Group #15683800 — subgroups

Subgroups — #15683800

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	#13092967	C₂	1	1	yes	maximal, characteristic
2	#13092787	C₂	1	2	yes	maximal, complemented
3	#13092783	C₂	1	2	yes	maximal, complemented
4	#13092774	C₂	1	4	yes	maximal, complemented
5	#13092416	C₂	1	4	yes	maximal, complemented
6	C₄×#67267	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#67288	C₂²	1	2	yes	—
9	C₂₃×#125	—	2	2	no	—
10	C₂₃×#125	C₂²	1	2	yes	—
11	C₂₃×#125	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	#10625101	—	2	2	no	—
16	C₂×#67267	C₂²	1	2	yes	—
17	#10625046	C₂²	1	2	yes	—
18	#10625045	C₂²	1	1	yes	characteristic
19	C₄×D₉₂	—	2	2	no	—
20	#10625040	C₂²	1	1	yes	characteristic
21	#10624992	C₂²	1	8	yes	—
22	#10624991	C₂²	1	8	yes	—
23	#10624985	C₂²	1	2	yes	—
24	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
25	C₂²×C₄×C₂₃	—	2	2	no	—
26	C₂²×C₄×C₂₃	D₄	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₂₃	—	2	2	no	—
29	#67288	—	2	8	no	—
30	C₂×D₉₂	—	2	2	no	—
31	C₂×D₉₂	C₂³	1	2	yes	—
32	C₂×C₄×D₂₃	—	2	2	no	—
33	C₂×Dic₄₆	C₂³	1	2	yes	—
34	C₄⋊C₄×C₂₃	—	2	4	no	—
35	C₄⋊C₄×C₂₃	—	2	4	no	—
36	C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
38	C₄²×C₂₃	D₄	1	2	yes	—
39	C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	(C₂×D₂₃)⋊C₄	—	2	4	no	—
42	C₄⋊Dic₂₃	—	2	2	no	—
43	#67267	—	2	4	no	—
44	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
45	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	—	4	2	no	—
49	C₂×C₄×C₂₃	—	4	4	no	—
50	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	#41	1	4	yes	—
54	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₂₃	—	4	4	no	—
56	C₂×Dic₂₃	—	2	2	no	—
57	D₉₂	—	2	4	no	—
58	D₉₂	—	4	4	no	—
59	C₄×D₂₃	—	4	4	no	—
60	Dic₄₆	—	2	4	no	—
61	C₂₃⋊C₈	—	2	4	no	—
62	#3142	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	#124	1	2	yes	—
65	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₂₃	—	4	2	no	—
67	C₂×D₂₃	—	4	4	no	—
68	C₄×C₂₃	—	2	2	no	—
69	C₄×C₂₃	—	2	2	no	—
70	C₄×C₂₃	—	4	4	no	—
71	C₄×C₂₃	#124	1	2	yes	—
72	C₄×C₂₃	C₂×#41	1	2	yes	—
73	Dic₂₃	—	4	2	no	—
74	#532	—	23	1	no	—
75	C₂×#125	D₂₃	1	1	yes	characteristic, complemented
76	#491	—	23	2	no	—
77	#490	—	23	2	no	—
78	#488	—	23	4	no	—
79	#417	—	23	4	no	—
80	C₄×M₁₆	—	23	1	no	—
81	C₂×C₂₃	—	2	2	no	—
82	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
83	C₂×C₂₃	#575	1	2	yes	—
84	D₂₃	—	8	2	no	—
85	C₂×#41	—	23	2	no	—
86	C₂×M₁₆	—	23	2	no	—
87	C₄⋊Q₈	—	23	1	no	—
88	#127	—	23	1	no	—
89	#125	—	2	2	no	—
90	#125	C₂×D₂₃	1	2	yes	—
91	#125	C₂×D₂₃	1	2	yes	—
92	#124	—	23	2	no	—
93	#121	—	46	2	no	—
94	C₄×D₄	—	46	2	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
97	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
98	#103	—	23	8	no	—
99	#102	—	23	8	no	—
100	#97	—	23	2	no	—
101	C₄×C₈	—	23	2	no	—
102	C₂₃	#3142	1	1	yes	characteristic, complemented, Sylow
103	#41	—	46	8	no	—
104	C₂×Q₈	—	23	2	no	—
105	C₂×D₄	—	23	2	no	—
106	C₂×D₄	—	46	2	no	—
107	C₂×D₄	—	46	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	46	2	no	—
110	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	M₁₆	—	46	4	no	—
113	C₂×C₈	—	23	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	46	2	no	—
118	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
119	C₂²⋊C₄	—	46	4	no	—
120	C₄²	C₂²⋊D₂₃	1	2	yes	—
121	C₄²	C₂²×D₂₃	1	1	yes	characteristic
122	C₄²	C₂²×D₂₃	1	1	yes	characteristic
123	C₂³	—	46	2	no	—
124	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
125	Q₈	—	46	4	no	—
126	D₄	—	46	4	no	—
127	D₄	—	92	4	no	—
128	D₄	—	92	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	46	2	no	—
133	C₂×C₄	—	92	4	no	—
134	C₂×C₄	#67292	1	4	yes	—
135	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
139	C₈	—	46	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	92	2	no	—
142	C₂²	—	92	4	no	—
143	C₂²	#10625118	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	92	2	no	—
149	C₄	#10625118	1	2	yes	—
150	C₄	C₂×#67292	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	184	2	no	—
153	C₂	C₂×#10625118	1	1	yes	characteristic
154	C₂	#13093215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092967

