Subgroups — #31885363

Order 18176.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁⋊C₁₂₈	C₂	1	2	yes	maximal, complemented
2	C₂×C₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₇₁	C₄	1	1	yes	characteristic
4	C₆₄×C₇₁	C₄	1	1	yes	characteristic
5	C₆₄×C₇₁	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₇₁	C₈	1	1	yes	characteristic
7	C₃₂×C₇₁	C₈	1	1	yes	characteristic
8	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₇₁	C₁₆	1	1	yes	characteristic
10	C₁₆×C₇₁	C₁₆	1	1	yes	characteristic
11	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
12	C₂×C₄×C₇₁	C₃₂	1	1	yes	characteristic
13	C₈×C₇₁	C₃₂	1	1	yes	characteristic
14	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
15	C₂²×C₇₁	C₆₄	1	1	yes	characteristic
16	C₄×C₇₁	C₆₄	1	1	yes	characteristic
17	C₄×C₇₁	C₂×C₃₂	1	1	yes	characteristic
18	M₂₅₆	—	71	1	no	maximal, Sylow
19	C₂×C₇₁	—	2	1	no	—
20	C₂×C₇₁	C₂×C₆₄	1	1	yes	characteristic
21	C₂×C₆₄	D₇₁	1	1	yes	characteristic
22	C₁₂₈	—	71	2	no	—
23	C₇₁	M₂₅₆	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₃₂	Dic₇₁	1	1	yes	characteristic
25	C₆₄	Dic₇₁	1	1	yes	characteristic
26	C₆₄	C₂×D₇₁	1	1	yes	characteristic
27	C₂×C₁₆	C₇₁⋊C₈	1	1	yes	characteristic
28	C₃₂	C₇₁⋊C₈	1	1	yes	characteristic
29	C₃₂	C₂×Dic₇₁	1	1	yes	characteristic
30	C₂×C₈	C₇₁⋊C₁₆	1	1	yes	characteristic
31	C₁₆	C₇₁⋊C₁₆	1	1	yes	characteristic
32	C₁₆	C₂×C₇₁⋊C₈	1	1	yes	characteristic
33	C₂×C₄	C₇₁⋊C₃₂	1	1	yes	characteristic
34	C₈	C₇₁⋊C₃₂	1	1	yes	characteristic
35	C₈	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
36	C₂²	C₇₁⋊C₆₄	1	1	yes	characteristic
37	C₄	C₇₁⋊C₆₄	1	1	yes	characteristic
38	C₄	C₂×C₇₁⋊C₃₂	1	1	yes	characteristic
39	C₂	—	2	1	no	—
40	C₂	C₂×C₇₁⋊C₆₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₆₄	C₂×D₇₁
commutator, socle	C₂×C₇₁	C₂×C₆₄
Fitting	C₂×C₆₄×C₇₁	C₂
prime radical	C₂²×C₇₁	C₆₄
cosocle	C₆₄×C₇₁	C₂²
hypercenter	C₂×C₆₄	D₇₁
hypocenter	C₇₁	M₂₅₆
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₆₄ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
1. C₇₁⋊C₁₂₈	C₂	semidirect
23. C₇₁	M₂₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₇₁⋊C₁₂₈	2
C₂×C₆₄×C₇₁	2
M₂₅₆	71