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
702	C₂²×C₄	—	2	1	no	—
703	C₂²×C₄	—	2	1	no	—
704	C₂²×C₄	—	2	1	no	—
705	C₂²×C₄	—	2	4	no	—
706	C₂²×C₄	—	2	2	no	—
707	C₂²×C₄	—	2	4	no	—
708	C₂²×C₄	—	2	2	no	—
709	C₂²×C₄	—	2	2	no	—
710	C₂²×C₄	—	4	2	no	—
711	C₂²×C₄	—	4	2	no	—
712	C₂²×C₄	—	4	4	no	—
713	C₂²×C₄	—	4	4	no	—
714	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
715	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
716	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
717	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
718	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
719	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
720	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
721	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
722	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
723	Q₁₆	—	2	8	no	—
724	Q₁₆	—	4	4	no	—
725	SD₁₆	—	2	8	no	—
726	SD₁₆	—	4	4	no	—
727	SD₁₆	—	4	4	no	—
728	D₈	—	4	4	no	—
729	M₁₆	—	4	4	no	—
730	C₂×C₈	—	4	4	no	—
731	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
732	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
733	C₄⋊C₄	—	2	1	no	—
734	C₄⋊C₄	—	2	4	no	—
735	C₄⋊C₄	—	2	2	no	—
736	C₄⋊C₄	—	2	2	no	—
737	C₄⋊C₄	—	2	2	no	—
738	C₄⋊C₄	—	2	2	no	—
739	C₄⋊C₄	—	2	1	no	—
740	C₄⋊C₄	—	2	1	no	—
741	C₄⋊C₄	—	2	2	no	—
742	C₄⋊C₄	—	2	1	no	—
743	C₄⋊C₄	—	2	1	no	—
744	C₄⋊C₄	—	2	2	no	—
745	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
746	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
747	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
748	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
749	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
750	C₂²⋊C₄	—	2	4	no	—
751	C₂²⋊C₄	—	2	2	no	—
752	C₂²⋊C₄	—	2	2	no	—
753	C₂²⋊C₄	—	2	2	no	—
754	C₂²⋊C₄	—	2	2	no	—
755	C₂²⋊C₄	—	2	2	no	—
756	C₂²⋊C₄	—	2	2	no	—
757	C₂²⋊C₄	—	2	2	no	—
758	C₂²⋊C₄	—	2	2	no	—
759	C₂²⋊C₄	—	2	2	no	—
760	C₂²⋊C₄	—	2	4	no	—
761	C₂²⋊C₄	—	2	2	no	—
762	C₂²⋊C₄	—	2	2	no	—
763	C₂²⋊C₄	—	2	2	no	—
764	C₂²⋊C₄	—	2	2	no	—
765	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
766	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
767	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
768	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
769	C₄²	—	2	2	no	—
770	C₄²	—	2	2	no	—
771	C₄²	—	2	2	no	—
772	C₄²	—	2	2	no	—
773	C₄²	—	2	2	no	—
774	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
775	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
776	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
777	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
778	C₂³	—	2	2	no	—
779	C₂³	—	2	2	no	—
780	C₂³	—	2	2	no	—
781	C₂³	—	2	2	no	—
782	C₂³	—	4	1	no	—
783	C₂³	—	4	1	no	—
784	C₂³	—	4	1	no	—
785	C₂³	—	4	1	no	—
786	C₂³	—	4	2	no	—
787	C₂³	—	4	2	no	—
788	C₂³	—	4	2	no	—
789	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
