Group #13131019 — subgroups

Subgroups — #13131019

Order 1664.

260 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 78 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	D₁₃×#125	C₂	1	1	yes	maximal, characteristic
2	#11719059	C₂	1	1	yes	maximal, characteristic
3	#11719047	C₂	1	1	yes	maximal, characteristic
4	#11718246	C₂	1	4	yes	maximal
5	C₂×C₄⋊#261	—	2	2	no	—
6	C₂×C₄⋊#261	C₂²	1	2	yes	—
7	C₂×D₁₃⋊C₈	C₂²	1	2	yes	—
8	C₁₃×#125	C₄	1	1	yes	characteristic, complemented
9	C₄⋊C₄×D₁₃	—	2	2	no	—
10	C₄⋊C₄×D₁₃	C₂²	1	2	yes	—
11	#88144	C₄	1	2	yes	complemented
12	#88143	—	2	2	no	—
13	C₄²×D₁₃	C₂²	1	1	yes	characteristic
14	#88124	C₄	1	1	yes	characteristic, complemented
15	#88110	—	2	2	no	—
16	C₂²×#261	—	2	2	no	—
17	C₂×C₄×D₁₃	—	2	2	no	—
18	C₂×C₄×D₁₃	D₄	1	2	yes	—
19	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
20	C₂×#1057	—	2	2	no	—
21	C₄⋊#261	—	2	4	no	—
22	C₄⋊#261	—	4	4	no	—
23	D₁₃⋊C₈	—	2	2	no	—
24	D₁₃⋊C₈	Q₈	1	4	yes	—
25	C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊Dic₁₃	—	2	2	no	—
29	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
30	#6101	—	2	2	no	—
31	#6101	—	2	2	no	—
32	#6101	—	2	2	no	—
33	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₃	D₄	1	2	yes	—
35	#1992	—	13	1	no	maximal, Sylow
36	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×#261	—	4	2	no	—
38	C₂×#261	—	4	4	no	—
39	C₂×C₄×C₁₃	—	2	2	no	—
40	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₁₃	—	2	2	no	—
43	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
45	C₄×D₁₃	—	2	2	no	—
46	C₄×D₁₃	—	2	2	no	—
47	C₄×D₁₃	—	4	2	no	—
48	C₄×D₁₃	—	4	2	no	—
49	C₄×D₁₃	C₂×Q₈	1	2	yes	—
50	C₄×D₁₃	#41	1	2	yes	—
51	#1057	—	2	4	no	—
52	C₂×#125	—	13	1	no	—
53	C₂×#97	—	13	1	no	—
54	#384	—	13	1	no	—
55	#340	—	13	4	no	—
56	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₁₃	#124	1	1	yes	characteristic
58	C₂×D₁₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×D₁₃	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×D₁₃	#137	1	2	yes	—
61	#261	—	8	2	no	—
62	C₄×C₁₃	—	2	2	no	—
63	C₄×C₁₃	—	4	2	no	—
64	C₄×C₁₃	C₄×Q₈	1	2	yes	—
65	Dic₁₃	—	2	2	no	—
66	Dic₁₃	—	4	2	no	—
67	Dic₁₃	C₂²⋊Q₈	1	2	yes	—
68	C₂²×C₈	—	13	2	no	—
69	#125	—	13	2	no	—
70	#125	—	13	1	no	—
71	#125	—	26	2	no	—
72	#125	#261	1	1	yes	characteristic, complemented
73	C₂×C₄⋊C₄	—	13	2	no	—
74	C₂×C₄⋊C₄	—	13	2	no	—
75	C₂×C₄⋊C₄	—	26	2	no	—
76	C₂×C₄⋊C₄	—	26	2	no	—
77	C₂×C₄²	—	13	1	no	—
78	#97	—	26	2	no	—
79	C₂×C₁₃	#391	1	1	yes	characteristic
80	C₂×C₁₃	#417	1	2	yes	—
81	D₁₃	#490	1	1	yes	characteristic
82	D₁₃	#491	1	1	yes	characteristic
83	D₁₃	#501	1	2	yes	—
84	C₂²×C₄	—	13	1	no	—
85	C₂²×C₄	—	13	2	no	—
86	C₂²×C₄	—	26	2	no	—
87	C₂²×C₄	—	26	2	no	—
88	C₂×C₈	—	13	4	no	—
89	C₂×C₈	—	26	2	no	—
90	C₂×C₈	—	26	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	13	2	no	—
93	C₄⋊C₄	—	26	2	no	—
94	C₄⋊C₄	—	26	2	no	—
95	C₄⋊C₄	—	26	2	no	—
96	C₄⋊C₄	—	26	4	no	—
97	C₄⋊C₄	—	26	2	no	—
98	C₄⋊C₄	—	52	4	no	—
99	C₄⋊C₄	C₂×#261	1	2	yes	—
100	C₄²	—	13	2	no	—
101	C₄²	—	13	1	no	—
102	C₄²	C₂×#261	1	1	yes	characteristic
103	C₁₃	#1992	1	1	yes	characteristic, complemented, Sylow
104	C₂³	—	13	1	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	13	1	no	—
107	C₂×C₄	—	13	2	no	—
108	C₂×C₄	—	13	2	no	—
109	C₂×C₄	—	13	2	no	—
110	C₂×C₄	—	26	2	no	—
111	C₂×C₄	—	26	2	no	—
112	C₂×C₄	—	26	2	no	—
113	C₂×C₄	—	52	2	no	—
114	C₂×C₄	—	52	2	no	—
115	C₂×C₄	—	52	2	no	—
116	C₂×C₄	—	52	4	no	—
117	C₂×C₄	C₂²⋊#261	1	2	yes	—
118	C₂×C₄	C₂²×#261	1	1	yes	characteristic
119	C₈	—	26	4	no	—
120	C₂²	—	13	1	no	—
121	C₂²	—	13	1	no	—
122	C₂²	—	13	2	no	—
123	C₂²	—	13	2	no	—
124	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	4	2	no	—
127	C₄	—	13	2	no	—
128	C₄	—	26	2	no	—
129	C₄	—	52	2	no	—
130	C₄	—	104	2	no	—
131	C₄	Q₈×#261	1	2	yes	—
132	C₂	—	13	1	no	—
133	C₂	—	13	1	no	—
134	C₂	—	13	2	no	—
135	C₂	(C₂×#261)⋊Q₈	1	1	yes	characteristic
136	C₂	#11719164	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₃×#125

