Group #13115053 — subgroups

Subgroups — #13115053

Order 1664.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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	#11719517	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#101	C₂	1	1	yes	maximal, characteristic
3	#11718891	C₂	1	1	yes	maximal, characteristic
4	#11718782	C₂	1	2	yes	maximal
5	#11718147	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
8	Q₈×Dic₁₃	—	2	2	no	—
9	Q₈×Dic₁₃	—	2	2	no	—
10	#88180	—	2	2	no	—
11	#88180	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#101	C₄	1	4	yes	complemented
15	#88071	—	2	2	no	—
16	#88058	—	2	2	no	—
17	#88057	—	2	2	no	—
18	C₂×Q₈×C₁₃	—	2	2	no	—
19	C₂×Q₈×C₁₃	—	2	2	no	—
20	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₁₃	Q₈	1	2	yes	—
22	C₂²×C₄×C₁₃	D₄	1	2	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₃	—	2	1	no	—
25	C₂²×Dic₁₃	—	2	1	no	—
26	C₁₃×M₁₆	—	2	2	no	—
27	C₁₃×M₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₃	—	2	2	no	—
29	C₂²⋊Dic₁₃	—	2	2	no	—
30	C₄⋊Dic₁₃	—	2	2	no	—
31	C₄⋊Dic₁₃	—	2	2	no	—
32	C₄⋊Dic₁₃	—	2	2	no	—
33	C₄⋊Dic₁₃	D₄	1	2	yes	—
34	C₄⋊Dic₁₃	D₄	1	2	yes	—
35	C₄×Dic₁₃	—	2	4	no	—
36	C₄×Dic₁₃	—	2	2	no	—
37	C₄×Dic₁₃	—	2	2	no	—
38	#1857	—	13	1	no	maximal, Sylow
39	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₃	—	2	4	no	—
41	Q₈×C₁₃	—	2	4	no	—
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
45	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#41	1	2	yes	—
49	C₂×Dic₁₃	—	2	2	no	—
50	C₂×Dic₁₃	—	2	2	no	—
51	C₂×Dic₁₃	—	2	2	no	—
52	C₂×Dic₁₃	—	4	1	no	—
53	C₂×Dic₁₃	—	4	1	no	—
54	C₂×Dic₁₃	—	4	2	no	—
55	C₂×Dic₁₃	—	4	1	no	—
56	C₈×C₁₃	—	2	4	no	—
57	#519	—	13	1	no	—
58	#428	—	13	2	no	—
59	C₂×#101	D₁₃	1	1	yes	characteristic
60	#391	—	13	1	no	—
61	#339	—	13	2	no	—
62	C₂²×C₁₃	—	2	1	no	—
63	C₂²×C₁₃	—	2	1	no	—
64	C₂²×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₁₃	#120	1	1	yes	characteristic
66	C₂²×C₁₃	#123	1	1	yes	characteristic
67	C₄×C₁₃	—	2	4	no	—
68	C₄×C₁₃	C₄×D₄	1	2	yes	—
69	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
70	Dic₁₃	—	4	2	no	—
71	Dic₁₃	—	4	2	no	—
72	Dic₁₃	—	4	2	no	—
73	Dic₁₃	—	8	1	no	—
74	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
75	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
76	C₄×Q₈	—	26	2	no	—
77	C₄×Q₈	—	26	2	no	—
78	#117	—	13	2	no	—
79	#117	—	26	2	no	—
80	C₂×C₄⋊C₄	—	13	1	no	—
81	C₂×C₄²	—	13	1	no	—
82	#107	—	26	2	no	—
83	#106	—	26	2	no	—
84	#101	Dic₁₃	1	4	yes	complemented
85	#95	—	26	2	no	—
86	C₂×C₁₃	—	2	1	no	—
87	C₂×C₁₃	—	2	1	no	—
88	C₂×C₁₃	#385	1	1	yes	characteristic
89	C₂×C₁₃	#454	1	1	yes	characteristic
90	C₂×C₁₃	#456	1	1	yes	characteristic
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	Dic₂₆	1	2	yes	—
94	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
95	C₂²×C₄	—	26	1	no	—
96	C₂²×C₄	—	26	1	no	—
97	C₂²×C₄	D₅₂	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	C₂×Dic₁₃	1	4	yes	—
101	C₂×C₈	—	2	2	no	—
102	C₄⋊C₄	—	13	2	no	—
103	C₄⋊C₄	—	13	2	no	—
104	C₄⋊C₄	—	26	2	no	—
105	C₄⋊C₄	—	26	2	no	—
106	C₄⋊C₄	—	26	2	no	—
107	C₂²⋊C₄	—	26	2	no	—
108	C₄²	—	26	4	no	—
109	C₄²	—	26	2	no	—
110	C₄²	—	26	2	no	—
111	C₁₃	#1857	1	1	yes	characteristic, complemented, Sylow
112	C₂³	C₂×D₅₂	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	26	2	no	—
118	C₂×C₄	—	26	2	no	—
119	C₂×C₄	—	26	2	no	—
120	C₂×C₄	—	52	1	no	—
121	C₂×C₄	—	52	1	no	—
122	C₂×C₄	—	52	2	no	—
123	C₂×C₄	—	52	1	no	—
124	C₂×C₄	C₄⋊Dic₁₃	1	4	yes	—
125	C₂×C₄	C₂×Dic₂₆	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₃	1	2	yes	—
127	C₂×C₄	#6129	1	2	yes	—
128	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
129	C₈	—	2	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#88136	1	1	yes	characteristic
133	C₂²	#88140	1	1	yes	characteristic
134	C₂²	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	52	2	no	—
137	C₄	—	52	2	no	—
138	C₄	—	52	2	no	—
139	C₄	—	104	1	no	—
140	C₄	#88132	1	2	yes	—
141	C₄	D₄×Dic₁₃	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#11718413	1	1	yes	characteristic
145	C₂	#11718414	1	1	yes	characteristic
146	C₂	#11718611	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719517

