Group #13092994 — subgroups

Subgroups — #13092994

Order 1472.

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

normal subgroups only

This group has 294 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	2	yes	maximal, complemented
2	C₂₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625110	C₂	1	2	yes	maximal, complemented
4	#10625110	C₂	1	2	yes	maximal, complemented
5	#10625101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#10625067	C₂	1	2	yes	maximal, complemented
8	#10625055	C₂	1	2	yes	maximal, complemented
9	#10625049	C₂	1	2	yes	maximal, complemented
10	C₂×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
16	C₂²×Dic₂₃	C₂²	1	2	yes	—
17	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
19	#67290	—	2	8	no	—
20	C₂×D₉₂	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₂₃	C₂²	1	2	yes	—
22	C₂×Dic₄₆	C₂²	1	2	yes	—
23	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
26	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
28	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	Dic₂₃⋊C₄	C₂²	1	2	yes	—
30	Dic₂₃⋊C₄	C₂²	1	2	yes	—
31	Dic₂₃⋊C₄	C₂²	1	2	yes	—
32	C₂³×C₂₃	C₂³	1	2	yes	—
33	C₂³×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×D₂₃	C₂³	1	2	yes	—
35	D₄×C₂₃	—	2	4	no	—
36	D₄×C₂₃	—	4	1	no	—
37	C₂×C₄×C₂₃	—	4	1	no	—
38	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²⋊D₂₃	—	2	4	no	—
42	C₂²⋊D₂₃	—	2	4	no	—
43	C₂²⋊D₂₃	—	4	2	no	—
44	C₂×Dic₂₃	—	2	4	no	—
45	C₂×Dic₂₃	—	4	1	no	—
46	C₂×Dic₂₃	D₄	1	4	yes	complemented
47	C₂×Dic₂₃	C₂³	1	2	yes	—
48	C₂×Dic₂₃	C₂³	1	2	yes	—
49	C₂×Dic₂₃	C₂³	1	2	yes	—
50	D₉₂	—	4	1	no	—
51	C₄×D₂₃	—	2	4	no	—
52	Dic₄₆	—	2	4	no	—
53	C₂²×C₂₃	—	2	1	no	—
54	C₂²×C₂₃	—	2	1	no	—
55	C₂²×C₂₃	—	2	2	no	—
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	—	2	2	no	—
58	C₂²×C₂₃	C₂×D₄	1	2	yes	—
59	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂×D₂₃	—	2	2	no	—
61	C₂×D₂₃	—	2	2	no	—
62	C₂×D₂₃	—	2	2	no	—
63	C₄×C₂₃	—	2	2	no	—
64	C₄×C₂₃	—	2	1	no	—
65	C₄×C₂₃	—	2	1	no	—
66	Dic₂₃	—	2	2	no	—
67	Dic₂₃	—	2	2	no	—
68	Dic₂₃	C₂×D₄	1	4	yes	—
69	#537	—	23	1	no	maximal, Sylow
70	C₂×C₂₃	—	2	2	no	—
71	C₂×C₂₃	—	4	2	no	—
72	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
74	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
75	D₂₃	—	4	2	no	—
76	C₂×#41	—	23	2	no	—
77	C₂²⋊Q₈	—	23	2	no	—
78	C₂²⋊Q₈	—	23	2	no	—
79	#121	—	23	2	no	—
80	#121	—	23	2	no	—
81	#121	—	23	1	no	—
82	#121	—	23	2	no	—
83	#121	D₂₃	1	1	yes	characteristic, complemented
84	C₂×C₄⋊C₄	—	23	1	no	—
85	C₂₃	#537	1	1	yes	characteristic, complemented, Sylow
86	#41	—	46	8	no	—
87	C₂×Q₈	—	23	2	no	—
88	C₂×D₄	—	23	2	no	—
89	C₂×D₄	—	23	2	no	—
90	C₂×D₄	—	23	2	no	—
91	C₂×D₄	—	23	1	no	—
92	C₂×D₄	C₂×D₂₃	1	2	yes	—
93	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
94	C₂²×C₄	—	23	1	no	—
95	C₂²×C₄	—	23	2	no	—
96	C₂²×C₄	—	23	1	no	—
97	C₂²×C₄	—	23	2	no	—
98	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
99	C₄⋊C₄	—	23	2	no	—
100	C₄⋊C₄	—	23	2	no	—
101	C₄⋊C₄	—	23	2	no	—
102	C₄⋊C₄	—	23	1	no	—
103	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
104	C₂²⋊C₄	—	23	2	no	—
105	C₂²⋊C₄	—	23	2	no	—
106	C₂²⋊C₄	—	23	2	no	—
107	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
108	C₂³	—	23	2	no	—
109	C₂³	C₂²×D₂₃	1	2	yes	—
110	C₂³	C₂²×D₂₃	1	1	yes	characteristic
111	Q₈	—	46	4	no	—
112	D₄	—	2	4	no	—
113	D₄	—	4	1	no	—
114	D₄	—	46	4	no	—
115	D₄	—	46	4	no	—
116	D₄	—	92	1	no	—
117	D₄	—	92	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	23	4	no	—
120	C₂×C₄	—	23	2	no	—
121	C₂×C₄	—	23	2	no	—
122	C₂×C₄	—	23	2	no	—
123	C₂×C₄	—	46	4	no	—
124	C₂×C₄	—	46	4	no	—
125	C₂×C₄	—	92	1	no	—
126	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
127	C₂×C₄	C₂²×D₂₃	1	2	yes	—
128	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	46	2	no	—
135	C₂²	—	46	2	no	—
136	C₂²	—	46	2	no	—
137	C₂²	D₄×D₂₃	1	2	yes	—
138	C₂²	C₂³×D₂₃	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	23	4	no	—
143	C₄	—	46	2	no	—
144	C₄	—	46	2	no	—
145	C₂	—	2	2	no	—
146	C₂	—	4	2	no	—
147	C₂	—	92	2	no	—
148	C₂	C₂×D₄×D₂₃	1	1	yes	characteristic
149	C₂	#10625155	1	1	yes	characteristic
150	C₂	#10625161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67290

