Subgroups — #31888170

Order 18176.

498 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556055	C₂	1	1	yes	maximal, characteristic
3	#22588085	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586621	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586177	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586172	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1476	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₇₁⋊C₈	C₄	1	2	yes	complemented
9	D₄×C₇₁⋊C₈	—	2	2	no	—
10	D₄×C₇₁⋊C₈	C₄	1	2	yes	complemented
11	#17986052	—	2	1	no	—
12	#17986052	C₂²	1	1	yes	characteristic
13	C₂×#14556055	C₄	1	2	yes	—
14	C₄×#11765849	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₁⋊C₈	—	2	1	no	—
16	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
18	#17985758	C₄	1	2	yes	—
19	#17985753	—	2	2	no	—
20	#17985721	—	2	1	no	—
21	C₈×C₇₁⋊C₈	—	2	1	no	—
22	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
23	#431×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
25	Q₈⋊C₈×C₇₁	C₄	1	2	yes	complemented
26	D₄⋊C₈×C₇₁	C₄	1	2	yes	complemented
27	C₂×#11765849	C₂×C₄	1	2	yes	—
28	C₂²×C₇₁⋊C₈	—	2	2	no	—
29	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
30	C₂×C₄×Dic₇₁	—	2	1	no	—
31	C₂×C₄×Dic₇₁	—	2	1	no	—
32	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
33	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
35	#14556055	C₈	1	8	yes	complemented
36	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
37	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
38	#14556039	—	2	1	no	—
39	C₄×C₇₁⋊C₈	—	2	2	no	—
40	C₄×C₇₁⋊C₈	—	2	1	no	—
41	C₄×C₇₁⋊C₈	—	2	1	no	—
42	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
43	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
44	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
45	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₁	—	2	1	no	—
47	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
49	C₄×D₄×C₇₁	—	2	1	no	—
50	C₄×D₄×C₇₁	—	2	1	no	—
51	C₄×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
53	#117×C₇₁	—	2	1	no	—
54	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₇₁	—	2	1	no	—
56	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
58	C₂²⋊C₈×C₇₁	—	2	1	no	—
59	C₄×C₈×C₇₁	—	2	1	no	—
60	C₄×C₈×C₇₁	—	2	1	no	—
61	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
62	C₂²×Dic₇₁	—	2	2	no	—
63	#11765849	—	2	2	no	—
64	#11765849	C₂×C₈	1	4	yes	complemented
65	C₂×C₇₁⋊C₈	—	2	2	no	—
66	C₂×C₇₁⋊C₈	—	2	2	no	—
67	C₂×C₇₁⋊C₈	—	4	2	no	—
68	C₂×C₇₁⋊C₈	—	4	4	no	—
69	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
71	C₄×Dic₇₁	—	2	2	no	—
72	C₄×Dic₇₁	—	4	2	no	—
73	C₄×Dic₇₁	—	4	2	no	—
74	C₄×Dic₇₁	C₂×C₈	1	4	yes	complemented
75	#41×C₇₁	—	2	2	no	—
76	#41×C₇₁	C₂×C₈	1	4	yes	—
77	C₂×Q₈×C₇₁	C₄²	1	1	yes	characteristic
78	C₂×D₄×C₇₁	—	2	1	no	—
79	C₂×D₄×C₇₁	C₄²	1	1	yes	characteristic
80	C₂²×C₄×C₇₁	—	2	1	no	—
81	C₂²×C₄×C₇₁	—	2	1	no	—
82	C₂²×C₄×C₇₁	—	2	1	no	—
83	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
86	C₂×C₈×C₇₁	—	2	2	no	—
87	C₂×C₈×C₇₁	—	2	2	no	—
88	C₂×C₈×C₇₁	—	4	2	no	—
89	C₄⋊C₄×C₇₁	—	2	1	no	—
90	C₄⋊C₄×C₇₁	—	2	1	no	—
91	C₄⋊C₄×C₇₁	C₄²	1	1	yes	characteristic
92	C₄⋊C₄×C₇₁	C₄²	1	1	yes	characteristic
