Group #13116331 — subgroups

Subgroups — #13116331

Order 1664.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂×#88058	C₂	1	1	yes	maximal, characteristic
2	C₂×#88054	C₂	1	1	yes	maximal, characteristic
3	#11718646	C₂	1	2	yes	maximal
4	C₁₃×#340	C₂	1	1	yes	maximal, characteristic
5	#11718121	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₃	—	2	2	no	—
9	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#1055	C₂²	1	1	yes	characteristic
12	#88071	—	2	2	no	—
13	#88058	—	2	2	no	—
14	#88054	—	2	2	no	—
15	C₂²×C₄×C₁₃	—	2	2	no	—
16	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₁₃	—	2	2	no	—
18	C₂²×Dic₁₃	D₄	1	1	yes	characteristic
19	C₂²×Dic₁₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₃	—	2	2	no	—
21	C₂×C₈×C₁₃	D₄	1	2	yes	—
22	C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₁₃×C₄⋊C₄	Q₈	1	4	yes	—
24	C₄⋊Dic₁₃	—	2	2	no	—
25	C₄⋊Dic₁₃	—	2	2	no	—
26	C₄⋊Dic₁₃	—	4	4	no	—
27	C₄×Dic₁₃	—	2	1	no	—
28	C₄×Dic₁₃	—	2	1	no	—
29	C₂×#1055	—	2	1	no	—
30	C₂×#1055	—	2	2	no	—
31	#2087	—	13	1	no	maximal, Sylow
32	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃	—	2	4	no	—
34	C₂×C₄×C₁₃	—	4	2	no	—
35	C₂×C₄×C₁₃	—	4	2	no	—
36	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₃	#41	1	2	yes	—
39	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
40	C₂×Dic₁₃	—	2	1	no	—
41	C₂×Dic₁₃	—	2	1	no	—
42	C₂×Dic₁₃	—	2	1	no	—
43	C₂×Dic₁₃	—	2	2	no	—
44	C₂×Dic₁₃	—	4	2	no	—
45	C₂×Dic₁₃	—	4	2	no	—
46	C₂×Dic₁₃	—	4	2	no	—
47	C₂×Dic₁₃	Q₁₆	1	2	yes	—
48	C₈×C₁₃	—	2	2	no	—
49	#1055	—	4	1	no	—
50	C₂×#107	—	13	1	no	—
51	C₂×C₄⋊C₈	—	13	1	no	—
52	#389	—	13	2	no	—
53	#340	—	13	2	no	—
54	#340	D₁₃	1	1	yes	characteristic
55	C₂²×C₁₃	#121	1	1	yes	characteristic
56	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
57	C₂²×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₃	#135	1	1	yes	characteristic
59	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
60	C₄×C₁₃	—	4	4	no	—
61	C₄×C₁₃	#123	1	1	yes	characteristic
62	C₄×C₁₃	#125	1	2	yes	—
63	C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
64	Dic₁₃	—	2	2	no	—
65	Dic₁₃	—	4	1	no	—
66	Dic₁₃	—	8	2	no	—
67	C₂²×C₈	—	13	1	no	—
68	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
69	C₂×C₄⋊C₄	—	13	2	no	—
70	C₂×C₄⋊C₄	—	26	2	no	—
71	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
72	C₂×C₄²	—	13	1	no	—
73	#107	—	26	2	no	—
74	C₄⋊C₈	—	26	2	no	—
75	#95	—	26	2	no	—
76	C₂×C₁₃	#398	1	1	yes	characteristic
77	C₂×C₁₃	#467	1	1	yes	characteristic
78	C₂×C₁₃	#469	1	1	yes	characteristic
79	C₂×C₁₃	#477	1	2	yes	—
80	C₂×C₁₃	#478	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	13	1	no	—
83	C₂²×C₄	—	13	1	no	—
84	C₂²×C₄	—	26	2	no	—
85	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	26	1	no	—
88	C₂×C₈	—	26	2	no	—
89	C₂×C₈	D₅₂	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	26	2	no	—
92	C₄⋊C₄	—	26	2	no	—
93	C₄⋊C₄	—	52	4	no	—
94	C₄⋊C₄	Dic₂₆	1	4	yes	—
95	C₄²	—	26	1	no	—
96	C₄²	—	26	1	no	—
97	C₁₃	#2087	1	1	yes	characteristic, complemented, Sylow
98	C₂³	D₄×D₁₃	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	13	2	no	—
103	C₂×C₄	—	26	1	no	—
104	C₂×C₄	—	26	1	no	—
105	C₂×C₄	—	26	1	no	—
106	C₂×C₄	—	26	2	no	—
107	C₂×C₄	—	52	2	no	—
108	C₂×C₄	—	52	2	no	—
109	C₂×C₄	—	52	2	no	—
110	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
111	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
112	C₂×C₄	#6129	1	2	yes	—
113	C₂×C₄	#6131	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	52	1	no	—
116	C₂²	#88132	1	2	yes	—
117	C₂²	#88149	1	1	yes	characteristic
118	C₂²	#88166	1	1	yes	characteristic
119	C₂²	#88168	1	2	yes	—
120	C₂²	Q₁₆×D₁₃	1	1	yes	characteristic
121	C₄	—	4	4	no	—
122	C₄	—	26	2	no	—
123	C₄	—	52	1	no	—
124	C₄	—	104	2	no	—
125	C₄	#88123	1	1	yes	characteristic
126	C₄	#88140	1	1	yes	characteristic
127	C₄	#88144	1	2	yes	—
128	C₂	#11718322	1	1	yes	characteristic
129	C₂	#11718427	1	2	yes	—
130	C₂	#11718455	1	2	yes	—
131	C₂	#11718526	1	1	yes	characteristic
132	C₂	#11718548	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88058

