Subgroups — Q₈×#15691333 #38744616

Order 25216.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15691333	C₂	1	3	yes	maximal
2	C₄×#15691333	C₂	1	3	yes	maximal
3	C₈×Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#15691333	C₂²	1	3	yes	—
5	C₂×#15691333	C₂²	1	1	yes	characteristic
6	C₄×Q₈×C₁₉₇	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₁₉₇	C₄	1	3	yes	—
8	C₄×C₈×C₁₉₇	C₂²	1	3	yes	—
9	#15691333	—	2	3	no	—
10	#15691333	Q₈	1	2	yes	direct product factor
11	C₂×Q₈×C₁₉₇	C₈	1	1	yes	characteristic
12	C₂×C₈×C₁₉₇	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₇	C₈	1	3	yes	—
15	C₄²×C₁₉₇	C₂×C₄	1	3	yes	—
16	Q₈×C₁₉₇	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₁₉₇	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₁₉₇	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
20	C₈×C₁₉₇	—	2	3	no	—
21	C₈×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
22	C₈×C₁₉₇	#41	1	1	yes	characteristic
23	C₂²×C₁₉₇	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₁₉₇	—	2	3	no	—
25	C₄×C₁₉₇	C₂×C₁₆	1	6	yes	—
26	C₄×C₁₉₇	C₄×Q₈	1	1	yes	characteristic
27	C₄×C₁₉₇	#131	1	1	yes	characteristic
28	C₂×C₁₉₇	C₈×Q₈	1	1	yes	characteristic
29	C₂×C₁₉₇	C₂²×C₁₆	1	1	yes	characteristic
30	C₂×C₁₉₇	#504	1	1	yes	characteristic
31	C₁₉₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
32	Q₈×C₁₆	—	197	1	no	maximal, Sylow
33	C₈×Q₈	D₁₉₇	1	1	yes	characteristic
34	C₄⋊C₁₆	—	197	3	no	—
35	C₄×C₁₆	—	197	3	no	—
36	C₄×Q₈	Dic₁₉₇	1	1	yes	characteristic
37	C₂×C₁₆	—	197	3	no	—
38	C₂×C₁₆	—	197	1	no	—
39	C₄⋊C₈	Dic₁₉₇	1	3	yes	—
40	C₄×C₈	C₂×D₁₉₇	1	3	yes	—
41	C₂×Q₈	#13096802	1	1	yes	characteristic
42	C₂×C₈	C₂×Dic₁₉₇	1	3	yes	—
43	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
44	C₄⋊C₄	#13096802	1	3	yes	—
45	C₄²	C₂×Dic₁₉₇	1	3	yes	—
46	C₁₆	—	394	3	no	—
47	C₁₆	—	197	2	no	—
48	Q₈	#15691333	1	4	yes	direct product factor
49	C₂×C₄	C₂×#13096802	1	3	yes	—
50	C₂×C₄	C₂×#13096802	1	3	yes	—
51	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
52	C₈	—	2	3	no	—
53	C₈	Q₈×D₁₉₇	1	1	yes	characteristic
54	C₈	#15691359	1	1	yes	characteristic
55	C₂²	C₂²×#13096802	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×#15691333	1	6	yes	—
58	C₄	Q₈×Dic₁₉₇	1	1	yes	characteristic
59	C₄	#20263333	1	1	yes	characteristic
60	C₂	Q₈×#13096802	1	1	yes	characteristic
61	C₂	C₂²×#15691333	1	1	yes	characteristic
62	C₂	#27161496	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₉₇
commutator	C₂×C₁₉₇	C₂²×C₁₆
Fitting	C₈×Q₈×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂²×C₈
cosocle	C₂×C₈×C₁₉₇	C₂³
hypercenter	C₈×Q₈	D₁₉₇
hypocenter	C₁₉₇	Q₈×C₁₆
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. #15691333	Q₈	direct (N × H)
16. Q₈×C₁₉₇	C₁₆	semidirect
31. C₁₉₇	Q₈×C₁₆	semidirect
48. Q₈	#15691333	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⋊#15691333	2
C₄×#15691333	2
C₈×Q₈×C₁₉₇	2
Q₈×C₁₆	197

Subgroups — Q8×#15691333 #38744616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#15691333 #38744616