Group #15673786 — subgroups

Subgroups — #15673786

Order 2944.

360 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13092892	C₂	1	1	yes	maximal, characteristic
2	#13092426	C₂	1	2	yes	maximal, complemented
3	C₂×#10625015	C₂	1	1	yes	maximal, characteristic
4	#13092177	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#124	—	2	2	no	—
9	C₂₃×#124	C₄	1	4	yes	complemented
10	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Dic₂₃	—	2	2	no	—
13	D₄×Dic₂₃	—	2	4	no	—
14	#10625097	C₂²	1	2	yes	—
15	C₂×#67267	C₂²	1	2	yes	—
16	#10625017	—	2	4	no	—
17	#10625015	—	2	2	no	—
18	C₂⁴×C₂₃	D₄	1	2	yes	—
19	C₂×Q₈×C₂₃	—	2	1	no	—
20	C₂×Q₈×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	—	2	2	no	—
22	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	complemented
23	C₂×D₄×C₂₃	—	2	2	no	—
24	C₂×D₄×C₂₃	—	2	2	no	—
25	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×D₄×C₂₃	D₄	1	2	yes	—
27	C₂²×C₄×C₂₃	—	2	1	no	—
28	C₂²×C₄×C₂₃	—	2	1	no	—
29	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₂₃	—	2	4	no	—
31	C₂²⋊C₄×C₂₃	—	2	8	no	—
32	C₄²×C₂₃	—	2	1	no	—
33	C₄²×C₂₃	C₂×C₄	1	2	yes	complemented
34	C₂²⋊Dic₂₃	—	2	4	no	—
35	C₂²⋊Dic₂₃	—	2	2	no	—
36	C₄⋊Dic₂₃	D₄	1	4	yes	—
37	C₄×Dic₂₃	—	2	2	no	—
38	#67267	—	2	2	no	—
39	#67267	—	2	2	no	—
40	C₂×C₂₃⋊C₈	—	2	2	no	—
41	C₂³×C₂₃	—	2	2	no	—
42	C₂³×C₂₃	—	2	2	no	—
43	C₂³×C₂₃	—	4	4	no	—
44	C₂³×C₂₃	C₂²⋊C₄	1	4	yes	—
45	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₂₃	—	2	4	no	—
47	Q₈×C₂₃	—	4	2	no	—
48	D₄×C₂₃	—	2	4	no	—
49	D₄×C₂₃	—	2	4	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	4	2	no	—
53	C₂×C₄×C₂₃	—	4	1	no	—
54	C₂×C₄×C₂₃	—	4	1	no	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	#41	1	2	yes	—
59	C₂×Dic₂₃	—	2	4	no	—
60	C₂×Dic₂₃	—	8	4	no	—
61	C₂₃⋊C₈	—	4	2	no	—
62	#1885	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	—	2	4	no	—
65	C₂²×C₂₃	—	4	4	no	—
66	C₂²×C₂₃	—	4	4	no	—
67	C₂²×C₂₃	—	8	2	no	—
68	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₃	#120	1	1	yes	characteristic
70	C₂²×C₂₃	#123	1	1	yes	characteristic
71	C₄×C₂₃	—	4	2	no	—
72	C₄×C₂₃	—	4	2	no	—
73	C₄×C₂₃	C₄×D₄	1	2	yes	—
74	C₄×C₂₃	#121	1	2	yes	—
75	Dic₂₃	—	4	4	no	—
76	C₂×#124	D₂₃	1	1	yes	characteristic
77	#518	—	23	1	no	—
78	#419	—	23	2	no	—
79	C₂×#100	—	23	1	no	—
80	#339	—	23	2	no	—
81	C₂×C₂₃	—	2	1	no	—
82	C₂×C₂₃	—	2	1	no	—
83	C₂×C₂₃	—	4	4	no	—
84	C₂×C₂₃	#386	1	1	yes	characteristic
85	C₂×C₂₃	#455	1	2	yes	—
86	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
87	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
88	C₂×M₁₆	—	23	2	no	—
89	#124	—	2	2	no	—
90	#124	Dic₂₃	1	4	yes	complemented
91	C₄×D₄	—	46	4	no	—
92	#117	—	23	2	no	—
93	C₂×C₂²⋊C₄	—	2	2	no	—
94	C₂×C₂²⋊C₄	—	46	2	no	—
95	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
96	#103	—	46	4	no	—
97	#100	—	46	2	no	—
98	C₂₃	#1885	1	1	yes	characteristic, complemented, Sylow
99	C₂⁴	C₂²⋊D₂₃	1	2	yes	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	C₂×Dic₂₃	1	2	yes	complemented
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
107	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	46	4	no	—
111	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	M₁₆	—	46	2	no	—
113	M₁₆	—	46	2	no	—
114	C₂×C₈	—	46	2	no	—
115	C₄⋊C₄	—	23	4	no	—
116	C₂²⋊C₄	—	2	8	no	—
117	C₂²⋊C₄	—	46	4	no	—
118	C₂²⋊C₄	—	46	2	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	46	2	no	—
121	C₄²	C₂×Dic₂₃	1	2	yes	complemented
122	C₂³	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	4	4	no	—
125	C₂³	C₂²⋊Dic₂₃	1	4	yes	—
126	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	4	2	no	—
129	D₄	—	2	4	no	—
130	D₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	46	4	no	—
137	C₂×C₄	—	184	4	no	—
138	C₂×C₄	D₄×D₂₃	1	2	yes	—
139	C₂×C₄	#67290	1	2	yes	—
140	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
142	C₈	—	92	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	4	no	—
145	C₂²	—	4	4	no	—
146	C₂²	—	4	4	no	—
147	C₂²	—	8	2	no	—
148	C₂²	#10625106	1	1	yes	characteristic
149	C₂²	#10625108	1	1	yes	characteristic
150	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	92	4	no	—
154	C₄	D₄×Dic₂₃	1	2	yes	—
155	C₄	#10625110	1	2	yes	—
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	—	4	4	no	—
159	C₂	#13092297	1	1	yes	characteristic
160	C₂	#13092594	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092892

