Subgroups — Q₈⋊(C₄₇⋊C₈) #15688854

Order 3008.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₄₇	C₄	1	1	yes	characteristic
5	C₄⋊C₄×C₄₇	—	2	1	no	—
6	C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
7	C₄²×C₄₇	—	2	1	no	—
8	C₄²×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄₇⋊C₈	—	2	1	no	—
10	C₂×C₄₇⋊C₈	—	2	1	no	—
11	Q₈×C₄₇	—	2	1	no	—
12	Q₈×C₄₇	C₈	1	2	yes	complemented
13	C₂×C₄×C₄₇	—	2	1	no	—
14	C₂×C₄×C₄₇	—	2	1	no	—
15	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
18	C₄₇⋊C₈	—	2	2	no	—
19	C₄₇⋊C₈	—	4	1	no	—
20	C₂²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₄₇	—	2	2	no	—
22	C₄×C₄₇	—	2	1	no	—
23	C₄×C₄₇	—	4	1	no	—
24	C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
25	C₄×C₄₇	M₁₆	1	1	yes	characteristic
26	C₄×C₄₇	SD₁₆	1	1	yes	characteristic
27	C₄×C₄₇	Q₁₆	1	1	yes	characteristic
28	C₂×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₄₇	#103	1	1	yes	characteristic
30	C₂×C₄₇	#104	1	1	yes	characteristic
31	Q₈⋊C₈	—	47	1	no	maximal, Sylow
32	C₄₇	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
33	C₄×Q₈	D₄₇	1	1	yes	characteristic
34	C₄⋊C₈	—	47	1	no	—
35	C₄×C₈	—	47	1	no	—
36	C₂×Q₈	Dic₄₇	1	1	yes	characteristic
37	C₂×C₈	—	94	1	no	—
38	C₂×C₈	—	94	1	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
41	C₄²	—	2	1	no	—
42	C₄²	C₂×D₄₇	1	1	yes	characteristic
43	Q₈	—	2	1	no	—
44	Q₈	C₄₇⋊C₈	1	2	yes	complemented
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
50	C₈	—	94	2	no	—
51	C₈	—	188	1	no	—
52	C₂²	C₂²⋊Dic₄₇	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	2	1	no	—
55	C₄	—	4	1	no	—
56	C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
57	C₄	#10625329	1	1	yes	characteristic
58	C₄	#10625336	1	1	yes	characteristic
59	C₄	#10625337	1	1	yes	characteristic
60	C₂	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
61	C₂	#13093948	1	1	yes	characteristic
62	C₂	#13093951	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂×C₈
Frattini	C₄²	C₂×D₄₇
Fitting	C₄×Q₈×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₂²⋊C₄
cosocle	C₄²×C₄₇	C₂²
hypercenter	C₄×Q₈	D₄₇
hypocenter	C₄₇	Q₈⋊C₈
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
12. Q₈×C₄₇	C₈	semidirect
32. C₄₇	Q₈⋊C₈	semidirect
44. Q₈	C₄₇⋊C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄⋊(C₄₇⋊C₈)	2
C₄×C₄₇⋊C₈	2
C₄×Q₈×C₄₇	2
Q₈⋊C₈	47

Subgroups — Q8⋊(C47⋊C8) #15688854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈⋊(C₄₇⋊C₈) #15688854