Subgroups — C₆₇×#61700 #30740951

Order 17152.

1842 conjugacy classes of subgroups.

The subgroup lattice has 660 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×#555×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3341	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3340	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3305	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3303	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3230	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3227	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3188	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3101	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3100	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3091	C₂	1	4	yes	maximal, complemented
13	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
14	#559×C₆₇	—	2	2	no	—
15	#556×C₆₇	C₂²	1	2	yes	complemented
16	#555×C₆₇	—	2	2	no	—
17	#555×C₆₇	—	2	2	no	—
18	#555×C₆₇	C₂²	1	4	yes	—
19	#555×C₆₇	C₂²	1	4	yes	—
20	#555×C₆₇	C₂²	1	2	yes	complemented
21	#553×C₆₇	—	2	2	no	—
22	#550×C₆₇	C₂²	1	2	yes	complemented
23	#547×C₆₇	C₂²	1	2	yes	complemented
24	#538×C₆₇	—	2	2	no	—
25	#535×C₆₇	—	2	1	no	—
26	#533×C₆₇	C₂²	1	1	yes	characteristic
27	#532×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#124×C₆₇	—	2	2	no	—
30	#525×C₆₇	C₂²	1	4	yes	—
31	C₂×#121×C₆₇	—	2	1	no	—
32	C₂×#121×C₆₇	C₂²	1	2	yes	—
33	#520×C₆₇	—	2	1	no	—
34	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#117×C₆₇	—	2	1	no	—
37	C₂×#117×C₆₇	C₂²	1	2	yes	—
38	#500×C₆₇	C₂²	1	2	yes	complemented
39	#498×C₆₇	C₂²	1	2	yes	complemented
40	#487×C₆₇	C₂²	1	2	yes	complemented
41	#486×C₆₇	C₂²	1	2	yes	complemented
42	#485×C₆₇	C₂²	1	8	yes	complemented
43	#484×C₆₇	C₂²	1	4	yes	—
44	#483×C₆₇	C₂²	1	2	yes	—
45	#482×C₆₇	C₂²	1	8	yes	complemented
46	#482×C₆₇	C₂²	1	8	yes	complemented
47	#481×C₆₇	C₂²	1	4	yes	—
48	#480×C₆₇	C₂²	1	2	yes	—
49	#477×C₆₇	C₂²	1	4	yes	complemented
50	#476×C₆₇	C₂²	1	4	yes	complemented
51	#476×C₆₇	C₂²	1	2	yes	complemented
52	#474×C₆₇	C₂²	1	2	yes	complemented
53	#466×C₆₇	C₂²	1	8	yes	complemented
54	#465×C₆₇	C₂²	1	8	yes	complemented
55	#461×C₆₇	C₂²	1	2	yes	complemented
56	#459×C₆₇	C₂²	1	2	yes	complemented
57	#450×C₆₇	C₂²	1	2	yes	—
58	#449×C₆₇	C₂²	1	8	yes	complemented
59	#447×C₆₇	C₂²	1	2	yes	—
60	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
61	C₄×D₈×C₆₇	C₂²	1	4	yes	complemented
62	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
63	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
64	#431×C₆₇	C₂²	1	1	yes	characteristic
65	#431×C₆₇	C₂²	1	1	yes	characteristic
66	#427×C₆₇	C₂²	1	8	yes	complemented
67	#416×C₆₇	C₂²	1	8	yes	complemented
68	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
69	C₂×#41×C₆₇	—	2	1	no	—
70	C₂×#41×C₆₇	C₂³	1	4	yes	—
71	C₂²×Q₈×C₆₇	—	2	1	no	—
72	C₂²×D₄×C₆₇	—	2	2	no	—
73	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
