Group #15675803 — subgroups

Subgroups — #15675803

Order 2944.

656 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 245 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×#10625063	C₂	1	2	yes	maximal
2	C₂×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092286	C₂	1	2	yes	maximal, complemented
4	#13092283	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092276	C₂	1	4	yes	maximal
6	#13092268	C₂	1	4	yes	maximal, complemented
7	C₂₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
9	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	C₂×Dic₂₃⋊C₄	C₂²	1	8	yes	—
15	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
16	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
18	#10625063	—	2	8	no	—
19	C₄²×D₂₃	C₄	1	8	yes	complemented
20	C₂₃×#95	C₂²	1	4	yes	—
21	C₂₃⋊#95	C₂²	1	4	yes	—
22	C₂³×D₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₃	C₂³	1	4	yes	—
27	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₂₃	C₂³	1	2	yes	—
29	C₂×C₄×D₂₃	—	2	8	no	—
30	C₂×C₄×D₂₃	C₂×C₄	1	8	yes	—
31	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
32	C₂×C₄×D₂₃	D₄	1	4	yes	—
33	C₂×C₄×D₂₃	Q₈	1	4	yes	—
34	C₄⋊C₄×C₂₃	—	2	4	no	—
35	C₄²×C₂₃	C₂×C₄	1	4	yes	—
36	(C₂×D₂₃)⋊C₄	—	2	8	no	—
37	C₄⋊Dic₂₃	—	2	4	no	—
38	Dic₂₃⋊C₄	—	2	16	no	—
39	C₄×Dic₂₃	—	2	8	no	—
40	C₄×Dic₂₃	C₂×C₄	1	8	yes	—
41	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
42	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
43	C₂²×D₂₃	—	2	4	no	—
44	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
45	C₂²×D₂₃	C₂×D₄	1	2	yes	—
46	C₂²×D₂₃	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	—	2	8	no	—
50	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
54	C₂×Dic₂₃	—	2	4	no	—
55	C₂×Dic₂₃	—	2	8	no	—
56	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
57	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
58	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
59	C₂×Dic₂₃	C₂×Q₈	1	2	yes	—
60	C₂×Dic₂₃	#41	1	8	yes	—
61	C₄×D₂₃	—	2	16	no	—
62	C₄×D₂₃	C₄⋊C₄	1	16	yes	complemented
63	#2285	—	23	1	no	maximal, Sylow
64	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
65	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
66	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₂₃	C₂×#41	1	4	yes	—
68	C₂×D₂₃	—	2	4	no	—
69	C₂×D₂₃	—	2	4	no	—
70	C₂×D₂₃	—	2	4	no	—
71	C₂×D₂₃	C₂×C₄⋊C₄	1	4	yes	—
72	C₄×C₂₃	—	2	4	no	—
73	C₄×C₂₃	—	4	4	no	—
74	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
75	Dic₂₃	—	4	4	no	—
76	Dic₂₃	C₂×C₄⋊C₄	1	4	yes	—
77	Dic₂₃	#117	1	8	yes	complemented
78	C₂×#117	—	23	2	no	—
79	C₂²×C₄²	—	23	1	no	—
80	#384	—	23	2	no	—
81	#382	—	23	1	no	—
82	#382	D₂₃	1	1	yes	characteristic, complemented
83	#380	—	23	4	no	—
84	C₄×C₄⋊C₄	—	23	4	no	—
85	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₂₃	C₂×#117	1	2	yes	—
87	C₂×C₂₃	#532	1	2	yes	—
88	C₂×C₂₃	#533	1	2	yes	—
89	D₂₃	—	2	4	no	—
90	C₂³×C₄	—	23	1	no	—
91	C₂³×C₄	—	23	2	no	—
92	#117	—	46	8	no	—
93	C₂×C₄⋊C₄	—	23	2	no	—
94	C₂×C₄⋊C₄	—	23	8	no	—
95	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
96	C₂×C₂²⋊C₄	—	23	4	no	—
97	C₂×C₄²	—	23	2	no	—
98	C₂×C₄²	—	23	1	no	—
99	C₂×C₄²	—	23	8	no	—
100	C₂×C₄²	—	23	4	no	—
101	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
102	#95	—	23	4	no	—
103	#95	C₂×D₂₃	1	4	yes	—
104	C₂₃	#2285	1	1	yes	characteristic, complemented, Sylow
105	C₂⁴	—	23	1	no	—
106	C₂²×C₄	—	23	1	no	—
107	C₂²×C₄	—	23	2	no	—
108	C₂²×C₄	—	23	4	no	—
109	C₂²×C₄	—	23	4	no	—
110	C₂²×C₄	—	23	4	no	—
111	C₂²×C₄	—	23	8	no	—
112	C₂²×C₄	—	23	4	no	—
113	C₂²×C₄	—	46	8	no	—
114	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
115	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
116	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	46	16	no	—
119	C₄⋊C₄	—	46	4	no	—
120	C₂²⋊C₄	—	46	8	no	—
121	C₄²	—	23	8	no	—
122	C₄²	—	23	4	no	—
123	C₄²	—	46	8	no	—
124	C₄²	C₄×D₂₃	1	4	yes	—
125	C₂³	—	23	2	no	—
126	C₂³	—	23	2	no	—
127	C₂³	—	23	2	no	—
128	C₂³	—	46	4	no	—
129	C₂³	C₂³×D₂₃	1	1	yes	characteristic
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	23	2	no	—
134	C₂×C₄	—	23	4	no	—
135	C₂×C₄	—	23	2	no	—
136	C₂×C₄	—	23	2	no	—
137	C₂×C₄	—	23	8	no	—
138	C₂×C₄	—	23	16	no	—
139	C₂×C₄	—	46	16	no	—
140	C₂×C₄	—	46	4	no	—
141	C₂×C₄	—	46	8	no	—
142	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
143	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
144	C₂×C₄	D₄×D₂₃	1	2	yes	—
145	C₂×C₄	Q₈×D₂₃	1	2	yes	—
146	C₂²	—	23	4	no	—
147	C₂²	—	46	4	no	—
148	C₂²	—	46	4	no	—
149	C₂²	—	46	4	no	—
150	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
151	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
152	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
153	C₂²	#41×D₂₃	1	4	yes	—
154	C₄	—	2	4	no	—
155	C₄	—	4	4	no	—
156	C₄	—	23	4	no	—
157	C₄	—	23	8	no	—
158	C₄	—	92	4	no	—
159	C₄	C₄⋊C₄×D₂₃	1	4	yes	—
160	C₂	—	46	4	no	—
161	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
162	C₂	#117×D₂₃	1	2	yes	—
163	C₂	#13092966	1	2	yes	—
164	C₂	#13092973	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625063

