Group #13122746 — subgroups

Subgroups — #13122746

Order 1664.

420 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₁₃×#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#11718338	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718337	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718333	C₂	1	3	yes	maximal
5	#11718331	C₂	1	3	yes	maximal, complemented
6	#11718330	C₂	1	3	yes	maximal, complemented
7	#11718308	C₂	1	3	yes	maximal, complemented
8	C₂²×D₅₂	C₂²	1	1	yes	characteristic
9	C₂×#6103	C₂²	1	3	yes	—
10	C₂×#6103	C₂²	1	3	yes	—
11	C₂×#6103	C₂²	1	3	yes	—
12	C₂×#6103	C₂²	1	3	yes	—
13	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
14	C₂×#6101	C₂²	1	3	yes	—
15	C₂×#6101	C₂²	1	3	yes	—
16	C₂×C₄×Dic₁₃	C₂²	1	3	yes	—
17	C₁₃×#95	C₂²	1	3	yes	—
18	C₁₃×#95	C₂²	1	3	yes	—
19	C₁₃×#95	C₂²	1	1	yes	characteristic
20	#88071	C₂²	1	3	yes	—
21	C₂³×D₁₃	C₂³	1	1	yes	characteristic
22	C₂³×D₁₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
26	C₂²×Dic₁₃	C₂³	1	3	yes	—
27	C₂²×Dic₁₃	C₂³	1	3	yes	—
28	C₂×D₅₂	—	4	3	no	—
29	#6103	—	2	6	no	—
30	#6103	—	2	6	no	—
31	#6103	—	4	3	no	—
32	#6103	—	4	3	no	—
33	C₄⋊Dic₁₃	—	4	3	no	—
34	#6101	—	2	6	no	—
35	#6101	—	4	3	no	—
36	C₄×Dic₁₃	—	4	3	no	—
37	#2801	—	13	1	no	maximal, Sylow
38	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
39	C₂²×D₁₃	—	2	3	no	—
40	C₂²×D₁₃	—	2	3	no	—
41	C₂²×D₁₃	—	2	3	no	—
42	C₂²×D₁₃	—	2	3	no	—
43	C₂²×D₁₃	—	2	1	no	—
44	C₂²×D₁₃	—	2	1	no	—
45	C₂×C₄×C₁₃	—	2	3	no	—
46	C₂×C₄×C₁₃	—	2	3	no	—
47	C₂×C₄×C₁₃	—	2	3	no	—
48	C₂×C₄×C₁₃	—	2	3	no	—
49	C₂×C₄×C₁₃	—	2	3	no	—
50	C₂×C₄×C₁₃	—	2	3	no	—
51	C₂×C₄×C₁₃	—	2	3	no	—
52	C₂×Dic₁₃	—	2	3	no	—
53	C₂×Dic₁₃	—	2	3	no	—
54	C₂×Dic₁₃	—	2	3	no	—
55	C₂×Dic₁₃	—	2	3	no	—
56	C₂×Dic₁₃	—	2	3	no	—
57	C₂×Dic₁₃	#41	1	6	yes	—
58	D₅₂	—	4	4	no	—
59	#401	D₁₃	1	1	yes	characteristic, complemented
60	#399	—	13	1	no	—
61	#396	—	13	1	no	—
62	#394	—	13	3	no	—
63	#389	—	13	3	no	—
64	#388	—	13	3	no	—
65	#388	—	13	3	no	—
66	C₂²×C₁₃	C₂×#41	1	3	yes	—
67	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
68	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
69	C₂×D₁₃	—	4	3	no	—
70	C₂×D₁₃	—	4	3	no	—
71	C₂×D₁₃	—	4	3	no	—
72	C₂×D₁₃	—	4	3	no	—
73	C₂×D₁₃	—	4	1	no	—
74	C₂×D₁₃	—	4	1	no	—
75	C₄×C₁₃	—	4	3	no	—
76	C₄×C₁₃	—	4	3	no	—
77	C₄×C₁₃	—	4	1	no	—
78	Dic₁₃	—	2	6	no	—
79	Dic₁₃	—	4	3	no	—
80	C₂²×D₄	—	13	1	no	—
81	C₂×C₄⋊C₄	—	13	3	no	—
82	C₂×C₄⋊C₄	—	13	3	no	—
83	C₂×C₄⋊C₄	—	13	3	no	—
84	C₂×C₂²⋊C₄	—	13	3	no	—
85	C₂×C₂²⋊C₄	—	13	3	no	—
86	C₂×C₂²⋊C₄	—	13	3	no	—
87	C₂×C₂²⋊C₄	—	13	3	no	—
88	C₂×C₄²	—	13	3	no	—
89	#95	—	13	3	no	—
90	#95	C₂×D₁₃	1	3	yes	—
91	#95	C₂×D₁₃	1	3	yes	—
92	#95	C₂×D₁₃	1	1	yes	characteristic
93	C₂×C₁₃	#542	1	3	yes	—
94	C₂×C₁₃	#553	1	3	yes	—
95	C₂×C₁₃	#560	1	1	yes	characteristic
96	D₁₃	—	8	1	no	—
97	D₁₃	—	8	1	no	—
98	C₂⁴	—	13	1	no	—
99	C₂⁴	—	13	1	no	—
100	C₂×D₄	—	52	3	no	—
101	C₂²×C₄	—	13	3	no	—
102	C₂²×C₄	—	13	3	no	—
103	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
104	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
105	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
106	C₄⋊C₄	—	26	6	no	—
107	C₄⋊C₄	—	52	3	no	—
108	C₄⋊C₄	—	52	3	no	—
109	C₂²⋊C₄	—	26	6	no	—
110	C₂²⋊C₄	—	26	6	no	—
111	C₂²⋊C₄	—	52	3	no	—
112	C₂²⋊C₄	—	52	3	no	—
113	C₄²	—	52	3	no	—
114	C₁₃	#2801	1	1	yes	characteristic, complemented, Sylow
115	C₂³	—	26	3	no	—
116	C₂³	—	26	3	no	—
117	C₂³	—	26	3	no	—
118	C₂³	—	26	3	no	—
119	C₂³	—	26	1	no	—
120	C₂³	—	26	1	no	—
121	C₂³	C₂³×D₁₃	1	1	yes	characteristic
122	D₄	—	52	4	no	—
123	C₂×C₄	—	2	3	no	—
124	C₂×C₄	—	2	3	no	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	13	6	no	—
131	C₂×C₄	—	26	3	no	—
132	C₂×C₄	—	26	3	no	—
133	C₂×C₄	—	26	3	no	—
134	C₂×C₄	—	26	3	no	—
135	C₂×C₄	—	26	3	no	—
136	C₂²	—	52	3	no	—
137	C₂²	—	52	3	no	—
138	C₂²	—	52	3	no	—
139	C₂²	—	52	3	no	—
140	C₂²	—	52	1	no	—
141	C₂²	—	52	1	no	—
142	C₂²	#41×D₁₃	1	3	yes	—
143	C₂²	#88256	1	3	yes	—
144	C₂²	#88257	1	1	yes	characteristic
145	C₄	—	4	3	no	—
146	C₄	—	4	3	no	—
147	C₄	—	4	1	no	—
148	C₄	—	26	6	no	—
149	C₄	—	52	3	no	—
150	C₂	—	104	1	no	—
151	C₂	—	104	1	no	—
152	C₂	#11719414	1	3	yes	—
153	C₂	#11719415	1	3	yes	—
154	C₂	#11719416	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#401

