Group #13127719 — subgroups

Subgroups — #13127719

Order 1664.

336 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 133 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	#11719427	C₂	1	2	yes	maximal
2	C₁₃×#500	C₂	1	1	yes	maximal, characteristic
3	#11718752	C₂	1	2	yes	maximal
4	#11718536	C₂	1	4	yes	maximal
5	#11718535	C₂	1	2	yes	maximal
6	#11718534	C₂	1	2	yes	maximal
7	C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
8	C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
9	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	#88198	—	2	4	no	—
11	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
12	#88142	—	2	2	no	—
13	#88142	C₂²	1	4	yes	—
14	#88141	—	2	2	no	—
15	#88141	C₂²	1	4	yes	—
16	C₄×Dic₂₆	C₂²	1	2	yes	—
17	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
18	C₁₃×#107	C₂²	1	2	yes	—
19	C₁₃×#107	C₂²	1	2	yes	—
20	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#88058	C₂²	1	1	yes	characteristic
22	#88056	C₂²	1	2	yes	—
23	#88054	C₂²	1	2	yes	—
24	C₈×Dic₁₃	C₂²	1	2	yes	—
25	#88050	C₂²	1	4	yes	—
26	#88047	C₂²	1	4	yes	—
27	C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₃	—	2	2	no	—
29	C₂×Dic₂₆	—	2	4	no	—
30	C₂×Dic₂₆	D₄	1	1	yes	characteristic
31	C₂×Dic₂₆	C₂³	1	2	yes	—
32	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
38	C₄⋊Dic₁₃	C₂³	1	2	yes	—
39	#6101	—	2	2	no	—
40	#6101	—	2	4	no	—
41	#6101	—	2	4	no	—
42	#6101	D₄	1	2	yes	—
43	C₄×Dic₁₃	—	2	2	no	—
44	C₄×Dic₁₃	—	2	4	no	—
45	C₄×Dic₁₃	C₂³	1	2	yes	—
46	C₂×#1055	C₂³	1	2	yes	—
47	Dic₅₂	Q₈	1	4	yes	complemented
48	Q₈×Q₁₆	—	13	1	no	maximal, Sylow
49	Q₈×C₁₃	—	8	2	no	—
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₁₃	—	2	2	no	—
55	C₂×Dic₁₃	—	2	2	no	—
56	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
57	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
58	Dic₂₆	—	2	2	no	—
59	Dic₂₆	—	8	4	no	—
60	Dic₂₆	Q₁₆	1	4	yes	complemented
61	Dic₂₆	C₂×Q₈	1	4	yes	—
62	C₈×C₁₃	—	2	1	no	—
63	C₈×C₁₃	C₂×Q₈	1	2	yes	—
64	#1055	—	2	2	no	—
65	Q₈²	—	13	2	no	—
66	#500	—	13	2	no	—
67	#500	D₁₃	1	1	yes	characteristic
68	#477	—	13	2	no	—
69	#477	—	13	4	no	—
70	C₈×Q₈	—	13	1	no	—
71	C₄×Q₁₆	—	13	2	no	—
72	C₄×Q₁₆	—	13	1	no	—
73	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₁₃	—	2	1	no	—
75	C₄×C₁₃	—	4	4	no	—
76	C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
77	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
78	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
79	Dic₁₃	—	2	4	no	—
80	Dic₁₃	—	2	2	no	—
81	Dic₁₃	—	4	2	no	—
82	Dic₁₃	C₂×Q₁₆	1	4	yes	—
83	C₂×Q₁₆	—	13	1	no	—
84	C₄⋊Q₈	—	13	4	no	—
85	C₄⋊Q₈	—	26	2	no	—
86	C₄⋊Q₈	—	26	4	no	—
87	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
88	C₄×Q₈	—	13	1	no	—
89	C₄×Q₈	—	13	4	no	—
90	C₄×Q₈	—	13	2	no	—
91	C₄×Q₈	—	26	2	no	—
92	#107	—	13	1	no	—
93	#107	—	13	4	no	—
94	#107	C₂×D₁₃	1	2	yes	—
95	#107	C₂×D₁₃	1	2	yes	—
96	C₄⋊C₈	—	13	2	no	—
97	C₄⋊C₈	—	13	1	no	—
98	#103	—	13	2	no	—
99	#103	—	13	4	no	—
100	C₄×C₈	—	13	2	no	—
101	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
102	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
103	C₂×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
104	C₂×C₁₃	#576	1	1	yes	characteristic
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	13	1	no	—
107	C₂×Q₈	—	13	2	no	—
108	C₂×Q₈	—	26	4	no	—
109	Q₁₆	—	13	4	no	—
110	C₂×C₈	—	13	2	no	—
111	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
112	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	13	2	no	—
115	C₄⋊C₄	—	13	1	no	—
116	C₄⋊C₄	—	13	2	no	—
117	C₄⋊C₄	—	26	4	no	—
118	C₄⋊C₄	—	26	4	no	—
119	C₄⋊C₄	—	26	2	no	—
120	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
121	C₄²	—	13	2	no	—
122	C₄²	—	26	4	no	—
123	C₄²	—	26	2	no	—
124	C₄²	C₂²×D₁₃	1	1	yes	characteristic
125	C₁₃	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
126	Q₈	—	8	2	no	—
127	Q₈	—	13	4	no	—
128	Q₈	—	13	4	no	—
129	Q₈	—	26	2	no	—
130	Q₈	—	104	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	13	2	no	—
133	C₂×C₄	—	13	2	no	—
134	C₂×C₄	—	26	2	no	—
135	C₂×C₄	—	26	2	no	—
136	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
137	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	26	2	no	—
141	C₈	Q₈×D₁₃	1	2	yes	—
142	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	—	13	4	no	—
146	C₄	—	26	4	no	—
147	C₄	—	26	2	no	—
148	C₄	—	52	2	no	—
149	C₄	Q₁₆×D₁₃	1	2	yes	—
150	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
151	C₄	#88254	1	1	yes	characteristic
152	C₂	#11719432	1	1	yes	characteristic
153	C₂	#11719462	1	1	yes	characteristic
154	C₂	C₂×Q₁₆×D₁₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719427

