Group #15673922 — subgroups

Subgroups — #15673922

Order 2944.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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	#13093184	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092981	C₂	1	1	yes	maximal, characteristic
3	#13092421	C₂	1	2	yes	maximal, complemented
4	#13092173	C₂	1	2	yes	maximal, complemented
5	C₂₃×#410	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×D₉₂	C₂²	1	1	yes	characteristic
8	C₂₃×#117	C₂²	1	2	yes	—
9	#10625119	—	2	4	no	—
10	#10625118	C₄	1	2	yes	complemented
11	#10625111	C₄	1	2	yes	complemented
12	#10625108	—	2	2	no	—
13	#10625101	—	2	2	no	—
14	#10625097	C₂²	1	1	yes	characteristic
15	C₂×#67267	—	2	2	no	—
16	C₂×#67267	C₂²	1	2	yes	—
17	C₂₃×#99	—	2	2	no	—
18	#10624991	—	2	4	no	—
19	C₂³×D₂₃	—	2	1	no	—
20	#41×C₂₃	—	2	2	no	—
21	#41×C₂₃	—	2	2	no	—
22	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	D₄	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊D₂₃	—	2	4	no	—
28	C₂×D₉₂	—	2	2	no	—
29	C₂×D₉₂	—	2	2	no	—
30	C₂×D₉₂	—	2	1	no	—
31	C₂×D₉₂	C₂×C₄	1	2	yes	—
32	C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂²⋊C₄×C₂₃	—	2	2	no	—
34	C₄²×C₂₃	—	2	2	no	—
35	C₂²⋊Dic₂₃	—	2	1	no	—
36	(C₂×D₂₃)⋊C₄	—	2	4	no	—
37	C₄⋊Dic₂₃	—	2	1	no	—
38	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
39	#67267	—	2	4	no	—
40	#67267	—	2	2	no	—
41	#67267	—	2	2	no	—
42	C₂×C₂₃⋊C₈	—	2	4	no	—
43	C₂×C₂₃⋊C₈	D₄	1	4	yes	complemented
44	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₂₃	—	2	2	no	—
47	C₂²×D₂₃	—	4	2	no	—
48	C₂²×D₂₃	—	8	2	no	—
49	Q₈×C₂₃	—	2	2	no	—
50	D₄×C₂₃	—	2	4	no	—
51	D₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	4	2	no	—
55	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₃	#41	1	2	yes	—
60	C₂²⋊D₂₃	—	8	4	no	—
61	C₂×Dic₂₃	—	2	2	no	—
62	D₉₂	—	2	4	no	—
63	D₉₂	—	4	1	no	—
64	D₉₂	—	4	1	no	—
65	C₂₃⋊C₈	—	2	4	no	—
66	#1910	—	23	1	no	maximal, Sylow
67	C₂²×C₂₃	—	2	2	no	—
68	C₂²×C₂₃	—	4	1	no	—
69	C₂²×C₂₃	—	4	2	no	—
70	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₂₃	#121	1	2	yes	—
72	C₂×D₂₃	—	4	4	no	—
73	C₂×D₂₃	—	4	2	no	—
74	C₂×D₂₃	—	8	4	no	—
75	C₄×C₂₃	—	2	2	no	—
76	C₄×C₂₃	—	8	2	no	—
77	C₄×C₂₃	C₄×D₄	1	2	yes	—
78	C₄×C₂₃	#120	1	1	yes	characteristic
79	C₄×C₂₃	#123	1	1	yes	characteristic
80	Dic₂₃	—	4	2	no	—
81	#568	—	23	1	no	—
82	#535	—	23	1	no	—
83	#418	—	23	2	no	—
84	#410	D₂₃	1	1	yes	characteristic, complemented
85	H(3,ℤ/4)	—	23	2	no	—
86	C₂×C₂₃	—	2	1	no	—
87	C₂×C₂₃	—	2	2	no	—
88	C₂×C₂₃	—	4	2	no	—
89	C₂×C₂₃	#386	1	1	yes	characteristic
90	D₂₃	—	8	2	no	—
91	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
92	C₂²×D₄	—	23	1	no	—
93	#131	—	46	4	no	—
94	C₂×M₁₆	—	23	2	no	—
95	C₂×M₁₆	—	46	2	no	—
96	#127	—	23	2	no	—
97	#124	—	23	2	no	—
98	#121	—	46	2	no	—
99	#120	—	46	2	no	—
100	#117	—	23	1	no	—
101	#117	C₂×D₂₃	1	2	yes	—
102	#102	—	46	4	no	—
103	#99	—	2	2	no	—
104	C₂₃	#1910	1	1	yes	characteristic, complemented, Sylow
105	C₂⁴	—	46	1	no	—
106	#41	—	2	2	no	—
107	#41	—	2	2	no	—
108	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
109	C₂×D₄	—	23	2	no	—
110	C₂×D₄	—	46	4	no	—
111	C₂×D₄	—	46	1	no	—
112	C₂×D₄	—	46	2	no	—
113	C₂×D₄	—	46	2	no	—
114	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
115	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
116	C₂²×C₄	D₉₂	1	2	yes	—
117	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
118	M₁₆	—	46	4	no	—
119	M₁₆	—	46	2	no	—
120	M₁₆	—	46	2	no	—
121	C₂×C₈	—	23	4	no	—
122	C₂×C₈	—	46	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	46	1	no	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	46	1	no	—
127	C₂²⋊C₄	—	46	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	23	2	no	—
130	C₂³	—	46	2	no	—
131	C₂³	—	92	2	no	—
132	C₂³	—	184	2	no	—
133	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
134	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
135	Q₈	—	2	2	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	2	no	—
138	D₄	—	46	4	no	—
139	D₄	—	92	1	no	—
140	D₄	—	92	1	no	—
141	D₄	—	184	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	46	2	no	—
146	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
147	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
148	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
149	C₂×C₄	D₄×D₂₃	1	2	yes	—
150	C₂×C₄	#67290	1	2	yes	—
151	C₈	—	46	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	92	4	no	—
156	C₂²	—	92	2	no	—
157	C₂²	—	184	4	no	—
158	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
159	C₂²	#10625110	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	92	2	no	—
163	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
164	C₄	#10625053	1	1	yes	characteristic
165	C₄	#10625057	1	1	yes	characteristic
166	C₂	—	2	2	no	—
167	C₂	—	2	1	no	—
168	C₂	—	4	2	no	—
169	C₂	—	184	2	no	—
170	C₂	#13092294	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093184

