Subgroups — #27163828

Order 12736.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263895	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263887	C₂	1	1	yes	maximal, characteristic, complemented
3	#135×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₁₉₉	—	2	1	no	—
5	C₁₉₉⋊C₁₆	—	2	1	no	—
6	#41×C₁₉₉	—	2	1	no	—
7	Q₁₆×C₁₉₉	C₄	1	1	yes	characteristic, complemented
8	SD₁₆×C₁₉₉	—	2	1	no	—
9	D₈×C₁₉₉	C₄	1	1	yes	characteristic, complemented
10	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₉	—	4	1	no	—
12	Q₈×C₁₉₉	—	2	1	no	—
13	D₄×C₁₉₉	—	2	1	no	—
14	D₄×C₁₉₉	—	4	1	no	—
15	C₂×C₄×C₁₉₉	—	4	1	no	—
16	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
17	C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₉₉	D₄	1	1	yes	characteristic
19	C₂²×C₁₉₉	—	4	1	no	—
20	C₂²×C₁₉₉	D₈	1	1	yes	characteristic
21	C₄×C₁₉₉	—	4	1	no	—
22	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
24	Dic₁₉₉	—	8	1	no	—
25	C₂×C₁₉₉	—	2	1	no	—
26	C₂×C₁₉₉	—	8	1	no	—
27	C₂×C₁₉₉	#102	1	1	yes	characteristic
28	C₁₉₉	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	199	1	no	maximal, Sylow
30	#135	D₁₉₉	1	1	yes	characteristic
31	M₃₂	—	199	1	no	—
32	#106	—	199	1	no	—
33	#41	—	2	1	no	—
34	Q₁₆	Dic₁₉₉	1	1	yes	characteristic, complemented
35	SD₁₆	—	2	1	no	—
36	D₈	Dic₁₉₉	1	1	yes	characteristic, complemented
37	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
38	C₄⋊C₄	—	398	1	no	—
39	C₁₆	—	398	1	no	—
40	Q₈	—	2	1	no	—
41	D₄	—	2	1	no	—
42	D₄	—	4	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	796	1	no	—
45	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
46	C₈	C₂×Dic₁₉₉	1	1	yes	characteristic
47	C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	#15691871	1	1	yes	characteristic
50	C₄	—	4	1	no	—
51	C₄	—	1592	1	no	—
52	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
53	C₄	#15691874	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	8	1	no	—
56	C₂	#20263876	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20263876
commutator	C₈×C₁₉₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₉₉
Fitting, prime radical	#135×C₁₉₉	C₂
socle	C₂×C₁₉₉	#102
cosocle	C₂×C₈×C₁₉₉	C₂²
hypercenter	#135	D₁₉₉
hypocenter	C₁₉₉	#360
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
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. #20263895	C₂	semidirect
2. #20263887	C₂	semidirect
7. Q₁₆×C₁₉₉	C₄	semidirect
9. D₈×C₁₉₉	C₄	semidirect
28. C₁₉₉	#360	semidirect
34. Q₁₆	Dic₁₉₉	semidirect
36. D₈	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263895	2
#20263887	2
#135×C₁₉₉	2
#360	199