Subgroups — Q₈×#14579261 #35203349

Order 20288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14579261	C₂	1	3	yes	maximal
2	C₄×#14579261	C₂	1	3	yes	maximal
3	C₄×Q₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579261	C₂²	1	3	yes	—
5	C₂×#14579261	C₂²	1	1	yes	characteristic
6	C₂×Q₈×C₃₁₇	C₄	1	1	yes	characteristic
7	C₄⋊C₄×C₃₁₇	C₄	1	3	yes	—
8	C₄²×C₃₁₇	C₂²	1	3	yes	—
9	#14579261	—	2	3	no	—
10	#14579261	Q₈	1	2	yes	direct product factor
11	Q₈×C₃₁₇	C₈	1	4	yes	complemented
12	C₂×C₄×C₃₁₇	C₂×C₄	1	3	yes	—
13	C₂×C₄×C₃₁₇	C₂×C₄	1	3	yes	—
14	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
15	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
16	C₄×C₃₁₇	—	2	3	no	—
17	C₄×C₃₁₇	C₂×C₈	1	6	yes	—
18	C₄×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
19	C₄×C₃₁₇	#41	1	1	yes	characteristic
20	C₂×C₃₁₇	C₄×Q₈	1	1	yes	characteristic
21	C₂×C₃₁₇	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₃₁₇	#131	1	1	yes	characteristic
23	C₃₁₇	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
24	C₈×Q₈	—	317	1	no	maximal, Sylow
25	C₄×Q₈	D₃₁₇	1	1	yes	characteristic
26	C₄⋊C₈	—	317	3	no	—
27	C₄×C₈	—	317	3	no	—
28	C₂×Q₈	Dic₃₁₇	1	1	yes	characteristic
29	C₂×C₈	—	317	1	no	—
30	C₂×C₈	—	317	3	no	—
31	C₄⋊C₄	Dic₃₁₇	1	3	yes	—
32	C₄²	C₂×D₃₁₇	1	3	yes	—
33	Q₈	#14579261	1	4	yes	direct product factor
34	C₂×C₄	C₂×Dic₃₁₇	1	3	yes	—
35	C₂×C₄	C₂×Dic₃₁₇	1	3	yes	—
36	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
37	C₈	—	317	2	no	—
38	C₈	—	634	3	no	—
39	C₂²	C₂²×Dic₃₁₇	1	1	yes	characteristic
40	C₄	—	2	3	no	—
41	C₄	C₂×#14579261	1	6	yes	—
42	C₄	Q₈×D₃₁₇	1	1	yes	characteristic
43	C₄	#19099575	1	1	yes	characteristic
44	C₂	Q₈×Dic₃₁₇	1	1	yes	characteristic
45	C₂	C₂²×#14579261	1	1	yes	characteristic
46	C₂	#23778787	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₁₇
commutator	C₂×C₃₁₇	C₂²×C₈
Fitting	C₄×Q₈×C₃₁₇	C₂
socle, prime radical	C₂²×C₃₁₇	C₂²×C₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₄×Q₈	D₃₁₇
hypocenter	C₃₁₇	C₈×Q₈
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. #14579261	Q₈	direct (N × H)
11. Q₈×C₃₁₇	C₈	semidirect
23. C₃₁₇	C₈×Q₈	semidirect
33. Q₈	#14579261	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⋊#14579261	2
C₄×#14579261	2
C₄×Q₈×C₃₁₇	2
C₈×Q₈	317

Subgroups — Q8×#14579261 #35203349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14579261 #35203349