Subgroups — #31912581

Order 18176.

382 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601442	C₂	1	1	yes	maximal, characteristic
2	#22589969	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985776	C₂	1	1	yes	maximal, characteristic, complemented
4	#22588186	C₂	1	1	yes	maximal, characteristic
5	#22586426	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556134	C₂²	1	1	yes	characteristic
9	#17986417	—	2	2	no	—
10	#17986415	C₄	1	4	yes	complemented
11	C₂×#14556068	C₂²	1	1	yes	characteristic
12	C₂×#14556068	C₂²	1	1	yes	characteristic
13	M₃₂×D₇₁	—	2	2	no	—
14	C₂×#14556054	C₄	1	1	yes	characteristic
15	#17985993	C₄	1	1	yes	characteristic
16	C₂×#14556038	C₂²	1	1	yes	characteristic
17	#17985784	C₄	1	4	yes	complemented
18	#17985776	—	2	2	no	—
19	#17985763	C₄	1	1	yes	characteristic
20	#17985758	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
23	#405×C₇₁	C₂²	1	1	yes	characteristic
24	#347×C₇₁	—	2	2	no	—
25	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
26	#14556134	—	2	1	no	—
27	#14556134	—	2	1	no	—
28	#14556134	C₂×C₄	1	2	yes	—
29	#14556134	C₂×C₄	1	2	yes	—
30	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
31	C₂×#11765848	—	2	1	no	—
32	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇₁	—	2	1	no	—
34	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
35	#14556068	—	2	1	no	—
36	#14556068	—	2	1	no	—
37	#14556068	—	2	1	no	—
38	#14556068	C₂×C₄	1	2	yes	—
39	#14556067	—	2	4	no	—
40	C₂×C₇₁⋊C₁₆	—	2	1	no	—
41	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
42	C₁₆×D₇₁	—	2	4	no	—
43	#14556054	—	2	2	no	—
44	#14556051	—	2	1	no	—
45	#14556048	—	2	1	no	—
46	#14556038	—	2	1	no	—
47	#14556038	—	2	1	no	—
48	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
50	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₇₁	—	2	1	no	—
52	M₃₂×C₇₁	C₂×C₄	1	2	yes	—
53	M₃₂×C₇₁	D₄	1	2	yes	—
54	M₃₂×C₇₁	D₄	1	2	yes	—
55	C₂×C₁₆×C₇₁	—	2	1	no	—
56	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
57	#97×C₇₁	—	2	1	no	—
58	C₄×C₈×C₇₁	—	2	1	no	—
59	#11765865	—	2	1	no	—
60	#11765865	—	2	1	no	—
61	#11765865	C₄²	1	2	yes	—
62	#11765865	M₁₆	1	2	yes	—
63	C₂×Dic₁₄₂	M₁₆	1	1	yes	characteristic
64	C₂×D₂₈₄	M₁₆	1	1	yes	characteristic
65	C₂×C₂²⋊D₇₁	—	2	1	no	—
66	C₂²×Dic₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	#11765849	—	2	1	no	—
69	#11765849	C₄²	1	2	yes	—
70	#11765848	—	2	2	no	—
71	#11765848	—	2	2	no	—
72	C₂×C₇₁⋊C₈	—	2	1	no	—
73	C₂×C₇₁⋊C₈	—	2	1	no	—
74	C₂×C₇₁⋊C₈	—	2	1	no	—
75	C₂×C₇₁⋊C₈	—	2	1	no	—
76	C₈×D₇₁	—	2	2	no	—
77	C₈×D₇₁	—	2	2	no	—
78	C₄×Dic₇₁	—	2	1	no	—
79	C₄×Dic₇₁	—	2	1	no	—
80	C₇₁⋊C₁₆	—	2	2	no	—
81	C₇₁⋊C₁₆	—	2	2	no	—
82	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
83	M₁₆×C₇₁	—	2	1	no	—
84	M₁₆×C₇₁	—	2	1	no	—
85	C₂×C₈×C₇₁	—	2	1	no	—
86	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
92	C₄⋊C₄×C₇₁	—	2	1	no	—
93	C₂²⋊C₄×C₇₁	—	2	1	no	—
94	C₄²×C₇₁	—	2	1	no	—
95	C₁₆×C₇₁	—	2	2	no	—
96	C₁₆×C₇₁	C₂²⋊C₄	1	4	yes	—
97	C₂³×C₇₁	#131	1	1	yes	characteristic
