Group #13131023 — subgroups

Subgroups — #13131023

Order 1664.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#11719057	C₂	1	1	yes	maximal, characteristic
3	#11719052	C₂	1	1	yes	maximal, characteristic
4	#11718247	C₂	1	4	yes	maximal
5	C₂×C₄×#261	C₂²	1	2	yes	—
6	C₂×#6118	C₂²	1	2	yes	—
7	C₁₃×#125	C₄	1	1	yes	characteristic, complemented
8	#88150	—	2	2	no	—
9	#88146	C₂²	1	2	yes	—
10	#88142	C₄	1	2	yes	complemented
11	#88141	—	2	2	no	—
12	C₄²×D₁₃	C₂²	1	1	yes	characteristic
13	#88124	C₄	1	1	yes	characteristic, complemented
14	(C₂×#261)⋊C₄	—	2	2	no	—
15	C₄⋊#1057	—	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	2	no	—
22	C₄×#261	Q₈	1	4	yes	complemented
23	#6118	—	2	4	no	—
24	C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
26	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	#6103	—	2	2	no	—
28	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
29	#6101	—	2	2	no	—
30	#6101	—	2	2	no	—
31	C₄×Dic₁₃	—	2	2	no	—
32	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₃	D₄	1	2	yes	—
34	#1995	—	13	1	no	maximal, Sylow
35	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×#261	—	2	4	no	—
37	C₂×#261	—	4	4	no	—
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₃	—	2	2	no	—
42	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
44	C₄×D₁₃	—	4	2	no	—
45	C₄×D₁₃	C₂×Q₈	1	2	yes	—
46	C₄×D₁₃	#41	1	2	yes	—
47	Dic₂₆	—	4	4	no	—
48	#1057	—	4	2	no	—
49	#533	—	13	1	no	—
50	#423	—	13	1	no	—
51	#376	—	13	1	no	—
52	#339	—	13	4	no	—
53	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×D₁₃	—	2	2	no	—
55	C₂×D₁₃	C₂²⋊Q₈	1	2	yes	—
56	#261	—	4	4	no	—
57	C₄×C₁₃	—	2	2	no	—
58	C₄×C₁₃	—	4	2	no	—
59	C₄×C₁₃	C₄×Q₈	1	2	yes	—
60	Dic₁₃	—	2	2	no	—
61	Dic₁₃	—	4	2	no	—
62	Dic₁₃	#124	1	1	yes	characteristic
63	Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×M₁₆	—	13	2	no	—
65	C₄⋊Q₈	—	13	2	no	—
66	#125	—	13	1	no	—
67	#125	#261	1	1	yes	characteristic, complemented
68	C₂²⋊Q₈	—	26	2	no	—
69	C₄×Q₈	—	26	2	no	—
70	#117	—	13	2	no	—
71	C₂×C₄²	—	13	1	no	—
72	C₂×C₄²	—	13	2	no	—
73	C₄⋊C₈	—	26	2	no	—
74	#95	—	26	2	no	—
75	C₂×C₁₃	#391	1	1	yes	characteristic
76	C₂×C₁₃	#421	1	2	yes	—
77	D₁₃	—	2	2	no	—
78	C₂×Q₈	—	26	2	no	—
79	C₂²×C₄	—	13	1	no	—
80	C₂²×C₄	—	13	2	no	—
81	C₂²×C₄	—	26	2	no	—
82	M₁₆	—	26	4	no	—
83	C₂×C₈	—	26	2	no	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	13	2	no	—
86	C₄⋊C₄	—	26	2	no	—
87	C₄⋊C₄	—	26	2	no	—
88	C₄⋊C₄	C₂×#261	1	2	yes	—
89	C₂²⋊C₄	—	26	2	no	—
90	C₄²	—	13	2	no	—
91	C₄²	—	13	1	no	—
92	C₄²	—	13	4	no	—
93	C₄²	—	26	2	no	—
94	C₄²	—	26	2	no	—
95	C₄²	C₂×#261	1	1	yes	characteristic
96	C₁₃	#1995	1	1	yes	characteristic, complemented, Sylow
97	C₂³	—	13	1	no	—
98	Q₈	—	52	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	13	1	no	—
101	C₂×C₄	—	13	2	no	—
102	C₂×C₄	—	13	2	no	—
103	C₂×C₄	—	13	2	no	—
104	C₂×C₄	—	26	4	no	—
105	C₂×C₄	—	26	2	no	—
106	C₂×C₄	—	52	2	no	—
107	C₂×C₄	—	52	4	no	—
108	C₂×C₄	C₂²⋊#261	1	2	yes	—
109	C₂×C₄	C₂²×#261	1	1	yes	characteristic
110	C₈	—	52	2	no	—
111	C₂²	—	13	2	no	—
112	C₂²	—	26	2	no	—
113	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	13	1	no	—
117	C₄	—	13	1	no	—
118	C₄	—	26	2	no	—
119	C₄	—	52	4	no	—
120	C₄	—	52	2	no	—
121	C₄	Q₈×#261	1	2	yes	—
122	C₂	—	26	2	no	—
123	C₂	(C₂×#261)⋊Q₈	1	1	yes	characteristic
124	C₂	#11719166	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719393

C₂

yes

maximal, characteristic

#11719057

C₂

yes

maximal, characteristic

#11719052

C₂

yes

maximal, characteristic

#11718247

C₂

yes

maximal

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

(C₂×#261)⋊C₄

—

C₄⋊#1057

—

C₂²×#261

—

C₂×C₄×D₁₃

D₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×#1057

—

C₄×#261

—

C₄×#261

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

—

#1995

—

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

—

—

—

—

—

—

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

—

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

—

C₄⋊C₈

—

#95

—

C₂×C₁₃

#391

yes

characteristic

C₂×C₁₃

#421

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

C₂×#261

yes

characteristic

C₁₃

#1995

yes

characteristic, complemented, Sylow

C₂³

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²⋊#261

yes

—

109

C₂×C₄

C₂²×#261

yes

characteristic

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₂²⋊#261

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×#261

yes

—

122

C₂

—

123

C₂

(C₂×#261)⋊Q₈

yes

characteristic

124

C₂

#11719166

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	#1995
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
7. C₁₃×#125	C₄	semidirect
10. #88142	C₄	semidirect
13. #88124	C₄	semidirect
22. C₄×#261	Q₈	semidirect
67. #125	#261	semidirect
96. C₁₃	#1995	semidirect

H ≅ G/N

type

7. C₁₃×#125

C₄

semidirect

10. #88142

C₄

semidirect

13. #88124

C₄

semidirect

22. C₄×#261

Q₈

semidirect

67. #125

#261

semidirect

96. C₁₃

#1995

semidirect

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

subgroup	index
#11719393	2
#11719057	2
#11719052	2
#11718247	2
#1995	13

subgroup

index

Subgroups — #13131023

Conjugacy classes of subgroups

Complements

Maximal subgroups