Group #15673379 — subgroups

Subgroups — #15673379

Order 2944.

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

normal subgroups only

This group has 312 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	C₂×#10625097	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#13092171	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₂₃	C₄	1	2	yes	—
9	#10625097	—	2	4	no	—
10	C₂×C₄⋊Dic₂₃	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₂₃×C₂²⋊C₈	—	2	2	no	—
15	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
16	Dic₂₃⋊C₈	—	2	4	no	—
17	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
25	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₂×C₈×C₂₃	—	2	2	no	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₂²⋊Dic₂₃	—	2	4	no	—
30	C₄⋊Dic₂₃	—	2	4	no	—
31	C₄×Dic₂₃	—	2	2	no	—
32	C₄×Dic₂₃	D₄	1	4	yes	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	—	2	2	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂³×C₂₃	—	2	1	no	—
37	C₂³×C₂₃	—	2	1	no	—
38	C₂³×C₂₃	—	2	2	no	—
39	C₂³×C₂₃	—	2	2	no	—
40	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
41	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	—	2	1	no	—
43	C₂×C₄×C₂₃	—	2	1	no	—
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₂₃	#41	1	2	yes	—
51	C₂×Dic₂₃	—	2	2	no	—
52	C₂×Dic₂₃	—	2	2	no	—
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	—	2	2	no	—
55	C₂×Dic₂₃	—	2	2	no	—
56	C₂×Dic₂₃	C₂²⋊C₄	1	4	yes	—
57	C₈×C₂₃	—	4	2	no	—
58	C₂₃⋊C₈	—	4	2	no	—
59	#1798	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	—	2	2	no	—
61	C₂²×C₂₃	—	2	2	no	—
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	—	4	2	no	—
64	C₂²×C₂₃	—	4	2	no	—
65	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₂₃	#117	1	1	yes	characteristic
68	C₂²×C₂₃	#131	1	2	yes	—
69	C₂²×C₂₃	#131	1	2	yes	—
70	C₄×C₂₃	—	2	2	no	—
71	C₄×C₂₃	#121	1	2	yes	—
72	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
73	Dic₂₃	—	2	4	no	—
74	Dic₂₃	—	4	2	no	—
75	C₂×#117	—	23	1	no	—
76	C₂×C₄⋊C₈	—	23	2	no	—
77	C₂×C₂²⋊C₈	—	23	1	no	—
78	C₂×C₂²⋊C₈	D₂₃	1	1	yes	characteristic
79	#336	—	23	2	no	—
80	C₂×C₂₃	—	4	2	no	—
81	C₂×C₂₃	#380	1	1	yes	characteristic
82	C₂×C₂₃	#408	1	2	yes	—
83	C₂×C₂₃	#424	1	1	yes	characteristic
84	C₂×C₂₃	#424	1	1	yes	characteristic
85	C₂×C₂₃	#433	1	2	yes	—
86	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
87	C₂²×C₈	—	23	2	no	—
88	C₂²×C₈	C₂×D₂₃	1	2	yes	—
89	#117	—	46	4	no	—
90	C₂×C₄⋊C₄	—	23	2	no	—
91	C₂×C₂²⋊C₄	—	23	2	no	—
92	C₂×C₄²	—	23	1	no	—
93	C₂×C₄²	—	23	1	no	—
94	C₄⋊C₈	—	46	4	no	—
95	C₂²⋊C₈	—	2	2	no	—
96	C₂²⋊C₈	—	46	2	no	—
97	C₂₃	#1798	1	1	yes	characteristic, complemented, Sylow
98	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	23	2	no	—
102	C₂²×C₄	—	23	2	no	—
103	C₂²×C₄	Dic₄₆	1	2	yes	—
104	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
106	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	46	2	no	—
111	C₂×C₈	—	46	2	no	—
112	C₂×C₈	—	46	2	no	—
113	C₄⋊C₄	—	46	4	no	—
114	C₂²⋊C₄	—	46	4	no	—
115	C₄²	—	23	4	no	—
116	C₄²	—	46	2	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	1	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	Dic₂₃⋊C₄	1	2	yes	—
122	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	23	4	no	—
128	C₂×C₄	—	46	2	no	—
129	C₂×C₄	—	46	2	no	—
130	C₂×C₄	—	46	2	no	—
131	C₂×C₄	—	46	2	no	—
132	C₂×C₄	—	46	2	no	—
133	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
134	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
135	C₂×C₄	D₄×D₂₃	1	2	yes	—
136	C₂×C₄	#67290	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	92	2	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	#10625048	1	1	yes	characteristic
146	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
147	C₂²	#10625072	1	2	yes	—
148	C₂²	#10625078	1	2	yes	—
149	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	46	4	no	—
152	C₄	—	92	2	no	—
153	C₄	#10625049	1	2	yes	—
154	C₄	#10625110	1	2	yes	—
155	C₂	—	4	2	no	—
156	C₂	#13092275	1	1	yes	characteristic
157	C₂	#13092382	1	2	yes	—
158	C₂	#13092443	1	1	yes	characteristic
159	C₂	#13092445	1	1	yes	characteristic
160	C₂	#13092475	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625097

