Group #13122540 — subgroups

Subgroups — #13122540

Order 1664.

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

normal subgroups only

This group has 428 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×#393	C₂	1	1	yes	maximal, characteristic, complemented
2	#11718892	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718890	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718620	C₂	1	3	yes	maximal
5	#11718606	C₂	1	3	yes	maximal, complemented
6	#11718330	C₂	1	3	yes	maximal, complemented
7	#11718311	C₂	1	3	yes	maximal, complemented
8	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
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	3	yes	—
18	#88071	C₂²	1	3	yes	—
19	#88071	C₂²	1	3	yes	—
20	#88071	C₂²	1	3	yes	—
21	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
22	C₂³×D₁₃	C₂³	1	1	yes	characteristic
23	C₂×Q₈×C₁₃	—	4	3	no	—
24	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
26	C₂×C₂²⋊D₁₃	—	4	3	no	—
27	C₂²×Dic₁₃	C₂³	1	3	yes	—
28	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₃	C₂³	1	3	yes	—
30	C₁₃×C₂²⋊C₄	—	4	3	no	—
31	C₂²⋊Dic₁₃	—	2	6	no	—
32	#6103	—	2	6	no	—
33	#6103	—	4	3	no	—
34	#6101	—	4	3	no	—
35	#6101	—	4	3	no	—
36	C₄×Dic₁₃	—	2	6	no	—
37	#2785	—	13	1	no	maximal, Sylow
38	C₂³×C₁₃	—	2	3	no	—
39	C₂³×C₁₃	—	2	3	no	—
40	C₂³×C₁₃	—	2	1	no	—
41	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
42	C₂²×D₁₃	—	2	3	no	—
43	C₂²×D₁₃	—	2	3	no	—
44	C₂²×D₁₃	—	2	1	no	—
45	Q₈×C₁₃	—	4	4	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	C₂²⋊D₁₃	—	4	4	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	—	2	3	no	—
58	C₂×Dic₁₃	—	2	3	no	—
59	C₂×Dic₁₃	#41	1	6	yes	—
60	#395	—	13	1	no	—
61	#394	—	13	3	no	—
62	#393	—	13	1	no	—
63	#393	D₁₃	1	1	yes	characteristic, complemented
64	#388	—	13	3	no	—
65	#388	—	13	3	no	—
66	#382	—	13	3	no	—
67	C₂²×C₁₃	—	4	3	no	—
68	C₂²×C₁₃	—	4	3	no	—
69	C₂²×C₁₃	—	4	1	no	—
70	C₂²×C₁₃	C₂×#41	1	3	yes	—
71	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
72	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
73	C₂×D₁₃	—	4	3	no	—
74	C₂×D₁₃	—	4	3	no	—
75	C₂×D₁₃	—	4	1	no	—
76	C₄×C₁₃	—	4	3	no	—
77	C₄×C₁₃	—	4	3	no	—
78	Dic₁₃	—	2	6	no	—
79	Dic₁₃	—	4	3	no	—
80	Dic₁₃	—	4	1	no	—
81	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
82	C₂²×D₄	—	13	1	no	—
83	C₂×C₄⋊C₄	—	13	3	no	—
84	C₂×C₄⋊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×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
89	C₂×C₄²	—	13	3	no	—
90	#95	—	13	3	no	—
91	#95	—	13	3	no	—
92	#95	—	13	3	no	—
93	#95	C₂×D₁₃	1	3	yes	—
94	C₂×C₁₃	—	8	1	no	—
95	C₂×C₁₃	#539	1	3	yes	—
96	C₂×C₁₃	#555	1	3	yes	—
97	C₂×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
98	D₁₃	—	8	1	no	—
99	C₂⁴	—	13	1	no	—
100	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
101	C₂×Q₈	—	4	3	no	—
102	C₂×D₄	—	52	3	no	—
103	C₂²×C₄	—	13	3	no	—
104	C₂²×C₄	—	13	3	no	—
105	C₂²×C₄	—	13	1	no	—
106	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
107	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
108	C₄⋊C₄	—	52	3	no	—
109	C₄⋊C₄	—	52	3	no	—
110	C₂²⋊C₄	—	4	3	no	—
111	C₂²⋊C₄	—	26	6	no	—
112	C₂²⋊C₄	—	26	6	no	—
113	C₂²⋊C₄	—	52	3	no	—
114	C₄²	—	26	6	no	—
115	C₁₃	#2785	1	1	yes	characteristic, complemented, Sylow
116	C₂³	—	2	3	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	3	no	—
119	C₂³	—	26	3	no	—
120	C₂³	—	26	1	no	—
121	C₂³	—	26	3	no	—
122	C₂³	C₂³×D₁₃	1	1	yes	characteristic
123	Q₈	—	4	4	no	—
124	D₄	—	52	4	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	—	2	3	no	—
131	C₂×C₄	—	13	6	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	C₂×C₄	—	26	3	no	—
137	C₂×C₄	—	26	3	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	—	52	1	no	—
142	C₂²	—	52	3	no	—
143	C₂²	—	52	3	no	—
144	C₂²	#88251	1	3	yes	—
145	C₂²	#88254	1	1	yes	characteristic
146	C₂²	#41×D₁₃	1	3	yes	—
147	C₄	—	4	3	no	—
148	C₄	—	4	3	no	—
149	C₄	—	26	6	no	—
150	C₄	—	52	1	no	—
151	C₄	—	52	3	no	—
152	C₂	—	8	1	no	—
153	C₂	—	104	1	no	—
154	C₂	#11719299	1	1	yes	characteristic
155	C₂	#11719343	1	3	yes	—
156	C₂	#11719375	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#393

