Subgroups — #31885125

Order 18176.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₇₁⋊C₃₂)	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇₁⋊C₃₂	—	2	2	no	—
4	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
6	C₈²×C₇₁	C₄	1	1	yes	characteristic
7	C₇₁⋊C₃₂	—	4	2	no	—
8	C₂×C₁₆×C₇₁	—	2	1	no	—
9	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
10	C₂×C₁₆×C₇₁	Q₈	1	1	yes	characteristic
11	C₄×C₈×C₇₁	C₈	1	1	yes	characteristic
12	C₄×C₈×C₇₁	C₈	1	1	yes	characteristic
13	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₇₁	—	2	1	no	—
15	C₂×C₈×C₇₁	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₇₁	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
19	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₇₁	—	2	2	no	—
21	C₁₆×C₇₁	—	2	1	no	—
22	C₁₆×C₇₁	D₈	1	1	yes	characteristic
23	C₁₆×C₇₁	Q₁₆	1	1	yes	characteristic
24	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
26	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
27	C₈×C₇₁	—	2	2	no	—
28	C₈×C₇₁	—	2	1	no	—
29	C₈×C₇₁	—	2	1	no	—
30	C₈×C₇₁	C₃₂	1	2	yes	complemented
31	C₈×C₇₁	#107	1	1	yes	characteristic
32	C₈×C₇₁	#108	1	1	yes	characteristic
33	C₂²×C₇₁	C₄⋊C₁₆	1	1	yes	characteristic
34	C₄×C₇₁	—	2	1	no	—
35	C₄×C₇₁	#335	1	1	yes	characteristic
36	C₄×C₇₁	#364	1	1	yes	characteristic
37	C₄×C₇₁	C₂×C₃₂	1	1	yes	characteristic
38	C₄×C₇₁	M₆₄	1	1	yes	characteristic
39	#7458	—	71	1	no	maximal, Sylow
40	C₂×C₇₁	#1370	1	1	yes	characteristic
41	C₂×C₇₁	C₄⋊C₃₂	1	1	yes	characteristic
42	C₂×C₇₁	#1422	1	1	yes	characteristic
43	C₄⋊C₃₂	—	71	2	no	—
44	C₈×C₁₆	D₇₁	1	1	yes	characteristic
45	C₇₁	#7458	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃₂	—	142	2	no	—
47	C₄×C₁₆	Dic₇₁	1	1	yes	characteristic
48	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
49	C₈²	Dic₇₁	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	Dic₁₄₂	1	1	yes	characteristic
52	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
53	C₄×C₈	C₇₁⋊C₈	1	1	yes	characteristic
54	C₄×C₈	C₇₁⋊C₈	1	1	yes	characteristic
55	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
56	C₃₂	—	284	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	C₇₁⋊C₁₆	1	1	yes	characteristic
59	C₂×C₈	C₇₁⋊C₁₆	1	1	yes	characteristic
60	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
61	C₂×C₈	#11765849	1	1	yes	characteristic
62	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	D₅₆₈	1	1	yes	characteristic
66	C₁₆	Dic₂₈₄	1	1	yes	characteristic
67	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
68	C₂×C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
69	C₂×C₄	#14556068	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	C₇₁⋊C₃₂	1	2	yes	complemented
74	C₈	#14556062	1	1	yes	characteristic
75	C₈	#14556064	1	1	yes	characteristic
76	C₂²	C₄⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#17985752	1	1	yes	characteristic
79	C₄	#17985820	1	1	yes	characteristic
80	C₄	C₂×C₇₁⋊C₃₂	1	1	yes	characteristic
81	C₄	#17985833	1	1	yes	characteristic
82	C₂	#22586410	1	1	yes	characteristic
83	C₂	C₄⋊(C₇₁⋊C₃₂)	1	1	yes	characteristic
84	C₂	#22586538	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₂₈₄
commutator	C₄×C₇₁	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₈×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	C₈×C₁₆	D₇₁
hypocenter	C₇₁	#7458
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
30. C₈×C₇₁	C₃₂	semidirect
45. C₇₁	#7458	semidirect
73. C₈	C₇₁⋊C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₇₁⋊C₃₂)	2
C₈×C₁₆×C₇₁	2
#7458	71