Subgroups — C₂₃×#7513 #19155139

Order 5888.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂₃×M₁₂₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
4	C₂₃×M₆₄	—	2	2	no	—
5	C₂×C₂₃×C₃₂	C₄	1	1	yes	characteristic
6	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
7	C₂₃×C₆₄	—	2	2	no	—
8	C₂²×C₈×C₂₃	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₂₃	—	2	1	no	—
10	C₂×C₁₆×C₂₃	—	2	1	no	—
11	C₂×C₁₆×C₂₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂₃×C₃₂	—	2	1	no	—
14	C₂₃×C₃₂	D₄	1	2	yes	—
15	C₂²×C₄×C₂₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₂₃	—	2	1	no	—
17	C₂×C₈×C₂₃	—	2	1	no	—
18	C₂×C₈×C₂₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₂₃	—	2	1	no	—
21	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₂₃	M₁₆	1	1	yes	characteristic
23	#7513	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₂₃	C₃₂	1	1	yes	characteristic
25	C₂×C₄×C₂₃	—	2	1	no	—
26	C₂×C₄×C₂₃	—	2	1	no	—
27	C₂×C₄×C₂₃	C₃₂	1	1	yes	characteristic
28	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
29	C₈×C₂₃	—	2	1	no	—
30	C₈×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
31	C₈×C₂₃	M₃₂	1	1	yes	characteristic
32	C₂×M₆₄	C₂×C₂₃	1	1	yes	characteristic
33	M₁₂₈	C₂×C₂₃	1	2	yes	complemented
34	C₂²×C₂₃	—	2	1	no	—
35	C₂²×C₂₃	—	4	1	no	—
36	C₂²×C₂₃	C₂×C₃₂	1	1	yes	characteristic
37	C₄×C₂₃	—	2	1	no	—
38	C₄×C₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
39	C₄×C₂₃	M₆₄	1	1	yes	characteristic
40	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
41	M₆₄	—	2	2	no	—
42	C₂×C₃₂	C₄×C₂₃	1	1	yes	characteristic
43	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
44	C₆₄	—	2	2	no	—
45	C₂×C₂₃	—	2	1	no	—
46	C₂×C₂₃	—	4	1	no	—
47	C₂×C₂₃	C₂²⋊C₃₂	1	1	yes	characteristic
48	C₂²×C₈	C₈×C₂₃	1	1	yes	characteristic
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	C₈×C₂₃	1	1	yes	characteristic
52	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
53	C₃₂	—	2	1	no	—
54	C₃₂	D₄×C₂₃	1	2	yes	—
55	C₂₃	#7513	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₄	C₁₆×C₂₃	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	C₁₆×C₂₃	1	1	yes	characteristic
60	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
63	C₁₆	M₁₆×C₂₃	1	1	yes	characteristic
64	C₂³	C₂₃×C₃₂	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	C₂₃×C₃₂	1	1	yes	characteristic
68	C₂×C₄	C₂×C₁₆×C₂₃	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
71	C₈	C₂₃×M₃₂	1	1	yes	characteristic
72	C₂²	—	2	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₂₃×C₃₂	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	C₂₃×C₂²⋊C₁₆	1	1	yes	characteristic
77	C₄	C₂₃×M₆₄	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	4	1	no	—
80	C₂	C₂₃×C₂²⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₂₃	C₂²⋊C₄
commutator	C₂²	C₂×C₂₃×C₃₂
Frattini, cosocle	C₂×C₃₂	C₂²×C₂₃
socle	C₂×C₂₃	C₂²⋊C₃₂
prime radical	C₂³×C₂₃	C₃₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₂₃ ◁ C₂×C₂₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×M₁₂₈	C₂	semidirect
23. #7513	C₂₃	direct (N × H)
33. M₁₂₈	C₂×C₂₃	semidirect
55. C₂₃	#7513	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×M₆₄	2
C₂₃×M₁₂₈	2
#7513	23

Subgroups — C23×#7513 #19155139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7513 #19155139