Subgroups — #32152955

Order 18176.

474 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	#22598873	C₂	1	2	yes	maximal
2	#22590624	C₂	1	2	yes	maximal, complemented
3	#22590623	C₂	1	2	yes	maximal, complemented
4	#22590619	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590617	C₂	1	1	yes	maximal, characteristic, complemented
6	#22590614	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590125	C₂	1	2	yes	maximal
8	#22590121	C₂	1	1	yes	maximal, characteristic
9	#366×D₇₁	C₂	1	1	yes	maximal, characteristic
10	#22589935	C₂	1	1	yes	maximal, characteristic
11	C₇₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986626	—	2	2	no	—
13	C₂×#14556067	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
15	#17986407	—	2	2	no	—
16	#17986405	—	2	2	no	—
17	#17986403	C₂²	1	2	yes	—
18	#17986401	C₂²	1	2	yes	—
19	#17986400	C₂²	1	2	yes	—
20	#17986040	—	2	2	no	—
21	#17986036	C₂²	1	2	yes	—
22	#107×D₇₁	C₂²	1	2	yes	—
23	#17985953	C₂²	1	1	yes	characteristic
24	#17985827	C₂²	1	2	yes	—
25	#17985826	C₂²	1	2	yes	—
26	#17985826	C₂²	1	2	yes	—
27	#17985825	C₂²	1	1	yes	characteristic
28	#17985825	C₂²	1	1	yes	characteristic
29	#17985824	C₂²	1	2	yes	—
30	#17985824	C₂²	1	2	yes	—
31	#17985775	C₂²	1	1	yes	characteristic
32	#17985774	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
34	#500×C₇₁	C₂²	1	2	yes	—
35	#366×C₇₁	C₂²	1	1	yes	characteristic
36	#366×C₇₁	C₂²	1	1	yes	characteristic
37	#366×C₇₁	C₂²	1	2	yes	—
38	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
39	C₂×#11765848	D₄	1	1	yes	characteristic
40	C₂×C₈×D₇₁	C₂³	1	1	yes	characteristic
41	#14556127	—	2	2	no	—
42	#14556126	—	4	2	no	—
43	#14556120	—	2	2	no	—
44	#14556119	—	2	2	no	—
45	#14556118	—	4	2	no	—
46	#14556106	—	4	2	no	—
47	#14556104	—	4	2	no	—
48	#14556086	—	2	2	no	—
49	#14556085	D₄	1	1	yes	characteristic
50	C₄⋊C₄×D₇₁	—	2	2	no	—
51	#14556067	Q₈	1	4	yes	complemented
52	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₁₆×D₇₁	Q₈	1	4	yes	complemented
55	#14556062	C₂³	1	2	yes	—
56	#14556062	C₂³	1	2	yes	—
57	#14556061	—	2	2	no	—
58	#14556061	—	2	2	no	—
59	#14556060	—	2	2	no	—
60	#14556059	—	2	2	no	—
61	#14556059	—	2	2	no	—
62	#14556039	D₄	1	1	yes	characteristic
63	#14556038	D₄	1	1	yes	characteristic
64	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₇₁	—	2	2	no	—
66	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
68	#107×C₇₁	—	2	2	no	—
69	#107×C₇₁	C₂³	1	2	yes	—
70	#107×C₇₁	C₂³	1	2	yes	—
71	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₄₂	—	4	2	no	—
73	C₂×C₄×D₇₁	—	4	2	no	—
74	C₂×C₄×D₇₁	C₂×D₄	1	1	yes	characteristic
75	#11765848	Q₁₆	1	4	yes	—
76	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
78	C₈×D₇₁	C₂×Q₈	1	2	yes	—
79	C₈×D₇₁	C₂×Q₈	1	2	yes	—
80	C₄⋊Dic₇₁	—	2	2	no	—
81	C₄⋊Dic₇₁	—	2	2	no	—
82	C₄⋊Dic₇₁	—	4	2	no	—
83	Dic₇₁⋊C₄	—	4	2	no	—
84	Dic₇₁⋊C₄	—	4	2	no	—
85	Dic₇₁⋊C₄	—	4	2	no	—
86	Dic₇₁⋊C₄	—	4	2	no	—
87	Dic₇₁⋊C₄	—	4	2	no	—
88	Dic₇₁⋊C₄	D₈	1	1	yes	characteristic
89	Dic₇₁⋊C₄	D₈	1	1	yes	characteristic
90	(C₂×D₇₁)⋊C₄	—	4	2	no	—
91	(C₂×D₇₁)⋊C₄	D₈	1	1	yes	characteristic