74	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	—	2	1	no	—
76	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
77	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
78	C₂×D₈×C₆₇	C₂³	1	4	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
80	C₄⋊Q₈×C₆₇	—	2	1	no	—
81	C₄⋊Q₈×C₆₇	—	2	1	no	—
82	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
85	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
87	#127×C₆₇	—	2	2	no	—
88	#127×C₆₇	C₂³	1	1	yes	characteristic
89	#126×C₆₇	—	2	4	no	—
90	#125×C₆₇	—	2	1	no	—
91	#124×C₆₇	—	2	2	no	—
92	#124×C₆₇	—	2	8	no	—
93	#124×C₆₇	—	2	8	no	—
94	#124×C₆₇	—	2	4	no	—
95	#124×C₆₇	—	2	1	no	—
96	#123×C₆₇	—	2	4	no	—
97	#123×C₆₇	—	2	4	no	—
98	C₂²⋊Q₈×C₆₇	—	2	1	no	—
99	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
100	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
101	#121×C₆₇	—	2	4	no	—
102	#121×C₆₇	—	2	4	no	—
103	#121×C₆₇	—	2	2	no	—
104	#121×C₆₇	—	2	2	no	—
105	#121×C₆₇	—	2	4	no	—
106	#121×C₆₇	—	2	1	no	—
107	#121×C₆₇	—	2	4	no	—
108	#121×C₆₇	C₂³	1	8	yes	—
109	#121×C₆₇	C₂³	1	4	yes	—
110	#121×C₆₇	C₂³	1	2	yes	—
111	#120×C₆₇	—	2	4	no	—
112	C₄×Q₈×C₆₇	—	2	2	no	—
113	C₄×Q₈×C₆₇	—	2	1	no	—
114	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
115	C₄×D₄×C₆₇	—	2	4	no	—
116	C₄×D₄×C₆₇	—	2	4	no	—
117	C₄×D₄×C₆₇	—	2	2	no	—
118	C₄×D₄×C₆₇	—	2	1	no	—
119	C₄×D₄×C₆₇	—	2	1	no	—
120	C₄×D₄×C₆₇	—	2	1	no	—
121	C₄×D₄×C₆₇	D₄	1	2	yes	—
122	C₄×D₄×C₆₇	D₄	1	2	yes	—
123	C₄×D₄×C₆₇	C₂³	1	2	yes	—
124	C₄×D₄×C₆₇	C₂³	1	2	yes	—
125	C₄×D₄×C₆₇	C₂³	1	2	yes	—
126	C₄×D₄×C₆₇	C₂³	1	2	yes	—
127	C₄×D₄×C₆₇	C₂³	1	2	yes	—
128	#117×C₆₇	—	2	4	no	—
129	#117×C₆₇	—	2	1	no	—
130	#117×C₆₇	—	2	4	no	—
131	#117×C₆₇	—	2	4	no	—
132	#117×C₆₇	—	2	2	no	—
133	#117×C₆₇	C₂³	1	4	yes	—
134	#117×C₆₇	C₂³	1	8	yes	—
135	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
138	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
140	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
141	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
143	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
144	C₂×C₄²×C₆₇	—	2	2	no	—
145	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
146	#107×C₆₇	C₂³	1	8	yes	—
147	#107×C₆₇	C₂³	1	4	yes	—
148	#106×C₆₇	C₂³	1	8	yes	—
149	#106×C₆₇	C₂³	1	4	yes	—
150	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
151	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
152	#103×C₆₇	C₂³	1	8	yes	—
153	#103×C₆₇	C₂³	1	4	yes	—
154	#102×C₆₇	C₂³	1	8	yes	—
155	#102×C₆₇	C₂³	1	4	yes	—
156	#102×C₆₇	C₂³	1	4	yes	—
157	#102×C₆₇	C₂³	1	4	yes	—
158	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
159	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
160	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
161	C₄×C₈×C₆₇	C₂³	1	2	yes	—
162	C₄×C₈×C₆₇	C₂³	1	2	yes	—
163	C₂⁴×C₆₇	—	2	1	no	—
164	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
165	#41×C₆₇	—	4	8	no	—
166	#41×C₆₇	—	4	4	no	—
167	C₂×Q₈×C₆₇	—	2	1	no	—
168	C₂×Q₈×C₆₇	—	2	1	no	—
