Subgroups — #31915132

Order 18176.

262 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589089	C₂	1	1	yes	maximal, characteristic
2	#22588221	C₂	1	1	yes	maximal, characteristic
3	#22586435	C₂	1	1	yes	maximal, characteristic
4	#22586435	C₂	1	1	yes	maximal, characteristic
5	#22586434	C₂	1	1	yes	maximal, characteristic
6	#22586417	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#14556064	C₂²	1	1	yes	characteristic
10	C₂×#14556064	C₂²	1	1	yes	characteristic
11	#17986026	—	2	1	no	—
12	#17985955	—	2	1	no	—
13	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
14	#17985758	—	2	1	no	—
15	#17985758	—	2	1	no	—
16	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	#427×C₇₁	C₂²	1	1	yes	characteristic
18	#427×C₇₁	C₂²	1	1	yes	characteristic
19	#367×C₇₁	C₄	1	1	yes	characteristic, complemented
20	#367×C₇₁	C₄	1	1	yes	characteristic, complemented
21	#366×C₇₁	C₄	1	1	yes	characteristic, complemented
22	#366×C₇₁	C₄	1	1	yes	characteristic, complemented
23	C₂×#11765849	—	2	1	no	—
24	C₂×#11765849	—	2	1	no	—
25	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₁₆	—	2	1	no	—
28	C₂×C₇₁⋊C₁₆	—	2	1	no	—
29	C₂×C₇₁⋊C₁₆	—	2	1	no	—
30	#14556064	—	2	1	no	—
31	#14556064	—	2	1	no	—
32	#14556064	—	2	1	no	—
33	#14556064	Q₈	1	2	yes	—
34	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
35	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
36	#14556039	—	4	1	no	—
37	C₄×C₇₁⋊C₈	—	4	1	no	—
38	C₈×Dic₇₁	—	2	1	no	—
39	C₈×Dic₇₁	—	2	1	no	—
40	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
41	#117×C₇₁	—	2	1	no	—
42	#117×C₇₁	—	2	1	no	—
43	C₂×C₁₆×C₇₁	—	2	1	no	—
44	C₂×C₁₆×C₇₁	—	2	1	no	—
45	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
47	#107×C₇₁	C₂×C₄	1	1	yes	characteristic
48	#107×C₇₁	C₂×C₄	1	1	yes	characteristic
49	#107×C₇₁	C₂×C₄	1	1	yes	characteristic
50	#107×C₇₁	C₂×C₄	1	1	yes	characteristic
51	#106×C₇₁	—	2	1	no	—
52	#106×C₇₁	—	2	1	no	—
53	C₂²×Dic₇₁	—	2	1	no	—
54	#11765849	—	2	2	no	—
55	#11765849	—	4	1	no	—
56	#11765849	—	4	1	no	—
57	#11765849	—	4	1	no	—
58	C₂×C₇₁⋊C₈	—	2	1	no	—
59	C₂×C₇₁⋊C₈	—	2	1	no	—
60	C₂×C₇₁⋊C₈	—	4	1	no	—
61	C₂×C₇₁⋊C₈	—	4	1	no	—
62	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
63	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
64	C₄×Dic₇₁	—	2	1	no	—
65	C₄×Dic₇₁	—	2	1	no	—
66	C₇₁⋊C₁₆	—	4	1	no	—
67	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
71	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
72	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
73	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
74	C₄⋊C₄×C₇₁	—	2	1	no	—
75	C₄⋊C₄×C₇₁	—	2	1	no	—
76	C₄⋊C₄×C₇₁	—	2	1	no	—
77	C₄⋊C₄×C₇₁	—	2	1	no	—
78	C₂²⋊C₄×C₇₁	—	4	1	no	—
79	C₂²⋊C₄×C₇₁	—	4	1	no	—
80	C₄²×C₇₁	—	4	1	no	—
81	C₄²×C₇₁	—	4	1	no	—
82	C₁₆×C₇₁	—	2	1	no	—
83	C₁₆×C₇₁	—	2	1	no	—
