Subgroups — #36322208

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	2	yes	maximal
2	C₈²×C₁₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆₇⋊C₁₆	—	2	2	no	—
4	C₄×C₈×C₁₆₇	C₄	1	1	yes	characteristic
5	C₄×C₈×C₁₆₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
7	C₁₆₇⋊C₁₆	—	4	2	no	—
8	C₂×C₈×C₁₆₇	—	2	1	no	—
9	C₂×C₈×C₁₆₇	C₈	1	1	yes	characteristic
10	C₂×C₈×C₁₆₇	C₈	1	1	yes	characteristic
11	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
12	C₂×C₈×C₁₆₇	Q₈	1	1	yes	characteristic
13	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₆₇	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₁₆₇	M₁₆	1	1	yes	characteristic
17	C₈×C₁₆₇	—	2	2	no	—
18	C₈×C₁₆₇	—	2	1	no	—
19	C₈×C₁₆₇	—	2	1	no	—
20	C₈×C₁₆₇	C₁₆	1	2	yes	complemented
21	C₈×C₁₆₇	D₈	1	1	yes	characteristic
22	C₈×C₁₆₇	Q₁₆	1	1	yes	characteristic
23	C₂²×C₁₆₇	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₁₆₇	—	2	1	no	—
25	C₄×C₁₆₇	#107	1	1	yes	characteristic
26	C₄×C₁₆₇	#108	1	1	yes	characteristic
27	C₄×C₁₆₇	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₁₆₇	M₃₂	1	1	yes	characteristic
29	C₂×C₁₆₇	#335	1	1	yes	characteristic
30	C₂×C₁₆₇	C₄⋊C₁₆	1	1	yes	characteristic
31	C₂×C₁₆₇	#364	1	1	yes	characteristic
32	C₁₆₇	#1370	1	1	yes	characteristic, complemented, Sylow
33	#1370	—	167	1	no	maximal, Sylow
34	C₄⋊C₁₆	—	167	2	no	—
35	C₈²	D₁₆₇	1	1	yes	characteristic
36	C₂×C₁₆	—	334	2	no	—
37	C₄×C₈	Dic₁₆₇	1	1	yes	characteristic
38	C₄×C₈	Dic₁₆₇	1	1	yes	characteristic
39	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	C₁₆₇⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₁₆₇⋊C₈	1	1	yes	characteristic
43	C₂×C₈	D₆₆₈	1	1	yes	characteristic
44	C₂×C₈	Dic₃₃₄	1	1	yes	characteristic
45	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
46	C₁₆	—	668	2	no	—
47	C₂×C₄	C₄⋊Dic₁₆₇	1	1	yes	characteristic
48	C₂×C₄	C₂×C₁₆₇⋊C₈	1	1	yes	characteristic
49	C₂×C₄	#14581934	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	2	1	no	—
52	C₈	—	2	1	no	—
53	C₈	C₁₆₇⋊C₁₆	1	2	yes	complemented
54	C₈	D₁₃₃₆	1	1	yes	characteristic
55	C₈	Dic₆₆₈	1	1	yes	characteristic
56	C₂²	C₄⋊(C₁₆₇⋊C₈)	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#19128766	1	1	yes	characteristic
59	C₄	#19128768	1	1	yes	characteristic
60	C₄	C₂×C₁₆₇⋊C₁₆	1	1	yes	characteristic
61	C₄	#19128772	1	1	yes	characteristic
62	C₂	#24923022	1	1	yes	characteristic
63	C₂	C₄⋊(C₁₆₇⋊C₁₆)	1	1	yes	characteristic
64	C₂	#24923090	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₆₆₈
commutator	C₄×C₁₆₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₆₇
Fitting	C₈²×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₁₆₇	C₂²
hypercenter	C₈²	D₁₆₇
hypocenter	C₁₆₇	#1370
LCS	G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₁₆₇	C₁₆	semidirect
32. C₁₆₇	#1370	semidirect
53. C₈	C₁₆₇⋊C₁₆	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
C₈²×C₁₆₇	2
#1370	167