C₂

yes

maximal, characteristic, complemented

#11718338

C₂

yes

maximal, characteristic, complemented

#11718337

C₂

yes

maximal, characteristic, complemented

#11718333

C₂

yes

maximal

#11718331

C₂

yes

maximal, complemented

#11718330

C₂

yes

maximal, complemented

#11718308

C₂

yes

maximal, complemented

C₂²×D₅₂

C₂²

yes

characteristic

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

C₁₃×#95

C₂²

yes

characteristic

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

characteristic

C₂³×D₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

—

—

—

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

#2801

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#41

yes

—

D₅₂

—

#401

D₁₃

yes

characteristic, complemented

—

—

—

—

—

—

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

C₂×D₁₃

yes

—

#95

C₂×D₁₃

yes

—

#95

C₂×D₁₃

yes

characteristic

C₂×C₁₃

#542

yes

—

C₂×C₁₃

#553

yes

—

C₂×C₁₃

#560

yes

characteristic

D₁₃

—

D₁₃

—

C₂⁴

—

C₂⁴

—

100

C₂×D₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×D₁₃

yes

—

104

C₂²×C₄

C₂²×D₁₃

yes

—

105

C₂²×C₄

C₂²×D₁₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₁₃

#2801

yes

characteristic, complemented, Sylow

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

C₂³×D₁₃

yes

characteristic

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

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

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#41×D₁₃

yes

—

143

C₂²

#88256

yes

—

144

C₂²

#88257

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₂

—

104

—

151

C₂

—

104

—

152

C₂

#11719414

yes

—

153

C₂

#11719415

yes

—

154

C₂

#11719416

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#401	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	#401	D₁₃
hypocenter	C₁₃	#2801
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #401

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#401

C₂

prime radical

C₂²×D₅₂

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #401

N	H ≅ G/N	type
1. C₁₃×#401	C₂	semidirect
2. #11718338	C₂	semidirect
3. #11718337	C₂	semidirect
5. #11718331	C₂	semidirect
6. #11718330	C₂	semidirect
7. #11718308	C₂	semidirect
59. #401	D₁₃	semidirect
114. C₁₃	#2801	semidirect

H ≅ G/N

type

1. C₁₃×#401

C₂

semidirect

2. #11718338

C₂

semidirect

3. #11718337

C₂

semidirect

5. #11718331

C₂

semidirect

6. #11718330

C₂

semidirect

7. #11718308

C₂

semidirect

59. #401

D₁₃

semidirect

114. C₁₃

#2801

semidirect

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

subgroup	index
C₁₃×#401	2
#11718338	2
#11718337	2
#11718333	2
#11718331	2
#11718330	2
#11718308	2
#2801	13

subgroup

index

C₁₃×#401

Subgroups — #13122746

Conjugacy classes of subgroups

Complements

Maximal subgroups