169	C₂×Q₈×C₆₇	—	2	2	no	—
170	C₂×Q₈×C₆₇	—	2	2	no	—
171	C₂×Q₈×C₆₇	—	8	2	no	—
172	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
173	C₂×D₄×C₆₇	—	2	2	no	—
174	C₂×D₄×C₆₇	—	2	4	no	—
175	C₂×D₄×C₆₇	—	2	4	no	—
176	C₂×D₄×C₆₇	—	2	8	no	—
177	C₂×D₄×C₆₇	—	2	4	no	—
178	C₂×D₄×C₆₇	—	2	1	no	—
179	C₂×D₄×C₆₇	—	2	1	no	—
180	C₂×D₄×C₆₇	—	2	1	no	—
181	C₂×D₄×C₆₇	—	4	2	no	—
182	C₂×D₄×C₆₇	—	8	4	no	—
183	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
184	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
185	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
186	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₆₇	—	2	1	no	—
188	C₂²×C₄×C₆₇	—	2	2	no	—
189	C₂²×C₄×C₆₇	—	2	2	no	—
190	C₂²×C₄×C₆₇	—	2	2	no	—
191	C₂²×C₄×C₆₇	—	2	2	no	—
192	C₂²×C₄×C₆₇	—	2	4	no	—
193	C₂²×C₄×C₆₇	—	4	4	no	—
194	C₂²×C₄×C₆₇	—	4	4	no	—
195	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
198	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
200	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
201	SD₁₆×C₆₇	—	4	4	no	—
202	D₈×C₆₇	—	4	4	no	—
203	C₂×C₈×C₆₇	—	4	8	no	—
204	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
205	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
206	C₄⋊C₄×C₆₇	—	2	1	no	—
207	C₄⋊C₄×C₆₇	—	2	2	no	—
208	C₄⋊C₄×C₆₇	—	2	2	no	—
209	C₄⋊C₄×C₆₇	—	2	4	no	—
210	C₄⋊C₄×C₆₇	—	2	1	no	—
211	C₄⋊C₄×C₆₇	—	2	2	no	—
212	C₄⋊C₄×C₆₇	—	2	1	no	—
213	C₄⋊C₄×C₆₇	—	2	2	no	—
214	C₄⋊C₄×C₆₇	—	2	1	no	—
215	C₄⋊C₄×C₆₇	—	2	1	no	—
216	C₄⋊C₄×C₆₇	—	2	2	no	—
217	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
218	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
219	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
220	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
221	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
222	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
223	C₂²⋊C₄×C₆₇	—	2	8	no	—
224	C₂²⋊C₄×C₆₇	—	2	2	no	—
225	C₂²⋊C₄×C₆₇	—	2	4	no	—
226	C₂²⋊C₄×C₆₇	—	2	4	no	—
227	C₂²⋊C₄×C₆₇	—	2	8	no	—
228	C₂²⋊C₄×C₆₇	—	2	2	no	—
229	C₂²⋊C₄×C₆₇	—	2	2	no	—
230	C₂²⋊C₄×C₆₇	—	2	4	no	—
231	C₂²⋊C₄×C₆₇	—	2	2	no	—
232	C₂²⋊C₄×C₆₇	—	2	2	no	—
233	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
234	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
235	C₄²×C₆₇	—	2	8	no	—
236	C₄²×C₆₇	—	2	4	no	—
237	C₄²×C₆₇	—	2	2	no	—
238	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
239	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
240	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
241	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
242	C₂³×C₆₇	—	2	4	no	—
243	C₂³×C₆₇	—	2	2	no	—
244	C₂³×C₆₇	—	2	2	no	—
245	C₂³×C₆₇	—	4	2	no	—
246	C₂³×C₆₇	—	4	2	no	—
247	C₂³×C₆₇	—	4	2	no	—
248	C₂³×C₆₇	—	4	2	no	—
249	C₂³×C₆₇	—	4	2	no	—
250	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
251	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
252	Q₈×C₆₇	—	2	4	no	—
253	Q₈×C₆₇	—	4	2	no	—
254	Q₈×C₆₇	—	8	1	no	—
255	Q₈×C₆₇	—	8	2	no	—
256	Q₈×C₆₇	—	8	2	no	—