C₂

yes

maximal, characteristic

#11719059

C₂

yes

maximal, characteristic

#11719047

C₂

yes

maximal, characteristic

#11718246

C₂

yes

maximal

C₂×C₄⋊#261

—

C₂×C₄⋊#261

C₂²

yes

—

C₂×D₁₃⋊C₈

C₂²

yes

—

C₁₃×#125

C₄

yes

characteristic, complemented

C₄⋊C₄×D₁₃

—

C₄⋊C₄×D₁₃

C₂²

yes

—

#88144

C₄

yes

complemented

#88143

—

C₄²×D₁₃

C₂²

yes

characteristic

#88124

C₄

yes

characteristic, complemented

#88110

—

C₂²×#261

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

D₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

—

—

—

D₁₃⋊C₈

—

D₁₃⋊C₈

Q₈

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

—

#6101

—

#6101

—

C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

—

#1992

—

maximal, Sylow

C₂²×D₁₃

C₂×D₄

yes

characteristic

C₂×#261

—

C₂×#261

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×Q₈

yes

—

C₄×D₁₃

#41

yes

—

#1057

—

C₂×#125

—

C₂×#97

—

#384

—

#340

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃

#124

yes

characteristic

C₂×D₁₃

C₄⋊Q₈

yes

characteristic

C₂×D₁₃

AGL(1,ℤ/8)

yes

—

C₂×D₁₃

#137

yes

—

#261

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

C₂²⋊Q₈

yes

—

C₂²×C₈

—

—

—

—

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#97

—

C₂×C₁₃

#391

yes

characteristic

C₂×C₁₃

#417

yes

—

D₁₃

yes

D₁₃

yes

D₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#261

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂×#261

yes

characteristic

103

C₁₃

#1992

yes

characteristic, complemented, Sylow

104

C₂³

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊#261

yes

—

118

C₂×C₄

C₂²×#261

yes

characteristic

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊#261

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

104

—

131

C₄

Q₈×#261

yes

—

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

(C₂×#261)⋊Q₈

yes

characteristic

136

C₂

#11719164

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₁₃×#125	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₃	C₂³
hypercenter	#125	#261
hypocenter	C₁₃	#1992
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#261

Fitting

C₁₃×#125

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃

C₂×D₄

cosocle

C₂×C₄×D₁₃

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
8. C₁₃×#125	C₄	semidirect
11. #88144	C₄	semidirect
14. #88124	C₄	semidirect
72. #125	#261	semidirect
103. C₁₃	#1992	semidirect

H ≅ G/N

type

8. C₁₃×#125

C₄

semidirect

11. #88144

C₄

semidirect

14. #88124

C₄

semidirect

72. #125

#261

semidirect

103. C₁₃

#1992

semidirect

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

subgroup	index
D₁₃×#125	2
#11719059	2
#11719047	2
#11718246	2
#1992	13

subgroup

index

D₁₃×#125

Subgroups — #13131019

Conjugacy classes of subgroups

Complements

Maximal subgroups