Subgroups — C₄⋊(C₁₆₇⋊C₃₂) #36322334

Order 21376.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆₇⋊C₃₂	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
3	C₁₆₇⋊C₃₂	—	2	2	no	—
4	C₂×C₁₆×C₁₆₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₆₇	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₆₇	C₈	1	1	yes	characteristic
8	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₁₆₇	C₈	1	1	yes	characteristic
10	C₁₆×C₁₆₇	—	2	1	no	—
11	C₁₆×C₁₆₇	D₄	1	1	yes	characteristic
12	C₁₆×C₁₆₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₁₆₇	C₁₆	1	1	yes	characteristic
14	C₂×C₄×C₁₆₇	C₁₆	1	1	yes	characteristic
15	C₂×C₄×C₁₆₇	C₂×C₈	1	1	yes	characteristic
16	C₈×C₁₆₇	—	2	1	no	—
17	C₈×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₁₆₇	M₁₆	1	1	yes	characteristic
19	C₂²×C₁₆₇	C₂×C₁₆	1	1	yes	characteristic
20	C₄×C₁₆₇	—	2	1	no	—
21	C₄×C₁₆₇	C₃₂	1	2	yes	complemented
22	C₄×C₁₆₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₆₇	M₃₂	1	1	yes	characteristic
24	C₂×C₁₆₇	C₄⋊C₁₆	1	1	yes	characteristic
25	C₂×C₁₆₇	C₂×C₃₂	1	1	yes	characteristic
26	C₂×C₁₆₇	M₆₄	1	1	yes	characteristic
27	C₁₆₇	C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₄⋊C₃₂	—	167	1	no	maximal, Sylow
29	C₂×C₃₂	—	167	2	no	—
30	C₄×C₁₆	D₁₆₇	1	1	yes	characteristic
31	C₂×C₁₆	Dic₁₆₇	1	1	yes	characteristic
32	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
33	C₄×C₈	Dic₁₆₇	1	1	yes	characteristic
34	C₃₂	—	334	2	no	—
35	C₂×C₈	C₁₆₇⋊C₈	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₁₆₇	1	1	yes	characteristic
37	C₄²	C₁₆₇⋊C₈	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	C₁₆	D₆₆₈	1	1	yes	characteristic
40	C₁₆	Dic₃₃₄	1	1	yes	characteristic
41	C₂×C₄	C₁₆₇⋊C₁₆	1	1	yes	characteristic
42	C₂×C₄	C₁₆₇⋊C₁₆	1	1	yes	characteristic
43	C₂×C₄	C₂×C₁₆₇⋊C₈	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	C₄⋊Dic₁₆₇	1	1	yes	characteristic
46	C₈	#14581934	1	1	yes	characteristic
47	C₂²	C₂×C₁₆₇⋊C₁₆	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	C₁₆₇⋊C₃₂	1	2	yes	complemented
50	C₄	C₄⋊(C₁₆₇⋊C₈)	1	1	yes	characteristic
51	C₄	#19128772	1	1	yes	characteristic
52	C₂	C₄⋊(C₁₆₇⋊C₁₆)	1	1	yes	characteristic
53	C₂	C₂×C₁₆₇⋊C₃₂	1	1	yes	characteristic
54	C₂	#24923103	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₁₆₇
commutator	C₂×C₁₆₇	C₂×C₃₂
Fitting	C₄×C₁₆×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₁₆₇	C₂²
hypercenter	C₄×C₁₆	D₁₆₇
hypocenter	C₁₆₇	C₄⋊C₃₂
LCS	G ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
21. C₄×C₁₆₇	C₃₂	semidirect
27. C₁₆₇	C₄⋊C₃₂	semidirect
49. C₄	C₁₆₇⋊C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆₇⋊C₃₂	2
C₄×C₁₆×C₁₆₇	2
C₄⋊C₃₂	167

Subgroups — C4⋊(C167⋊C32) #36322334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(C₁₆₇⋊C₃₂) #36322334