C₂

yes

maximal, complemented

C₂₃×#121

C₂

yes

maximal, characteristic, complemented

#10625110

C₂

yes

maximal, complemented

#10625110

C₂

yes

maximal, complemented

#10625101

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃⋊C₄

C₂

yes

maximal, characteristic, complemented

#10625067

C₂

yes

maximal, complemented

#10625055

C₂

yes

maximal, complemented

#10625049

C₂

yes

maximal, complemented

C₂×D₄×C₂₃

C₂²

yes

—

C₂×D₄×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

characteristic

#67290

—

C₂×D₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₂³×C₂₃

C₂³

yes

—

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

C₂³

yes

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

D₄

yes

complemented

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

D₉₂

—

C₄×D₂₃

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×D₄

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₂×D₄

yes

—

#537

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₃

H(5,𝔽₂)

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

D₂₃

—

C₂×#41

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂₃

#537

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₂₃

yes

—

C₂×D₄

C₂×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂×D₂₃

yes

characteristic

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

C₂×D₂₃

yes

—

108

C₂³

—

109

C₂³

C₂²×D₂₃

yes

—

110

C₂³

C₂²×D₂₃

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂²×D₂₃

yes

characteristic

127

C₂×C₄

C₂²×D₂₃

yes

—

128

C₂×C₄

C₂²×D₂₃

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

D₄×D₂₃

yes

—

138

C₂²

C₂³×D₂₃

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₂×D₄×D₂₃

yes

characteristic

149

C₂

#10625155

yes

characteristic

150

C₂

#10625161

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#121	C₂
hypercenter	#121	D₂₃
hypocenter	C₂₃	#537
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#121

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. C₂×#67290	C₂	semidirect
2. C₂₃×#121	C₂	semidirect
3. #10625110	C₂	semidirect
4. #10625110	C₂	semidirect
5. #10625101	C₂	semidirect
6. C₂×Dic₂₃⋊C₄	C₂	semidirect
7. #10625067	C₂	semidirect
8. #10625055	C₂	semidirect
9. #10625049	C₂	semidirect
46. C₂×Dic₂₃	D₄	semidirect
83. #121	D₂₃	semidirect
85. C₂₃	#537	semidirect

H ≅ G/N

type

1. C₂×#67290

C₂

semidirect

2. C₂₃×#121

C₂

semidirect

3. #10625110

C₂

semidirect

4. #10625110

C₂

semidirect

5. #10625101

C₂

semidirect

6. C₂×Dic₂₃⋊C₄

C₂

semidirect

7. #10625067

C₂

semidirect

8. #10625055

C₂

semidirect

9. #10625049

C₂

semidirect

46. C₂×Dic₂₃

D₄

semidirect

83. #121

D₂₃

semidirect

85. C₂₃

#537

semidirect

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

subgroup	index
C₂×#67290	2
C₂₃×#121	2
#10625110	2
#10625101	2
C₂×Dic₂₃⋊C₄	2
#10625067	2
#10625055	2
#10625049	2
#537	23

subgroup

index

C₂₃×#121

C₂×Dic₂₃⋊C₄

Subgroups — #13092994

Conjugacy classes of subgroups

Complements

Maximal subgroups