Subgroups — C₆₇×#22550 #30701801

Order 17152.

1150 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3480	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1980	C₂	1	3	yes	maximal, complemented
5	C₆₇×#1962	C₂	1	3	yes	maximal, complemented
6	C₆₇×#1806	C₂	1	3	yes	maximal, complemented
7	C₄×#104×C₆₇	C₂	1	3	yes	maximal, complemented
8	#559×C₆₇	C₄	1	3	yes	complemented
9	#557×C₆₇	C₄	1	1	yes	characteristic, complemented
10	#556×C₆₇	C₄	1	3	yes	complemented
11	#553×C₆₇	—	2	3	no	—
12	#550×C₆₇	C₄	1	1	yes	characteristic, complemented
13	#548×C₆₇	—	2	1	no	—
14	#532×C₆₇	C₂²	1	3	yes	—
15	#529×C₆₇	—	2	3	no	—
16	C₄×#41×C₆₇	—	2	1	no	—
17	C₄×#41×C₆₇	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	#446×C₆₇	—	2	3	no	—
21	C₈×Q₈×C₆₇	—	2	1	no	—
22	#423×C₆₇	C₂²	1	3	yes	—
23	C₂×#104×C₆₇	C₂²	1	3	yes	—
24	C₂×#104×C₆₇	C₂²	1	3	yes	—
25	C₂×#104×C₆₇	C₂²	1	6	yes	—
26	C₄×M₁₆×C₆₇	C₂²	1	3	yes	—
27	C₄×C₄⋊C₄×C₆₇	—	2	3	no	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
29	C₄³×C₆₇	C₂²	1	3	yes	—
30	#339×C₆₇	C₂²	1	1	yes	characteristic
31	#339×C₆₇	C₂²	1	3	yes	—
32	C₂×#41×C₆₇	—	2	1	no	—
33	C₂×#41×C₆₇	C₂³	1	3	yes	—
34	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	C₂³	1	3	yes	—
36	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₆₇	C₂×C₄	1	3	yes	—
38	#127×C₆₇	C₂×C₄	1	3	yes	—
39	#126×C₆₇	—	2	6	no	—
40	#125×C₆₇	—	2	3	no	—
41	#124×C₆₇	—	2	3	no	—
42	#124×C₆₇	C₂×C₄	1	6	yes	—
43	#123×C₆₇	—	2	6	no	—
44	C₂²⋊Q₈×C₆₇	—	2	3	no	—
45	#121×C₆₇	—	2	3	no	—
46	#121×C₆₇	—	2	3	no	—
47	#121×C₆₇	—	2	3	no	—
48	C₄×Q₈×C₆₇	—	2	8	no	—
49	C₄×Q₈×C₆₇	—	2	1	no	—
50	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
51	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
52	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
54	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
55	C₄×Q₈×C₆₇	D₄	1	3	yes	—
56	C₄×D₄×C₆₇	—	2	3	no	—
57	C₄×D₄×C₆₇	—	2	3	no	—
58	C₄×D₄×C₆₇	—	2	3	no	—
59	C₄×D₄×C₆₇	—	2	3	no	—
60	C₄×D₄×C₆₇	—	2	3	no	—
61	C₄×D₄×C₆₇	C₂×C₄	1	3	yes	—
62	C₄×D₄×C₆₇	C₂×C₄	1	3	yes	—
63	#117×C₆₇	—	2	3	no	—
64	#117×C₆₇	—	2	3	no	—
65	#117×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
67	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
68	C₂×C₄²×C₆₇	—	2	3	no	—
69	C₂×C₄²×C₆₇	—	2	3	no	—
70	C₂×C₄²×C₆₇	—	2	3	no	—
71	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
74	C₄⋊C₈×C₆₇	—	2	3	no	—
75	C₄⋊C₈×C₆₇	—	2	3	no	—
76	#104×C₆₇	—	2	6	no	—
77	#104×C₆₇	—	2	6	no	—
78	#104×C₆₇	—	4	6	no	—
79	#97×C₆₇	—	2	3	no	—
80	C₄×C₈×C₆₇	—	2	3	no	—
81	#95×C₆₇	—	2	3	no	—
82	#41×C₆₇	—	2	3	no	—
83	#41×C₆₇	—	2	3	no	—
84	#41×C₆₇	—	2	3	no	—
85	#41×C₆₇	—	2	3	no	—
86	#41×C₆₇	—	4	4	no	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	C₂²⋊C₄	1	3	yes	—
