Group #15671719 — subgroups

Subgroups — #15671719

Order 2944.

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

normal subgroups only

This group has 248 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×#10625058	C₂	1	1	yes	maximal, characteristic
2	#13092467	C₂	1	2	yes	maximal
3	#13092143	C₂	1	1	yes	maximal, characteristic
4	#13092142	C₂	1	2	yes	maximal
5	C₂₃×#324	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₆	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×Dic₂₃⋊C₄	C₄	1	1	yes	characteristic
9	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₂₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₂₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
13	#10625058	—	2	2	no	—
14	#10625058	—	2	2	no	—
15	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
17	Dic₂₃⋊C₈	—	2	2	no	—
18	C₈×Dic₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₄₆	—	2	2	no	—
27	C₂×Dic₄₆	—	2	2	no	—
28	C₂×Dic₄₆	C₈	1	4	yes	complemented
29	C₂×C₈×C₂₃	—	2	2	no	—
30	C₄⋊C₄×C₂₃	—	2	2	no	—
31	Dic₂₃⋊C₄	—	2	2	no	—
32	Dic₂₃⋊C₄	—	2	2	no	—
33	C₄×Dic₂₃	—	2	1	no	—
34	C₄×Dic₂₃	—	2	1	no	—
35	C₄×Dic₂₃	—	2	1	no	—
36	C₄×Dic₂₃	—	2	1	no	—
37	C₄×Dic₂₃	D₄	1	2	yes	—
38	C₄×Dic₂₃	D₄	1	2	yes	—
39	C₂×C₂₃⋊C₈	—	2	2	no	—
40	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	—	2	1	no	—
42	C₂×C₄×C₂₃	—	2	1	no	—
43	C₂×C₄×C₂₃	—	2	1	no	—
44	C₂×C₄×C₂₃	—	4	1	no	—
45	C₂×C₄×C₂₃	—	4	1	no	—
46	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂×Dic₂₃	—	2	1	no	—
49	C₂×Dic₂₃	—	2	1	no	—
50	C₂×Dic₂₃	—	2	2	no	—
51	C₂×Dic₂₃	—	2	1	no	—
52	C₂×Dic₂₃	—	4	1	no	—
53	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₂₃	C₂²⋊C₄	1	2	yes	—
55	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
57	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
58	C₂×Dic₂₃	M₁₆	1	2	yes	—
59	Dic₄₆	—	2	8	no	—
60	C₈×C₂₃	—	4	2	no	—
61	C₂₃⋊C₈	—	4	2	no	—
62	#1461	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	2	1	no	—
64	C₂²×C₂₃	—	2	1	no	—
65	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
67	C₂²×C₂₃	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₂₃	—	2	2	no	—
69	C₄×C₂₃	—	2	2	no	—
70	C₄×C₂₃	—	4	1	no	—
71	Dic₂₃	—	2	2	no	—
72	Dic₂₃	—	2	1	no	—
73	Dic₂₃	—	4	1	no	—
74	Dic₂₃	C₂²⋊C₈	1	4	yes	—
75	Dic₂₃	#101	1	2	yes	—
76	C₂×C₄×Q₈	—	23	1	no	—
77	#431	—	23	2	no	—
78	#324	—	23	1	no	—
79	#324	—	23	2	no	—
80	#324	D₂₃	1	1	yes	characteristic
81	C₂×C₂₃	—	2	1	no	—
82	C₂×C₂₃	—	2	1	no	—
83	C₂×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₂₃	#410	1	1	yes	characteristic
85	C₂×C₂₃	C₂×#101	1	1	yes	characteristic
86	C₂²×Q₈	—	23	1	no	—
87	C₄×Q₈	—	46	2	no	—
88	C₄×Q₈	—	46	2	no	—
89	C₂×C₄⋊C₄	—	23	1	no	—
90	C₂×C₄⋊C₄	—	23	1	no	—
91	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
92	C₂×C₄²	—	23	1	no	—
93	C₂×C₄²	—	23	1	no	—
94	C₂×C₄²	—	23	1	no	—
95	C₄⋊C₈	—	46	2	no	—
96	C₂²⋊C₈	—	23	2	no	—
97	C₂²⋊C₈	C₂×D₂₃	1	2	yes	—
98	C₄×C₈	—	46	2	no	—
99	C₂₃	#1461	1	1	yes	characteristic, complemented, Sylow
100	C₂×Q₈	—	23	4	no	—
101	C₂×Q₈	—	46	2	no	—
102	C₂×Q₈	—	46	2	no	—
103	C₂²×C₄	—	23	1	no	—
104	C₂²×C₄	—	23	1	no	—
105	C₂²×C₄	—	23	1	no	—
106	C₂²×C₄	—	23	1	no	—
107	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	46	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	46	2	no	—
114	C₄⋊C₄	—	46	2	no	—
115	C₄²	—	23	2	no	—
116	C₄²	—	23	2	no	—
117	C₄²	—	46	1	no	—
118	C₄²	—	46	1	no	—
119	C₄²	—	46	1	no	—
120	C₄²	—	46	1	no	—
121	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
122	Q₈	—	46	8	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	23	1	no	—
129	C₂×C₄	—	23	2	no	—
130	C₂×C₄	—	23	2	no	—
131	C₂×C₄	—	23	2	no	—
132	C₂×C₄	—	23	1	no	—
133	C₂×C₄	—	23	2	no	—
134	C₂×C₄	—	46	2	no	—
135	C₂×C₄	—	46	1	no	—
136	C₂×C₄	—	46	1	no	—
137	C₂×C₄	—	46	1	no	—
138	C₂×C₄	—	92	1	no	—
139	C₂×C₄	C₈×D₂₃	1	2	yes	—
140	C₂×C₄	D₄×D₂₃	1	2	yes	—
141	C₈	—	4	2	no	—
142	C₈	—	92	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
146	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
147	C₂²	M₁₆×D₂₃	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	1	no	—
151	C₄	—	23	4	no	—
152	C₄	—	23	2	no	—
153	C₄	—	46	2	no	—
154	C₄	—	46	1	no	—
155	C₄	—	92	1	no	—
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	C₂²⋊C₈×D₂₃	1	1	yes	characteristic
159	C₂	#13092389	1	1	yes	characteristic
160	C₂	#101×D₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625058

