Subgroups — #31912573

Order 18176.

382 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986417	C₂	1	1	yes	maximal, characteristic
2	#22589960	C₂	1	1	yes	maximal, characteristic
3	#22589944	C₂	1	1	yes	maximal, characteristic, complemented
4	#22588141	C₂	1	1	yes	maximal, characteristic
5	#22586430	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586429	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	#17986417	—	2	2	no	—
11	C₂×#14556068	C₂²	1	1	yes	characteristic
12	C₂×#14556067	C₄	1	2	yes	complemented
13	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₁	C₄	1	2	yes	complemented
15	#17985981	C₄	1	1	yes	characteristic
16	#17985972	C₄	1	1	yes	characteristic
17	#17985954	C₂²	1	1	yes	characteristic
18	#17985780	C₄	1	4	yes	complemented
19	#17985776	—	2	2	no	—
20	#17985771	C₄	1	1	yes	characteristic
21	#17985769	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
24	#405×C₇₁	C₂²	1	1	yes	characteristic
25	#347×C₇₁	—	2	2	no	—
26	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
27	#14556134	—	2	1	no	—
28	#14556134	—	2	1	no	—
29	#14556134	—	2	1	no	—
30	#14556134	—	2	1	no	—
31	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
32	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
33	C₂×#11765848	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
35	#14556087	C₂×C₄	1	1	yes	characteristic
36	#14556068	—	2	1	no	—
37	#14556068	C₂×C₄	1	2	yes	—
38	#14556067	—	2	4	no	—
39	C₂×C₇₁⋊C₁₆	—	2	1	no	—
40	C₂×C₇₁⋊C₁₆	—	2	1	no	—
41	C₂×C₇₁⋊C₁₆	—	2	1	no	—
42	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₇₁	—	2	4	no	—
45	#14556046	—	2	1	no	—
46	#14556044	—	2	1	no	—
47	#14556042	—	2	2	no	—
48	#14556038	—	2	1	no	—
49	C₈×Dic₇₁	—	2	1	no	—
50	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
52	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₇₁	—	2	1	no	—
54	M₃₂×C₇₁	C₂×C₄	1	2	yes	—
55	M₃₂×C₇₁	D₄	1	2	yes	—
56	M₃₂×C₇₁	D₄	1	2	yes	—
57	C₂×C₁₆×C₇₁	—	2	1	no	—
58	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
59	#97×C₇₁	—	2	1	no	—
60	C₄×C₈×C₇₁	—	2	1	no	—
61	#11765865	—	2	1	no	—
62	#11765865	—	2	1	no	—
63	#11765865	—	2	1	no	—
64	#11765865	—	2	1	no	—
65	C₂×Dic₁₄₂	M₁₆	1	1	yes	characteristic
66	C₂×D₂₈₄	M₁₆	1	1	yes	characteristic
67	C₂×C₂²⋊D₇₁	M₁₆	1	2	yes	—
68	C₂×C₄×D₇₁	C₄²	1	2	yes	—
69	#11765849	—	2	1	no	—
70	#11765849	—	2	1	no	—
71	#11765849	—	2	1	no	—
72	#11765849	—	2	1	no	—
73	#11765848	—	2	2	no	—
74	#11765848	—	2	2	no	—
75	C₂×C₇₁⋊C₈	—	2	1	no	—
76	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
77	C₈×D₇₁	—	2	2	no	—
78	C₈×D₇₁	—	2	2	no	—
79	C₄⋊Dic₇₁	—	2	1	no	—
80	C₂²⋊Dic₇₁	—	2	1	no	—
81	C₄×Dic₇₁	—	2	1	no	—
82	C₇₁⋊C₁₆	—	2	1	no	—
83	C₇₁⋊C₁₆	—	2	1	no	—
84	C₇₁⋊C₁₆	—	2	2	no	—
85	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
86	M₁₆×C₇₁	—	2	1	no	—
87	M₁₆×C₇₁	—	2	1	no	—
88	C₂×C₈×C₇₁	—	2	1	no	—
89	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
94	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