257	D₄×C₆₇	—	2	4	no	—
258	D₄×C₆₇	—	2	4	no	—
259	D₄×C₆₇	—	2	2	no	—
260	D₄×C₆₇	—	2	4	no	—
261	D₄×C₆₇	—	2	2	no	—
262	D₄×C₆₇	—	4	2	no	—
263	D₄×C₆₇	—	4	4	no	—
264	D₄×C₆₇	—	4	2	no	—
265	D₄×C₆₇	—	8	1	no	—
266	D₄×C₆₇	—	8	4	no	—
267	D₄×C₆₇	—	8	8	no	—
268	D₄×C₆₇	—	8	4	no	—
269	C₂×C₄×C₆₇	—	2	8	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	4	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	4	4	no	—
277	C₂×C₄×C₆₇	—	4	2	no	—
278	C₂×C₄×C₆₇	—	4	1	no	—
279	C₂×C₄×C₆₇	—	4	4	no	—
280	C₂×C₄×C₆₇	—	4	2	no	—
281	C₂×C₄×C₆₇	—	4	4	no	—
282	C₂×C₄×C₆₇	—	4	4	no	—
283	C₂×C₄×C₆₇	—	4	4	no	—
284	C₂×C₄×C₆₇	—	8	4	no	—
285	C₂×C₄×C₆₇	#135	1	4	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
287	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
293	C₈×C₆₇	—	2	4	no	—
294	C₈×C₆₇	—	2	4	no	—
295	C₂²×C₆₇	—	2	1	no	—
296	C₂²×C₆₇	—	2	1	no	—
297	C₂²×C₆₇	—	2	4	no	—
298	C₂²×C₆₇	—	2	4	no	—
299	C₂²×C₆₇	—	2	4	no	—
300	C₂²×C₆₇	—	4	2	no	—
301	C₂²×C₆₇	—	4	2	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	—	4	2	no	—
304	C₂²×C₆₇	—	4	2	no	—
305	C₂²×C₆₇	—	4	2	no	—
306	C₂²×C₆₇	—	8	4	no	—
307	C₂²×C₆₇	—	8	4	no	—
308	C₂²×C₆₇	C₂×#135	1	2	yes	—
309	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
310	C₄×C₆₇	—	2	1	no	—
311	C₄×C₆₇	—	2	1	no	—
312	C₄×C₆₇	—	2	1	no	—
313	C₄×C₆₇	—	2	4	no	—
314	C₄×C₆₇	—	4	4	no	—
315	C₄×C₆₇	—	4	2	no	—
316	C₄×C₆₇	—	4	2	no	—
317	C₄×C₆₇	—	4	2	no	—
318	C₄×C₆₇	—	4	2	no	—
319	C₄×C₆₇	C₂×#135	1	2	yes	—
320	C₄×C₆₇	C₂×#135	1	2	yes	—
321	C₄×C₆₇	#585	1	2	yes	—
322	#61700	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
323	C₂×C₆₇	—	2	2	no	—
324	C₂×C₆₇	—	4	4	no	—
325	C₂×C₆₇	—	8	2	no	—
326	C₂×C₆₇	—	8	2	no	—
327	C₂×C₆₇	#3485	1	1	yes	characteristic
328	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
329	C₂×C₆₇	#3585	1	1	yes	characteristic
330	#3506	C₂×C₆₇	1	1	yes	characteristic, complemented
331	C₂×#555	C₂×C₆₇	1	1	yes	characteristic, complemented
332	#3341	C₂×C₆₇	1	4	yes	complemented
333	#3340	C₂×C₆₇	1	4	yes	complemented
334	#3305	C₂×C₆₇	1	4	yes	complemented
335	#3303	C₂×C₆₇	1	4	yes	complemented
336	#3230	C₂×C₆₇	1	1	yes	characteristic, complemented
337	#3227	C₂×C₆₇	1	1	yes	characteristic, complemented
338	#3188	C₂×C₆₇	1	4	yes	complemented
339	#3101	C₂×C₆₇	1	1	yes	characteristic, complemented
340	#3100	C₂×C₆₇	1	1	yes	characteristic, complemented
341	#3091	C₂×C₆₇	1	4	yes	complemented
342	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
343	C₆₇	#61700	1	1	yes	characteristic, direct product factor, Sylow
344	#559	—	2	2	no	—
345	#556	C₂²×C₆₇	1	2	yes	complemented
346	#555	—	2	2	no	—
347	#555	—	2	2	no	—
348	#555	C₂²×C₆₇	1	4	yes	—
349	#555	C₂²×C₆₇	1	4	yes	—
350	#555	C₂²×C₆₇	1	2	yes	complemented
351	#553	—	2	2	no	—
352	#550	C₂²×C₆₇	1	2	yes	complemented
353	#547	C₂²×C₆₇	1	2	yes	complemented
354	#538	—	2	2	no	—
355	#535	—	2	1	no	—
356	#533	C₂²×C₆₇	1	1	yes	characteristic
357	#532	C₂²×C₆₇	1	1	yes	characteristic
