Subgroups — #32155321

Order 18176.

494 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#22601192	C₂	1	1	yes	maximal, characteristic
3	#22590621	C₂	1	2	yes	maximal, complemented
4	#22590620	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590488	C₂	1	1	yes	maximal, characteristic
6	#22590486	C₂	1	2	yes	maximal
7	#22590480	C₂	1	2	yes	maximal, complemented
8	#22590242	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14556071	C₂	1	1	yes	maximal, characteristic
10	#22590189	C₂	1	2	yes	maximal
11	C₇₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₄₂	—	2	1	no	—
13	#17986664	—	2	1	no	—
14	C₂×#14556071	C₂²	1	1	yes	characteristic
15	#17986404	C₂²	1	1	yes	characteristic
16	#17986403	C₂²	1	2	yes	—
17	#17986403	C₂²	1	2	yes	—
18	#17986294	—	2	1	no	—
19	#17986292	—	2	2	no	—
20	#17986278	—	2	2	no	—
21	#17986276	—	2	1	no	—
22	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₇₁	C₂²	1	2	yes	—
24	C₄×D₅₆₈	C₂²	1	1	yes	characteristic
25	#17986081	C₂²	1	2	yes	—
26	#17985827	C₂²	1	2	yes	—
27	#17985826	C₂²	1	2	yes	—
28	#17985826	C₂²	1	2	yes	—
29	#17985826	C₂²	1	1	yes	characteristic
30	Dic₇₁⋊C₁₆	C₂²	1	2	yes	—
31	#17985803	C₂²	1	1	yes	characteristic
32	#17985801	C₂²	1	2	yes	—
33	#17985799	C₂²	1	1	yes	characteristic
34	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇₁	C₂²	1	2	yes	—
36	#500×C₇₁	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
38	#367×C₇₁	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
40	#358×C₇₁	C₂²	1	2	yes	—
41	C₂×Q₈×D₇₁	—	4	1	no	—
42	C₂×D₅₆₈	C₂³	1	1	yes	characteristic
43	#14556128	—	2	1	no	—
44	#14556128	—	4	1	no	—
45	#14556127	—	2	2	no	—
46	#14556127	—	2	2	no	—
47	#14556127	—	4	2	no	—
48	#14556106	—	4	2	no	—
49	#14556104	—	4	1	no	—
50	Q₈×Dic₇₁	—	2	2	no	—
51	C₄×Dic₁₄₂	—	4	1	no	—
52	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
53	C₄×D₂₈₄	—	2	1	no	—
54	#14556090	—	2	2	no	—
55	#14556071	Q₈	1	4	yes	complemented
56	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
58	#14556062	—	2	1	no	—
59	#14556062	C₂³	1	2	yes	—
60	#14556062	C₂³	1	1	yes	characteristic
61	#14556061	—	2	2	no	—
62	#14556061	—	2	2	no	—
63	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
64	Dic₇₁⋊C₈	D₄	1	2	yes	—
65	#14556052	—	2	2	no	—
66	#14556050	—	2	1	no	—
67	#14556048	—	2	2	no	—
68	C₈×Dic₇₁	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₇₁	—	2	1	no	—
71	C₄×Q₈×C₇₁	—	2	1	no	—
72	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
73	#107×C₇₁	—	2	1	no	—
74	#107×C₇₁	C₂³	1	1	yes	characteristic
75	#107×C₇₁	C₂³	1	2	yes	—
76	#103×C₇₁	—	2	1	no	—
77	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₄₂	—	4	2	no	—
79	C₂×Dic₁₄₂	—	4	1	no	—
80	C₂×Dic₁₄₂	—	4	2	no	—
81	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
82	Q₈×D₇₁	—	8	4	no	—
83	C₂×D₂₈₄	—	2	1	no	—
84	C₂×C₄×D₇₁	—	4	1	no	—
85	C₂×C₄×D₇₁	—	4	2	no	—
86	D₅₆₈	—	2	1	no	—
87	D₅₆₈	C₂×Q₈	1	2	yes	—
88	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
89	C₄⋊Dic₇₁	—	2	1	no	—
90	C₄⋊Dic₇₁	—	2	2	no	—
91	C₄⋊Dic₇₁	—	4	1	no	—
92	C₄⋊Dic₇₁	—	4	2	no	—
93	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
94	Dic₇₁⋊C₄	—	4	2	no	—
95	Dic₇₁⋊C₄	—	4	2	no	—
96	Dic₇₁⋊C₄	D₈	1	2	yes	—
97	(C₂×D₇₁)⋊C₄	—	4	1	no	—
98	(C₂×D₇₁)⋊C₄	—	4	2	no	—
99	C₄×Dic₇₁	—	4	2	no	—
