Group #15676123 — subgroups

Subgroups — #15676123

Order 2944.

740 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 237 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×D₄×Dic₂₃	C₂	1	2	yes	maximal, complemented
2	#13092320	C₂	1	2	yes	maximal, complemented
3	#13092297	C₂	1	8	yes	maximal, complemented
4	#13092283	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092263	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
8	C₂²×D₄×C₂₃	C₂²	1	4	yes	—
9	C₂³×Dic₂₃	C₂²	1	4	yes	—
10	C₂₃×#127	C₄	1	8	yes	complemented
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊Dic₂₃	C₂²	1	8	yes	—
14	D₄×Dic₂₃	—	2	8	no	—
15	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂₃⋊#95	C₂²	1	4	yes	—
19	C₂₃⋊#95	C₂²	1	4	yes	—
20	C₂⁴×C₂₃	C₂³	1	4	yes	—
21	C₂×D₄×C₂₃	—	2	4	no	—
22	C₂×D₄×C₂₃	—	2	4	no	—
23	C₂×D₄×C₂₃	—	4	8	no	—
24	C₂×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
25	C₂×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
26	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₂₃	—	2	8	no	—
29	C₂²×Dic₂₃	—	4	8	no	—
30	C₂²×Dic₂₃	C₂³	1	2	yes	—
31	C₂²×Dic₂₃	C₂³	1	4	yes	—
32	C₂²×Dic₂₃	C₂³	1	2	yes	—
33	C₄²×C₂₃	C₂×C₄	1	4	yes	complemented
34	C₂²⋊Dic₂₃	—	2	8	no	—
35	C₂²⋊Dic₂₃	—	4	8	no	—
36	C₄⋊Dic₂₃	—	2	4	no	—
37	C₄×Dic₂₃	—	2	2	no	—
38	C₄×Dic₂₃	D₄	1	4	yes	complemented
39	C₂³×C₂₃	—	2	4	no	—
40	C₂³×C₂₃	—	2	4	no	—
41	C₂³×C₂₃	—	2	8	no	—
42	C₂³×C₂₃	—	2	8	no	—
43	C₂³×C₂₃	C₂²×C₄	1	8	yes	—
44	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
45	D₄×C₂₃	—	2	16	no	—
46	D₄×C₂₃	—	4	16	no	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₃	#41	1	2	yes	—
52	C₂×Dic₂₃	—	2	4	no	—
53	C₂×Dic₂₃	—	2	4	no	—
54	C₂×Dic₂₃	—	2	2	no	—
55	C₂×Dic₂₃	—	2	4	no	—
56	C₂×Dic₂₃	—	2	8	no	—
57	C₂×Dic₂₃	—	4	16	no	—
58	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
59	#2328	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	—	2	8	no	—
61	C₂²×C₂₃	—	2	16	no	—
62	C₂²×C₂₃	—	4	4	no	—
63	C₂²×C₂₃	—	4	4	no	—
64	C₂²×C₂₃	—	4	8	no	—
65	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
66	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
67	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
68	C₂²×C₂₃	H(5,𝔽₂)	1	4	yes	—
69	C₄×C₂₃	—	2	4	no	—
70	C₄×C₂₃	C₄×D₄	1	4	yes	—
71	Dic₂₃	—	2	4	no	—
72	Dic₂₃	—	4	4	no	—
73	Dic₂₃	—	4	2	no	—
