Group #13092989 — subgroups

Subgroups — #13092989

Order 1472.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#67290	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#123	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625106	C₂	1	1	yes	maximal, characteristic, complemented
5	#10625100	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625067	C₂	1	2	yes	maximal, complemented
7	#10625059	C₂	1	2	yes	maximal, complemented
8	#10625057	C₂	1	1	yes	maximal, characteristic, complemented
9	#10625056	C₂	1	2	yes	maximal, complemented
10	#10625049	C₂	1	2	yes	maximal, complemented
11	#10625048	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
19	#67290	—	2	4	no	—
20	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₄₆	—	2	4	no	—
22	C₂×Dic₄₆	C₂²	1	2	yes	—
23	C₂×Dic₄₆	C₂²	1	2	yes	—
24	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
30	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
31	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
32	C₄⋊Dic₂₃	C₂²	1	2	yes	—
33	Dic₂₃⋊C₄	C₂²	1	2	yes	—
34	Dic₂₃⋊C₄	C₂²	1	2	yes	—
35	Dic₂₃⋊C₄	C₂²	1	2	yes	—
36	C₄×Dic₂₃	C₂²	1	2	yes	—
37	C₂³×C₂₃	C₂³	1	1	yes	characteristic
38	C₂³×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×D₂₃	C₂³	1	1	yes	characteristic
40	D₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²⋊D₂₃	—	2	2	no	—
46	C₂²⋊D₂₃	—	2	2	no	—
47	C₂×Dic₂₃	—	2	2	no	—
48	C₂×Dic₂₃	—	2	2	no	—
49	C₂×Dic₂₃	D₄	1	4	yes	—
50	C₂×Dic₂₃	C₂³	1	2	yes	—
51	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₃	C₂³	1	2	yes	—
53	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
54	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
55	C₄×D₂₃	—	2	2	no	—
56	Dic₄₆	—	2	4	no	—
57	Dic₄₆	—	2	4	no	—
58	Dic₄₆	—	2	2	no	—
59	C₂²×C₂₃	—	2	2	no	—
60	C₂²×C₂₃	—	2	1	no	—
61	C₂²×C₂₃	—	2	2	no	—
62	C₂²×C₂₃	C₂×D₄	1	2	yes	—
63	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
64	C₂×D₂₃	—	2	1	no	—
65	C₂×D₂₃	—	2	2	no	—
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	—	2	1	no	—
68	C₄×C₂₃	—	2	2	no	—
69	Dic₂₃	—	2	2	no	—
70	Dic₂₃	—	2	1	no	—
71	Dic₂₃	—	2	2	no	—
72	Dic₂₃	C₂×D₄	1	2	yes	—
73	Dic₂₃	C₂×D₄	1	2	yes	—
74	#536	—	23	1	no	maximal, Sylow
75	C₂×C₂₃	—	2	2	no	—
76	C₂×C₂₃	—	4	1	no	—
77	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₂₃	2₋¹⁺⁴	1	2	yes	—
79	D₂₃	—	4	1	no	—
80	C₂×#41	—	23	1	no	—
81	C₂²×Q₈	—	23	1	no	—
82	C₄⋊Q₈	—	23	2	no	—
83	#124	—	23	2	no	—
84	#123	—	23	1	no	—
85	#123	—	23	1	no	—
86	#123	—	23	1	no	—
87	#123	D₂₃	1	1	yes	characteristic, complemented
88	C₂²⋊Q₈	—	23	2	no	—
89	C₂²⋊Q₈	—	23	2	no	—
90	#117	—	23	1	no	—
91	C₂₃	#536	1	1	yes	characteristic, complemented, Sylow
92	#41	—	46	4	no	—
93	C₂×Q₈	—	23	2	no	—
94	C₂×Q₈	—	23	2	no	—
95	C₂×Q₈	—	23	1	no	—
96	C₂×Q₈	—	46	4	no	—
97	C₂×D₄	—	23	1	no	—
98	C₂×D₄	—	23	1	no	—
99	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
100	C₂²×C₄	—	23	1	no	—
101	C₂²×C₄	—	23	1	no	—
102	C₂²×C₄	—	23	1	no	—
103	C₂²×C₄	—	23	1	no	—
104	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
105	C₄⋊C₄	—	23	2	no	—
106	C₄⋊C₄	—	23	2	no	—
107	C₄⋊C₄	—	23	2	no	—
108	C₄⋊C₄	—	23	2	no	—
109	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
110	C₂²⋊C₄	—	23	1	no	—
111	C₂²⋊C₄	—	23	2	no	—
112	C₂²⋊C₄	—	23	2	no	—
113	C₂²⋊C₄	—	23	2	no	—
114	C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
115	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
116	C₄²	—	23	2	no	—
117	C₂³	—	23	1	no	—
118	C₂³	C₂²×D₂₃	1	1	yes	characteristic
119	C₂³	C₂²×D₂₃	1	1	yes	characteristic
120	Q₈	—	46	4	no	—
121	Q₈	—	46	4	no	—
122	Q₈	—	46	2	no	—
123	D₄	—	2	2	no	—
124	D₄	—	46	2	no	—
125	D₄	—	46	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	23	4	no	—
128	C₂×C₄	—	23	1	no	—
129	C₂×C₄	—	23	1	no	—
130	C₂×C₄	—	23	2	no	—
131	C₂×C₄	—	23	2	no	—
132	C₂×C₄	—	23	1	no	—
133	C₂×C₄	—	46	2	no	—
134	C₂×C₄	—	46	2	no	—
135	C₂×C₄	—	46	2	no	—
136	C₂×C₄	C₂²×D₂₃	1	2	yes	—
137	C₂×C₄	C₂²×D₂₃	1	2	yes	—
138	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	46	1	no	—
143	C₂²	—	46	2	no	—
144	C₂²	D₄×D₂₃	1	2	yes	—
145	C₂²	C₂³×D₂₃	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	23	2	no	—
150	C₄	—	23	2	no	—
151	C₄	—	46	1	no	—
152	C₄	—	46	2	no	—
153	C₄	—	46	2	no	—
154	C₂	—	2	2	no	—
155	C₂	—	4	1	no	—
156	C₂	—	92	1	no	—
157	C₂	C₂×D₄×D₂₃	1	1	yes	characteristic
158	C₂	#10625161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67290

