Subgroups — Q₈×C₁₉₉⋊C₈ #27164144

Order 12736.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₉₉⋊C₈)	C₂	1	3	yes	maximal
2	C₄×C₁₉₉⋊C₈	C₂	1	3	yes	maximal
3	C₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉₉⋊C₈	C₂²	1	3	yes	—
5	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
6	C₂×Q₈×C₁₉₉	C₄	1	1	yes	characteristic
7	C₄⋊C₄×C₁₉₉	C₄	1	3	yes	—
8	C₄²×C₁₉₉	C₂²	1	3	yes	—
9	C₁₉₉⋊C₈	—	2	3	no	—
10	C₁₉₉⋊C₈	Q₈	1	2	yes	direct product factor
11	Q₈×C₁₉₉	C₈	1	4	yes	complemented
12	C₂×C₄×C₁₉₉	C₂×C₄	1	3	yes	—
13	C₂×C₄×C₁₉₉	C₂×C₄	1	3	yes	—
14	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
15	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
16	C₄×C₁₉₉	—	2	3	no	—
17	C₄×C₁₉₉	C₂×C₈	1	6	yes	—
18	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
19	C₄×C₁₉₉	#41	1	1	yes	characteristic
20	C₂×C₁₉₉	C₄×Q₈	1	1	yes	characteristic
21	C₂×C₁₉₉	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₁₉₉	#131	1	1	yes	characteristic
23	C₁₉₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
24	C₈×Q₈	—	199	1	no	maximal, Sylow
25	C₄×Q₈	D₁₉₉	1	1	yes	characteristic
26	C₄⋊C₈	—	199	3	no	—
27	C₄×C₈	—	199	3	no	—
28	C₂×Q₈	Dic₁₉₉	1	1	yes	characteristic
29	C₂×C₈	—	199	3	no	—
30	C₂×C₈	—	199	1	no	—
31	C₄⋊C₄	Dic₁₉₉	1	3	yes	—
32	C₄²	C₂×D₁₉₉	1	3	yes	—
33	Q₈	C₁₉₉⋊C₈	1	4	yes	direct product factor
34	C₂×C₄	C₂×Dic₁₉₉	1	3	yes	—
35	C₂×C₄	C₂×Dic₁₉₉	1	3	yes	—
36	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
37	C₈	—	398	3	no	—
38	C₈	—	199	2	no	—
39	C₂²	C₂²×Dic₁₉₉	1	1	yes	characteristic
40	C₄	—	2	3	no	—
41	C₄	C₂×C₁₉₉⋊C₈	1	6	yes	—
42	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
43	C₄	#15691887	1	1	yes	characteristic
44	C₂	Q₈×Dic₁₉₉	1	1	yes	characteristic
45	C₂	C₂²×C₁₉₉⋊C₈	1	1	yes	characteristic
46	C₂	#20263962	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₉
commutator	C₂×C₁₉₉	C₂²×C₈
Fitting	C₄×Q₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂²×C₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₄×Q₈	D₁₉₉
hypocenter	C₁₉₉	C₈×Q₈
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. C₁₉₉⋊C₈	Q₈	direct (N × H)
11. Q₈×C₁₉₉	C₈	semidirect
23. C₁₉₉	C₈×Q₈	semidirect
33. Q₈	C₁₉₉⋊C₈	direct (N × H)

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
C₈×Q₈	199

Subgroups — Q8×C199⋊C8 #27164144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₉₉⋊C₈ #27164144