Group #15673917 — subgroups

Subgroups — #15673917

Order 2944.

312 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 85 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	#13093184	C₂	1	1	yes	maximal, characteristic
2	#13092988	C₂	1	1	yes	maximal, characteristic
3	#13092425	C₂	1	2	yes	maximal, complemented
4	#13092173	C₂	1	2	yes	maximal, complemented
5	C₂₃×#410	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
8	C₂₃×#117	C₂²	1	2	yes	—
9	#10625119	—	2	4	no	—
10	#10625115	C₄	1	2	yes	complemented
11	#10625107	C₄	1	2	yes	complemented
12	#10625106	—	2	2	no	—
13	#10625097	C₂²	1	1	yes	characteristic
14	#10625095	—	2	2	no	—
15	C₂×#67267	—	2	2	no	—
16	C₂×#67267	C₂²	1	2	yes	—
17	C₂₃×#99	—	2	2	no	—
18	#10624992	—	2	4	no	—
19	#41×C₂₃	—	2	2	no	—
20	#41×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₂₃	D₄	1	2	yes	—
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₃	—	2	1	no	—
27	C₂×Dic₄₆	—	2	2	no	—
28	C₂×Dic₄₆	—	2	2	no	—
29	C₂×Dic₄₆	—	2	1	no	—
30	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
31	C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₄²×C₂₃	—	2	2	no	—
34	C₂²⋊Dic₂₃	—	2	4	no	—
35	C₂²⋊Dic₂₃	—	2	1	no	—
36	C₄⋊Dic₂₃	—	2	1	no	—
37	Dic₂₃⋊C₄	—	2	4	no	—
38	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
39	#67267	—	2	4	no	—
40	#67267	—	2	2	no	—
41	#67267	—	2	2	no	—
42	C₂×C₂₃⋊C₈	—	2	4	no	—
43	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
44	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₂₃	—	2	2	no	—
47	D₄×C₂₃	—	2	4	no	—
48	D₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	4	2	no	—
52	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₂₃	#41	1	2	yes	—
57	C₂×Dic₂₃	—	2	2	no	—
58	C₂×Dic₂₃	—	2	2	no	—
59	C₂×Dic₂₃	—	4	2	no	—
60	Dic₄₆	—	2	4	no	—
61	Dic₄₆	—	4	1	no	—
62	Dic₄₆	—	4	1	no	—
63	C₂₃⋊C₈	—	2	4	no	—
64	#1909	—	23	1	no	maximal, Sylow
65	C₂²×C₂₃	—	2	2	no	—
66	C₂²×C₂₃	—	4	1	no	—
67	C₂²×C₂₃	—	4	2	no	—
68	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₃	#121	1	2	yes	—
70	C₄×C₂₃	—	2	2	no	—
71	C₄×C₂₃	—	8	2	no	—
72	C₄×C₂₃	C₄×D₄	1	2	yes	—
73	C₄×C₂₃	#120	1	1	yes	characteristic
74	C₄×C₂₃	#123	1	1	yes	characteristic
75	Dic₂₃	—	4	2	no	—
76	Dic₂₃	—	4	2	no	—
77	#568	—	23	1	no	—
78	#536	—	23	1	no	—
79	#419	—	23	2	no	—
80	#410	D₂₃	1	1	yes	characteristic
81	H(3,ℤ/4)	—	23	2	no	—
82	C₂×C₂₃	—	2	1	no	—
83	C₂×C₂₃	—	2	2	no	—
84	C₂×C₂₃	—	4	2	no	—
85	C₂×C₂₃	#386	1	1	yes	characteristic
86	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
87	C₂²×Q₈	—	23	1	no	—
88	#131	—	46	4	no	—
89	C₂×M₁₆	—	23	2	no	—
90	C₂×M₁₆	—	46	2	no	—
91	C₄⋊Q₈	—	23	2	no	—
92	#124	—	23	2	no	—
93	#123	—	46	2	no	—
94	C₂²⋊Q₈	—	46	2	no	—
95	#117	—	23	1	no	—
96	#117	C₂×D₂₃	1	2	yes	—
97	#103	—	46	4	no	—
98	#99	—	2	2	no	—
99	C₂₃	#1909	1	1	yes	characteristic, complemented, Sylow
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	—	23	2	no	—
103	C₂×Q₈	—	46	1	no	—
104	C₂×Q₈	—	46	2	no	—
105	C₂×Q₈	—	46	2	no	—
106	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
107	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
108	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
109	C₂²×C₄	—	46	1	no	—
110	C₂²×C₄	D₉₂	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	M₁₆	—	46	4	no	—
113	M₁₆	—	46	2	no	—
114	M₁₆	—	46	2	no	—
115	C₂×C₈	—	23	4	no	—
116	C₂×C₈	—	46	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	46	4	no	—
119	C₄⋊C₄	—	46	1	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	46	4	no	—
122	C₂²⋊C₄	—	46	1	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	23	2	no	—
125	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
126	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	46	4	no	—
129	Q₈	—	92	1	no	—
130	Q₈	—	92	1	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	46	2	no	—
137	C₂×C₄	—	46	2	no	—
138	C₂×C₄	—	92	2	no	—
139	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
141	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
142	C₂×C₄	D₄×D₂₃	1	2	yes	—
143	C₂×C₄	#67290	1	2	yes	—
144	C₈	—	46	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
149	C₂²	#10625110	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	92	2	no	—
153	C₄	—	92	2	no	—
154	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
155	C₄	#10625053	1	1	yes	characteristic
156	C₄	#10625057	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	2	1	no	—
159	C₂	—	4	2	no	—
160	C₂	#13092294	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093184

