Subgroups — #27164422

Order 12736.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263954	C₂	1	2	yes	maximal
2	#20263889	C₂	1	1	yes	maximal, characteristic
3	#20263888	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×Dic₃₉₈	—	2	2	no	—
7	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
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	Dic₃₉₈	—	4	4	no	—
14	C₂×Dic₁₉₉	—	2	2	no	—
15	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
16	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
17	C₁₉₉⋊C₈	Q₈	1	2	yes	complemented
18	C₂×C₄×C₁₉₉	—	2	2	no	—
19	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
20	C₈×C₁₉₉	Q₈	1	2	yes	complemented
21	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₉₉	—	4	2	no	—
23	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
24	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
25	Dic₁₉₉	—	4	2	no	—
26	Dic₁₉₉	D₈	1	2	yes	—
27	Dic₁₉₉	Q₁₆	1	2	yes	—
28	C₂×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
29	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
30	C₂×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
31	C₁₉₉	#500	1	1	yes	characteristic, complemented, Sylow
32	#500	—	199	1	no	maximal, Sylow
33	C₄⋊Q₈	—	199	2	no	—
34	#107	—	199	1	no	—
35	#107	—	199	2	no	—
36	#107	D₁₉₉	1	1	yes	characteristic
37	C₄×C₈	—	199	1	no	—
38	C₂×Q₈	—	398	2	no	—
39	C₂×C₈	—	199	1	no	—
40	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
41	C₄⋊C₄	—	398	2	no	—
42	C₄⋊C₄	—	199	2	no	—
43	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
44	C₄²	—	199	1	no	—
45	Q₈	—	796	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	398	2	no	—
48	C₂×C₄	—	199	1	no	—
49	C₂×C₄	—	199	1	no	—
50	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
51	C₈	—	199	2	no	—
52	C₈	Dic₃₉₈	1	2	yes	complemented
53	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
54	C₄	—	4	2	no	—
55	C₄	—	796	2	no	—
56	C₄	—	199	2	no	—
57	C₄	—	199	2	no	—
58	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
59	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
60	C₂	#20263954	1	1	yes	characteristic
61	C₂	D₈×D₁₉₉	1	1	yes	characteristic
62	C₂	Q₁₆×D₁₉₉	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	#500
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
17. C₁₉₉⋊C₈	Q₈	semidirect
20. C₈×C₁₉₉	Q₈	semidirect
31. C₁₉₉	#500	semidirect
52. C₈	Dic₃₉₈	semidirect

Maximal subgroups

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

subgroup	index
#20263954	2
#20263889	2
#20263888	2
C₈×Dic₁₉₉	2
#107×C₁₉₉	2
#500	199