Subgroups — #32155079

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22600772	C₂	1	1	yes	maximal, characteristic
2	#22600734	C₂	1	1	yes	maximal, characteristic
3	#22590622	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590546	C₂	1	2	yes	maximal, complemented
5	#22590536	C₂	1	2	yes	maximal, complemented
6	#22590492	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590482	C₂	1	1	yes	maximal, characteristic, complemented
8	#22590445	C₂	1	2	yes	maximal, complemented
9	#22590173	C₂	1	1	yes	maximal, characteristic
10	#22590125	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986722	—	2	1	no	—
13	#17986709	—	2	1	no	—
14	C₂×#14556067	C₂²	1	2	yes	—
15	#17986404	C₂²	1	1	yes	characteristic
16	#17986404	C₂²	1	1	yes	characteristic
17	#17986339	—	2	2	no	—
18	#17986333	—	2	2	no	—
19	#17986294	—	2	1	no	—
20	#17986284	—	2	1	no	—
21	#17986237	C₂²	1	2	yes	—
22	#17986237	C₂²	1	2	yes	—
23	C₈×D₂₈₄	C₂²	1	1	yes	characteristic
24	#17986036	C₂²	1	2	yes	—
25	#107×D₇₁	C₂²	1	2	yes	—
26	#17985827	C₂²	1	2	yes	—
27	#17985827	C₂²	1	2	yes	—
28	#17985826	C₂²	1	2	yes	—
29	#17985824	C₂²	1	2	yes	—
30	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
31	#17985803	C₂²	1	2	yes	—
32	#17985803	C₂²	1	2	yes	—
33	(C₂×D₇₁)⋊C₁₆	C₂²	1	2	yes	—
34	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
37	#366×C₇₁	C₂²	1	1	yes	characteristic
38	#358×C₇₁	C₂²	1	1	yes	characteristic
39	#358×C₇₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
41	C₂×C₈×D₇₁	C₂³	1	2	yes	—
42	#14556128	—	2	1	no	—
43	#14556128	—	2	1	no	—
44	#14556123	—	4	2	no	—
45	#14556120	—	4	1	no	—
46	#14556116	—	4	1	no	—
47	#14556106	—	4	2	no	—
48	#14556104	—	2	2	no	—
49	#14556104	—	2	2	no	—
50	C₄×Dic₁₄₂	—	4	1	no	—
51	C₄×D₂₈₄	—	4	1	no	—
52	C₄×D₂₈₄	D₄	1	1	yes	characteristic
53	#14556086	—	2	2	no	—
54	C₄⋊C₄×D₇₁	—	2	2	no	—
55	#14556067	—	2	4	no	—
56	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
57	#14556062	—	2	1	no	—
58	#14556062	—	2	1	no	—
59	#14556062	C₂³	1	2	yes	—
60	#14556062	C₂³	1	2	yes	—
61	#14556061	—	2	2	no	—
62	#14556059	—	2	2	no	—
63	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
64	#14556052	—	2	2	no	—
65	#14556052	—	2	2	no	—
66	(C₂×D₇₁)⋊C₈	D₄	1	2	yes	—
67	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
68	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₇₁	—	2	1	no	—
70	C₄×Q₈×C₇₁	—	2	1	no	—
71	Q₃₂×C₇₁	—	2	2	no	—
72	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
74	#107×C₇₁	C₂³	1	1	yes	characteristic
75	#107×C₇₁	C₂³	1	1	yes	characteristic
76	#103×C₇₁	—	2	1	no	—
77	#103×C₇₁	—	2	1	no	—
78	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₄₂	—	4	1	no	—
80	C₂×Dic₁₄₂	—	4	1	no	—
81	C₂×D₂₈₄	D₈	1	1	yes	characteristic
82	C₂×C₄×D₇₁	—	4	2	no	—
83	C₂×C₄×D₇₁	C₂×D₄	1	2	yes	—
84	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
85	C₈×D₇₁	—	2	2	no	—
86	C₈×D₇₁	—	2	2	no	—
87	C₄⋊Dic₇₁	—	2	2	no	—
88	C₄⋊Dic₇₁	—	2	2	no	—
89	C₄⋊Dic₇₁	—	4	1	no	—
90	C₄⋊Dic₇₁	—	4	1	no	—
91	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
92	Dic₇₁⋊C₄	—	4	2	no	—
93	Dic₇₁⋊C₄	—	4	2	no	—
94	Dic₇₁⋊C₄	—	4	2	no	—
95	(C₂×D₇₁)⋊C₄	—	4	2	no	—
96	(C₂×D₇₁)⋊C₄	—	4	2	no	—
97	(C₂×D₇₁)⋊C₄	—	4	2	no	—
98	(C₂×D₇₁)⋊C₄	D₈	1	2	yes	—