C₂

yes

maximal

C₁₃×#500

C₂

yes

maximal, characteristic

#11718752

C₂

yes

maximal

#11718536

C₂

yes

maximal

#11718535

C₂

yes

maximal

#11718534

C₂

yes

maximal

C₄×Dic₅₂

C₂

yes

maximal, characteristic

C₈×Dic₂₆

C₂

yes

maximal, characteristic

C₁₃×C₄⋊Q₈

C₂²

yes

—

#88198

—

C₂×Dic₅₂

C₂²

yes

characteristic

#88142

—

#88142

C₂²

yes

—

#88141

—

#88141

C₂²

yes

—

C₄×Dic₂₆

C₂²

yes

—

C₄×Dic₂₆

C₂²

yes

characteristic

C₁₃×#107

C₂²

yes

—

C₁₃×#107

C₂²

yes

—

C₄×C₈×C₁₃

C₂²

yes

characteristic

#88058

C₂²

yes

characteristic

#88056

C₂²

yes

—

#88054

C₂²

yes

—

C₈×Dic₁₃

C₂²

yes

—

#88050

C₂²

yes

—

#88047

C₂²

yes

—

C₄⋊#1055

C₂²

yes

characteristic

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₄

yes

characteristic

C₂×Dic₂₆

C₂³

yes

—

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄⋊Dic₁₃

D₄

yes

characteristic

C₄⋊Dic₁₃

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂³

yes

—

C₂×#1055

C₂³

yes

—

Dic₅₂

Q₈

yes

complemented

Q₈×Q₁₆

—

maximal, Sylow

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

Q₁₆

yes

complemented

Dic₂₆

C₂×Q₈

yes

—

C₈×C₁₃

—

C₈×C₁₃

C₂×Q₈

yes

—

#1055

—

Q₈²

—

#500

—

#500

D₁₃

yes

characteristic

#477

—

#477

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×Q₁₆

yes

—

C₄×C₁₃

C₂²×Q₈

yes

characteristic

C₄×C₁₃

2₋¹⁺⁴

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

C₂×Q₁₆

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#107

—

#107

—

#107

C₂×D₁₃

yes

—

#107

C₂×D₁₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₁₃

yes

characteristic

102

C₂×C₁₃

D₄×Q₈

yes

characteristic

103

C₂×C₁₃

C₂²×Q₁₆

yes

characteristic

104

C₂×C₁₃

#576

yes

characteristic

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

Q₁₆

—

110

C₂×C₈

—

111

C₂×C₈

C₂²×D₁₃

yes

characteristic

112

C₂×C₈

C₂²×D₁₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

C₂²×D₁₃

yes

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²×D₁₃

yes

characteristic

125

C₁₃

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

104

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

D₄×D₁₃

yes

characteristic

137

C₂×C₄

D₄×D₁₃

yes

characteristic

138

C₂×C₄

C₂³×D₁₃

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

Q₈×D₁₃

yes

—

142

C₂²

C₂×D₄×D₁₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

Q₁₆×D₁₃

yes

—

150

C₄

C₂×Q₈×D₁₃

yes

characteristic

151

C₄

#88254

yes

characteristic

152

C₂

#11719432

yes

characteristic

153

C₂

#11719462

yes

characteristic

154

C₂

C₂×Q₁₆×D₁₃

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#500	C₂
socle, prime radical	C₂²×C₁₃	C₂²×D₄
hypercenter	#500	D₁₃
hypocenter	C₁₃	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₁₃×#500

C₂

socle, prime radical

C₂²×C₁₃

C₂²×D₄

hypercenter

#500

D₁₃

hypocenter

C₁₃

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
47. Dic₅₂	Q₈	semidirect
60. Dic₂₆	Q₁₆	semidirect
125. C₁₃	Q₈×Q₁₆	semidirect

H ≅ G/N

type

47. Dic₅₂

Q₈

semidirect

60. Dic₂₆

Q₁₆

semidirect

125. C₁₃

Q₈×Q₁₆

semidirect

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

subgroup	index
#11719427	2
C₁₃×#500	2
#11718752	2
#11718536	2
#11718535	2
#11718534	2
C₄×Dic₅₂	2
C₈×Dic₂₆	2
Q₈×Q₁₆	13

subgroup

index

C₁₃×#500

C₄×Dic₅₂

C₈×Dic₂₆

Q₈×Q₁₆

Subgroups — #13127719

Conjugacy classes of subgroups

Complements

Maximal subgroups