Subgroups — #27164388

Order 12736.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263955	C₂	1	1	yes	maximal, characteristic
2	#20263878	C₂	1	4	yes	maximal
3	#20263866	C₂	1	1	yes	maximal, characteristic
4	#125×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₉₈	C₂²	1	2	yes	—
6	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₁₉₉	—	2	2	no	—
8	C₄⋊C₄×C₁₉₉	—	2	2	no	—
9	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
10	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
11	Dic₃₉₈	—	2	4	no	—
12	C₂×Dic₁₉₉	—	2	2	no	—
13	C₁₉₉⋊C₈	—	2	2	no	—
14	C₂×C₄×C₁₉₉	—	2	2	no	—
15	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
16	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
17	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
18	C₄×C₁₉₉	—	2	2	no	—
19	C₄×C₁₉₉	—	4	2	no	—
20	C₄×C₁₉₉	#41	1	2	yes	—
21	Dic₁₉₉	—	4	2	no	—
22	C₂×C₁₉₉	#124	1	1	yes	characteristic
23	C₂×C₁₉₉	#137	1	2	yes	—
24	C₁₉₉	#491	1	1	yes	characteristic, complemented, Sylow
25	#491	—	199	1	no	maximal, Sylow
26	C₄⋊Q₈	—	199	1	no	—
27	#125	D₁₉₉	1	1	yes	characteristic
28	#103	—	199	4	no	—
29	#97	—	199	1	no	—
30	C₂×Q₈	—	199	2	no	—
31	C₂×C₈	—	199	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	—	398	2	no	—
34	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
35	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
36	Q₈	—	398	4	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	398	2	no	—
39	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
40	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
41	C₈	—	398	2	no	—
42	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	4	2	no	—
45	C₄	—	796	2	no	—
46	C₄	#15691887	1	2	yes	—
47	C₂	#20263944	1	1	yes	characteristic
48	C₂	#20263985	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	#491
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
24. C₁₉₉	#491	semidirect

Maximal subgroups

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

subgroup	index
#20263955	2
#20263878	2
#20263866	2
#125×C₁₉₉	2
#491	199