Subgroups — Q₈×#13096802 #27161205

Order 12608.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13096802	C₂	1	3	yes	maximal
2	C₄×#13096802	C₂	1	3	yes	maximal
3	C₄×Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13096802	C₂²	1	3	yes	—
5	C₂×#13096802	C₂²	1	1	yes	characteristic
6	C₂×Q₈×C₁₉₇	C₄	1	1	yes	characteristic
7	C₄⋊C₄×C₁₉₇	C₄	1	3	yes	—
8	C₄²×C₁₉₇	C₂²	1	3	yes	—
9	#13096802	—	2	3	no	—
10	#13096802	Q₈	1	2	yes	direct product factor
11	Q₈×C₁₉₇	C₈	1	4	yes	complemented
12	C₂×C₄×C₁₉₇	C₂×C₄	1	3	yes	—
13	C₂×C₄×C₁₉₇	C₂×C₄	1	3	yes	—
14	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
15	C₂²×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
16	C₄×C₁₉₇	—	2	3	no	—
17	C₄×C₁₉₇	C₂×C₈	1	6	yes	—
18	C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
19	C₄×C₁₉₇	#41	1	1	yes	characteristic
20	C₂×C₁₉₇	C₄×Q₈	1	1	yes	characteristic
21	C₂×C₁₉₇	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₁₉₇	#131	1	1	yes	characteristic
23	C₁₉₇	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
24	C₈×Q₈	—	197	1	no	maximal, Sylow
25	C₄×Q₈	D₁₉₇	1	1	yes	characteristic
26	C₄⋊C₈	—	197	3	no	—
27	C₄×C₈	—	197	3	no	—
28	C₂×Q₈	Dic₁₉₇	1	1	yes	characteristic
29	C₂×C₈	—	197	3	no	—
30	C₂×C₈	—	197	1	no	—
31	C₄⋊C₄	Dic₁₉₇	1	3	yes	—
32	C₄²	C₂×D₁₉₇	1	3	yes	—
33	Q₈	#13096802	1	4	yes	direct product factor
34	C₂×C₄	C₂×Dic₁₉₇	1	3	yes	—
35	C₂×C₄	C₂×Dic₁₉₇	1	3	yes	—
36	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
37	C₈	—	394	3	no	—
38	C₈	—	197	2	no	—
39	C₂²	C₂²×Dic₁₉₇	1	1	yes	characteristic
40	C₄	—	2	3	no	—
41	C₄	C₂×#13096802	1	6	yes	—
42	C₄	Q₈×D₁₉₇	1	1	yes	characteristic
43	C₄	#15691359	1	1	yes	characteristic
44	C₂	Q₈×Dic₁₉₇	1	1	yes	characteristic
45	C₂	C₂²×#13096802	1	1	yes	characteristic
46	C₂	#20263333	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₇
commutator	C₂×C₁₉₇	C₂²×C₈
Fitting	C₄×Q₈×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂²×C₄
cosocle	C₂×C₄×C₁₉₇	C₂³
hypercenter	C₄×Q₈	D₁₉₇
hypocenter	C₁₉₇	C₈×Q₈
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. #13096802	Q₈	direct (N × H)
11. Q₈×C₁₉₇	C₈	semidirect
23. C₁₉₇	C₈×Q₈	semidirect
33. Q₈	#13096802	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⋊#13096802	2
C₄×#13096802	2
C₄×Q₈×C₁₉₇	2
C₈×Q₈	197

Subgroups — Q8×#13096802 #27161205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13096802 #27161205