Group #15673421 — subgroups

Subgroups — #15673421

Order 2944.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624990	C₂	1	2	yes	maximal, complemented
3	#13092274	C₂	1	2	yes	maximal
4	#13092180	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₂×(C₂×D₂₃)⋊C₄	—	2	2	no	—
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₂₃	—	2	4	no	—
13	C₂×C₈×D₂₃	C₄	1	4	yes	complemented
14	C₂₃⋊#95	—	2	2	no	—
15	#10624990	—	2	4	no	—
16	C₂³×D₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	—	2	2	no	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₃	—	2	2	no	—
20	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×D₂₃	—	2	2	no	—
22	C₂×C₄×D₂₃	—	2	2	no	—
23	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₂₃	Q₈	1	2	yes	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₄⋊C₄×C₂₃	—	2	2	no	—
29	(C₂×D₂₃)⋊C₄	—	4	4	no	—
30	C₄⋊Dic₂₃	—	2	2	no	—
31	C₄⋊Dic₂₃	—	2	2	no	—
32	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
34	C₈×D₂₃	—	2	8	no	—
35	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×D₂₃	—	2	2	no	—
37	C₂²×D₂₃	—	2	1	no	—
38	C₂²×D₂₃	—	2	1	no	—
39	C₂²×D₂₃	C₄⋊C₄	1	2	yes	—
40	C₂²×D₂₃	SD₁₆	1	4	yes	—
41	C₂×C₄×C₂₃	—	4	2	no	—
42	C₂×C₄×C₂₃	—	4	2	no	—
43	C₂×C₄×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#41	1	2	yes	—
48	C₂×Dic₂₃	—	2	2	no	—
49	C₂×Dic₂₃	—	4	2	no	—
50	C₂×Dic₂₃	—	4	2	no	—
51	C₂×Dic₂₃	—	4	2	no	—
52	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
53	C₄×D₂₃	—	2	2	no	—
54	C₄×D₂₃	—	2	2	no	—
55	C₄×D₂₃	—	2	4	no	—
56	C₈×C₂₃	—	2	2	no	—
57	C₂₃⋊C₈	C₂²⋊C₄	1	4	yes	complemented
58	#1807	—	23	1	no	maximal, Sylow
59	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₃	#121	1	2	yes	—
61	C₂²×C₂₃	C₂×SD₁₆	1	2	yes	—
62	C₂²×C₂₃	#135	1	1	yes	characteristic
63	C₂²×C₂₃	#135	1	1	yes	characteristic
64	C₂×D₂₃	—	2	2	no	—
65	C₂×D₂₃	—	2	2	no	—
66	C₂×D₂₃	—	2	4	no	—
67	C₂×D₂₃	—	2	4	no	—
68	C₂×D₂₃	#106	1	4	yes	complemented
69	C₄×C₂₃	—	8	2	no	—
70	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₂₃	#117	1	1	yes	characteristic
72	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
73	Dic₂₃	—	2	2	no	—
74	Dic₂₃	—	8	2	no	—
75	C₂³×C₈	—	23	1	no	—
76	C₂×#106	—	23	2	no	—
77	#380	—	23	2	no	—
78	#340	—	23	1	no	—
79	#340	D₂₃	1	1	yes	characteristic, complemented
80	C₂×C₂₃	#380	1	1	yes	characteristic
81	C₂×C₂₃	C₂×#106	1	1	yes	characteristic
82	C₂×C₂₃	#427	1	1	yes	characteristic
83	C₂×C₂₃	#465	1	2	yes	—
84	C₂×C₂₃	#465	1	2	yes	—
85	D₂₃	—	2	4	no	—
86	C₂³×C₄	—	23	1	no	—
87	C₂²×C₈	—	23	1	no	—
88	C₂²×C₈	—	23	4	no	—
89	C₂²×C₈	—	46	4	no	—
90	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	23	2	no	—
92	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
93	C₂×C₂²⋊C₄	—	46	2	no	—
94	#106	—	46	4	no	—
95	#95	—	46	2	no	—
96	C₂₃	#1807	1	1	yes	characteristic, complemented, Sylow
97	C₂⁴	—	23	1	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	23	1	no	—
100	C₂²×C₄	—	23	2	no	—
101	C₂²×C₄	—	23	2	no	—
102	C₂²×C₄	—	46	2	no	—
103	C₂²×C₄	—	46	2	no	—
104	C₂²×C₄	—	46	2	no	—
105	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	23	2	no	—
108	C₂×C₈	—	23	4	no	—
109	C₂×C₈	—	46	8	no	—
110	C₂×C₈	C₄×D₂₃	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	46	2	no	—
114	C₄⋊C₄	—	46	2	no	—
115	C₂²⋊C₄	—	92	4	no	—
116	C₂³	—	23	2	no	—
117	C₂³	—	23	4	no	—
118	C₂³	—	46	2	no	—
119	C₂³	—	46	1	no	—
120	C₂³	—	46	1	no	—
121	C₂³	D₄×D₂₃	1	1	yes	characteristic
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	23	2	no	—
126	C₂×C₄	—	46	2	no	—
127	C₂×C₄	—	46	2	no	—
128	C₂×C₄	—	46	2	no	—
129	C₂×C₄	—	46	4	no	—
130	C₂×C₄	—	92	2	no	—
131	C₂×C₄	—	92	2	no	—
132	C₂×C₄	—	92	2	no	—
133	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
134	C₂×C₄	#67288	1	2	yes	—
135	C₂×C₄	D₄×D₂₃	1	2	yes	—
136	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	23	4	no	—
139	C₂²	—	23	4	no	—
140	C₂²	—	46	4	no	—
141	C₂²	—	46	4	no	—
142	C₂²	—	46	2	no	—
143	C₂²	—	46	2	no	—
144	C₂²	#10625055	1	2	yes	—
145	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
146	C₂²	#10625083	1	1	yes	characteristic
147	C₂²	SD₁₆×D₂₃	1	2	yes	—
148	C₂²	#10625090	1	1	yes	characteristic
149	C₄	—	8	2	no	—
150	C₄	—	46	2	no	—
151	C₄	—	184	2	no	—
152	C₄	#10625043	1	1	yes	characteristic
153	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
154	C₄	#10625067	1	2	yes	—
155	C₂	—	46	4	no	—
156	C₂	#13092276	1	1	yes	characteristic
157	C₂	#106×D₂₃	1	1	yes	characteristic
158	C₂	#13092454	1	1	yes	characteristic
159	C₂	#13092645	1	2	yes	—
160	C₂	#13092649	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₂₃

