Group #13130291 — subgroups

Subgroups — #13130291

Order 1664.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 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	#11719421	C₂	1	1	yes	maximal, characteristic
2	C₄×#6118	C₂	1	2	yes	maximal, complemented
3	#11718293	C₂	1	4	yes	maximal, complemented
4	C₂×#6129	C₄	1	2	yes	—
5	C₂×#6118	—	2	2	no	—
6	C₁₃×#127	C₄	1	1	yes	characteristic
7	#88192	—	2	2	no	—
8	#88190	C₂²	1	2	yes	—
9	D₄×Dic₁₃	—	2	2	no	—
10	C₄²×D₁₃	C₂²	1	1	yes	characteristic
11	#88123	C₄	1	1	yes	characteristic
12	#88110	C₂²	1	4	yes	—
13	C₄×#1057	—	2	2	no	—
14	C₂×D₄×C₁₃	—	2	2	no	—
15	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
16	C₂×C₂²⋊D₁₃	—	2	2	no	—
17	C₂²×Dic₁₃	—	2	2	no	—
18	#6129	—	4	4	no	—
19	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₃	D₄	1	2	yes	—
21	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
22	C₂×#1057	—	2	4	no	—
23	#6118	—	2	8	no	—
24	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊Dic₁₃	—	2	4	no	—
26	C₄⋊Dic₁₃	—	2	2	no	—
27	C₄×Dic₁₃	D₄	1	2	yes	—
28	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
29	#1525	—	13	1	no	maximal, Sylow
30	C₂³×C₁₃	—	2	2	no	—
31	C₂²×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
32	D₄×C₁₃	—	4	2	no	—
33	D₄×C₁₃	—	4	4	no	—
34	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²⋊D₁₃	—	4	4	no	—
37	C₂×Dic₁₃	—	2	2	no	—
38	C₂×Dic₁₃	—	4	4	no	—
39	C₂×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
41	C₄×D₁₃	—	2	4	no	—
42	C₄×D₁₃	—	2	2	no	—
43	Dic₂₆	—	4	2	no	—
44	#1057	—	2	8	no	—
45	#532	—	13	1	no	—
46	C₄×M₁₆	—	13	2	no	—
47	#329	—	13	4	no	—
48	C₂²×C₁₃	—	4	2	no	—
49	C₂²×C₁₃	—	4	4	no	—
50	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×D₁₃	—	2	1	no	—
52	C₂×D₁₃	—	2	2	no	—
53	C₄×C₁₃	—	2	1	no	—
54	C₄×C₁₃	#104	1	4	yes	—
55	Dic₁₃	—	2	2	no	—
56	Dic₁₃	—	4	2	no	—
57	Dic₁₃	#100	1	2	yes	—
58	C₂×#41	—	13	2	no	—
59	C₂×M₁₆	—	26	2	no	—
60	C₄⋊Q₈	—	13	1	no	—
61	#127	#261	1	1	yes	characteristic
62	#124	—	13	2	no	—
63	#121	—	26	2	no	—
64	C₄×D₄	—	26	2	no	—
65	C₂×C₄²	—	13	1	no	—
66	#97	—	13	4	no	—
67	C₄×C₈	—	26	2	no	—
68	C₂×C₁₃	—	8	2	no	—
69	C₂×C₁₃	C₂×#100	1	1	yes	characteristic
70	C₂×C₁₃	C₂×#104	1	2	yes	—
71	D₁₃	—	4	1	no	—
72	#41	—	52	4	no	—
73	C₂×Q₈	—	13	2	no	—
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	—	26	2	no	—
76	C₂×D₄	C₂×#261	1	2	yes	—
77	C₂²×C₄	—	13	2	no	—
78	C₂²×C₄	—	13	1	no	—
79	C₂²×C₄	—	26	2	no	—
80	M₁₆	—	26	8	no	—
81	C₂×C₈	—	26	4	no	—
82	C₄⋊C₄	—	26	2	no	—
83	C₂²⋊C₄	—	26	4	no	—
84	C₄²	—	13	2	no	—
85	C₄²	—	13	1	no	—
86	C₄²	C₂×#261	1	1	yes	characteristic
87	C₁₃	#1525	1	1	yes	characteristic, complemented, Sylow
88	C₂³	—	2	2	no	—
89	C₂³	—	13	1	no	—
90	Q₈	—	52	2	no	—
91	D₄	—	4	2	no	—
92	D₄	—	4	4	no	—
93	D₄	—	52	4	no	—
94	C₂×C₄	—	13	1	no	—
95	C₂×C₄	—	13	2	no	—
96	C₂×C₄	—	26	2	no	—
97	C₂×C₄	—	26	4	no	—
98	C₂×C₄	—	26	2	no	—
99	C₂×C₄	—	52	4	no	—
100	C₂×C₄	C₂²⋊#261	1	2	yes	—
101	C₂×C₄	C₂²×#261	1	1	yes	characteristic
102	C₈	—	26	8	no	—
103	C₂²	—	4	4	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	26	2	no	—
106	C₂²	—	26	1	no	—
107	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	13	2	no	—
110	C₄	—	26	2	no	—
111	C₄	—	52	2	no	—
112	C₄	#88116	1	4	yes	—
113	C₂	—	8	2	no	—
114	C₂	—	52	1	no	—
115	C₂	C₂×#88116	1	2	yes	—
116	C₂	D₁₃⋊#100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719421

C₂

yes

maximal, characteristic

C₄×#6118

C₂

yes

maximal, complemented

#11718293

C₂

yes

maximal, complemented

C₂×#6129

C₄

yes

—

C₂×#6118

—

C₁₃×#127

C₄

yes

characteristic

#88192

—

#88190

C₂²

yes

—

D₄×Dic₁₃

—

C₄²×D₁₃

C₂²

yes

characteristic

#88123

C₄

yes

characteristic

#88110

C₂²

yes

—

C₄×#1057

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

#6129

—

C₂×C₄×D₁₃

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃

D₄

yes

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×#1057

—

#6118

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

D₄

yes

—

C₄×Dic₁₃

C₂³

yes

characteristic

#1525

—

maximal, Sylow

C₂³×C₁₃

—

C₂²×D₁₃

C₂²⋊C₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

#1057

—

#532

—

C₄×M₁₆

—

#329

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#104

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#100

yes

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

yes

—

—

C₄×D₄

—

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#100

yes

characteristic

C₂×C₁₃

C₂×#104

yes

—

D₁₃

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×#261

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#261

yes

characteristic

C₁₃

#1525

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊#261

yes

—

101

C₂×C₄

C₂²×#261

yes

characteristic

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₂²⋊#261

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#88116

yes

—

113

C₂

—

114

C₂

—

115

C₂

C₂×#88116

yes

—

116

C₂

D₁₃⋊#100

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₁₃×#127	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	#11719421	C₂
cosocle	C₄×Dic₁₃	C₂³
hypercenter	#127	#261
hypocenter	C₁₃	#1525
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#261

Fitting

C₁₃×#127

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

#11719421

C₂

cosocle

C₄×Dic₁₃

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. C₄×#6118	C₂	semidirect
3. #11718293	C₂	semidirect
87. C₁₃	#1525	semidirect

H ≅ G/N

type

2. C₄×#6118

C₂

semidirect

3. #11718293

C₂

semidirect

87. C₁₃

#1525

semidirect

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

subgroup	index
#11719421	2
C₄×#6118	2
#11718293	2
#1525	13

subgroup

index

Subgroups — #13130291

Conjugacy classes of subgroups

Complements

Maximal subgroups