Group #15675193 — subgroups

Subgroups — #15675193

Order 2944.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
2	Dic₂₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₂₃	C₂	1	2	yes	maximal, complemented
5	C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₃	C₄	1	1	yes	characteristic
8	C₂²×C₂₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₂₃	C₂²	1	2	yes	—
10	C₈×Dic₂₃	C₄	1	1	yes	characteristic
11	C₈×Dic₂₃	C₄	1	1	yes	characteristic
12	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₂₃	C₈	1	1	yes	characteristic
17	C₂²×Dic₂₃	C₈	1	1	yes	characteristic
18	C₂×C₈×C₂₃	—	2	2	no	—
19	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₄×Dic₂₃	C₈	1	1	yes	characteristic
22	C₄×Dic₂₃	C₈	1	1	yes	characteristic
23	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₂₃⋊C₈	—	2	2	no	—
26	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₂₃	—	2	2	no	—
29	C₂₃⋊C₁₆	—	2	2	no	—
30	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₂₃	—	2	1	no	—
32	C₂×C₄×C₂₃	—	2	1	no	—
33	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₂₃	—	2	2	no	—
36	C₂×Dic₂₃	C₁₆	1	4	yes	complemented
37	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
41	C₈×C₂₃	—	2	1	no	—
42	C₈×C₂₃	#41	1	2	yes	—
43	C₂₃⋊C₈	—	2	1	no	—
44	C₂₃⋊C₈	M₁₆	1	2	yes	—
45	#2162	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	—	2	1	no	—
47	C₂²×C₂₃	—	2	1	no	—
48	C₂²×C₂₃	C₂×C₁₆	1	2	yes	—
49	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₂₃	—	2	1	no	—
51	C₄×C₂₃	#117	1	1	yes	characteristic
52	C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
53	Dic₂₃	—	2	1	no	—
54	Dic₂₃	C₂×C₁₆	1	2	yes	—
55	Dic₂₃	C₂×C₁₆	1	2	yes	—
56	Dic₂₃	M₃₂	1	2	yes	—
57	C₂×C₄×C₈	—	23	1	no	—
58	C₄⋊C₁₆	—	23	2	no	—
59	C₂²⋊C₁₆	—	23	1	no	—
60	C₂²⋊C₁₆	D₂₃	1	1	yes	characteristic
61	C₄×C₁₆	—	23	2	no	—
62	C₂×C₂₃	—	2	2	no	—
63	C₂×C₂₃	#431	1	1	yes	characteristic
64	C₂×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
65	C₂×C₂₃	C₂×M₃₂	1	1	yes	characteristic
66	C₂²×C₈	—	23	1	no	—
67	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
68	C₂×C₄²	—	23	1	no	—
69	C₂×C₁₆	—	23	2	no	—
70	C₂×C₁₆	C₂×D₂₃	1	2	yes	—
71	C₄×C₈	—	23	1	no	—
72	C₄×C₈	—	23	1	no	—
73	C₄×C₈	—	23	1	no	—
74	C₄×C₈	—	23	1	no	—
75	C₂₃	#2162	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₄	—	23	1	no	—
77	C₂²×C₄	—	23	1	no	—
78	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	23	1	no	—
81	C₂×C₈	—	23	1	no	—
82	C₂×C₈	—	46	2	no	—
83	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
85	C₄²	—	23	1	no	—
86	C₄²	—	23	1	no	—
87	C₄²	—	23	1	no	—
88	C₄²	—	23	1	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	46	2	no	—
91	C₂³	C₈×D₂₃	1	1	yes	characteristic
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	23	1	no	—
95	C₂×C₄	—	23	1	no	—
96	C₂×C₄	—	23	1	no	—
97	C₂×C₄	—	23	4	no	—
98	C₂×C₄	—	23	1	no	—
99	C₂×C₄	—	46	2	no	—
100	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	23	2	no	—
104	C₈	—	46	1	no	—
105	C₈	#67290	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	C₁₆×D₂₃	1	2	yes	—
109	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	23	2	no	—
112	C₄	—	23	2	no	—
113	C₄	—	23	2	no	—
114	C₄	—	46	1	no	—
115	C₄	#10625048	1	1	yes	characteristic
116	C₄	M₁₆×D₂₃	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#13092467	1	1	yes	characteristic
119	C₂	C₂×C₁₆×D₂₃	1	1	yes	characteristic
120	C₂	M₃₂×D₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