C₂

yes

maximal, characteristic, complemented

#11718892

C₂

yes

maximal, characteristic, complemented

#11718890

C₂

yes

maximal, characteristic, complemented

#11718620

C₂

yes

maximal

#11718606

C₂

yes

maximal, complemented

#11718330

C₂

yes

maximal, complemented

#11718311

C₂

yes

maximal, complemented

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×C₂²⋊D₁₃

C₂²

yes

characteristic

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

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂³×D₁₃

C₂³

yes

characteristic

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

—

—

—

—

C₄×Dic₁₃

—

#2785

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

Q₈×C₁₃

—

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

#388

—

#388

—

#382

—

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

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

—

—

—

C₂×D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

#539

yes

—

C₂×C₁₃

#555

yes

—

C₂×C₁₃

H(3,𝔽₄)

yes

characteristic

D₁₃

—

C₂⁴

—

100

C₂⁴

C₂²×D₁₃

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₁₃

yes

—

107

C₂²×C₄

C₂²×D₁₃

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₁₃

#2785

yes

characteristic, complemented, Sylow

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂³×D₁₃

yes

characteristic

123

Q₈

—

124

D₄

—

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

C₂×C₄

—

137

C₂×C₄

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#88251

yes

—

145

C₂²

#88254

yes

characteristic

146

C₂²

#41×D₁₃

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₂

—

153

C₂

—

104

—

154

C₂

#11719299

yes

characteristic

155

C₂

#11719343

yes

—

156

C₂

#11719375

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#393	C₂
prime radical	C₂²×C₂²⋊D₁₃	C₂²
hypercenter	#393	D₁₃
hypocenter	C₁₃	#2785
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#393

C₂

prime radical

C₂²×C₂²⋊D₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
1. C₁₃×#393	C₂	semidirect
2. #11718892	C₂	semidirect
3. #11718890	C₂	semidirect
5. #11718606	C₂	semidirect
6. #11718330	C₂	semidirect
7. #11718311	C₂	semidirect
63. #393	D₁₃	semidirect
115. C₁₃	#2785	semidirect

H ≅ G/N

type

1. C₁₃×#393

C₂

semidirect

2. #11718892

C₂

semidirect

3. #11718890

C₂

semidirect

5. #11718606

C₂

semidirect

6. #11718330

C₂

semidirect

7. #11718311

C₂

semidirect

63. #393

D₁₃

semidirect

115. C₁₃

#2785

semidirect

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

subgroup	index
C₁₃×#393	2
#11718892	2
#11718890	2
#11718620	2
#11718606	2
#11718330	2
#11718311	2
#2785	13

subgroup

index

C₁₃×#393

Subgroups — #13122540

Conjugacy classes of subgroups

Complements

Maximal subgroups