Subgroups — #31884433

Order 18176.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590139	C₂	1	2	yes	maximal
2	C₇₁×#1452	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
4	C₂²⋊(C₇₁⋊C₁₆)	C₄	1	4	yes	—
5	C₄×C₇₁⋊C₁₆	—	2	2	no	—
6	#431×C₇₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
8	#322×C₇₁	—	2	2	no	—
9	C₂×C₇₁⋊C₁₆	—	2	4	no	—
10	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
12	C₄⋊C₈×C₇₁	C₈	1	4	yes	—
13	C₂²⋊C₈×C₇₁	—	2	2	no	—
14	C₄×C₈×C₇₁	—	2	2	no	—
15	C₄×C₈×C₇₁	D₄	1	2	yes	—
16	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
18	C₇₁⋊C₁₆	—	4	4	no	—
19	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
21	C₂×C₈×C₇₁	—	2	4	no	—
22	C₂×C₈×C₇₁	—	4	2	no	—
23	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
24	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
25	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
27	C₄²×C₇₁	C₂×C₈	1	2	yes	—
28	C₂³×C₇₁	#97	1	1	yes	characteristic
29	C₂×C₄×C₇₁	—	2	2	no	—
30	C₂×C₄×C₇₁	—	2	4	no	—
31	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
34	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
35	C₈×C₇₁	—	2	2	no	—
36	C₈×C₇₁	—	2	2	no	—
37	C₈×C₇₁	—	4	4	no	—
38	C₂²×C₇₁	—	2	1	no	—
39	C₂²×C₇₁	—	2	2	no	—
40	C₂²×C₇₁	#336	1	1	yes	characteristic
41	C₄×C₇₁	—	2	1	no	—
42	C₄×C₇₁	—	2	2	no	—
43	C₄×C₇₁	#344	1	2	yes	—
44	C₄×C₇₁	#350	1	4	yes	—
45	#7198	—	71	1	no	maximal, Sylow
46	C₂×C₇₁	—	4	1	no	—
47	C₂×C₇₁	#1281	1	1	yes	characteristic
48	C₂×C₇₁	#1379	1	2	yes	—
49	#2162	—	71	2	no	—
50	#1452	D₇₁	1	1	yes	characteristic
51	C₇₁	#7198	1	1	yes	characteristic, complemented, Sylow
52	#431	Dic₇₁	1	2	yes	—
53	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
54	C₄⋊C₁₆	—	142	2	no	—
55	C₂²⋊C₁₆	—	71	4	no	—
56	C₄×C₁₆	—	142	2	no	—
57	#322	—	2	2	no	—
58	C₂²×C₈	C₄×D₇₁	1	2	yes	—
59	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
60	C₂×C₁₆	—	142	4	no	—
61	C₄⋊C₈	C₇₁⋊C₈	1	4	yes	—
62	C₂²⋊C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
65	C₄×C₈	D₂₈₄	1	1	yes	characteristic
66	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
67	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
68	C₂²×C₄	#11765849	1	2	yes	—
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
72	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
73	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
74	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
75	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
76	C₁₆	—	284	4	no	—
77	C₂³	#14556039	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
81	C₂×C₄	C₄×C₇₁⋊C₈	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
83	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
84	C₈	—	2	2	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	4	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	1	no	—
89	C₂²	#17985754	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	#17985770	1	2	yes	—
93	C₄	#17985783	1	4	yes	—
94	C₂	—	4	1	no	—
95	C₂	#22586185	1	1	yes	characteristic
96	C₂	#22586429	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#1452	C₂
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	#97
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#1452	D₇₁
hypocenter	C₇₁	#7198
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

Complements

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

N	H ≅ G/N	type
51. C₇₁	#7198	semidirect

Maximal subgroups

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

subgroup	index
#22590139	2
C₇₁×#1452	2
#7198	71