100	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
101	C₇₁⋊C₁₆	—	2	1	no	—
102	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	complemented
103	C₂×Q₈×C₇₁	—	2	1	no	—
104	C₂×Q₈×C₇₁	—	4	1	no	—
105	Q₁₆×C₇₁	—	2	1	no	—
106	Q₁₆×C₇₁	C₂×Q₈	1	2	yes	complemented
107	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₇₁	—	2	1	no	—
110	C₄⋊C₄×C₇₁	—	2	2	no	—
111	C₄⋊C₄×C₇₁	—	4	1	no	—
112	C₄⋊C₄×C₇₁	—	4	1	no	—
113	C₄²×C₇₁	—	4	1	no	—
114	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₇₁	—	2	2	no	—
116	C₂²×D₇₁	—	4	1	no	—
117	Q₈×C₇₁	—	2	2	no	—
118	Q₈×C₇₁	—	4	1	no	—
119	Q₈×C₇₁	—	8	2	no	—
120	C₂×C₄×C₇₁	—	4	1	no	—
121	C₂×C₄×C₇₁	—	4	1	no	—
122	C₂×C₄×C₇₁	—	4	2	no	—
123	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₇₁	—	4	2	no	—
127	C₂×Dic₇₁	—	4	1	no	—
128	C₂×Dic₇₁	C₂×D₈	1	2	yes	—
129	C₂×Dic₇₁	C₂×D₈	1	2	yes	—
130	D₂₈₄	—	2	2	no	—
131	D₂₈₄	—	4	1	no	—
132	C₄×D₇₁	—	8	2	no	—
133	C₄×D₇₁	—	8	4	no	—
134	Dic₁₄₂	—	8	4	no	—
135	Dic₁₄₂	—	16	2	no	—
136	Dic₁₄₂	—	16	1	no	—
137	Dic₁₄₂	SD₃₂	1	4	yes	complemented
138	C₈×C₇₁	—	2	1	no	—
139	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
140	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
141	C₇₁⋊C₈	—	2	2	no	—
142	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
143	C₂×D₇₁	—	4	2	no	—
144	C₂×D₇₁	—	8	1	no	—
145	C₂×D₇₁	—	8	1	no	—
146	C₄×C₇₁	—	2	1	no	—
147	C₄×C₇₁	—	4	2	no	—
148	C₄×C₇₁	—	8	2	no	—
149	C₄×C₇₁	—	8	1	no	—
150	C₄×C₇₁	C₂×SD₃₂	1	2	yes	—
151	C₄×C₇₁	D₄×Q₈	1	1	yes	characteristic
152	C₄×C₇₁	#578	1	1	yes	characteristic
153	Dic₇₁	—	2	2	no	—
154	Dic₇₁	—	8	1	no	—
155	Dic₇₁	—	8	2	no	—
156	Dic₇₁	C₂×SD₃₂	1	4	yes	—
157	Q₈×SD₃₂	—	71	1	no	maximal, Sylow
158	C₂×C₇₁	Q₈×D₈	1	1	yes	characteristic
159	C₂×C₇₁	C₂²×SD₃₂	1	1	yes	characteristic
160	C₂×C₇₁	#3416	1	1	yes	characteristic
161	D₇₁	—	8	2	no	—
162	Q₈×Q₁₆	—	71	1	no	—
163	Q₈×D₈	—	71	1	no	—
164	#2254	—	71	1	no	—
165	#2254	—	71	2	no	—
166	#2226	—	71	2	no	—
167	#2226	—	71	1	no	—
168	#2225	—	71	2	no	—
169	#2225	D₇₁	1	1	yes	characteristic, complemented
170	Q₈×C₁₆	—	71	1	no	—
171	C₄×SD₃₂	—	71	1	no	—
172	C₄×SD₃₂	—	71	2	no	—
173	C₇₁	Q₈×SD₃₂	1	1	yes	characteristic, complemented, Sylow
174	Q₈²	—	142	1	no	—
175	D₄×Q₈	—	142	1	no	—
176	C₂×SD₃₂	—	71	1	no	—
177	#500	—	71	2	no	—
178	#500	—	71	2	no	—
179	#500	—	71	1	no	—
180	#500	C₂×D₇₁	1	1	yes	characteristic
181	#477	—	142	2	no	—
182	#477	—	142	1	no	—
183	#474	—	142	2	no	—
184	#474	—	142	1	no	—
185	C₈×Q₈	—	71	1	no	—
186	C₄×Q₁₆	—	71	2	no	—
187	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
188	C₄×D₈	—	71	2	no	—
189	C₄×D₈	—	71	1	no	—
190	#367	—	71	2	no	—
191	#367	—	71	2	no	—
192	#367	—	71	1	no	—
193	#367	—	71	2	no	—
194	#367	C₂×D₇₁	1	2	yes	—
195	C₄⋊C₁₆	—	71	2	no	—
196	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
197	#358	—	71	1	no	—
198	#358	C₂×D₇₁	1	2	yes	—
199	#357	—	71	2	no	—
200	#357	—	71	1	no	—
201	C₄×C₁₆	—	71	2	no	—
202	C₄×C₁₆	—	71	1	no	—
203	C₂²×Q₈	—	284	1	no	—
204	C₂×Q₁₆	C₂²×D₇₁	1	1	yes	characteristic
205	C₂×D₈	—	71	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	284	1	no	—
208	C₄⋊Q₈	—	284	2	no	—
209	C₄⋊Q₈	—	142	2	no	—
210	C₄⋊Q₈	—	142	2	no	—
211	C₄⋊Q₈	—	142	1	no	—
212	C₂²⋊Q₈	—	284	2	no	—
213	C₂²⋊Q₈	—	284	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	284	1	no	—
216	C₄×Q₈	—	71	1	no	—
