Group #15673489 — subgroups

Subgroups — #15673489

Order 2944.

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

normal subgroups only

This group has 300 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	C₂×#10625097	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#13092179	C₂	1	2	yes	maximal, complemented
4	#13092179	C₂	1	2	yes	maximal, complemented
5	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
9	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
10	#10625097	—	2	2	no	—
11	#10625097	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₃	—	2	1	no	—
15	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
16	C₂₃×#95	—	2	1	no	—
17	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
18	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	—	2	4	no	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₃	D₄	1	2	yes	—
23	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₂₃	—	2	1	no	—
26	C₂²×Dic₂₃	—	2	1	no	—
27	C₄⋊C₄×C₂₃	—	2	4	no	—
28	C₂²⋊C₄×C₂₃	—	4	2	no	—
29	C₂²⋊Dic₂₃	—	2	4	no	—
30	C₄⋊Dic₂₃	—	2	1	no	—
31	C₄⋊Dic₂₃	—	2	1	no	—
32	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
34	C₄×Dic₂₃	—	2	4	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂×C₂₃⋊C₈	—	2	4	no	—
37	C₂³×C₂₃	—	2	2	no	—
38	C₂³×C₂₃	—	2	2	no	—
39	C₂³×C₂₃	—	2	1	no	—
40	C₂³×C₂₃	—	2	1	no	—
41	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	—	4	2	no	—
47	C₂×C₄×C₂₃	—	4	2	no	—
48	C₂×C₄×C₂₃	—	4	2	no	—
49	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃	#41	1	4	yes	—
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	—	2	2	no	—
55	C₂×Dic₂₃	—	4	2	no	—
56	C₂×Dic₂₃	—	4	1	no	—
57	C₂×Dic₂₃	—	4	1	no	—
58	C₂₃⋊C₈	—	4	2	no	—
59	#1825	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	—	2	2	no	—
61	C₂²×C₂₃	—	2	2	no	—
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	—	4	2	no	—
64	C₂²×C₂₃	—	4	2	no	—
65	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₂₃	#123	1	2	yes	—
67	C₂²×C₂₃	#135	1	2	yes	—
68	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂²×C₂₃	#137	1	1	yes	characteristic
70	C₄×C₂₃	—	2	2	no	—
71	C₄×C₂₃	—	8	2	no	—
72	C₄×C₂₃	#117	1	2	yes	—
73	C₄×C₂₃	#123	1	2	yes	—
74	Dic₂₃	—	4	2	no	—
75	Dic₂₃	—	4	2	no	—
76	C₂×#117	—	23	1	no	—
77	C₂×C₂²⋊C₈	—	23	1	no	—
78	#380	D₂₃	1	1	yes	characteristic
79	#340	—	23	2	no	—
80	#340	—	23	2	no	—
81	C₂×C₂₃	—	4	2	no	—
82	C₂×C₂₃	#381	1	1	yes	characteristic
83	C₂×C₂₃	#416	1	1	yes	characteristic
84	C₂×C₂₃	#417	1	1	yes	characteristic
85	C₂×C₂₃	#482	1	2	yes	—
86	C₂×C₂₃	#485	1	2	yes	—
87	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
88	C₂²×C₈	—	23	2	no	—
89	#117	—	23	4	no	—
90	#117	—	46	2	no	—
91	C₂×C₄⋊C₄	—	23	1	no	—
92	C₂×C₄⋊C₄	—	23	1	no	—
93	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
94	C₂×C₂²⋊C₄	—	2	1	no	—
95	C₂×C₂²⋊C₄	—	46	1	no	—
96	C₂×C₄²	—	46	1	no	—
97	C₂²⋊C₈	—	46	2	no	—
98	#95	—	2	1	no	—
99	C₂₃	#1825	1	1	yes	characteristic, complemented, Sylow
100	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	46	1	no	—
104	C₂²×C₄	—	46	1	no	—
105	C₂²×C₄	C₄×D₂₃	1	2	yes	—
106	C₂²×C₄	D₉₂	1	2	yes	—
107	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
109	C₂×C₈	—	46	2	no	—
110	C₂×C₈	—	46	4	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	23	2	no	—
113	C₄⋊C₄	—	23	2	no	—
114	C₄⋊C₄	—	46	1	no	—
115	C₄⋊C₄	—	46	1	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	—	46	4	no	—
118	C₄²	—	46	4	no	—
119	C₂³	—	2	1	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	2	1	no	—
122	C₂³	—	2	2	no	—
123	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
124	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	46	2	no	—
132	C₂×C₄	—	46	2	no	—
133	C₂×C₄	—	92	1	no	—
134	C₂×C₄	—	92	1	no	—
135	C₂×C₄	—	92	2	no	—
136	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
139	C₂×C₄	#67290	1	4	yes	—
140	C₈	—	92	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	2	no	—
146	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
147	C₂²	#10625103	1	1	yes	characteristic
148	C₂²	#10625106	1	2	yes	—
149	C₂²	#10625113	1	1	yes	characteristic
150	C₂²	#10625121	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	92	2	no	—
154	C₄	—	92	2	no	—
155	C₄	#10625048	1	2	yes	—
156	C₄	#10625057	1	2	yes	—
157	C₂	—	4	2	no	—
158	C₂	#13092278	1	1	yes	characteristic
159	C₂	#13092411	1	1	yes	characteristic
160	C₂	#13092416	1	1	yes	characteristic
161	C₂	#13092738	1	2	yes	—
162	C₂	#13092755	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625097