98	C₂²×D₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	4	1	no	—
100	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
106	C₂²⋊D₇₁	—	2	2	no	—
107	C₂²⋊D₇₁	—	2	2	no	—
108	C₂×Dic₇₁	—	2	1	no	—
109	C₂×Dic₇₁	—	4	1	no	—
110	C₂×Dic₇₁	—	8	1	no	—
111	D₂₈₄	—	2	1	no	—
112	D₂₈₄	#97	1	2	yes	—
113	C₄×D₇₁	—	2	2	no	—
114	C₄×D₇₁	—	2	2	no	—
115	Dic₁₄₂	—	2	1	no	—
116	Dic₁₄₂	#97	1	2	yes	—
117	C₈×C₇₁	—	4	1	no	—
118	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₈×C₇₁	#117	1	1	yes	characteristic
120	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
121	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
122	C₇₁⋊C₈	—	2	2	no	—
123	C₇₁⋊C₈	—	4	1	no	—
124	C₂²×C₇₁	—	4	1	no	—
125	C₂²×C₇₁	#405	1	1	yes	characteristic
126	C₂²×C₇₁	#435	1	1	yes	characteristic
127	C₂²×C₇₁	#435	1	1	yes	characteristic
128	C₂×D₇₁	—	2	2	no	—
129	C₂×D₇₁	—	8	1	no	—
130	C₄×C₇₁	—	8	1	no	—
131	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
133	C₄×C₇₁	#435	1	1	yes	characteristic
134	C₄×C₇₁	#435	1	1	yes	characteristic
135	Dic₇₁	—	2	2	no	—
136	Dic₇₁	—	8	1	no	—
137	#12618	—	71	1	no	maximal, Sylow
138	C₂×C₇₁	—	2	1	no	—
139	C₂×C₇₁	—	2	1	no	—
140	C₂×C₇₁	—	2	1	no	—
141	C₂×C₇₁	#1752	1	1	yes	characteristic
142	D₇₁	—	4	2	no	—
143	#3407	—	71	1	no	—
144	#2117	—	71	1	no	—
145	#2110	D₇₁	1	1	yes	characteristic, complemented
146	C₂×#347	—	71	1	no	—
147	#1765	—	71	1	no	—
148	#1378	—	71	1	no	—
149	#1377	—	71	1	no	—
150	C₇₁	#12618	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	71	1	no	—
152	#504	—	71	4	no	—
153	#504	—	142	2	no	—
154	C₂×M₃₂	—	71	1	no	—
155	C₂×M₃₂	—	71	1	no	—
156	C₂×M₃₂	—	142	2	no	—
157	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
158	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
159	C₂×#104	—	71	1	no	—
160	#416	—	71	1	no	—
161	#405	C₂×D₇₁	1	1	yes	characteristic
162	C₂×#97	—	71	1	no	—
163	#350	—	71	4	no	—
164	#347	—	2	2	no	—
165	#347	—	142	2	no	—
166	#341	—	71	1	no	—
167	#339	—	71	1	no	—
168	C₂×#41	—	71	1	no	—
169	#131	—	71	2	no	—
170	#131	—	71	2	no	—
171	#131	—	142	1	no	—
172	#131	—	142	1	no	—
173	C₂×M₁₆	—	71	1	no	—
174	C₂×M₁₆	—	142	1	no	—
175	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
176	C₂²×C₈	—	71	1	no	—
177	C₂²×C₈	—	142	1	no	—
178	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
179	#117	C₄×D₇₁	1	1	yes	characteristic
180	C₂×C₄²	—	71	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	71	2	no	—
183	M₃₂	—	142	1	no	—
184	M₃₂	—	142	1	no	—
185	M₃₂	—	142	1	no	—
186	M₃₂	—	142	4	no	—
187	M₃₂	C₄×D₇₁	1	2	yes	—
188	M₃₂	D₂₈₄	1	2	yes	—
189	M₃₂	C₂²⋊D₇₁	1	2	yes	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	71	2	no	—
192	C₂×C₁₆	—	142	1	no	—
193	C₂×C₁₆	—	142	4	no	—
194	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
195	#104	—	142	2	no	—
196	#103	—	142	1	no	—
197	#102	—	142	1	no	—
198	#97	—	2	1	no	—
199	#97	—	142	1	no	—
200	#97	—	142	1	no	—
201	C₄×C₈	—	2	1	no	—
202	#41	—	71	2	no	—
203	#41	—	71	2	no	—
204	#41	—	142	1	no	—
205	#41	—	142	1	no	—
206	C₂×Q₈	—	71	1	no	—
207	C₂×D₄	—	71	1	no	—
