Group #13120798 — subgroups

Subgroups — #13120798

Order 1664.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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×#396	C₂	1	1	yes	maximal, characteristic, complemented
2	#11718654	C₂	1	3	yes	maximal, complemented
3	#11718633	C₂	1	3	yes	maximal, complemented
4	#11718618	C₂	1	3	yes	maximal
5	#11718613	C₂	1	3	yes	maximal, complemented
6	#11718337	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	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
11	C₂×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
12	C₂×#6103	C₂²	1	3	yes	—
13	C₂×#6103	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	1	yes	characteristic
18	#88071	C₂²	1	3	yes	—
19	#88071	C₂²	1	6	yes	—
20	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
21	C₂³×D₁₃	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
24	C₂×C₂²⋊D₁₃	—	2	6	no	—
25	C₂²×Dic₁₃	C₂³	1	3	yes	—
26	C₂²×Dic₁₃	C₂³	1	3	yes	—
27	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×C₄⋊C₄	—	4	3	no	—
29	C₁₃×C₂²⋊C₄	—	4	3	no	—
30	C₂²⋊Dic₁₃	—	4	3	no	—
31	#6103	—	4	3	no	—
32	#6103	—	4	3	no	—
33	#6101	—	2	6	no	—
34	#6101	—	4	3	no	—
35	C₄×Dic₁₃	—	2	6	no	—
36	#2644	—	13	1	no	maximal, Sylow
37	C₂³×C₁₃	—	2	3	no	—
38	C₂³×C₁₃	—	2	3	no	—
39	C₂³×C₁₃	—	2	1	no	—
40	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂²×D₁₃	—	2	3	no	—
42	C₂²×D₁₃	—	2	3	no	—
43	C₂²×D₁₃	—	2	1	no	—
44	C₂×C₄×C₁₃	—	2	3	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	C₂²⋊D₁₃	—	4	4	no	—
51	C₂×Dic₁₃	—	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	#397	—	13	3	no	—
59	#397	—	13	3	no	—
60	#396	—	13	1	no	—
61	#396	D₁₃	1	1	yes	characteristic, complemented
62	#390	—	13	3	no	—
63	#389	—	13	3	no	—
64	#376	—	13	1	no	—
65	C₂²×C₁₃	—	4	1	no	—
66	C₂²×C₁₃	—	4	3	no	—
67	C₂²×C₁₃	—	4	3	no	—
68	C₂²×C₁₃	C₂×#41	1	3	yes	—
69	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
70	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
71	C₂×D₁₃	—	4	1	no	—
72	C₂×D₁₃	—	4	3	no	—
73	C₂×D₁₃	—	4	3	no	—
74	C₄×C₁₃	—	4	3	no	—
75	C₄×C₁₃	—	4	3	no	—
76	Dic₁₃	—	4	3	no	—
77	Dic₁₃	—	4	3	no	—
78	Dic₁₃	#126	1	4	yes	—
79	C₂²×D₄	—	13	1	no	—
80	C₂×C₄⋊C₄	—	13	3	no	—
81	C₂×C₄⋊C₄	—	13	3	no	—
82	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
83	C₂×C₂²⋊C₄	—	13	3	no	—
84	C₂×C₂²⋊C₄	—	13	3	no	—
85	C₂×C₂²⋊C₄	—	13	3	no	—
86	C₂×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
87	C₂×C₄²	—	13	3	no	—
88	#95	—	13	3	no	—
89	#95	—	13	6	no	—
90	#95	C₂×D₁₃	1	1	yes	characteristic
91	C₂×C₁₃	—	8	1	no	—
92	C₂×C₁₃	C₂×#126	1	1	yes	characteristic
93	C₂×C₁₃	#540	1	3	yes	—
94	C₂×C₁₃	#542	1	3	yes	—
95	D₁₃	—	8	1	no	—
96	C₂⁴	—	13	1	no	—
97	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
98	C₂×D₄	—	26	6	no	—
99	C₂²×C₄	—	13	1	no	—
100	C₂²×C₄	—	13	3	no	—
101	C₂²×C₄	—	13	3	no	—
102	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
103	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
104	C₄⋊C₄	—	4	3	no	—
105	C₄⋊C₄	—	26	6	no	—
106	C₄⋊C₄	—	52	3	no	—
107	C₂²⋊C₄	—	4	3	no	—
108	C₂²⋊C₄	—	52	3	no	—
109	C₂²⋊C₄	—	52	3	no	—
110	C₂²⋊C₄	—	52	3	no	—
111	C₄²	—	26	6	no	—
112	C₁₃	#2644	1	1	yes	characteristic, complemented, Sylow
113	C₂³	—	2	3	no	—
114	C₂³	—	2	1	no	—
115	C₂³	—	2	3	no	—
116	C₂³	—	26	3	no	—
117	C₂³	—	26	1	no	—
118	C₂³	—	26	3	no	—
119	C₂³	C₂³×D₁₃	1	1	yes	characteristic
120	D₄	—	52	4	no	—
121	C₂×C₄	—	2	3	no	—
122	C₂×C₄	—	2	3	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	—	13	6	no	—
128	C₂×C₄	—	26	3	no	—
129	C₂×C₄	—	26	3	no	—
130	C₂×C₄	—	26	3	no	—
131	C₂×C₄	—	26	3	no	—
132	C₂×C₄	—	26	3	no	—
133	C₂×C₄	—	26	3	no	—
134	C₂²	—	4	3	no	—
135	C₂²	—	4	3	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	52	3	no	—
138	C₂²	—	52	3	no	—
139	C₂²	—	52	1	no	—
140	C₂²	#88251	1	3	yes	—
141	C₂²	#88254	1	1	yes	characteristic
142	C₂²	#41×D₁₃	1	3	yes	—
143	C₄	—	4	3	no	—
144	C₄	—	4	3	no	—
145	C₄	—	13	4	no	—
146	C₄	—	52	3	no	—
147	C₄	—	52	3	no	—
148	C₂	—	8	1	no	—
149	C₂	—	104	1	no	—
150	C₂	#11719320	1	3	yes	—
151	C₂	#11719345	1	3	yes	—
152	C₂	D₁₃×#126	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#396

