Subgroups — #32151182

Order 18176.

484 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	#22590249	C₂	1	2	yes	maximal, complemented
2	#22590172	C₂	1	2	yes	maximal, complemented
3	#22590166	C₂	1	2	yes	maximal, complemented
4	#22590166	C₂	1	2	yes	maximal, complemented
5	#22590147	C₂	1	2	yes	maximal, complemented
6	#22590143	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590099	C₂	1	2	yes	maximal, complemented
8	#22590096	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₁×#2966	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556068	C₂²	1	1	yes	characteristic
11	C₂×#14556068	C₂²	1	1	yes	characteristic
12	C₂×#14556068	C₂²	1	2	yes	—
13	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
14	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
16	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
17	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
18	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
20	C₂²⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
21	C₂²⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
22	C₂²⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
23	#17985775	C₂²	1	2	yes	—
24	#17985775	C₂²	1	2	yes	—
25	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
26	#529×C₇₁	C₄	1	1	yes	characteristic
27	#446×C₇₁	C₄	1	1	yes	characteristic
28	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
29	#436×C₇₁	C₄	1	1	yes	characteristic
30	#435×C₇₁	C₄	1	2	yes	—
31	C₈×D₄×C₇₁	C₂²	1	1	yes	characteristic
32	C₈×D₄×C₇₁	C₂²	1	2	yes	—
33	#433×C₇₁	C₄	1	2	yes	—
34	#432×C₇₁	C₄	1	1	yes	characteristic
35	#405×C₇₁	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
37	#14556068	—	4	2	no	—
38	#14556068	—	4	1	no	—
39	#14556068	—	4	1	no	—
40	C₂×C₇₁⋊C₁₆	—	4	2	no	—
41	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
42	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
43	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
44	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
45	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
49	#126×C₇₁	C₈	1	2	yes	—
50	#125×C₇₁	C₈	1	1	yes	characteristic
51	#124×C₇₁	C₈	1	1	yes	characteristic
52	#123×C₇₁	C₈	1	2	yes	—
53	C₂²⋊Q₈×C₇₁	C₈	1	1	yes	characteristic
54	#121×C₇₁	C₈	1	1	yes	characteristic
55	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
56	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
57	C₄×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
58	#117×C₇₁	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
63	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
64	C₂²⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
67	#97×C₇₁	C₂×C₄	1	2	yes	—
68	#97×C₇₁	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₇₁	C₂³	1	2	yes	—
73	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
74	C₇₁⋊C₁₆	—	2	2	no	—
75	C₇₁⋊C₁₆	—	2	2	no	—
76	C₇₁⋊C₁₆	—	2	2	no	—
77	C₇₁⋊C₁₆	—	2	2	no	—
78	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
79	C₂×D₄×C₇₁	C₂×C₈	1	2	yes	—
80	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
85	M₁₆×C₇₁	—	4	2	no	—
86	C₂×C₈×C₇₁	—	2	2	no	—
87	C₂×C₈×C₇₁	—	2	1	no	—
88	C₂×C₈×C₇₁	—	2	2	no	—
89	C₂×C₈×C₇₁	—	2	1	no	—
90	C₂×C₈×C₇₁	—	4	1	no	—
91	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
98	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
99	C₄⋊C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
101	C₄⋊C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
102	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
103	C₂²⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
108	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
109	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
110	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
113	C₂³×C₇₁	C₂²×C₈	1	2	yes	—
114	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
115	Q₈×C₇₁	—	4	1	no	—
116	D₄×C₇₁	—	4	1	no	—
117	D₄×C₇₁	—	4	2	no	—
118	C₂×C₄×C₇₁	—	2	2	no	—
119	C₂×C₄×C₇₁	—	2	1	no	—
120	C₂×C₄×C₇₁	—	2	1	no	—
121	C₂×C₄×C₇₁	—	2	2	no	—
122	C₂×C₄×C₇₁	—	4	1	no	—
123	C₂×C₄×C₇₁	—	4	1	no	—
124	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₇₁	—	2	2	no	—
134	C₈×C₇₁	—	2	1	no	—
135	C₈×C₇₁	—	2	1	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₈×C₇₁	—	2	2	no	—
138	C₈×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
139	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂²×C₇₁	—	2	1	no	—
141	C₂²×C₇₁	—	2	1	no	—
142	C₂²×C₇₁	—	2	2	no	—
143	C₂²×C₇₁	—	2	2	no	—
144	C₂²×C₇₁	—	2	1	no	—
145	C₂²×C₇₁	—	2	2	no	—
146	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
147	C₄×C₇₁	—	2	1	no	—
148	C₄×C₇₁	—	2	1	no	—
149	C₄×C₇₁	—	2	1	no	—
150	C₄×C₇₁	—	2	1	no	—
151	C₄×C₇₁	—	2	2	no	—
152	C₄×C₇₁	—	2	2	no	—
153	C₄×C₇₁	—	2	1	no	—
154	C₄×C₇₁	—	2	2	no	—
155	C₄×C₇₁	#520	1	1	yes	characteristic
156	C₄×C₇₁	#568	1	1	yes	characteristic
157	#33509	—	71	1	no	maximal, Sylow
158	C₂×C₇₁	—	4	1	no	—
159	C₂×C₇₁	—	4	2	no	—
160	C₂×C₇₁	#2988	1	1	yes	characteristic
161	C₂×C₇₁	#3407	1	1	yes	characteristic
162	C₂×C₇₁	#3407	1	1	yes	characteristic
163	#2966	D₇₁	1	1	yes	characteristic