92	(C₂×D₇₁)⋊C₄	D₈	1	1	yes	characteristic
93	C₄×Dic₇₁	—	4	2	no	—
94	C₄×Dic₇₁	C₂×D₄	1	1	yes	characteristic
95	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	—
96	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	—
97	C₂×Q₈×C₇₁	—	4	2	no	—
98	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	—	2	2	no	—
101	C₄⋊C₄×C₇₁	—	2	2	no	—
102	C₄⋊C₄×C₇₁	—	4	2	no	—
103	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
104	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
105	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
106	C₂²×D₇₁	C₂×D₈	1	1	yes	characteristic
107	Q₈×C₇₁	—	16	2	no	—
108	C₂×C₄×C₇₁	—	4	2	no	—
109	C₂×C₄×C₇₁	—	4	2	no	—
110	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×Dic₇₁	—	4	2	no	—
114	C₂×Dic₇₁	—	4	2	no	—
115	C₂×Dic₇₁	C₂×D₈	1	1	yes	characteristic
116	C₂×Dic₇₁	C₂×D₈	1	1	yes	characteristic
117	C₂×Dic₇₁	C₂×D₈	1	1	yes	characteristic
118	C₄×D₇₁	—	16	2	no	—
119	C₄×D₇₁	C₄⋊Q₈	1	2	yes	—
120	C₄×D₇₁	C₂×Q₁₆	1	2	yes	—
121	Dic₁₄₂	—	16	2	no	—
122	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
123	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
125	C₇₁⋊C₈	C₄⋊Q₈	1	2	yes	—
126	C₇₁⋊C₈	C₂×Q₁₆	1	2	yes	—
127	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
128	C₂×D₇₁	—	2	1	no	—
129	C₂×D₇₁	—	2	1	no	—
130	C₂×D₇₁	#500	1	2	yes	—
131	C₄×C₇₁	—	2	1	no	—
132	C₄×C₇₁	—	2	1	no	—
133	C₄×C₇₁	—	8	2	no	—
134	C₄×C₇₁	—	8	2	no	—
135	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
137	Dic₇₁	—	2	1	no	—
138	Dic₇₁	—	2	1	no	—
139	Dic₇₁	—	8	2	no	—
140	Dic₇₁	—	8	2	no	—
141	Dic₇₁	#500	1	2	yes	—
142	#33729	—	71	1	no	maximal, Sylow
143	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
144	C₂×C₇₁	#3415	1	1	yes	characteristic
145	C₂×C₇₁	#3417	1	1	yes	characteristic
146	D₇₁	—	2	2	no	—
147	#3164	—	71	2	no	—
148	#2255	—	71	2	no	—
149	#2255	—	71	2	no	—
150	#2253	—	71	1	no	—
151	#2253	—	71	1	no	—
152	#2252	—	71	1	no	—
153	#2252	D₇₁	1	1	yes	characteristic, complemented
154	#2159	—	71	2	no	—
155	#2158	—	71	1	no	—
156	C₂×#366	—	71	1	no	—
157	#2108	—	71	1	no	—
158	C₇₁	#33729	1	1	yes	characteristic, complemented, Sylow
159	#544	—	142	2	no	—
160	C₂×M₃₂	—	71	1	no	—
161	C₂²×C₁₆	—	71	1	no	—
162	#501	—	142	2	no	—
163	#501	—	142	2	no	—
164	#500	—	71	2	no	—
165	#500	C₂×D₇₁	1	2	yes	—
166	#499	—	71	2	no	—
167	#499	—	71	2	no	—
168	#428	—	142	2	no	—
169	#427	—	71	2	no	—
170	C₂×#107	—	71	2	no	—
171	#405	—	71	1	no	—
172	#367	—	71	2	no	—
173	#367	—	71	2	no	—
174	#367	—	71	2	no	—
175	#366	—	71	1	no	—
176	#366	—	71	1	no	—
177	#366	—	71	2	no	—
178	#366	—	71	2	no	—
179	#366	C₂×D₇₁	1	1	yes	characteristic
180	#366	C₂×D₇₁	1	1	yes	characteristic
181	#366	C₂×D₇₁	1	2	yes	—
182	#346	—	71	1	no	—
183	#346	—	71	1	no	—
184	C₄×C₁₆	—	71	1	no	—
185	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
186	C₂×M₁₆	—	71	1	no	—
187	C₂²×C₈	—	71	1	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	284	2	no	—
190	C₄⋊Q₈	—	142	2	no	—
191	#125	—	284	2	no	—
192	#125	—	142	2	no	—
193	#125	—	142	2	no	—
194	C₂²⋊Q₈	—	284	2	no	—
195	C₂²⋊Q₈	—	284	2	no	—
196	#117	—	71	1	no	—
