Subgroups — C₂₃×#7514 #19155140

Order 5888.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2258	C₂	1	1	yes	maximal, characteristic
2	C₂₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#504	C₄	1	1	yes	characteristic
5	C₂₃×M₆₄	—	2	1	no	—
6	C₂₃×M₆₄	C₄	1	1	yes	characteristic
7	C₂×C₂₃×C₃₂	—	2	1	no	—
8	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
9	C₂₃×C₆₄	—	2	1	no	—
10	C₂₃×C₆₄	—	2	1	no	—
11	C₂₃×#131	C₈	1	1	yes	characteristic
12	C₂₃×M₃₂	—	2	1	no	—
13	C₂₃×M₃₂	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₂₃	—	2	1	no	—
15	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂₃×C₃₂	—	2	1	no	—
17	C₂₃×C₃₂	D₄	1	1	yes	characteristic
18	C₂₃×C₃₂	D₄	1	1	yes	characteristic
19	#41×C₂₃	C₁₆	1	1	yes	characteristic
20	M₁₆×C₂₃	—	2	1	no	—
21	M₁₆×C₂₃	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₂₃	—	2	1	no	—
23	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₂₃	—	2	1	no	—
25	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₂₃	M₁₆	1	1	yes	characteristic
27	#7514	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	Q₈×C₂₃	C₃₂	1	1	yes	characteristic
29	D₄×C₂₃	—	2	1	no	—
30	D₄×C₂₃	C₃₂	1	1	yes	characteristic
31	C₂×C₄×C₂₃	—	2	1	no	—
32	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
33	C₈×C₂₃	—	2	1	no	—
34	C₈×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
35	C₈×C₂₃	M₃₂	1	1	yes	characteristic
36	#2258	C₂×C₂₃	1	1	yes	characteristic
37	M₁₂₈	C₂×C₂₃	1	1	yes	characteristic, complemented
38	C₂×C₆₄	C₂×C₂₃	1	1	yes	characteristic, complemented
39	C₂²×C₂₃	—	2	1	no	—
40	C₂²×C₂₃	M₆₄	1	1	yes	characteristic
41	C₄×C₂₃	—	2	1	no	—
42	C₄×C₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
43	C₄×C₂₃	C₂×C₃₂	1	1	yes	characteristic
44	#504	C₄×C₂₃	1	1	yes	characteristic
45	M₆₄	—	2	1	no	—
46	M₆₄	C₄×C₂₃	1	1	yes	characteristic
47	C₂×C₃₂	—	2	1	no	—
48	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
49	C₆₄	—	2	1	no	—
50	C₆₄	—	2	1	no	—
51	C₂×C₂₃	—	2	1	no	—
52	C₂×C₂₃	—	4	1	no	—
53	C₂×C₂₃	C₂²⋊C₃₂	1	1	yes	characteristic
54	#131	C₈×C₂₃	1	1	yes	characteristic
55	M₃₂	—	2	1	no	—
56	M₃₂	C₈×C₂₃	1	1	yes	characteristic
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
59	C₃₂	—	2	1	no	—
60	C₃₂	D₄×C₂₃	1	1	yes	characteristic
61	C₃₂	D₄×C₂₃	1	1	yes	characteristic
62	C₂₃	#7514	1	1	yes	characteristic, direct product factor, Sylow
63	#41	C₁₆×C₂₃	1	1	yes	characteristic
64	M₁₆	—	2	1	no	—
65	M₁₆	C₁₆×C₂₃	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
70	C₁₆	M₁₆×C₂₃	1	1	yes	characteristic
71	Q₈	C₂₃×C₃₂	1	1	yes	characteristic
72	D₄	—	2	1	no	—
73	D₄	C₂₃×C₃₂	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×C₁₆×C₂₃	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
78	C₈	C₂₃×M₃₂	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	C₂₃×M₆₄	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	C₂₃×C₂²⋊C₁₆	1	1	yes	characteristic
83	C₄	C₂×C₂₃×C₃₂	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	C₂₃×C₂²⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂₃×C₃₂	D₄
commutator	C₄	C₂×C₂₃×C₃₂
Frattini, cosocle	C₂×C₃₂	C₂²×C₂₃
socle	C₂×C₂₃	C₂²⋊C₃₂
prime radical	#41×C₂₃	C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂₃×C₃₂ ◁ 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
3. C₂×C₂₃×C₆₄	C₂	semidirect
27. #7514	C₂₃	direct (N × H)
37. M₁₂₈	C₂×C₂₃	semidirect
38. C₂×C₆₄	C₂×C₂₃	semidirect
62. C₂₃	#7514	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×#2258	2
C₂₃×M₁₂₈	2
C₂×C₂₃×C₆₄	2
#7514	23

Subgroups — C23×#7514 #19155140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7514 #19155140