Subgroups — C₆₇×#18873 #30698124

Order 17152.

900 conjugacy classes of subgroups.

The subgroup lattice has 554 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
2	C₆₇×#3249	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3209	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3076	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1645	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1644	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1633	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1632	C₂	1	2	yes	maximal, complemented
9	C₂×#333×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1481	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1477	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₆₇	C₂²	1	2	yes	—
13	#557×C₆₇	—	2	1	no	—
14	#554×C₆₇	—	2	2	no	—
15	#554×C₆₇	C₂²	1	2	yes	—
16	D₄×Q₈×C₆₇	—	2	2	no	—
17	D₄×Q₈×C₆₇	—	2	1	no	—
18	#544×C₆₇	—	2	2	no	—
19	#544×C₆₇	—	2	1	no	—
20	#533×C₆₇	—	2	1	no	—
21	#533×C₆₇	C₂²	1	1	yes	characteristic
22	#533×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
24	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	#520×C₆₇	—	2	1	no	—
26	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
27	#494×C₆₇	C₂²	1	2	yes	—
28	#487×C₆₇	C₂²	1	2	yes	—
29	#479×C₆₇	—	2	1	no	—
30	#478×C₆₇	—	2	1	no	—
31	#477×C₆₇	—	2	1	no	—
32	#477×C₆₇	C₂²	1	1	yes	characteristic
33	#477×C₆₇	C₂²	1	2	yes	—
34	#476×C₆₇	C₂²	1	1	yes	characteristic
35	#475×C₆₇	C₂²	1	1	yes	characteristic
36	#474×C₆₇	C₂²	1	1	yes	characteristic
37	#467×C₆₇	—	2	2	no	—
38	#462×C₆₇	—	2	2	no	—
39	#462×C₆₇	—	2	1	no	—
40	#462×C₆₇	C₂²	1	2	yes	—
41	#451×C₆₇	—	2	2	no	—
42	#431×C₆₇	C₂²	1	1	yes	characteristic
43	#427×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	#416×C₆₇	—	2	1	no	—
47	#333×C₆₇	C₂²	1	2	yes	—
48	#333×C₆₇	C₂²	1	2	yes	—
49	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
50	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
51	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
52	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
53	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×Q₈×C₆₇	—	2	2	no	—
55	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
56	C₂×Q₁₆×C₆₇	—	2	4	no	—
57	C₂×Q₁₆×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	—	2	1	no	—
59	C₂²×C₈×C₆₇	—	2	1	no	—
60	C₄⋊Q₈×C₆₇	—	2	2	no	—
61	C₄⋊Q₈×C₆₇	—	2	2	no	—
62	C₄⋊Q₈×C₆₇	—	2	4	no	—
63	C₄⋊Q₈×C₆₇	—	2	2	no	—
64	C₄⋊Q₈×C₆₇	—	2	2	no	—
65	C₄⋊Q₈×C₆₇	—	2	1	no	—
66	C₄⋊Q₈×C₆₇	—	2	1	no	—
67	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
68	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
69	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
73	#125×C₆₇	—	2	2	no	—
74	#125×C₆₇	—	2	4	no	—
75	#125×C₆₇	—	2	2	no	—
76	#125×C₆₇	—	2	1	no	—
77	#125×C₆₇	—	2	1	no	—
78	C₂²⋊Q₈×C₆₇	—	2	2	no	—
79	C₂²⋊Q₈×C₆₇	—	2	4	no	—
80	C₂²⋊Q₈×C₆₇	—	2	4	no	—
81	C₂²⋊Q₈×C₆₇	—	2	2	no	—
82	C₂²⋊Q₈×C₆₇	—	2	4	no	—
83	C₂²⋊Q₈×C₆₇	—	2	1	no	—
84	C₂²⋊Q₈×C₆₇	—	2	1	no	—
85	C₂²⋊Q₈×C₆₇	—	2	1	no	—
86	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
87	C₄×Q₈×C₆₇	—	2	2	no	—
88	C₄×Q₈×C₆₇	—	2	2	no	—
89	C₄×Q₈×C₆₇	—	2	2	no	—
90	C₄×Q₈×C₆₇	—	2	1	no	—
91	C₄×Q₈×C₆₇	—	2	1	no	—
92	C₄×Q₈×C₆₇	—	2	1	no	—
