Group #15683742 — subgroups

Subgroups — #15683742

Order 2944.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#13092974	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092785	C₂	1	2	yes	maximal, complemented
3	#13092781	C₂	1	2	yes	maximal, complemented
4	#13092776	C₂	1	4	yes	maximal, complemented
5	#13092413	C₂	1	4	yes	maximal, complemented
6	C₂×#10624985	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#532	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₂₃	C₂²	1	2	yes	—
9	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂₃×#127	C₂²	1	1	yes	characteristic
11	C₂₃×#124	C₂²	1	2	yes	—
12	C₂₃×#121	—	2	2	no	—
13	C₄×D₄×C₂₃	—	2	2	no	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	#10625097	C₂²	1	2	yes	—
16	#10625095	—	2	2	no	—
17	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
18	#10625041	C₂²	1	2	yes	—
19	#10625040	C₂²	1	2	yes	—
20	C₄×Dic₄₆	—	2	2	no	—
21	#10625017	C₂²	1	8	yes	—
22	#10625014	C₂²	1	8	yes	—
23	#10624985	C₂²	1	2	yes	—
24	#10624985	C₂²	1	2	yes	—
25	#41×C₂₃	—	2	8	no	—
26	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
27	C₂×D₄×C₂₃	—	2	2	no	—
28	C₂×D₄×C₂₃	—	2	2	no	—
29	C₂×D₄×C₂₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₂₃	—	2	2	no	—
31	C₂²×C₄×C₂₃	D₄	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₄₆	—	2	2	no	—
34	C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂²⋊C₄×C₂₃	—	2	4	no	—
36	C₄²×C₂₃	D₄	1	2	yes	—
37	C₄²×C₂₃	C₂³	1	1	yes	characteristic
38	C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₂₃	—	2	2	no	—
40	C₄⋊Dic₂₃	—	2	2	no	—
41	C₄⋊Dic₂₃	C₂³	1	4	yes	—
42	Dic₂₃⋊C₄	—	2	4	no	—
43	C₄×Dic₂₃	—	2	2	no	—
44	C₂×C₂₃⋊C₈	—	2	4	no	—
45	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
46	C₂³×C₂₃	—	2	2	no	—
47	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₂₃	—	2	4	no	—
49	D₄×C₂₃	—	2	4	no	—
50	D₄×C₂₃	—	4	4	no	—
51	D₄×C₂₃	—	4	4	no	—
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	4	2	no	—
54	C₂×C₄×C₂₃	—	4	4	no	—
55	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
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	4	yes	—
59	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₂₃	—	2	4	no	—
61	Dic₄₆	—	4	4	no	—
62	C₂₃⋊C₈	—	2	4	no	—
63	#3135	—	23	1	no	maximal, Sylow
64	C₂²×C₂₃	—	2	2	no	—
65	C₂²×C₂₃	—	4	2	no	—
66	C₂²×C₂₃	—	4	4	no	—
67	C₂²×C₂₃	#124	1	2	yes	—
68	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₂₃	—	2	2	no	—
70	C₄×C₂₃	—	2	2	no	—
71	C₄×C₂₃	—	4	2	no	—
72	C₄×C₂₃	#124	1	2	yes	—
73	C₄×C₂₃	C₂×#41	1	2	yes	—
74	Dic₂₃	—	4	4	no	—
75	#533	—	23	1	no	—
76	#532	D₂₃	1	1	yes	characteristic
77	#491	—	23	2	no	—
78	#490	—	23	2	no	—
79	#489	—	23	4	no	—
80	#416	—	23	4	no	—
81	C₂×#97	—	23	1	no	—
82	C₂×C₂₃	—	2	2	no	—
83	C₂×C₂₃	—	8	2	no	—
84	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
85	C₂×C₂₃	#575	1	2	yes	—
86	C₂×#41	C₂×D₂₃	1	2	yes	—
87	C₂²×C₈	—	23	2	no	—
88	C₄⋊Q₈	—	23	2	no	—
89	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
90	#127	C₂×D₂₃	1	1	yes	characteristic
91	#125	—	23	2	no	—
92	#124	C₂×D₂₃	1	2	yes	—
93	C₂²⋊Q₈	—	46	2	no	—
94	#121	—	2	2	no	—
95	C₄×Q₈	—	46	2	no	—
96	C₄×D₄	—	2	2	no	—
97	#117	—	23	2	no	—
98	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
99	#103	—	23	8	no	—
100	#102	—	23	8	no	—
101	#97	—	23	2	no	—
102	#97	—	23	2	no	—
103	C₂₃	#3135	1	1	yes	characteristic, complemented, Sylow
104	#41	—	2	8	no	—
105	C₂×Q₈	—	46	2	no	—
106	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂²×D₂₃	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
113	C₂×C₈	—	23	4	no	—
114	C₂×C₈	—	46	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	23	4	no	—
117	C₄⋊C₄	—	46	4	no	—
118	C₄⋊C₄	—	46	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	46	2	no	—
121	C₄²	—	46	2	no	—
122	C₄²	C₂²⋊D₂₃	1	2	yes	—
123	C₄²	C₂²×D₂₃	1	1	yes	characteristic
124	C₄²	C₂²×D₂₃	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	92	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	46	4	no	—
136	C₂×C₄	#67290	1	4	yes	—
137	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
141	C₈	—	46	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	4	no	—
145	C₂²	#10625107	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	92	4	no	—
151	C₄	#10625107	1	2	yes	—
152	C₄	C₂×#67290	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	8	2	no	—
155	C₂	C₂×#10625107	1	1	yes	characteristic
156	C₂	#13093215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal, characteristic, complemented

