Subgroups — Q₈×C₅₂₃⋊C₈ #45690331

Order 33472.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₅₂₃⋊C₈)	C₂	1	3	yes	maximal
2	C₄×C₅₂₃⋊C₈	C₂	1	3	yes	maximal
3	C₄×Q₈×C₅₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₂₃⋊C₈	C₂²	1	3	yes	—
5	C₂×C₅₂₃⋊C₈	C₂²	1	1	yes	characteristic
6	C₂×Q₈×C₅₂₃	C₄	1	1	yes	characteristic
7	C₄⋊C₄×C₅₂₃	C₄	1	3	yes	—
8	C₄²×C₅₂₃	C₂²	1	3	yes	—
9	C₅₂₃⋊C₈	—	2	3	no	—
10	C₅₂₃⋊C₈	Q₈	1	2	yes	direct product factor
11	Q₈×C₅₂₃	C₈	1	4	yes	complemented
12	C₂×C₄×C₅₂₃	C₂×C₄	1	3	yes	—
13	C₂×C₄×C₅₂₃	C₂×C₄	1	3	yes	—
14	C₂×C₄×C₅₂₃	C₂³	1	1	yes	characteristic
15	C₂²×C₅₂₃	C₂²×C₄	1	1	yes	characteristic
16	C₄×C₅₂₃	—	2	3	no	—
17	C₄×C₅₂₃	C₂×C₈	1	6	yes	—
18	C₄×C₅₂₃	C₂×Q₈	1	1	yes	characteristic
19	C₄×C₅₂₃	#41	1	1	yes	characteristic
20	C₂×C₅₂₃	C₄×Q₈	1	1	yes	characteristic
21	C₂×C₅₂₃	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₅₂₃	#131	1	1	yes	characteristic
23	C₅₂₃	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
24	C₈×Q₈	—	523	1	no	maximal, Sylow
25	C₄×Q₈	D₅₂₃	1	1	yes	characteristic
26	C₄⋊C₈	—	523	3	no	—
27	C₄×C₈	—	523	3	no	—
28	C₂×Q₈	Dic₅₂₃	1	1	yes	characteristic
29	C₂×C₈	—	523	1	no	—
30	C₂×C₈	—	523	3	no	—
31	C₄⋊C₄	Dic₅₂₃	1	3	yes	—
32	C₄²	C₂×D₅₂₃	1	3	yes	—
33	Q₈	C₅₂₃⋊C₈	1	4	yes	direct product factor
34	C₂×C₄	C₂×Dic₅₂₃	1	3	yes	—
35	C₂×C₄	C₂×Dic₅₂₃	1	3	yes	—
36	C₂×C₄	C₂²×D₅₂₃	1	1	yes	characteristic
37	C₈	—	523	2	no	—
38	C₈	—	1046	3	no	—
39	C₂²	C₂²×Dic₅₂₃	1	1	yes	characteristic
40	C₄	—	2	3	no	—
41	C₄	C₂×C₅₂₃⋊C₈	1	6	yes	—
42	C₄	Q₈×D₅₂₃	1	1	yes	characteristic
43	C₄	#22552642	1	1	yes	characteristic
44	C₂	Q₈×Dic₅₂₃	1	1	yes	characteristic
45	C₂	C₂²×C₅₂₃⋊C₈	1	1	yes	characteristic
46	C₂	#30660380	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₂₃
commutator	C₂×C₅₂₃	C₂²×C₈
Fitting	C₄×Q₈×C₅₂₃	C₂
socle, prime radical	C₂²×C₅₂₃	C₂²×C₄
cosocle	C₂×C₄×C₅₂₃	C₂³
hypercenter	C₄×Q₈	D₅₂₃
hypocenter	C₅₂₃	C₈×Q₈
LCS	G ▷ C₂×C₅₂₃ ▷ C₅₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. C₅₂₃⋊C₈	Q₈	direct (N × H)
11. Q₈×C₅₂₃	C₈	semidirect
23. C₅₂₃	C₈×Q₈	semidirect
33. Q₈	C₅₂₃⋊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⋊(C₅₂₃⋊C₈)	2
C₄×C₅₂₃⋊C₈	2
C₄×Q₈×C₅₂₃	2
C₈×Q₈	523

Subgroups — Q8×C523⋊C8 #45690331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₅₂₃⋊C₈ #45690331