Subgroups — C₂²×C₇₁⋊C₆₄ #31918486

Order 18176.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂²×C₃₂	D₇₁
commutator, hypocenter	C₇₁	C₂²×C₆₄
Frattini	C₃₂	C₂²×D₇₁
Fitting	C₂²×C₃₂×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₃₂
cosocle	C₃₂×C₇₁	C₂³
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₇₁⋊C₆₄	C₂	direct (N × H)
3. C₇₁⋊C₆₄	C₂²	direct (N × H)
22. C₂²×C₇₁	C₆₄	semidirect
27. C₂×C₇₁	C₂×C₆₄	semidirect
31. C₇₁	C₂²×C₆₄	semidirect
51. C₂²	C₇₁⋊C₆₄	direct (N × H)
55. C₂	C₂×C₇₁⋊C₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇₁⋊C₆₄	2
C₂²×C₃₂×C₇₁	2
C₂²×C₆₄	71

Subgroups — C22×C71⋊C64 #31918486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇₁⋊C₆₄ #31918486