C₂

yes

maximal

C₂×C₄²×D₂₃

C₂

yes

maximal, characteristic

#13092286

C₂

yes

maximal, complemented

#13092283

C₂

yes

maximal, characteristic, complemented

#13092276

C₂

yes

maximal

#13092268

C₂

yes

maximal, complemented

C₂₃×#382

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₃

C₂²

yes

—

C₂²×C₄×D₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

#10625063

—

C₄²×D₂₃

C₄

yes

complemented

C₂₃×#95

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂³×D₂₃

C₂³

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂³

yes

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

D₄

yes

—

C₂×C₄×D₂₃

Q₈

yes

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

C₂²×C₄

yes

—

C₂²×D₂₃

C₂×D₄

yes

—

C₂²×D₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²×C₄

yes

—

C₂×Dic₂₃

C₂²×C₄

yes

—

C₂×Dic₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

C₂×Q₈

yes

—

C₂×Dic₂₃

#41

yes

—

C₄×D₂₃

—

C₄×D₂₃

C₄⋊C₄

yes

complemented

#2285

—

maximal, Sylow

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂²×C₂₃

C₂²×Q₈

yes

characteristic

C₂²×C₂₃

C₂×#41

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

C₂×C₄⋊C₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

—

Dic₂₃

—

Dic₂₃

C₂×C₄⋊C₄

yes

—

Dic₂₃

#117

yes

complemented

C₂×#117

—

C₂²×C₄²

—

#384

—

#382

—

#382

D₂₃

yes

characteristic, complemented

#380

—

C₄×C₄⋊C₄

—

C₂×C₂₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₂₃

C₂×#117

yes

—

C₂×C₂₃

#532

yes

—

C₂×C₂₃

#533

yes

—

D₂₃

—

C₂³×C₄

—

C₂³×C₄

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₂₃

yes

characteristic

102

#95

—

103

#95

C₂×D₂₃

yes

—

104

C₂₃

#2285

yes

characteristic, complemented, Sylow

105

C₂⁴

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²×D₂₃

yes

—

115

C₂²×C₄

C₂²×D₂₃

yes

—

116

C₂²×C₄

C₂²×D₂₃

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₄×D₂₃

yes

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂³×D₂₃

yes

characteristic

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

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×C₄×D₂₃

yes

—

143

C₂×C₄

C₂×C₄×D₂₃

yes

—

144

C₂×C₄

D₄×D₂₃

yes

—

145

C₂×C₄

Q₈×D₂₃

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₂₃

yes

characteristic

151

C₂²

C₂×D₄×D₂₃

yes

characteristic

152

C₂²

C₂×Q₈×D₂₃

yes

characteristic

153

C₂²

#41×D₂₃

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

C₄⋊C₄×D₂₃

yes

—

160

C₂

—

161

C₂

C₂×C₄⋊C₄×D₂₃

yes

characteristic

162

C₂

#117×D₂₃

yes

—

163

C₂

#13092966

yes

—

164

C₂

#13092973

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#382	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂³×D₂₃	C₂³
hypercenter	#382	D₂₃
hypocenter	C₂₃	#2285
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₃

commutator

C₂²×C₂₃

C₂³×C₄

Fitting

C₂₃×#382

C₂

socle, cosocle

C₂³×C₂₃

C₂⁴

prime radical

C₂³×D₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
3. #13092286	C₂	semidirect
4. #13092283	C₂	semidirect
6. #13092268	C₂	semidirect
7. C₂₃×#382	C₂	semidirect
19. C₄²×D₂₃	C₄	semidirect
62. C₄×D₂₃	C₄⋊C₄	semidirect
77. Dic₂₃	#117	semidirect
82. #382	D₂₃	semidirect
104. C₂₃	#2285	semidirect

H ≅ G/N

type

3. #13092286

C₂

semidirect

4. #13092283

C₂

semidirect

6. #13092268

C₂

semidirect

7. C₂₃×#382

C₂

semidirect

19. C₄²×D₂₃

C₄

semidirect

62. C₄×D₂₃

C₄⋊C₄

semidirect

77. Dic₂₃

#117

semidirect

82. #382

D₂₃

semidirect

104. C₂₃

#2285

semidirect

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

subgroup	index
C₂×#10625063	2
C₂×C₄²×D₂₃	2
#13092286	2
#13092283	2
#13092276	2
#13092268	2
C₂₃×#382	2
#2285	23

subgroup

index

C₂×C₄²×D₂₃

C₂₃×#382

Subgroups — #15675803

Conjugacy classes of subgroups

Complements

Maximal subgroups