Subgroups — #27163819

Order 12736.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263889	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₁₉₉	—	2	1	no	—
5	C₁₉₉⋊C₁₆	—	2	1	no	—
6	C₂×D₄×C₁₉₉	—	2	1	no	—
7	D₈×C₁₉₉	—	2	1	no	—
8	D₈×C₁₉₉	C₄	1	2	yes	complemented
9	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₉₉	—	4	1	no	—
11	C₂³×C₁₉₉	—	4	1	no	—
12	D₄×C₁₉₉	—	2	2	no	—
13	D₄×C₁₉₉	—	4	1	no	—
14	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
15	C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₉₉	D₄	1	1	yes	characteristic
17	C₂²×C₁₉₉	—	4	2	no	—
18	C₂²×C₁₉₉	—	8	1	no	—
19	C₂²×C₁₉₉	—	8	1	no	—
20	C₂²×C₁₉₉	D₈	1	1	yes	characteristic
21	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
23	Dic₁₉₉	—	8	1	no	—
24	C₂×C₁₉₉	—	8	2	no	—
25	C₂×C₁₉₉	#102	1	1	yes	characteristic
26	C₂×C₁₉₉	D₁₆	1	1	yes	characteristic
27	C₂×C₁₉₉	SD₃₂	1	1	yes	characteristic
28	C₁₉₉	#357	1	1	yes	characteristic, complemented, Sylow
29	#357	—	199	1	no	maximal, Sylow
30	C₂×D₈	D₁₉₉	1	1	yes	characteristic
31	C₂×C₁₆	—	199	1	no	—
32	#107	—	199	1	no	—
33	C₂×D₄	—	2	1	no	—
34	D₈	—	2	1	no	—
35	D₈	Dic₁₉₉	1	2	yes	complemented
36	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
37	C₄⋊C₄	—	398	1	no	—
38	C₁₆	—	398	1	no	—
39	C₂³	—	4	1	no	—
40	D₄	—	2	2	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	796	1	no	—
43	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
44	C₈	C₂×Dic₁₉₉	1	1	yes	characteristic
45	C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
46	C₂²	—	4	2	no	—
47	C₂²	—	8	1	no	—
48	C₂²	—	8	1	no	—
49	C₂²	#15691871	1	1	yes	characteristic
50	C₄	—	1592	1	no	—
51	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
52	C₄	#15691874	1	1	yes	characteristic
53	C₂	—	8	2	no	—
54	C₂	#20263876	1	1	yes	characteristic
55	C₂	#20263896	1	1	yes	characteristic
56	C₂	#20263899	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15691871
commutator	C₈×C₁₉₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₉₉
Fitting, prime radical	C₂×D₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	D₈
cosocle	C₂×C₈×C₁₉₉	C₂²
hypercenter	C₂×D₈	D₁₉₉
hypocenter	C₁₉₉	#357
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉₉⋊C₁₆	C₂	semidirect
2. #20263889	C₂	semidirect
8. D₈×C₁₉₉	C₄	semidirect
28. C₁₉₉	#357	semidirect
35. D₈	Dic₁₉₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₉₉⋊C₁₆	2
#20263889	2
C₂×D₈×C₁₉₉	2
#357	199