Subgroups — #24923528

Order 10688.

110 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, complemented
2	C₂×#14581933	C₂	1	1	yes	maximal, characteristic
3	#19128812	C₂	1	2	yes	maximal
4	#19128755	C₂	1	2	yes	maximal, complemented
5	#106×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₃₄	C₂²	1	2	yes	—
7	C₂×C₄×D₁₆₇	C₂²	1	1	yes	characteristic
8	Dic₆₆₈	—	2	2	no	—
9	#14581933	—	2	2	no	—
10	C₂×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₆₇	—	2	2	no	—
12	(C₂×D₁₆₇)⋊C₄	—	2	2	no	—
13	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
15	C₂²×D₁₆₇	D₄	1	1	yes	characteristic
16	Dic₃₃₄	—	2	2	no	—
17	Dic₃₃₄	—	2	2	no	—
18	C₂×Dic₁₆₇	—	2	2	no	—
19	C₂×Dic₁₆₇	D₄	1	1	yes	characteristic
20	C₄×D₁₆₇	—	2	1	no	—
21	C₄×D₁₆₇	—	2	1	no	—
22	C₁₆₇⋊C₈	—	2	1	no	—
23	C₂×C₄×C₁₆₇	—	2	2	no	—
24	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
25	C₈×C₁₆₇	D₄	1	2	yes	—
26	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₆₇	—	2	1	no	—
28	C₂×D₁₆₇	—	2	2	no	—
29	C₄×C₁₆₇	—	4	2	no	—
30	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₆₇	#41	1	1	yes	characteristic
32	Dic₁₆₇	—	2	1	no	—
33	Dic₁₆₇	—	4	2	no	—
34	C₂×C₁₆₇	#121	1	1	yes	characteristic
35	C₂×C₁₆₇	#137	1	2	yes	—
36	D₁₆₇	—	4	1	no	—
37	C₁₆₇	#470	1	1	yes	characteristic, complemented, Sylow
38	#470	—	167	1	no	maximal, Sylow
39	C₂×Q₁₆	—	167	1	no	—
40	C₂×M₁₆	—	167	1	no	—
41	C₂²⋊Q₈	—	167	2	no	—
42	#106	D₁₆₇	1	1	yes	characteristic, complemented
43	#103	—	167	2	no	—
44	C₂×Q₈	—	167	2	no	—
45	C₂²×C₄	—	167	1	no	—
46	Q₁₆	—	334	2	no	—
47	M₁₆	—	334	2	no	—
48	C₂×C₈	—	167	1	no	—
49	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
50	C₄⋊C₄	—	334	2	no	—
51	C₄⋊C₄	C₂×D₁₆₇	1	2	yes	—
52	C₂²⋊C₄	—	334	2	no	—
53	C₂³	—	167	1	no	—
54	Q₈	—	334	2	no	—
55	Q₈	—	334	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	334	1	no	—
58	C₂×C₄	—	334	1	no	—
59	C₂×C₄	—	334	2	no	—
60	C₂×C₄	—	167	1	no	—
61	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
62	C₈	—	334	1	no	—
63	C₈	D₆₆₈	1	2	yes	—
64	C₂²	—	334	2	no	—
65	C₂²	—	334	1	no	—
66	C₂²	D₄×D₁₆₇	1	1	yes	characteristic
67	C₄	—	4	2	no	—
68	C₄	—	334	1	no	—
69	C₄	—	668	2	no	—
70	C₄	C₂×D₆₆₈	1	1	yes	characteristic
71	C₄	#14581951	1	1	yes	characteristic
72	C₂	—	668	1	no	—
73	C₂	#19128807	1	1	yes	characteristic
74	C₂	#19128861	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₆₇
Fitting	#106×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₄
prime radical	C₂²×D₁₆₇	D₄
hypercenter	#106	D₁₆₇
hypocenter	C₁₆₇	#470
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₆₆₈	C₂	semidirect
4. #19128755	C₂	semidirect
5. #106×C₁₆₇	C₂	semidirect
37. C₁₆₇	#470	semidirect
42. #106	D₁₆₇	semidirect