Subgroups — C₂³×C₁₆₇⋊C₁₆ #36337231

Order 21376.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆₇⋊C₁₆	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆₇⋊C₁₆	C₂²	1	28	yes	direct product factor
4	C₂³×C₄×C₁₆₇	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₁₆₇	C₄	1	7	yes	—
6	C₂²×C₈×C₁₆₇	C₂²	1	7	yes	—
7	C₁₆₇⋊C₁₆	C₂³	1	8	yes	direct product factor
8	C₂⁴×C₁₆₇	C₈	1	1	yes	characteristic
9	C₂²×C₄×C₁₆₇	C₈	1	7	yes	—
10	C₂²×C₄×C₁₆₇	C₂×C₄	1	7	yes	—
11	C₂×C₈×C₁₆₇	C₂×C₄	1	21	yes	—
12	C₂×C₈×C₁₆₇	C₂³	1	7	yes	—
13	C₂³×C₁₆₇	C₁₆	1	8	yes	complemented
14	C₂³×C₁₆₇	C₂×C₈	1	7	yes	—
15	C₂×C₄×C₁₆₇	C₂×C₈	1	21	yes	—
16	C₂×C₄×C₁₆₇	C₂²×C₄	1	7	yes	—
17	C₈×C₁₆₇	C₂²×C₄	1	7	yes	—
18	C₈×C₁₆₇	C₂⁴	1	1	yes	characteristic
19	C₂²×C₁₆₇	C₂×C₁₆	1	28	yes	complemented
20	C₂²×C₁₆₇	C₂²×C₈	1	7	yes	—
21	C₄×C₁₆₇	C₂²×C₈	1	7	yes	—
22	C₄×C₁₆₇	C₂³×C₄	1	1	yes	characteristic
23	C₂×C₁₆₇	C₂²×C₁₆	1	14	yes	complemented
24	C₂×C₁₆₇	C₂³×C₈	1	1	yes	characteristic
25	C₁₆₇	C₂³×C₁₆	1	1	yes	characteristic, complemented, Sylow
26	C₂³×C₁₆	—	167	1	no	maximal, Sylow
27	C₂³×C₈	D₁₆₇	1	1	yes	characteristic
28	C₂²×C₁₆	—	167	14	no	—
29	C₂³×C₄	Dic₁₆₇	1	1	yes	characteristic
30	C₂²×C₈	Dic₁₆₇	1	7	yes	—
31	C₂²×C₈	C₂×D₁₆₇	1	7	yes	—
32	C₂×C₁₆	—	167	28	no	—
33	C₂⁴	C₁₆₇⋊C₈	1	1	yes	characteristic
34	C₂²×C₄	C₁₆₇⋊C₈	1	7	yes	—
35	C₂²×C₄	C₂×Dic₁₆₇	1	7	yes	—
36	C₂×C₈	C₂×Dic₁₆₇	1	21	yes	—
37	C₂×C₈	C₂²×D₁₆₇	1	7	yes	—
38	C₁₆	—	167	8	no	—
39	C₂³	C₁₆₇⋊C₁₆	1	8	yes	direct product factor
40	C₂³	C₂×C₁₆₇⋊C₈	1	7	yes	—
41	C₂×C₄	C₂×C₁₆₇⋊C₈	1	21	yes	—
42	C₂×C₄	C₂²×Dic₁₆₇	1	7	yes	—
43	C₈	C₂²×Dic₁₆₇	1	7	yes	—
44	C₈	C₂³×D₁₆₇	1	1	yes	characteristic
45	C₂²	C₂×C₁₆₇⋊C₁₆	1	28	yes	direct product factor
46	C₂²	C₂²×C₁₆₇⋊C₈	1	7	yes	—
47	C₄	C₂²×C₁₆₇⋊C₈	1	7	yes	—
48	C₄	C₂³×Dic₁₆₇	1	1	yes	characteristic
49	C₂	C₂²×C₁₆₇⋊C₁₆	1	14	yes	direct product factor
50	C₂	C₂³×C₁₆₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	D₁₆₇
commutator, hypocenter	C₁₆₇	C₂³×C₁₆
Frattini	C₈	C₂³×D₁₆₇
Fitting	C₂³×C₈×C₁₆₇	C₂
socle, prime radical	C₂⁴×C₁₆₇	C₈
cosocle	C₈×C₁₆₇	C₂⁴
LCS	G ▷ C₁₆₇
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆₇⋊C₁₆	C₂	direct (N × H)
3. C₂×C₁₆₇⋊C₁₆	C₂²	direct (N × H)
7. C₁₆₇⋊C₁₆	C₂³	direct (N × H)
13. C₂³×C₁₆₇	C₁₆	semidirect
19. C₂²×C₁₆₇	C₂×C₁₆	semidirect
23. C₂×C₁₆₇	C₂²×C₁₆	semidirect
25. C₁₆₇	C₂³×C₁₆	semidirect
39. C₂³	C₁₆₇⋊C₁₆	direct (N × H)
45. C₂²	C₂×C₁₆₇⋊C₁₆	direct (N × H)
49. C₂	C₂²×C₁₆₇⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₆₇⋊C₁₆	2
C₂³×C₈×C₁₆₇	2
C₂³×C₁₆	167

Subgroups — C23×C167⋊C16 #36337231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₆₇⋊C₁₆ #36337231