93	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
95	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₄×D₄×C₆₇	—	2	4	no	—
97	C₄×D₄×C₆₇	—	2	2	no	—
98	C₄×D₄×C₆₇	—	2	1	no	—
99	C₄×D₄×C₆₇	—	2	1	no	—
100	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
101	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
102	#117×C₆₇	—	2	2	no	—
103	#117×C₆₇	—	2	1	no	—
104	#117×C₆₇	—	2	1	no	—
105	#117×C₆₇	—	2	1	no	—
106	#117×C₆₇	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
109	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
110	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₂×C₄²×C₆₇	—	2	1	no	—
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	#107×C₆₇	—	2	2	no	—
114	#107×C₆₇	—	2	1	no	—
115	#107×C₆₇	—	2	1	no	—
116	#106×C₆₇	—	2	1	no	—
117	#106×C₆₇	—	2	1	no	—
118	C₄⋊C₈×C₆₇	—	2	1	no	—
119	C₄⋊C₈×C₆₇	—	2	1	no	—
120	C₄⋊C₈×C₆₇	—	2	1	no	—
121	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
122	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
123	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
124	#103×C₆₇	—	2	4	no	—
125	#103×C₆₇	—	2	2	no	—
126	#103×C₆₇	—	2	1	no	—
127	#103×C₆₇	—	2	1	no	—
128	#103×C₆₇	—	2	2	no	—
129	#102×C₆₇	—	2	1	no	—
130	#102×C₆₇	—	2	1	no	—
131	C₂²⋊C₈×C₆₇	—	2	1	no	—
132	C₄×C₈×C₆₇	C₂³	1	2	yes	—
133	#41×C₆₇	—	2	2	no	—
134	#41×C₆₇	Q₁₆	1	4	yes	complemented
135	C₂×Q₈×C₆₇	—	2	4	no	—
136	C₂×Q₈×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	—	2	1	no	—
138	C₂×Q₈×C₆₇	—	2	1	no	—
139	C₂×Q₈×C₆₇	—	2	1	no	—
140	C₂×Q₈×C₆₇	—	8	4	no	—
141	C₂×Q₈×C₆₇	—	8	2	no	—
142	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
145	C₂×D₄×C₆₇	—	2	1	no	—
146	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	1	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	Q₁₆×C₆₇	—	8	2	no	—
156	Q₁₆×C₆₇	—	8	4	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	4	2	no	—
162	C₂×C₈×C₆₇	—	8	1	no	—
163	C₄⋊C₄×C₆₇	—	2	4	no	—
164	C₄⋊C₄×C₆₇	—	2	4	no	—
165	C₄⋊C₄×C₆₇	—	2	4	no	—
166	C₄⋊C₄×C₆₇	—	2	4	no	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₄⋊C₄×C₆₇	—	2	1	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	1	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	1	no	—
173	C₄⋊C₄×C₆₇	—	2	2	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	—	2	1	no	—
177	C₄⋊C₄×C₆₇	—	2	2	no	—
178	C₄⋊C₄×C₆₇	—	2	1	no	—
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	2	1	no	—
181	C₄⋊C₄×C₆₇	—	2	1	no	—
182	C₄⋊C₄×C₆₇	—	4	1	no	—
183	C₄⋊C₄×C₆₇	—	4	1	no	—
184	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂²⋊C₄×C₆₇	—	2	4	no	—
188	C₂²⋊C₄×C₆₇	—	2	2	no	—
189	C₂²⋊C₄×C₆₇	—	2	1	no	—
190	C₂²⋊C₄×C₆₇	—	2	1	no	—
191	C₂²⋊C₄×C₆₇	—	2	1	no	—
192	C₂²⋊C₄×C₆₇	—	4	1	no	—
193	C₄²×C₆₇	—	2	4	no	—
194	C₄²×C₆₇	—	2	1	no	—
195	C₄²×C₆₇	—	2	1	no	—
196	C₄²×C₆₇	—	2	1	no	—
197	C₄²×C₆₇	—	2	2	no	—
198	C₄²×C₆₇	—	4	1	no	—
199	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
202	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
203	C₂³×C₆₇	—	2	1	no	—
204	C₂³×C₆₇	#120	1	1	yes	characteristic
205	Q₈×C₆₇	—	2	1	no	—
206	Q₈×C₆₇	—	2	2	no	—
207	Q₈×C₆₇	—	2	2	no	—
208	Q₈×C₆₇	—	4	4	no	—
209	Q₈×C₆₇	—	4	4	no	—
210	Q₈×C₆₇	—	4	2	no	—
211	Q₈×C₆₇	—	4	2	no	—
212	Q₈×C₆₇	—	4	1	no	—
213	Q₈×C₆₇	—	4	1	no	—
214	Q₈×C₆₇	—	8	4	no	—
