Group #15673479 — subgroups

Subgroups — #15673479

Order 2944.

272 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 77 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	1	yes	maximal, characteristic
2	#13092380	C₂	1	2	yes	maximal
3	#13092186	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092185	C₂	1	2	yes	maximal, complemented
5	C₂₃×#343	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₂₃)⋊C₄	C₄	1	2	yes	—
10	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
12	C₂×#67267	C₂²	1	2	yes	—
13	#10625063	—	2	4	no	—
14	(C₂×D₂₃)⋊C₈	—	2	4	no	—
15	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₂₃	—	2	2	no	—
21	C₂×C₄×D₂₃	—	2	1	no	—
22	C₂×C₄×D₂₃	—	2	1	no	—
23	C₂×C₄×D₂₃	D₄	1	4	yes	—
24	M₁₆×C₂₃	—	2	2	no	—
25	M₁₆×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₃	—	2	2	no	—
28	(C₂×D₂₃)⋊C₄	—	2	4	no	—
29	C₄⋊Dic₂₃	—	2	2	no	—
30	C₄×Dic₂₃	—	2	4	no	—
31	#67267	—	2	2	no	—
32	#67267	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×D₂₃	—	2	1	no	—
36	C₂²×D₂₃	—	2	1	no	—
37	C₂²×D₂₃	—	4	1	no	—
38	C₂²×D₂₃	—	4	1	no	—
39	C₂²×D₂₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	4	2	no	—
42	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₃	#41	1	2	yes	—
44	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
46	C₂×Dic₂₃	—	2	1	no	—
47	C₂×Dic₂₃	—	2	1	no	—
48	C₂×Dic₂₃	—	2	2	no	—
49	C₂×Dic₂₃	—	4	2	no	—
50	C₂×Dic₂₃	C₂²⋊C₄	1	2	yes	—
51	C₄×D₂₃	—	2	4	no	—
52	C₄×D₂₃	—	2	4	no	—
53	C₈×C₂₃	—	4	2	no	—
54	C₂₃⋊C₈	—	4	2	no	—
55	#1822	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	—	2	1	no	—
57	C₂²×C₂₃	—	2	1	no	—
58	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₃	#117	1	1	yes	characteristic
60	C₂²×C₂₃	#117	1	1	yes	characteristic
61	C₂×D₂₃	—	2	2	no	—
62	C₂×D₂₃	—	2	2	no	—
63	C₂×D₂₃	—	2	2	no	—
64	C₂×D₂₃	—	4	2	no	—
65	C₂×D₂₃	—	4	2	no	—
66	C₄×C₂₃	—	4	2	no	—
67	C₄×C₂₃	#123	1	2	yes	—
68	C₄×C₂₃	#123	1	2	yes	—
69	Dic₂₃	—	2	2	no	—
70	Dic₂₃	—	4	2	no	—
71	C₂×#117	—	23	1	no	—
72	#407	—	23	2	no	—
73	#343	—	23	2	no	—
74	#343	—	23	1	no	—
75	#343	D₂₃	1	1	yes	characteristic, complemented
76	C₂×C₂₃	—	2	1	no	—
77	C₂×C₂₃	—	2	1	no	—
78	C₂×C₂₃	#381	1	1	yes	characteristic
79	C₂×C₂₃	#413	1	1	yes	characteristic
80	C₂×C₂₃	#413	1	1	yes	characteristic
81	D₂₃	—	4	2	no	—
82	C₂³×C₄	—	23	1	no	—
83	C₂×M₁₆	—	23	2	no	—
84	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
85	#117	—	46	4	no	—
86	C₂×C₄⋊C₄	—	23	1	no	—
87	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
88	C₂×C₂²⋊C₄	—	23	2	no	—
89	C₂×C₄²	—	23	2	no	—
90	C₂²⋊C₈	—	46	4	no	—
91	C₂₃	#1822	1	1	yes	characteristic, complemented, Sylow
92	C₂⁴	—	23	1	no	—
93	C₂²×C₄	—	23	1	no	—
94	C₂²×C₄	—	23	4	no	—
95	C₂²×C₄	—	23	2	no	—
96	C₂²×C₄	—	46	2	no	—
97	C₂²×C₄	—	46	1	no	—
98	C₂²×C₄	—	46	1	no	—
99	C₂²×C₄	C₄×D₂₃	1	2	yes	—
100	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	46	2	no	—
104	M₁₆	—	46	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	46	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	46	2	no	—
109	C₂²⋊C₄	—	46	4	no	—
110	C₄²	—	46	4	no	—
111	C₂³	—	23	2	no	—
112	C₂³	—	46	1	no	—
113	C₂³	—	46	1	no	—
114	C₂³	—	92	1	no	—
115	C₂³	—	92	1	no	—
116	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	23	2	no	—
120	C₂×C₄	—	46	1	no	—
121	C₂×C₄	—	46	1	no	—
122	C₂×C₄	—	46	4	no	—
123	C₂×C₄	—	46	4	no	—
124	C₂×C₄	—	46	2	no	—
125	C₂×C₄	—	92	2	no	—
126	C₂×C₄	#67288	1	2	yes	—
127	C₂×C₄	D₄×D₂₃	1	2	yes	—
128	C₂×C₄	#67290	1	1	yes	characteristic
129	C₂×C₄	#67292	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	92	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	46	2	no	—
135	C₂²	—	46	2	no	—
136	C₂²	—	46	2	no	—
137	C₂²	—	92	2	no	—
138	C₂²	—	92	2	no	—
139	C₂²	#10625043	1	1	yes	characteristic
140	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
141	C₂²	#10625063	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	—	46	2	no	—
144	C₄	—	92	2	no	—
145	C₄	#10625054	1	2	yes	—
146	C₄	#10625065	1	2	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	—	92	2	no	—
150	C₂	#13092279	1	1	yes	characteristic
151	C₂	#13092398	1	1	yes	characteristic
152	C₂	#13092400	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625063

