Group #13113571 — subgroups

Subgroups — #13113571

Order 1664.

254 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 85 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×#532	C₂	1	1	yes	maximal, characteristic
2	#11718585	C₂	1	2	yes	maximal, complemented
3	#11718138	C₂	1	2	yes	maximal, complemented
4	#11718136	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#41	C₄	1	2	yes	—
6	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₁₃×#127	C₂²	1	1	yes	characteristic
8	C₁₃×#124	C₄	1	2	yes	—
9	C₁₃×#121	—	2	2	no	—
10	C₄×D₄×C₁₃	—	2	2	no	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#6099	—	2	2	no	—
13	C₄⋊#1055	—	2	2	no	—
14	C₄⋊#1055	C₂²	1	4	yes	—
15	C₁₃×#41	—	4	4	no	—
16	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₁₃	—	2	2	no	—
18	C₂×D₄×C₁₃	—	2	2	no	—
19	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₃	—	2	2	no	—
21	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	D₄	1	2	yes	—
23	C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₁₃×C₂²⋊C₄	—	2	4	no	—
25	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₃	D₄	1	2	yes	—
27	C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	#6099	—	4	4	no	—
29	C₂×#1055	—	2	4	no	—
30	#1560	—	13	1	no	maximal, Sylow
31	C₂³×C₁₃	—	2	2	no	—
32	C₂³×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	Q₈×C₁₃	—	4	2	no	—
34	D₄×C₁₃	—	4	2	no	—
35	D₄×C₁₃	—	4	4	no	—
36	D₄×C₁₃	—	4	2	no	—
37	D₄×C₁₃	—	4	2	no	—
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	2	2	no	—
40	C₂×C₄×C₁₃	—	2	2	no	—
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	4	4	no	—
43	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
47	#1055	—	4	4	no	—
48	#532	D₁₃	1	1	yes	characteristic
49	#423	—	13	2	no	—
50	#333	—	13	2	no	—
51	#331	—	13	2	no	—
52	C₂²×C₁₃	—	2	1	no	—
53	C₂²×C₁₃	—	2	2	no	—
54	C₂²×C₁₃	—	4	2	no	—
55	C₂²×C₁₃	—	4	4	no	—
56	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₃	—	2	1	no	—
58	C₄×C₁₃	—	2	2	no	—
59	C₄×C₁₃	—	4	2	no	—
60	C₄×C₁₃	#100	1	2	yes	—
61	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₃	#137	1	2	yes	—
63	C₂×#41	Dic₁₃	1	2	yes	—
64	C₂×M₁₆	—	26	2	no	—
65	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
66	#127	C₂×D₁₃	1	1	yes	characteristic
67	#124	Dic₁₃	1	2	yes	—
68	#121	—	2	2	no	—
69	C₄×D₄	—	2	2	no	—
70	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
71	C₄⋊C₈	—	13	4	no	—
72	C₄⋊C₈	—	26	2	no	—
73	C₂×C₁₃	—	4	1	no	—
74	C₂×C₁₃	—	8	2	no	—
75	C₂×C₁₃	C₂×#100	1	1	yes	characteristic
76	C₂×C₁₃	#417	1	2	yes	—
77	#41	—	4	4	no	—
78	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
85	M₁₆	—	52	4	no	—
86	C₂×C₈	—	26	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₁₃	1	2	yes	—
91	C₄²	C₂²×D₁₃	1	1	yes	characteristic
92	C₁₃	#1560	1	1	yes	characteristic, complemented, Sylow
93	C₂³	—	2	2	no	—
94	C₂³	C₂²⋊Dic₁₃	1	1	yes	characteristic
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	4	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
106	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
109	C₈	—	52	4	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	#88073	1	2	yes	—
119	C₄	#88203	1	2	yes	—
120	C₄	#88205	1	2	yes	—
121	C₂	—	4	1	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#88073	1	1	yes	characteristic
124	C₂	#11718893	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#532

C₂

yes

maximal, characteristic

#11718585

C₂

yes

maximal, complemented

#11718138

C₂

yes

maximal, complemented

#11718136

C₂

yes

maximal, complemented

C₂×C₁₃×#41

C₄

yes

—

C₁₃×C₄⋊Q₈

C₂²

yes

characteristic

C₁₃×#127

C₂²

yes

characteristic

C₁₃×#124

C₄

yes

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

#1055

—

#532

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#100

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

#137

yes

—

C₂×#41

Dic₁₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₁₃

yes

characteristic

#127

C₂×D₁₃

yes

characteristic

#124

Dic₁₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#100

yes

characteristic

C₂×C₁₃

#417

yes

—

#41

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₁₃

yes

characteristic

C₄²

C₂²⋊D₁₃

yes

—

C₄²

C₂²×D₁₃

yes

characteristic

C₁₃

#1560

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

C₂²⋊Dic₁₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₁₃

yes

—

106

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

107

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#88073

yes

—

119

C₄

#88203

yes

—

120

C₄

#88205

yes

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#88073

yes

characteristic

124

C₂

#11718893

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃
Fitting, prime radical	C₁₃×#532	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
cosocle	C₄²×C₁₃	C₂³
hypercenter	#532	D₁₃
hypocenter	C₁₃	#1560
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃

Fitting, prime radical

C₁₃×#532

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

cosocle

C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
2. #11718585	C₂	semidirect
3. #11718138	C₂	semidirect
4. #11718136	C₂	semidirect
92. C₁₃	#1560	semidirect

H ≅ G/N

type

2. #11718585

C₂

semidirect

3. #11718138

C₂

semidirect

4. #11718136

C₂

semidirect

92. C₁₃

#1560

semidirect

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

subgroup	index
C₁₃×#532	2
#11718585	2
#11718138	2
#11718136	2
#1560	13

subgroup

index

C₁₃×#532

Subgroups — #13113571

Conjugacy classes of subgroups

Complements

Maximal subgroups