74	C₂×#127	D₂₃	1	1	yes	characteristic
75	C₂×C₄×D₄	—	23	2	no	—
76	#390	—	23	2	no	—
77	#386	—	23	8	no	—
78	#382	—	23	1	no	—
79	#376	—	23	1	no	—
80	C₂×C₂₃	—	4	8	no	—
81	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
82	C₂×C₂₃	#518	1	2	yes	—
83	C₂×C₂₃	#535	1	2	yes	—
84	C₂×C₂₃	#540	1	2	yes	—
85	C₂²×D₄	C₂×D₂₃	1	2	yes	—
86	C₂²×D₄	C₂×D₂₃	1	4	yes	—
87	C₂³×C₄	—	23	4	no	—
88	#127	Dic₂₃	1	8	yes	complemented
89	C₄×D₄	—	46	8	no	—
90	C₂×C₄⋊C₄	—	23	2	no	—
91	C₂×C₂²⋊C₄	—	23	4	no	—
92	C₂×C₂²⋊C₄	—	23	8	no	—
93	C₂×C₄²	—	23	1	no	—
94	C₂×C₄²	—	23	1	no	—
95	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
96	#95	—	23	4	no	—
97	#95	—	23	4	no	—
98	C₂₃	#2328	1	1	yes	characteristic, complemented, Sylow
99	C₂⁴	C₂²×D₂₃	1	4	yes	—
100	C₂×D₄	—	2	4	no	—
101	C₂×D₄	—	2	4	no	—
102	C₂×D₄	—	4	8	no	—
103	C₂×D₄	C₂×Dic₂₃	1	8	yes	complemented
104	C₂×D₄	C₂×Dic₂₃	1	16	yes	complemented
105	C₂²×C₄	—	23	2	no	—
106	C₂²×C₄	—	23	2	no	—
107	C₂²×C₄	—	23	4	no	—
108	C₂²×C₄	—	46	8	no	—
109	C₂²×C₄	—	92	8	no	—
110	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	C₄⋊C₄	—	46	4	no	—
113	C₂²⋊C₄	—	46	8	no	—
114	C₂²⋊C₄	—	92	8	no	—
115	C₄²	—	23	4	no	—
116	C₄²	—	46	2	no	—
117	C₄²	C₂×Dic₂₃	1	4	yes	complemented
118	C₂³	—	2	4	no	—
119	C₂³	—	2	4	no	—
120	C₂³	—	2	8	no	—
121	C₂³	—	2	8	no	—
122	C₂³	C₂²×Dic₂₃	1	8	yes	—
123	C₂³	C₂³×D₂₃	1	1	yes	characteristic
124	D₄	—	2	16	no	—
125	D₄	—	4	16	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	23	4	no	—
128	C₂×C₄	—	46	4	no	—
129	C₂×C₄	—	46	4	no	—
130	C₂×C₄	—	46	2	no	—
131	C₂×C₄	—	46	4	no	—
132	C₂×C₄	—	46	8	no	—
133	C₂×C₄	—	92	16	no	—
134	C₂×C₄	D₄×D₂₃	1	2	yes	—
135	C₂×C₄	#67290	1	2	yes	—
136	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
137	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
138	C₂²	—	2	8	no	—
139	C₂²	—	2	16	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	8	no	—
143	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
144	C₂²	C₂×#67290	1	1	yes	characteristic
145	C₂²	#10625155	1	4	yes	—
146	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	46	4	no	—
149	C₄	—	92	4	no	—
150	C₄	—	92	2	no	—
151	C₄	D₄×Dic₂₃	1	4	yes	—
152	C₂	—	4	8	no	—
153	C₂	C₂×D₄×Dic₂₃	1	1	yes	characteristic
154	C₂	#13092892	1	2	yes	—
155	C₂	#13092980	1	2	yes	—
156	C₂	#13093021	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×Dic₂₃

