Subgroups — #15671432

Order 2944.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092195	C₂	1	2	yes	maximal
2	C₂₃×#491	C₂	1	1	yes	maximal, characteristic
3	C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
4	C₂₃×#125	C₄	1	1	yes	characteristic
5	C₂₃×#103	—	2	2	no	—
6	C₂₃×#97	C₂²	1	1	yes	characteristic
7	#10624994	—	2	2	no	—
8	C₂×Q₈×C₂₃	—	2	1	no	—
9	C₂×C₈×C₂₃	D₄	1	2	yes	—
10	C₄⋊C₄×C₂₃	—	2	1	no	—
11	C₄⋊C₄×C₂₃	—	4	1	no	—
12	C₄⋊C₄×C₂₃	—	4	1	no	—
13	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂₃⋊C₁₆	—	2	4	no	—
15	Q₈×C₂₃	—	4	2	no	—
16	C₂×C₄×C₂₃	—	2	1	no	—
17	C₂×C₄×C₂₃	—	4	1	no	—
18	C₂×C₄×C₂₃	—	4	1	no	—
19	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₂₃	—	2	2	no	—
21	#1356	—	23	1	no	maximal, Sylow
22	C₂²×C₂₃	#100	1	1	yes	characteristic
23	C₄×C₂₃	—	4	1	no	—
24	C₄×C₂₃	—	8	1	no	—
25	C₄×C₂₃	—	8	1	no	—
26	C₄×C₂₃	#104	1	2	yes	—
27	#491	D₂₃	1	1	yes	characteristic
28	#347	—	23	2	no	—
29	C₂×C₂₃	—	2	1	no	—
30	C₂×C₂₃	#329	1	1	yes	characteristic
31	C₄⋊Q₈	Dic₂₃	1	1	yes	characteristic
32	#125	Dic₂₃	1	1	yes	characteristic
33	M₃₂	—	46	2	no	—
34	#103	—	2	2	no	—
35	#97	C₂×D₂₃	1	1	yes	characteristic
36	C₂₃	#1356	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₈	—	2	1	no	—
38	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	4	1	no	—
41	C₄⋊C₄	—	4	1	no	—
42	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
43	C₁₆	—	46	4	no	—
44	Q₈	—	4	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₂²	#10625015	1	1	yes	characteristic
51	C₄	—	4	1	no	—
52	C₄	—	8	1	no	—
53	C₄	—	8	1	no	—
54	C₄	#10625019	1	2	yes	—
55	C₂	—	2	1	no	—
56	C₂	#13092157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092157
commutator	C₄²×C₂₃	C₂×C₄
Frattini	#97	C₂×D₂₃
Fitting	C₂₃×#491	C₂
socle	C₂×C₂₃	#329
prime radical	C₂²×C₂₃	#100
cosocle	C₂₃×#97	C₂²
hypercenter	#491	D₂₃
hypocenter	C₂₃	#1356
LCS	G ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #491

Complements

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

N	H ≅ G/N	type
36. C₂₃	#1356	semidirect

Maximal subgroups

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

subgroup	index
#13092195	2
C₂₃×#491	2
#1356	23