Group #13115708 — subgroups

Subgroups — #13115708

Order 1664.

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

normal subgroups only

This group has 304 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	C₂²×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#11718891	C₂	1	2	yes	maximal
3	C₂×#88075	C₂	1	2	yes	maximal
4	C₂×#88057	C₂	1	1	yes	maximal, characteristic
5	C₂×#88055	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₁₃×Q₁₆	—	2	2	no	—
8	C₂×C₁₃×Q₁₆	—	2	2	no	—
9	C₂×C₁₃×Q₁₆	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂²×#1055	C₂²	1	1	yes	characteristic
14	#88075	—	2	4	no	—
15	#88071	—	2	4	no	—
16	#88057	—	2	2	no	—
17	#88055	—	2	2	no	—
18	C₂×Q₈×C₁₃	—	2	2	no	—
19	C₂×Q₈×C₁₃	—	2	2	no	—
20	C₂×Q₈×C₁₃	—	2	2	no	—
21	C₂×Q₈×C₁₃	—	2	2	no	—
22	C₂×Q₈×C₁₃	—	2	2	no	—
23	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₃	—	2	2	no	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₃	—	2	2	no	—
27	C₂²×Dic₁₃	D₄	1	2	yes	—
28	C₁₃×Q₁₆	—	2	8	no	—
29	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₃	D₄	1	2	yes	—
31	C₂×C₈×C₁₃	D₄	1	2	yes	—
32	C₄⋊Dic₁₃	—	2	2	no	—
33	C₄⋊Dic₁₃	—	2	2	no	—
34	C₄×Dic₁₃	—	2	1	no	—
35	C₄×Dic₁₃	—	2	1	no	—
36	C₂×#1055	—	2	1	no	—
37	C₂×#1055	—	2	1	no	—
38	C₂×#1055	D₄	1	2	yes	—
39	#1971	—	13	1	no	maximal, Sylow
40	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₁₃	—	2	4	no	—
42	Q₈×C₁₃	—	2	4	no	—
43	Q₈×C₁₃	—	4	2	no	—
44	Q₈×C₁₃	—	4	2	no	—
45	C₂×C₄×C₁₃	—	2	4	no	—
46	C₂×C₄×C₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	—	4	2	no	—
48	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
53	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
54	C₂×Dic₁₃	—	2	2	no	—
55	C₂×Dic₁₃	—	2	4	no	—
56	C₂×Dic₁₃	—	4	2	no	—
57	C₂×Dic₁₃	—	4	4	no	—
58	C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
59	#1055	—	2	2	no	—
60	C₂²×Q₁₆	D₁₃	1	1	yes	characteristic
61	C₂×#106	—	13	1	no	—
62	C₂×#103	—	13	2	no	—
63	C₂×#97	—	13	1	no	—
64	#391	—	13	2	no	—
65	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
66	C₂²×C₁₃	#121	1	1	yes	characteristic
67	C₂²×C₁₃	#127	1	1	yes	characteristic
68	C₂²×C₁₃	#137	1	4	yes	—
69	C₄×C₁₃	—	4	4	no	—
70	C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
72	C₄×C₁₃	#121	1	1	yes	characteristic
73	C₄×C₁₃	#124	1	1	yes	characteristic
74	Dic₁₃	—	4	2	no	—
75	Dic₁₃	—	8	2	no	—
76	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	Dic₁₃	1	4	yes	complemented
80	C₂²×C₈	—	13	1	no	—
81	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	13	2	no	—
83	C₂×C₄²	—	13	1	no	—
84	#106	—	26	2	no	—
85	#103	—	26	4	no	—
86	#97	—	26	2	no	—
87	#95	—	26	4	no	—
88	C₂×C₁₃	#390	1	1	yes	characteristic
89	C₂×C₁₃	#441	1	2	yes	—
90	C₂×C₁₃	#470	1	2	yes	—
91	C₂×C₁₃	#497	1	2	yes	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	C₂×Dic₁₃	1	4	yes	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	13	2	no	—
100	C₂²×C₄	—	26	2	no	—
101	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
102	Q₁₆	—	2	8	no	—
103	C₂×C₈	—	13	2	no	—
104	C₂×C₈	—	26	1	no	—
105	C₂×C₈	—	26	1	no	—
106	C₂×C₈	C₂×Dic₁₃	1	2	yes	—
107	C₂×C₈	C₂²⋊D₁₃	1	2	yes	—
108	C₂×C₈	C₂²⋊D₁₃	1	2	yes	—
109	C₄⋊C₄	—	26	2	no	—
110	C₄⋊C₄	—	26	2	no	—
111	C₄²	—	26	1	no	—
112	C₄²	—	26	1	no	—
113	C₁₃	#1971	1	1	yes	characteristic, complemented, Sylow
114	C₂³	D₄×D₁₃	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	26	4	no	—
123	C₂×C₄	—	26	2	no	—
124	C₂×C₄	—	52	4	no	—
125	C₂×C₄	—	52	2	no	—
126	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
127	C₂×C₄	#6129	1	1	yes	characteristic
128	C₂×C₄	#6129	1	1	yes	characteristic
129	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
132	C₈	—	26	2	no	—
133	C₈	C₂²⋊Dic₁₃	1	4	yes	—
134	C₂²	#88172	1	4	yes	—
135	C₂²	D₄×Dic₁₃	1	1	yes	characteristic
136	C₂²	#88192	1	1	yes	characteristic
137	C₂²	#88194	1	1	yes	characteristic
138	C₄	—	4	4	no	—
139	C₄	—	52	2	no	—
140	C₄	—	104	2	no	—
141	C₄	D₄×Dic₁₃	1	1	yes	characteristic
142	C₄	#88190	1	1	yes	characteristic
143	C₄	#88192	1	1	yes	characteristic
144	C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
145	C₂	#11718845	1	2	yes	—
146	C₂	#11718850	1	2	yes	—
147	C₂	#11718851	1	2	yes	—
148	C₂	#11718882	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₃×Q₁₆