358	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×#124	—	2	2	no	—
360	#525	C₂²×C₆₇	1	4	yes	—
361	C₂×#121	—	2	1	no	—
362	C₂×#121	C₂²×C₆₇	1	2	yes	—
363	#520	—	2	1	no	—
364	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
365	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
366	C₂×#117	—	2	1	no	—
367	C₂×#117	C₂²×C₆₇	1	2	yes	—
368	#500	C₂²×C₆₇	1	2	yes	complemented
369	#498	C₂²×C₆₇	1	2	yes	complemented
370	#487	C₂²×C₆₇	1	2	yes	complemented
371	#486	C₂²×C₆₇	1	2	yes	complemented
372	#485	C₂²×C₆₇	1	8	yes	complemented
373	#484	C₂²×C₆₇	1	4	yes	—
374	#483	C₂²×C₆₇	1	2	yes	—
375	#482	C₂²×C₆₇	1	8	yes	complemented
376	#482	C₂²×C₆₇	1	8	yes	complemented
377	#481	C₂²×C₆₇	1	4	yes	—
378	#480	C₂²×C₆₇	1	2	yes	—
379	#477	C₂²×C₆₇	1	4	yes	complemented
380	#476	C₂²×C₆₇	1	2	yes	complemented
381	#476	C₂²×C₆₇	1	4	yes	complemented
382	#474	C₂²×C₆₇	1	2	yes	complemented
383	#466	C₂²×C₆₇	1	8	yes	complemented
384	#465	C₂²×C₆₇	1	8	yes	complemented
385	#461	C₂²×C₆₇	1	2	yes	complemented
386	#459	C₂²×C₆₇	1	2	yes	complemented
387	#450	C₂²×C₆₇	1	2	yes	—
388	#449	C₂²×C₆₇	1	8	yes	complemented
389	#447	C₂²×C₆₇	1	2	yes	—
390	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
391	C₄×D₈	C₂²×C₆₇	1	4	yes	complemented
392	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
393	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
394	#431	C₂²×C₆₇	1	1	yes	characteristic
395	#431	C₂²×C₆₇	1	1	yes	characteristic
396	#427	C₂²×C₆₇	1	8	yes	complemented
397	#416	C₂²×C₆₇	1	8	yes	complemented
398	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
399	C₂×#41	—	2	1	no	—
400	C₂×#41	C₂³×C₆₇	1	4	yes	—
401	C₂²×Q₈	—	2	1	no	—
402	C₂²×D₄	—	2	2	no	—
403	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
404	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
405	C₂³×C₄	—	2	1	no	—
406	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
407	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
408	C₂×D₈	C₂³×C₆₇	1	4	yes	—
409	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
410	C₄⋊Q₈	—	2	1	no	—
411	C₄⋊Q₈	—	2	1	no	—
412	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
413	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
414	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
415	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
416	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
417	#127	—	2	2	no	—
418	#127	C₂³×C₆₇	1	1	yes	characteristic
419	#126	—	2	4	no	—
420	#125	—	2	1	no	—
421	#124	—	2	2	no	—
422	#124	—	2	8	no	—
423	#124	—	2	8	no	—
424	#124	—	2	4	no	—
425	#124	—	2	1	no	—
426	#123	—	2	4	no	—
427	#123	—	2	4	no	—
428	C₂²⋊Q₈	—	2	1	no	—
429	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
430	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
431	#121	—	2	4	no	—
432	#121	—	2	4	no	—
433	#121	—	2	2	no	—
434	#121	—	2	2	no	—
435	#121	—	2	4	no	—
436	#121	—	2	1	no	—
437	#121	—	2	4	no	—
438	#121	C₂³×C₆₇	1	8	yes	—
439	#121	C₂³×C₆₇	1	2	yes	—
