Subgroups — #13092791

Order 1472.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#67274	C₂	1	4	yes	maximal, complemented
3	#10625045	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₂₃	C₂²	1	2	yes	—
6	C₂×D₉₂	—	2	2	no	—
7	C₂×D₉₂	C₂²	1	2	yes	—
8	C₄⋊C₄×C₂₃	—	2	2	no	—
9	C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	#67274	—	2	8	no	—
11	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
12	C₂²×D₂₃	—	2	2	no	—
13	Q₈×C₂₃	—	2	4	no	—
14	C₂×C₄×C₂₃	—	2	2	no	—
15	C₂×C₄×C₂₃	D₄	1	2	yes	—
16	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	D₉₂	—	2	4	no	—
18	D₉₂	—	4	4	no	—
19	C₂₃⋊C₈	D₄	1	4	yes	complemented
20	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
21	C₂×D₂₃	—	4	2	no	—
22	C₂×D₂₃	—	4	4	no	—
23	C₄×C₂₃	—	4	2	no	—
24	C₄×C₂₃	SD₁₆	1	4	yes	complemented
25	C₄×C₂₃	C₂×D₄	1	2	yes	—
26	#492	—	23	1	no	maximal, Sylow
27	C₂×C₂₃	#127	1	1	yes	characteristic
28	C₂×C₂₃	C₂×SD₁₆	1	2	yes	—
29	D₂₃	—	8	2	no	—
30	C₂×SD₁₆	—	23	4	no	—
31	C₄⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
32	#127	—	23	1	no	—
33	C₄×C₈	—	23	1	no	—
34	C₂₃	#492	1	1	yes	characteristic, complemented, Sylow
35	C₂×Q₈	C₂×D₂₃	1	2	yes	—
36	C₂×D₄	—	23	2	no	—
37	C₂×D₄	—	46	2	no	—
38	SD₁₆	—	46	8	no	—
39	C₂×C₈	—	23	2	no	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄²	C₂×D₂₃	1	1	yes	characteristic
42	C₂³	—	46	2	no	—
43	Q₈	—	2	4	no	—
44	D₄	—	46	4	no	—
45	D₄	—	92	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
48	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
49	C₈	—	23	4	no	—
50	C₂²	—	92	2	no	—
51	C₂²	—	92	4	no	—
52	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	#67274	1	4	yes	complemented
55	C₄	D₄×D₂₃	1	2	yes	—
56	C₂	—	184	2	no	—
57	C₂	#10625111	1	1	yes	characteristic
58	C₂	C₂×#67274	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×C₄⋊Q₈	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	#10625045	C₂
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#492
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂₃×C₄⋊Q₈	C₂	semidirect
2. C₂×#67274	C₂	semidirect
4. C₄×C₂₃⋊C₈	C₂	semidirect
19. C₂₃⋊C₈	D₄	semidirect
24. C₄×C₂₃	SD₁₆	semidirect
31. C₄⋊Q₈	D₂₃	semidirect
34. C₂₃	#492	semidirect
54. C₄	#67274	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×C₄⋊Q₈	2
C₂×#67274	2
#10625045	2
C₄×C₂₃⋊C₈	2
#492	23