Subgroups — #24923086

Order 10688.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₆₆₈	C₂	1	1	yes	maximal, characteristic
2	#19128768	C₂	1	1	yes	maximal, characteristic
3	M₃₂×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₃₄	—	2	1	no	—
5	Dic₆₆₈	—	2	1	no	—
6	Dic₆₆₈	C₄	1	2	yes	—
7	#14581934	—	2	1	no	—
8	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
9	C₁₆×C₁₆₇	—	2	1	no	—
10	Dic₃₃₄	—	2	2	no	—
11	Dic₃₃₄	—	4	1	no	—
12	C₂×Dic₁₆₇	—	4	1	no	—
13	C₁₆₇⋊C₈	—	4	1	no	—
14	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
15	C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₆₇	D₄	1	1	yes	characteristic
17	C₂²×C₁₆₇	SD₁₆	1	1	yes	characteristic
18	C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₆₇	D₈	1	1	yes	characteristic
20	Dic₁₆₇	—	4	2	no	—
21	C₂×C₁₆₇	—	2	1	no	—
22	C₂×C₁₆₇	#102	1	1	yes	characteristic
23	C₁₆₇	#362	1	1	yes	characteristic, complemented, Sylow
24	#362	—	167	1	no	maximal, Sylow
25	C₂×Q₁₆	—	167	1	no	—
26	M₃₂	D₁₆₇	1	1	yes	characteristic
27	#108	—	167	1	no	—
28	C₂×Q₈	—	334	1	no	—
29	Q₁₆	—	334	1	no	—
30	Q₁₆	—	167	2	no	—
31	M₁₆	—	334	1	no	—
32	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
33	C₁₆	—	2	1	no	—
34	Q₈	—	334	2	no	—
35	Q₈	—	668	1	no	—
36	C₂×C₄	—	668	1	no	—
37	C₂×C₄	D₆₆₈	1	1	yes	characteristic
38	C₈	—	668	1	no	—
39	C₈	C₄×D₁₆₇	1	1	yes	characteristic
40	C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
41	C₂²	#14581939	1	1	yes	characteristic
42	C₄	—	668	2	no	—
43	C₄	(C₂×D₁₆₇)⋊C₄	1	1	yes	characteristic
44	C₄	D₁₃₃₆	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#19128754	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19128754
commutator	C₈×C₁₆₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₆₇
Fitting	M₃₂×C₁₆₇	C₂
socle	C₂×C₁₆₇	#102
prime radical	C₂²×C₁₆₇	SD₁₆
cosocle	C₂×C₈×C₁₆₇	C₂²
hypercenter	M₃₂	D₁₆₇
hypocenter	C₁₆₇	#362
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
23. C₁₆₇	#362	semidirect