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal

C₂×#88075

C₂

yes

maximal

C₂×#88057

C₂

yes

maximal, characteristic

C₂×#88055

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

C₄

yes

complemented

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

D₄

yes

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

D₄

yes

—

#1971

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

C₂²⋊C₄

yes

—

#1055

—

C₂²×Q₁₆

D₁₃

yes

characteristic

C₂×#106

—

C₂×#103

—

C₂×#97

—

#391

—

C₂²×C₁₃

C₄×D₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

C₄×D₄

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₁₃

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

—

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

#441

yes

—

C₂×C₁₃

#470

yes

—

C₂×C₁₃

#497

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₁₃

yes

characteristic

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×Dic₁₃

yes

—

107

C₂×C₈

C₂²⋊D₁₃

yes

—

108

C₂×C₈

C₂²⋊D₁₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₁₃

#1971

yes

characteristic, complemented, Sylow

114

C₂³

D₄×D₁₃

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₄×D₁₃

yes

characteristic

127

C₂×C₄

#6129

yes

characteristic

128

C₂×C₄

#6129

yes

characteristic

129

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

132

C₈

—

133

C₈

C₂²⋊Dic₁₃

yes

—

134

C₂²

#88172

yes

—

135

C₂²

D₄×Dic₁₃

yes

characteristic

136

C₂²

#88192

yes

characteristic

137

C₂²

#88194

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

104

—

141

C₄

D₄×Dic₁₃

yes

characteristic

142

C₄

#88190

yes

characteristic

143

C₄

#88192

yes

characteristic

144

C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

145

C₂

#11718845

yes

—

146

C₂

#11718850

yes

—

147

C₂

#11718851

yes

—

148

C₂

#11718882

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂²×C₁₃×Q₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂²×Q₁₆	D₁₃
hypocenter	C₁₃	#1971
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂²×C₁₃×Q₁₆

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂²×Q₁₆

D₁₃

hypocenter

C₁₃

#1971

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
9. C₂×C₁₃×Q₁₆	C₄	semidirect
79. C₂×Q₁₆	Dic₁₃	semidirect
113. C₁₃	#1971	semidirect

H ≅ G/N

type

9. C₂×C₁₃×Q₁₆

C₄

semidirect

79. C₂×Q₁₆

Dic₁₃

semidirect

113. C₁₃

#1971

semidirect

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

subgroup	index
C₂²×C₁₃×Q₁₆	2
#11718891	2
C₂×#88075	2
C₂×#88057	2
C₂×#88055	2
#1971	13

subgroup

index

C₂²×C₁₃×Q₁₆

Subgroups — #13115708

Conjugacy classes of subgroups

Complements

Maximal subgroups