Subgroups — #31907171

Order 18176.

394 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14556068	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586298	C₂	1	4	yes	maximal, complemented
4	C₇₁×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	—	2	1	no	—
6	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
9	#17985775	C₂²	1	2	yes	—
10	C₄×C₇₁⋊C₁₆	—	2	1	no	—
11	C₂×#131×C₇₁	—	2	1	no	—
12	#532×C₇₁	C₄	1	1	yes	characteristic
13	#436×C₇₁	—	2	1	no	—
14	#436×C₇₁	C₂²	1	2	yes	—
15	C₈×D₄×C₇₁	—	2	1	no	—
16	C₈×D₄×C₇₁	C₄	1	2	yes	—
17	#423×C₇₁	C₄	1	1	yes	characteristic
18	#408×C₇₁	—	2	1	no	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	#14556068	—	2	4	no	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×C₇₁⋊C₁₆	—	4	2	no	—
24	C₂×#41×C₇₁	—	2	1	no	—
25	#131×C₇₁	—	4	2	no	—
26	#131×C₇₁	—	4	2	no	—
27	C₂×M₁₆×C₇₁	—	2	1	no	—
28	C₂×M₁₆×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₇₁	C₈	1	1	yes	characteristic
33	#127×C₇₁	C₈	1	1	yes	characteristic
34	#124×C₇₁	—	2	1	no	—
35	#121×C₇₁	—	2	1	no	—
36	#121×C₇₁	C₈	1	2	yes	—
37	C₄×D₄×C₇₁	—	2	1	no	—
38	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₇₁	—	2	1	no	—
41	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₇₁	—	2	2	no	—
43	C₂²⋊C₈×C₇₁	—	2	2	no	—
44	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
48	C₇₁⋊C₁₆	—	2	4	no	—
49	C₇₁⋊C₁₆	—	4	2	no	—
50	#41×C₇₁	—	4	2	no	—
51	#41×C₇₁	—	4	2	no	—
52	C₂×Q₈×C₇₁	—	2	1	no	—
53	C₂×D₄×C₇₁	—	2	2	no	—
54	C₂×D₄×C₇₁	—	2	1	no	—
55	C₂×D₄×C₇₁	—	2	1	no	—
56	C₂×D₄×C₇₁	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₇₁	—	2	2	no	—
58	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
61	M₁₆×C₇₁	—	4	2	no	—
62	M₁₆×C₇₁	—	4	2	no	—
63	M₁₆×C₇₁	—	4	2	no	—
64	C₂×C₈×C₇₁	—	2	2	no	—
65	C₂×C₈×C₇₁	—	2	2	no	—
66	C₂×C₈×C₇₁	—	4	4	no	—
67	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	D₈	1	2	yes	—
70	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
71	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₇₁	—	2	1	no	—
74	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
75	C₂²⋊C₄×C₇₁	—	2	2	no	—
76	C₂²⋊C₄×C₇₁	—	2	2	no	—
77	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
80	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
81	C₂³×C₇₁	—	2	2	no	—
82	C₂³×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
83	Q₈×C₇₁	—	4	2	no	—
84	D₄×C₇₁	—	4	2	no	—
85	D₄×C₇₁	—	4	4	no	—
86	D₄×C₇₁	—	4	2	no	—
87	D₄×C₇₁	—	4	2	no	—
88	C₂×C₄×C₇₁	—	2	2	no	—
89	C₂×C₄×C₇₁	—	2	2	no	—
90	C₂×C₄×C₇₁	—	4	1	no	—
91	C₂×C₄×C₇₁	—	4	2	no	—
92	C₂×C₄×C₇₁	—	4	2	no	—
93	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₇₁	#102	1	2	yes	—
97	C₂×C₄×C₇₁	#104	1	2	yes	—
98	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
101	C₈×C₇₁	—	2	1	no	—
102	C₈×C₇₁	—	2	1	no	—
103	C₈×C₇₁	—	4	2	no	—
104	C₈×C₇₁	#102	1	2	yes	—
105	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
106	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×C₇₁	—	2	1	no	—
108	C₂²×C₇₁	—	2	1	no	—
109	C₂²×C₇₁	—	4	2	no	—
110	C₂²×C₇₁	—	4	2	no	—
111	C₂²×C₇₁	—	4	2	no	—
112	C₂²×C₇₁	D₄⋊C₈	1	2	yes	—
113	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₄×C₇₁	—	2	1	no	—
115	C₄×C₇₁	—	2	1	no	—
116	C₄×C₇₁	—	2	1	no	—
117	C₄×C₇₁	—	4	2	no	—
118	C₄×C₇₁	D₄⋊C₈	1	2	yes	—
119	C₄×C₇₁	#350	1	2	yes	—
120	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
121	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
122	#11586	—	71	1	no	maximal, Sylow
123	C₂×C₇₁	—	2	2	no	—
124	C₂×C₇₁	—	8	2	no	—
125	C₂×C₇₁	C₂×D₄⋊C₈	1	1	yes	characteristic