89	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Q₈×C₆₇	C₂²×C₄	1	3	yes	—
91	C₂×D₄×C₆₇	—	2	3	no	—
92	C₂×D₄×C₆₇	—	2	3	no	—
93	C₂×D₄×C₆₇	—	2	3	no	—
94	C₂×D₄×C₆₇	C₂²×C₄	1	3	yes	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	3	no	—
97	C₂²×C₄×C₆₇	—	2	3	no	—
98	C₂²×C₄×C₆₇	—	2	3	no	—
99	C₂²×C₄×C₆₇	—	2	3	no	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
102	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₆₇	—	2	1	no	—
104	M₁₆×C₆₇	—	2	3	no	—
105	M₁₆×C₆₇	—	2	3	no	—
106	M₁₆×C₆₇	#41	1	2	yes	complemented
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	—	2	3	no	—
109	C₄⋊C₄×C₆₇	—	2	12	no	—
110	C₄⋊C₄×C₆₇	—	2	3	no	—
111	C₄⋊C₄×C₆₇	—	2	3	no	—
112	C₄⋊C₄×C₆₇	—	2	3	no	—
113	C₄⋊C₄×C₆₇	—	2	3	no	—
114	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
115	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
116	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
117	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²⋊C₄×C₆₇	—	2	1	no	—
120	C₂²⋊C₄×C₆₇	—	2	6	no	—
121	C₂²⋊C₄×C₆₇	—	2	3	no	—
122	C₂²⋊C₄×C₆₇	—	2	3	no	—
123	C₂²⋊C₄×C₆₇	—	2	6	no	—
124	C₄²×C₆₇	—	2	12	no	—
125	C₄²×C₆₇	—	2	3	no	—
126	C₄²×C₆₇	—	2	3	no	—
127	C₄²×C₆₇	—	2	1	no	—
128	C₄²×C₆₇	—	2	1	no	—
129	C₄²×C₆₇	—	2	3	no	—
130	C₄²×C₆₇	—	2	3	no	—
131	C₄²×C₆₇	—	4	6	no	—
132	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
133	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
134	C₄²×C₆₇	C₂×D₄	1	3	yes	—
135	C₄²×C₆₇	C₂×D₄	1	3	yes	—
136	C₄²×C₆₇	#41	1	2	yes	complemented
137	C₂³×C₆₇	—	2	3	no	—
138	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	Q₈×C₆₇	—	2	6	no	—
140	Q₈×C₆₇	—	2	3	no	—
141	Q₈×C₆₇	—	2	3	no	—
142	Q₈×C₆₇	#104	1	4	yes	complemented
143	D₄×C₆₇	—	2	3	no	—
144	D₄×C₆₇	—	2	3	no	—
145	D₄×C₆₇	—	4	3	no	—
146	D₄×C₆₇	—	4	3	no	—
147	D₄×C₆₇	—	4	6	no	—
148	D₄×C₆₇	—	8	3	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	6	no	—
153	C₂×C₄×C₆₇	—	2	6	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	3	no	—
157	C₂×C₄×C₆₇	—	4	3	no	—
158	C₂×C₄×C₆₇	—	4	3	no	—
159	C₂×C₄×C₆₇	—	4	3	no	—
160	C₂×C₄×C₆₇	—	4	3	no	—
161	C₂×C₄×C₆₇	—	4	3	no	—
162	C₂×C₄×C₆₇	—	4	6	no	—
163	C₂×C₄×C₆₇	—	8	3	no	—
164	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
165	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
166	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
167	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
168	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
172	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
174	C₈×C₆₇	—	2	2	no	—
175	C₈×C₆₇	—	4	3	no	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	—	2	1	no	—
178	C₂²×C₆₇	—	4	3	no	—