440	#121	C₂³×C₆₇	1	4	yes	—
441	#120	—	2	4	no	—
442	C₄×Q₈	—	2	2	no	—
443	C₄×Q₈	—	2	1	no	—
444	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
445	C₄×D₄	—	2	4	no	—
446	C₄×D₄	—	2	4	no	—
447	C₄×D₄	—	2	2	no	—
448	C₄×D₄	—	2	1	no	—
449	C₄×D₄	—	2	1	no	—
450	C₄×D₄	—	2	1	no	—
451	C₄×D₄	D₄×C₆₇	1	2	yes	—
452	C₄×D₄	D₄×C₆₇	1	2	yes	—
453	C₄×D₄	C₂³×C₆₇	1	2	yes	—
454	C₄×D₄	C₂³×C₆₇	1	2	yes	—
455	C₄×D₄	C₂³×C₆₇	1	2	yes	—
456	C₄×D₄	C₂³×C₆₇	1	2	yes	—
457	C₄×D₄	C₂³×C₆₇	1	2	yes	—
458	#117	—	2	4	no	—
459	#117	—	2	1	no	—
460	#117	—	2	4	no	—
461	#117	—	2	4	no	—
462	#117	—	2	2	no	—
463	#117	C₂³×C₆₇	1	8	yes	—
464	#117	C₂³×C₆₇	1	4	yes	—
465	C₂×C₄⋊C₄	—	2	1	no	—
466	C₂×C₄⋊C₄	—	2	2	no	—
467	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
468	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
469	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
470	C₂×C₂²⋊C₄	—	2	2	no	—
471	C₂×C₂²⋊C₄	—	2	2	no	—
472	C₂×C₂²⋊C₄	—	2	1	no	—
473	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
474	C₂×C₄²	—	2	2	no	—
475	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
476	#107	C₂³×C₆₇	1	4	yes	—
477	#107	C₂³×C₆₇	1	8	yes	—
478	#106	C₂³×C₆₇	1	4	yes	—
479	#106	C₂³×C₆₇	1	8	yes	—
480	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
481	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
482	#103	C₂³×C₆₇	1	4	yes	—
483	#103	C₂³×C₆₇	1	8	yes	—
484	#102	C₂³×C₆₇	1	4	yes	—
485	#102	C₂³×C₆₇	1	8	yes	—
486	#102	C₂³×C₆₇	1	4	yes	—
487	#102	C₂³×C₆₇	1	4	yes	—
488	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
489	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
490	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
491	C₄×C₈	C₂³×C₆₇	1	2	yes	—
492	C₄×C₈	C₂³×C₆₇	1	2	yes	—
493	C₂⁴	—	2	1	no	—
494	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
495	#41	—	4	8	no	—
496	#41	—	4	4	no	—
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	—	2	2	no	—
500	C₂×Q₈	—	2	2	no	—
501	C₂×Q₈	—	8	2	no	—
502	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	2	8	no	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	—	2	1	no	—
511	C₂×D₄	—	4	2	no	—
512	C₂×D₄	—	8	4	no	—
513	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
514	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
515	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
516	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	2	no	—
519	C₂²×C₄	—	2	2	no	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	4	no	—
523	C₂²×C₄	—	4	4	no	—
524	C₂²×C₄	—	4	4	no	—
525	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
526	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
527	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
528	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
529	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
530	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
531	SD₁₆	—	4	4	no	—
532	D₈	—	4	4	no	—
533	C₂×C₈	—	4	8	no	—
534	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
