Group #13116374 — subgroups

Subgroups — #13116374

Order 1664.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₁₃×#391	C₂	1	1	yes	maximal, characteristic
2	C₂×#88075	C₂	1	2	yes	maximal
3	C₂×#88047	C₂	1	2	yes	maximal
4	#11718596	C₂	1	1	yes	maximal, characteristic
5	#11718116	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃	—	2	2	no	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂²×#1055	C₂²	1	2	yes	—
12	C₁₃×#95	—	2	2	no	—
13	#88075	—	2	4	no	—
14	#88047	—	2	4	no	—
15	C₂×Q₈×C₁₃	—	2	1	no	—
16	C₂×Q₈×C₁₃	—	2	2	no	—
17	C₂×Q₈×C₁₃	—	2	2	no	—
18	C₂×Q₈×C₁₃	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₃	—	2	1	no	—
20	C₂²×C₄×C₁₃	—	2	1	no	—
21	C₂²×C₄×C₁₃	D₄	1	2	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₃	—	2	2	no	—
24	C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₄²×C₁₃	—	2	2	no	—
26	C₄⋊Dic₁₃	—	2	2	no	—
27	C₄⋊Dic₁₃	—	2	2	no	—
28	C₄⋊Dic₁₃	—	4	4	no	—
29	C₂×#1055	—	2	2	no	—
30	C₂×#1055	—	2	2	no	—
31	C₂×#1055	D₄	1	4	yes	—
32	#2095	—	13	1	no	maximal, Sylow
33	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₃	—	2	4	no	—
35	Q₈×C₁₃	—	4	2	no	—
36	C₂×C₄×C₁₃	—	2	2	no	—
37	C₂×C₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	4	1	no	—
40	C₂×C₄×C₁₃	—	4	1	no	—
41	C₂×C₄×C₁₃	—	4	1	no	—
42	C₂×C₄×C₁₃	—	4	2	no	—
43	C₂×C₄×C₁₃	Q₁₆	1	4	yes	—
44	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#41	1	2	yes	—
47	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
48	C₂×Dic₁₃	—	4	2	no	—
49	C₂×Dic₁₃	—	4	4	no	—
50	#1055	—	2	4	no	—
51	C₂×#107	—	13	2	no	—
52	C₂×#103	—	13	2	no	—
53	#391	D₁₃	1	1	yes	characteristic
54	#384	—	13	1	no	—
55	#336	—	13	1	no	—
56	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂²×C₁₃	#123	1	1	yes	characteristic
58	C₂²×C₁₃	#127	1	1	yes	characteristic
59	C₂²×C₁₃	C₂×Q₁₆	1	2	yes	—
60	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₄×C₁₃	—	2	4	no	—
62	C₄×C₁₃	—	4	2	no	—
63	C₄×C₁₃	—	8	1	no	—
64	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₃	#123	1	2	yes	—
66	Dic₁₃	—	8	2	no	—
67	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
68	C₂²×C₈	—	13	2	no	—
69	C₂×C₄⋊C₄	—	13	2	no	—
70	C₂×C₄⋊C₄	—	26	2	no	—
71	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
72	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
73	#107	—	26	4	no	—
74	#103	—	26	4	no	—
75	#95	—	2	2	no	—
76	C₂×C₁₃	#399	1	1	yes	characteristic
77	C₂×C₁₃	#477	1	2	yes	—
78	C₂×C₁₃	#484	1	2	yes	—
79	C₂×C₁₃	#494	1	1	yes	characteristic
80	C₂×C₁₃	#496	1	1	yes	characteristic
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	Dic₂₆	1	2	yes	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	26	2	no	—
88	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
90	C₂×C₈	—	13	4	no	—
91	C₂×C₈	—	26	2	no	—
92	C₂×C₈	—	26	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	26	2	no	—
95	C₄⋊C₄	—	26	2	no	—
96	C₄⋊C₄	—	52	4	no	—
97	C₄²	—	2	2	no	—
98	C₁₃	#2095	1	1	yes	characteristic, complemented, Sylow
99	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	52	2	no	—
110	C₂×C₄	—	52	4	no	—
111	C₂×C₄	#6107	1	4	yes	—
112	C₂×C₄	C₂×Dic₂₆	1	1	yes	characteristic
113	C₂×C₄	#6127	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₃	1	2	yes	—
115	C₂×C₄	#6129	1	2	yes	—
116	C₈	—	26	4	no	—
117	C₂²	#88178	1	1	yes	characteristic
118	C₂²	#88189	1	1	yes	characteristic
119	C₂²	#88194	1	1	yes	characteristic
120	C₂²	C₂×#6107	1	2	yes	—
121	C₂²	#88203	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	104	2	no	—
126	C₄	#88132	1	2	yes	—
127	C₄	#88137	1	2	yes	—
128	C₂	#11718610	1	1	yes	characteristic
129	C₂	#11718682	1	2	yes	—
130	C₂	#11718705	1	2	yes	—
131	C₂	#11718719	1	1	yes	characteristic
132	C₂	#11718753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#391

C₂

yes

maximal, characteristic

C₂×#88075

C₂

yes

maximal

C₂×#88047

C₂

yes

maximal

#11718596

C₂

yes

maximal, characteristic

#11718116

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂²×#1055

C₂²

yes

—

C₁₃×#95

—

#88075

—

#88047

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

Q₈

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

D₄

yes

—

#2095

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

Q₁₆

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

—

—

—

D₁₃

yes

characteristic

#384

—

#336

—

C₂²×C₁₃

C₂²⋊Q₈

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

C₂×Q₁₆

yes

—

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

#123

yes

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

#399

yes

characteristic

C₂×C₁₃

#477

yes

—

C₂×C₁₃

#484

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₂₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₁₃

#2095

yes

characteristic, complemented, Sylow

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#6107

yes

—

112

C₂×C₄

C₂×Dic₂₆

yes

characteristic

113

C₂×C₄

#6127

yes

characteristic

114

C₂×C₄

D₄×D₁₃

yes

—

115

C₂×C₄

#6129

yes

—

116

C₈

—

117

C₂²

#88178

yes

characteristic

118

C₂²

#88189

yes

characteristic

119

C₂²

#88194

yes

characteristic

120

C₂²

C₂×#6107

yes

—

121

C₂²

#88203

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

104

—

126

C₄

#88132

yes

—

127

C₄

#88137

yes

—

128

C₂

#11718610

yes

characteristic

129

C₂

#11718682

yes

—

130

C₂

#11718705

yes

—

131

C₂

#11718719

yes

characteristic

132

C₂

#11718753

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#391	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#391	D₁₃
hypocenter	C₁₃	#2095
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#391

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
98. C₁₃	#2095	semidirect

H ≅ G/N

type

98. C₁₃

#2095

semidirect

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

subgroup	index
C₁₃×#391	2
C₂×#88075	2
C₂×#88047	2
#11718596	2
#11718116	2
#2095	13

subgroup

index

C₁₃×#391

Subgroups — #13116374

Conjugacy classes of subgroups

Complements

Maximal subgroups