Subgroups — #31912651

Order 18176.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590025	C₂	1	1	yes	maximal, characteristic
2	#22590005	C₂	1	1	yes	maximal, characteristic
3	#22588140	C₂	1	2	yes	maximal, complemented
4	#22586469	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2932	C₂	1	1	yes	maximal, characteristic
6	#17985983	C₂²	1	2	yes	—
7	#17985973	C₂²	1	2	yes	—
8	#17985954	—	2	2	no	—
9	#17985798	C₄	1	2	yes	complemented
10	#17985796	C₄	1	2	yes	complemented
11	#17985794	C₄	1	2	yes	—
12	#17985792	C₄	1	2	yes	—
13	#17985770	—	2	2	no	—
14	C₂×#131×C₇₁	C₄	1	1	yes	characteristic, complemented
15	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
16	#532×C₇₁	C₂²	1	1	yes	characteristic
17	#435×C₇₁	—	2	2	no	—
18	#435×C₇₁	—	2	2	no	—
19	#423×C₇₁	C₄	1	1	yes	characteristic, complemented
20	#408×C₇₁	C₄	1	1	yes	characteristic, complemented
21	#408×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×#97×C₇₁	C₄	1	1	yes	characteristic, complemented
23	C₂×#11765849	—	2	2	no	—
24	#14556087	—	2	2	no	—
25	#14556047	C₂×C₄	1	4	yes	—
26	#14556045	—	2	2	no	—
27	#14556043	—	2	2	no	—
28	#14556043	C₂×C₄	1	4	yes	—
29	#14556038	—	4	2	no	—
30	C₈×Dic₇₁	—	4	2	no	—
31	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
33	#131×C₇₁	—	4	2	no	—
34	#131×C₇₁	—	4	2	no	—
35	C₂×M₁₆×C₇₁	—	2	1	no	—
36	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₇₁	D₄	1	2	yes	—
39	C₂²×C₈×C₇₁	—	2	1	no	—
40	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₁	D₄	1	2	yes	—
43	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
44	#127×C₇₁	C₂×C₄	1	1	yes	characteristic
45	#124×C₇₁	C₂×C₄	1	1	yes	characteristic
46	#124×C₇₁	C₂×C₄	1	1	yes	characteristic
47	#121×C₇₁	—	2	2	no	—
48	C₄×D₄×C₇₁	—	2	2	no	—
49	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₇₁	—	2	2	no	—
51	C₂²⋊C₈×C₇₁	—	2	2	no	—
52	C₂²⋊C₈×C₇₁	—	2	2	no	—
53	#97×C₇₁	—	2	1	no	—
54	#97×C₇₁	—	2	1	no	—
55	#11765849	—	2	4	no	—
56	#11765849	—	4	2	no	—
57	C₂×C₇₁⋊C₈	—	4	2	no	—
58	C₄⋊Dic₇₁	—	4	2	no	—
59	C₂²⋊Dic₇₁	—	2	4	no	—
60	C₄×Dic₇₁	—	4	2	no	—
61	#41×C₇₁	—	4	1	no	—
62	#41×C₇₁	—	4	1	no	—
63	#41×C₇₁	—	4	1	no	—
64	#41×C₇₁	—	4	1	no	—
65	C₂×Q₈×C₇₁	C₄²	1	1	yes	characteristic
66	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×D₄×C₇₁	—	2	1	no	—
68	C₂×D₄×C₇₁	—	2	2	no	—
69	C₂×D₄×C₇₁	C₄²	1	1	yes	characteristic
70	C₂×D₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×D₄×C₇₁	#41	1	2	yes	—
72	C₂²×C₄×C₇₁	—	2	1	no	—
73	C₂²×C₄×C₇₁	—	2	1	no	—
74	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
76	M₁₆×C₇₁	—	4	1	no	—
77	M₁₆×C₇₁	—	4	2	no	—
78	M₁₆×C₇₁	—	4	2	no	—
79	M₁₆×C₇₁	—	4	1	no	—
80	C₂×C₈×C₇₁	—	2	2	no	—
81	C₂×C₈×C₇₁	—	2	2	no	—
82	C₂×C₈×C₇₁	—	2	2	no	—
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	1	no	—
85	C₂×C₈×C₇₁	—	4	2	no	—
86	C₂×C₈×C₇₁	—	4	1	no	—
87	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₇₁	—	2	2	no	—
91	C₂²⋊C₄×C₇₁	—	2	2	no	—
92	C₂²⋊C₄×C₇₁	—	2	2	no	—
93	C₄²×C₇₁	—	2	1	no	—
94	C₄²×C₇₁	C₄²	1	1	yes	characteristic
95	C₄²×C₇₁	C₄²	1	1	yes	characteristic
96	C₂³×C₇₁	—	2	1	no	—
97	C₂³×C₇₁	#117	1	1	yes	characteristic
98	C₂³×C₇₁	C₄×D₄	1	2	yes	—
99	Q₈×C₇₁	—	4	1	no	—
100	Q₈×C₇₁	—	4	1	no	—
101	D₄×C₇₁	—	4	1	no	—
102	D₄×C₇₁	—	4	2	no	—
103	D₄×C₇₁	—	4	1	no	—
104	D₄×C₇₁	—	4	2	no	—
105	D₄×C₇₁	—	8	2	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	2	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	2	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	4	1	no	—
114	C₂×C₄×C₇₁	—	4	1	no	—
115	C₂×C₄×C₇₁	—	8	1	no	—
116	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
119	C₂×Dic₇₁	—	4	4	no	—
120	C₈×C₇₁	—	2	1	no	—
121	C₈×C₇₁	—	2	2	no	—
122	C₈×C₇₁	—	4	1	no	—
123	C₈×C₇₁	—	4	1	no	—
124	C₈×C₇₁	#99	1	2	yes	—
125	C₇₁⋊C₈	—	4	4	no	—
126	C₂²×C₇₁	—	2	2	no	—
127	C₂²×C₇₁	—	2	1	no	—
128	C₂²×C₇₁	—	4	1	no	—
129	C₂²×C₇₁	—	4	2	no	—
130	C₂²×C₇₁	—	4	2	no	—
131	C₂²×C₇₁	—	4	1	no	—
132	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₇₁	—	2	2	no	—
134	C₄×C₇₁	—	2	1	no	—
135	C₄×C₇₁	—	4	1	no	—
136	C₄×C₇₁	—	4	1	no	—
137	C₄×C₇₁	—	4	1	no	—
138	C₄×C₇₁	C₂×#99	1	1	yes	characteristic
139	C₄×C₇₁	#410	1	1	yes	characteristic
140	Dic₇₁	—	8	4	no	—
141	#12634	—	71	1	no	maximal, Sylow
142	C₂×C₇₁	—	2	1	no	—
143	C₂×C₇₁	—	4	2	no	—
144	C₂×C₇₁	—	4	1	no	—