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

C₂

yes

maximal, characteristic, complemented

C₂₃×#123

C₂

yes

maximal, characteristic, complemented

#10625106

C₂

yes

maximal, characteristic, complemented

#10625100

C₂

yes

maximal, characteristic, complemented

#10625067

C₂

yes

maximal, complemented

#10625059

C₂

yes

maximal, complemented

#10625057

C₂

yes

maximal, characteristic, complemented

#10625056

C₂

yes

maximal, complemented

#10625049

C₂

yes

maximal, complemented

#10625048

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

characteristic

#67290

—

C₂×C₄×D₂₃

C₂²

yes

characteristic

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₂³×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

C₂³

yes

characteristic

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

D₄

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₄×D₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×D₄

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₂×D₄

yes

—

Dic₂₃

C₂×D₄

yes

—

#536

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

—

D₂₃

—

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#117

—

C₂₃

#536

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₂₃

yes

characteristic

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×D₂₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂×D₂₃

yes

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

C₂×D₂₃

yes

characteristic

115

C₂²⋊C₄

C₂×D₂₃

yes

—

116

C₄²

—

117

C₂³

—

118

C₂³

C₂²×D₂₃

yes

characteristic

119

C₂³

C₂²×D₂₃

yes

characteristic

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

D₄

—

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

—

135

C₂×C₄

—

136

C₂×C₄

C₂²×D₂₃

yes

—

137

C₂×C₄

C₂²×D₂₃

yes

—

138

C₂×C₄

C₂²×D₂₃

yes

characteristic

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

D₄×D₂₃

yes

—

145

C₂²

C₂³×D₂₃

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₂

—

155

C₂

—

156

C₂

—

157

C₂

C₂×D₄×D₂₃

yes

characteristic

158

C₂

#10625161

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#123	C₂
prime radical	C₂×#67290	C₂
hypercenter	#123	D₂₃
hypocenter	C₂₃	#536
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#123

C₂

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
2. C₂²×Dic₄₆	C₂	semidirect
3. C₂₃×#123	C₂	semidirect
4. #10625106	C₂	semidirect
5. #10625100	C₂	semidirect
6. #10625067	C₂	semidirect
7. #10625059	C₂	semidirect
8. #10625057	C₂	semidirect
9. #10625056	C₂	semidirect
10. #10625049	C₂	semidirect
11. #10625048	C₂	semidirect
87. #123	D₂₃	semidirect
91. C₂₃	#536	semidirect

H ≅ G/N

type

2. C₂²×Dic₄₆

C₂

semidirect

3. C₂₃×#123

C₂

semidirect

4. #10625106

C₂

semidirect

5. #10625100

C₂

semidirect

6. #10625067

C₂

semidirect

7. #10625059

C₂

semidirect

8. #10625057

C₂

semidirect

9. #10625056

C₂

semidirect

10. #10625049

C₂

semidirect

11. #10625048

C₂

semidirect

87. #123

D₂₃

semidirect

91. C₂₃

#536

semidirect

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

subgroup	index
C₂×#67290	2
C₂²×Dic₄₆	2
C₂₃×#123	2
#10625106	2
#10625100	2
#10625067	2
#10625059	2
#10625057	2
#10625056	2
#10625049	2
#10625048	2
#536	23

subgroup

index

C₂×#67290

C₂²×Dic₄₆

C₂₃×#123

Subgroups — #13092989

Conjugacy classes of subgroups

Complements

Maximal subgroups