Dic₂₃⋊C₁₆

C₂

yes

maximal, complemented

C₂²⋊(C₂₃⋊C₁₆)

C₂

yes

maximal, characteristic

C₁₆×Dic₂₃

C₂

yes

maximal, complemented

C₂₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

characteristic

C₂²×C₂₃⋊C₈

C₄

yes

characteristic

C₂×C₁₆×C₂₃

C₂²

yes

—

C₈×Dic₂₃

C₄

yes

characteristic

C₈×Dic₂₃

C₄

yes

characteristic

C₈×Dic₂₃

C₂²

yes

characteristic

C₈×Dic₂₃

C₂²

yes

characteristic

C₂×C₂₃⋊C₁₆

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₈

yes

characteristic

C₂²×Dic₂₃

C₈

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄×Dic₂₃

C₈

yes

characteristic

C₄×Dic₂₃

C₈

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₁₆×C₂₃

—

C₂₃⋊C₁₆

—

C₂³×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

C₁₆

yes

complemented

C₂×Dic₂₃

C₂×C₈

yes

characteristic

C₂×Dic₂₃

C₂×C₈

yes

characteristic

C₂×Dic₂₃

C₂×C₈

yes

characteristic

C₂×Dic₂₃

C₂×C₈

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

#41

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

M₁₆

yes

—

#2162

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₁₆

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#117

yes

characteristic

C₄×C₂₃

C₂×M₁₆

yes

characteristic

Dic₂₃

—

Dic₂₃

C₂×C₁₆

yes

—

Dic₂₃

C₂×C₁₆

yes

—

Dic₂₃

M₃₂

yes

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₂₃

yes

characteristic

C₄×C₁₆

—

C₂×C₂₃

—

C₂×C₂₃

#431

yes

characteristic

C₂×C₂₃

C₂²×C₁₆

yes

characteristic

C₂×C₂₃

C₂×M₃₂

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₂₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#2162

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

C₈×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₈×D₂₃

yes

characteristic

101

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

#67290

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₁₆×D₂₃

yes

—

109

C₂²

C₂×C₈×D₂₃

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#10625048

yes

characteristic

116

C₄

M₁₆×D₂₃

yes

characteristic

117

C₂

—

118

C₂

#13092467

yes

characteristic

119

C₂

C₂×C₁₆×D₂₃

yes

characteristic

120

C₂

M₃₂×D₂₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃
commutator	C₂×C₂₃	C₂²×C₁₆
Fitting	C₂₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₂₃	C₂²×C₈
prime radical	C₂³×C₂₃	C₂×C₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₂²⋊C₁₆	D₂₃
hypocenter	C₂₃	#2162
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₃

commutator

C₂×C₂₃

C₂²×C₁₆

Fitting

C₂₃×C₂²⋊C₁₆

C₂

socle

C₂²×C₂₃

C₂²×C₈

prime radical

C₂³×C₂₃

C₂×C₈

cosocle

C₂×C₈×C₂₃

C₂³

hypercenter

C₂²⋊C₁₆

D₂₃

C₂₃

G ▷ C₂×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. Dic₂₃⋊C₁₆	C₂	semidirect
4. C₁₆×Dic₂₃	C₂	semidirect
36. C₂×Dic₂₃	C₁₆	semidirect
75. C₂₃	#2162	semidirect

H ≅ G/N

type

2. Dic₂₃⋊C₁₆

C₂

semidirect

4. C₁₆×Dic₂₃

C₂

semidirect

36. C₂×Dic₂₃

C₁₆

semidirect

75. C₂₃

#2162

semidirect

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

subgroup	index
C₂×C₈×Dic₂₃	2
Dic₂₃⋊C₁₆	2
C₂²⋊(C₂₃⋊C₁₆)	2
C₁₆×Dic₂₃	2
C₂₃×C₂²⋊C₁₆	2
#2162	23

subgroup

index

C₂×C₈×Dic₂₃

Dic₂₃⋊C₁₆

C₂²⋊(C₂₃⋊C₁₆)

C₁₆×Dic₂₃

C₂₃×C₂²⋊C₁₆

#2162

Subgroups — #15675193

Conjugacy classes of subgroups

Complements

Maximal subgroups