164	#2183	—	71	2	no	—
165	#2169	—	71	2	no	—
166	#2168	—	71	2	no	—
167	#2168	—	71	2	no	—
168	#2164	—	71	2	no	—
169	#2163	—	71	1	no	—
170	#2151	—	71	2	no	—
171	#2150	—	71	1	no	—
172	C₇₁	#33509	1	1	yes	characteristic, complemented, Sylow
173	#529	Dic₇₁	1	1	yes	characteristic
174	C₂×M₃₂	—	71	1	no	—
175	C₂×M₃₂	—	71	1	no	—
176	C₂×M₃₂	—	71	2	no	—
177	C₂²×C₁₆	—	71	2	no	—
178	#446	Dic₇₁	1	1	yes	characteristic
179	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
180	#436	Dic₇₁	1	1	yes	characteristic
181	#435	Dic₇₁	1	2	yes	—
182	C₈×D₄	C₂×D₇₁	1	1	yes	characteristic
183	C₈×D₄	C₂×D₇₁	1	2	yes	—
184	#433	Dic₇₁	1	2	yes	—
185	#432	Dic₇₁	1	1	yes	characteristic
186	#405	C₂×D₇₁	1	2	yes	—
187	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
188	C₄⋊C₁₆	—	71	2	no	—
189	C₄⋊C₁₆	—	71	2	no	—
190	C₄⋊C₁₆	—	71	1	no	—
191	C₄⋊C₁₆	—	71	2	no	—
192	C₄⋊C₁₆	—	71	2	no	—
193	C₄⋊C₁₆	—	71	1	no	—
194	C₂²⋊C₁₆	—	71	2	no	—
195	C₂²⋊C₁₆	—	71	2	no	—
196	C₂²⋊C₁₆	—	71	2	no	—
197	#346	—	71	2	no	—
198	#346	—	71	2	no	—
199	C₄×C₁₆	—	71	2	no	—
200	C₂×M₁₆	C₂×Dic₇₁	1	2	yes	—
201	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₇₁	1	2	yes	—
204	#126	C₇₁⋊C₈	1	2	yes	—
205	#125	C₇₁⋊C₈	1	1	yes	characteristic
206	#124	C₇₁⋊C₈	1	1	yes	characteristic
207	#123	C₇₁⋊C₈	1	2	yes	—
208	C₂²⋊Q₈	C₇₁⋊C₈	1	1	yes	characteristic
209	#121	C₇₁⋊C₈	1	1	yes	characteristic
210	C₄×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
211	C₄×D₄	C₂×Dic₇₁	1	2	yes	—
212	C₄×D₄	C₂×Dic₇₁	1	1	yes	characteristic
213	#117	C₂×Dic₇₁	1	2	yes	—
214	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
215	M₃₂	—	284	1	no	—
216	M₃₂	—	284	2	no	—
217	M₃₂	—	284	1	no	—
218	C₂×C₁₆	—	284	2	no	—
219	C₂×C₁₆	—	71	2	no	—
220	C₂×C₁₆	—	71	2	no	—
221	C₂×C₁₆	—	71	2	no	—
222	C₂×C₁₆	—	71	2	no	—
223	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
224	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
225	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
226	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
227	C₂²⋊C₈	C₂×Dic₇₁	1	2	yes	—
228	C₂²⋊C₈	C₂×Dic₇₁	1	2	yes	—
229	C₂²⋊C₈	C₂×Dic₇₁	1	2	yes	—
230	#97	C₂×Dic₇₁	1	2	yes	—
231	#97	C₂×Dic₇₁	1	2	yes	—
232	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
233	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₇₁	1	2	yes	—
235	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
237	C₂×Q₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
238	C₂×D₄	C₂×C₇₁⋊C₈	1	2	yes	—
239	C₂×D₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
243	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
253	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
254	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
255	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
257	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
258	C₄⋊C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
259	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
260	C₄⋊C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
261	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
262	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
263	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
264	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
265	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
266	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
267	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
268	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
269	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
270	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
271	C₄²	C₂²×Dic₇₁	1	2	yes	—
272	C₁₆	—	142	2	no	—
273	C₁₆	—	142	2	no	—
274	C₁₆	—	142	2	no	—
275	C₁₆	—	142	2	no	—
276	C₂³	C₂²×C₇₁⋊C₈	1	2	yes	—
277	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
278	Q₈	—	4	1	no	—
279	D₄	—	4	1	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
293	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
294	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
295	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	#14556174	1	1	yes	characteristic
302	C₈	#14556175	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	#17986482	1	1	yes	characteristic
319	C₄	#17986766	1	1	yes	characteristic
320	C₂	—	4	1	no	—
321	C₂	—	4	2	no	—
322	C₂	#22597451	1	1	yes	characteristic
323	C₂	#22601443	1	1	yes	characteristic
324	C₂	#22601443	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₇₁×#2966	C₂
prime radical	#121×C₇₁	C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2966	D₇₁
hypocenter	C₇₁	#33509
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #22590249	C₂	semidirect
2. #22590172	C₂	semidirect
3. #22590166	C₂	semidirect
4. #22590166	C₂	semidirect
5. #22590147	C₂	semidirect
6. #22590143	C₂	semidirect
7. #22590099	C₂	semidirect
8. #22590096	C₂	semidirect
172. C₇₁	#33509	semidirect

Maximal subgroups

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

subgroup	index
#22590249	2
#22590172	2
#22590166	2
#22590147	2
#22590143	2
#22590099	2
#22590096	2
C₇₁×#2966	2
#33509	71