C₂

yes

maximal, characteristic

#13092426

C₂

yes

maximal, complemented

C₂×#10625015

C₂

yes

maximal, characteristic

#13092177

C₂

yes

maximal, complemented

C₂×C₂₃×#124

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂₃×#124

—

C₂₃×#124

C₄

yes

complemented

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₂₃

—

D₄×Dic₂₃

—

#10625097

C₂²

yes

—

C₂×#67267

C₂²

yes

—

#10625017

—

#10625015

—

C₂⁴×C₂₃

D₄

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

complemented

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

complemented

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

C₄×Dic₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

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×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1885

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

#121

yes

—

Dic₂₃

—

C₂×#124

D₂₃

yes

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#386

yes

characteristic

C₂×C₂₃

#455

yes

—

C₂²×Q₈

C₂×D₂₃

yes

characteristic

C₂²×D₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

#124

—

#124

Dic₂₃

yes

complemented

C₄×D₄

—

#117

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#103

—

#100

—

C₂₃

#1885

yes

characteristic, complemented, Sylow

C₂⁴

C₂²⋊D₂₃

yes

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

C₂×Dic₂₃

yes

complemented

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

C₂×Dic₂₃

yes

—

107

C₂×D₄

C₂²⋊D₂₃

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₂₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₂×Dic₂₃

yes

complemented

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂²⋊Dic₂₃

yes

—

126

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

184

—

138

C₂×C₄

D₄×D₂₃

yes

—

139

C₂×C₄

#67290

yes

—

140

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

142

C₈

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#10625106

yes

characteristic

149

C₂²

#10625108

yes

characteristic

150

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

D₄×Dic₂₃

yes

—

155

C₄

#10625110

yes

—

156

C₂

—

157

C₂

—

158

C₂

—

159

C₂

#13092297

yes

characteristic

160

C₂

#13092594

yes

—

	subgroup	G/H
center	C₂²	#10625108
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#124	C₂
socle	C₂²×C₂₃	#120
prime radical	C₂²×D₄×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#124	D₂₃
hypocenter	C₂₃	#1885
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#124

subgroup

G/H

center

C₂²

#10625108

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#124

C₂

socle

C₂²×C₂₃

#120

prime radical

C₂²×D₄×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#124

N	H ≅ G/N	type
2. #13092426	C₂	semidirect
4. #13092177	C₂	semidirect
9. C₂₃×#124	C₄	semidirect
22. C₂×Q₈×C₂₃	C₂×C₄	semidirect
33. C₄²×C₂₃	C₂×C₄	semidirect
90. #124	Dic₂₃	semidirect
98. C₂₃	#1885	semidirect
103. C₂×Q₈	C₂×Dic₂₃	semidirect
121. C₄²	C₂×Dic₂₃	semidirect

H ≅ G/N

type

2. #13092426

C₂

semidirect

4. #13092177

C₂

semidirect

9. C₂₃×#124

C₄

semidirect

22. C₂×Q₈×C₂₃

C₂×C₄

semidirect

33. C₄²×C₂₃

C₂×C₄

semidirect

90. #124

Dic₂₃

semidirect

98. C₂₃

#1885

semidirect

103. C₂×Q₈

C₂×Dic₂₃

semidirect

121. C₄²

C₂×Dic₂₃

semidirect

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

subgroup	index
#13092892	2
#13092426	2
C₂×#10625015	2
#13092177	2
C₂×C₂₃×#124	2
#1885	23

subgroup

index

C₂×C₂₃×#124

Subgroups — #15673786

Conjugacy classes of subgroups

Complements

Maximal subgroups