Group #15674259 — subgroups

Subgroups — #15674259

Order 2944.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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×#10625000	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625017	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092324	C₂	1	2	yes	maximal
5	C₂²×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
7	C₂×Q₁₆×C₂₃	—	2	2	no	—
8	C₂×Q₁₆×C₂₃	—	2	2	no	—
9	C₂×Q₁₆×C₂₃	C₄	1	4	yes	complemented
10	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	#10625017	—	2	4	no	—
15	C₂₃⋊#95	—	2	4	no	—
16	#10625000	—	2	2	no	—
17	C₈×Dic₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	—	2	2	no	—
19	C₂×Q₈×C₂₃	—	2	2	no	—
20	C₂×Q₈×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	—	2	2	no	—
22	C₂×Q₈×C₂₃	—	2	2	no	—
23	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₃	—	2	2	no	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₃	—	2	2	no	—
27	C₂²×Dic₂₃	D₄	1	2	yes	—
28	Q₁₆×C₂₃	—	2	8	no	—
29	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₂₃	D₄	1	2	yes	—
31	C₂×C₈×C₂₃	D₄	1	2	yes	—
32	C₄⋊Dic₂₃	—	2	2	no	—
33	C₄⋊Dic₂₃	—	2	2	no	—
34	C₄×Dic₂₃	—	2	1	no	—
35	C₄×Dic₂₃	—	2	1	no	—
36	C₂×C₂₃⋊C₈	—	2	1	no	—
37	C₂×C₂₃⋊C₈	—	2	1	no	—
38	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
39	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₂₃	—	2	4	no	—
41	Q₈×C₂₃	—	2	4	no	—
42	Q₈×C₂₃	—	4	2	no	—
43	Q₈×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	—	4	2	no	—
47	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	—	2	2	no	—
55	C₂×Dic₂₃	—	4	2	no	—
56	C₂×Dic₂₃	—	4	4	no	—
57	C₂×Dic₂₃	Q₁₆	1	4	yes	complemented
58	C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
59	C₂₃⋊C₈	—	2	2	no	—
60	#1969	—	23	1	no	maximal, Sylow
61	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₂₃	#121	1	1	yes	characteristic
63	C₂²×C₂₃	#127	1	1	yes	characteristic
64	C₂²×C₂₃	C₂×Q₁₆	1	2	yes	—
65	C₂²×C₂₃	#135	1	2	yes	—
66	C₄×C₂₃	—	4	4	no	—
67	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
69	C₄×C₂₃	#121	1	1	yes	characteristic
70	C₄×C₂₃	#124	1	1	yes	characteristic
71	Dic₂₃	—	2	4	no	—
72	Dic₂₃	—	8	2	no	—
73	C₂²×Q₁₆	D₂₃	1	1	yes	characteristic
74	C₂×#107	—	23	1	no	—
75	C₂×#103	—	23	2	no	—
76	C₂×C₄×C₈	—	23	1	no	—
77	#391	—	23	2	no	—
78	C₂×C₂₃	#390	1	1	yes	characteristic
79	C₂×C₂₃	C₄×Q₁₆	1	2	yes	—
80	C₂×C₂₃	#467	1	2	yes	—
81	C₂×C₂₃	#494	1	1	yes	characteristic
82	C₂×C₂₃	#495	1	1	yes	characteristic
83	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
84	C₂×Q₁₆	—	2	2	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×Q₁₆	Dic₂₃	1	4	yes	complemented
87	C₂²×C₈	—	23	1	no	—
88	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	23	2	no	—
90	C₂×C₄²	—	23	1	no	—
91	#107	—	46	2	no	—
92	#103	—	46	4	no	—
93	C₄×C₈	—	46	2	no	—
94	#95	—	46	4	no	—
95	C₂₃	#1969	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₂₃	1	4	yes	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	23	2	no	—
104	C₂²×C₄	—	46	2	no	—
105	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
106	Q₁₆	—	2	8	no	—
107	C₂×C₈	—	23	2	no	—
108	C₂×C₈	—	46	1	no	—
109	C₂×C₈	—	46	1	no	—
110	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
111	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
112	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
113	C₄⋊C₄	—	46	2	no	—
114	C₄⋊C₄	—	46	2	no	—
115	C₄²	—	46	1	no	—
116	C₄²	—	46	1	no	—
117	C₂³	D₄×D₂₃	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	23	4	no	—
126	C₂×C₄	—	46	2	no	—
127	C₂×C₄	—	46	2	no	—
128	C₂×C₄	—	92	4	no	—
129	C₂×C₄	—	92	2	no	—
130	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
131	C₂×C₄	#67290	1	1	yes	characteristic
132	C₂×C₄	#67290	1	1	yes	characteristic
133	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
136	C₈	—	46	2	no	—
137	C₈	C₂²⋊Dic₂₃	1	4	yes	—
138	C₂²	Q₁₆×D₂₃	1	2	yes	—
139	C₂²	#10625090	1	2	yes	—
140	C₂²	D₄×Dic₂₃	1	1	yes	characteristic
141	C₂²	#10625109	1	1	yes	characteristic
142	C₂²	#10625111	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	46	4	no	—
145	C₄	—	184	2	no	—
146	C₄	D₄×Dic₂₃	1	1	yes	characteristic
147	C₄	#10625107	1	1	yes	characteristic
148	C₄	#10625109	1	1	yes	characteristic
149	C₄	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
150	C₂	#13092320	1	1	yes	characteristic
151	C₂	Q₁₆×Dic₂₃	1	2	yes	—
152	C₂	#13092655	1	2	yes	—
153	C₂	#13092795	1	1	yes	characteristic
154	C₂	#13092802	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625000