C₂

yes

maximal, characteristic

#13092787

C₂

yes

maximal, complemented

#13092783

C₂

yes

maximal, complemented

#13092774

C₂

yes

maximal, complemented

#13092416

C₂

yes

maximal, complemented

C₄×#67267

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#125

C₂

yes

maximal, characteristic, complemented

C₂×#67288

C₂²

yes

—

C₂₃×#125

—

C₂₃×#125

C₂²

yes

—

C₂₃×#125

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625101

—

C₂×#67267

C₂²

yes

—

#10625046

C₂²

yes

—

#10625045

C₂²

yes

characteristic

C₄×D₉₂

—

#10625040

C₂²

yes

characteristic

#10624992

C₂²

yes

—

#10624991

C₂²

yes

—

#10624985

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

—

#67288

—

C₂×D₉₂

—

C₂×D₉₂

C₂³

yes

—

C₂×C₄×D₂₃

—

C₂×Dic₄₆

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

#67267

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

—

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

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

Dic₄₆

—

C₂₃⋊C₈

—

#3142

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

C₂×#41

yes

—

Dic₂₃

—

#532

—

C₂×#125

D₂₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#124

yes

characteristic

C₂×C₂₃

#575

yes

—

D₂₃

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#125

—

#125

C₂×D₂₃

yes

—

#125

C₂×D₂₃

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#103

—

#102

—

100

#97

—

101

C₄×C₈

—

102

C₂₃

#3142

yes

characteristic, complemented, Sylow

103

#41

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₂₃

yes

—

111

C₂²×C₄

C₂²×D₂₃

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂²×D₂₃

yes

—

119

C₂²⋊C₄

—

120

C₄²

C₂²⋊D₂₃

yes

—

121

C₄²

C₂²×D₂₃

yes

characteristic

122

C₄²

C₂²×D₂₃

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#67292

yes

—

135

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

136

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

137

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

138

C₂×C₄

C₂³×D₂₃

yes

characteristic

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#10625118

yes

—

144

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#10625118

yes

—

150

C₄

C₂×#67292

yes

—

151

C₂

—

152

C₂

—

184

—

153

C₂

C₂×#10625118

yes

characteristic

154

C₂

#13093215

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#125	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	#13092967	C₂
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	#3142
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₂₃×#125

C₂

socle

C₂²×C₂₃

C₂²×D₄

C₂

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #13092787	C₂	semidirect
3. #13092783	C₂	semidirect
4. #13092774	C₂	semidirect
5. #13092416	C₂	semidirect
6. C₄×#67267	C₂	semidirect
7. C₂×C₂₃×#125	C₂	semidirect
75. C₂×#125	D₂₃	semidirect
102. C₂₃	#3142	semidirect

H ≅ G/N

type

2. #13092787

C₂

semidirect

3. #13092783

C₂

semidirect

4. #13092774

C₂

semidirect

5. #13092416

C₂

semidirect

6. C₄×#67267

C₂

semidirect

7. C₂×C₂₃×#125

C₂

semidirect

75. C₂×#125

D₂₃

semidirect

102. C₂₃

#3142

semidirect

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

subgroup	index
#13092967	2
#13092787	2
#13092783	2
#13092774	2
#13092416	2
C₄×#67267	2
C₂×C₂₃×#125	2
#3142	23

subgroup

index

C₂×C₂₃×#125

Subgroups — #15683800

Conjugacy classes of subgroups

Complements

Maximal subgroups