Subgroups — #36322335

Order 21376.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923103	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₆₇	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆₇⋊C₁₆	C₄	1	1	yes	characteristic
5	C₈×Dic₁₆₇	C₄	1	1	yes	characteristic
6	C₁₆₇⋊C₃₂	—	2	1	no	—
7	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
8	C₃₂×C₁₆₇	—	2	1	no	—
9	C₂×C₁₆₇⋊C₈	C₈	1	1	yes	characteristic
10	C₄×Dic₁₆₇	C₈	1	1	yes	characteristic
11	C₁₆₇⋊C₁₆	—	2	1	no	—
12	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₆₇	D₄	1	1	yes	characteristic
14	C₁₆×C₁₆₇	Q₈	1	1	yes	characteristic
15	C₂×Dic₁₆₇	—	2	1	no	—
16	C₁₆₇⋊C₈	C₁₆	1	2	yes	—
17	C₂×C₄×C₁₆₇	C₂×C₈	1	1	yes	characteristic
18	C₈×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
19	C₈×C₁₆₇	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₆₇	M₃₂	1	1	yes	characteristic
21	C₄×C₁₆₇	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₆₇	C₂×C₁₆	1	1	yes	characteristic
23	Dic₁₆₇	—	2	2	no	—
24	C₂×C₁₆₇	—	2	1	no	—
25	C₂×C₁₆₇	C₄⋊C₁₆	1	1	yes	characteristic
26	C₁₆₇	#1422	1	1	yes	characteristic, complemented, Sylow
27	#1422	—	167	1	no	maximal, Sylow
28	M₆₄	—	167	1	no	—
29	M₆₄	D₁₆₇	1	1	yes	characteristic
30	C₄×C₁₆	—	167	1	no	—
31	C₂×C₁₆	—	167	1	no	—
32	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
33	C₄×C₈	—	167	1	no	—
34	C₃₂	—	2	1	no	—
35	C₃₂	—	334	1	no	—
36	C₂×C₈	—	167	1	no	—
37	C₂×C₈	C₄×D₁₆₇	1	1	yes	characteristic
38	C₄²	—	167	1	no	—
39	C₁₆	—	334	1	no	—
40	C₁₆	C₂²⋊D₁₆₇	1	1	yes	characteristic
41	C₁₆	Dic₃₃₄	1	1	yes	characteristic
42	C₂×C₄	—	334	1	no	—
43	C₂×C₄	C₈×D₁₆₇	1	1	yes	characteristic
44	C₈	—	167	2	no	—
45	C₈	Dic₁₆₇⋊C₄	1	1	yes	characteristic
46	C₈	#14581933	1	1	yes	characteristic
47	C₂²	#19128771	1	1	yes	characteristic
48	C₄	—	334	2	no	—
49	C₄	Dic₁₆₇⋊C₈	1	1	yes	characteristic
50	C₄	C₁₆×D₁₆₇	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	Dic₁₆₇⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₆₇
commutator	C₄×C₁₆₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₆₇
Fitting	M₆₄×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₄⋊C₁₆
prime radical	C₂²×C₁₆₇	M₃₂
cosocle	C₂×C₁₆×C₁₆₇	C₂²
hypercenter	M₆₄	D₁₆₇
hypocenter	C₁₆₇	#1422
LCS	G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
26. C₁₆₇	#1422	semidirect

Maximal subgroups

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

subgroup	index
#24923103	2
C₁₆×Dic₁₆₇	2
M₆₄×C₁₆₇	2
#1422	167