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal, complemented

#13092179

C₂

yes

maximal, complemented

C₂₃×#380

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₂²⋊Dic₂₃

—

#10625097

—

#10625097

C₄

yes

complemented

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×#95

—

C₂²⋊(C₂₃⋊C₈)

—

C₂⁴×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1825

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#123

yes

—

C₂²×C₂₃

#135

yes

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

#123

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×#117

—

C₂×C₂²⋊C₈

—

#380

D₂₃

yes

characteristic

#340

—

#340

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#485

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

#95

—

C₂₃

#1825

yes

characteristic, complemented, Sylow

100

C₂⁴

C₂²⋊D₂₃

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₄×D₂₃

yes

—

106

C₂²×C₄

D₉₂

yes

—

107

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

108

C₂²×C₄

C₂²×D₂₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

(C₂×D₂₃)⋊C₄

yes

—

124

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

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

(C₂×D₂₃)⋊C₄

yes

—

137

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

138

C₂×C₄

C₂×D₉₂

yes

characteristic

139

C₂×C₄

#67290

yes

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

147

C₂²

#10625103

yes

characteristic

148

C₂²

#10625106

yes

—

149

C₂²

#10625113

yes

characteristic

150

C₂²

#10625121

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#10625048

yes

—

156

C₄

#10625057

yes

—

157

C₂

—

158

C₂

#13092278

yes

characteristic

159

C₂

#13092411

yes

characteristic

160

C₂

#13092416

yes

characteristic

161

C₂

#13092738

yes

—

162

C₂

#13092755

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#380	C₂
socle	C₂³×C₂₃	C₂×D₄
prime radical	C₂⁴×C₂₃	D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#380	D₂₃
hypocenter	C₂₃	#1825
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#380

C₂

socle

C₂³×C₂₃

C₂×D₄

prime radical

C₂⁴×C₂₃

D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
3. #13092179	C₂	semidirect
4. #13092179	C₂	semidirect
11. #10625097	C₄	semidirect
99. C₂₃	#1825	semidirect

H ≅ G/N

type

3. #13092179

C₂

semidirect

4. #13092179

C₂

semidirect

11. #10625097

C₄

semidirect

99. C₂₃

#1825

semidirect

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

subgroup	index
C₂×#10625097	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
#13092179	2
C₂₃×#380	2
#1825	23

subgroup

index

C₂×#10625097

C₂×C₂²⋊(C₂₃⋊C₈)

#13092179

C₂₃×#380

#1825

Subgroups — #15673489

Conjugacy classes of subgroups

Complements

Maximal subgroups