Group #15673433 — subgroups

Subgroups — #15673433

Order 2944.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625004	C₂	1	2	yes	maximal, complemented
3	#13092381	C₂	1	1	yes	maximal, characteristic
4	#13092181	C₂	1	2	yes	maximal, complemented
5	C₂₃×#407	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×#67267	C₂²	1	2	yes	—
9	C₂²×C₂₃⋊C₈	—	2	4	no	—
10	C₂²×C₂₃⋊C₈	C₄	1	4	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂₃×C₂²⋊C₈	—	2	2	no	—
14	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
15	#10625004	—	2	4	no	—
16	C₂⁴×C₂₃	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	—	2	2	no	—
18	C₂²×C₄×C₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₃	—	2	2	no	—
24	M₁₆×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	#67267	—	2	2	no	—
27	#67267	—	2	2	no	—
28	C₂×C₂₃⋊C₈	—	2	2	no	—
29	C₂×C₂₃⋊C₈	—	2	8	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
34	C₂³×C₂₃	—	2	4	no	—
35	C₂³×C₂₃	—	2	2	no	—
36	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₈×C₂₃	—	4	2	no	—
48	C₂₃⋊C₈	—	2	2	no	—
49	C₂₃⋊C₈	—	4	2	no	—
50	C₂₃⋊C₈	C₂²⋊C₄	1	4	yes	—
51	#1810	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	—	2	8	no	—
53	C₂²×C₂₃	—	2	2	no	—
54	C₂²×C₂₃	—	2	2	no	—
55	C₂²×C₂₃	—	2	2	no	—
56	C₂²×C₂₃	#108	1	4	yes	complemented
57	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₂₃	—	2	2	no	—
60	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₂₃	#117	1	1	yes	characteristic
62	C₄×C₂₃	#121	1	2	yes	—
63	C₂³×C₈	—	23	1	no	—
64	C₂×#108	—	23	2	no	—
65	#407	—	23	1	no	—
66	#407	D₂₃	1	1	yes	characteristic
67	#341	—	23	2	no	—
68	C₂×C₂₃	—	2	4	no	—
69	C₂×C₂₃	—	2	2	no	—
70	C₂×C₂₃	#380	1	1	yes	characteristic
71	C₂×C₂₃	C₂×#108	1	2	yes	—
72	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
73	C₂×M₁₆	—	23	2	no	—
74	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
75	C₂²×C₈	—	23	1	no	—
76	C₂²×C₈	—	23	4	no	—
77	C₂²×C₈	—	23	1	no	—
78	C₂²×C₈	—	46	4	no	—
79	#108	—	46	4	no	—
80	C₂²⋊C₈	—	2	2	no	—
81	C₂²⋊C₈	—	46	2	no	—
82	C₂₃	#1810	1	1	yes	characteristic, complemented, Sylow
83	C₂⁴	Dic₄₆	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₂₃	1	2	yes	—
88	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	46	2	no	—
93	M₁₆	—	46	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	23	2	no	—
96	C₂×C₈	—	23	4	no	—
97	C₂×C₈	—	23	2	no	—
98	C₂×C₈	—	46	8	no	—
99	C₂×C₈	—	46	2	no	—
100	C₂×C₈	—	46	2	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	2	4	no	—
103	C₂³	Dic₂₃⋊C₄	1	2	yes	—
104	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₃	1	2	yes	—
112	C₂×C₄	#67290	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	23	4	no	—
116	C₈	—	46	2	no	—
117	C₈	—	92	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#10625004	1	4	yes	complemented
123	C₂²	#10625049	1	2	yes	—
124	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	#10625048	1	1	yes	characteristic
127	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
128	C₄	#10625110	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	4	no	—
131	C₂	#13092275	1	1	yes	characteristic
132	C₂	C₂×#10625004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₂₃⋊C₈

C₂

yes

maximal, characteristic

C₂×#10625004

C₂

yes

maximal, complemented

#13092381

C₂

yes

maximal, characteristic

#13092181

C₂

yes

maximal, complemented

C₂₃×#407

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

—

C₂²×C₂₃⋊C₈

C₄

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×C₂²⋊C₈

—

C₂²⋊(C₂₃⋊C₈)

—

#10625004

—

C₂⁴×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₂²⋊C₄

yes

—

#1810

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#108

yes

complemented

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂³×C₈

—

C₂×#108

—

#407

—

#407

D₂₃

yes

characteristic

#341

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#380

yes

characteristic

C₂×C₂₃

C₂×#108

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂₃

#1810

yes

characteristic, complemented, Sylow

C₂⁴

Dic₄₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₄₆

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂³

—

102

C₂³

—

103

C₂³

Dic₂₃⋊C₄

yes

—

104

C₂³

C₂×Dic₄₆

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₂₃⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

111

C₂×C₄

D₄×D₂₃

yes

—

112

C₂×C₄

#67290

yes

—

113

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#10625004

yes

complemented

123

C₂²

#10625049

yes

—

124

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

125

C₄

—

126

C₄

#10625048

yes

characteristic

127

C₄

C₂²⋊C₄×D₂₃

yes

characteristic

128

C₄

#10625110

yes

—

129

C₂

—

130

C₂

—

131

C₂

#13092275

yes

characteristic

132

C₂

C₂×#10625004

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#407	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#407	D₂₃
hypocenter	C₂₃	#1810
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#407

C₂

socle

C₂²×C₂₃

C₂×C₄⋊C₄

prime radical

C₂⁴×C₂₃

Q₈

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#10625004	C₂	semidirect
4. #13092181	C₂	semidirect
56. C₂²×C₂₃	#108	semidirect
82. C₂₃	#1810	semidirect
122. C₂²	#10625004	semidirect

H ≅ G/N

type

2. C₂×#10625004

C₂

semidirect

4. #13092181

C₂

semidirect

56. C₂²×C₂₃

#108

semidirect

82. C₂₃

#1810

semidirect

122. C₂²

#10625004

semidirect

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

subgroup	index
C₂³×C₂₃⋊C₈	2
C₂×#10625004	2
#13092381	2
#13092181	2
C₂₃×#407	2
#1810	23

subgroup

index

C₂³×C₂₃⋊C₈

C₂₃×#407

Subgroups — #15673433

Conjugacy classes of subgroups

Complements

Maximal subgroups