99	C₄×Dic₇₁	—	4	2	no	—
100	C₇₁⋊C₁₆	—	2	2	no	—
101	C₂×Q₈×C₇₁	—	2	1	no	—
102	C₂×Q₈×C₇₁	—	2	1	no	—
103	Q₁₆×C₇₁	—	2	1	no	—
104	Q₁₆×C₇₁	—	2	1	no	—
105	Q₁₆×C₇₁	—	2	1	no	—
106	Q₁₆×C₇₁	—	2	1	no	—
107	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₇₁	—	2	1	no	—
110	C₄⋊C₄×C₇₁	—	2	1	no	—
111	C₄⋊C₄×C₇₁	—	4	1	no	—
112	C₄⋊C₄×C₇₁	—	4	1	no	—
113	C₄²×C₇₁	—	4	1	no	—
114	C₄²×C₇₁	—	4	1	no	—
115	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₇₁	—	2	1	no	—
117	C₁₆×C₇₁	#41	1	2	yes	—
118	C₂²×D₇₁	C₂×D₈	1	2	yes	—
119	Q₈×C₇₁	—	4	1	no	—
120	Q₈×C₇₁	—	4	1	no	—
121	Q₈×C₇₁	—	4	1	no	—
122	Q₈×C₇₁	—	4	1	no	—
123	C₂×C₄×C₇₁	—	4	1	no	—
124	C₂×C₄×C₇₁	—	4	1	no	—
125	C₂×C₄×C₇₁	—	4	1	no	—
126	C₂×C₄×C₇₁	—	4	1	no	—
127	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₇₁	—	4	2	no	—
131	C₂×Dic₇₁	—	4	2	no	—
132	C₂×Dic₇₁	C₂×D₈	1	2	yes	—
133	D₂₈₄	—	2	2	no	—
134	C₄×D₇₁	—	2	2	no	—
135	C₄×D₇₁	—	2	2	no	—
136	C₄×D₇₁	—	16	2	no	—
137	Dic₁₄₂	—	8	2	no	—
138	Dic₁₄₂	—	16	1	no	—
139	C₈×C₇₁	—	2	1	no	—
140	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
141	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
142	C₇₁⋊C₈	—	2	2	no	—
143	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₇₁	—	2	2	no	—
145	C₂×D₇₁	—	2	2	no	—
146	C₂×D₇₁	—	2	2	no	—
147	C₄×C₇₁	—	2	1	no	—
148	C₄×C₇₁	—	8	1	no	—
149	C₄×C₇₁	—	8	1	no	—
150	C₄×C₇₁	—	8	1	no	—
151	C₄×C₇₁	—	8	1	no	—
152	C₄×C₇₁	#510	1	2	yes	—
153	C₄×C₇₁	#547	1	1	yes	characteristic
154	C₄×C₇₁	#578	1	1	yes	characteristic
155	Dic₇₁	—	2	2	no	—
156	Dic₇₁	—	8	2	no	—
157	Dic₇₁	—	8	2	no	—
158	#33923	—	71	1	no	maximal, Sylow
159	C₂×C₇₁	#3334	1	1	yes	characteristic
160	C₂×C₇₁	C₂×#510	1	1	yes	characteristic
161	C₂×C₇₁	#3416	1	1	yes	characteristic
162	D₇₁	—	4	2	no	—
163	#3342	—	71	1	no	—
164	#3338	—	71	1	no	—
165	#2254	—	71	1	no	—
166	#2237	—	71	2	no	—
167	#2235	—	71	2	no	—
168	#2227	—	71	1	no	—
169	#2225	—	71	1	no	—
170	#2219	—	71	2	no	—
171	C₄×Q₃₂	D₇₁	1	1	yes	characteristic, complemented
172	#2169	—	71	1	no	—
173	#2159	—	71	2	no	—
174	C₇₁	#33923	1	1	yes	characteristic, complemented, Sylow
175	#556	—	142	1	no	—
176	#554	—	142	1	no	—
177	C₂×Q₃₂	C₂×D₇₁	1	1	yes	characteristic
178	C₂×M₃₂	—	71	2	no	—
179	#500	—	71	1	no	—
180	#500	—	71	1	no	—
181	#485	—	142	2	no	—
182	#484	—	142	2	no	—
183	#477	—	142	1	no	—
184	#475	—	142	1	no	—
185	#467	—	71	2	no	—
186	#467	—	71	2	no	—
187	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
188	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
189	C₈×D₄	—	71	1	no	—
190	#427	—	71	2	no	—
191	C₂×#107	—	71	2	no	—
192	#367	—	71	2	no	—
193	#367	—	71	2	no	—
194	#367	—	71	2	no	—
195	#366	—	71	2	no	—
196	#366	C₂×D₇₁	1	1	yes	characteristic
197	C₄⋊C₁₆	—	71	1	no	—
198	#358	—	71	2	no	—
199	#358	—	71	2	no	—
200	#358	C₂×D₇₁	1	1	yes	characteristic
201	#358	C₂×D₇₁	1	1	yes	characteristic
202	C₂²⋊C₁₆	—	71	2	no	—
203	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
204	C₂×Q₁₆	C₂²×D₇₁	1	1	yes	characteristic
205	C₂×Q₁₆	C₂²×D₇₁	1	1	yes	characteristic
206	C₂²×C₈	—	71	2	no	—
207	C₄⋊Q₈	—	142	1	no	—
208	C₄⋊Q₈	—	142	1	no	—
209	#126	—	284	2	no	—
210	#125	—	284	1	no	—
211	#124	—	284	1	no	—
212	C₂²⋊Q₈	—	284	2	no	—
213	C₂²⋊Q₈	—	142	2	no	—