197	#117	—	142	2	no	—
198	C₂×C₄⋊C₄	—	142	2	no	—
199	M₃₂	—	71	4	no	—
200	C₂×C₁₆	—	71	1	no	—
201	C₂×C₁₆	—	71	1	no	—
202	C₂×C₁₆	—	71	4	no	—
203	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
204	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
205	#107	—	2	2	no	—
206	#107	—	71	2	no	—
207	#107	—	71	2	no	—
208	#107	—	142	2	no	—
209	#107	—	142	2	no	—
210	#107	C₂²×D₇₁	1	2	yes	—
211	#107	C₂²×D₇₁	1	2	yes	—
212	#106	—	142	2	no	—
213	#106	—	142	2	no	—
214	#106	—	142	2	no	—
215	#97	—	71	1	no	—
216	#97	—	71	1	no	—
217	C₄×C₈	—	71	1	no	—
218	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	284	2	no	—
221	C₂²×C₄	—	284	2	no	—
222	C₂²×C₄	—	71	1	no	—
223	M₁₆	—	71	4	no	—
224	C₂×C₈	—	71	1	no	—
225	C₂×C₈	—	71	1	no	—
226	C₂×C₈	—	71	2	no	—
227	C₂×C₈	—	71	2	no	—
228	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	284	2	no	—
234	C₄⋊C₄	—	284	2	no	—
235	C₄⋊C₄	—	284	2	no	—
236	C₄⋊C₄	—	284	2	no	—
237	C₄⋊C₄	—	284	2	no	—
238	C₄⋊C₄	—	284	2	no	—
239	C₄⋊C₄	—	71	1	no	—
240	C₄⋊C₄	—	71	1	no	—
241	C₄⋊C₄	—	142	2	no	—
242	C₄⋊C₄	—	142	2	no	—
243	C₂²⋊C₄	—	284	2	no	—
244	C₂²⋊C₄	—	71	1	no	—
245	C₂²⋊C₄	—	71	1	no	—
246	C₄²	—	284	2	no	—
247	C₄²	—	71	1	no	—
248	C₄²	D₄×D₇₁	1	1	yes	characteristic
249	C₁₆	—	71	2	no	—
250	C₁₆	—	71	2	no	—
251	C₁₆	Q₈×D₇₁	1	2	yes	—
252	C₁₆	Q₈×D₇₁	1	2	yes	—
253	C₂³	—	71	1	no	—
254	Q₈	—	16	2	no	—
255	Q₈	—	1136	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	284	2	no	—
259	C₂×C₄	—	284	2	no	—
260	C₂×C₄	—	71	1	no	—
261	C₂×C₄	—	71	2	no	—
262	C₂×C₄	—	71	2	no	—
263	C₂×C₄	—	71	1	no	—
264	C₂×C₄	—	71	1	no	—
265	C₂×C₄	—	1136	2	no	—
266	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
267	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
269	C₈	—	71	2	no	—
270	C₈	—	71	2	no	—
271	C₈	Q₁₆×D₇₁	1	2	yes	—
272	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
274	C₂²	—	71	2	no	—
275	C₂²	—	142	1	no	—
276	C₂²	—	142	1	no	—
277	C₂²	C₂×D₈×D₇₁	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	568	2	no	—
283	C₄	—	568	2	no	—
284	C₄	—	71	2	no	—
285	C₄	—	142	1	no	—
286	C₄	—	142	1	no	—
287	C₄	C₄⋊Q₈×D₇₁	1	1	yes	characteristic
288	C₄	C₂×Q₁₆×D₇₁	1	1	yes	characteristic
289	C₂	—	142	2	no	—
290	C₂	#500×D₇₁	1	1	yes	characteristic
291	C₂	#22601477	1	1	yes	characteristic
292	C₂	#22601489	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₁
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2252	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂²×D₇₁	C₂×D₈
hypercenter	#2252	D₇₁
hypocenter	C₇₁	#33729
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #22590624	C₂	semidirect
3. #22590623	C₂	semidirect
4. #22590619	C₂	semidirect
5. #22590617	C₂	semidirect
6. #22590614	C₂	semidirect
11. C₇₁×#2252	C₂	semidirect
51. #14556067	Q₈	semidirect
54. C₁₆×D₇₁	Q₈	semidirect
153. #2252	D₇₁	semidirect
158. C₇₁	#33729	semidirect

Maximal subgroups

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

subgroup	index
#22598873	2
#22590624	2
#22590623	2
#22590619	2
#22590617	2
#22590614	2
#22590125	2
#22590121	2
#366×D₇₁	2
#22589935	2
C₇₁×#2252	2
#33729	71