179	C₂²×C₆₇	—	4	3	no	—
180	C₂²×C₆₇	—	4	3	no	—
181	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₆₇	#548	1	1	yes	characteristic
183	C₂²×C₆₇	#550	1	1	yes	characteristic
184	C₄×C₆₇	—	2	4	no	—
185	C₄×C₆₇	—	2	3	no	—
186	C₄×C₆₇	—	2	3	no	—
187	C₄×C₆₇	—	2	3	no	—
188	C₄×C₆₇	—	4	6	no	—
189	C₄×C₆₇	—	4	3	no	—
190	C₄×C₆₇	—	4	1	no	—
191	C₄×C₆₇	—	4	1	no	—
192	C₄×C₆₇	C₂×#104	1	6	yes	—
193	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
194	C₄×C₆₇	#520	1	1	yes	characteristic
195	C₄×C₆₇	#548	1	1	yes	characteristic
196	C₄×C₆₇	#550	1	1	yes	characteristic
197	#22550	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₆₇	—	2	1	no	—
199	C₂×C₆₇	—	2	1	no	—
200	C₂×C₆₇	—	8	3	no	—
201	C₂×C₆₇	#2341	1	1	yes	characteristic
202	C₂×C₆₇	C₂²×#104	1	1	yes	characteristic
203	C₂×C₆₇	#2901	1	1	yes	characteristic
204	#3480	C₂×C₆₇	1	1	yes	characteristic
205	Q₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
206	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#1980	C₂×C₆₇	1	3	yes	complemented
208	#1962	C₂×C₆₇	1	3	yes	complemented
209	#1806	C₂×C₆₇	1	3	yes	complemented
210	C₄×#104	C₂×C₆₇	1	3	yes	complemented
211	C₆₇	#22550	1	1	yes	characteristic, direct product factor, Sylow
212	#559	C₄×C₆₇	1	3	yes	complemented
213	#557	C₄×C₆₇	1	1	yes	characteristic, complemented
214	#556	C₄×C₆₇	1	3	yes	complemented
215	#553	—	2	3	no	—
216	#550	C₄×C₆₇	1	1	yes	characteristic, complemented
217	#548	—	2	1	no	—
218	#532	C₂²×C₆₇	1	3	yes	—
219	#529	—	2	3	no	—
220	C₄×#41	—	2	1	no	—
221	C₄×#41	C₂²×C₆₇	1	3	yes	—
222	C₂×C₄×Q₈	—	2	1	no	—
223	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
224	#446	—	2	3	no	—
225	C₈×Q₈	—	2	1	no	—
226	#423	C₂²×C₆₇	1	3	yes	—
227	C₂×#104	C₂²×C₆₇	1	3	yes	—
228	C₂×#104	C₂²×C₆₇	1	3	yes	—
229	C₂×#104	C₂²×C₆₇	1	6	yes	—
230	C₄×M₁₆	C₂²×C₆₇	1	3	yes	—
231	C₄×C₄⋊C₄	—	2	3	no	—
232	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
233	C₄³	C₂²×C₆₇	1	3	yes	—
234	#339	C₂²×C₆₇	1	3	yes	—
235	#339	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#41	—	2	1	no	—
237	C₂×#41	C₂³×C₆₇	1	3	yes	—
238	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
239	C₂×M₁₆	C₂³×C₆₇	1	3	yes	—
240	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
241	C₄⋊Q₈	C₂×C₄×C₆₇	1	3	yes	—
242	#127	C₂×C₄×C₆₇	1	3	yes	—
243	#126	—	2	6	no	—
244	#125	—	2	3	no	—
245	#124	—	2	3	no	—
246	#124	C₂×C₄×C₆₇	1	6	yes	—
247	#123	—	2	6	no	—
248	C₂²⋊Q₈	—	2	3	no	—
249	#121	—	2	3	no	—
250	#121	—	2	3	no	—
251	#121	—	2	3	no	—
252	C₄×Q₈	—	2	8	no	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×Q₈	C₂×C₄×C₆₇	1	3	yes	—
255	C₄×Q₈	C₂×C₄×C₆₇	1	3	yes	—
256	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