C₂

yes

maximal, characteristic

#13092988

C₂

yes

maximal, characteristic

#13092425

C₂

yes

maximal, complemented

#13092173

C₂

yes

maximal, complemented

C₂₃×#410

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂²×Dic₄₆

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

#10625119

—

#10625115

C₄

yes

complemented

#10625107

C₄

yes

complemented

#10625106

—

#10625097

C₂²

yes

—

—

C₂²

yes

—

C₂₃×#99

—

#10624992

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

C₂×C₄

yes

—

#67267

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₂₃⋊C₈

—

#1909

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

—

—

—

D₂₃

yes

characteristic

H(3,ℤ/4)

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#386

yes

characteristic

C₂×#41

C₂×D₂₃

yes

characteristic

C₂²×Q₈

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₂₃

yes

—

#103

—

#99

—

C₂₃

#1909

yes

characteristic, complemented, Sylow

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₄×D₂₃

yes

characteristic

107

C₂×D₄

C₄×D₂₃

yes

characteristic

108

C₂×D₄

C₂²⋊D₂₃

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

D₉₂

yes

—

111

C₂²×C₄

C₂²×D₂₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₂³

(C₂×D₂₃)⋊C₄

yes

—

126

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

140

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

141

C₂×C₄

C₂×D₉₂

yes

characteristic

142

C₂×C₄

D₄×D₂₃

yes

—

143

C₂×C₄

#67290

yes

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

149

C₂²

#10625110

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

(C₂²⋊D₂₃)⋊C₄

yes

—

155

C₄

#10625053

yes

characteristic

156

C₄

#10625057

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#13092294

yes

characteristic

	subgroup	G/H
center	C₂	#13092294
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#410	C₂
socle	C₂×C₂₃	#386
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#410	D₂₃
hypocenter	C₂₃	#1909
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₂

#13092294

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#410

C₂

socle

C₂×C₂₃

#386

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
3. #13092425	C₂	semidirect
4. #13092173	C₂	semidirect
10. #10625115	C₄	semidirect
11. #10625107	C₄	semidirect
99. C₂₃	#1909	semidirect

H ≅ G/N

type

3. #13092425

C₂

semidirect

4. #13092173

C₂

semidirect

10. #10625115

C₄

semidirect

11. #10625107

C₄

semidirect

99. C₂₃

#1909

semidirect

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

subgroup	index
#13093184	2
#13092988	2
#13092425	2
#13092173	2
C₂₃×#410	2
#1909	23

subgroup

index

C₂₃×#410

Subgroups — #15673917

Conjugacy classes of subgroups

Complements

Maximal subgroups