95	C₄⋊C₄×C₇₁	—	2	1	no	—
96	C₂²⋊C₄×C₇₁	—	2	1	no	—
97	C₄²×C₇₁	—	2	1	no	—
98	C₁₆×C₇₁	—	2	2	no	—
99	C₁₆×C₇₁	C₂²⋊C₄	1	4	yes	—
100	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₇₁	#97	1	2	yes	—
102	C₂×C₄×C₇₁	—	4	1	no	—
103	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
109	C₂²⋊D₇₁	—	2	2	no	—
110	C₂²⋊D₇₁	—	2	2	no	—
111	C₂×Dic₇₁	—	4	1	no	—
112	C₂×Dic₇₁	#97	1	2	yes	—
113	D₂₈₄	—	2	1	no	—
114	D₂₈₄	—	2	1	no	—
115	C₄×D₇₁	—	2	2	no	—
116	C₄×D₇₁	—	2	2	no	—
117	Dic₁₄₂	—	2	1	no	—
118	Dic₁₄₂	—	2	1	no	—
119	C₈×C₇₁	—	4	1	no	—
120	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₈×C₇₁	#117	1	1	yes	characteristic
122	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
123	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
124	C₇₁⋊C₈	—	2	2	no	—
125	C₇₁⋊C₈	—	4	1	no	—
126	C₂²×C₇₁	—	4	1	no	—
127	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
128	C₂²×C₇₁	#435	1	1	yes	characteristic
129	C₂²×C₇₁	#435	1	1	yes	characteristic
130	C₂×D₇₁	—	2	2	no	—
131	C₂×D₇₁	—	4	2	no	—
132	C₄×C₇₁	—	8	1	no	—
133	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₇₁	#405	1	1	yes	characteristic
135	C₄×C₇₁	#435	1	1	yes	characteristic
136	C₄×C₇₁	#435	1	1	yes	characteristic
137	Dic₇₁	—	2	2	no	—
138	Dic₇₁	—	8	1	no	—
139	#12617	—	71	1	no	maximal, Sylow
140	C₂×C₇₁	—	2	1	no	—
141	C₂×C₇₁	—	2	1	no	—
142	C₂×C₇₁	—	2	1	no	—
143	C₂×C₇₁	#1752	1	1	yes	characteristic
144	D₇₁	—	4	2	no	—
145	C₂×#504	—	71	1	no	—
146	#2115	—	71	1	no	—
147	#2110	—	71	1	no	—
148	#2110	D₇₁	1	1	yes	characteristic, complemented
149	#1757	—	71	1	no	—
150	#1379	—	71	1	no	—
151	#1379	—	71	1	no	—
152	C₇₁	#12617	1	1	yes	characteristic, complemented, Sylow
153	C₂×#131	—	71	1	no	—
154	#504	—	142	2	no	—
155	#504	—	142	2	no	—
156	C₂×M₃₂	—	71	2	no	—
157	C₂×M₃₂	—	71	1	no	—
158	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
159	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
160	C₂²×C₁₆	—	71	1	no	—
161	C₂²×C₁₆	—	71	2	no	—
162	#413	—	71	1	no	—
163	#410	—	71	1	no	—
164	#405	—	71	1	no	—
165	#405	C₂×D₇₁	1	1	yes	characteristic
166	#349	—	71	4	no	—
167	#347	—	2	2	no	—
168	#347	—	142	2	no	—
169	#344	—	71	1	no	—
170	#344	—	71	1	no	—
171	C₂×#41	—	71	1	no	—
172	#131	—	142	1	no	—
173	#131	—	142	1	no	—
174	#131	—	142	1	no	—
175	#131	—	142	1	no	—
176	C₂×M₁₆	—	71	1	no	—
177	C₂×M₁₆	—	71	2	no	—
178	C₂×M₁₆	—	71	1	no	—
179	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
180	C₂²×C₈	—	71	2	no	—
181	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
182	#117	—	71	1	no	—
183	#117	C₄×D₇₁	1	1	yes	characteristic
184	M₃₂	—	2	1	no	—
185	M₃₂	—	71	2	no	—
186	M₃₂	—	142	1	no	—
187	M₃₂	—	142	4	no	—
188	M₃₂	C₄×D₇₁	1	2	yes	—
189	M₃₂	D₂₈₄	1	2	yes	—
190	M₃₂	C₂²⋊D₇₁	1	2	yes	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	71	1	no	—
193	C₂×C₁₆	—	71	1	no	—
194	C₂×C₁₆	—	142	1	no	—
195	C₂×C₁₆	—	142	1	no	—
196	C₂×C₁₆	—	142	1	no	—
197	C₂×C₁₆	—	142	4	no	—
198	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