790	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
791	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
792	Q₈	—	2	4	no	—
793	Q₈	—	2	2	no	—
794	Q₈	—	2	4	no	—
795	Q₈	—	2	4	no	—
796	Q₈	—	4	8	no	—
797	Q₈	—	4	2	no	—
798	Q₈	—	4	2	no	—
799	Q₈	—	8	1	no	—
800	Q₈	H(5,𝔽₂)×C₆₇	1	4	yes	—
801	D₄	—	2	4	no	—
802	D₄	—	2	4	no	—
803	D₄	—	4	2	no	—
804	D₄	—	4	2	no	—
805	D₄	—	4	8	no	—
806	D₄	—	4	2	no	—
807	D₄	—	4	2	no	—
808	D₄	—	4	8	no	—
809	D₄	—	4	4	no	—
810	D₄	—	8	1	no	—
811	D₄	—	8	2	no	—
812	D₄	—	8	2	no	—
813	D₄	—	8	2	no	—
814	D₄	—	8	2	no	—
815	C₂×C₄	—	2	4	no	—
816	C₂×C₄	—	2	2	no	—
817	C₂×C₄	—	2	2	no	—
818	C₂×C₄	—	2	4	no	—
819	C₂×C₄	—	2	4	no	—
820	C₂×C₄	—	2	2	no	—
821	C₂×C₄	—	2	2	no	—
822	C₂×C₄	—	2	2	no	—
823	C₂×C₄	—	2	2	no	—
824	C₂×C₄	—	2	2	no	—
825	C₂×C₄	—	2	2	no	—
826	C₂×C₄	—	4	1	no	—
827	C₂×C₄	—	4	2	no	—
828	C₂×C₄	—	4	1	no	—
829	C₂×C₄	—	4	2	no	—
830	C₂×C₄	—	4	1	no	—
831	C₂×C₄	—	4	8	no	—
832	C₂×C₄	—	4	8	no	—
833	C₂×C₄	—	4	4	no	—
834	C₂×C₄	—	4	2	no	—
835	C₂×C₄	—	4	2	no	—
836	C₂×C₄	—	8	2	no	—
837	C₂×C₄	—	8	2	no	—
838	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
839	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
840	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
841	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
842	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
843	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
844	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
845	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
846	C₈	—	2	4	no	—
847	C₈	—	2	4	no	—
848	C₂²	—	2	1	no	—
849	C₂²	—	2	1	no	—
850	C₂²	—	2	2	no	—
851	C₂²	—	2	2	no	—
852	C₂²	—	2	2	no	—
853	C₂²	—	2	2	no	—
854	C₂²	—	2	2	no	—
855	C₂²	—	2	2	no	—
856	C₂²	—	4	4	no	—
857	C₂²	—	4	2	no	—
858	C₂²	—	4	2	no	—
859	C₂²	—	4	2	no	—
860	C₂²	—	4	2	no	—
861	C₂²	—	4	2	no	—
862	C₂²	—	4	2	no	—
863	C₂²	—	8	2	no	—
864	C₂²	—	8	4	no	—
865	C₂²	#575×C₆₇	1	2	yes	—
866	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
867	C₄	—	2	1	no	—
868	C₄	—	2	1	no	—
869	C₄	—	2	1	no	—
870	C₄	—	2	1	no	—
871	C₄	—	2	4	no	—
872	C₄	—	2	2	no	—
873	C₄	—	2	1	no	—
874	C₄	—	2	2	no	—
875	C₄	—	4	2	no	—
876	C₄	—	4	2	no	—
877	C₄	—	4	2	no	—
878	C₄	—	4	2	no	—
879	C₄	—	4	2	no	—
880	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
881	C₂	—	2	2	no	—
882	C₂	—	4	2	no	—
883	C₂	—	4	2	no	—
884	C₂	—	8	2	no	—
885	C₂	—	8	2	no	—
886	C₂	C₆₇×#3484	1	1	yes	characteristic
887	C₂	C₂×#575×C₆₇	1	1	yes	characteristic
888	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 ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3506	C₂	semidirect
2. C₂×#548×C₆₇	C₂	semidirect
3. C₆₇×#3321	C₂	semidirect
4. C₆₇×#3313	C₂	semidirect
5. C₆₇×#3271	C₂	semidirect
6. C₆₇×#3269	C₂	semidirect
7. C₆₇×#3215	C₂	semidirect
8. C₆₇×#3213	C₂	semidirect
9. C₆₇×#3194	C₂	semidirect
10. C₆₇×#3190	C₂	semidirect
11. C₆₇×#3113	C₂	semidirect
12. C₆₇×#3112	C₂	semidirect
13. C₆₇×#3004	C₂	semidirect
