Group #15674550 — subgroups

Subgroups — #15674550

Order 2944.

282 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 71 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	#13092421	C₂	1	2	yes	maximal
2	#13092318	C₂	1	1	yes	maximal, characteristic
3	#13092186	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092176	C₂	1	2	yes	maximal, complemented
5	C₂₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₉₂	C₂²	1	1	yes	characteristic
7	C₂₃×C₄⋊Q₈	—	2	2	no	—
8	C₂₃×#125	—	2	2	no	—
9	C₂₃×C₂²⋊Q₈	—	2	2	no	—
10	C₂₃×#117	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₂₃)⋊C₄	—	2	2	no	—
13	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×#67267	C₂²	1	2	yes	—
15	#10624991	—	2	2	no	—
16	#10624991	—	2	2	no	—
17	C₂³×D₂₃	—	2	1	no	—
18	C₂×Q₈×C₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₃	D₄	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₂₃	—	2	1	no	—
22	C₂×D₉₂	—	2	2	no	—
23	C₂×D₉₂	—	2	2	no	—
24	C₂×D₉₂	—	2	1	no	—
25	C₂×D₉₂	Q₈	1	2	yes	complemented
26	C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₃	—	2	4	no	—
28	C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
30	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
31	C₂²⋊C₄×C₂₃	—	2	2	no	—
32	C₄²×C₂₃	—	2	2	no	—
33	(C₂×D₂₃)⋊C₄	—	4	4	no	—
34	C₄×Dic₂₃	—	2	2	no	—
35	#67267	—	2	2	no	—
36	#67267	—	2	2	no	—
37	C₂×C₂₃⋊C₈	—	2	2	no	—
38	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×D₂₃	—	2	2	no	—
40	C₂²×D₂₃	—	4	2	no	—
41	C₂²×D₂₃	—	4	2	no	—
42	C₂²×D₂₃	—	4	1	no	—
43	C₂²×D₂₃	—	4	1	no	—
44	Q₈×C₂₃	—	8	2	no	—
45	C₂×C₄×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	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#41	1	2	yes	—
52	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	—	4	1	no	—
55	C₂×Dic₂₃	—	4	1	no	—
56	D₉₂	—	2	4	no	—
57	D₉₂	—	4	2	no	—
58	C₂₃⋊C₈	—	4	2	no	—
59	#2027	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	—	2	1	no	—
61	C₂²×C₂₃	—	2	1	no	—
62	C₂²×C₂₃	#120	1	1	yes	characteristic
63	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₂₃	#124	1	1	yes	characteristic
65	C₂×D₂₃	—	4	4	no	—
66	C₂×D₂₃	—	4	2	no	—
67	C₂×D₂₃	—	8	2	no	—
68	C₂×D₂₃	—	8	1	no	—
69	C₂×D₂₃	—	8	1	no	—
70	C₄×C₂₃	—	4	2	no	—
71	C₄×C₂₃	—	4	2	no	—
72	C₄×C₂₃	—	4	2	no	—
73	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₂₃	#124	1	2	yes	—
75	Dic₂₃	—	4	2	no	—
76	#544	D₂₃	1	1	yes	characteristic, complemented
77	#418	—	23	2	no	—
78	#390	—	23	1	no	—
79	#343	—	23	1	no	—
80	#339	—	23	2	no	—
81	C₂×C₂₃	—	2	1	no	—
82	C₂×C₂₃	—	2	1	no	—
83	C₂×C₂₃	#393	1	1	yes	characteristic
84	C₂×C₂₃	#453	1	1	yes	characteristic
85	C₂×C₂₃	#454	1	1	yes	characteristic
86	D₂₃	—	8	2	no	—
87	C₂²×D₄	—	23	1	no	—
88	C₂×M₁₆	—	23	2	no	—
89	C₄⋊Q₈	—	2	2	no	—
90	#125	—	2	2	no	—
91	C₂²⋊Q₈	—	2	2	no	—
92	#117	C₂×D₂₃	1	2	yes	—
93	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
94	C₂×C₂²⋊C₄	—	46	2	no	—
95	C₂×C₄²	—	23	1	no	—
96	#102	—	46	2	no	—
97	#102	—	46	2	no	—
98	C₂₃	#2027	1	1	yes	characteristic, complemented, Sylow
99	C₂⁴	—	46	1	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×D₄	—	23	2	no	—
102	C₂×D₄	—	46	2	no	—
103	C₂×D₄	—	46	1	no	—
104	C₂×D₄	—	46	2	no	—
105	C₂²×C₄	—	46	1	no	—
106	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
107	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
108	M₁₆	—	46	2	no	—
109	M₁₆	—	46	2	no	—
110	C₂×C₈	—	46	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	C₂²⋊D₂₃	1	2	yes	—
115	C₄⋊C₄	C₂²⋊D₂₃	1	2	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	92	4	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	46	2	no	—
120	C₂³	—	46	2	no	—
121	C₂³	—	92	2	no	—
122	C₂³	—	92	2	no	—
123	C₂³	—	92	1	no	—
124	C₂³	—	92	1	no	—
125	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
126	Q₈	—	8	2	no	—
127	D₄	—	46	4	no	—
128	D₄	—	92	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	46	2	no	—
134	C₂×C₄	—	92	1	no	—
135	C₂×C₄	—	92	1	no	—
136	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
137	C₂×C₄	#67292	1	2	yes	—
138	C₂×C₄	#67292	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
140	C₈	—	92	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	92	4	no	—
144	C₂²	—	92	2	no	—
145	C₂²	—	184	2	no	—
146	C₂²	—	184	1	no	—
147	C₂²	—	184	1	no	—
148	C₂²	#10625117	1	1	yes	characteristic
149	C₂²	#10625118	1	1	yes	characteristic
150	C₂²	#10625124	1	1	yes	characteristic
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	92	2	no	—
155	C₄	#10625117	1	2	yes	—
156	C₄	#10625118	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	184	2	no	—
160	C₂	#13092328	1	1	yes	characteristic
161	C₂	#13092585	1	1	yes	characteristic
162	C₂	#13092590	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092421