C₂

yes

maximal, characteristic, complemented

#13092981

C₂

yes

maximal, characteristic

#13092421

C₂

yes

maximal, complemented

#13092173

C₂

yes

maximal, complemented

C₂₃×#410

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂²×D₉₂

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

#10625119

—

#10625118

C₄

yes

complemented

#10625111

C₄

yes

—

—

C₂²

yes

characteristic

C₂×#67267

—

C₂×#67267

C₂²

yes

—

C₂₃×#99

—

#10624991

—

C₂³×D₂₃

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂²⋊Dic₂₃

—

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

—

#67267

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

complemented

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×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

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

D₉₂

—

C₂₃⋊C₈

—

#1910

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

—

—

—

D₂₃

yes

characteristic, complemented

H(3,ℤ/4)

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#386

yes

characteristic

D₂₃

—

C₂×#41

C₂×D₂₃

yes

characteristic

C₂²×D₄

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

—

—

—

—

100

#117

—

101

#117

C₂×D₂₃

yes

—

102

#102

—

103

#99

—

104

C₂₃

#1910

yes

characteristic, complemented, Sylow

105

C₂⁴

—

106

#41

—

107

#41

—

108

C₂×Q₈

C₄×D₂₃

yes

characteristic

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

C₄×D₂₃

yes

characteristic

115

C₂×D₄

C₂²⋊D₂₃

yes

—

116

C₂²×C₄

D₉₂

yes

—

117

C₂²×C₄

C₂²×D₂₃

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

184

—

133

C₂³

(C₂×D₂₃)⋊C₄

yes

—

134

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

135

Q₈

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

184

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

147

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

148

C₂×C₄

C₂×D₉₂

yes

characteristic

149

C₂×C₄

D₄×D₂₃

yes

—

150

C₂×C₄

#67290

yes

—

151

C₈

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

184

—

158

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

159

C₂²

#10625110

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

(C₂²⋊D₂₃)⋊C₄

yes

—

164

C₄

#10625053

yes

characteristic

165

C₄

#10625057

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

184

—

170

C₂

#13092294

yes

characteristic

	subgroup	G/H
center	C₂	#13092294
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#410	C₂
socle	C₂×C₂₃	#386
prime radical	#13092981	C₂
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#410	D₂₃
hypocenter	C₂₃	#1910
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₂

#13092294

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#410

C₂

C₂×C₂₃

C₂

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
1. #13093184	C₂	semidirect
3. #13092421	C₂	semidirect
4. #13092173	C₂	semidirect
5. C₂₃×#410	C₂	semidirect
10. #10625118	C₄	semidirect
11. #10625111	C₄	semidirect
43. C₂×C₂₃⋊C₈	D₄	semidirect
84. #410	D₂₃	semidirect
104. C₂₃	#1910	semidirect

H ≅ G/N

type

1. #13093184

C₂

semidirect

3. #13092421

C₂

semidirect

4. #13092173

C₂

semidirect

5. C₂₃×#410

C₂

semidirect

10. #10625118

C₄

semidirect

11. #10625111

C₄

semidirect

43. C₂×C₂₃⋊C₈

D₄

semidirect

84. #410

D₂₃

semidirect

104. C₂₃

#1910

semidirect

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

subgroup	index
#13093184	2
#13092981	2
#13092421	2
#13092173	2
C₂₃×#410	2
#1910	23

subgroup

index

C₂₃×#410

Subgroups — #15673922

Conjugacy classes of subgroups

Complements

Maximal subgroups