Subgroups — #27164428

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	#20263888	C₂	1	2	yes	maximal
3	#20263886	C₂	1	2	yes	maximal
4	#20263866	C₂	1	1	yes	maximal, characteristic
5	#125×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	—	2	1	no	—
7	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₉₉	—	2	1	no	—
9	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₉₉	—	2	1	no	—
11	C₄⋊C₄×C₁₉₉	—	2	1	no	—
12	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	Dic₃₉₈	—	4	2	no	—
16	C₂×Dic₁₉₉	—	2	2	no	—
17	C₁₉₉⋊C₈	Q₈	1	4	yes	complemented
18	C₂×C₄×C₁₉₉	—	2	1	no	—
19	C₂×C₄×C₁₉₉	—	2	1	no	—
20	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
23	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₉₉	—	2	1	no	—
25	C₄×C₁₉₉	—	2	1	no	—
26	C₄×C₁₉₉	—	4	1	no	—
27	C₄×C₁₉₉	—	4	1	no	—
28	C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
29	Dic₁₉₉	—	4	2	no	—
30	C₂×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₁₉₉	#137	1	1	yes	characteristic
33	C₁₉₉	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	199	1	no	maximal, Sylow
35	C₄⋊Q₈	—	199	1	no	—
36	#125	D₁₉₉	1	1	yes	characteristic
37	#107	—	199	2	no	—
38	#106	—	199	2	no	—
39	#97	—	199	1	no	—
40	C₂×Q₈	—	398	1	no	—
41	C₂×C₈	—	199	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	398	1	no	—
45	C₄⋊C₄	—	199	2	no	—
46	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
47	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
48	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
49	Q₈	—	796	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	398	2	no	—
53	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
56	C₈	—	199	4	no	—
57	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	796	2	no	—
63	C₄	Q₈×D₁₉₉	1	2	yes	—
64	C₂	#20263953	1	1	yes	characteristic
65	C₂	#20263980	1	1	yes	characteristic
66	C₂	#20263985	1	1	yes	characteristic

	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	#501
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
17. C₁₉₉⋊C₈	Q₈	semidirect
33. C₁₉₉	#501	semidirect

Maximal subgroups

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

subgroup	index
#20263955	2
#20263888	2
#20263886	2
#20263866	2
#125×C₁₉₉	2
#501	199