93	C₂²⋊C₄×C₇₁	—	2	1	no	—
94	C₂²⋊C₄×C₇₁	—	2	1	no	—
95	C₂²⋊C₄×C₇₁	—	2	1	no	—
96	C₄²×C₇₁	—	2	1	no	—
97	C₄²×C₇₁	—	2	1	no	—
98	C₄²×C₇₁	—	2	1	no	—
99	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₁	#41	1	1	yes	characteristic
103	C₂³×C₇₁	—	2	1	no	—
104	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₇₁	—	2	1	no	—
106	Q₈×C₇₁	C₄×C₈	1	2	yes	—
107	D₄×C₇₁	—	2	4	no	—
108	D₄×C₇₁	—	2	1	no	—
109	D₄×C₇₁	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	2	no	—
112	C₂×C₄×C₇₁	—	2	4	no	—
113	C₂×C₄×C₇₁	—	2	2	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	4	2	no	—
117	C₂×C₄×C₇₁	—	4	2	no	—
118	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₇₁	#131	1	2	yes	—
126	C₂×Dic₇₁	—	2	4	no	—
127	C₂×Dic₇₁	—	4	2	no	—
128	C₂×Dic₇₁	—	4	2	no	—
129	C₈×C₇₁	—	4	2	no	—
130	C₈×C₇₁	—	4	2	no	—
131	C₇₁⋊C₈	—	2	2	no	—
132	C₇₁⋊C₈	—	2	2	no	—
133	C₇₁⋊C₈	—	2	4	no	—
134	C₇₁⋊C₈	—	4	2	no	—
135	C₂²×C₇₁	—	2	1	no	—
136	C₂²×C₇₁	—	2	1	no	—
137	C₂²×C₇₁	—	2	2	no	—
138	C₂²×C₇₁	—	4	1	no	—
139	C₂²×C₇₁	—	4	1	no	—
140	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₇₁	C₈×D₄	1	2	yes	—
142	C₄×C₇₁	—	2	2	no	—
143	C₄×C₇₁	—	2	1	no	—
144	C₄×C₇₁	—	2	1	no	—
145	C₄×C₇₁	—	4	1	no	—
146	C₄×C₇₁	—	4	1	no	—
147	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
148	C₄×C₇₁	#405	1	1	yes	characteristic
149	C₄×C₇₁	#421	1	2	yes	—
150	C₄×C₇₁	C₈×D₄	1	2	yes	—
151	C₄×C₇₁	#444	1	2	yes	—
152	Dic₇₁	—	4	4	no	—
153	#8191	—	71	1	no	maximal, Sylow
154	C₂×C₇₁	—	2	2	no	—
155	C₂×C₇₁	—	4	2	no	—
156	C₂×C₇₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₇₁	#1759	1	1	yes	characteristic
158	C₂×C₇₁	#1765	1	1	yes	characteristic
159	C₈×#41	—	71	1	no	—
160	C₄×#104	—	71	1	no	—
161	#1744	—	71	1	no	—
162	#1476	D₇₁	1	1	yes	characteristic
163	C₈×M₁₆	—	71	1	no	—
164	#1278	—	71	1	no	—
165	#1276	—	71	1	no	—
166	C₇₁	#8191	1	1	yes	characteristic, complemented, Sylow
167	C₄×#41	C₂×D₇₁	1	1	yes	characteristic
168	C₈×Q₈	—	71	2	no	—
169	C₈×D₄	—	71	2	no	—
170	C₈×D₄	—	142	2	no	—
171	#431	—	71	1	no	—
172	#431	—	142	1	no	—
173	#431	C₂×D₇₁	1	1	yes	characteristic
174	C₂×#104	—	71	2	no	—
175	C₄×M₁₆	—	71	1	no	—
176	C₂×C₄×C₈	—	71	1	no	—
177	C₂×C₄×C₈	—	142	1	no	—
178	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
179	C₄³	—	71	1	no	—
180	#339	—	71	2	no	—
181	#336	—	142	2	no	—
182	Q₈⋊C₈	Dic₇₁	1	2	yes	complemented
183	D₄⋊C₈	Dic₇₁	1	2	yes	complemented
184	#322	—	142	1	no	—
185	C₈²	—	142	1	no	—
186	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
187	C₂×M₁₆	—	71	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	71	2	no	—
190	C₂²×C₈	—	142	2	no	—
191	C₄×Q₈	C₄×D₇₁	1	1	yes	characteristic
192	C₄×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	C₄×D₇₁	1	1	yes	characteristic
196	C₄×D₄	C₂×Dic₇₁	1	1	yes	characteristic
197	#117	—	2	1	no	—
198	#117	C₄×D₇₁	1	1	yes	characteristic
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	71	2	no	—
201	C₂×C₄²	—	142	1	no	—
202	C₂×C₄²	—	142	1	no	—
203	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
204	C₄⋊C₈	—	71	2	no	—
205	C₄⋊C₈	—	142	2	no	—
206	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
207	#104	—	71	8	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	142	2	no	—
