Subgroups — C₂₃×#12514 #19160140

Order 5888.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2241	C₂	1	4	yes	maximal
2	C₂₃×#1713	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1710	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1311	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#500	—	2	2	no	—
6	C₂₃×#500	C₂²	1	2	yes	—
7	C₂₃×#494	C₂²	1	2	yes	—
8	C₂₃×#487	—	2	2	no	—
9	C₂₃×#358	—	2	8	no	—
10	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
11	C₈²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₂₃	—	2	4	no	—
13	C₂₃×C₄⋊Q₈	—	2	2	no	—
14	C₂₃×C₄⋊Q₈	—	2	2	no	—
15	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
16	C₂₃×#107	—	2	4	no	—
17	C₂₃×#107	—	4	4	no	—
18	C₂₃×#103	—	4	4	no	—
19	C₄×C₈×C₂₃	D₄	1	2	yes	—
20	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₂₃	—	4	2	no	—
22	C₂×Q₈×C₂₃	—	4	2	no	—
23	Q₁₆×C₂₃	—	4	8	no	—
24	C₂×C₈×C₂₃	D₈	1	4	yes	—
25	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
26	C₄⋊C₄×C₂₃	—	4	2	no	—
27	C₄⋊C₄×C₂₃	—	4	2	no	—
28	C₄⋊C₄×C₂₃	—	4	4	no	—
29	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₁₆×C₂₃	—	2	4	no	—
31	#12514	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	Q₈×C₂₃	—	8	4	no	—
33	Q₈×C₂₃	—	8	4	no	—
34	C₂×C₄×C₂₃	—	8	2	no	—
35	C₂×C₄×C₂₃	—	8	2	no	—
36	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
38	C₈×C₂₃	—	2	4	no	—
39	C₈×C₂₃	#135	1	4	yes	—
40	#2241	C₂×C₂₃	1	4	yes	—
41	#1713	C₂×C₂₃	1	1	yes	characteristic
42	#1710	C₂×C₂₃	1	1	yes	characteristic
43	#1311	C₂×C₂₃	1	1	yes	characteristic
44	C₂²×C₂₃	#493	1	1	yes	characteristic
45	C₄×C₂₃	—	16	2	no	—
46	C₄×C₂₃	—	16	2	no	—
47	C₄×C₂₃	#495	1	2	yes	—
48	C₄×C₂₃	#510	1	4	yes	—
49	#500	—	2	2	no	—
50	#500	C₂²×C₂₃	1	2	yes	—
51	#494	C₂²×C₂₃	1	2	yes	—
52	#487	—	2	2	no	—
53	#358	—	2	8	no	—
54	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
55	C₈²	C₂²×C₂₃	1	1	yes	characteristic
56	C₂×C₂₃	#1709	1	1	yes	characteristic
57	C₂×C₂₃	#2251	1	2	yes	—
58	C₂×Q₁₆	—	2	4	no	—
59	C₄⋊Q₈	—	2	2	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
62	#107	—	2	4	no	—
63	#107	—	4	4	no	—
64	#103	—	4	4	no	—
65	C₄×C₈	D₄×C₂₃	1	2	yes	—
66	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
67	C₂₃	#12514	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	4	2	no	—
70	Q₁₆	—	4	8	no	—
71	C₂×C₈	D₈×C₂₃	1	4	yes	—
72	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	4	4	no	—
76	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
77	C₁₆	—	2	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	8	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	8	2	no	—
82	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
83	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	C₂₃×#135	1	4	yes	—
86	C₂²	C₂₃×#493	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	16	2	no	—
89	C₄	C₂₃×#495	1	2	yes	—
90	C₄	C₂₃×#510	1	4	yes	—
91	C₂	C₂₃×#1709	1	1	yes	characteristic
92	C₂	C₂₃×#2251	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
31. #12514	C₂₃	direct (N × H)
67. C₂₃	#12514	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×#2241	2
C₂₃×#1713	2
C₂₃×#1710	2
C₂₃×#1311	2
#12514	23

Subgroups — C23×#12514 #19160140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12514 #19160140