#13092785

C₂

yes

maximal, complemented

#13092781

C₂

yes

maximal, complemented

#13092776

C₂

yes

maximal, complemented

#13092413

C₂

yes

maximal, complemented

C₂×#10624985

C₂

yes

maximal, characteristic, complemented

C₂₃×#532

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#124

C₂²

yes

—

C₂₃×#121

—

C₄×D₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

#10625095

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625041

C₂²

yes

—

#10625040

C₂²

yes

—

C₄×Dic₄₆

—

#10625017

C₂²

yes

—

#10625014

C₂²

yes

—

#10624985

C₂²

yes

—

#10624985

C₂²

yes

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂³

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

—

#3135

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

C₂×#41

yes

—

Dic₂₃

—

#533

—

#532

D₂₃

yes

—

—

—

—

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#124

yes

characteristic

C₂×C₂₃

#575

yes

—

C₂×#41

C₂×D₂₃

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

#127

C₂×D₂₃

yes

characteristic

#125

—

#124

C₂×D₂₃

yes

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

#97

—

103

C₂₃

#3135

yes

characteristic, complemented, Sylow

104

#41

—

105

C₂×Q₈

—

106

C₂×Q₈

C₂²×D₂₃

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₂₃

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₂₃

yes

—

112

C₂²×C₄

C₂²×D₂₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

C₂²⋊D₂₃

yes

—

123

C₄²

C₂²×D₂₃

yes

characteristic

124

C₄²

C₂²×D₂₃

yes

characteristic

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#67290

yes

—

137

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

138

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

139

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

140

C₂×C₄

C₂³×D₂₃

yes

characteristic

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#10625107

yes

—

146

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#10625107

yes

—

152

C₄

C₂×#67290

yes

—

153

C₂

—

154

C₂

—

155

C₂

C₂×#10625107

yes

characteristic

156

C₂

#13093215

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting, prime radical	C₂₃×#532	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	#532	D₂₃
hypocenter	C₂₃	#3135
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting, prime radical

C₂₃×#532

C₂

socle

C₂²×C₂₃

C₂²×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #13092974	C₂	semidirect
2. #13092785	C₂	semidirect
3. #13092781	C₂	semidirect
4. #13092776	C₂	semidirect
5. #13092413	C₂	semidirect
6. C₂×#10624985	C₂	semidirect
103. C₂₃	#3135	semidirect

H ≅ G/N

type

1. #13092974

C₂

semidirect

2. #13092785

C₂

semidirect

3. #13092781

C₂

semidirect

4. #13092776

C₂

semidirect

5. #13092413

C₂

semidirect

6. C₂×#10624985

C₂

semidirect

103. C₂₃

#3135

semidirect

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

subgroup	index
#13092974	2
#13092785	2
#13092781	2
#13092776	2
#13092413	2
C₂×#10624985	2
C₂₃×#532	2
#3135	23

subgroup

index

C₂₃×#532

Subgroups — #15683742

Conjugacy classes of subgroups

Complements

Maximal subgroups