C₂

yes

maximal, characteristic

#13092380

C₂

yes

maximal

#13092186

C₂

yes

maximal, characteristic, complemented

#13092185

C₂

yes

maximal, complemented

C₂₃×#343

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×(C₂×D₂₃)⋊C₄

C₄

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×#67267

C₂²

yes

—

#10625063

—

(C₂×D₂₃)⋊C₈

—

C₂³×D₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

D₄

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²⋊C₄

yes

—

C₄×D₂₃

—

C₄×D₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1822

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#123

yes

—

C₄×C₂₃

#123

yes

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

D₂₃

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂₃

#1822

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

—

100

C₂²×C₄

C₂²×D₂₃

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

C₂×C₄×D₂₃

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#67288

yes

—

127

C₂×C₄

D₄×D₂₃

yes

—

128

C₂×C₄

#67290

yes

characteristic

129

C₂×C₄

#67292

yes

characteristic

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

#10625043

yes

characteristic

140

C₂²

C₂²⋊C₄×D₂₃

yes

characteristic

141

C₂²

#10625063

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#10625054

yes

—

146

C₄

#10625065

yes

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

#13092279

yes

characteristic

151

C₂

#13092398

yes

characteristic

152

C₂

#13092400

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₃
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#343	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂³×D₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#343	D₂₃
hypocenter	C₂₃	#1822
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₃

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#343

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂³×D₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
3. #13092186	C₂	semidirect
4. #13092185	C₂	semidirect
5. C₂₃×#343	C₂	semidirect
75. #343	D₂₃	semidirect
91. C₂₃	#1822	semidirect

H ≅ G/N

type

3. #13092186

C₂

semidirect

4. #13092185

C₂

semidirect

5. C₂₃×#343

C₂

semidirect

75. #343

D₂₃

semidirect

91. C₂₃

#1822

semidirect

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

subgroup	index
C₂×#10625063	2
#13092380	2
#13092186	2
#13092185	2
C₂₃×#343	2
#1822	23

subgroup

index

C₂₃×#343

Subgroups — #15673479

Conjugacy classes of subgroups

Complements

Maximal subgroups