Subgroups — #24923510

Order 10688.

114 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	#19128811	C₂	1	2	yes	maximal
3	#19128766	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128757	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₄	C₂²	1	2	yes	—
7	Dic₆₆₈	—	2	2	no	—
8	C₄⋊Dic₁₆₇	C₂²	1	2	yes	—
9	Dic₁₆₇⋊C₄	—	2	2	no	—
10	C₂²⋊Dic₁₆₇	—	2	2	no	—
11	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₆₇	—	2	2	no	—
13	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
15	Dic₃₃₄	—	2	2	no	—
16	Dic₃₃₄	—	2	2	no	—
17	C₂×Dic₁₆₇	—	2	2	no	—
18	C₂×Dic₁₆₇	—	2	2	no	—
19	C₂³×C₁₆₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₆₇	—	2	1	no	—
21	C₂×C₄×C₁₆₇	—	2	1	no	—
22	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
24	C₈×C₁₆₇	—	2	1	no	—
25	C₈×C₁₆₇	D₄	1	2	yes	—
26	C₂²×C₁₆₇	—	2	1	no	—
27	C₂²×C₁₆₇	—	2	1	no	—
28	C₂²×C₁₆₇	Q₁₆	1	2	yes	complemented
29	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₁₆₇	—	2	1	no	—
31	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₆₇	#41	1	1	yes	characteristic
33	Dic₁₆₇	—	4	2	no	—
34	Dic₁₆₇	—	4	2	no	—
35	C₂×C₁₆₇	—	2	2	no	—
36	C₂×C₁₆₇	#121	1	1	yes	characteristic
37	C₂×C₁₆₇	C₂×Q₁₆	1	1	yes	characteristic
38	C₂×C₁₆₇	#135	1	1	yes	characteristic
39	C₁₆₇	#467	1	1	yes	characteristic, complemented, Sylow
40	#467	—	167	1	no	maximal, Sylow
41	C₂×Q₁₆	—	167	1	no	—
42	C₂²×C₈	D₁₆₇	1	1	yes	characteristic
43	C₂²⋊Q₈	—	167	2	no	—
44	#107	—	167	1	no	—
45	#103	—	167	2	no	—
46	C₂×Q₈	—	167	2	no	—
47	C₂²×C₄	C₂×D₁₆₇	1	1	yes	characteristic
48	Q₁₆	—	334	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
51	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
52	C₄⋊C₄	—	334	2	no	—
53	C₄⋊C₄	—	167	2	no	—
54	C₂²⋊C₄	—	334	2	no	—
55	C₂³	D₆₆₈	1	1	yes	characteristic
56	Q₈	—	334	2	no	—
57	Q₈	—	334	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	334	2	no	—
61	C₂×C₄	—	334	2	no	—
62	C₂×C₄	D₆₆₈	1	1	yes	characteristic
63	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	C₂²⋊D₁₆₇	1	2	yes	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	Dic₆₆₈	1	2	yes	complemented
69	C₂²	C₂×D₆₆₈	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	668	2	no	—
72	C₄	—	668	2	no	—
73	C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
74	C₄	#14581950	1	1	yes	characteristic
75	C₂	—	2	2	no	—
76	C₂	#19128806	1	1	yes	characteristic
77	C₂	C₂×Dic₆₆₈	1	1	yes	characteristic
78	C₂	#19128854	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₆₆₈
commutator, cosocle	C₂×C₄×C₁₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₆₇
Fitting	C₂²×C₈×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₄
prime radical	C₂³×C₁₆₇	D₄
hypercenter	C₂²×C₈	D₁₆₇
hypocenter	C₁₆₇	#467
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₆₆₈	C₂	semidirect
3. #19128766	C₂	semidirect
4. #19128757	C₂	semidirect
28. C₂²×C₁₆₇	Q₁₆	semidirect
39. C₁₆₇	#467	semidirect
68. C₂²	Dic₆₆₈	semidirect