84	C₂³×C₇₁	#135	1	1	yes	characteristic
85	C₂×C₄×C₇₁	—	4	1	no	—
86	C₂×C₄×C₇₁	—	4	1	no	—
87	C₂×C₄×C₇₁	—	4	1	no	—
88	C₂×C₄×C₇₁	—	4	1	no	—
89	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
94	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
95	C₂×Dic₇₁	—	2	2	no	—
96	C₂×Dic₇₁	—	4	1	no	—
97	C₂×Dic₇₁	—	4	1	no	—
98	C₈×C₇₁	#117	1	1	yes	characteristic
99	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
100	C₈×C₇₁	#125	1	1	yes	characteristic
101	C₈×C₇₁	#126	1	1	yes	characteristic
102	C₇₁⋊C₈	—	2	1	no	—
103	C₇₁⋊C₈	—	2	1	no	—
104	C₇₁⋊C₈	—	4	2	no	—
105	C₇₁⋊C₈	—	8	1	no	—
106	C₂²×C₇₁	—	2	1	no	—
107	C₂²×C₇₁	—	2	1	no	—
108	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
109	C₂²×C₇₁	#479	1	1	yes	characteristic
110	C₂²×C₇₁	#485	1	1	yes	characteristic
111	C₄×C₇₁	—	8	1	no	—
112	C₄×C₇₁	—	8	1	no	—
113	C₄×C₇₁	—	8	1	no	—
114	C₄×C₇₁	—	8	1	no	—
115	C₄×C₇₁	#387	1	1	yes	characteristic
116	C₄×C₇₁	#441	1	1	yes	characteristic
117	C₄×C₇₁	#474	1	1	yes	characteristic
118	C₄×C₇₁	#480	1	1	yes	characteristic
119	Dic₇₁	—	4	2	no	—
120	#13113	—	71	1	no	maximal, Sylow
121	C₂×C₇₁	—	2	1	no	—
122	C₂×C₇₁	—	2	1	no	—
123	C₂×C₇₁	#1918	1	1	yes	characteristic
124	C₂×C₇₁	#2178	1	1	yes	characteristic
125	C₂×C₇₁	#2179	1	1	yes	characteristic
126	#2158	D₇₁	1	1	yes	characteristic
127	#1945	—	71	1	no	—
128	#1772	—	71	1	no	—
129	#1382	—	71	1	no	—
130	#1382	—	71	1	no	—
131	#1381	—	71	1	no	—
132	#1374	—	71	1	no	—
133	C₇₁	#13113	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₁₆	—	71	1	no	—
135	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
136	C₂×#108	—	71	1	no	—
137	C₂×#108	—	71	1	no	—
138	#427	C₂×D₇₁	1	1	yes	characteristic
139	#427	C₂×D₇₁	1	1	yes	characteristic
140	#424	—	142	1	no	—
141	#405	—	142	1	no	—
142	C₂×C₄×C₈	—	71	1	no	—
143	#367	Dic₇₁	1	1	yes	characteristic, complemented
144	#367	Dic₇₁	1	1	yes	characteristic, complemented
145	#366	Dic₇₁	1	1	yes	characteristic, complemented
146	#366	Dic₇₁	1	1	yes	characteristic, complemented
147	#339	—	142	1	no	—
148	#339	—	142	1	no	—
149	C₂×M₁₆	—	142	1	no	—
150	C₂×M₁₆	—	142	1	no	—
151	C₂²×C₈	—	71	1	no	—
152	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
153	#117	—	2	1	no	—
154	#117	—	2	1	no	—
155	C₂×C₄²	—	71	1	no	—
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	142	1	no	—
159	C₂×C₁₆	—	142	1	no	—
160	C₂×C₁₆	—	142	1	no	—
161	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
162	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
163	#108	—	71	2	no	—
164	#108	—	142	1	no	—
165	#108	—	142	1	no	—
166	#108	—	142	1	no	—
167	#107	C₂×Dic₇₁	1	1	yes	characteristic
168	#107	C₂×Dic₇₁	1	1	yes	characteristic
169	#107	C₂×Dic₇₁	1	1	yes	characteristic
170	#107	C₂×Dic₇₁	1	1	yes	characteristic
171	#106	—	2	1	no	—
172	#106	—	2	1	no	—
173	C₄⋊C₈	—	284	1	no	—
174	C₄⋊C₈	—	284	1	no	—