14. C₆₇×#3002	C₂	semidirect
15. C₆₇×#2973	C₂	semidirect
18. C₂×#137×C₆₇	C₂²	semidirect
19. C₂×#135×C₆₇	C₂²	semidirect
21. #556×C₆₇	C₂²	semidirect
22. #555×C₆₇	C₂²	semidirect
24. #550×C₆₇	C₂²	semidirect
29. #547×C₆₇	C₂²	semidirect
47. #497×C₆₇	C₂²	semidirect
48. #496×C₆₇	C₂²	semidirect
49. #489×C₆₇	C₂²	semidirect
50. #489×C₆₇	C₂²	semidirect
51. #485×C₆₇	C₂²	semidirect
53. #482×C₆₇	C₂²	semidirect
55. #477×C₆₇	C₂²	semidirect
56. #475×C₆₇	C₂²	semidirect
57. #469×C₆₇	C₂²	semidirect
58. #468×C₆₇	C₂²	semidirect
59. #467×C₆₇	C₂²	semidirect
60. #465×C₆₇	C₂²	semidirect
61. #464×C₆₇	C₂²	semidirect
62. #463×C₆₇	C₂²	semidirect
63. #462×C₆₇	C₂²	semidirect
64. #460×C₆₇	C₂²	semidirect
65. #452×C₆₇	C₂²	semidirect
66. #452×C₆₇	C₂²	semidirect
69. #451×C₆₇	C₂²	semidirect
71. #449×C₆₇	C₂²	semidirect
72. #449×C₆₇	C₂²	semidirect
75. #448×C₆₇	C₂²	semidirect
77. #441×C₆₇	C₂²	semidirect
78. #440×C₆₇	C₂²	semidirect
79. #435×C₆₇	C₂²	semidirect
80. #435×C₆₇	C₂²	semidirect
83. #419×C₆₇	C₂²	semidirect
84. #416×C₆₇	C₂²	semidirect
436. #61578	C₆₇	direct (N × H)
445. #3506	C₂×C₆₇	semidirect
446. C₂×#548	C₂×C₆₇	semidirect
447. #3321	C₂×C₆₇	semidirect
448. #3313	C₂×C₆₇	semidirect
449. #3271	C₂×C₆₇	semidirect
450. #3269	C₂×C₆₇	semidirect
451. #3215	C₂×C₆₇	semidirect
452. #3213	C₂×C₆₇	semidirect
453. #3194	C₂×C₆₇	semidirect
454. #3190	C₂×C₆₇	semidirect
455. #3113	C₂×C₆₇	semidirect
456. #3112	C₂×C₆₇	semidirect
457. #3004	C₂×C₆₇	semidirect
458. #3002	C₂×C₆₇	semidirect
459. #2973	C₂×C₆₇	semidirect
460. C₆₇	#61578	direct (N × H)
463. C₂×#137	C₂²×C₆₇	semidirect
464. C₂×#135	C₂²×C₆₇	semidirect
466. #556	C₂²×C₆₇	semidirect
467. #555	C₂²×C₆₇	semidirect
469. #550	C₂²×C₆₇	semidirect
474. #547	C₂²×C₆₇	semidirect
492. #497	C₂²×C₆₇	semidirect
493. #496	C₂²×C₆₇	semidirect
494. #489	C₂²×C₆₇	semidirect
495. #489	C₂²×C₆₇	semidirect
496. #485	C₂²×C₆₇	semidirect
498. #482	C₂²×C₆₇	semidirect
500. #477	C₂²×C₆₇	semidirect
501. #475	C₂²×C₆₇	semidirect
502. #469	C₂²×C₆₇	semidirect
503. #468	C₂²×C₆₇	semidirect
504. #467	C₂²×C₆₇	semidirect
505. #465	C₂²×C₆₇	semidirect
506. #464	C₂²×C₆₇	semidirect
507. #463	C₂²×C₆₇	semidirect
508. #462	C₂²×C₆₇	semidirect
509. #460	C₂²×C₆₇	semidirect
510. #452	C₂²×C₆₇	semidirect
511. #452	C₂²×C₆₇	semidirect
514. #451	C₂²×C₆₇	semidirect
516. #449	C₂²×C₆₇	semidirect
517. #449	C₂²×C₆₇	semidirect
520. #448	C₂²×C₆₇	semidirect
522. #441	C₂²×C₆₇	semidirect
523. #440	C₂²×C₆₇	semidirect
524. #435	C₂²×C₆₇	semidirect
525. #435	C₂²×C₆₇	semidirect
528. #419	C₂²×C₆₇	semidirect
529. #416	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3506	2
C₂×#548×C₆₇	2
C₆₇×#3321	2
C₆₇×#3313	2
C₆₇×#3271	2
C₆₇×#3269	2
C₆₇×#3215	2
C₆₇×#3213	2
C₆₇×#3194	2
C₆₇×#3190	2
C₆₇×#3113	2
C₆₇×#3112	2
C₆₇×#3004	2
C₆₇×#3002	2
C₆₇×#2973	2
#61578	67

Subgroups — C67×#61578 #30740829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61578 #30740829