217	C₄×Q₈	—	142	2	no	—
218	C₄×D₄	—	142	2	no	—
219	C₄×D₄	—	142	1	no	—
220	SD₃₂	—	71	4	no	—
221	C₂×C₁₆	—	71	1	no	—
222	C₂×C₁₆	—	71	1	no	—
223	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
224	#107	—	2	1	no	—
225	#107	—	71	2	no	—
226	#107	—	71	1	no	—
227	#107	—	142	2	no	—
228	#107	—	142	2	no	—
229	#107	—	142	1	no	—
230	#107	C₂²×D₇₁	1	1	yes	characteristic
231	#107	C₂²×D₇₁	1	2	yes	—
232	C₄⋊C₈	—	71	2	no	—
233	C₄⋊C₈	—	71	1	no	—
234	#103	—	2	1	no	—
235	#103	—	142	2	no	—
236	#102	—	142	2	no	—
237	#102	—	142	1	no	—
238	C₄×C₈	—	71	2	no	—
239	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	284	2	no	—
243	C₂×Q₈	—	284	2	no	—
244	C₂×Q₈	—	284	1	no	—
245	C₂×Q₈	—	568	4	no	—
246	C₂×Q₈	—	71	1	no	—
247	C₂×D₄	—	142	1	no	—
248	C₂²×C₄	—	284	2	no	—
249	C₂²×C₄	—	284	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	Q₈×D₇₁	1	2	yes	complemented
252	D₈	—	71	2	no	—
253	D₈	—	142	1	no	—
254	C₂×C₈	—	71	2	no	—
255	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	284	2	no	—
262	C₄⋊C₄	—	284	2	no	—
263	C₄⋊C₄	—	284	2	no	—
264	C₄⋊C₄	—	284	1	no	—
265	C₄⋊C₄	—	71	2	no	—
266	C₄⋊C₄	—	71	1	no	—
267	C₄⋊C₄	—	142	1	no	—
268	C₄⋊C₄	—	142	2	no	—
269	C₂²⋊C₄	—	284	2	no	—
270	C₂²⋊C₄	—	284	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	284	2	no	—
273	C₄²	—	71	2	no	—
274	C₄²	D₄×D₇₁	1	1	yes	characteristic
275	C₁₆	—	2	2	no	—
276	C₁₆	—	71	2	no	—
277	C₁₆	—	142	1	no	—
278	C₂³	—	284	1	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	4	1	no	—
281	Q₈	—	8	2	no	—
282	Q₈	—	568	4	no	—
283	Q₈	—	71	4	no	—
284	Q₈	—	1136	1	no	—
285	Q₈	—	1136	2	no	—
286	D₄	—	284	1	no	—
287	D₄	—	142	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	284	2	no	—
292	C₂×C₄	—	284	1	no	—
293	C₂×C₄	—	568	4	no	—
294	C₂×C₄	—	568	2	no	—
295	C₂×C₄	—	71	2	no	—
296	C₂×C₄	—	71	2	no	—
297	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
298	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	142	2	no	—
302	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
303	C₈	#14556175	1	1	yes	characteristic
304	C₂²	—	284	2	no	—
305	C₂²	—	568	1	no	—
306	C₂²	—	568	1	no	—
307	C₂²	C₂×D₈×D₇₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	568	2	no	—
313	C₄	—	568	1	no	—
314	C₄	—	71	4	no	—
315	C₄	—	142	2	no	—
316	C₄	SD₃₂×D₇₁	1	2	yes	—
317	C₄	#17986662	1	1	yes	characteristic
318	C₄	#17986815	1	1	yes	characteristic
319	C₂	—	568	2	no	—
320	C₂	#22601190	1	1	yes	characteristic
321	C₂	C₂×SD₃₂×D₇₁	1	1	yes	characteristic
322	C₂	#22601485	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×#2225	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂×D₅₆₈	C₂³
hypercenter	#2225	D₇₁
hypocenter	C₇₁	Q₈×SD₃₂
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
1. Q₁₆×Dic₁₄₂	C₂	semidirect
3. #22590621	C₂	semidirect
4. #22590620	C₂	semidirect
7. #22590480	C₂	semidirect
8. #22590242	C₂	semidirect
11. C₇₁×#2225	C₂	semidirect
55. #14556071	Q₈	semidirect
102. C₇₁⋊C₁₆	C₂×Q₈	semidirect
106. Q₁₆×C₇₁	C₂×Q₈	semidirect
137. Dic₁₄₂	SD₃₂	semidirect
169. #2225	D₇₁	semidirect
173. C₇₁	Q₈×SD₃₂	semidirect
251. Q₁₆	Q₈×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×Dic₁₄₂	2
#22601192	2
#22590621	2
#22590620	2
#22590488	2
#22590486	2
#22590480	2
#22590242	2
C₄×#14556071	2
#22590189	2
C₇₁×#2225	2
Q₈×SD₃₂	71