Subgroups — #27164417

Order 12736.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263947	C₂	1	1	yes	maximal, characteristic
2	#20263888	C₂	1	2	yes	maximal
3	#20263886	C₂	1	2	yes	maximal
4	C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#125×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
9	Dic₁₉₉⋊C₄	—	2	2	no	—
10	C₄⋊C₄×C₁₉₉	—	2	2	no	—
11	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
12	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉₉	—	2	2	no	—
14	C₁₉₉⋊C₈	Q₈	1	2	yes	—
15	C₁₉₉⋊C₈	Q₈	1	2	yes	—
16	C₂×C₄×C₁₉₉	—	2	2	no	—
17	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
18	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
19	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₉₉	—	2	2	no	—
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	—	4	2	no	—
25	C₂×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₁₉₉	#135	1	2	yes	—
27	C₁₉₉	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	199	1	no	maximal, Sylow
29	#125	—	199	1	no	—
30	#125	D₁₉₉	1	1	yes	characteristic
31	#107	—	199	2	no	—
32	#106	—	199	2	no	—
33	C₄×C₈	—	199	1	no	—
34	C₂×C₈	—	199	1	no	—
35	C₂×C₈	—	199	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	398	2	no	—
38	C₄⋊C₄	—	199	2	no	—
39	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
40	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	398	2	no	—
43	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
44	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
45	C₈	—	199	2	no	—
46	C₈	—	199	2	no	—
47	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	796	2	no	—
51	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
52	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
53	C₂	#20263953	1	1	yes	characteristic
54	C₂	#20263975	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#125×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	#125	D₁₉₉
hypocenter	C₁₉₉	#499
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

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

Maximal subgroups

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

subgroup	index
#20263947	2
#20263888	2
#20263886	2
C₄×C₁₉₉⋊C₈	2
#125×C₁₉₉	2
#499	199