145	C₂×C₇₁	—	8	1	no	—
146	C₂×C₇₁	C₄×#99	1	1	yes	characteristic
147	#2932	D₇₁	1	1	yes	characteristic
148	#2132	—	71	1	no	—
149	#2126	—	71	1	no	—
150	#1757	—	71	2	no	—
151	#1395	—	71	2	no	—
152	C₇₁	#12634	1	1	yes	characteristic, complemented, Sylow
153	C₂×#131	Dic₇₁	1	1	yes	characteristic, complemented
154	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
155	#532	C₂×D₇₁	1	1	yes	characteristic
156	#435	—	2	2	no	—
157	#435	—	2	2	no	—
158	#423	Dic₇₁	1	1	yes	characteristic, complemented
159	#413	—	71	2	no	—
160	#410	—	71	2	no	—
161	#408	Dic₇₁	1	1	yes	characteristic, complemented
162	#408	C₂×D₇₁	1	1	yes	characteristic
163	#405	—	142	2	no	—
164	C₂×#97	Dic₇₁	1	1	yes	characteristic, complemented
165	#356	—	71	2	no	—
166	#355	—	71	2	no	—
167	#354	—	71	2	no	—
168	#353	—	71	2	no	—
169	#344	—	142	2	no	—
170	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
171	C₂×#41	C₂²×D₇₁	1	1	yes	characteristic
172	#131	—	4	2	no	—
173	#131	—	4	2	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	142	2	no	—
176	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
177	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
178	C₂×M₁₆	C₂²⋊D₇₁	1	2	yes	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
183	C₄⋊Q₈	C₄×D₇₁	1	1	yes	characteristic
184	#127	C₄×D₇₁	1	1	yes	characteristic
185	#124	C₄×D₇₁	1	1	yes	characteristic
186	#124	C₄×D₇₁	1	1	yes	characteristic
187	#121	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	#117	—	142	2	no	—
190	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	#101	—	71	4	no	—
193	#100	—	142	2	no	—
194	#99	—	71	4	no	—
195	#99	—	142	2	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	284	2	no	—
201	C₄×C₈	—	284	2	no	—
202	#41	—	4	1	no	—
203	#41	—	4	1	no	—
204	#41	—	4	1	no	—
205	#41	—	4	1	no	—
206	C₂×Q₈	C₄×Dic₇₁	1	1	yes	characteristic
207	C₂×Q₈	C₂×C₄×D₇₁	1	1	yes	characteristic
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₄×Dic₇₁	1	1	yes	characteristic
211	C₂×D₄	C₂×C₄×D₇₁	1	1	yes	characteristic
212	C₂×D₄	#11765865	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	284	2	no	—
222	M₁₆	—	142	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	284	2	no	—
231	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
233	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	284	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	142	4	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	284	2	no	—
241	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
242	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
243	C₂³	—	2	1	no	—
244	C₂³	#14556087	1	1	yes	characteristic
245	C₂³	C₄×C₂²⋊D₇₁	1	2	yes	—
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	4	1	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	284	4	no	—
264	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
266	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
267	C₈	—	2	1	no	—
268	C₈	—	2	2	no	—
269	C₈	—	4	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	284	4	no	—
272	C₈	#14556043	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	C₄×C₂²⋊Dic₇₁	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	568	4	no	—
286	C₄	C₂×#14556043	1	1	yes	characteristic
287	C₄	#17985973	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	4	2	no	—
290	C₂	—	4	1	no	—
291	C₂	—	8	1	no	—
292	C₂	C₄×#14556043	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14556043
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×#41	C₂²×D₇₁
Fitting	C₇₁×#2932	C₂
socle	C₂×C₇₁	C₄×#99
prime radical	#532×C₇₁	C₂²
cosocle	C₂×#41×C₇₁	C₂³
hypercenter	#2932	D₇₁
hypocenter	C₇₁	#12634
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ #2932

Complements

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

N	H ≅ G/N	type
3. #22588140	C₂	semidirect
4. #22586469	C₂	semidirect
9. #17985798	C₄	semidirect
10. #17985796	C₄	semidirect
14. C₂×#131×C₇₁	C₄	semidirect
19. #423×C₇₁	C₄	semidirect
20. #408×C₇₁	C₄	semidirect
22. C₂×#97×C₇₁	C₄	semidirect
152. C₇₁	#12634	semidirect
153. C₂×#131	Dic₇₁	semidirect
158. #423	Dic₇₁	semidirect
161. #408	Dic₇₁	semidirect
164. C₂×#97	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22590025	2
#22590005	2
#22588140	2
#22586469	2
C₇₁×#2932	2
#12634	71