C₂

yes

maximal, characteristic

#13092467

C₂

yes

maximal

#13092143

C₂

yes

maximal, characteristic

#13092142

C₂

yes

maximal

C₂₃×#324

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

C₄

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×Dic₂₃⋊C₄

C₄

yes

characteristic

C₂×Dic₂₃⋊C₄

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

#10625058

—

#10625058

—

C₂₃×C₂²⋊C₈

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

C₂²

yes

—

Dic₂₃⋊C₈

—

C₈×Dic₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₈

yes

complemented

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₃

C₂²⋊C₄

yes

—

C₂×Dic₂₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₃

C₂×C₈

yes

—

C₂×Dic₂₃

C₂×C₈

yes

—

C₂×Dic₂₃

M₁₆

yes

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1461

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₂²⋊C₈

yes

—

Dic₂₃

#101

yes

—

C₂×C₄×Q₈

—

—

—

—

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₂²⋊C₈

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₂₃

yes

—

C₄×C₈

—

C₂₃

#1461

yes

characteristic, complemented, Sylow

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₂₃

yes

characteristic

108

C₂²×C₄

C₄×D₂₃

yes

characteristic

109

C₂²×C₄

C₂²×D₂₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₂³

C₂×C₄×D₂₃

yes

characteristic

122

Q₈

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₈×D₂₃

yes

—

140

C₂×C₄

D₄×D₂₃

yes

—

141

C₈

—

142

C₈

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂²⋊C₄×D₂₃

yes

characteristic

146

C₂²

C₂×C₈×D₂₃

yes

characteristic

147

C₂²

M₁₆×D₂₃

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₂

—

157

C₂

—

158

C₂

C₂²⋊C₈×D₂₃

yes

characteristic

159

C₂

#13092389

yes

characteristic

160

C₂

#101×D₂₃

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₃
commutator	C₂²×C₂₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#324	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#324	D₂₃
hypocenter	C₂₃	#1461
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #324

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₃

commutator

C₂²×C₂₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#324

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #324

N	H ≅ G/N	type
28. C₂×Dic₄₆	C₈	semidirect
99. C₂₃	#1461	semidirect

H ≅ G/N

type

28. C₂×Dic₄₆

C₈

semidirect

99. C₂₃

#1461

semidirect

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

subgroup	index
C₂×#10625058	2
#13092467	2
#13092143	2
#13092142	2
C₂₃×#324	2
#1461	23

subgroup

index

C₂₃×#324

Subgroups — #15671719

Conjugacy classes of subgroups

Complements

Maximal subgroups