Subgroups — #36322344

Order 21376.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923099	C₂	1	1	yes	maximal, characteristic
2	#24923097	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	#19128768	—	2	1	no	—
5	#19128766	—	2	1	no	—
6	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
7	C₃₂×C₁₆₇	C₄	1	2	yes	complemented
8	#14581934	—	4	1	no	—
9	C₄⋊Dic₁₆₇	—	4	1	no	—
10	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
11	C₁₆×C₁₆₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₆₇	Q₈	1	1	yes	characteristic
13	C₂×Dic₁₆₇	—	8	1	no	—
14	C₁₆₇⋊C₈	—	8	1	no	—
15	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
16	C₈×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₆₇	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₆₇	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₆₇	#107	1	1	yes	characteristic
20	C₄×C₁₆₇	SD₃₂	1	1	yes	characteristic
21	Dic₁₆₇	—	16	1	no	—
22	C₂×C₁₆₇	—	2	1	no	—
23	C₂×C₁₆₇	#367	1	1	yes	characteristic
24	C₁₆₇	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	167	1	no	maximal, Sylow
26	M₆₄	D₁₆₇	1	1	yes	characteristic
27	#368	—	167	1	no	—
28	#367	—	167	1	no	—
29	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
30	#108	—	334	1	no	—
31	#107	—	334	1	no	—
32	C₃₂	Dic₁₆₇	1	2	yes	complemented
33	M₁₆	—	668	1	no	—
34	C₂×C₈	D₆₆₈	1	1	yes	characteristic
35	C₄⋊C₄	—	668	1	no	—
36	C₁₆	C₂×Dic₁₆₇	1	1	yes	characteristic
37	C₁₆	Dic₃₃₄	1	1	yes	characteristic
38	C₂×C₄	—	1336	1	no	—
39	C₂×C₄	D₁₃₃₆	1	1	yes	characteristic
40	C₈	—	1336	1	no	—
41	C₈	C₄⋊Dic₁₆₇	1	1	yes	characteristic
42	C₈	Dic₆₆₈	1	1	yes	characteristic
43	C₂²	#19128777	1	1	yes	characteristic
44	C₄	—	2672	1	no	—
45	C₄	#19128766	1	1	yes	characteristic
46	C₄	#19128777	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#24923097	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#24923097
commutator	C₁₆×C₁₆₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₆₇
Fitting	M₆₄×C₁₆₇	C₂
socle	C₂×C₁₆₇	#367
prime radical	C₂²×C₁₆₇	SD₃₂
cosocle	C₂×C₁₆×C₁₆₇	C₂²
hypercenter	M₆₄	D₁₆₇
hypocenter	C₁₆₇	#1426
LCS	G ▷ C₁₆×C₁₆₇ ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₁₆₇	C₄	semidirect
24. C₁₆₇	#1426	semidirect
32. C₃₂	Dic₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#24923099	2
#24923097	2
M₆₄×C₁₆₇	2
#1426	167