Subgroups — #13092373

Order 1472.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#67267	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10624985	C₂	1	2	yes	maximal, complemented
5	C₄×C₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₄⋊C₄×C₂₃	C₄	1	2	yes	—
8	C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
9	C₄²×C₂₃	C₂²	1	2	yes	—
10	#67267	C₄	1	4	yes	complemented
11	C₂×C₂₃⋊C₈	C₄	1	4	yes	complemented
12	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂³×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	4	yes	—
19	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₂₃⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₂₃	—	2	2	no	—
25	C₂²×C₂₃	C₄²	1	2	yes	—
26	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₂₃	—	2	4	no	—
28	C₄×C₂₃	C₄²	1	2	yes	—
29	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	#405	—	23	1	no	maximal, Sylow
32	C₂×C₂₃	—	2	2	no	—
33	C₂×C₂₃	C₂×C₄²	1	1	yes	characteristic
34	C₂×C₂₃	#131	1	2	yes	—
35	C₂×M₁₆	—	23	1	no	—
36	C₂²×C₈	—	23	1	no	—
37	#117	D₂₃	1	1	yes	characteristic
38	#97	—	23	2	no	—
39	C₄×C₈	—	23	2	no	—
40	C₂₃	#405	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
42	M₁₆	—	23	4	no	—
43	C₂×C₈	—	23	4	no	—
44	C₂×C₈	—	23	1	no	—
45	C₂×C₈	—	23	2	no	—
46	C₂×C₈	—	23	1	no	—
47	C₄⋊C₄	Dic₂₃	1	2	yes	—
48	C₂²⋊C₄	Dic₂₃	1	2	yes	—
49	C₄²	C₂×D₂₃	1	2	yes	—
50	C₂³	C₂×Dic₂₃	1	1	yes	characteristic
51	C₂×C₄	C₄×D₂₃	1	2	yes	—
52	C₂×C₄	C₄×D₂₃	1	2	yes	—
53	C₂×C₄	C₂×Dic₂₃	1	4	yes	—
54	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
56	C₈	—	23	2	no	—
57	C₈	—	23	2	no	—
58	C₈	—	23	4	no	—
59	C₂²	—	2	2	no	—
60	C₂²	C₄×Dic₂₃	1	2	yes	—
61	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
62	C₄	—	2	4	no	—
63	C₄	C₄×Dic₂₃	1	2	yes	—
64	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
65	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	C₂×C₄×Dic₂₃	1	1	yes	characteristic
68	C₂	#10625119	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator	C₂×C₂₃	C₂×C₄²
Fitting	C₂₃×#117	C₂
socle	C₂²×C₂₃	C₂²×C₄
prime radical	C₂³×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	#117	D₂₃
hypocenter	C₂₃	#405
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #10624985	C₂	semidirect
5. C₄×C₂₃⋊C₈	C₂	semidirect
10. #67267	C₄	semidirect
11. C₂×C₂₃⋊C₈	C₄	semidirect
40. C₂₃	#405	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#117	2
C₂×#67267	2
C₂²×C₂₃⋊C₈	2
#10624985	2
C₄×C₂₃⋊C₈	2
#405	23