Subgroups — #24923554

Order 10688.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128832	C₂	1	1	yes	maximal, characteristic
2	#19128763	C₂	1	2	yes	maximal
3	C₄⋊(C₁₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#19128756	C₂	1	2	yes	maximal
5	C₄×Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₄	—	2	1	no	—
7	C₂×C₁₆₇⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₆₇	—	2	1	no	—
9	C₄⋊Dic₁₆₇	C₂²	1	2	yes	—
10	C₂×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₁₆₇	—	2	1	no	—
12	C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₄²×C₁₆₇	—	2	1	no	—
14	C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
15	Dic₃₃₄	—	4	2	no	—
16	C₂×Dic₁₆₇	—	2	2	no	—
17	C₁₆₇⋊C₈	—	2	2	no	—
18	Q₈×C₁₆₇	—	2	1	no	—
19	Q₈×C₁₆₇	Q₈	1	2	yes	complemented
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	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
26	C₄×C₁₆₇	—	2	2	no	—
27	C₄×C₁₆₇	—	2	1	no	—
28	C₄×C₁₆₇	—	4	1	no	—
29	C₄×C₁₆₇	SD₁₆	1	2	yes	—
30	C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
31	C₄×C₁₆₇	#41	1	1	yes	characteristic
32	Dic₁₆₇	—	4	2	no	—
33	C₂×C₁₆₇	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₁₆₇	#137	1	1	yes	characteristic
36	C₁₆₇	#475	1	1	yes	characteristic, complemented, Sylow
37	#475	—	167	1	no	maximal, Sylow
38	C₄⋊Q₈	—	167	1	no	—
39	C₄×Q₈	D₁₆₇	1	1	yes	characteristic
40	#106	—	167	2	no	—
41	C₄⋊C₈	—	167	1	no	—
42	#103	—	167	2	no	—
43	C₂×Q₈	—	334	1	no	—
44	C₂×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
45	C₂×C₈	—	167	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	334	1	no	—
48	C₄⋊C₄	—	167	2	no	—
49	C₄⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
50	C₄²	—	2	1	no	—
51	C₄²	C₂×D₁₆₇	1	1	yes	characteristic
52	Q₈	—	2	1	no	—
53	Q₈	—	668	2	no	—
54	Q₈	Dic₃₃₄	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	334	2	no	—
58	C₂×C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
61	C₈	—	334	2	no	—
62	C₂²	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	668	2	no	—
67	C₄	#14581937	1	2	yes	—
68	C₄	C₂×Dic₃₃₄	1	1	yes	characteristic
69	C₄	#14581950	1	1	yes	characteristic
70	C₂	#19128811	1	1	yes	characteristic
71	C₂	C₂×#14581937	1	1	yes	characteristic
72	C₂	#19128862	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₆₇
Fitting	C₄×Q₈×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×D₄
hypercenter	C₄×Q₈	D₁₆₇
hypocenter	C₁₆₇	#475
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
19. Q₈×C₁₆₇	Q₈	semidirect
36. C₁₆₇	#475	semidirect
54. Q₈	Dic₃₃₄	semidirect