C₂

yes

maximal

#13092318

C₂

yes

maximal, characteristic

#13092186

C₂

yes

maximal, characteristic, complemented

#13092176

C₂

yes

maximal, complemented

C₂₃×#544

C₂

yes

maximal, characteristic, complemented

C₂²×D₉₂

C₂²

yes

characteristic

C₂₃×C₄⋊Q₈

—

C₂₃×#125

—

C₂₃×C₂²⋊Q₈

—

C₂₃×#117

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×(C₂×D₂₃)⋊C₄

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

#10624991

—

#10624991

—

C₂³×D₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

Q₈

yes

complemented

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

(C₂×D₂₃)⋊C₄

—

C₄×Dic₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

C₂₃⋊C₈

—

#2027

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃

#124

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

—

Dic₂₃

—

#544

D₂₃

yes

characteristic, complemented

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

D₂₃

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₂₃

yes

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#102

—

#102

—

C₂₃

#2027

yes

characteristic, complemented, Sylow

C₂⁴

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²⋊D₂₃

yes

—

107

C₂²×C₄

C₂²×D₂₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂²⋊D₂₃

yes

—

115

C₄⋊C₄

C₂²⋊D₂₃

yes

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

126

Q₈

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

Q₈×D₂₃

yes

characteristic

137

C₂×C₄

#67292

yes

—

138

C₂×C₄

#67292

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

184

—

146

C₂²

—

184

—

147

C₂²

—

184

—

148

C₂²

#10625117

yes

characteristic

149

C₂²

#10625118

yes

characteristic

150

C₂²

#10625124

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#10625117

yes

—

156

C₄

#10625118

yes

—

157

C₂

—

158

C₂

—

159

C₂

—

184

—

160

C₂

#13092328

yes

characteristic

161

C₂

#13092585

yes

characteristic

162

C₂

#13092590

yes

characteristic

	subgroup	G/H
center	C₂²	#10625124
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#544	C₂
socle	C₂²×C₂₃	#120
prime radical	C₂²×D₉₂	C₂²
hypercenter	#544	D₂₃
hypocenter	C₂₃	#2027
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#10625124

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#544

C₂

socle

C₂²×C₂₃

#120

prime radical

C₂²×D₉₂

C₂²

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
3. #13092186	C₂	semidirect
4. #13092176	C₂	semidirect
5. C₂₃×#544	C₂	semidirect
25. C₂×D₉₂	Q₈	semidirect
76. #544	D₂₃	semidirect
98. C₂₃	#2027	semidirect

H ≅ G/N

type

3. #13092186

C₂

semidirect

4. #13092176

C₂

semidirect

5. C₂₃×#544

C₂

semidirect

25. C₂×D₉₂

Q₈

semidirect

76. #544

D₂₃

semidirect

98. C₂₃

#2027

semidirect

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

subgroup	index
#13092421	2
#13092318	2
#13092186	2
#13092176	2
C₂₃×#544	2
#2027	23

subgroup

index

C₂₃×#544

Subgroups — #15674550

Conjugacy classes of subgroups

Complements

Maximal subgroups