C₂

yes

maximal, complemented

#13092320

C₂

yes

maximal, complemented

#13092297

C₂

yes

maximal, complemented

#13092283

C₂

yes

maximal, characteristic, complemented

#13092263

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

C₂²

yes

—

C₂²×D₄×C₂₃

C₂²

yes

—

C₂³×Dic₂₃

C₂²

yes

—

C₂₃×#127

C₄

yes

complemented

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₂²

yes

—

D₄×Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂₃⋊#95

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂⁴×C₂₃

C₂³

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

complemented

C₂×D₄×C₂₃

C₂×C₄

yes

complemented

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂²×Dic₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₄²×C₂₃

C₂×C₄

yes

complemented

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

complemented

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×D₄

yes

—

#2328

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂²×C₂₃

C₂×#41

yes

characteristic

C₂²×C₂₃

H(5,𝔽₂)

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

C₂×#127

D₂₃

yes

characteristic

C₂×C₄×D₄

—

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄×D₄

yes

characteristic

C₂×C₂₃

#518

yes

—

C₂×C₂₃

#535

yes

—

C₂×C₂₃

#540

yes

—

C₂²×D₄

C₂×D₂₃

yes

—

C₂²×D₄

C₂×D₂₃

yes

—

C₂³×C₄

—

#127

Dic₂₃

yes

complemented

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#95

—

#95

—

C₂₃

#2328

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₂₃

yes

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

C₂×Dic₂₃

yes

complemented

104

C₂×D₄

C₂×Dic₂₃

yes

complemented

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₂₃

yes

—

111

C₂²×C₄

C₂²×D₂₃

yes

characteristic

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₂×Dic₂₃

yes

complemented

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂²×Dic₂₃

yes

—

123

C₂³

C₂³×D₂₃

yes

characteristic

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₂₃

yes

—

135

C₂×C₄

#67290

yes

—

136

C₂×C₄

C₂²×Dic₂₃

yes

—

137

C₂×C₄

C₂²×Dic₂₃

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×D₄×D₂₃

yes

characteristic

144

C₂²

C₂×#67290

yes

characteristic

145

C₂²

#10625155

yes

—

146

C₂²

C₂³×Dic₂₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

D₄×Dic₂₃

yes

—

152

C₂

—

153

C₂

C₂×D₄×Dic₂₃

yes

characteristic

154

C₂

#13092892

yes

—

155

C₂

#13092980

yes

—

156

C₂

#13093021

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting, prime radical	C₂×C₂₃×#127	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₂×#127	D₂₃
hypocenter	C₂₃	#2328
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₃

commutator

C₂²×C₂₃

C₂³×C₄

Fitting, prime radical

C₂×C₂₃×#127

C₂

socle, cosocle

C₂³×C₂₃

C₂⁴

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×D₄×Dic₂₃	C₂	semidirect
2. #13092320	C₂	semidirect
3. #13092297	C₂	semidirect
4. #13092283	C₂	semidirect
5. #13092263	C₂	semidirect
10. C₂₃×#127	C₄	semidirect
24. C₂×D₄×C₂₃	C₂×C₄	semidirect
25. C₂×D₄×C₂₃	C₂×C₄	semidirect
33. C₄²×C₂₃	C₂×C₄	semidirect
38. C₄×Dic₂₃	D₄	semidirect
88. #127	Dic₂₃	semidirect
98. C₂₃	#2328	semidirect
103. C₂×D₄	C₂×Dic₂₃	semidirect
104. C₂×D₄	C₂×Dic₂₃	semidirect
117. C₄²	C₂×Dic₂₃	semidirect

H ≅ G/N

type

1. C₂×D₄×Dic₂₃

C₂

semidirect

2. #13092320

C₂

semidirect

3. #13092297

C₂

semidirect

4. #13092283

C₂

semidirect

5. #13092263

C₂

semidirect

10. C₂₃×#127

C₄

semidirect

24. C₂×D₄×C₂₃

C₂×C₄

semidirect

25. C₂×D₄×C₂₃

C₂×C₄

semidirect

33. C₄²×C₂₃

C₂×C₄

semidirect

38. C₄×Dic₂₃

D₄

semidirect

88. #127

Dic₂₃

semidirect

98. C₂₃

#2328

semidirect

103. C₂×D₄

C₂×Dic₂₃

semidirect

104. C₂×D₄

C₂×Dic₂₃

semidirect

117. C₄²

C₂×Dic₂₃

semidirect

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

subgroup	index
C₂×D₄×Dic₂₃	2
#13092320	2
#13092297	2
#13092283	2
#13092263	2
C₂×C₂₃×#127	2
#2328	23

subgroup

index

C₂×D₄×Dic₂₃

C₂×C₂₃×#127

Subgroups — #15676123

Conjugacy classes of subgroups

Complements

Maximal subgroups