C₂

yes

maximal, characteristic

C₂×#10624990

C₂

yes

maximal, complemented

#13092274

C₂

yes

maximal

#13092180

C₂

yes

maximal, characteristic, complemented

C₂₃×#340

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×(C₂×D₂₃)⋊C₄

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₂₃

—

C₂×C₈×D₂₃

C₄

yes

complemented

C₂₃⋊#95

—

#10624990

—

C₂³×D₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

D₄

yes

characteristic

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

Q₈

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₈×D₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

C₄⋊C₄

yes

—

C₂²×D₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

C₂²⋊C₄

yes

complemented

#1807

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₂²×C₂₃

C₂×SD₁₆

yes

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

#106

yes

complemented

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

#117

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

C₂³×C₈

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#465

yes

—

D₂₃

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₂²⋊C₄

—

#106

—

#95

—

C₂₃

#1807

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₂₃

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₄×D₂₃

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

D₄×D₂₃

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

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

C₂×C₄×D₂₃

yes

characteristic

134

C₂×C₄

#67288

yes

—

135

C₂×C₄

D₄×D₂₃

yes

—

136

C₂×C₄

Q₈×D₂₃

yes

characteristic

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#10625055

yes

—

145

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

146

C₂²

#10625083

yes

characteristic

147

C₂²

SD₁₆×D₂₃

yes

—

148

C₂²

#10625090

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

184

—

152

C₄

#10625043

yes

characteristic

153

C₄

C₂²⋊C₄×D₂₃

yes

characteristic

154

C₄

#10625067

yes

—

155

C₂

—

156

C₂

#13092276

yes

characteristic

157

C₂

#106×D₂₃

yes

characteristic

158

C₂

#13092454

yes

characteristic

159

C₂

#13092645

yes

—

160

C₂

#13092649

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#340	C₂
socle	C₂³×C₂₃	C₂×D₄
prime radical	C₂³×D₂₃	D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#340	D₂₃
hypocenter	C₂₃	#1807
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#340

C₂

socle

C₂³×C₂₃

C₂×D₄

prime radical

C₂³×D₂₃

D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
2. C₂×#10624990	C₂	semidirect
4. #13092180	C₂	semidirect
5. C₂₃×#340	C₂	semidirect
13. C₂×C₈×D₂₃	C₄	semidirect
57. C₂₃⋊C₈	C₂²⋊C₄	semidirect
68. C₂×D₂₃	#106	semidirect
79. #340	D₂₃	semidirect
96. C₂₃	#1807	semidirect

H ≅ G/N

type

2. C₂×#10624990

C₂

semidirect

4. #13092180

C₂

semidirect

5. C₂₃×#340

C₂

semidirect

13. C₂×C₈×D₂₃

C₄

semidirect

57. C₂₃⋊C₈

C₂²⋊C₄

semidirect

68. C₂×D₂₃

#106

semidirect

79. #340

D₂₃

semidirect

96. C₂₃

#1807

semidirect

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

subgroup	index
C₂²×C₈×D₂₃	2
C₂×#10624990	2
#13092274	2
#13092180	2
C₂₃×#340	2
#1807	23

subgroup

index

C₂²×C₈×D₂₃

C₂₃×#340

Subgroups — #15673421

Conjugacy classes of subgroups

Complements

Maximal subgroups