C₂

yes

maximal, characteristic

C₂×Dic₂₃⋊C₈

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal, complemented

C₂×C₂₃×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₄

yes

—

#10625097

—

C₂×C₄⋊Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×C₂²⋊C₈

—

C₂²⋊(C₂₃⋊C₈)

—

Dic₂₃⋊C₈

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²⋊C₄

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1798

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×#117

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

D₂₃

yes

characteristic

#336

—

C₂×C₂₃

—

C₂×C₂₃

#380

yes

characteristic

C₂×C₂₃

#408

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂₃

#1798

yes

characteristic, complemented, Sylow

C₂⁴

C₄×D₂₃

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

Dic₄₆

yes

—

104

C₂²×C₄

C₄×D₂₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₂₃

yes

—

106

C₂²×C₄

C₂²×D₂₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

Dic₂₃⋊C₄

yes

—

122

C₂³

C₂×C₄×D₂₃

yes

characteristic

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

Dic₂₃⋊C₄

yes

—

134

C₂×C₄

C₂×Dic₄₆

yes

characteristic

135

C₂×C₄

D₄×D₂₃

yes

—

136

C₂×C₄

#67290

yes

—

137

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

138

C₈

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#10625048

yes

characteristic

146

C₂²

C₂²⋊C₄×D₂₃

yes

characteristic

147

C₂²

#10625072

yes

—

148

C₂²

#10625078

yes

—

149

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#10625049

yes

—

154

C₄

#10625110

yes

—

155

C₂

—

156

C₂

#13092275

yes

characteristic

157

C₂

#13092382

yes

—

158

C₂

#13092443

yes

characteristic

159

C₂

#13092445

yes

characteristic

160

C₂

#13092475

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle	C₂³×C₂₃	C₂²×C₄
Fitting	C₂×C₂₃×C₂²⋊C₈	C₂
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₂²⋊C₈	D₂₃
hypocenter	C₂₃	#1798
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle

C₂³×C₂₃

C₂²×C₄

Fitting

C₂×C₂₃×C₂²⋊C₈

C₂

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₂²⋊C₈

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×Dic₂₃⋊C₈	C₂	semidirect
4. #13092171	C₂	semidirect
97. C₂₃	#1798	semidirect

H ≅ G/N

type

2. C₂×Dic₂₃⋊C₈

C₂

semidirect

4. #13092171

C₂

semidirect

97. C₂₃

#1798

semidirect

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

subgroup	index
C₂×#10625097	2
C₂×Dic₂₃⋊C₈	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
#13092171	2
C₂×C₂₃×C₂²⋊C₈	2
#1798	23

subgroup

index

C₂×#10625097

C₂×Dic₂₃⋊C₈

C₂×C₂²⋊(C₂₃⋊C₈)

#13092171

C₂×C₂₃×C₂²⋊C₈

#1798

Subgroups — #15673379

Conjugacy classes of subgroups

Complements

Maximal subgroups