Group #13131022 — subgroups

Subgroups — #13131022

Order 1664.

284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 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	#11719393	C₂	1	1	yes	maximal, characteristic
2	C₄²×#261	C₂	1	1	yes	maximal, characteristic
3	#11719053	C₂	1	1	yes	maximal, characteristic
4	#11718247	C₂	1	4	yes	maximal
5	C₂×C₄×#261	—	2	2	no	—
6	C₂×C₄×#261	C₂²	1	2	yes	—
7	C₂×#6118	C₂²	1	2	yes	—
8	C₁₃×#125	C₄	1	1	yes	characteristic, complemented
9	#88150	—	2	2	no	—
10	#88146	C₂²	1	2	yes	—
11	#88142	C₄	1	2	yes	complemented
12	#88141	—	2	2	no	—
13	C₄²×D₁₃	C₂²	1	1	yes	characteristic
14	#88124	C₄	1	1	yes	characteristic, complemented
15	#88112	—	2	2	no	—
16	C₂²×#261	—	2	2	no	—
17	C₂×C₄×D₁₃	D₄	1	2	yes	—
18	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₆	—	2	2	no	—
20	C₂×#1057	—	2	2	no	—
21	C₄×#261	—	2	8	no	—
22	C₄×#261	—	2	2	no	—
23	C₄×#261	Q₈	1	4	yes	complemented
24	#6118	—	2	4	no	—
25	C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	#6103	—	2	2	no	—
29	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
30	#6101	—	2	2	no	—
31	#6101	—	2	2	no	—
32	C₄×Dic₁₃	—	2	2	no	—
33	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₃	D₄	1	2	yes	—
35	#1994	—	13	1	no	maximal, Sylow
36	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×#261	—	2	8	no	—
38	C₂×#261	—	2	4	no	—
39	C₂×C₄×C₁₃	—	2	2	no	—
40	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₁₃	—	2	2	no	—
43	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
45	C₄×D₁₃	—	2	4	no	—
46	C₄×D₁₃	C₂×Q₈	1	2	yes	—
47	C₄×D₁₃	#41	1	2	yes	—
48	Dic₂₆	—	4	4	no	—
49	#1057	—	4	2	no	—
50	#533	—	13	1	no	—
51	#423	—	13	1	no	—
52	C₄³	—	13	1	no	—
53	#339	—	13	4	no	—
54	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×D₁₃	—	2	2	no	—
56	C₂×D₁₃	C₂²⋊Q₈	1	2	yes	—
57	#261	—	2	8	no	—
58	C₄×C₁₃	—	2	2	no	—
59	C₄×C₁₃	—	4	2	no	—
60	C₄×C₁₃	C₄×Q₈	1	2	yes	—
61	Dic₁₃	—	4	2	no	—
62	Dic₁₃	#104	1	4	yes	—
63	Dic₁₃	#124	1	1	yes	characteristic
64	Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×M₁₆	—	13	2	no	—
66	C₄⋊Q₈	—	13	2	no	—
67	#125	—	13	1	no	—
68	#125	#261	1	1	yes	characteristic, complemented
69	C₂²⋊Q₈	—	26	2	no	—
70	C₄×Q₈	—	26	2	no	—
71	#117	—	13	2	no	—
72	C₂×C₄²	—	13	1	no	—
73	C₂×C₄²	—	13	2	no	—
74	C₂×C₄²	—	26	2	no	—
75	C₄⋊C₈	—	26	2	no	—
76	C₂×C₁₃	#391	1	1	yes	characteristic
77	C₂×C₁₃	C₂×#104	1	2	yes	—
78	D₁₃	—	2	2	no	—
79	C₂×Q₈	—	26	2	no	—
80	C₂²×C₄	—	13	1	no	—
81	C₂²×C₄	—	13	2	no	—
82	C₂²×C₄	—	26	2	no	—
83	M₁₆	—	26	4	no	—
84	C₂×C₈	—	26	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	13	2	no	—
87	C₄⋊C₄	—	26	2	no	—
88	C₄⋊C₄	—	26	2	no	—
89	C₄⋊C₄	C₂×#261	1	2	yes	—
90	C₂²⋊C₄	—	26	2	no	—
91	C₄²	—	13	2	no	—
92	C₄²	—	13	1	no	—
93	C₄²	—	13	4	no	—
94	C₄²	—	26	2	no	—
95	C₄²	—	26	2	no	—
96	C₄²	—	26	8	no	—
97	C₄²	C₂×#261	1	1	yes	characteristic
98	C₁₃	#1994	1	1	yes	characteristic, complemented, Sylow
99	C₂³	—	13	1	no	—
100	Q₈	—	52	4	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	13	1	no	—
103	C₂×C₄	—	13	2	no	—
104	C₂×C₄	—	13	2	no	—
105	C₂×C₄	—	13	2	no	—
106	C₂×C₄	—	26	4	no	—
107	C₂×C₄	—	26	4	no	—
108	C₂×C₄	—	26	8	no	—
109	C₂×C₄	—	26	2	no	—
110	C₂×C₄	C₂²⋊#261	1	2	yes	—
111	C₂×C₄	C₂²×#261	1	1	yes	characteristic
112	C₈	—	52	2	no	—
113	C₂²	—	13	2	no	—
114	C₂²	—	26	2	no	—
115	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	13	1	no	—
119	C₄	—	13	1	no	—
120	C₄	—	13	4	no	—
121	C₄	—	26	8	no	—
122	C₄	—	52	2	no	—
123	C₄	Q₈×#261	1	2	yes	—
124	C₂	—	26	2	no	—
125	C₂	(C₂×#261)⋊Q₈	1	1	yes	characteristic
126	C₂	D₁₃⋊#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719393

