Group #13115559 — subgroups

Subgroups — #13115559

Order 1664.

248 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#88054	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
5	#11718142	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
11	C₂²×#1055	C₂²	1	2	yes	—
12	C₁₃×#97	—	2	2	no	—
13	#88071	C₄	1	1	yes	characteristic
14	#88071	C₄	1	1	yes	characteristic
15	#88054	—	2	4	no	—
16	C₄⋊#1055	—	2	2	no	—
17	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
21	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₁₃	—	2	1	no	—
23	C₂×C₈×C₁₃	—	2	1	no	—
24	C₂×C₈×C₁₃	—	2	1	no	—
25	C₂×C₈×C₁₃	—	2	1	no	—
26	C₂×C₈×C₁₃	D₄	1	2	yes	—
27	C₂×C₈×C₁₃	Q₈	1	2	yes	—
28	C₄²×C₁₃	D₄	1	2	yes	—
29	C₄²×C₁₃	Q₈	1	2	yes	—
30	C₄⋊Dic₁₃	—	2	4	no	—
31	C₄×Dic₁₃	—	2	2	no	—
32	C₄×Dic₁₃	—	2	2	no	—
33	C₂×#1055	—	2	2	no	—
34	C₂×#1055	—	2	2	no	—
35	C₂×#1055	—	2	2	no	—
36	#1940	—	13	1	no	maximal, Sylow
37	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	2	2	no	—
40	C₂×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
48	C₂×Dic₁₃	—	2	2	no	—
49	C₂×Dic₁₃	—	2	2	no	—
50	C₂×Dic₁₃	—	2	2	no	—
51	C₂×Dic₁₃	—	2	2	no	—
52	C₂×Dic₁₃	—	2	2	no	—
53	C₂×Dic₁₃	M₁₆	1	4	yes	—
54	C₈×C₁₃	—	2	2	no	—
55	C₈×C₁₃	—	2	2	no	—
56	#1055	—	4	2	no	—
57	C₂×C₄⋊C₈	—	13	2	no	—
58	C₂×C₄⋊C₈	—	13	1	no	—
59	C₂×#97	D₁₃	1	1	yes	characteristic
60	C₄×C₄⋊C₄	—	13	1	no	—
61	#336	—	13	2	no	—
62	C₂²×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₁₃	C₄×Q₈	1	1	yes	characteristic
65	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
66	C₂²×C₁₃	#131	1	2	yes	—
67	C₄×C₁₃	—	2	2	no	—
68	C₄×C₁₃	—	2	2	no	—
69	C₄×C₁₃	#121	1	1	yes	characteristic
70	C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₁₃	#125	1	1	yes	characteristic
72	C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₁₃	—	2	4	no	—
74	Dic₁₃	—	4	2	no	—
75	C₂²×C₈	—	13	2	no	—
76	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	—	13	2	no	—
79	C₂×C₄²	—	13	2	no	—
80	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
81	C₄⋊C₈	—	26	4	no	—
82	C₄⋊C₈	—	26	2	no	—
83	#97	—	2	2	no	—
84	#95	—	13	1	no	—
85	#95	—	13	1	no	—
86	C₂×C₁₃	#389	1	1	yes	characteristic
87	C₂×C₁₃	#423	1	1	yes	characteristic
88	C₂×C₁₃	#424	1	1	yes	characteristic
89	C₂×C₁₃	#435	1	2	yes	—
90	C₂×C₁₃	#446	1	2	yes	—
91	C₂²×C₄	—	13	2	no	—
92	C₂²×C₄	—	13	2	no	—
93	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
94	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	26	2	no	—
101	C₂×C₈	—	26	2	no	—
102	C₂×C₈	—	26	2	no	—
103	C₂×C₈	Dic₂₆	1	2	yes	—
104	C₂×C₈	D₅₂	1	2	yes	—
105	C₄⋊C₄	—	26	4	no	—
106	C₄²	—	26	2	no	—
107	C₄²	—	26	2	no	—
108	C₄²	Dic₂₆	1	2	yes	—
109	C₄²	C₂²⋊D₁₃	1	2	yes	—
110	C₁₃	#1940	1	1	yes	characteristic, complemented, Sylow
111	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	13	4	no	—
115	C₂×C₄	—	26	2	no	—
116	C₂×C₄	—	26	2	no	—
117	C₂×C₄	—	26	2	no	—
118	C₂×C₄	—	26	2	no	—
119	C₂×C₄	—	26	2	no	—
120	C₂×C₄	#6101	1	2	yes	—
121	C₂×C₄	#6101	1	2	yes	—
122	C₂×C₄	C₂×Dic₂₆	1	1	yes	characteristic
123	C₂×C₄	C₂×Dic₂₆	1	1	yes	characteristic
124	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
125	C₂×C₄	#6127	1	1	yes	characteristic
126	C₂×C₄	#6127	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	52	2	no	—
131	C₂²	C₄×Dic₂₆	1	1	yes	characteristic
132	C₂²	C₄×D₅₂	1	1	yes	characteristic
133	C₂²	M₁₆×D₁₃	1	2	yes	—
134	C₂²	#88161	1	2	yes	—
135	C₂²	C₂×#6101	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	26	4	no	—
139	C₄	—	52	2	no	—
140	C₄	#88123	1	1	yes	characteristic
141	C₄	#88124	1	1	yes	characteristic
142	C₄	#88178	1	1	yes	characteristic
143	C₄	#88184	1	1	yes	characteristic
144	C₂	#11718362	1	2	yes	—
145	C₂	#11718367	1	2	yes	—
146	C₂	#11718589	1	1	yes	characteristic
147	C₂	#11718779	1	1	yes	characteristic
148	C₂	#11718780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#97

