Group #15673441 — subgroups

Subgroups — #15673441

Order 2944.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#10625119	C₂	1	1	yes	maximal, characteristic
2	#13092453	C₂	1	2	yes	maximal, complemented
3	#13092391	C₂	1	1	yes	maximal, characteristic
4	#13092173	C₂	1	2	yes	maximal, complemented
5	C₂₃×#410	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#117	C₂²	1	2	yes	—
8	#10625119	—	2	2	no	—
9	#10625119	—	2	2	no	—
10	#10625097	C₂²	1	2	yes	—
11	C₂×#67267	C₄	1	2	yes	complemented
12	C₂×#67267	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₄	1	2	yes	complemented
14	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂₃×#99	—	2	2	no	—
16	#10625016	—	2	2	no	—
17	#10624990	—	2	2	no	—
18	#10624989	—	2	2	no	—
19	#41×C₂₃	—	2	2	no	—
20	#41×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	Q₈	1	1	yes	characteristic
22	C₂×D₄×C₂₃	D₄	1	2	yes	—
23	C₂×D₄×C₂₃	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂²⋊C₄×C₂₃	—	2	2	no	—
28	C₄²×C₂₃	—	2	2	no	—
29	C₂²⋊Dic₂₃	—	2	2	no	—
30	C₄⋊Dic₂₃	—	2	2	no	—
31	C₄×Dic₂₃	—	2	2	no	—
32	#67267	—	2	2	no	—
33	#67267	—	2	4	no	—
34	C₂×C₂₃⋊C₈	—	2	2	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
40	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
41	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
42	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₂₃	—	2	2	no	—
44	D₄×C₂₃	—	2	4	no	—
45	D₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	4	2	no	—
49	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	#41	1	2	yes	—
54	C₂×Dic₂₃	—	4	2	no	—
55	C₂₃⋊C₈	—	2	2	no	—
56	C₂₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂₃⋊C₈	C₂²⋊C₄	1	2	yes	—
59	#1812	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	—	2	2	no	—
61	C₂²×C₂₃	—	4	1	no	—
62	C₂²×C₂₃	—	4	2	no	—
63	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₂₃	#121	1	2	yes	—
65	C₄×C₂₃	—	2	2	no	—
66	C₄×C₂₃	—	8	2	no	—
67	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₂₃	#117	1	1	yes	characteristic
69	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
70	Dic₂₃	—	8	2	no	—
71	C₂×#131	—	23	1	no	—
72	#427	—	23	2	no	—
73	#410	—	23	1	no	—
74	#410	D₂₃	1	1	yes	characteristic
75	H(3,ℤ/4)	—	23	2	no	—
76	C₂×C₂₃	—	2	2	no	—
77	C₂×C₂₃	—	2	1	no	—
78	C₂×C₂₃	—	4	2	no	—
79	C₂×C₂₃	#380	1	1	yes	characteristic
80	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
81	#131	—	46	2	no	—
82	#131	—	46	2	no	—
83	C₂×M₁₆	—	23	2	no	—
84	C₂×M₁₆	—	23	1	no	—
85	C₂²×C₈	—	23	1	no	—
86	C₂²×C₈	—	23	2	no	—
87	#117	—	23	2	no	—
88	#117	C₂×D₂₃	1	2	yes	—
89	#107	—	46	2	no	—
90	#106	—	46	2	no	—
91	#99	—	2	2	no	—
92	#99	—	46	2	no	—
93	C₂₃	#1812	1	1	yes	characteristic, complemented, Sylow
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₄₆	1	1	yes	characteristic
97	C₂×D₄	Dic₄₆	1	1	yes	characteristic
98	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
99	C₂²×C₄	C₄×D₂₃	1	2	yes	—
100	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
101	M₁₆	—	46	4	no	—
102	M₁₆	—	46	2	no	—
103	C₂×C₈	—	23	1	no	—
104	C₂×C₈	—	23	1	no	—
105	C₂×C₈	—	23	2	no	—
106	C₂×C₈	—	23	2	no	—
107	C₂×C₈	—	23	2	no	—
108	C₂×C₈	—	46	2	no	—
109	C₂×C₈	—	46	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	46	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	46	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	46	2	no	—
116	C₂³	Dic₂₃⋊C₄	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	92	2	no	—
125	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
128	C₂×C₄	D₄×D₂₃	1	2	yes	—
129	C₂×C₄	#67290	1	2	yes	—
130	C₈	—	23	1	no	—
131	C₈	—	23	1	no	—
132	C₈	—	23	2	no	—
133	C₈	—	46	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
138	C₂²	#10625110	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	184	2	no	—
142	C₄	#10625048	1	1	yes	characteristic
143	C₄	#10625049	1	2	yes	—
144	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#13092275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625119

C₂

yes

maximal, characteristic

#13092453

C₂

yes

maximal, complemented

#13092391

C₂

yes

maximal, characteristic

#13092173

C₂

yes

maximal, complemented

C₂₃×#410

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

#10625119

—

#10625119

—

#10625097

C₂²

yes

—

C₂×#67267

C₄

yes

complemented

C₂×#67267

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₄

yes

complemented

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#99

—

#10625016

—

#10624990

—

#10624989

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

Q₈

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×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×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₃⋊C₈

C₂²⋊C₄

yes

—

#1812

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

#117

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

—

—

—

D₂₃

yes

characteristic

H(3,ℤ/4)

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#380

yes

characteristic

C₂×#41

C₂×D₂₃

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₂₃

yes

—

—

—

—

—

C₂₃

#1812

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

Dic₄₆

yes

characteristic

C₂×D₄

Dic₄₆

yes

characteristic

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₄×D₂₃

yes

—

100

C₂²×C₄

C₂²×D₂₃

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₂³

Dic₂₃⋊C₄

yes

—

117

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

Dic₂₃⋊C₄

yes

—

126

C₂×C₄

C₂×Dic₄₆

yes

characteristic

127

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

128

C₂×C₄

D₄×D₂₃

yes

—

129

C₂×C₄

#67290

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

138

C₂²

#10625110

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

184

—

142

C₄

#10625048

yes

characteristic

143

C₄

#10625049

yes

—

144

C₄

C₂²⋊C₄×D₂₃

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#13092275

yes

characteristic

	subgroup	G/H
center	C₄	#10625048
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#410	C₂
socle	C₂×C₂₃	#380
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#410	D₂₃
hypocenter	C₂₃	#1812
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#10625048

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#410

C₂

socle

C₂×C₂₃

#380

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #13092453	C₂	semidirect
4. #13092173	C₂	semidirect
11. C₂×#67267	C₄	semidirect
13. C₂²×C₂₃⋊C₈	C₄	semidirect
93. C₂₃	#1812	semidirect

H ≅ G/N

type

2. #13092453

C₂

semidirect

4. #13092173

C₂

semidirect

11. C₂×#67267

C₄

semidirect

13. C₂²×C₂₃⋊C₈

C₄

semidirect

93. C₂₃

#1812

semidirect

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

subgroup	index
C₂×#10625119	2
#13092453	2
#13092391	2
#13092173	2
C₂₃×#410	2
#1812	23

subgroup

index

C₂₃×#410

Subgroups — #15673441

Conjugacy classes of subgroups

Complements

Maximal subgroups