Group #13115680 — subgroups

Subgroups — #13115680

Order 1664.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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×#88123	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#88064	C₂	1	4	yes	maximal
4	#11718585	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆	C₂²	1	2	yes	—
6	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃	C₄	1	4	yes	—
9	C₂×#6099	C₂²	1	2	yes	—
10	#88123	—	2	2	no	—
11	#88123	—	2	1	no	—
12	#88123	—	2	1	no	—
13	C₁₃×#97	—	2	1	no	—
14	C₄×C₈×C₁₃	—	2	1	no	—
15	#88064	—	2	8	no	—
16	C₄⋊#1055	—	2	2	no	—
17	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₃	C₂×C₄	1	4	yes	—
20	C₂×Dic₂₆	—	2	4	no	—
21	C₂×Dic₂₆	—	2	2	no	—
22	C₂×Dic₂₆	—	2	2	no	—
23	C₂×Dic₂₆	—	2	4	no	—
24	C₁₃×M₁₆	—	2	2	no	—
25	C₁₃×M₁₆	D₄	1	4	yes	—
26	C₂×C₈×C₁₃	—	2	2	no	—
27	C₄²×C₁₃	D₄	1	2	yes	—
28	C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₄⋊Dic₁₃	—	2	4	no	—
31	C₄⋊Dic₁₃	—	2	4	no	—
32	#6099	—	2	4	no	—
33	C₂×#1055	—	2	2	no	—
34	#1966	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃	#41	1	2	yes	—
43	C₂×Dic₁₃	—	2	4	no	—
44	C₂×Dic₁₃	—	2	8	no	—
45	Dic₂₆	—	4	4	no	—
46	Dic₂₆	—	4	2	no	—
47	Dic₂₆	—	4	2	no	—
48	C₈×C₁₃	—	2	4	no	—
49	#1055	—	4	2	no	—
50	C₂×C₄⋊Q₈	—	13	1	no	—
51	#423	—	13	1	no	—
52	C₂×#101	—	13	4	no	—
53	C₄×M₁₆	D₁₃	1	1	yes	characteristic
54	C₂²×C₁₃	—	2	2	no	—
55	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₃	C₄×D₄	1	2	yes	—
57	C₄×C₁₃	—	2	2	no	—
58	C₄×C₁₃	#101	1	4	yes	complemented
59	C₄×C₁₃	#121	1	2	yes	—
60	C₄×C₁₃	#124	1	1	yes	characteristic
61	C₄×C₁₃	#127	1	1	yes	characteristic
62	Dic₁₃	—	4	4	no	—
63	C₂²×Q₈	—	13	2	no	—
64	C₂×M₁₆	—	13	2	no	—
65	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
66	C₄⋊Q₈	—	26	2	no	—
67	C₄⋊Q₈	—	26	1	no	—
68	C₄⋊Q₈	—	26	1	no	—
69	C₂×C₄⋊C₄	—	13	4	no	—
70	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
71	C₄⋊C₈	—	26	2	no	—
72	#101	—	26	8	no	—
73	#97	—	2	1	no	—
74	C₄×C₈	—	2	1	no	—
75	C₂×C₁₃	—	2	2	no	—
76	C₂×C₁₃	#390	1	1	yes	characteristic
77	C₂×C₁₃	C₂×#101	1	2	yes	—
78	C₂×Q₈	—	26	4	no	—
79	C₂×Q₈	—	26	2	no	—
80	C₂×Q₈	—	26	4	no	—
81	C₂×Q₈	—	26	2	no	—
82	C₂²×C₄	—	13	4	no	—
83	C₂²×C₄	C₄×D₁₃	1	2	yes	—
84	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	26	4	no	—
87	M₁₆	D₅₂	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	26	2	no	—
90	C₄⋊C₄	—	26	4	no	—
91	C₄⋊C₄	—	26	4	no	—
92	C₄²	D₅₂	1	2	yes	—
93	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
95	C₁₃	#1966	1	1	yes	characteristic, complemented, Sylow
96	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
97	Q₈	—	52	2	no	—
98	Q₈	—	52	2	no	—
99	Q₈	—	52	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	26	8	no	—
102	C₂×C₄	—	26	4	no	—
103	C₂×C₄	#6103	1	2	yes	—
104	C₂×C₄	#6103	1	2	yes	—
105	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
106	C₂×C₄	C₂×D₅₂	1	2	yes	—
107	C₂×C₄	#6127	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	52	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₄×D₅₂	1	2	yes	—
113	C₂²	C₂×#6103	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	52	4	no	—
116	C₄	#88064	1	4	yes	complemented
117	C₄	#88128	1	1	yes	characteristic
118	C₄	#88129	1	1	yes	characteristic
119	C₄	#88184	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	#11718600	1	1	yes	characteristic
122	C₂	C₂×#88064	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88123

C₂

yes

maximal, characteristic

C₄×C₁₃×M₁₆

C₂

yes

maximal, characteristic

C₂×#88064

C₂

yes

maximal

#11718585

C₂

yes

maximal, characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×M₁₆

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₄

yes

—

C₂×#6099

C₂²

yes

—

#88123

—

#88123

—

#88123

—

C₁₃×#97

—

C₄×C₈×C₁₃

—

#88064

—

C₄⋊#1055

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

D₄

yes

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊Q₈

—

#423

—

C₂×#101

—

C₄×M₁₆

D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

—

C₄×C₁₃

#101

yes

complemented

C₄×C₁₃

#121

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

#101

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

C₂×#101

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

D₅₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₅₂

yes

—

C₄²

C₂²⋊D₁₃

yes

characteristic

C₄²

C₂²⋊D₁₃

yes

characteristic

C₁₃

#1966

yes

characteristic, complemented, Sylow

C₂³

C₂×C₄×D₁₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#6103

yes

—

104

C₂×C₄

#6103

yes

—

105

C₂×C₄

C₂×D₅₂

yes

characteristic

106

C₂×C₄

C₂×D₅₂

yes

—

107

C₂×C₄

#6127

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

C₄×D₅₂

yes

—

113

C₂²

C₂×#6103

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

#88064

yes

complemented

117

C₄

#88128

yes

characteristic

118

C₄

#88129

yes

characteristic

119

C₄

#88184

yes

—

120

C₂

—

121

C₂

#11718600

yes

characteristic

122

C₂

C₂×#88064

yes

—

	subgroup	G/H
center	C₂²	C₂×#6103
commutator, prime radical	C₂³×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₄×M₁₆	D₁₃
hypocenter	C₁₃	#1966
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#6103

commutator, prime radical

C₂³×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₄×C₁₃×M₁₆

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₄×M₁₆

D₁₃

hypocenter

C₁₃

#1966

LCS

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
58. C₄×C₁₃	#101	semidirect
95. C₁₃	#1966	semidirect
116. C₄	#88064	semidirect

H ≅ G/N

type

58. C₄×C₁₃

#101

semidirect

95. C₁₃

#1966

semidirect

116. C₄

#88064

semidirect

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

subgroup	index
C₂×#88123	2
C₄×C₁₃×M₁₆	2
C₂×#88064	2
#11718585	2
#1966	13

subgroup

index

C₄×C₁₃×M₁₆

Subgroups — #13115680

Conjugacy classes of subgroups

Complements

Maximal subgroups