Group #13116531 — subgroups

Subgroups — #13116531

Order 1664.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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×#536	C₂	1	1	yes	maximal, characteristic
2	#11718898	C₂	1	2	yes	maximal
3	#11718140	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
5	C₂²×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₃×#124	C₄	1	2	yes	—
8	C₁₃×#123	—	2	2	no	—
9	C₁₃×C₂²⋊Q₈	—	2	2	no	—
10	C₁₃×#117	C₄	1	1	yes	characteristic
11	C₂×#6099	—	2	2	no	—
12	#88076	C₂²	1	4	yes	—
13	#88073	—	2	2	no	—
14	C₁₃×#41	—	4	2	no	—
15	C₂×Q₈×C₁₃	—	2	1	no	—
16	C₂×Q₈×C₁₃	—	4	2	no	—
17	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₁₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₃	—	2	1	no	—
22	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	D₄	1	2	yes	—
24	C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₁₃×C₄⋊C₄	—	2	4	no	—
26	C₁₃×C₂²⋊C₄	—	2	1	no	—
27	C₁₃×C₂²⋊C₄	—	2	4	no	—
28	C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	#6099	—	2	4	no	—
30	#6099	—	4	2	no	—
31	C₂×#1055	—	4	2	no	—
32	#2135	—	13	1	no	maximal, Sylow
33	C₂³×C₁₃	C₂²⋊C₄	1	2	yes	—
34	C₂³×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₁₃	—	4	1	no	—
36	Q₈×C₁₃	—	4	2	no	—
37	Q₈×C₁₃	—	4	2	no	—
38	D₄×C₁₃	—	4	1	no	—
39	D₄×C₁₃	—	4	2	no	—
40	C₂×C₄×C₁₃	—	2	2	no	—
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	4	2	no	—
46	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	#1055	—	4	4	no	—
50	#536	D₁₃	1	1	yes	characteristic
51	#413	—	13	2	no	—
52	#356	—	13	4	no	—
53	C₂²×C₁₃	—	2	1	no	—
54	C₂²×C₁₃	—	2	2	no	—
55	C₂²×C₁₃	—	4	2	no	—
56	C₂²×C₁₃	—	4	1	no	—
57	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₁₃	—	2	2	no	—
59	C₄×C₁₃	—	2	1	no	—
60	C₄×C₁₃	—	4	2	no	—
61	C₄×C₁₃	—	4	2	no	—
62	C₄×C₁₃	#99	1	2	yes	—
63	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
64	C₂²×Q₈	Dic₁₃	1	1	yes	characteristic
65	C₂×M₁₆	—	26	2	no	—
66	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
67	#124	Dic₁₃	1	2	yes	—
68	#123	—	2	2	no	—
69	C₂²⋊Q₈	—	2	2	no	—
70	#117	Dic₁₃	1	1	yes	characteristic
71	#101	—	13	4	no	—
72	#100	—	26	2	no	—
73	C₂×C₁₃	—	2	1	no	—
74	C₂×C₁₃	—	4	1	no	—
75	C₂×C₁₃	—	4	2	no	—
76	C₂×C₁₃	C₂×#99	1	1	yes	characteristic
77	#41	—	4	2	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	4	2	no	—
80	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
81	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
82	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
83	C₂×D₄	C₂²⋊D₁₃	1	2	yes	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
87	M₁₆	—	26	4	no	—
88	M₁₆	—	52	2	no	—
89	C₂×C₈	—	52	2	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₂²⋊C₄	—	2	1	no	—
93	C₂²⋊C₄	—	2	4	no	—
94	C₄²	C₂×Dic₁₃	1	2	yes	—
95	C₁₃	#2135	1	1	yes	characteristic, complemented, Sylow
96	C₂³	C₂²⋊Dic₁₃	1	2	yes	—
97	C₂³	C₂²⋊Dic₁₃	1	1	yes	characteristic
98	Q₈	—	4	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	4	1	no	—
102	D₄	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
110	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
112	C₈	—	52	4	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	1	no	—
117	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	#88074	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	4	1	no	—
125	C₂	—	4	2	no	—
126	C₂	C₂×#88074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#536

C₂

yes

maximal, characteristic

#11718898

C₂

yes

maximal

#11718140

C₂

yes

maximal, complemented

C₂×C₁₃×#41

C₂²

yes

characteristic

C₂²×Q₈×C₁₃

C₄

yes

characteristic

C₁₃×C₄⋊Q₈

C₂²

yes

—

C₁₃×#124

C₄

yes

—

C₁₃×#123

—

C₁₃×C₂²⋊Q₈

—

C₁₃×#117

C₄

yes

characteristic

C₂×#6099

—

#88076

C₂²

yes

—

#88073

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×Q₈×C₁₃

C₂³

yes

characteristic

C₂×D₄×C₁₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

—

—

—

—

—

C₂³×C₁₃

C₂²⋊C₄

yes

—

C₂³×C₁₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

#1055

—

#536

D₁₃

yes

characteristic

#413

—

#356

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#99

yes

—

C₂×#41

C₂×D₁₃

yes

characteristic

C₂²×Q₈

Dic₁₃

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₁₃

yes

—

#124

Dic₁₃

yes

—

#123

—

C₂²⋊Q₈

—

#117

Dic₁₃

yes

characteristic

#101

—

#100

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#99

yes

characteristic

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂×Q₈

C₂²×D₁₃

yes

characteristic

C₂×D₄

C₂×Dic₁₃

yes

characteristic

C₂×D₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₁₃

yes

—

C₁₃

#2135

yes

characteristic, complemented, Sylow

C₂³

C₂²⋊Dic₁₃

yes

—

C₂³

C₂²⋊Dic₁₃

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

110

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#88074

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂×#88074

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#88074
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₁₃
Fitting	C₁₃×#536	C₂
socle	C₂×C₁₃	C₂×#99
prime radical	C₂×C₁₃×#41	C₂²
cosocle	C₂×Q₈×C₁₃	C₂³
hypercenter	#536	D₁₃
hypocenter	C₁₃	#2135
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

subgroup

G/H

center

C₂

C₂×#88074

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₁₃

Fitting

C₁₃×#536

C₂

socle

C₂×C₁₃

C₂×#99

prime radical

C₂×C₁₃×#41

C₂²

cosocle

C₂×Q₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

N	H ≅ G/N	type
3. #11718140	C₂	semidirect
95. C₁₃	#2135	semidirect

H ≅ G/N

type

3. #11718140

C₂

semidirect

95. C₁₃

#2135

semidirect

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

subgroup	index
C₁₃×#536	2
#11718898	2
#11718140	2
#2135	13

subgroup

index

C₁₃×#536

#11718898

#11718140

#2135

Subgroups — #13116531

Conjugacy classes of subgroups

Complements

Maximal subgroups