Subgroups — #15671350

Order 2944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092199	C₂	1	2	yes	maximal
2	C₂₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
4	C₂²×C₈×C₂₃	C₄	1	1	yes	characteristic
5	C₂₃×#117	C₄	1	1	yes	characteristic
6	C₂₃×C₄⋊C₈	—	2	2	no	—
7	#10624994	—	2	2	no	—
8	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
9	M₁₆×C₂₃	—	4	1	no	—
10	C₂×C₈×C₂₃	—	2	1	no	—
11	C₂×C₈×C₂₃	—	4	1	no	—
12	C₂×C₈×C₂₃	D₄	1	2	yes	—
13	C₄⋊C₄×C₂₃	—	2	1	no	—
14	C₂²⋊C₄×C₂₃	C₈	1	2	yes	—
15	C₄²×C₂₃	—	2	1	no	—
16	C₂₃⋊C₁₆	—	4	2	no	—
17	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₂₃	—	2	1	no	—
19	C₂×C₄×C₂₃	—	2	1	no	—
20	C₂×C₄×C₂₃	—	2	2	no	—
21	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	M₁₆	1	1	yes	characteristic
23	C₈×C₂₃	—	2	2	no	—
24	C₈×C₂₃	—	4	1	no	—
25	#1328	—	23	1	no	maximal, Sylow
26	C₂²×C₂₃	—	2	1	no	—
27	C₂²×C₂₃	—	4	1	no	—
28	C₂²×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
29	C₄×C₂₃	—	2	1	no	—
30	C₄×C₂₃	—	4	2	no	—
31	C₄×C₂₃	#99	1	1	yes	characteristic
32	C₄×C₂₃	#101	1	1	yes	characteristic
33	#424	D₂₃	1	1	yes	characteristic
34	#349	—	23	2	no	—
35	C₂×C₂₃	—	2	1	no	—
36	C₂×C₂₃	—	4	1	no	—
37	C₂×C₂₃	#324	1	1	yes	characteristic
38	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
39	C₂²×C₈	Dic₂₃	1	1	yes	characteristic
40	#117	Dic₂₃	1	1	yes	characteristic
41	M₃₂	—	46	2	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₂₃	#1328	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
45	M₁₆	—	4	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
49	C₄⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	C₂₃⋊C₈	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₁₆	—	92	2	no	—
53	C₂³	C₂×C₂₃⋊C₈	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	#67267	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	4	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#10625016	1	1	yes	characteristic
67	C₄	#10625018	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#13092143	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10625016
commutator, prime radical	C₂³×C₂₃	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₂₃
Fitting	C₂₃×#424	C₂
socle	C₂×C₂₃	#324
cosocle	C₂×M₁₆×C₂₃	C₂²
hypercenter	#424	D₂₃
hypocenter	C₂₃	#1328
LCS	G ▷ C₂³×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #424

Complements

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

N	H ≅ G/N	type
43. C₂₃	#1328	semidirect

Maximal subgroups

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

subgroup	index
#13092199	2
C₂₃×#424	2
#1328	23