Subgroups — #31918491

Order 18176.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586551	C₂	1	3	yes	maximal, complemented
2	C₂×C₇₁⋊C₆₄	C₂	1	3	yes	maximal, complemented
3	C₇₁×#2258	C₂	1	1	yes	maximal, characteristic
4	C₇₁⋊C₆₄	C₂²	1	1	yes	characteristic
5	C₇₁⋊C₆₄	C₂²	1	3	yes	—
6	#504×C₇₁	C₄	1	1	yes	characteristic
7	M₆₄×C₇₁	C₄	1	3	yes	—
8	C₂×C₃₂×C₇₁	C₂²	1	3	yes	—
9	#131×C₇₁	C₈	1	1	yes	characteristic
10	M₃₂×C₇₁	C₈	1	3	yes	—
11	C₂×C₁₆×C₇₁	C₂×C₄	1	3	yes	—
12	C₃₂×C₇₁	C₂×C₄	1	3	yes	—
13	C₃₂×C₇₁	C₂³	1	1	yes	characteristic
14	#41×C₇₁	C₁₆	1	1	yes	characteristic
15	M₁₆×C₇₁	C₁₆	1	3	yes	—
16	C₂×C₈×C₇₁	C₂×C₈	1	3	yes	—
17	C₁₆×C₇₁	C₂×C₈	1	3	yes	—
18	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
19	Q₈×C₇₁	C₃₂	1	1	yes	characteristic
20	D₄×C₇₁	C₃₂	1	3	yes	—
21	C₂×C₄×C₇₁	C₂×C₁₆	1	3	yes	—
22	C₈×C₇₁	C₂×C₁₆	1	3	yes	—
23	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
24	C₂²×C₇₁	C₂×C₃₂	1	3	yes	—
25	C₄×C₇₁	C₂×C₃₂	1	3	yes	—
26	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
27	#13737	—	71	1	no	maximal, Sylow
28	C₂×C₇₁	—	2	3	no	—
29	C₂×C₇₁	C₂²×C₃₂	1	1	yes	characteristic
30	#2258	D₇₁	1	1	yes	characteristic
31	M₁₂₈	—	71	3	no	—
32	C₂×C₆₄	—	71	3	no	—
33	C₇₁	#13737	1	1	yes	characteristic, complemented, Sylow
34	#504	Dic₇₁	1	1	yes	characteristic
35	M₆₄	Dic₇₁	1	3	yes	—
36	C₂×C₃₂	C₂×D₇₁	1	3	yes	—
37	C₆₄	—	71	1	no	—
38	C₆₄	—	71	3	no	—
39	#131	C₇₁⋊C₈	1	1	yes	characteristic
40	M₃₂	C₇₁⋊C₈	1	3	yes	—
41	C₂×C₁₆	C₂×Dic₇₁	1	3	yes	—
42	C₃₂	C₂×Dic₇₁	1	3	yes	—
43	C₃₂	C₂²×D₇₁	1	1	yes	characteristic
44	#41	C₇₁⋊C₁₆	1	1	yes	characteristic
45	M₁₆	C₇₁⋊C₁₆	1	3	yes	—
46	C₂×C₈	C₂×C₇₁⋊C₈	1	3	yes	—
47	C₁₆	C₂×C₇₁⋊C₈	1	3	yes	—
48	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
49	Q₈	C₇₁⋊C₃₂	1	1	yes	characteristic
50	D₄	C₇₁⋊C₃₂	1	3	yes	—
51	C₂×C₄	C₂×C₇₁⋊C₁₆	1	3	yes	—
52	C₈	C₂×C₇₁⋊C₁₆	1	3	yes	—
53	C₈	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
54	C₂²	C₂×C₇₁⋊C₃₂	1	3	yes	—
55	C₄	C₂×C₇₁⋊C₃₂	1	3	yes	—
56	C₄	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₇₁⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₇₁
commutator, socle	C₂×C₇₁	C₂²×C₃₂
Fitting	C₇₁×#2258	C₂
prime radical	#41×C₇₁	C₁₆
cosocle	C₃₂×C₇₁	C₂³
hypercenter	#2258	D₇₁
hypocenter	C₇₁	#13737
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₃₂ ◁ #2258

Complements

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

N	H ≅ G/N	type
1. #22586551	C₂	semidirect
2. C₂×C₇₁⋊C₆₄	C₂	semidirect
33. C₇₁	#13737	semidirect

Maximal subgroups

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

subgroup	index
#22586551	2
C₂×C₇₁⋊C₆₄	2
C₇₁×#2258	2
#13737	71