126	C₂×C₇₁	#2115	1	1	yes	characteristic
127	C₂×C₇₁	C₂×#350	1	1	yes	characteristic
128	#2931	D₇₁	1	1	yes	characteristic
129	C₂×C₄⋊C₁₆	—	71	1	no	—
130	C₄×M₃₂	—	71	1	no	—
131	#1330	—	71	4	no	—
132	C₇₁	#11586	1	1	yes	characteristic, complemented, Sylow
133	C₂×#131	—	2	1	no	—
134	#532	Dic₇₁	1	1	yes	characteristic
135	C₂×M₃₂	—	142	1	no	—
136	C₂²×C₁₆	—	142	1	no	—
137	#436	—	2	1	no	—
138	#436	C₂×D₇₁	1	2	yes	—
139	C₈×D₄	—	2	1	no	—
140	C₈×D₄	Dic₇₁	1	2	yes	—
141	#423	Dic₇₁	1	1	yes	characteristic
142	#408	—	2	1	no	—
143	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
144	C₄⋊C₁₆	—	71	2	no	—
145	C₄⋊C₁₆	—	142	1	no	—
146	#346	—	71	2	no	—
147	C₄×C₁₆	—	142	1	no	—
148	C₂×#41	—	2	1	no	—
149	#131	—	4	2	no	—
150	#131	—	4	2	no	—
151	C₂×M₁₆	—	2	1	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
155	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
156	C₄⋊Q₈	C₇₁⋊C₈	1	1	yes	characteristic
157	#127	C₇₁⋊C₈	1	1	yes	characteristic
158	#124	—	2	1	no	—
159	#121	—	2	1	no	—
160	#121	C₇₁⋊C₈	1	2	yes	—
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	C₂×Dic₇₁	1	2	yes	—
163	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
164	M₃₂	—	142	4	no	—
165	C₂×C₁₆	—	284	2	no	—
166	C₂×C₁₆	—	142	2	no	—
167	C₂×C₁₆	—	142	2	no	—
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	C₂×C₇₁⋊C₈	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
186	C₂²×C₄	#11765849	1	1	yes	characteristic
187	M₁₆	—	4	2	no	—
188	M₁₆	—	4	2	no	—
189	M₁₆	—	4	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
194	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
195	C₂×C₈	#11765850	1	2	yes	—
196	C₂×C₈	#11765853	1	2	yes	—
197	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
198	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
204	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
205	C₄²	#11765849	1	1	yes	characteristic
206	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
207	C₁₆	—	284	2	no	—
208	C₁₆	—	142	4	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
211	Q₈	—	4	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	4	no	—
214	D₄	—	4	2	no	—
215	D₄	—	4	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
222	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
223	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
224	C₂×C₄	#14556049	1	2	yes	—
225	C₂×C₄	#14556055	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
228	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	4	2	no	—
232	C₈	#14556049	1	2	yes	—
233	C₈	C₂×#11765850	1	1	yes	characteristic
234	C₈	C₂×#11765853	1	1	yes	characteristic
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	D₄⋊(C₇₁⋊C₈)	1	2	yes	—
241	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	2	no	—
246	C₄	D₄⋊(C₇₁⋊C₈)	1	2	yes	—
247	C₄	#17985783	1	2	yes	—
248	C₄	C₂×#14556049	1	1	yes	characteristic
249	C₄	C₂×#14556055	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	8	2	no	—
252	C₂	C₂×D₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
253	C₂	#22589961	1	1	yes	characteristic
254	C₂	C₂×#17985783	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2931	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	#532×C₇₁	C₄
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#2931	D₇₁
hypocenter	C₇₁	#11586
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
2. C₄×#14556068	C₂	semidirect
3. #22586298	C₂	semidirect
132. C₇₁	#11586	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₄×#14556068	2
#22586298	2
C₇₁×#2931	2
#11586	71