C₂

yes

maximal, characteristic

C₂×C₁₃×#101

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal, characteristic

#11718782

C₂

yes

maximal

#11718147

C₂

yes

maximal

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×M₁₆

C₂²

yes

—

Q₈×Dic₁₃

—

Q₈×Dic₁₃

—

#88180

—

#88180

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₁₃×#101

C₄

yes

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×Q₈×C₁₃

Q₈

yes

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₁₃×M₁₆

—

C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₈×C₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

C₄⋊Dic₁₃

D₄

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

#1857

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

#519

—

#428

—

C₂×#101

D₁₃

yes

characteristic

#391

—

#339

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₄⋊C₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂×M₁₆

C₂×D₁₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

#106

—

#101

Dic₁₃

yes

complemented

#95

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₂₆

yes

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₅₂

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

100

M₁₆

C₂×Dic₁₃

yes

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₁₃

#1857

yes

characteristic, complemented, Sylow

112

C₂³

C₂×D₅₂

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₄⋊Dic₁₃

yes

—

125

C₂×C₄

C₂×Dic₂₆

yes

characteristic

126

C₂×C₄

D₄×D₁₃

yes

—

127

C₂×C₄

#6129

yes

—

128

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

#88136

yes

characteristic

133

C₂²

#88140

yes

characteristic

134

C₂²

C₂×C₄⋊Dic₁₃

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

104

—

140

C₄

#88132

yes

—

141

C₄

D₄×Dic₁₃

yes

—

142

C₂

—

143

C₂

—

144

C₂

#11718413

yes

characteristic

145

C₂

#11718414

yes

characteristic

146

C₂

#11718611

yes

characteristic

	subgroup	G/H
center	C₂²	#88136
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#101	C₂
socle	C₂²×C₁₃	#120
prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×#101	D₁₃
hypocenter	C₁₃	#1857
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#88136

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×#101

C₂

socle

C₂²×C₁₃

#120

prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
14. C₁₃×#101	C₄	semidirect
84. #101	Dic₁₃	semidirect
111. C₁₃	#1857	semidirect

H ≅ G/N

type

14. C₁₃×#101

C₄

semidirect

84. #101

Dic₁₃

semidirect

111. C₁₃

#1857

semidirect

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

subgroup	index
#11719517	2
C₂×C₁₃×#101	2
#11718891	2
#11718782	2
#11718147	2
#1857	13

subgroup

index

C₂×C₁₃×#101

Subgroups — #13115053

Conjugacy classes of subgroups

Complements

Maximal subgroups