175	#97	—	284	1	no	—
176	C₄×C₈	—	284	1	no	—
177	C₄×C₈	—	142	1	no	—
178	C₄×C₈	—	142	1	no	—
179	C₂²×C₄	—	142	1	no	—
180	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
181	M₁₆	—	284	1	no	—
182	M₁₆	—	284	1	no	—
183	M₁₆	—	284	1	no	—
184	M₁₆	—	142	2	no	—
185	C₂×C₈	—	284	1	no	—
186	C₂×C₈	—	284	1	no	—
187	C₂×C₈	—	71	1	no	—
188	C₂×C₈	—	71	1	no	—
189	C₂×C₈	—	142	1	no	—
190	C₂×C₈	—	142	1	no	—
191	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
192	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
193	C₂×C₈	#11765864	1	1	yes	characteristic
194	C₂×C₈	#11765865	1	1	yes	characteristic
195	C₂×C₈	#11765865	1	1	yes	characteristic
196	C₂×C₈	#11765866	1	1	yes	characteristic
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	1	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	142	1	no	—
206	C₄²	—	142	1	no	—
207	C₁₆	—	2	1	no	—
208	C₁₆	—	2	1	no	—
209	C₁₆	—	284	1	no	—
210	C₂³	#14556147	1	1	yes	characteristic
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	284	1	no	—
216	C₂×C₄	—	284	1	no	—
217	C₂×C₄	—	142	2	no	—
218	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
219	C₂×C₄	#14556106	1	1	yes	characteristic
220	C₂×C₄	#14556113	1	1	yes	characteristic
221	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
222	C₂×C₄	#14556160	1	1	yes	characteristic
223	C₂×C₄	#14556160	1	1	yes	characteristic
224	C₈	—	284	2	no	—
225	C₈	—	568	1	no	—
226	C₈	—	142	1	no	—
227	C₈	—	142	1	no	—
228	C₈	#14556087	1	1	yes	characteristic
229	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
230	C₈	#14556118	1	1	yes	characteristic
231	C₈	#14556123	1	1	yes	characteristic
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	D₈×Dic₇₁	1	1	yes	characteristic
235	C₂²	#17986305	1	1	yes	characteristic
236	C₂²	#17986342	1	1	yes	characteristic
237	C₄	—	8	1	no	—
238	C₄	—	8	1	no	—
239	C₄	—	8	1	no	—
240	C₄	—	8	1	no	—
241	C₄	—	284	2	no	—
242	C₄	#17985885	1	1	yes	characteristic
243	C₄	#17986106	1	1	yes	characteristic
244	C₄	#17986278	1	1	yes	characteristic
245	C₄	#17986314	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	#22588913	1	1	yes	characteristic
249	C₂	#22590221	1	1	yes	characteristic
250	C₂	#22590226	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2158	C₂
socle	C₂²×C₇₁	C₄×D₈
prime radical	C₂³×C₇₁	#135
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2158	D₇₁
hypocenter	C₇₁	#13113
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
19. #367×C₇₁	C₄	semidirect
20. #367×C₇₁	C₄	semidirect
21. #366×C₇₁	C₄	semidirect
22. #366×C₇₁	C₄	semidirect
133. C₇₁	#13113	semidirect
143. #367	Dic₇₁	semidirect
144. #367	Dic₇₁	semidirect
145. #366	Dic₇₁	semidirect
146. #366	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22589089	2
#22588221	2
#22586435	2
#22586434	2
#22586417	2
C₇₁×#2158	2
#13113	71