215	Q₈×C₆₇	C₂×Q₁₆	1	2	yes	—
216	D₄×C₆₇	—	2	4	no	—
217	D₄×C₆₇	—	2	1	no	—
218	D₄×C₆₇	C₂×Q₁₆	1	2	yes	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	4	1	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	4	1	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	—	4	1	no	—
233	C₂×C₄×C₆₇	—	4	1	no	—
234	C₂×C₄×C₆₇	—	8	2	no	—
235	C₂×C₄×C₆₇	—	8	4	no	—
236	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
237	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
238	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
240	C₂×C₄×C₆₇	#137	1	2	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
244	C₈×C₆₇	—	4	2	no	—
245	C₈×C₆₇	—	4	1	no	—
246	C₈×C₆₇	—	4	1	no	—
247	C₈×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	—	2	1	no	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	2	no	—
251	C₂²×C₆₇	—	4	1	no	—
252	C₂²×C₆₇	—	4	1	no	—
253	C₂²×C₆₇	#451	1	2	yes	—
254	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
255	C₄×C₆₇	—	2	2	no	—
256	C₄×C₆₇	—	2	1	no	—
257	C₄×C₆₇	—	2	1	no	—
258	C₄×C₆₇	—	2	1	no	—
259	C₄×C₆₇	—	4	1	no	—
260	C₄×C₆₇	—	4	2	no	—
261	C₄×C₆₇	—	4	1	no	—
262	C₄×C₆₇	—	4	4	no	—
263	C₄×C₆₇	—	4	2	no	—
264	C₄×C₆₇	—	8	1	no	—
265	C₄×C₆₇	—	8	1	no	—
266	C₄×C₆₇	#451	1	2	yes	—
267	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
268	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
269	C₄×C₆₇	#577	1	1	yes	characteristic
270	C₄×C₆₇	#578	1	1	yes	characteristic
271	#18873	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂×C₆₇	—	2	2	no	—
273	C₂×C₆₇	—	4	2	no	—
274	C₂×C₆₇	C₂×#451	1	1	yes	characteristic
275	C₂×C₆₇	#3013	1	1	yes	characteristic
276	C₂×C₆₇	#3019	1	1	yes	characteristic
277	#3503	C₂×C₆₇	1	1	yes	characteristic
278	#3249	C₂×C₆₇	1	1	yes	characteristic, complemented
279	#3209	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#3076	C₂×C₆₇	1	1	yes	characteristic
281	#1645	C₂×C₆₇	1	2	yes	complemented
282	#1644	C₂×C₆₇	1	2	yes	complemented
283	#1633	C₂×C₆₇	1	2	yes	complemented
284	#1632	C₂×C₆₇	1	2	yes	complemented
285	C₂×#333	C₂×C₆₇	1	1	yes	characteristic, complemented
286	#1481	C₂×C₆₇	1	1	yes	characteristic, complemented
287	#1477	C₂×C₆₇	1	1	yes	characteristic
288	C₆₇	#18873	1	1	yes	characteristic, direct product factor, Sylow
289	Q₈²	C₂²×C₆₇	1	2	yes	—
290	#557	—	2	1	no	—
291	#554	—	2	2	no	—
292	#554	C₂²×C₆₇	1	2	yes	—
293	D₄×Q₈	—	2	2	no	—
294	D₄×Q₈	—	2	1	no	—
295	#544	—	2	2	no	—
296	#544	—	2	1	no	—
297	#533	—	2	1	no	—
298	#533	C₂²×C₆₇	1	1	yes	characteristic
299	#533	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×C₄⋊Q₈	—	2	1	no	—
301	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
302	#520	—	2	1	no	—
303	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
304	#494	C₂²×C₆₇	1	2	yes	—
305	#487	C₂²×C₆₇	1	2	yes	—
306	#479	—	2	1	no	—
307	#478	—	2	1	no	—
308	#477	—	2	1	no	—
309	#477	C₂²×C₆₇	1	1	yes	characteristic
310	#477	C₂²×C₆₇	1	2	yes	—
311	#476	C₂²×C₆₇	1	1	yes	characteristic
312	#475	C₂²×C₆₇	1	1	yes	characteristic
313	#474	C₂²×C₆₇	1	1	yes	characteristic
314	#467	—	2	2	no	—
315	#462	—	2	2	no	—
316	#462	—	2	1	no	—
317	#462	C₂²×C₆₇	1	2	yes	—
318	#451	—	2	2	no	—
319	#431	C₂²×C₆₇	1	1	yes	characteristic
320	#427	—	2	1	no	—
