Group #13115808 — subgroups

Subgroups — #13115808

Order 1664.

276 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₁₃×#382	C₂	1	1	yes	maximal, characteristic
3	C₂×#88043	C₂	1	1	yes	maximal, characteristic
4	#11718121	C₂	1	4	yes	maximal
5	C₂²×Dic₂₆	—	2	1	no	—
6	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃	—	2	1	no	—
9	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	C₂²×#1055	C₂²	1	2	yes	—
11	#88123	—	2	2	no	—
12	#88123	C₄	1	4	yes	complemented
13	C₁₃×#95	—	2	1	no	—
14	C₁₃×#95	—	2	1	no	—
15	#88043	—	2	2	no	—
16	C₂²×C₄×C₁₃	—	2	2	no	—
17	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	—	2	2	no	—
21	C₂×Dic₂₆	—	2	4	no	—
22	C₂×Dic₂₆	—	4	4	no	—
23	C₁₃×C₄⋊C₄	—	2	4	no	—
24	C₄²×C₁₃	C₂×C₄	1	2	yes	—
25	C₄²×C₁₃	D₄	1	2	yes	—
26	C₄⋊Dic₁₃	—	2	4	no	—
27	C₄⋊Dic₁₃	—	2	2	no	—
28	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
29	C₂×#1055	—	2	2	no	—
30	C₂×#1055	—	2	2	no	—
31	C₂×#1055	Q₈	1	4	yes	complemented
32	#1993	—	13	1	no	maximal, Sylow
33	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃	—	2	2	no	—
35	C₂×C₄×C₁₃	—	2	2	no	—
36	C₂×C₄×C₁₃	—	4	2	no	—
37	C₂×C₄×C₁₃	—	4	2	no	—
38	C₂×C₄×C₁₃	—	4	2	no	—
39	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₃	#41	1	2	yes	—
45	C₂×Dic₁₃	—	2	4	no	—
46	C₂×Dic₁₃	—	4	4	no	—
47	Dic₂₆	—	4	8	no	—
48	#1055	—	2	4	no	—
49	C₂×C₄⋊Q₈	—	13	1	no	—
50	C₂×#97	—	13	1	no	—
51	#382	D₁₃	1	1	yes	characteristic
52	#340	—	13	4	no	—
53	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₃	#124	1	1	yes	characteristic
55	C₂²×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
57	C₂²×C₁₃	#137	1	2	yes	—
58	C₄×C₁₃	—	2	2	no	—
59	C₄×C₁₃	—	2	2	no	—
60	C₄×C₁₃	—	8	2	no	—
61	C₄×C₁₃	C₄×Q₈	1	2	yes	—
62	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
63	Dic₁₃	—	4	4	no	—
64	C₂²×Q₈	—	26	1	no	—
65	C₂²×C₈	—	13	2	no	—
66	C₄⋊Q₈	—	13	4	no	—
67	C₄⋊Q₈	—	26	2	no	—
68	C₂×C₄⋊C₄	—	13	2	no	—
69	C₂×C₄⋊C₄	—	26	1	no	—
70	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
71	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
72	#97	—	26	2	no	—
73	#95	—	2	1	no	—
74	#95	—	2	1	no	—
75	C₂×C₁₃	#391	1	1	yes	characteristic
76	C₂×C₁₃	#418	1	1	yes	characteristic
77	C₂×C₁₃	#419	1	1	yes	characteristic
78	C₂×C₁₃	#489	1	2	yes	—
79	C₂×C₁₃	#501	1	2	yes	—
80	C₂×Q₈	—	26	4	no	—
81	C₂×Q₈	—	52	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	26	2	no	—
84	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
85	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
87	C₂×C₈	—	13	4	no	—
88	C₂×C₈	—	26	2	no	—
89	C₂×C₈	—	26	2	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	13	4	no	—
92	C₄⋊C₄	—	26	4	no	—
93	C₄⋊C₄	—	26	2	no	—
94	C₄²	C₄×D₁₃	1	2	yes	—
95	C₄²	D₅₂	1	2	yes	—
96	C₁₃	#1993	1	1	yes	characteristic, complemented, Sylow
97	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
98	Q₈	—	52	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	26	4	no	—
105	C₂×C₄	—	52	4	no	—
106	C₂×C₄	#6103	1	2	yes	—
107	C₂×C₄	#6103	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
109	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
110	C₂×C₄	#6129	1	2	yes	—
111	C₂×C₄	Q₈×D₁₃	1	2	yes	—
112	C₈	—	26	4	no	—
113	C₂²	C₂×#6103	1	1	yes	characteristic
114	C₂²	#88190	1	1	yes	characteristic
115	C₂²	#88198	1	1	yes	characteristic
116	C₂²	#88203	1	2	yes	—
117	C₂²	#88205	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	2	no	—
121	C₄	—	52	4	no	—
122	C₄	#88141	1	2	yes	—
123	C₄	#88151	1	2	yes	—
124	C₂	#11718635	1	1	yes	characteristic
125	C₂	#11718662	1	1	yes	characteristic
126	C₂	#11718665	1	1	yes	characteristic
127	C₂	#11718716	1	2	yes	—
128	C₂	#11718726	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88123

C₂

yes

maximal, characteristic

C₁₃×#382

C₂

yes

maximal, characteristic

C₂×#88043

C₂

yes

maximal, characteristic

#11718121

C₂

yes

maximal

C₂²×Dic₂₆

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂²×#1055

C₂²

yes

—

#88123

—

#88123

C₄

yes

complemented

C₁₃×#95

—

C₁₃×#95

—

#88043

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

Q₈

yes

complemented

#1993

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

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

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

#1055

—

C₂×C₄⋊Q₈

—

C₂×#97

—

#382

D₁₃

yes

characteristic

#340

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

#124

yes

characteristic

C₂²×C₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃

#137

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₈

yes

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×C₁₃

#501

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₁₃

yes

—

C₄²

D₅₂

yes

—

C₁₃

#1993

yes

characteristic, complemented, Sylow

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#6103

yes

—

107

C₂×C₄

#6103

yes

—

108

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

109

C₂×C₄

C₂×D₅₂

yes

characteristic

110

C₂×C₄

#6129

yes

—

111

C₂×C₄

Q₈×D₁₃

yes

—

112

C₈

—

113

C₂²

C₂×#6103

yes

characteristic

114

C₂²

#88190

yes

characteristic

115

C₂²

#88198

yes

characteristic

116

C₂²

#88203

yes

—

117

C₂²

#88205

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#88141

yes

—

123

C₄

#88151

yes

—

124

C₂

#11718635

yes

characteristic

125

C₂

#11718662

yes

characteristic

126

C₂

#11718665

yes

characteristic

127

C₂

#11718716

yes

—

128

C₂

#11718726

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#382	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#382	D₁₃
hypocenter	C₁₃	#1993
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#382

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
12. #88123	C₄	semidirect
31. C₂×#1055	Q₈	semidirect
96. C₁₃	#1993	semidirect

H ≅ G/N

type

12. #88123

C₄

semidirect

31. C₂×#1055

Q₈

semidirect

96. C₁₃

#1993

semidirect

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

subgroup	index
C₂×#88123	2
C₁₃×#382	2
C₂×#88043	2
#11718121	2
#1993	13

subgroup

index

C₁₃×#382

Subgroups — #13115808

Conjugacy classes of subgroups

Complements

Maximal subgroups