Subgroups — Q₈⋊(C₁₉₉⋊C₈) #27163749

Order 12736.

70 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	C₄	1	1	yes	characteristic
7	C₄⋊C₄×C₁₉₉	—	2	1	no	—
8	C₄⋊C₄×C₁₉₉	C₄	1	1	yes	characteristic
9	C₄²×C₁₉₉	—	2	1	no	—
10	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₁₉₉⋊C₈	—	2	2	no	—
12	C₁₉₉⋊C₈	—	4	1	no	—
13	Q₈×C₁₉₉	—	2	1	no	—
14	Q₈×C₁₉₉	C₈	1	2	yes	complemented
15	C₂×C₄×C₁₉₉	—	2	1	no	—
16	C₂×C₄×C₁₉₉	—	2	1	no	—
17	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
20	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₉₉	—	2	2	no	—
22	C₄×C₁₉₉	—	2	1	no	—
23	C₄×C₁₉₉	—	4	1	no	—
24	C₄×C₁₉₉	C₂×C₈	1	1	yes	characteristic
25	C₄×C₁₉₉	M₁₆	1	1	yes	characteristic
26	C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
27	C₄×C₁₉₉	Q₁₆	1	1	yes	characteristic
28	C₂×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₁₉₉	#103	1	1	yes	characteristic
30	C₂×C₁₉₉	#104	1	1	yes	characteristic
31	C₁₉₉	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	199	1	no	maximal, Sylow
33	C₄×Q₈	D₁₉₉	1	1	yes	characteristic
34	C₄⋊C₈	—	199	1	no	—
35	C₄×C₈	—	199	1	no	—
36	C₂×Q₈	Dic₁₉₉	1	1	yes	characteristic
37	C₂×C₈	—	398	1	no	—
38	C₂×C₈	—	398	1	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	Dic₁₉₉	1	1	yes	characteristic
41	C₄²	—	2	1	no	—
42	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
43	Q₈	—	2	1	no	—
44	Q₈	C₁₉₉⋊C₈	1	2	yes	complemented
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
50	C₈	—	796	1	no	—
51	C₈	—	398	2	no	—
52	C₂²	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	2	1	no	—
55	C₄	—	4	1	no	—
56	C₄	C₂×C₁₉₉⋊C₈	1	1	yes	characteristic
57	C₄	#15691870	1	1	yes	characteristic
58	C₄	#15691873	1	1	yes	characteristic
59	C₄	#15691876	1	1	yes	characteristic
60	C₂	C₂²⋊(C₁₉₉⋊C₈)	1	1	yes	characteristic
61	C₂	#20263879	1	1	yes	characteristic
62	C₂	#20263882	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₉₉
commutator	C₄×C₁₉₉	C₂×C₈
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	Q₈⋊C₈
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
14. Q₈×C₁₉₉	C₈	semidirect
31. C₁₉₉	Q₈⋊C₈	semidirect
44. Q₈	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×C₁₉₉⋊C₈	2
C₄×Q₈×C₁₉₉	2
Q₈⋊C₈	199

Subgroups — Q8⋊(C199⋊C8) #27163749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈⋊(C₁₉₉⋊C₈) #27163749