C₂

yes

maximal, characteristic

C₂×#10625017

C₂

yes

maximal

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

#13092324

C₂

yes

maximal

C₂²×Q₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

C₄

yes

complemented

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

#10625017

—

C₂₃⋊#95

—

#10625000

—

C₈×Dic₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

D₄

yes

—

Q₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

Q₁₆

yes

complemented

C₈×C₂₃

C₂²⋊C₄

yes

—

C₂₃⋊C₈

—

#1969

—

maximal, Sylow

C₂²×C₂₃

C₄×D₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

C₂×Q₁₆

yes

—

C₂²×C₂₃

#135

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

C₂²×Q₁₆

D₂₃

yes

characteristic

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

—

#391

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

C₄×Q₁₆

yes

—

C₂×C₂₃

#467

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

C₂×D₂₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₂₃

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

#103

—

C₄×C₈

—

#95

—

C₂₃

#1969

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₂₃

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₂₃

yes

characteristic

106

Q₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂×Dic₂₃

yes

—

111

C₂×C₈

C₂²⋊D₂₃

yes

—

112

C₂×C₈

C₂²⋊D₂₃

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₂³

D₄×D₂₃

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

D₄×D₂₃

yes

characteristic

131

C₂×C₄

#67290

yes

characteristic

132

C₂×C₄

#67290

yes

characteristic

133

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

136

C₈

—

137

C₈

C₂²⋊Dic₂₃

yes

—

138

C₂²

Q₁₆×D₂₃

yes

—

139

C₂²

#10625090

yes

—

140

C₂²

D₄×Dic₂₃

yes

characteristic

141

C₂²

#10625109

yes

characteristic

142

C₂²

#10625111

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

184

—

146

C₄

D₄×Dic₂₃

yes

characteristic

147

C₄

#10625107

yes

characteristic

148

C₄

#10625109

yes

characteristic

149

C₄

C₂×C₂²⋊Dic₂₃

yes

characteristic

150

C₂

#13092320

yes

characteristic

151

C₂

Q₁₆×Dic₂₃

yes

—

152

C₂

#13092655

yes

—

153

C₂

#13092795

yes

characteristic

154

C₂

#13092802

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂²×Q₁₆×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂²×Q₁₆	D₂₃
hypocenter	C₂₃	#1969
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂²×Q₁₆×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂²×Q₁₆

D₂₃

hypocenter

C₂₃

#1969

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
9. C₂×Q₁₆×C₂₃	C₄	semidirect
57. C₂×Dic₂₃	Q₁₆	semidirect
86. C₂×Q₁₆	Dic₂₃	semidirect
95. C₂₃	#1969	semidirect

H ≅ G/N

type

9. C₂×Q₁₆×C₂₃

C₄

semidirect

57. C₂×Dic₂₃

Q₁₆

semidirect

86. C₂×Q₁₆

Dic₂₃

semidirect

95. C₂₃

#1969

semidirect

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

subgroup	index
C₂×#10625000	2
C₂×#10625017	2
C₂×C₈×Dic₂₃	2
#13092324	2
C₂²×Q₁₆×C₂₃	2
#1969	23

subgroup

index

C₂×C₈×Dic₂₃

C₂²×Q₁₆×C₂₃

Subgroups — #15674259

Conjugacy classes of subgroups

Complements

Maximal subgroups