199	#101	—	142	1	no	—
200	#100	—	142	1	no	—
201	#99	—	142	2	no	—
202	#97	—	2	1	no	—
203	#97	—	142	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	142	1	no	—
206	#41	—	142	1	no	—
207	#41	—	142	1	no	—
208	#41	—	142	1	no	—
209	#41	—	142	1	no	—
210	C₂×Q₈	—	71	1	no	—
211	C₂×D₄	—	71	2	no	—
212	C₂×D₄	—	71	1	no	—
213	C₂²×C₄	—	71	2	no	—
214	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	1	no	—
217	M₁₆	—	142	1	no	—
218	M₁₆	—	142	1	no	—
219	M₁₆	—	142	1	no	—
220	M₁₆	—	142	1	no	—
221	M₁₆	—	142	2	no	—
222	M₁₆	—	142	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	71	2	no	—
225	C₂×C₈	—	142	1	no	—
226	C₂×C₈	—	142	2	no	—
227	C₂×C₈	—	142	2	no	—
228	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
232	C₂×C₈	#11765865	1	1	yes	characteristic
233	C₂×C₈	#11765865	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	142	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	142	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	142	1	no	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	142	1	no	—
242	C₁₆	—	142	1	no	—
243	C₁₆	—	142	2	no	—
244	C₁₆	(C₂×D₇₁)⋊C₄	1	4	yes	—
245	C₂³	—	71	2	no	—
246	C₂³	M₁₆×D₇₁	1	1	yes	characteristic
247	Q₈	—	142	1	no	—
248	Q₈	—	142	1	no	—
249	D₄	—	142	2	no	—
250	D₄	—	142	2	no	—
251	D₄	—	142	1	no	—
252	D₄	—	142	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	284	1	no	—
255	C₂×C₄	—	71	2	no	—
256	C₂×C₄	—	142	2	no	—
257	C₂×C₄	—	142	2	no	—
258	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
261	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
262	C₂×C₄	#14556136	1	1	yes	characteristic
263	C₂×C₄	#14556136	1	1	yes	characteristic
264	C₈	—	4	1	no	—
265	C₈	—	284	1	no	—
266	C₈	—	142	2	no	—
267	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
268	C₈	#14556085	1	1	yes	characteristic
269	C₈	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
270	C₈	C₄×D₂₈₄	1	1	yes	characteristic
271	C₂²	—	4	1	no	—
272	C₂²	—	284	2	no	—
273	C₂²	—	142	2	no	—
274	C₂²	#97×D₇₁	1	1	yes	characteristic
275	C₂²	#17986068	1	1	yes	characteristic
276	C₂²	#17986072	1	1	yes	characteristic
277	C₄	—	8	1	no	—
278	C₄	—	568	1	no	—
279	C₄	—	142	2	no	—
280	C₄	C₄×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
281	C₄	#17985956	1	1	yes	characteristic
282	C₄	#17986068	1	1	yes	characteristic
283	C₄	#17986072	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	284	2	no	—
288	C₂	#22588117	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556085
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	#12617
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #22589944	C₂	semidirect
5. #22586430	C₂	semidirect
6. #22586429	C₂	semidirect
7. C₇₁×#2110	C₂	semidirect
12. C₂×#14556067	C₄	semidirect
14. C₂×C₁₆×D₇₁	C₄	semidirect
18. #17985780	C₄	semidirect
148. #2110	D₇₁	semidirect
152. C₇₁	#12617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986417	2
#22589960	2
#22589944	2
#22588141	2
#22586430	2
#22586429	2
C₇₁×#2110	2
#12617	71