257	C₄×Q₈	C₂×C₄×C₆₇	1	3	yes	—
258	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
259	C₄×Q₈	D₄×C₆₇	1	3	yes	—
260	C₄×D₄	—	2	3	no	—
261	C₄×D₄	—	2	3	no	—
262	C₄×D₄	—	2	3	no	—
263	C₄×D₄	—	2	3	no	—
264	C₄×D₄	—	2	3	no	—
265	C₄×D₄	C₂×C₄×C₆₇	1	3	yes	—
266	C₄×D₄	C₂×C₄×C₆₇	1	3	yes	—
267	#117	—	2	3	no	—
268	#117	—	2	3	no	—
269	#117	C₂³×C₆₇	1	1	yes	characteristic
270	C₂×C₄⋊C₄	—	2	3	no	—
271	C₂×C₄⋊C₄	D₄×C₆₇	1	3	yes	—
272	C₂×C₄²	—	2	3	no	—
273	C₂×C₄²	—	2	3	no	—
274	C₂×C₄²	—	2	3	no	—
275	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
276	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
278	C₄⋊C₈	—	2	3	no	—
279	C₄⋊C₈	—	2	3	no	—
280	#104	—	2	6	no	—
281	#104	—	2	6	no	—
282	#104	—	4	6	no	—
283	#97	—	2	3	no	—
284	C₄×C₈	—	2	3	no	—
285	#95	—	2	3	no	—
286	#41	—	2	3	no	—
287	#41	—	2	3	no	—
288	#41	—	2	3	no	—
289	#41	—	2	3	no	—
290	#41	—	4	4	no	—
291	C₂×Q₈	—	2	4	no	—
292	C₂×Q₈	C₂²⋊C₄×C₆₇	1	3	yes	—
293	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
294	C₂×Q₈	C₂²×C₄×C₆₇	1	3	yes	—
295	C₂×D₄	—	2	3	no	—
296	C₂×D₄	—	2	3	no	—
297	C₂×D₄	—	2	3	no	—
298	C₂×D₄	C₂²×C₄×C₆₇	1	3	yes	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	3	no	—
301	C₂²×C₄	—	2	3	no	—
302	C₂²×C₄	—	2	3	no	—
303	C₂²×C₄	—	2	3	no	—
304	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
305	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
306	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
307	M₁₆	—	2	1	no	—
308	M₁₆	—	2	3	no	—
309	M₁₆	—	2	3	no	—
310	M₁₆	#41×C₆₇	1	2	yes	complemented
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	3	no	—
313	C₄⋊C₄	—	2	12	no	—
314	C₄⋊C₄	—	2	3	no	—
315	C₄⋊C₄	—	2	3	no	—
316	C₄⋊C₄	—	2	3	no	—
317	C₄⋊C₄	—	2	3	no	—
318	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	3	yes	—
319	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	3	yes	—
320	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	6	yes	—
321	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
322	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
323	C₂²⋊C₄	—	2	1	no	—
324	C₂²⋊C₄	—	2	6	no	—
325	C₂²⋊C₄	—	2	3	no	—
326	C₂²⋊C₄	—	2	3	no	—
327	C₂²⋊C₄	—	2	6	no	—
328	C₄²	—	2	12	no	—
329	C₄²	—	2	3	no	—
330	C₄²	—	2	3	no	—
331	C₄²	—	2	1	no	—
332	C₄²	—	2	1	no	—
333	C₄²	—	2	3	no	—
334	C₄²	—	2	3	no	—
335	C₄²	—	4	6	no	—
336	C₄²	C₂²×C₄×C₆₇	1	3	yes	—
337	C₄²	C₂²×C₄×C₆₇	1	3	yes	—
338	C₄²	C₂×D₄×C₆₇	1	3	yes	—
339	C₄²	C₂×D₄×C₆₇	1	3	yes	—
340	C₄²	#41×C₆₇	1	2	yes	complemented
341	C₂³	—	2	3	no	—
342	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
343	Q₈	—	2	6	no	—
344	Q₈	—	2	3	no	—
345	Q₈	—	2	3	no	—
346	Q₈	#104×C₆₇	1	4	yes	complemented
347	D₄	—	2	3	no	—
348	D₄	—	2	3	no	—
349	D₄	—	4	3	no	—
350	D₄	—	4	3	no	—
