Group #13117559 — subgroups

Subgroups — #13117559

Order 1664.

740 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 197 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×#88147	C₂	1	2	yes	maximal, complemented
2	C₂×#88128	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718649	C₂	1	2	yes	maximal, complemented
5	#11718593	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718326	C₂	1	8	yes	maximal, complemented
7	C₂²×D₅₂	C₂²	1	2	yes	—
8	C₂²×D₅₂	C₂²	1	4	yes	—
9	C₂²×C₄×D₁₃	C₂²	1	4	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#6103	C₂²	1	4	yes	—
13	C₂×#6103	C₂²	1	8	yes	—
14	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
16	#88147	—	2	8	no	—
17	#88128	C₄	1	8	yes	complemented
18	C₁₃×#95	C₂²	1	4	yes	—
19	#88071	C₂²	1	4	yes	—
20	C₂³×D₁₃	C₂³	1	4	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₃	C₂³	1	2	yes	—
25	C₂²×Dic₁₃	C₂³	1	2	yes	—
26	C₂×D₅₂	—	2	4	no	—
27	C₂×D₅₂	—	2	4	no	—
28	C₂×D₅₂	—	4	8	no	—
29	C₂×D₅₂	C₂×C₄	1	8	yes	complemented
30	C₂×D₅₂	C₂×C₄	1	16	yes	complemented
31	C₂×C₄×D₁₃	—	2	8	no	—
32	C₂×C₄×D₁₃	—	4	8	no	—
33	C₁₃×C₄⋊C₄	—	2	4	no	—
34	C₄²×C₁₃	C₂×C₄	1	4	yes	complemented
35	#6103	—	2	8	no	—
36	#6103	—	4	8	no	—
37	C₄×Dic₁₃	—	2	2	no	—
38	C₄×Dic₁₃	D₄	1	4	yes	complemented
39	#2328	—	13	1	no	maximal, Sylow
40	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂²×D₁₃	—	2	4	no	—
42	C₂²×D₁₃	—	2	4	no	—
43	C₂²×D₁₃	—	2	8	no	—
44	C₂²×D₁₃	—	2	8	no	—
45	C₂²×D₁₃	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₁₃	—	2	4	no	—
47	C₂×C₄×C₁₃	—	2	4	no	—
48	C₂×C₄×C₁₃	—	2	8	no	—
49	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃	#41	1	2	yes	—
53	C₂×Dic₁₃	—	2	4	no	—
54	C₂×Dic₁₃	—	2	2	no	—
55	C₂×Dic₁₃	—	2	4	no	—
56	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
57	D₅₂	—	2	16	no	—
58	D₅₂	—	4	16	no	—
59	C₄×D₁₃	—	4	16	no	—
60	C₂×#127	—	13	1	no	—
61	C₂×C₄×D₄	—	13	2	no	—
62	#390	—	13	2	no	—
63	#386	—	13	8	no	—
64	#382	D₁₃	1	1	yes	characteristic, complemented
65	#376	—	13	1	no	—
66	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
67	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
69	C₂²×C₁₃	H(5,𝔽₂)	1	4	yes	—
70	C₂×D₁₃	—	2	8	no	—
71	C₂×D₁₃	—	2	16	no	—
72	C₂×D₁₃	—	4	4	no	—
73	C₂×D₁₃	—	4	4	no	—
74	C₂×D₁₃	—	4	8	no	—
75	C₄×C₁₃	—	2	4	no	—
76	C₄×C₁₃	—	4	4	no	—
77	C₄×C₁₃	C₄×D₄	1	4	yes	—
78	Dic₁₃	—	2	4	no	—
79	Dic₁₃	—	4	2	no	—
80	C₂²×D₄	—	13	2	no	—
81	C₂²×D₄	—	13	4	no	—
82	C₂³×C₄	—	13	4	no	—
83	#127	—	13	8	no	—
84	C₄×D₄	—	26	8	no	—
85	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
86	C₂×C₂²⋊C₄	—	13	4	no	—
87	C₂×C₂²⋊C₄	—	13	8	no	—
88	C₂×C₄²	—	13	1	no	—
89	C₂×C₄²	—	13	1	no	—
90	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
91	#95	—	13	4	no	—
92	#95	C₂×D₁₃	1	4	yes	—
93	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
94	C₂×C₁₃	#518	1	2	yes	—
95	C₂×C₁₃	#535	1	2	yes	—
96	C₂×C₁₃	#540	1	2	yes	—
97	D₁₃	—	4	8	no	—
98	C₂⁴	—	13	4	no	—
99	C₂×D₄	—	13	8	no	—
100	C₂×D₄	—	13	16	no	—
101	C₂×D₄	—	26	4	no	—
102	C₂×D₄	—	26	4	no	—
103	C₂×D₄	—	52	8	no	—
104	C₂²×C₄	—	13	2	no	—
105	C₂²×C₄	—	13	2	no	—
106	C₂²×C₄	—	26	8	no	—
107	C₂²×C₄	—	52	8	no	—
108	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
109	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
111	C₄⋊C₄	—	2	4	no	—
112	C₂²⋊C₄	—	26	8	no	—
113	C₂²⋊C₄	—	52	8	no	—
114	C₄²	—	13	4	no	—
115	C₄²	—	26	2	no	—
116	C₄²	C₄×D₁₃	1	4	yes	complemented
117	C₁₃	#2328	1	1	yes	characteristic, complemented, Sylow
118	C₂³	—	13	8	no	—
119	C₂³	—	26	4	no	—
120	C₂³	—	26	4	no	—
121	C₂³	—	26	8	no	—
122	C₂³	—	26	8	no	—
123	C₂³	C₂³×D₁₃	1	1	yes	characteristic
124	D₄	—	26	16	no	—
125	D₄	—	52	16	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	8	no	—
129	C₂×C₄	—	13	4	no	—
130	C₂×C₄	—	26	4	no	—
131	C₂×C₄	—	26	2	no	—
132	C₂×C₄	—	26	4	no	—
133	C₂×C₄	—	52	16	no	—
134	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
136	C₂×C₄	D₄×D₁₃	1	2	yes	—
137	C₂×C₄	#6131	1	2	yes	—
138	C₂²	—	26	8	no	—
139	C₂²	—	26	16	no	—
140	C₂²	—	52	4	no	—
141	C₂²	—	52	4	no	—
142	C₂²	—	52	8	no	—
143	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
144	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
145	C₂²	C₂×#6131	1	1	yes	characteristic
146	C₂²	#88256	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	4	4	no	—
149	C₄	—	26	4	no	—
150	C₄	—	52	2	no	—
151	C₄	#88147	1	4	yes	—
152	C₂	—	52	8	no	—
153	C₂	C₂×#88147	1	1	yes	characteristic
154	C₂	#11719226	1	2	yes	—
155	C₂	#11719379	1	2	yes	—
156	C₂	#11719403	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88147

