Subgroups — #36322207

Order 21376.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₈×C₁₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆₇⋊C₁₆	—	2	1	no	—
5	C₄×C₁₆₇⋊C₈	C₄	1	1	yes	characteristic
6	C₄×C₁₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₆₇	—	2	1	no	—
8	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆₇⋊C₈	—	2	1	no	—
10	C₂×C₁₆₇⋊C₈	C₈	1	1	yes	characteristic
11	C₂×C₁₆₇⋊C₈	C₈	1	1	yes	characteristic
12	C₁₆₇⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₆₇	Q₈	1	1	yes	characteristic
15	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₆₇	—	4	1	no	—
17	C₁₆₇⋊C₈	—	2	2	no	—
18	C₁₆₇⋊C₈	—	2	1	no	—
19	C₁₆₇⋊C₈	C₁₆	1	2	yes	complemented
20	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₆₇	C₂×C₈	1	1	yes	characteristic
22	C₂×C₄×C₁₆₇	M₁₆	1	1	yes	characteristic
23	C₈×C₁₆₇	—	2	1	no	—
24	C₈×C₁₆₇	D₈	1	1	yes	characteristic
25	C₈×C₁₆₇	Q₁₆	1	1	yes	characteristic
26	C₂²×C₁₆₇	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₁₆₇	—	2	1	no	—
28	C₄×C₁₆₇	#107	1	1	yes	characteristic
29	C₄×C₁₆₇	#108	1	1	yes	characteristic
30	C₄×C₁₆₇	C₂×C₁₆	1	1	yes	characteristic
31	C₄×C₁₆₇	M₃₂	1	1	yes	characteristic
32	C₂×C₁₆₇	#335	1	1	yes	characteristic
33	C₂×C₁₆₇	C₄⋊C₁₆	1	1	yes	characteristic
34	C₂×C₁₆₇	#364	1	1	yes	characteristic
35	C₁₆₇	#1370	1	1	yes	characteristic, complemented, Sylow
36	#1370	—	167	1	no	maximal, Sylow
37	C₄⋊C₁₆	—	167	1	no	—
38	C₄⋊C₁₆	D₁₆₇	1	1	yes	characteristic
39	C₈²	—	167	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	334	1	no	—
42	C₄×C₈	—	167	1	no	—
43	C₄×C₈	—	167	1	no	—
44	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
45	C₂×C₈	—	334	1	no	—
46	C₂×C₈	—	167	1	no	—
47	C₂×C₈	—	167	1	no	—
48	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
49	C₂×C₈	Dic₃₃₄	1	1	yes	characteristic
50	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
51	C₁₆	—	4	1	no	—
52	C₁₆	—	668	1	no	—
53	C₂×C₄	Dic₁₆₇⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₈×D₁₆₇	1	1	yes	characteristic
55	C₂×C₄	#14581933	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	—	334	2	no	—
58	C₈	—	334	1	no	—
59	C₈	—	167	2	no	—
60	C₈	#14581935	1	1	yes	characteristic
61	C₈	#14581940	1	1	yes	characteristic
62	C₂²	Dic₁₆₇⋊C₈	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	#19128765	1	1	yes	characteristic
65	C₄	#19128767	1	1	yes	characteristic
66	C₄	C₁₆×D₁₆₇	1	1	yes	characteristic
67	C₄	#19128771	1	1	yes	characteristic
68	C₂	#24923021	1	1	yes	characteristic
69	C₂	Dic₁₆₇⋊C₁₆	1	1	yes	characteristic
70	C₂	#24923089	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₁₆₇
commutator	C₄×C₁₆₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₆₇
Fitting	C₄⋊C₁₆×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₁₆₇	C₂²
hypercenter	C₄⋊C₁₆	D₁₆₇
hypocenter	C₁₆₇	#1370
LCS	G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
19. C₁₆₇⋊C₈	C₁₆	semidirect
35. C₁₆₇	#1370	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₆₇⋊C₁₆)	2
C₈×C₁₆₇⋊C₈	2
C₄⋊C₁₆×C₁₆₇	2
#1370	167