Subgroups — #24923709

Order 10688.

82 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	#19128778	C₂	1	2	yes	maximal, complemented
3	#19128776	C₂	1	2	yes	maximal, complemented
4	#19128772	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₄	—	2	1	no	—
7	Dic₆₆₈	—	2	1	no	—
8	Dic₆₆₈	C₂²	1	2	yes	—
9	C₁₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#41×C₁₆₇	—	2	1	no	—
11	Q₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
12	SD₁₆×C₁₆₇	—	2	1	no	—
13	D₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
15	Dic₃₃₄	—	2	2	no	—
16	Dic₃₃₄	—	4	1	no	—
17	C₂×Dic₁₆₇	—	4	1	no	—
18	Q₈×C₁₆₇	—	2	1	no	—
19	D₄×C₁₆₇	—	2	1	no	—
20	D₄×C₁₆₇	—	4	1	no	—
21	C₂×C₄×C₁₆₇	—	4	1	no	—
22	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
23	C₈×C₁₆₇	D₄	1	1	yes	characteristic
24	C₈×C₁₆₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₆₇	—	4	1	no	—
26	C₂²×C₁₆₇	D₈	1	1	yes	characteristic
27	C₄×C₁₆₇	—	4	1	no	—
28	C₄×C₁₆₇	D₈	1	1	yes	characteristic
29	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
30	Dic₁₆₇	—	4	2	no	—
31	C₂×C₁₆₇	—	2	1	no	—
32	C₂×C₁₆₇	—	8	1	no	—
33	C₂×C₁₆₇	C₂×D₈	1	1	yes	characteristic
34	C₁₆₇	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	167	1	no	maximal, Sylow
36	#135	D₁₆₇	1	1	yes	characteristic
37	C₂×Q₁₆	—	167	1	no	—
38	Q₃₂	—	167	2	no	—
39	SD₃₂	—	167	2	no	—
40	M₃₂	—	167	1	no	—
41	#41	—	2	1	no	—
42	C₂×Q₈	—	334	1	no	—
43	Q₁₆	—	334	1	no	—
44	Q₁₆	—	167	2	no	—
45	Q₁₆	C₂×D₁₆₇	1	1	yes	characteristic
46	SD₁₆	—	2	1	no	—
47	D₈	C₂×D₁₆₇	1	1	yes	characteristic
48	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
49	C₁₆	—	167	2	no	—
50	Q₈	—	2	1	no	—
51	Q₈	—	334	2	no	—
52	Q₈	—	668	1	no	—
53	D₄	—	2	1	no	—
54	D₄	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	668	1	no	—
57	C₂×C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
58	C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
59	C₈	C₂²×D₁₆₇	1	1	yes	characteristic
60	C₂²	—	4	1	no	—
61	C₂²	#14581935	1	1	yes	characteristic
62	C₄	—	4	1	no	—
63	C₄	—	668	2	no	—
64	C₄	#14581935	1	1	yes	characteristic
65	C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	8	1	no	—
68	C₂	C₂×#14581935	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14581935
commutator, cosocle	C₈×C₁₆₇	C₂³
Frattini	C₈	C₂²×D₁₆₇
Fitting, prime radical	#135×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₂×D₈
hypercenter	#135	D₁₆₇
hypocenter	C₁₆₇	#510
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₆₆₈	C₂	semidirect
2. #19128778	C₂	semidirect
3. #19128776	C₂	semidirect
4. #19128772	C₂	semidirect
34. C₁₆₇	#510	semidirect