Group #13115656 — subgroups

Subgroups — #13115656

Order 1664.

308 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×#88056	C₂	1	4	yes	maximal
4	#11718597	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆	C₂²	1	2	yes	—
6	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃	—	2	2	no	—
9	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	#88123	—	2	2	no	—
11	#88123	C₄	1	4	yes	complemented
12	C₁₃×#97	—	2	2	no	—
13	#88071	—	2	2	no	—
14	#88056	—	2	8	no	—
15	C₂²×C₄×C₁₃	D₄	1	2	yes	—
16	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₁₃	—	2	2	no	—
18	C₂²×Dic₁₃	—	2	2	no	—
19	C₂×Dic₂₆	—	2	4	no	—
20	C₂×Dic₂₆	—	2	4	no	—
21	C₂×Dic₂₆	—	2	2	no	—
22	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₁₃	—	2	2	no	—
24	C₂×C₈×C₁₃	—	2	2	no	—
25	C₂×C₈×C₁₃	D₄	1	4	yes	—
26	C₄²×C₁₃	C₂×C₄	1	2	yes	—
27	C₄²×C₁₃	D₄	1	2	yes	—
28	C₄⋊Dic₁₃	—	2	8	no	—
29	C₄⋊Dic₁₃	—	2	4	no	—
30	#1960	—	13	1	no	maximal, Sylow
31	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃	—	2	4	no	—
33	C₂×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃	#41	1	2	yes	—
38	C₂×Dic₁₃	—	2	4	no	—
39	C₂×Dic₁₃	—	4	2	no	—
40	C₂×Dic₁₃	—	4	2	no	—
41	C₂×Dic₁₃	—	4	2	no	—
42	C₂×Dic₁₃	—	4	4	no	—
43	Dic₂₆	—	2	8	no	—
44	Dic₂₆	—	4	4	no	—
45	C₈×C₁₃	—	2	4	no	—
46	C₂×C₄⋊Q₈	—	13	1	no	—
47	C₂×#103	—	13	4	no	—
48	C₂×#97	D₁₃	1	1	yes	characteristic
49	#382	—	13	1	no	—
50	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₃	#124	1	1	yes	characteristic
52	C₂²×C₁₃	#127	1	1	yes	characteristic
53	C₂²×C₁₃	#137	1	4	yes	—
54	C₄×C₁₃	—	2	4	no	—
55	C₄×C₁₃	C₄×D₄	1	2	yes	—
56	C₄×C₁₃	#121	1	2	yes	—
57	Dic₁₃	—	4	4	no	—
58	Dic₁₃	—	8	2	no	—
59	C₂²×Q₈	—	13	2	no	—
60	C₂²×C₈	C₂×D₁₃	1	2	yes	—
61	C₄⋊Q₈	—	13	4	no	—
62	C₄⋊Q₈	—	26	2	no	—
63	C₂×C₄⋊C₄	—	13	2	no	—
64	C₂×C₄⋊C₄	—	26	2	no	—
65	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
66	#103	—	26	8	no	—
67	#97	—	2	2	no	—
68	#95	—	26	2	no	—
69	C₂×C₁₃	#390	1	1	yes	characteristic
70	C₂×C₁₃	#419	1	2	yes	—
71	C₂×C₁₃	#491	1	2	yes	—
72	C₂×C₁₃	#497	1	2	yes	—
73	C₂×Q₈	—	13	4	no	—
74	C₂×Q₈	—	26	2	no	—
75	C₂×Q₈	—	26	4	no	—
76	C₂×Q₈	—	26	4	no	—
77	C₂²×C₄	—	26	2	no	—
78	C₂²×C₄	—	26	2	no	—
79	C₂²×C₄	D₅₂	1	2	yes	—
80	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	D₅₂	1	4	yes	—
84	C₄⋊C₄	—	26	8	no	—
85	C₄⋊C₄	—	26	4	no	—
86	C₄²	C₄×D₁₃	1	2	yes	—
87	C₄²	C₂²⋊D₁₃	1	2	yes	—
88	C₁₃	#1960	1	1	yes	characteristic, complemented, Sylow
89	C₂³	C₂×D₅₂	1	1	yes	characteristic
90	Q₈	—	26	8	no	—
91	Q₈	—	52	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	26	4	no	—
94	C₂×C₄	—	52	2	no	—
95	C₂×C₄	—	52	2	no	—
96	C₂×C₄	—	52	2	no	—
97	C₂×C₄	—	52	4	no	—
98	C₂×C₄	#6103	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
100	C₂×C₄	C₂×D₅₂	1	2	yes	—
101	C₂×C₄	#6127	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	#88128	1	1	yes	characteristic
105	C₂²	#88129	1	1	yes	characteristic
106	C₂²	#88163	1	4	yes	—
107	C₂²	C₂×#6103	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	52	4	no	—
110	C₄	—	104	2	no	—
111	C₄	C₄×D₅₂	1	2	yes	—
112	C₄	#88184	1	2	yes	—
113	C₂	#11718364	1	2	yes	—
114	C₂	#11718376	1	2	yes	—
115	C₂	#11718600	1	1	yes	characteristic
116	C₂	#11718781	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88123

C₂

yes

maximal, characteristic

C₂×C₁₃×#97

C₂

yes

maximal, characteristic

C₂×#88056

C₂

yes

maximal

#11718597

C₂

yes

maximal, characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂²×C₈×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

#88123

—

#88123

C₄

yes

complemented

C₁₃×#97

—

#88071

—

#88056

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#1960

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#97

D₁₃

yes

characteristic

#382

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

#121

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

#419

yes

—

C₂×C₁₃

#491

yes

—

C₂×C₁₃

#497

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₅₂

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₅₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₁₃

yes

—

C₄²

C₂²⋊D₁₃

yes

—

C₁₃

#1960

yes

characteristic, complemented, Sylow

C₂³

C₂×D₅₂

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6103

yes

—

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

100

C₂×C₄

C₂×D₅₂

yes

—

101

C₂×C₄

#6127

yes

—

102

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

103

C₈

—

104

C₂²

#88128

yes

characteristic

105

C₂²

#88129

yes

characteristic

106

C₂²

#88163

yes

—

107

C₂²

C₂×#6103

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

104

—

111

C₄

C₄×D₅₂

yes

—

112

C₄

#88184

yes

—

113

C₂

#11718364

yes

—

114

C₂

#11718376

yes

—

115

C₂

#11718600

yes

characteristic

116

C₂

#11718781

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#97	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×#97	D₁₃
hypocenter	C₁₃	#1960
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×#97

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂×#97

D₁₃

hypocenter

C₁₃

#1960

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
11. #88123	C₄	semidirect
88. C₁₃	#1960	semidirect

H ≅ G/N

type

11. #88123

C₄

semidirect

88. C₁₃

#1960

semidirect

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

subgroup	index
C₂×#88123	2
C₂×C₁₃×#97	2
C₂×#88056	2
#11718597	2
#1960	13

subgroup

index

C₂×C₁₃×#97

Subgroups — #13115656

Conjugacy classes of subgroups

Complements

Maximal subgroups