Subgroups — #24923081

Order 10688.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃₃₆	C₂	1	1	yes	maximal, characteristic
2	#19128772	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₆₆₈	—	2	1	no	—
5	D₁₃₃₆	—	2	1	no	—
6	D₁₃₃₆	C₄	1	2	yes	—
7	C₁₆₇⋊C₁₆	—	2	1	no	—
8	M₁₆×C₁₆₇	—	2	1	no	—
9	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
10	C₂²×D₁₆₇	—	4	1	no	—
11	D₆₆₈	—	2	2	no	—
12	D₆₆₈	—	4	1	no	—
13	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
14	C₈×C₁₆₇	—	4	1	no	—
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	C₂×D₁₆₇	—	4	2	no	—
19	C₂×D₁₆₇	—	8	2	no	—
20	C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₆₇	D₈	1	1	yes	characteristic
22	C₂×C₁₆₇	—	2	1	no	—
23	C₂×C₁₆₇	#102	1	1	yes	characteristic
24	D₁₆₇	—	8	2	no	—
25	C₁₆₇	#361	1	1	yes	characteristic, complemented, Sylow
26	#361	—	167	1	no	maximal, Sylow
27	C₂×D₈	—	167	1	no	—
28	M₃₂	—	167	1	no	—
29	#108	D₁₆₇	1	1	yes	characteristic, complemented
30	C₂×D₄	—	334	1	no	—
31	D₈	—	334	1	no	—
32	D₈	—	167	2	no	—
33	M₁₆	—	2	1	no	—
34	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
35	C₁₆	—	334	1	no	—
36	C₂³	—	668	1	no	—
37	D₄	—	334	2	no	—
38	D₄	—	668	1	no	—
39	C₂×C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
40	C₈	—	4	1	no	—
41	C₈	C₄×D₁₆₇	1	1	yes	characteristic
42	C₈	D₆₆₈	1	1	yes	characteristic
43	C₂²	—	1336	2	no	—
44	C₂²	—	668	2	no	—
45	C₂²	#14581937	1	1	yes	characteristic
46	C₄	(C₂×D₁₆₇)⋊C₄	1	1	yes	characteristic
47	C₄	#14581935	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	1336	2	no	—
50	C₂	#19128752	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19128752
commutator	C₈×C₁₆₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₆₇
Fitting	#108×C₁₆₇	C₂
socle	C₂×C₁₆₇	#102
prime radical	C₂×D₁₃₃₆	C₂
cosocle	C₂×C₈×C₁₆₇	C₂²
hypercenter	#108	D₁₆₇
hypocenter	C₁₆₇	#361
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #19128772	C₂	semidirect
3. #108×C₁₆₇	C₂	semidirect
25. C₁₆₇	#361	semidirect
29. #108	D₁₆₇	semidirect