535	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	2	no	—
547	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
548	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
549	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
550	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
551	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
552	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
553	C₂²⋊C₄	—	2	8	no	—
554	C₂²⋊C₄	—	2	2	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	8	no	—
558	C₂²⋊C₄	—	2	2	no	—
559	C₂²⋊C₄	—	2	2	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	—	2	2	no	—
562	C₂²⋊C₄	—	2	2	no	—
563	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
564	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
565	C₄²	—	2	8	no	—
566	C₄²	—	2	4	no	—
567	C₄²	—	2	2	no	—
568	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
569	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
570	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
571	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
572	C₂³	—	2	4	no	—
573	C₂³	—	2	2	no	—
574	C₂³	—	2	2	no	—
575	C₂³	—	4	2	no	—
576	C₂³	—	4	2	no	—
577	C₂³	—	4	2	no	—
578	C₂³	—	4	2	no	—
579	C₂³	—	4	2	no	—
580	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
581	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
582	Q₈	—	2	4	no	—
583	Q₈	—	4	2	no	—
584	Q₈	—	8	2	no	—
585	Q₈	—	8	2	no	—
586	Q₈	—	8	1	no	—
587	D₄	—	2	4	no	—
588	D₄	—	2	4	no	—
589	D₄	—	2	2	no	—
590	D₄	—	2	4	no	—
591	D₄	—	2	2	no	—
592	D₄	—	4	2	no	—
593	D₄	—	4	4	no	—
594	D₄	—	4	2	no	—
595	D₄	—	8	1	no	—
596	D₄	—	8	8	no	—
597	D₄	—	8	4	no	—
598	D₄	—	8	4	no	—
599	C₂×C₄	—	2	8	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	4	4	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	4	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	4	no	—
612	C₂×C₄	—	4	4	no	—
613	C₂×C₄	—	4	4	no	—
614	C₂×C₄	—	8	4	no	—
615	C₂×C₄	#135×C₆₇	1	4	yes	—
616	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
617	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
618	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
619	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
620	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
621	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
622	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
623	C₈	—	2	4	no	—
624	C₈	—	2	4	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	2	1	no	—
627	C₂²	—	2	4	no	—
628	C₂²	—	2	4	no	—
629	C₂²	—	2	4	no	—
630	C₂²	—	4	2	no	—
631	C₂²	—	4	2	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	8	4	no	—
637	C₂²	—	8	4	no	—
638	C₂²	C₂×#135×C₆₇	1	2	yes	—
639	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
640	C₄	—	2	1	no	—
641	C₄	—	2	1	no	—
642	C₄	—	2	1	no	—
643	C₄	—	2	4	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	4	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	2	no	—