351	D₄	—	4	6	no	—
352	D₄	—	8	3	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	6	no	—
357	C₂×C₄	—	2	6	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	3	no	—
361	C₂×C₄	—	4	3	no	—
362	C₂×C₄	—	4	3	no	—
363	C₂×C₄	—	4	3	no	—
364	C₂×C₄	—	4	3	no	—
365	C₂×C₄	—	4	3	no	—
366	C₂×C₄	—	4	6	no	—
367	C₂×C₄	—	8	3	no	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
369	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
370	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
371	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
372	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
376	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
377	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
378	C₈	—	2	2	no	—
379	C₈	—	4	3	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	4	3	no	—
383	C₂²	—	4	3	no	—
384	C₂²	—	4	3	no	—
385	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
386	C₂²	#548×C₆₇	1	1	yes	characteristic
387	C₂²	#550×C₆₇	1	1	yes	characteristic
388	C₄	—	2	4	no	—
389	C₄	—	2	3	no	—
390	C₄	—	2	3	no	—
391	C₄	—	2	3	no	—
392	C₄	—	4	6	no	—
393	C₄	—	4	3	no	—
394	C₄	—	4	1	no	—
395	C₄	—	4	1	no	—
396	C₄	C₂×#104×C₆₇	1	6	yes	—
397	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
398	C₄	#520×C₆₇	1	1	yes	characteristic
399	C₄	#548×C₆₇	1	1	yes	characteristic
400	C₄	#550×C₆₇	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	8	3	no	—
404	C₂	C₆₇×#2341	1	1	yes	characteristic
405	C₂	C₂²×#104×C₆₇	1	1	yes	characteristic
406	C₂	C₆₇×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₆₇×#3480	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₆₇	C₂	semidirect
3. C₄²×Q₈×C₆₇	C₂	semidirect
4. C₆₇×#1980	C₂	semidirect
5. C₆₇×#1962	C₂	semidirect
6. C₆₇×#1806	C₂	semidirect
7. C₄×#104×C₆₇	C₂	semidirect
8. #559×C₆₇	C₄	semidirect
9. #557×C₆₇	C₄	semidirect
10. #556×C₆₇	C₄	semidirect
12. #550×C₆₇	C₄	semidirect
106. M₁₆×C₆₇	#41	semidirect
136. C₄²×C₆₇	#41	semidirect
142. Q₈×C₆₇	#104	semidirect
197. #22550	C₆₇	direct (N × H)
205. Q₈×M₁₆	C₂×C₆₇	semidirect
206. C₄²×Q₈	C₂×C₆₇	semidirect
207. #1980	C₂×C₆₇	semidirect
208. #1962	C₂×C₆₇	semidirect
209. #1806	C₂×C₆₇	semidirect
210. C₄×#104	C₂×C₆₇	semidirect
211. C₆₇	#22550	direct (N × H)
212. #559	C₄×C₆₇	semidirect
213. #557	C₄×C₆₇	semidirect
214. #556	C₄×C₆₇	semidirect
216. #550	C₄×C₆₇	semidirect
310. M₁₆	#41×C₆₇	semidirect
340. C₄²	#41×C₆₇	semidirect
346. Q₈	#104×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3480	2
Q₈×M₁₆×C₆₇	2
C₄²×Q₈×C₆₇	2
C₆₇×#1980	2
C₆₇×#1962	2
C₆₇×#1806	2
C₄×#104×C₆₇	2
#22550	67

Subgroups — C67×#22550 #30701801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22550 #30701801