Subgroups — #27163769

Order 12736.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₉₉⋊C₈)	C₂	1	2	yes	maximal
2	C₄×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉₉⋊C₈	—	2	2	no	—
4	C₂×C₈×C₁₉₉	C₄	1	1	yes	characteristic
5	C₂×C₈×C₁₉₉	C₄	1	1	yes	characteristic
6	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
7	C₁₉₉⋊C₈	—	4	2	no	—
8	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
10	C₂×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
11	C₈×C₁₉₉	—	2	1	no	—
12	C₈×C₁₉₉	C₈	1	2	yes	complemented
13	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₁₉₉	—	2	1	no	—
15	C₄×C₁₉₉	C₂×C₈	1	1	yes	characteristic
16	C₄×C₁₉₉	M₁₆	1	1	yes	characteristic
17	C₄×C₁₉₉	SD₁₆	1	2	yes	—
18	C₂×C₁₉₉	C₄⋊C₈	1	1	yes	characteristic
19	C₂×C₁₉₉	#106	1	1	yes	characteristic
20	C₂×C₁₉₉	#108	1	1	yes	characteristic
21	C₁₉₉	#334	1	1	yes	characteristic, complemented, Sylow
22	#334	—	199	1	no	maximal, Sylow
23	C₄⋊C₈	—	199	2	no	—
24	C₄×C₈	D₁₉₉	1	1	yes	characteristic
25	C₂×C₈	—	398	2	no	—
26	C₂×C₈	Dic₁₉₉	1	1	yes	characteristic
27	C₂×C₈	Dic₁₉₉	1	1	yes	characteristic
28	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
29	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
30	C₂×C₄	D₇₉₆	1	1	yes	characteristic
31	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	—	796	2	no	—
34	C₈	C₁₉₉⋊C₈	1	2	yes	complemented
35	C₂²	C₄⋊Dic₁₉₉	1	1	yes	characteristic
36	C₄	—	2	1	no	—
37	C₄	C₂×C₁₉₉⋊C₈	1	1	yes	characteristic
38	C₄	#15691870	1	1	yes	characteristic
39	C₄	#15691875	1	2	yes	—
40	C₂	C₄⋊(C₁₉₉⋊C₈)	1	1	yes	characteristic
41	C₂	#20263887	1	1	yes	characteristic
42	C₂	#20263891	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₇₉₆
commutator	C₄×C₁₉₉	C₂×C₈
Frattini	C₄²	C₂×D₁₉₉
Fitting	C₄×C₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₄⋊C₄
cosocle	C₄²×C₁₉₉	C₂²
hypercenter	C₄×C₈	D₁₉₉
hypocenter	C₁₉₉	#334
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₁₉₉	C₈	semidirect
21. C₁₉₉	#334	semidirect
34. C₈	C₁₉₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₉₉⋊C₈)	2
C₄×C₈×C₁₉₉	2
#334	199