208	C₂×D₄	—	142	1	no	—
209	C₂²×C₄	—	142	1	no	—
210	C₂²×C₄	—	142	1	no	—
211	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	1	no	—
214	M₁₆	—	71	2	no	—
215	M₁₆	—	142	1	no	—
216	M₁₆	—	142	2	no	—
217	M₁₆	—	142	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	142	1	no	—
220	C₂×C₈	—	142	1	no	—
221	C₂×C₈	—	142	1	no	—
222	C₂×C₈	—	142	1	no	—
223	C₂×C₈	—	142	2	no	—
224	C₂×C₈	—	142	2	no	—
225	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
227	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
228	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
229	C₂×C₈	#11765865	1	1	yes	characteristic
230	C₂×C₈	#11765865	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	142	1	no	—
235	C₄²	—	142	1	no	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	142	2	no	—
238	C₁₆	—	142	2	no	—
239	C₁₆	(C₂×D₇₁)⋊C₄	1	4	yes	—
240	C₂³	—	142	1	no	—
241	C₂³	#14556136	1	1	yes	characteristic
242	Q₈	—	71	2	no	—
243	Q₈	—	142	1	no	—
244	D₄	—	71	2	no	—
245	D₄	—	142	2	no	—
246	D₄	—	142	2	no	—
247	D₄	—	142	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	284	1	no	—
250	C₂×C₄	—	568	1	no	—
251	C₂×C₄	—	142	1	no	—
252	C₂×C₄	—	142	2	no	—
253	C₂×C₄	—	142	2	no	—
254	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
255	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
256	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
257	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
258	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
259	C₂×C₄	#14556136	1	1	yes	characteristic
260	C₈	—	4	1	no	—
261	C₈	—	284	1	no	—
262	C₈	—	142	2	no	—
263	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
264	C₈	#14556085	1	1	yes	characteristic
265	C₈	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
266	C₈	C₄×D₂₈₄	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	568	1	no	—
269	C₂²	—	142	2	no	—
270	C₂²	#17985956	1	1	yes	characteristic
271	C₂²	#17986068	1	1	yes	characteristic
272	C₂²	#17986072	1	1	yes	characteristic
273	C₄	—	8	1	no	—
274	C₄	—	568	1	no	—
275	C₄	—	142	2	no	—
276	C₄	C₄×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
277	C₄	#97×D₇₁	1	1	yes	characteristic
278	C₄	#17986068	1	1	yes	characteristic
279	C₄	#17986072	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	284	2	no	—
284	C₂	#22588117	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2110	C₂
socle	C₂×C₇₁	#1752
prime radical	C₂×#11765865	C₂×C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2110	D₇₁
hypocenter	C₇₁	#12618
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×M₃₂ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. C₂×#17985776	C₂	semidirect
5. #22586426	C₂	semidirect
6. #22586423	C₂	semidirect
7. C₇₁×#2110	C₂	semidirect
10. #17986415	C₄	semidirect
17. #17985784	C₄	semidirect
145. #2110	D₇₁	semidirect
150. C₇₁	#12618	semidirect

Maximal subgroups

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

subgroup	index
#22601442	2
#22589969	2
C₂×#17985776	2
#22588186	2
#22586426	2
#22586423	2
C₇₁×#2110	2
#12618	71