Group #15673800 — subgroups

Subgroups — #15673800

Order 2944.

344 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 93 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×#10625118	C₂	1	1	yes	maximal, characteristic
2	#13092421	C₂	1	2	yes	maximal
3	C₂×#10625018	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092176	C₂	1	2	yes	maximal, complemented
5	C₂₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×D₉₂	C₂²	1	1	yes	characteristic
8	C₄×Q₈×C₂₃	—	2	4	no	—
9	C₂₃×#117	—	2	2	no	—
10	C₂₃×#117	C₂²	1	2	yes	—
11	#10625118	—	2	2	no	—
12	#10625118	C₄	1	4	yes	complemented
13	C₂×(C₂×D₂₃)⋊C₄	—	2	2	no	—
14	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂×#67267	C₂²	1	2	yes	—
16	#10625018	—	2	2	no	—
17	#10624991	—	2	4	no	—
18	C₂³×D₂₃	—	2	1	no	—
19	C₂×Q₈×C₂₃	—	2	2	no	—
20	C₂×Q₈×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×Q₈×C₂₃	D₄	1	2	yes	—
23	C₂²×C₄×C₂₃	D₄	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₂₃	—	2	1	no	—
26	C₂×D₉₂	—	2	2	no	—
27	C₂×D₉₂	—	2	2	no	—
28	C₂×D₉₂	—	2	1	no	—
29	C₂×D₉₂	C₂×C₄	1	2	yes	—
30	C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₄⋊C₄×C₂₃	D₄	1	4	yes	—
32	C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₄²×C₂₃	—	2	4	no	—
34	C₄²×C₂₃	—	2	2	no	—
35	(C₂×D₂₃)⋊C₄	—	2	8	no	—
36	C₄×Dic₂₃	—	2	1	no	—
37	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
38	#67267	—	2	2	no	—
39	#67267	—	2	2	no	—
40	C₂×C₂₃⋊C₈	—	2	2	no	—
41	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₂₃	—	2	2	no	—
43	C₂²×D₂₃	—	4	4	no	—
44	C₂²×D₂₃	—	4	1	no	—
45	C₂²×D₂₃	—	4	1	no	—
46	Q₈×C₂₃	—	2	4	no	—
47	Q₈×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₃	#41	1	2	yes	—
56	C₂×Dic₂₃	—	2	2	no	—
57	C₂×Dic₂₃	—	4	2	no	—
58	D₉₂	—	2	4	no	—
59	D₉₂	—	4	2	no	—
60	C₂₃⋊C₈	—	4	2	no	—
61	#1888	—	23	1	no	maximal, Sylow
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₂₃	#120	1	1	yes	characteristic
65	C₂²×C₂₃	#123	1	1	yes	characteristic
66	C₂×D₂₃	—	4	4	no	—
67	C₂×D₂₃	—	4	2	no	—
68	C₂×D₂₃	—	8	2	no	—
69	C₂×D₂₃	—	8	2	no	—
70	C₄×C₂₃	—	2	4	no	—
71	C₄×C₂₃	—	4	4	no	—
72	C₄×C₂₃	C₄×D₄	1	2	yes	—
73	C₄×C₂₃	#121	1	2	yes	—
74	Dic₂₃	—	4	2	no	—
75	C₂×#124	—	23	1	no	—
76	#519	D₂₃	1	1	yes	characteristic, complemented
77	#418	—	23	2	no	—
78	C₂×#101	—	23	1	no	—
79	#339	—	23	2	no	—
80	C₂×C₂₃	—	2	1	no	—
81	C₂×C₂₃	—	2	1	no	—
82	C₂×C₂₃	#386	1	1	yes	characteristic
83	C₂×C₂₃	#454	1	2	yes	—
84	D₂₃	—	8	2	no	—
85	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
86	C₂²×D₄	—	23	1	no	—
87	C₂×M₁₆	—	23	2	no	—
88	#124	—	23	4	no	—
89	#124	—	46	2	no	—
90	C₄×Q₈	—	2	4	no	—
91	#117	—	2	2	no	—
92	#117	C₂×D₂₃	1	2	yes	—
93	C₂×C₂²⋊C₄	—	46	2	no	—
94	C₂×C₄²	—	23	1	no	—
95	#102	—	46	4	no	—
96	#101	—	46	2	no	—
97	C₂₃	#1888	1	1	yes	characteristic, complemented, Sylow
98	C₂⁴	—	46	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₄×D₂₃	1	2	yes	—
102	C₂×Q₈	D₉₂	1	2	yes	—
103	C₂×D₄	—	23	2	no	—
104	C₂×D₄	—	46	1	no	—
105	C₂×D₄	—	46	2	no	—
106	C₂×D₄	—	46	2	no	—
107	C₂²×C₄	—	46	1	no	—
108	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
110	M₁₆	—	46	2	no	—
111	M₁₆	—	46	2	no	—
112	C₂×C₈	—	46	2	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	C₂²⋊D₂₃	1	4	yes	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	46	8	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	23	2	no	—
120	C₄²	—	46	1	no	—
121	C₂³	—	46	2	no	—
122	C₂³	—	92	4	no	—
123	C₂³	—	92	1	no	—
124	C₂³	—	92	1	no	—
125	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	46	4	no	—
129	D₄	—	92	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	46	2	no	—
134	C₂×C₄	—	92	2	no	—
135	C₂×C₄	(C₂×D₂₃)⋊C₄	1	4	yes	—
136	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
137	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
138	C₂×C₄	#67288	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
140	C₈	—	92	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	92	4	no	—
143	C₂²	—	92	2	no	—
144	C₂²	—	184	2	no	—
145	C₂²	—	184	2	no	—
146	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
147	C₂²	#10625100	1	1	yes	characteristic
148	C₂²	#10625124	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	4	4	no	—
151	C₄	—	92	2	no	—
152	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
153	C₄	#10625101	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	184	2	no	—
157	C₂	#13092296	1	1	yes	characteristic
158	C₂	#13092590	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625118

