Subgroups — C₂₃×#12515 #19160141

Order 5888.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2246	C₂	1	4	yes	maximal, complemented
2	C₂₃×#1712	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×D₁₆	—	2	8	no	—
5	C₂₃×#493	—	2	4	no	—
6	C₂₃×#493	C₂²	1	4	yes	—
7	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
8	C₈²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×D₈×C₂₃	—	2	8	no	—
10	C₂×D₈×C₂₃	—	4	8	no	—
11	C₂₃×#127	—	2	4	no	—
12	C₂₃×D₁₆	—	4	16	no	—
13	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
14	C₄×C₈×C₂₃	D₄	1	2	yes	—
15	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×D₄×C₂₃	—	4	4	no	—
17	C₂×D₄×C₂₃	—	4	8	no	—
18	D₈×C₂₃	—	4	16	no	—
19	D₈×C₂₃	—	8	16	no	—
20	C₂×C₈×C₂₃	D₈	1	4	yes	—
21	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
22	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
23	C₁₆×C₂₃	D₈	1	8	yes	complemented
24	#12515	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₂₃	—	8	4	no	—
26	D₄×C₂₃	—	8	8	no	—
27	D₄×C₂₃	—	8	16	no	—
28	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
29	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
30	C₈×C₂₃	D₁₆	1	8	yes	complemented
31	C₈×C₂₃	C₂×D₈	1	4	yes	—
32	#2246	C₂×C₂₃	1	4	yes	complemented
33	#1712	C₂×C₂₃	1	2	yes	complemented
34	C₈×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
35	C₂²×C₂₃	—	16	4	no	—
36	C₂²×C₂₃	—	16	8	no	—
37	C₂²×C₂₃	#493	1	1	yes	characteristic
38	C₄×C₂₃	#493	1	2	yes	—
39	C₄×C₂₃	C₂×D₁₆	1	4	yes	—
40	C₂×D₁₆	—	2	8	no	—
41	#493	—	2	4	no	—
42	#493	C₂²×C₂₃	1	4	yes	—
43	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
44	C₈²	C₂²×C₂₃	1	1	yes	characteristic
45	C₂×C₂₃	—	32	4	no	—
46	C₂×C₂₃	#1712	1	1	yes	characteristic
47	C₂×C₂₃	#2246	1	2	yes	—
48	C₂×D₈	—	2	8	no	—
49	C₂×D₈	—	4	8	no	—
50	#127	—	2	4	no	—
51	D₁₆	—	4	16	no	—
52	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
53	C₄×C₈	D₄×C₂₃	1	2	yes	—
54	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
55	C₂₃	#12515	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×D₄	—	4	4	no	—
57	C₂×D₄	—	4	8	no	—
58	D₈	—	4	16	no	—
59	D₈	—	8	16	no	—
60	C₂×C₈	D₈×C₂₃	1	4	yes	—
61	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
62	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
63	C₁₆	D₈×C₂₃	1	8	yes	complemented
64	C₂³	—	8	4	no	—
65	D₄	—	8	8	no	—
66	D₄	—	8	16	no	—
67	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
68	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
69	C₈	C₂₃×D₁₆	1	8	yes	complemented
70	C₈	C₂×D₈×C₂₃	1	4	yes	—
71	C₂²	—	16	4	no	—
72	C₂²	—	16	8	no	—
73	C₂²	C₂₃×#493	1	1	yes	characteristic
74	C₄	C₂₃×#493	1	2	yes	—
75	C₄	C₂×C₂₃×D₁₆	1	4	yes	—
76	C₂	—	32	4	no	—
77	C₂	C₂₃×#1712	1	1	yes	characteristic
78	C₂	C₂₃×#2246	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₂₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ C₈²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2246	C₂	semidirect
2. C₂₃×#1712	C₂	semidirect
3. C₈×C₁₆×C₂₃	C₂	semidirect
23. C₁₆×C₂₃	D₈	semidirect
24. #12515	C₂₃	direct (N × H)
30. C₈×C₂₃	D₁₆	semidirect
32. #2246	C₂×C₂₃	semidirect
33. #1712	C₂×C₂₃	semidirect
34. C₈×C₁₆	C₂×C₂₃	semidirect
55. C₂₃	#12515	direct (N × H)
63. C₁₆	D₈×C₂₃	semidirect
69. C₈	C₂₃×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2246	2
C₂₃×#1712	2
C₈×C₁₆×C₂₃	2
#12515	23

Subgroups — C23×#12515 #19160141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12515 #19160141