C₂

yes

maximal, characteristic

C₄²×#261

C₂

yes

maximal, characteristic

#11719053

C₂

yes

maximal, characteristic

#11718247

C₂

yes

maximal

C₂×C₄×#261

—

C₂×C₄×#261

C₂²

yes

—

C₂×#6118

C₂²

yes

—

C₁₃×#125

C₄

yes

characteristic, complemented

#88150

—

#88146

C₂²

yes

—

#88142

C₄

yes

complemented

#88141

—

C₄²×D₁₃

C₂²

yes

characteristic

#88124

C₄

yes

characteristic, complemented

#88112

—

C₂²×#261

—

C₂×C₄×D₁₃

D₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

—

—

—

Q₈

yes

complemented

#6118

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

#6103

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

—

#1994

—

maximal, Sylow

C₂²×D₁₃

C₂×D₄

yes

characteristic

C₂×#261

—

C₂×#261

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

C₂×Q₈

yes

—

C₄×D₁₃

#41

yes

—

Dic₂₆

—

#1057

—

#533

—

#423

—

C₄³

—

#339

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

C₂²⋊Q₈

yes

—

#261

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₈

yes

—

Dic₁₃

—

Dic₁₃

#104

yes

—

Dic₁₃

#124

yes

characteristic

Dic₁₃

C₄⋊Q₈

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

#261

yes

characteristic, complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂×C₁₃

#391

yes

characteristic

C₂×C₁₃

C₂×#104

yes

—

D₁₃

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#261

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#261

yes

characteristic

C₁₃

#1994

yes

characteristic, complemented, Sylow

C₂³

—

100

Q₈

—

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

C₂²⋊#261

yes

—

111

C₂×C₄

C₂²×#261

yes

characteristic

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₂²⋊#261

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

Q₈×#261

yes

—

124

C₂

—

125

C₂

(C₂×#261)⋊Q₈

yes

characteristic

126

C₂

D₁₃⋊#104

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₁₃×#125	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₃	C₂³
hypercenter	#125	#261
hypocenter	C₁₃	#1994
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#261

Fitting

C₁₃×#125

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃

C₂×D₄

cosocle

C₂×C₄×D₁₃

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
8. C₁₃×#125	C₄	semidirect
11. #88142	C₄	semidirect
14. #88124	C₄	semidirect
23. C₄×#261	Q₈	semidirect
68. #125	#261	semidirect
98. C₁₃	#1994	semidirect

H ≅ G/N

type

8. C₁₃×#125

C₄

semidirect

11. #88142

C₄

semidirect

14. #88124

C₄

semidirect

23. C₄×#261

Q₈

semidirect

68. #125

#261

semidirect

98. C₁₃

#1994

semidirect

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

subgroup	index
#11719393	2
C₄²×#261	2
#11719053	2
#11718247	2
#1994	13

subgroup

index

C₄²×#261

Subgroups — #13131022

Conjugacy classes of subgroups

Complements

Maximal subgroups