Subgroups — #24923998

Order 10688.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128832	C₂	1	3	yes	maximal
2	#19128823	C₂	1	6	yes	maximal
3	Q₈×Dic₁₆₇	C₂	1	2	yes	maximal
4	C₄×Dic₃₃₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₄	C₂²	1	3	yes	—
7	C₄⋊Dic₁₆₇	C₂²	1	6	yes	—
8	C₄⋊Dic₁₆₇	C₂²	1	6	yes	—
9	C₄⋊Dic₁₆₇	C₂²	1	1	yes	characteristic
10	Dic₁₆₇⋊C₄	C₂²	1	6	yes	—
11	C₄×Dic₁₆₇	C₂²	1	6	yes	—
12	C₂×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₆₇	C₂²	1	3	yes	—
14	C₄²×C₁₆₇	C₂²	1	3	yes	—
15	Dic₃₃₄	—	2	6	no	—
16	C₂×Dic₁₆₇	C₂³	1	2	yes	—
17	C₂×Dic₁₆₇	C₂³	1	6	yes	—
18	Q₈×C₁₆₇	Q₈	1	4	yes	complemented
19	C₂×C₄×C₁₆₇	C₂³	1	3	yes	—
20	C₂×C₄×C₁₆₇	C₂³	1	3	yes	—
21	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
22	C₂²×C₁₆₇	C₂⁴	1	1	yes	characteristic
23	C₄×C₁₆₇	—	2	3	no	—
24	C₄×C₁₆₇	C₂×Q₈	1	6	yes	—
25	C₄×C₁₆₇	#41	1	2	yes	—
26	Dic₁₆₇	—	2	6	no	—
27	Dic₁₆₇	—	2	2	no	—
28	C₂×C₁₆₇	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₁₆₇	C₂×#41	1	1	yes	characteristic
30	C₂×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
31	C₁₆₇	#557	1	1	yes	characteristic, complemented, Sylow
32	#557	—	167	1	no	maximal, Sylow
33	C₄⋊Q₈	—	167	3	no	—
34	#125	—	167	6	no	—
35	C₄×Q₈	—	167	2	no	—
36	C₄×Q₈	—	167	3	no	—
37	C₄×Q₈	D₁₆₇	1	1	yes	characteristic
38	C₂×Q₈	—	167	3	no	—
39	C₂×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
40	C₄⋊C₄	—	167	1	no	—
41	C₄⋊C₄	—	167	6	no	—
42	C₄⋊C₄	—	167	6	no	—
43	C₄⋊C₄	—	167	6	no	—
44	C₄⋊C₄	C₂×D₁₆₇	1	3	yes	—
45	C₄²	—	167	6	no	—
46	C₄²	C₂×D₁₆₇	1	3	yes	—
47	Q₈	—	334	6	no	—
48	Q₈	Dic₃₃₄	1	4	yes	complemented
49	C₂×C₄	—	167	2	no	—
50	C₂×C₄	—	167	6	no	—
51	C₂×C₄	C₂²×D₁₆₇	1	3	yes	—
52	C₂×C₄	C₂²×D₁₆₇	1	3	yes	—
53	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
54	C₂²	C₂³×D₁₆₇	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	334	6	no	—
57	C₄	—	334	2	no	—
58	C₄	C₂×Dic₃₃₄	1	6	yes	—
59	C₄	#14581951	1	2	yes	—
60	C₂	C₂²×Dic₃₃₄	1	1	yes	characteristic
61	C₂	C₂×#14581951	1	1	yes	characteristic
62	C₂	#19128879	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₆₇
commutator, socle, prime radical, cosocle	C₂²×C₁₆₇	C₂⁴
Fitting	C₄×Q₈×C₁₆₇	C₂
hypercenter	C₄×Q₈	D₁₆₇
hypocenter	C₁₆₇	#557
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
18. Q₈×C₁₆₇	Q₈	semidirect
31. C₁₆₇	#557	semidirect
48. Q₈	Dic₃₃₄	semidirect