649	C₄	C₂×#135×C₆₇	1	2	yes	—
650	C₄	C₂×#135×C₆₇	1	2	yes	—
651	C₄	#585×C₆₇	1	2	yes	—
652	C₂	—	2	2	no	—
653	C₂	—	4	4	no	—
654	C₂	—	8	2	no	—
655	C₂	—	8	2	no	—
656	C₂	C₆₇×#3485	1	1	yes	characteristic
657	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
658	C₂	C₆₇×#3585	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×#555×C₆₇	C₂	semidirect
3. C₆₇×#3341	C₂	semidirect
4. C₆₇×#3340	C₂	semidirect
5. C₆₇×#3305	C₂	semidirect
6. C₆₇×#3303	C₂	semidirect
7. C₆₇×#3230	C₂	semidirect
8. C₆₇×#3227	C₂	semidirect
9. C₆₇×#3188	C₂	semidirect
10. C₆₇×#3101	C₂	semidirect
11. C₆₇×#3100	C₂	semidirect
12. C₆₇×#3091	C₂	semidirect
13. C₆₇×#2972	C₂	semidirect
15. #556×C₆₇	C₂²	semidirect
20. #555×C₆₇	C₂²	semidirect
22. #550×C₆₇	C₂²	semidirect
23. #547×C₆₇	C₂²	semidirect
38. #500×C₆₇	C₂²	semidirect
39. #498×C₆₇	C₂²	semidirect
40. #487×C₆₇	C₂²	semidirect
41. #486×C₆₇	C₂²	semidirect
42. #485×C₆₇	C₂²	semidirect
45. #482×C₆₇	C₂²	semidirect
46. #482×C₆₇	C₂²	semidirect
49. #477×C₆₇	C₂²	semidirect
50. #476×C₆₇	C₂²	semidirect
51. #476×C₆₇	C₂²	semidirect
52. #474×C₆₇	C₂²	semidirect
53. #466×C₆₇	C₂²	semidirect
54. #465×C₆₇	C₂²	semidirect
55. #461×C₆₇	C₂²	semidirect
56. #459×C₆₇	C₂²	semidirect
58. #449×C₆₇	C₂²	semidirect
60. C₄×SD₁₆×C₆₇	C₂²	semidirect
61. C₄×D₈×C₆₇	C₂²	semidirect
62. C₈×D₄×C₆₇	C₂²	semidirect
63. C₈×D₄×C₆₇	C₂²	semidirect
66. #427×C₆₇	C₂²	semidirect
67. #416×C₆₇	C₂²	semidirect
322. #61700	C₆₇	direct (N × H)
330. #3506	C₂×C₆₇	semidirect
331. C₂×#555	C₂×C₆₇	semidirect
332. #3341	C₂×C₆₇	semidirect
333. #3340	C₂×C₆₇	semidirect
334. #3305	C₂×C₆₇	semidirect
335. #3303	C₂×C₆₇	semidirect
336. #3230	C₂×C₆₇	semidirect
337. #3227	C₂×C₆₇	semidirect
338. #3188	C₂×C₆₇	semidirect
339. #3101	C₂×C₆₇	semidirect
340. #3100	C₂×C₆₇	semidirect
341. #3091	C₂×C₆₇	semidirect
342. #2972	C₂×C₆₇	semidirect
343. C₆₇	#61700	direct (N × H)
345. #556	C₂²×C₆₇	semidirect
350. #555	C₂²×C₆₇	semidirect
352. #550	C₂²×C₆₇	semidirect
353. #547	C₂²×C₆₇	semidirect
368. #500	C₂²×C₆₇	semidirect
369. #498	C₂²×C₆₇	semidirect
370. #487	C₂²×C₆₇	semidirect
371. #486	C₂²×C₆₇	semidirect
372. #485	C₂²×C₆₇	semidirect
375. #482	C₂²×C₆₇	semidirect
376. #482	C₂²×C₆₇	semidirect
379. #477	C₂²×C₆₇	semidirect
380. #476	C₂²×C₆₇	semidirect
381. #476	C₂²×C₆₇	semidirect
382. #474	C₂²×C₆₇	semidirect
383. #466	C₂²×C₆₇	semidirect
384. #465	C₂²×C₆₇	semidirect
385. #461	C₂²×C₆₇	semidirect
386. #459	C₂²×C₆₇	semidirect
388. #449	C₂²×C₆₇	semidirect
390. C₄×SD₁₆	C₂²×C₆₇	semidirect
391. C₄×D₈	C₂²×C₆₇	semidirect
392. C₈×D₄	C₂²×C₆₇	semidirect
393. C₈×D₄	C₂²×C₆₇	semidirect
396. #427	C₂²×C₆₇	semidirect
397. #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×#555×C₆₇	2
C₆₇×#3341	2
C₆₇×#3340	2
C₆₇×#3305	2
C₆₇×#3303	2
C₆₇×#3230	2
C₆₇×#3227	2
C₆₇×#3188	2
C₆₇×#3101	2
C₆₇×#3100	2
C₆₇×#3091	2
C₆₇×#2972	2
#61700	67

Subgroups — C67×#61700 #30740951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61700 #30740951