Subgroups — Q₈×C₂₃⋊C₁₆ #15675301

Order 2944.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₃⋊C₁₆)	C₂	1	3	yes	maximal
2	C₄×C₂₃⋊C₁₆	C₂	1	3	yes	maximal
3	C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
5	C₂₃×C₄⋊C₈	C₄	1	3	yes	—
6	C₄×C₈×C₂₃	C₂²	1	3	yes	—
7	C₂×C₂₃⋊C₁₆	C₂²	1	3	yes	—
8	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₂₃	C₈	1	1	yes	characteristic
10	C₂×C₈×C₂₃	C₂×C₄	1	3	yes	—
11	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
12	C₄⋊C₄×C₂₃	C₈	1	3	yes	—
13	C₄²×C₂₃	C₂×C₄	1	3	yes	—
14	C₂₃⋊C₁₆	—	2	3	no	—
15	C₂₃⋊C₁₆	Q₈	1	2	yes	direct product factor
16	Q₈×C₂₃	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₂₃	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₂₃	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
20	C₈×C₂₃	—	2	3	no	—
21	C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
22	C₈×C₂₃	#41	1	1	yes	characteristic
23	Q₈×C₁₆	—	23	1	no	maximal, Sylow
24	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
25	C₄×C₂₃	—	2	3	no	—
26	C₄×C₂₃	C₂×C₁₆	1	6	yes	—
27	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
28	C₄×C₂₃	#131	1	1	yes	characteristic
29	C₈×Q₈	D₂₃	1	1	yes	characteristic
30	C₄⋊C₁₆	—	23	3	no	—
31	C₄×C₁₆	—	23	3	no	—
32	C₂×C₂₃	C₈×Q₈	1	1	yes	characteristic
33	C₂×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
34	C₂×C₂₃	#504	1	1	yes	characteristic
35	C₄×Q₈	Dic₂₃	1	1	yes	characteristic
36	C₂×C₁₆	—	23	1	no	—
37	C₂×C₁₆	—	23	3	no	—
38	C₄⋊C₈	Dic₂₃	1	3	yes	—
39	C₄×C₈	C₂×D₂₃	1	3	yes	—
40	C₂₃	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
41	C₂×Q₈	C₂₃⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₂×Dic₂₃	1	3	yes	—
43	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
44	C₄⋊C₄	C₂₃⋊C₈	1	3	yes	—
45	C₄²	C₂×Dic₂₃	1	3	yes	—
46	C₁₆	—	23	2	no	—
47	C₁₆	—	46	3	no	—
48	Q₈	C₂₃⋊C₁₆	1	4	yes	direct product factor
49	C₂×C₄	C₂×C₂₃⋊C₈	1	3	yes	—
50	C₂×C₄	C₂×C₂₃⋊C₈	1	3	yes	—
51	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
52	C₈	—	2	3	no	—
53	C₈	Q₈×D₂₃	1	1	yes	characteristic
54	C₈	#67292	1	1	yes	characteristic
55	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×C₂₃⋊C₁₆	1	6	yes	—
58	C₄	Q₈×Dic₂₃	1	1	yes	characteristic
59	C₄	#10625119	1	1	yes	characteristic
60	C₂	Q₈×C₂₃⋊C₈	1	1	yes	characteristic
61	C₂	C₂²×C₂₃⋊C₁₆	1	1	yes	characteristic
62	C₂	#13092834	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃
commutator	C₂×C₂₃	C₂²×C₁₆
Fitting	C₈×Q₈×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	C₂²×C₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₈×Q₈	D₂₃
hypocenter	C₂₃	Q₈×C₁₆
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₂₃⋊C₁₆	Q₈	direct (N × H)
16. Q₈×C₂₃	C₁₆	semidirect
40. C₂₃	Q₈×C₁₆	semidirect
48. Q₈	C₂₃⋊C₁₆	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⋊(C₂₃⋊C₁₆)	2
C₄×C₂₃⋊C₁₆	2
C₈×Q₈×C₂₃	2
Q₈×C₁₆	23

Subgroups — Q8×C23⋊C16 #15675301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂₃⋊C₁₆ #15675301