214	C₂²⋊Q₈	—	142	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	284	1	no	—
218	C₄×D₄	—	284	1	no	—
219	C₄×D₄	—	71	1	no	—
220	#117	—	142	2	no	—
221	C₂×C₄⋊C₄	—	142	2	no	—
222	Q₃₂	—	2	2	no	—
223	M₃₂	—	142	4	no	—
224	C₂×C₁₆	—	71	2	no	—
225	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
227	#107	—	71	2	no	—
228	#107	—	71	2	no	—
229	#107	—	142	2	no	—
230	#107	—	142	1	no	—
231	#107	—	142	1	no	—
232	#107	C₂²×D₇₁	1	1	yes	characteristic
233	#107	C₂²×D₇₁	1	1	yes	characteristic
234	#106	—	142	2	no	—
235	C₄⋊C₈	—	71	1	no	—
236	#103	—	2	1	no	—
237	#103	—	2	1	no	—
238	#103	—	142	2	no	—
239	#103	—	142	2	no	—
240	C₂²⋊C₈	—	71	2	no	—
241	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	284	1	no	—
245	C₂×Q₈	—	284	1	no	—
246	C₂×D₄	—	71	1	no	—
247	C₂²×C₄	—	284	2	no	—
248	C₂²×C₄	—	71	2	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	1	no	—
253	C₂×C₈	—	71	2	no	—
254	C₂×C₈	—	142	2	no	—
255	C₂×C₈	—	142	2	no	—
256	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	284	2	no	—
263	C₄⋊C₄	—	284	2	no	—
264	C₄⋊C₄	—	284	2	no	—
265	C₄⋊C₄	—	284	1	no	—
266	C₄⋊C₄	—	284	1	no	—
267	C₄⋊C₄	—	71	1	no	—
268	C₄⋊C₄	—	142	2	no	—
269	C₄⋊C₄	—	142	2	no	—
270	C₂²⋊C₄	—	284	2	no	—
271	C₂²⋊C₄	—	284	2	no	—
272	C₂²⋊C₄	—	284	2	no	—
273	C₂²⋊C₄	—	71	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	284	2	no	—
277	C₄²	D₄×D₇₁	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	142	2	no	—
280	C₁₆	#11765865	1	2	yes	—
281	C₂³	—	71	2	no	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	568	2	no	—
287	Q₈	—	1136	1	no	—
288	D₄	—	142	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	284	2	no	—
294	C₂×C₄	—	284	2	no	—
295	C₂×C₄	—	71	2	no	—
296	C₂×C₄	—	1136	2	no	—
297	C₂×C₄	—	142	2	no	—
298	C₂×C₄	—	142	2	no	—
299	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
300	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	142	2	no	—
304	C₈	C₂×#11765865	1	1	yes	characteristic
305	C₈	#14556175	1	1	yes	characteristic
306	C₂²	—	142	2	no	—
307	C₂²	—	142	2	no	—
308	C₂²	—	142	2	no	—
309	C₂²	C₂×D₈×D₇₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	1	no	—
315	C₄	—	568	2	no	—
316	C₄	—	568	2	no	—
317	C₄	—	142	2	no	—
318	C₄	#17986441	1	2	yes	—
319	C₄	#17986654	1	1	yes	characteristic
320	C₄	#17986813	1	1	yes	characteristic
321	C₂	—	284	2	no	—
322	C₂	#22600689	1	1	yes	characteristic
323	C₂	C₂×#17986441	1	1	yes	characteristic
324	C₂	#22601483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₁
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₄×Q₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂×D₂₈₄	D₈
hypercenter	C₄×Q₃₂	D₇₁
hypocenter	C₇₁	#33923
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #22590622	C₂	semidirect
4. #22590546	C₂	semidirect
5. #22590536	C₂	semidirect
6. #22590492	C₂	semidirect
7. #22590482	C₂	semidirect
8. #22590445	C₂	semidirect
10. #22590125	C₂	semidirect
11. C₄×Q₃₂×C₇₁	C₂	semidirect
171. C₄×Q₃₂	D₇₁	semidirect
174. C₇₁	#33923	semidirect

Maximal subgroups

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

subgroup	index
#22600772	2
#22600734	2
#22590622	2
#22590546	2
#22590536	2
#22590492	2
#22590482	2
#22590445	2
#22590173	2
#22590125	2
C₄×Q₃₂×C₇₁	2
#33923	71