Subgroups — #27164419

Order 12736.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263946	C₂	1	2	yes	maximal
2	#20263889	C₂	1	1	yes	maximal, characteristic
3	#20263886	C₂	1	2	yes	maximal
4	C₈×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
5	#107×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₁₉₉	—	2	2	no	—
8	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
9	Dic₁₉₉⋊C₄	—	2	2	no	—
10	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
13	C₂×Dic₁₉₉	—	2	2	no	—
14	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
15	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
16	C₁₉₉⋊C₈	Q₈	1	2	yes	—
17	C₂×C₄×C₁₉₉	—	2	2	no	—
18	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
19	C₈×C₁₉₉	Q₈	1	2	yes	—
20	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
21	C₄×C₁₉₉	—	4	2	no	—
22	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₉₉	—	2	1	no	—
25	Dic₁₉₉	—	2	1	no	—
26	Dic₁₉₉	—	4	2	no	—
27	C₂×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₁₉₉	#135	1	1	yes	characteristic
29	C₂×C₁₉₉	#135	1	1	yes	characteristic
30	C₁₉₉	#499	1	1	yes	characteristic, complemented, Sylow
31	#499	—	199	1	no	maximal, Sylow
32	#125	—	199	2	no	—
33	#107	—	199	1	no	—
34	#107	D₁₉₉	1	1	yes	characteristic
35	#106	—	199	2	no	—
36	C₄×C₈	—	199	1	no	—
37	C₂×C₈	—	199	1	no	—
38	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
39	C₄⋊C₄	—	398	2	no	—
40	C₄⋊C₄	—	398	2	no	—
41	C₄⋊C₄	—	199	2	no	—
42	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
43	C₄²	—	199	1	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	398	2	no	—
46	C₂×C₄	—	199	1	no	—
47	C₂×C₄	—	199	1	no	—
48	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
49	C₈	—	199	2	no	—
50	C₈	Dic₃₉₈	1	2	yes	—
51	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
52	C₄	—	4	2	no	—
53	C₄	—	796	2	no	—
54	C₄	—	398	1	no	—
55	C₄	—	398	1	no	—
56	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
57	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
58	C₂	#20263954	1	1	yes	characteristic
59	C₂	#20263974	1	1	yes	characteristic
60	C₂	#20263978	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#107×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	#107	D₁₉₉
hypocenter	C₁₉₉	#499
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
30. C₁₉₉	#499	semidirect

Maximal subgroups

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

subgroup	index
#20263946	2
#20263889	2
#20263886	2
C₈×Dic₁₉₉	2
#107×C₁₉₉	2
#499	199