Subgroups — #13092548

Order 1472.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
3	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10624985	C₂	1	2	yes	maximal
5	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₂₃	C₄	1	1	yes	characteristic
7	C₄⋊C₄×C₂₃	C₄	1	2	yes	—
8	C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
9	C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₄²×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	Q₈×C₂₃	—	2	2	no	—
15	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂₃⋊C₈	—	2	1	no	—
21	C₂₃⋊C₈	—	2	2	no	—
22	C₂₃⋊C₈	Q₈	1	2	yes	complemented
23	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₃	—	2	2	no	—
25	C₄×C₂₃	—	2	1	no	—
26	C₄×C₂₃	—	2	2	no	—
27	C₄×C₂₃	M₁₆	1	2	yes	—
28	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₂₃	#41	1	1	yes	characteristic
30	#446	—	23	1	no	maximal, Sylow
31	C₂×C₂₃	C₄×Q₈	1	1	yes	characteristic
32	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
33	C₂×C₂₃	#131	1	1	yes	characteristic
34	C₄×Q₈	D₂₃	1	1	yes	characteristic
35	C₄⋊C₈	—	23	1	no	—
36	C₄⋊C₈	—	23	2	no	—
37	#97	—	23	2	no	—
38	C₄×C₈	—	23	1	no	—
39	C₂₃	#446	1	1	yes	characteristic, complemented, Sylow
40	C₂×Q₈	Dic₂₃	1	1	yes	characteristic
41	C₂×C₈	—	23	1	no	—
42	C₂×C₈	—	23	1	no	—
43	C₂×C₈	—	23	2	no	—
44	C₄⋊C₄	Dic₂₃	1	2	yes	—
45	C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
46	C₄²	C₂×D₂₃	1	1	yes	characteristic
47	C₄²	C₂×D₂₃	1	2	yes	—
48	Q₈	—	2	2	no	—
49	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
50	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
51	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
52	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
54	C₈	—	23	2	no	—
55	C₈	—	46	1	no	—
56	C₈	—	46	2	no	—
57	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	1	no	—
60	C₄	—	2	2	no	—
61	C₄	#67267	1	2	yes	—
62	C₄	Q₈×D₂₃	1	1	yes	characteristic
63	C₄	#67292	1	1	yes	characteristic
64	C₂	C₂×#67267	1	1	yes	characteristic
65	C₂	Q₈×Dic₂₃	1	1	yes	characteristic
66	C₂	#10625119	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle, prime radical	C₂²×C₂₃	C₂²×C₄
Fitting	C₄×Q₈×C₂₃	C₂
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₄×Q₈	D₂₃
hypocenter	C₂₃	#446
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
22. C₂₃⋊C₈	Q₈	semidirect
39. C₂₃	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₂₃	2
C₄⋊(C₂₃⋊C₈)	2
#10624985	2
C₄×C₂₃⋊C₈	2
#446	23