Subgroups — #27164415

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	#20263887	C₂	1	1	yes	maximal, characteristic
3	#20263886	C₂	1	2	yes	maximal
4	C₈×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
5	#106×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	2	yes	—
16	C₁₉₉⋊C₈	Q₈	1	2	yes	complemented
17	C₂×C₄×C₁₉₉	—	2	2	no	—
18	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
19	C₈×C₁₉₉	Q₈	1	2	yes	complemented
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	—	4	2	no	—
25	Dic₁₉₉	SD₁₆	1	4	yes	—
26	C₂×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₁₉₉	C₂×SD₁₆	1	2	yes	—
28	C₁₉₉	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	199	1	no	maximal, Sylow
30	C₄⋊Q₈	—	199	2	no	—
31	#106	—	199	1	no	—
32	#106	—	199	2	no	—
33	#106	D₁₉₉	1	1	yes	characteristic
34	C₄×C₈	—	199	1	no	—
35	C₂×Q₈	—	398	2	no	—
36	C₂×C₈	—	199	1	no	—
37	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
38	C₄⋊C₄	—	398	2	no	—
39	C₄⋊C₄	—	199	2	no	—
40	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
41	C₄²	—	199	1	no	—
42	Q₈	—	796	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	398	2	no	—
45	C₂×C₄	—	199	2	no	—
46	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
47	C₈	—	199	2	no	—
48	C₈	Dic₃₉₈	1	2	yes	complemented
49	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	796	2	no	—
52	C₄	—	199	4	no	—
53	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
54	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
55	C₂	#20263954	1	1	yes	characteristic
56	C₂	SD₁₆×D₁₉₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#106×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	#106	D₁₉₉
hypocenter	C₁₉₉	#498
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
16. C₁₉₉⋊C₈	Q₈	semidirect
19. C₈×C₁₉₉	Q₈	semidirect
28. C₁₉₉	#498	semidirect
48. 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
#20263887	2
#20263886	2
C₈×Dic₁₉₉	2
#106×C₁₉₉	2
#498	199