C₂

yes

maximal, characteristic

C₂×#88054

C₂

yes

maximal, characteristic

#11718646

C₂

yes

maximal

C₁₃×#340

C₂

yes

maximal, characteristic

#11718121

C₂

yes

maximal

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

D₄

yes

characteristic

C₂²×Dic₁₃

D₄

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

Q₁₆

yes

—

C₈×C₁₃

—

#1055

—

C₂×#107

—

C₂×C₄⋊C₈

—

#389

—

#340

—

#340

D₁₃

yes

characteristic

C₂²×C₁₃

#121

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

—

C₂²×C₁₃

C₂×Q₁₆

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

#123

yes

characteristic

C₄×C₁₃

#125

yes

—

C₄×C₁₃

C₄⋊Q₈

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

#107

—

C₄⋊C₈

—

#95

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×C₁₃

#478

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₅₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Dic₂₆

yes

—

C₄²

—

C₄²

—

C₁₃

#2087

yes

characteristic, complemented, Sylow

C₂³

D₄×D₁₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

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

C₂×Dic₂₆

yes

—

111

C₂×C₄

C₂×D₅₂

yes

characteristic

112

C₂×C₄

#6129

yes

—

113

C₂×C₄

#6131

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#88132

yes

—

117

C₂²

#88149

yes

characteristic

118

C₂²

#88166

yes

characteristic

119

C₂²

#88168

yes

—

120

C₂²

Q₁₆×D₁₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

104

—

125

C₄

#88123

yes

characteristic

126

C₄

#88140

yes

characteristic

127

C₄

#88144

yes

—

128

C₂

#11718322

yes

characteristic

129

C₂

#11718427

yes

—

130

C₂

#11718455

yes

—

131

C₂

#11718526

yes

characteristic

132

C₂

#11718548

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#340	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#340	D₁₃
hypocenter	C₁₃	#2087
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#340

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
97. C₁₃	#2087	semidirect

H ≅ G/N

type

97. C₁₃

#2087

semidirect

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

subgroup	index
C₂×#88058	2
C₂×#88054	2
#11718646	2
C₁₃×#340	2
#11718121	2
#2087	13

subgroup

index

C₁₃×#340

Subgroups — #13116331

Conjugacy classes of subgroups

Complements

Maximal subgroups