Group #13120945 — subgroups

Subgroups — #13120945

Order 1664.

612 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 153 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×#88142	C₂	1	3	yes	maximal, complemented
2	C₂×#88139	C₂	1	3	yes	maximal
3	C₁₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718653	C₂	1	3	yes	maximal, complemented
5	#11718610	C₂	1	3	yes	maximal, complemented
6	#11718338	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718309	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₆	C₂²	1	3	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
12	C₂×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
13	C₂×#6103	C₂²	1	3	yes	—
14	C₂×#6103	C₂²	1	3	yes	—
15	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
16	C₂×#6101	C₂²	1	6	yes	—
17	C₂×C₄×Dic₁₃	C₂²	1	3	yes	—
18	#88142	—	2	12	no	—
19	#88139	—	2	12	no	—
20	C₁₃×#95	C₂²	1	1	yes	characteristic
21	#88071	C₂²	1	3	yes	—
22	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×D₁₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
26	C₂×C₂²⋊D₁₃	—	2	6	no	—
27	C₂²×Dic₁₃	C₂³	1	3	yes	—
28	C₂²×Dic₁₃	C₂³	1	3	yes	—
29	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₂₆	—	2	6	no	—
31	C₂×Dic₂₆	—	2	12	no	—
32	C₁₃×C₄⋊C₄	—	2	6	no	—
33	C₁₃×C₂²⋊C₄	—	2	6	no	—
34	C₂²⋊Dic₁₃	—	2	6	no	—
35	#6103	—	2	6	no	—
36	#6103	—	2	6	no	—
37	C₄⋊Dic₁₃	—	2	6	no	—
38	#6101	—	2	12	no	—
39	C₄×Dic₁₃	D₄	1	12	yes	—
40	#2658	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	—	2	3	no	—
42	C₂³×C₁₃	—	2	1	no	—
43	C₂³×C₁₃	—	2	3	no	—
44	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
45	C₂²×D₁₃	—	2	3	no	—
46	C₂²×D₁₃	—	2	1	no	—
47	C₂²×D₁₃	—	2	3	no	—
48	C₂×C₄×C₁₃	—	2	6	no	—
49	C₂×C₄×C₁₃	—	2	6	no	—
50	C₂×C₄×C₁₃	—	2	3	no	—
51	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
52	C₂²⋊D₁₃	—	4	4	no	—
53	C₂×Dic₁₃	—	2	3	no	—
54	C₂×Dic₁₃	—	2	3	no	—
55	C₂×Dic₁₃	—	2	6	no	—
56	C₂×Dic₁₃	—	2	3	no	—
57	C₂×Dic₁₃	C₂×D₄	1	6	yes	—
58	C₂×Dic₁₃	C₂×D₄	1	6	yes	—
59	Dic₂₆	—	4	12	no	—
60	C₂×C₄⋊Q₈	—	13	3	no	—
61	C₂×#124	—	13	3	no	—
62	#399	—	13	3	no	—
63	#399	—	13	3	no	—
64	#399	—	13	1	no	—
65	#399	D₁₃	1	1	yes	characteristic, complemented
66	#376	—	13	1	no	—
67	C₂²×C₁₃	—	4	1	no	—
68	C₂²×C₁₃	—	4	3	no	—
69	C₂²×C₁₃	—	4	3	no	—
70	C₂²×C₁₃	C₂²×D₄	1	3	yes	—
71	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
72	C₂²×C₁₃	2₋¹⁺⁴	1	3	yes	—
73	C₂×D₁₃	—	4	1	no	—
74	C₂×D₁₃	—	4	3	no	—
75	C₂×D₁₃	—	4	3	no	—
76	C₄×C₁₃	—	2	6	no	—
77	C₄×C₁₃	—	4	3	no	—
78	Dic₁₃	—	2	6	no	—
79	Dic₁₃	—	4	3	no	—
80	Dic₁₃	#127	1	4	yes	—
81	C₂²×Q₈	—	13	3	no	—
82	C₂²×D₄	—	13	1	no	—
83	C₄⋊Q₈	—	26	12	no	—
84	#124	—	26	12	no	—
85	C₂×C₄⋊C₄	—	13	6	no	—
86	C₂×C₄⋊C₄	—	13	3	no	—
87	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
88	C₂×C₂²⋊C₄	—	13	3	no	—
89	C₂×C₂²⋊C₄	—	13	3	no	—
90	C₂×C₂²⋊C₄	—	13	3	no	—
91	C₂×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
92	C₂×C₄²	—	13	3	no	—
93	#95	—	13	3	no	—
94	#95	C₂×D₁₃	1	1	yes	characteristic
95	C₂×C₁₃	—	8	1	no	—
96	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
97	C₂×C₁₃	#536	1	3	yes	—
98	C₂×C₁₃	#536	1	3	yes	—
99	D₁₃	—	8	1	no	—
100	C₂⁴	—	13	1	no	—
101	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
102	C₂×Q₈	—	26	6	no	—
103	C₂×Q₈	—	26	12	no	—
104	C₂×D₄	—	26	6	no	—
105	C₂²×C₄	—	13	1	no	—
106	C₂²×C₄	—	13	3	no	—
107	C₂²×C₄	—	13	3	no	—
108	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
109	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
110	C₄⋊C₄	—	2	6	no	—
111	C₄⋊C₄	—	26	12	no	—
112	C₄⋊C₄	—	26	6	no	—
113	C₂²⋊C₄	—	2	6	no	—
114	C₂²⋊C₄	—	26	6	no	—
115	C₂²⋊C₄	—	26	6	no	—
116	C₂²⋊C₄	—	26	6	no	—
117	C₄²	—	13	12	no	—
118	C₁₃	#2658	1	1	yes	characteristic, complemented, Sylow
119	C₂³	—	2	3	no	—
120	C₂³	—	2	1	no	—
121	C₂³	—	2	3	no	—
122	C₂³	—	26	3	no	—
123	C₂³	—	26	1	no	—
124	C₂³	—	26	3	no	—
125	C₂³	C₂³×D₁₃	1	1	yes	characteristic
126	Q₈	—	52	12	no	—
127	D₄	—	52	4	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	6	no	—
131	C₂×C₄	—	13	6	no	—
132	C₂×C₄	—	13	6	no	—
133	C₂×C₄	—	26	3	no	—
134	C₂×C₄	—	26	3	no	—
135	C₂×C₄	—	26	3	no	—
136	C₂×C₄	—	26	6	no	—
137	C₂×C₄	D₄×D₁₃	1	6	yes	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	52	3	no	—
142	C₂²	—	52	3	no	—
143	C₂²	—	52	1	no	—
144	C₂²	C₂×D₄×D₁₃	1	3	yes	—
145	C₂²	#88254	1	1	yes	characteristic
146	C₂²	#88257	1	3	yes	—
147	C₄	—	2	6	no	—
148	C₄	—	4	3	no	—
149	C₄	—	13	4	no	—
150	C₄	—	26	6	no	—
151	C₄	—	52	3	no	—
152	C₂	—	8	1	no	—
153	C₂	—	104	1	no	—
154	C₂	#11719333	1	3	yes	—
155	C₂	#11719353	1	3	yes	—
156	C₂	D₁₃×#127	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88142