321	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
323	#416	—	2	1	no	—
324	#333	C₂²×C₆₇	1	2	yes	—
325	#333	C₂²×C₆₇	1	2	yes	—
326	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
327	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
328	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
329	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
330	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
331	C₂²×Q₈	—	2	2	no	—
332	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
333	C₂×Q₁₆	—	2	4	no	—
334	C₂×Q₁₆	—	2	2	no	—
335	C₂²×C₈	—	2	1	no	—
336	C₂²×C₈	—	2	1	no	—
337	C₄⋊Q₈	—	2	2	no	—
338	C₄⋊Q₈	—	2	2	no	—
339	C₄⋊Q₈	—	2	4	no	—
340	C₄⋊Q₈	—	2	2	no	—
341	C₄⋊Q₈	—	2	2	no	—
342	C₄⋊Q₈	—	2	1	no	—
343	C₄⋊Q₈	—	2	1	no	—
344	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
345	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
346	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
347	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
348	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
349	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
350	#125	—	2	2	no	—
351	#125	—	2	4	no	—
352	#125	—	2	2	no	—
353	#125	—	2	1	no	—
354	#125	—	2	1	no	—
355	C₂²⋊Q₈	—	2	2	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	C₂²⋊Q₈	—	2	2	no	—
359	C₂²⋊Q₈	—	2	4	no	—
360	C₂²⋊Q₈	—	2	1	no	—
361	C₂²⋊Q₈	—	2	1	no	—
362	C₂²⋊Q₈	—	2	1	no	—
363	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
364	C₄×Q₈	—	2	2	no	—
365	C₄×Q₈	—	2	2	no	—
366	C₄×Q₈	—	2	2	no	—
367	C₄×Q₈	—	2	1	no	—
368	C₄×Q₈	—	2	1	no	—
369	C₄×Q₈	—	2	1	no	—
370	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
371	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
372	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
373	C₄×D₄	—	2	4	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	1	no	—
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
378	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
379	#117	—	2	2	no	—
380	#117	—	2	1	no	—
381	#117	—	2	1	no	—
382	#117	—	2	1	no	—
383	#117	D₄×C₆₇	1	1	yes	characteristic
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₄²	—	2	1	no	—
389	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
390	#107	—	2	1	no	—
391	#107	—	2	1	no	—
392	#107	—	2	2	no	—
393	#106	—	2	1	no	—
394	#106	—	2	1	no	—
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	—	2	1	no	—
397	C₄⋊C₈	—	2	1	no	—
398	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
399	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
400	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
401	#103	—	2	1	no	—
402	#103	—	2	1	no	—
403	#103	—	2	4	no	—
404	#103	—	2	2	no	—
405	#103	—	2	2	no	—
406	#102	—	2	1	no	—
407	#102	—	2	1	no	—
408	C₂²⋊C₈	—	2	1	no	—
409	C₄×C₈	C₂³×C₆₇	1	2	yes	—
410	#41	—	2	2	no	—
411	#41	Q₁₆×C₆₇	1	4	yes	complemented
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	1	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	8	4	no	—
418	C₂×Q₈	—	8	2	no	—
419	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
420	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
421	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
422	C₂×D₄	—	2	1	no	—
423	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
