Subgroups — #13092813

Order 1472.

98 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
2	#10625040	C₂	1	1	yes	maximal, characteristic
3	#10624990	C₂	1	4	yes	maximal
4	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₂₃	—	2	1	no	—
6	C₂×Dic₄₆	—	2	1	no	—
7	C₄⋊C₄×C₂₃	—	2	1	no	—
8	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₃	—	2	1	no	—
11	C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
13	Q₈×C₂₃	—	4	2	no	—
14	C₂×C₄×C₂₃	—	2	2	no	—
15	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₂₃	—	2	2	no	—
19	Dic₄₆	—	4	2	no	—
20	C₂₃⋊C₈	Q₈	1	4	yes	complemented
21	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
22	C₄×C₂₃	—	4	2	no	—
23	C₄×C₂₃	SD₁₆	1	2	yes	—
24	C₄×C₂₃	SD₁₆	1	2	yes	—
25	C₄×C₂₃	C₂×Q₈	1	2	yes	—
26	Dic₂₃	—	4	2	no	—
27	#498	—	23	1	no	maximal, Sylow
28	C₂×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
29	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
30	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
31	C₄⋊Q₈	—	23	1	no	—
32	C₄⋊Q₈	D₂₃	1	1	yes	characteristic
33	#106	—	23	4	no	—
34	C₄×C₈	—	23	1	no	—
35	C₂₃	#498	1	1	yes	characteristic, complemented, Sylow
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	46	1	no	—
38	C₂×C₈	—	23	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	23	2	no	—
41	C₄⋊C₄	—	46	1	no	—
42	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
43	C₄²	C₂×D₂₃	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	92	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	46	2	no	—
48	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
51	C₈	—	23	4	no	—
52	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	92	2	no	—
55	C₄	#67273	1	2	yes	—
56	C₄	#67274	1	2	yes	—
57	C₄	Q₈×D₂₃	1	2	yes	—
58	C₂	C₂×#67273	1	1	yes	characteristic
59	C₂	C₂×#67274	1	1	yes	characteristic
60	C₂	#10625115	1	1	yes	characteristic

	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, prime radical	C₂²×C₂₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#498
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
20. C₂₃⋊C₈	Q₈	semidirect
35. C₂₃	#498	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
#10625040	2
#10624990	2
C₄×C₂₃⋊C₈	2
#498	23