C₂

yes

maximal, characteristic

#13092421

C₂

yes

maximal

C₂×#10625018

C₂

yes

maximal, characteristic, complemented

#13092176

C₂

yes

maximal, complemented

C₂₃×#519

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×D₉₂

C₂²

yes

characteristic

C₄×Q₈×C₂₃

—

C₂₃×#117

—

C₂₃×#117

C₂²

yes

—

#10625118

—

#10625118

C₄

yes

complemented

C₂×(C₂×D₂₃)⋊C₄

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

#10625018

—

#10624991

—

C₂³×D₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

(C₂×D₂₃)⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

Q₈×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

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

C₂₃⋊C₈

—

#1888

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

#121

yes

—

Dic₂₃

—

C₂×#124

—

#519

D₂₃

yes

characteristic, complemented

#418

—

C₂×#101

—

#339

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#386

yes

characteristic

C₂×C₂₃

#454

yes

—

D₂₃

—

C₂²×Q₈

C₂×D₂₃

yes

characteristic

C₂²×D₄

—

C₂×M₁₆

—

#124

—

#124

—

C₄×Q₈

—

#117

—

#117

C₂×D₂₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#102

—

#101

—

C₂₃

#1888

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₄×D₂₃

yes

—

102

C₂×Q₈

D₉₂

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₂₃

yes

—

109

C₂²×C₄

C₂²×D₂₃

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂²⋊D₂₃

yes

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

136

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

137

C₂×C₄

C₂×D₉₂

yes

characteristic

138

C₂×C₄

#67288

yes

—

139

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

184

—

145

C₂²

—

184

—

146

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

147

C₂²

#10625100

yes

characteristic

148

C₂²

#10625124

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₄×C₂²⋊D₂₃

yes

—

153

C₄

#10625101

yes

—

154

C₂

—

155

C₂

—

156

C₂

—

184

—

157

C₂

#13092296

yes

characteristic

158

C₂

#13092590

yes

—

	subgroup	G/H
center	C₂²	#10625124
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#519	C₂
socle	C₂²×C₂₃	#120
prime radical	C₂²×D₉₂	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#519	D₂₃
hypocenter	C₂₃	#1888
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #519

subgroup

G/H

center

C₂²

#10625124

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#519

C₂

socle

C₂²×C₂₃

#120

prime radical

C₂²×D₉₂

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #519

N	H ≅ G/N	type
3. C₂×#10625018	C₂	semidirect
4. #13092176	C₂	semidirect
5. C₂₃×#519	C₂	semidirect
12. #10625118	C₄	semidirect
76. #519	D₂₃	semidirect
97. C₂₃	#1888	semidirect

H ≅ G/N

type

3. C₂×#10625018

C₂

semidirect

4. #13092176

C₂

semidirect

5. C₂₃×#519

C₂

semidirect

12. #10625118

C₄

semidirect

76. #519

D₂₃

semidirect

97. C₂₃

#1888

semidirect

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

subgroup	index
C₂×#10625118	2
#13092421	2
C₂×#10625018	2
#13092176	2
C₂₃×#519	2
#1888	23

subgroup

index

C₂₃×#519

Subgroups — #15673800

Conjugacy classes of subgroups

Complements

Maximal subgroups