210	C₂²⋊C₈	—	142	1	no	—
211	#97	—	142	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	71	2	no	—
215	C₄×C₈	—	71	1	no	—
216	C₄×C₈	—	71	1	no	—
217	C₄×C₈	—	142	2	no	—
218	C₄×C₈	—	142	1	no	—
219	C₄×C₈	—	142	1	no	—
220	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
221	#41	—	2	2	no	—
222	#41	C₈×D₇₁	1	4	yes	—
223	C₂×Q₈	C₄×Dic₇₁	1	1	yes	characteristic
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	C₄×Dic₇₁	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	142	2	no	—
230	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
232	C₂²×C₄	#11765864	1	1	yes	characteristic
233	M₁₆	—	71	4	no	—
234	M₁₆	—	142	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	284	4	no	—
239	C₂×C₈	—	284	2	no	—
240	C₂×C₈	—	71	2	no	—
241	C₂×C₈	—	71	2	no	—
242	C₂×C₈	—	142	2	no	—
243	C₂×C₈	—	142	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	C₄×Dic₇₁	1	1	yes	characteristic
247	C₄⋊C₄	C₄×Dic₇₁	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	284	2	no	—
255	C₄²	—	284	2	no	—
256	C₄²	—	71	4	no	—
257	C₄²	—	142	2	no	—
258	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
259	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
260	C₄²	D₄×D₇₁	1	1	yes	characteristic
261	C₄²	#11765864	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	C₈×Dic₇₁	1	2	yes	—
266	D₄	—	2	4	no	—
267	D₄	—	2	1	no	—
268	D₄	C₈×Dic₇₁	1	2	yes	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	284	2	no	—
278	C₂×C₄	—	284	2	no	—
279	C₂×C₄	—	142	4	no	—
280	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
282	C₂×C₄	#14556084	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
285	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
287	C₂×C₄	#14556136	1	2	yes	—
288	C₈	—	4	2	no	—
289	C₈	—	4	2	no	—
290	C₈	—	284	2	no	—
291	C₈	—	142	4	no	—
292	C₈	—	142	2	no	—
293	C₈	—	142	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₂²⋊C₄×Dic₇₁	1	1	yes	characteristic
300	C₂²	(C₂²⋊D₇₁)⋊C₈	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	284	4	no	—
307	C₄	C₂×C₈×Dic₇₁	1	1	yes	characteristic
308	C₄	#17985957	1	1	yes	characteristic
309	C₄	#17986014	1	2	yes	—
310	C₄	(C₂²⋊D₇₁)⋊C₈	1	2	yes	—
311	C₄	#17986121	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	4	2	no	—
314	C₂	#22588114	1	1	yes	characteristic
315	C₂	#22588152	1	1	yes	characteristic
316	C₂	#22588187	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1476	C₂
socle	C₂²×C₇₁	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1476	D₇₁
hypocenter	C₇₁	#8191
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1476

Complements

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

N	H ≅ G/N	type
3. #22588085	C₂	semidirect
4. #22586621	C₂	semidirect
5. #22586177	C₂	semidirect
6. #22586172	C₂	semidirect
8. Q₈×C₇₁⋊C₈	C₄	semidirect
10. D₄×C₇₁⋊C₈	C₄	semidirect
25. Q₈⋊C₈×C₇₁	C₄	semidirect
26. D₄⋊C₈×C₇₁	C₄	semidirect
35. #14556055	C₈	semidirect
64. #11765849	C₂×C₈	semidirect
74. C₄×Dic₇₁	C₂×C₈	semidirect
166. C₇₁	#8191	semidirect
182. Q₈⋊C₈	Dic₇₁	semidirect
183. D₄⋊C₈	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#41×C₇₁⋊C₈	2
C₄×#14556055	2
#22588085	2
#22586621	2
#22586177	2
#22586172	2
C₇₁×#1476	2
#8191	71