Subgroups — C₂₃×C₂²⋊C₆₄ #19155138

Order 5888.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₆₄	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
4	C₂×C₂₃×C₃₂	—	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	C₂²⋊C₆₄	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	C₂²×C₃₂	C₂×C₂₃	1	1	yes	characteristic
33	C₂×C₆₄	C₂×C₂₃	1	2	yes	complemented
34	C₂²×C₂₃	—	2	1	no	—
35	C₂²×C₂₃	—	2	1	no	—
36	C₂²×C₂₃	C₆₄	1	2	yes	complemented
37	C₂²×C₂₃	C₂×C₃₂	1	1	yes	characteristic
38	C₄×C₂₃	—	2	1	no	—
39	C₄×C₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
40	C₄×C₂₃	M₆₄	1	1	yes	characteristic
41	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
42	C₂×C₃₂	—	2	2	no	—
43	C₂×C₃₂	C₄×C₂₃	1	1	yes	characteristic
44	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
45	C₆₄	—	2	2	no	—
46	C₂×C₂₃	—	2	2	no	—
47	C₂×C₂₃	C₂²⋊C₃₂	1	1	yes	characteristic
48	C₂×C₂₃	C₂×C₆₄	1	1	yes	characteristic
49	C₂×C₂₃	M₁₂₈	1	1	yes	characteristic
50	C₂²×C₈	C₈×C₂₃	1	1	yes	characteristic
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	C₈×C₂₃	1	1	yes	characteristic
54	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
55	C₃₂	—	2	1	no	—
56	C₃₂	D₄×C₂₃	1	2	yes	—
57	C₂₃	C₂²⋊C₆₄	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₄	C₁₆×C₂₃	1	1	yes	characteristic
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	C₁₆×C₂₃	1	1	yes	characteristic
62	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
65	C₁₆	M₁₆×C₂₃	1	1	yes	characteristic
66	C₂³	C₂₃×C₃₂	1	1	yes	characteristic
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂₃×C₃₂	1	1	yes	characteristic
70	C₂×C₄	C₂×C₁₆×C₂₃	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
73	C₈	C₂₃×M₃₂	1	1	yes	characteristic
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	C₂₃×C₆₄	1	2	yes	complemented
77	C₂²	C₂×C₂₃×C₃₂	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	C₂₃×C₂²⋊C₁₆	1	1	yes	characteristic
80	C₄	C₂₃×M₆₄	1	1	yes	characteristic
81	C₂	—	2	2	no	—
82	C₂	C₂₃×C₂²⋊C₃₂	1	1	yes	characteristic
83	C₂	C₂×C₂₃×C₆₄	1	1	yes	characteristic
84	C₂	C₂₃×M₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₂₃×C₃₂	C₂²
commutator	C₂	C₂×C₂₃×C₆₄
Frattini, cosocle	C₂×C₃₂	C₂²×C₂₃
socle	C₂²×C₂₃	C₂×C₃₂
prime radical	C₂³×C₂₃	C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×C₆₄	C₂	semidirect
23. C₂²⋊C₆₄	C₂₃	direct (N × H)
33. C₂×C₆₄	C₂×C₂₃	semidirect
36. C₂²×C₂₃	C₆₄	semidirect
57. C₂₃	C₂²⋊C₆₄	direct (N × H)
76. C₂²	C₂₃×C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×C₃₂	2
C₂×C₂₃×C₆₄	2
C₂²⋊C₆₄	23

Subgroups — C23×C22⋊C64 #19155138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×C₂²⋊C₆₄ #19155138