C₂

yes

maximal, complemented

C₂×#88128

C₂

yes

maximal, characteristic

C₁₃×#382

C₂

yes

maximal, characteristic, complemented

#11718649

C₂

yes

maximal, complemented

#11718593

C₂

yes

maximal, characteristic, complemented

#11718326

C₂

yes

maximal, complemented

C₂²×D₅₂

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

—

C₂²×C₄×D₁₃

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88147

—

#88128

C₄

yes

complemented

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

C₂×C₄

yes

complemented

C₂×D₅₂

C₂×C₄

yes

complemented

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

complemented

#6103

—

#6103

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

complemented

#2328

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₂×#127

—

C₂×C₄×D₄

—

#390

—

#386

—

#382

D₁₃

yes

characteristic, complemented

#376

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

#127

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

#95

C₂×D₁₃

yes

—

C₂×C₁₃

C₂×C₄×D₄

yes

characteristic

C₂×C₁₃

#518

yes

—

C₂×C₁₃

#535

yes

—

C₂×C₁₃

#540

yes

—

D₁₃

—

C₂⁴

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²×D₁₃

yes

—

109

C₂²×C₄

C₂²×D₁₃

yes

—

110

C₂²×C₄

C₂²×D₁₃

yes

characteristic

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₄×D₁₃

yes

complemented

117

C₁₃

#2328

yes

characteristic, complemented, Sylow

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

C₂³×D₁₃

yes

characteristic

124

D₄

—

125

D₄

—

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

C₂×C₄×D₁₃

yes

—

135

C₂×C₄

C₂×C₄×D₁₃

yes

—

136

C₂×C₄

D₄×D₁₃

yes

—

137

C₂×C₄

#6131

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂²×C₄×D₁₃

yes

characteristic

144

C₂²

C₂×D₄×D₁₃

yes

characteristic

145

C₂²

C₂×#6131

yes

characteristic

146

C₂²

#88256

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#88147

yes

—

152

C₂

—

153

C₂

C₂×#88147

yes

characteristic

154

C₂

#11719226

yes

—

155

C₂

#11719379

yes

—

156

C₂

#11719403

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#382	C₂
socle, cosocle	C₂³×C₁₃	C₂⁴
prime radical	C₂×#88128	C₂
hypercenter	#382	D₁₃
hypocenter	C₁₃	#2328
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator

C₂²×C₁₃

C₂³×C₄

Fitting

C₁₃×#382

C₂

socle, cosocle

C₂³×C₁₃

C₂⁴

C₂

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
1. C₂×#88147	C₂	semidirect
3. C₁₃×#382	C₂	semidirect
4. #11718649	C₂	semidirect
5. #11718593	C₂	semidirect
6. #11718326	C₂	semidirect
17. #88128	C₄	semidirect
29. C₂×D₅₂	C₂×C₄	semidirect
30. C₂×D₅₂	C₂×C₄	semidirect
34. C₄²×C₁₃	C₂×C₄	semidirect
38. C₄×Dic₁₃	D₄	semidirect
64. #382	D₁₃	semidirect
116. C₄²	C₄×D₁₃	semidirect
117. C₁₃	#2328	semidirect

H ≅ G/N

type

1. C₂×#88147

C₂

semidirect

3. C₁₃×#382

C₂

semidirect

4. #11718649

C₂

semidirect

5. #11718593

C₂

semidirect

6. #11718326

C₂

semidirect

17. #88128

C₄

semidirect

29. C₂×D₅₂

C₂×C₄

semidirect

30. C₂×D₅₂

C₂×C₄

semidirect

34. C₄²×C₁₃

C₂×C₄

semidirect

38. C₄×Dic₁₃

D₄

semidirect

64. #382

D₁₃

semidirect

116. C₄²

C₄×D₁₃

semidirect

117. C₁₃

#2328

semidirect

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

subgroup	index
C₂×#88147	2
C₂×#88128	2
C₁₃×#382	2
#11718649	2
#11718593	2
#11718326	2
#2328	13

subgroup

index

C₁₃×#382

Subgroups — #13117559

Conjugacy classes of subgroups

Complements

Maximal subgroups