Subgroups — C₂₃×#12522 #19160148

Order 5888.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2254	C₂	1	2	yes	maximal
2	C₂₃×#2252	C₂	1	2	yes	maximal
3	C₂₃×#1710	C₂	1	2	yes	maximal
4	C₈×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#500	C₂²	1	4	yes	—
6	C₂₃×#487	—	2	4	no	—
7	C₂₃×#367	—	2	4	no	—
8	C₂₃×#366	—	2	4	no	—
9	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₈²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×C₄⋊Q₈	—	2	4	no	—
13	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
14	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
15	C₂₃×#107	—	2	4	no	—
16	C₂₃×#107	—	2	4	no	—
17	C₂₃×#103	—	4	8	no	—
18	C₄×C₈×C₂₃	D₄	1	2	yes	—
19	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×Q₈×C₂₃	—	4	4	no	—
21	C₂×C₈×C₂₃	D₈	1	4	yes	—
22	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₂₃	—	4	4	no	—
25	C₄⋊C₄×C₂₃	—	4	4	no	—
26	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
27	C₁₆×C₂₃	Q₁₆	1	4	yes	—
28	C₁₆×C₂₃	Q₁₆	1	4	yes	—
29	#12522	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	Q₈×C₂₃	—	8	8	no	—
31	C₂×C₄×C₂₃	—	8	4	no	—
32	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
34	C₈×C₂₃	—	2	4	no	—
35	C₈×C₂₃	C₂×Q₁₆	1	2	yes	—
36	C₈×C₂₃	C₂×Q₁₆	1	2	yes	—
37	#2254	C₂×C₂₃	1	2	yes	—
38	#2252	C₂×C₂₃	1	2	yes	—
39	#1710	C₂×C₂₃	1	2	yes	—
40	C₈×C₁₆	C₂×C₂₃	1	1	yes	characteristic
41	C₂²×C₂₃	#493	1	1	yes	characteristic
42	C₄×C₂₃	—	16	4	no	—
43	C₄×C₂₃	#494	1	1	yes	characteristic
44	C₄×C₂₃	#494	1	1	yes	characteristic
45	C₄×C₂₃	#508	1	4	yes	—
46	#500	C₂²×C₂₃	1	4	yes	—
47	#487	—	2	4	no	—
48	#367	—	2	4	no	—
49	#366	—	2	4	no	—
50	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
51	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
52	C₈²	C₂²×C₂₃	1	1	yes	characteristic
53	C₂×C₂₃	#1713	1	1	yes	characteristic
54	C₂×C₂₃	#2249	1	2	yes	—
55	C₄⋊Q₈	—	2	4	no	—
56	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
57	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
58	#107	—	2	4	no	—
59	#107	—	2	4	no	—
60	#103	—	4	8	no	—
61	C₄×C₈	D₄×C₂₃	1	2	yes	—
62	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
63	C₂₃	#12522	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×Q₈	—	4	4	no	—
65	C₂×C₈	D₈×C₂₃	1	4	yes	—
66	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
67	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
68	C₄⋊C₄	—	4	4	no	—
69	C₄⋊C₄	—	4	4	no	—
70	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
71	C₁₆	Q₁₆×C₂₃	1	4	yes	—
72	C₁₆	Q₁₆×C₂₃	1	4	yes	—
73	Q₈	—	8	8	no	—
74	C₂×C₄	—	8	4	no	—
75	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
76	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
77	C₈	—	2	4	no	—
78	C₈	C₂×Q₁₆×C₂₃	1	2	yes	—
79	C₈	C₂×Q₁₆×C₂₃	1	2	yes	—
80	C₂²	C₂₃×#493	1	1	yes	characteristic
81	C₄	—	16	4	no	—
82	C₄	C₂₃×#494	1	1	yes	characteristic
83	C₄	C₂₃×#494	1	1	yes	characteristic
84	C₄	C₂₃×#508	1	4	yes	—
85	C₂	C₂₃×#1713	1	1	yes	characteristic
86	C₂	C₂₃×#2249	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	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
29. #12522	C₂₃	direct (N × H)
63. C₂₃	#12522	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2254	2
C₂₃×#2252	2
C₂₃×#1710	2
C₈×C₁₆×C₂₃	2
#12522	23

Subgroups — C23×#12522 #19160148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12522 #19160148