Subgroups — C₂₃×#8131 #19155757

Order 5888.

574 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈×C₂₃	C₂	1	3	yes	maximal
2	C₂₃×#1273	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₂₃	C₄	1	6	yes	—
4	C₂×C₄×C₈×C₂₃	C₄	1	6	yes	—
5	C₂×C₄×C₈×C₂₃	C₂²	1	6	yes	—
6	C₄³×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂×C₄	1	12	yes	—
8	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
9	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
10	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
11	C₄×C₈×C₂₃	—	2	12	no	—
12	C₄×C₈×C₂₃	—	2	12	no	—
13	C₄×C₈×C₂₃	C₈	1	24	yes	complemented
14	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	C₄²	1	3	yes	—
16	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
17	C₂×C₈×C₂₃	—	2	12	no	—
18	C₂×C₈×C₂₃	—	2	12	no	—
19	C₂×C₈×C₂₃	C₂×C₈	1	24	yes	—
20	C₄²×C₂₃	C₂×C₈	1	12	yes	—
21	C₄²×C₂₃	M₁₆	1	12	yes	—
22	C₄²×C₂₃	#41	1	4	yes	—
23	#8131	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
25	C₂×C₄×C₂₃	C₄×C₈	1	6	yes	—
26	C₂×C₄×C₂₃	C₄×C₈	1	6	yes	—
27	C₂×C₄×C₂₃	#97	1	12	yes	—
28	C₂×C₄×C₂₃	#117	1	6	yes	—
29	C₂×C₄×C₂₃	C₂²×C₈	1	6	yes	—
30	C₂×C₄×C₂₃	C₂×M₁₆	1	6	yes	—
31	C₈×C₂₃	—	2	24	no	—
32	C₄²×C₈	C₂×C₂₃	1	3	yes	—
33	#1273	C₂×C₂₃	1	4	yes	—
34	C₂²×C₂₃	#376	1	1	yes	characteristic
35	C₂²×C₂₃	C₂×C₄×C₈	1	3	yes	—
36	C₂²×C₂₃	C₂×#97	1	3	yes	—
37	C₄×C₂₃	C₈²	1	4	yes	—
38	C₄×C₂₃	#322	1	12	yes	—
39	C₄×C₂₃	#431	1	12	yes	—
40	C₂×C₄×C₈	C₄×C₂₃	1	6	yes	—
41	C₂×C₄×C₈	C₄×C₂₃	1	6	yes	—
42	C₂×C₄×C₈	C₂²×C₂₃	1	6	yes	—
43	C₄³	C₂²×C₂₃	1	1	yes	characteristic
44	C₂×C₂₃	C₂×C₈²	1	1	yes	characteristic
45	C₂×C₂₃	C₂×#322	1	3	yes	—
46	C₂×C₂₃	#1744	1	3	yes	—
47	C₂²×C₈	C₂×C₄×C₂₃	1	12	yes	—
48	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
49	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
50	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
51	C₄×C₈	—	2	12	no	—
52	C₄×C₈	—	2	12	no	—
53	C₄×C₈	C₈×C₂₃	1	24	yes	complemented
54	C₂₃	#8131	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
56	C₂²×C₄	C₄²×C₂₃	1	3	yes	—
57	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
58	C₂×C₈	—	2	12	no	—
59	C₂×C₈	—	2	12	no	—
60	C₂×C₈	C₂×C₈×C₂₃	1	24	yes	—
61	C₄²	C₂×C₈×C₂₃	1	12	yes	—
62	C₄²	M₁₆×C₂₃	1	12	yes	—
63	C₄²	#41×C₂₃	1	4	yes	—
64	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
65	C₂×C₄	C₄×C₈×C₂₃	1	6	yes	—
66	C₂×C₄	C₄×C₈×C₂₃	1	6	yes	—
67	C₂×C₄	C₂₃×#97	1	12	yes	—
68	C₂×C₄	C₂₃×#117	1	6	yes	—
69	C₂×C₄	C₂²×C₈×C₂₃	1	6	yes	—
70	C₂×C₄	C₂×M₁₆×C₂₃	1	6	yes	—
71	C₈	—	2	24	no	—
72	C₂²	C₂₃×#376	1	1	yes	characteristic
73	C₂²	C₂×C₄×C₈×C₂₃	1	3	yes	—
74	C₂²	C₂×C₂₃×#97	1	3	yes	—
75	C₄	C₈²×C₂₃	1	4	yes	—
76	C₄	C₂₃×#322	1	12	yes	—
77	C₄	C₂₃×#431	1	12	yes	—
78	C₂	C₂×C₈²×C₂₃	1	1	yes	characteristic
79	C₂	C₂×C₂₃×#322	1	3	yes	—
80	C₂	C₂₃×#1744	1	3	yes	—

	subgroup	G/H
center	C₄³×C₂₃	C₂²
commutator	C₂	C₂×C₈²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₈×C₂₃	C₈	semidirect
23. #8131	C₂₃	direct (N × H)
53. C₄×C₈	C₈×C₂₃	semidirect
54. C₂₃	#8131	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₈×C₂₃	2
C₂₃×#1273	2
#8131	23

Subgroups — C23×#8131 #19155757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8131 #19155757