C₂

yes

maximal, complemented

C₂×#88139

C₂

yes

maximal

C₁₃×#399

C₂

yes

maximal, characteristic, complemented

#11718653

C₂

yes

maximal, complemented

#11718610

C₂

yes

maximal, complemented

#11718338

C₂

yes

maximal, characteristic, complemented

#11718309

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₁₃

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88142

—

#88139

—

C₁₃×#95

C₂²

yes

characteristic

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂³×D₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

#6103

—

#6103

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

D₄

yes

—

#2658

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

—

—

—

—

D₁₃

yes

characteristic, complemented

#376

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#127

yes

—

C₂²×Q₈

—

C₂²×D₄

—

C₄⋊Q₈

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

#95

—

#95

C₂×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

C₂×#127

yes

characteristic

C₂×C₁₃

#536

yes

—

C₂×C₁₃

#536

yes

—

D₁₃

—

100

C₂⁴

—

101

C₂⁴

C₂²×D₁₃

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²×D₁₃

yes

—

109

C₂²×C₄

C₂²×D₁₃

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₁₃

#2658

yes

characteristic, complemented, Sylow

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂³×D₁₃

yes

characteristic

126

Q₈

—

127

D₄

—

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

—

137

C₂×C₄

D₄×D₁₃

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×D₄×D₁₃

yes

—

145

C₂²

#88254

yes

characteristic

146

C₂²

#88257

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₂

—

153

C₂

—

104

—

154

C₂

#11719333

yes

—

155

C₂

#11719353

yes

—

156

C₂

D₁₃×#127

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#399	C₂
prime radical	C₂²×C₂²⋊D₁₃	C₂²
hypercenter	#399	D₁₃
hypocenter	C₁₃	#2658
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#399

C₂

prime radical

C₂²×C₂²⋊D₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
1. C₂×#88142	C₂	semidirect
3. C₁₃×#399	C₂	semidirect
4. #11718653	C₂	semidirect
5. #11718610	C₂	semidirect
6. #11718338	C₂	semidirect
7. #11718309	C₂	semidirect
65. #399	D₁₃	semidirect
118. C₁₃	#2658	semidirect

H ≅ G/N

type

1. C₂×#88142

C₂

semidirect

3. C₁₃×#399

C₂

semidirect

4. #11718653

C₂

semidirect

5. #11718610

C₂

semidirect

6. #11718338

C₂

semidirect

7. #11718309

C₂

semidirect

65. #399

D₁₃

semidirect

118. C₁₃

#2658

semidirect

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

subgroup	index
C₂×#88142	2
C₂×#88139	2
C₁₃×#399	2
#11718653	2
#11718610	2
#11718338	2
#11718309	2
#2658	13

subgroup

index

C₁₃×#399

Subgroups — #13120945

Conjugacy classes of subgroups

Complements

Maximal subgroups