C₂

yes

maximal, characteristic, complemented

#11718654

C₂

yes

maximal, complemented

#11718633

C₂

yes

maximal, complemented

#11718618

C₂

yes

maximal

#11718613

C₂

yes

maximal, complemented

#11718337

C₂

yes

maximal, characteristic, complemented

#11718309

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₁₃

C₂²

yes

characteristic

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×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₁₃×#95

C₂²

yes

characteristic

#88071

C₂²

yes

—

#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

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

—

—

—

—

C₄×Dic₁₃

—

#2644

—

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×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#41

yes

—

—

—

—

D₁₃

yes

characteristic, complemented

#390

—

#389

—

#376

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#126

yes

—

C₂²×D₄

—

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

—

#95

C₂×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

C₂×#126

yes

characteristic

C₂×C₁₃

#540

yes

—

C₂×C₁₃

#542

yes

—

D₁₃

—

C₂⁴

—

C₂⁴

C₂²×D₁₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₁₃

yes

—

103

C₂²×C₄

C₂²×D₁₃

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₁₃

#2644

yes

characteristic, complemented, Sylow

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

C₂³×D₁₃

yes

characteristic

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

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

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#88251

yes

—

141

C₂²

#88254

yes

characteristic

142

C₂²

#41×D₁₃

yes

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₂

—

149

C₂

—

104

—

150

C₂

#11719320

yes

—

151

C₂

#11719345

yes

—

152

C₂

D₁₃×#126

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#396	C₂
prime radical	C₂²×C₂²⋊D₁₃	C₂²
hypercenter	#396	D₁₃
hypocenter	C₁₃	#2644
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #396

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#396

C₂

prime radical

C₂²×C₂²⋊D₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #396

N	H ≅ G/N	type
1. C₁₃×#396	C₂	semidirect
2. #11718654	C₂	semidirect
3. #11718633	C₂	semidirect
5. #11718613	C₂	semidirect
6. #11718337	C₂	semidirect
7. #11718309	C₂	semidirect
61. #396	D₁₃	semidirect
112. C₁₃	#2644	semidirect

H ≅ G/N

type

1. C₁₃×#396

C₂

semidirect

2. #11718654

C₂

semidirect

3. #11718633

C₂

semidirect

5. #11718613

C₂

semidirect

6. #11718337

C₂

semidirect

7. #11718309

C₂

semidirect

61. #396

D₁₃

semidirect

112. C₁₃

#2644

semidirect

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

subgroup	index
C₁₃×#396	2
#11718654	2
#11718633	2
#11718618	2
#11718613	2
#11718337	2
#11718309	2
#2644	13

subgroup

index

C₁₃×#396

Subgroups — #13120798

Conjugacy classes of subgroups

Complements

Maximal subgroups