432	Q₁₆	—	8	2	no	—
433	Q₁₆	—	8	4	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	4	2	no	—
439	C₂×C₈	—	8	1	no	—
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	4	1	no	—
460	C₄⋊C₄	—	4	1	no	—
461	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
462	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	—	2	1	no	—
467	C₂²⋊C₄	—	2	1	no	—
468	C₂²⋊C₄	—	2	1	no	—
469	C₂²⋊C₄	—	4	1	no	—
470	C₄²	—	2	4	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	—	2	1	no	—
474	C₄²	—	2	2	no	—
475	C₄²	—	4	1	no	—
476	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
478	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
479	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
480	C₂³	—	2	1	no	—
481	C₂³	#120×C₆₇	1	1	yes	characteristic
482	Q₈	—	2	1	no	—
483	Q₈	—	2	2	no	—
484	Q₈	—	2	2	no	—
485	Q₈	—	4	4	no	—
486	Q₈	—	4	4	no	—
487	Q₈	—	4	2	no	—
488	Q₈	—	4	2	no	—
489	Q₈	—	4	1	no	—
490	Q₈	—	4	1	no	—
491	Q₈	—	8	4	no	—
492	Q₈	C₂×Q₁₆×C₆₇	1	2	yes	—
493	D₄	—	2	4	no	—
494	D₄	—	2	1	no	—
495	D₄	C₂×Q₁₆×C₆₇	1	2	yes	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	8	2	no	—
512	C₂×C₄	—	8	4	no	—
513	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
514	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
515	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
517	C₂×C₄	#137×C₆₇	1	2	yes	—
518	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
521	C₈	—	4	2	no	—
522	C₈	—	4	1	no	—
523	C₈	—	4	1	no	—
524	C₈	—	4	2	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	1	no	—
527	C₂²	—	2	2	no	—
528	C₂²	—	4	1	no	—
529	C₂²	—	4	1	no	—
530	C₂²	#451×C₆₇	1	2	yes	—
531	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
532	C₄	—	2	1	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	2	no	—
536	C₄	—	4	1	no	—
537	C₄	—	4	2	no	—
538	C₄	—	4	1	no	—
539	C₄	—	4	4	no	—
540	C₄	—	4	2	no	—
541	C₄	—	8	1	no	—
542	C₄	—	8	1	no	—
543	C₄	#451×C₆₇	1	2	yes	—
544	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
545	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
546	C₄	#577×C₆₇	1	1	yes	characteristic
547	C₄	#578×C₆₇	1	1	yes	characteristic
548	C₂	—	2	2	no	—
549	C₂	—	4	2	no	—
550	C₂	C₂×#451×C₆₇	1	1	yes	characteristic
551	C₂	C₆₇×#3013	1	1	yes	characteristic
552	C₂	C₆₇×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×#41×C₆₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3249	C₂	semidirect
3. C₆₇×#3209	C₂	semidirect
5. C₆₇×#1645	C₂	semidirect
6. C₆₇×#1644	C₂	semidirect
7. C₆₇×#1633	C₂	semidirect
8. C₆₇×#1632	C₂	semidirect
9. C₂×#333×C₆₇	C₂	semidirect
10. C₆₇×#1481	C₂	semidirect
134. #41×C₆₇	Q₁₆	semidirect
271. #18873	C₆₇	direct (N × H)
278. #3249	C₂×C₆₇	semidirect
279. #3209	C₂×C₆₇	semidirect
281. #1645	C₂×C₆₇	semidirect
282. #1644	C₂×C₆₇	semidirect
283. #1633	C₂×C₆₇	semidirect
284. #1632	C₂×C₆₇	semidirect
285. C₂×#333	C₂×C₆₇	semidirect
286. #1481	C₂×C₆₇	semidirect
288. C₆₇	#18873	direct (N × H)
411. #41	Q₁₆×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×#3249	2
C₆₇×#3209	2
C₆₇×#3076	2
C₆₇×#1645	2
C₆₇×#1644	2
C₆₇×#1633	2
C₆₇×#1632	2
C₂×#333×C₆₇	2
C₆₇×#1481	2
C₆₇×#1477	2
#18873	67

Subgroups — C67×#18873 #30698124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18873 #30698124