C₂

yes

maximal, characteristic

C₂×#88054

C₂

yes

maximal

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#1055

C₂

yes

maximal, characteristic

#11718142

C₂

yes

maximal

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₄

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂²×#1055

C₂²

yes

—

C₁₃×#97

—

#88071

C₄

yes

characteristic

#88071

C₄

yes

characteristic

#88054

—

C₄⋊#1055

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

Q₈

yes

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

Q₈

yes

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

M₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

D₁₃

yes

characteristic

C₄×C₄⋊C₄

—

#336

—

C₂²×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

characteristic

C₂²×C₁₃

C₄×Q₈

yes

characteristic

C₂²×C₁₃

C₂×M₁₆

yes

—

C₂²×C₁₃

#131

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#121

yes

characteristic

C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃

#125

yes

characteristic

C₄×C₁₃

C₄⋊Q₈

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

—

#95

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×C₁₃

#446

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₃

yes

characteristic

C₂²×C₄

C₄×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

Dic₂₆

yes

—

104

C₂×C₈

D₅₂

yes

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

Dic₂₆

yes

—

109

C₄²

C₂²⋊D₁₃

yes

—

110

C₁₃

#1940

yes

characteristic, complemented, Sylow

111

C₂³

C₂×C₄×D₁₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#6101

yes

—

121

C₂×C₄

#6101

yes

—

122

C₂×C₄

C₂×Dic₂₆

yes

characteristic

123

C₂×C₄

C₂×Dic₂₆

yes

characteristic

124

C₂×C₄

C₂×D₅₂

yes

characteristic

125

C₂×C₄

#6127

yes

characteristic

126

C₂×C₄

#6127

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

C₄×Dic₂₆

yes

characteristic

132

C₂²

C₄×D₅₂

yes

characteristic

133

C₂²

M₁₆×D₁₃

yes

—

134

C₂²

#88161

yes

—

135

C₂²

C₂×#6101

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#88123

yes

characteristic

141

C₄

#88124

yes

characteristic

142

C₄

#88178

yes

characteristic

143

C₄

#88184

yes

characteristic

144

C₂

#11718362

yes

—

145

C₂

#11718367

yes

—

146

C₂

#11718589

yes

characteristic

147

C₂

#11718779

yes

characteristic

148

C₂

#11718780

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂²×C₄
Fitting	C₂×C₁₃×#97	C₂
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×#97	D₁₃
hypocenter	C₁₃	#1940
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃

commutator, socle, prime radical

C₂³×C₁₃

C₂²×C₄

Fitting

C₂×C₁₃×#97

C₂

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂×#97

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
110. C₁₃	#1940	semidirect

H ≅ G/N

type

110. C₁₃

#1940

semidirect

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

subgroup	index
C₂×C₁₃×#97	2
C₂×#88054	2
C₄×C₄⋊Dic₁₃	2
C₂×C₄⋊#1055	2
#11718142	2
#1940	13

subgroup

index

C₂×C₁₃×#97

C₂×#88054

C₄×C₄⋊Dic₁₃

C₂×C₄⋊#1055

#11718142

#1940

Subgroups — #13115559

Conjugacy classes of subgroups

Complements

Maximal subgroups