Subgroups — #27163764

Order 12736.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉₉⋊C₈	—	2	1	no	—
5	C₂×C₁₉₉⋊C₈	—	2	1	no	—
6	C₂×Q₈×C₁₉₉	—	2	1	no	—
7	C₄⋊C₄×C₁₉₉	—	2	1	no	—
8	C₄⋊C₄×C₁₉₉	C₄	1	2	yes	—
9	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₁₉₉⋊C₈	—	4	1	no	—
11	C₁₉₉⋊C₈	—	4	1	no	—
12	Q₈×C₁₉₉	—	4	1	no	—
13	Q₈×C₁₉₉	—	4	1	no	—
14	C₂×C₄×C₁₉₉	—	2	2	no	—
15	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
18	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₉	—	2	1	no	—
20	C₄×C₁₉₉	—	4	2	no	—
21	C₄×C₁₉₉	D₈	1	1	yes	characteristic
22	C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
23	C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
24	C₄×C₁₉₉	Q₁₆	1	1	yes	characteristic
25	C₂×C₁₉₉	#101	1	1	yes	characteristic
26	C₂×C₁₉₉	#102	1	1	yes	characteristic
27	C₂×C₁₉₉	#103	1	1	yes	characteristic
28	C₁₉₉	#332	1	1	yes	characteristic, complemented, Sylow
29	#332	—	199	1	no	maximal, Sylow
30	C₄⋊Q₈	D₁₉₉	1	1	yes	characteristic
31	C₄⋊C₈	—	199	1	no	—
32	C₄⋊C₈	—	199	1	no	—
33	C₂×Q₈	—	2	1	no	—
34	C₂×C₈	—	398	1	no	—
35	C₂×C₈	—	398	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	Dic₁₉₉	1	2	yes	—
38	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
39	Q₈	—	4	1	no	—
40	Q₈	—	4	1	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
43	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
44	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
45	C₈	—	796	1	no	—
46	C₈	—	796	1	no	—
47	C₂²	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	4	2	no	—
50	C₄	#15691871	1	1	yes	characteristic
51	C₄	#15691873	1	1	yes	characteristic
52	C₄	#15691874	1	1	yes	characteristic
53	C₄	#15691876	1	1	yes	characteristic
54	C₂	#20263874	1	1	yes	characteristic
55	C₂	#20263876	1	1	yes	characteristic
56	C₂	#20263879	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₁₉₉
commutator	C₂×C₄×C₁₉₉	C₂×C₄
Frattini	C₄²	C₂×D₁₉₉
Fitting	C₄⋊Q₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂²⋊C₄
cosocle	C₄²×C₁₉₉	C₂²
hypercenter	C₄⋊Q₈	D₁₉₉
hypocenter	C₁₉₉	#332
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
28. C₁₉₉	#332	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⋊Q₈×C₁₉₉	2
#332	199