Group #13123921 — subgroups

Subgroups — #13123921

Order 1664.

356 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 217 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×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	#11718685	C₂	1	8	yes	maximal, complemented
3	#11718587	C₂	1	4	yes	maximal
4	#11718586	C₂	1	2	yes	maximal
5	C₂²×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	C₄×Q₈×C₁₃	C₄	1	4	yes	—
7	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
12	C₂²⋊#1055	C₂²	1	4	yes	—
13	C₄⋊#1055	C₂²	1	4	yes	—
14	C₄⋊#1055	C₂²	1	8	yes	—
15	#88043	C₂²	1	8	yes	—
16	C₄×#1055	C₂²	1	4	yes	—
17	C₂×Q₈×C₁₃	—	2	4	no	—
18	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
25	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₄²×C₁₃	C₂×C₄	1	4	yes	—
27	C₄²×C₁₃	C₂×C₄	1	2	yes	—
28	C₄²×C₁₃	C₂³	1	2	yes	—
29	C₄²×C₁₃	C₂³	1	4	yes	—
30	C₂×#1055	C₂³	1	8	yes	—
31	#2983	—	13	1	no	maximal, Sylow
32	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
33	Q₈×C₁₃	—	2	8	no	—
34	C₂×C₄×C₁₃	—	2	4	no	—
35	C₂×C₄×C₁₃	—	2	2	no	—
36	C₂×C₄×C₁₃	—	4	1	no	—
37	C₂×C₄×C₁₃	—	4	2	no	—
38	C₂×C₄×C₁₃	M₁₆	1	4	yes	—
39	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
42	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
43	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
45	#1055	—	2	8	no	—
46	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic
47	#446	—	13	8	no	—
48	#432	—	13	4	no	—
49	#431	—	13	2	no	—
50	C₂²×C₁₃	—	2	1	no	—
51	C₂²×C₁₃	—	2	1	no	—
52	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
53	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₃	—	2	1	no	—
55	C₄×C₁₃	—	2	4	no	—
56	C₄×C₁₃	—	2	2	no	—
57	C₄×C₁₃	—	2	4	no	—
58	C₄×C₁₃	C₂×M₁₆	1	4	yes	—
59	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
60	C₂²×Q₈	Dic₁₃	1	1	yes	characteristic
61	C₄×Q₈	Dic₁₃	1	4	yes	—
62	C₄×Q₈	C₂×D₁₃	1	4	yes	—
63	C₂×C₄⋊C₄	Dic₁₃	1	2	yes	—
64	C₂×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₁₃	1	2	yes	—
67	C₄⋊C₈	—	13	4	no	—
68	C₄⋊C₈	—	13	8	no	—
69	C₂²⋊C₈	—	13	4	no	—
70	#97	—	13	8	no	—
71	C₄×C₈	—	13	4	no	—
72	C₂×C₁₃	—	2	2	no	—
73	C₂×C₁₃	#519	1	1	yes	characteristic
74	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
75	C₂×C₁₃	#568	1	1	yes	characteristic
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₃	1	4	yes	—
78	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
79	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
80	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
81	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
83	C₂×C₈	—	13	8	no	—
84	C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
85	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
86	C₄²	C₂×Dic₁₃	1	4	yes	—
87	C₄²	C₂×Dic₁₃	1	2	yes	—
88	C₄²	C₂²×D₁₃	1	2	yes	—
89	C₄²	C₂²×D₁₃	1	4	yes	—
90	C₁₃	#2983	1	1	yes	characteristic, complemented, Sylow
91	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	#6099	1	4	yes	—
98	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
99	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
100	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
101	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
102	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
103	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
104	C₈	—	26	8	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×#6099	1	2	yes	—
108	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
109	C₄	—	2	4	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	4	no	—
113	C₄	C₂×#6099	1	4	yes	—
114	C₄	#88254	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	C₂²×#6099	1	1	yes	characteristic
117	C₂	#11719517	1	1	yes	characteristic
118	C₂	#11719524	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₃

C₂

yes

maximal, characteristic

#11718685

C₂

yes

maximal, complemented

#11718587

C₂

yes

maximal

#11718586

C₂

yes

maximal

C₂²×Q₈×C₁₃

C₄

yes

characteristic

C₄×Q₈×C₁₃

C₄

yes

—

C₄×Q₈×C₁₃

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₄

yes

—

C₂×C₁₃×C₄⋊C₄

C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

—

C₂²⋊#1055

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

#88043

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

C₂³

yes

—

C₄²×C₁₃

C₂³

yes

—

C₂×#1055

C₂³

yes

—

#2983

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

M₁₆

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

#1055

—

C₂×C₄×Q₈

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×M₁₆

yes

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×M₁₆

yes

—

C₄×C₁₃

2₋¹⁺⁴

yes

—

C₂²×Q₈

Dic₁₃

yes

characteristic

C₄×Q₈

Dic₁₃

yes

—

C₄×Q₈

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

Dic₁₃

yes

—

C₂×C₄⋊C₄

Dic₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#519

yes

characteristic

C₂×C₁₃

C₂²×M₁₆

yes

characteristic

C₂×C₁₃

#568

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₁₃

yes

characteristic

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄²

C₂×Dic₁₃

yes

—

C₄²

C₂×Dic₁₃

yes

—

C₄²

C₂²×D₁₃

yes

—

C₄²

C₂²×D₁₃

yes

—

C₁₃

#2983

yes

characteristic, complemented, Sylow

C₂³

C₂²×Dic₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6099

yes

—

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

C₂×C₄

C₂²×Dic₁₃

yes

—

100

C₂×C₄

C₂²×Dic₁₃

yes

—

101

C₂×C₄

C₂²×Dic₁₃

yes

—

102

C₂×C₄

C₂²×Dic₁₃

yes

—

103

C₂×C₄

C₂³×D₁₃

yes

characteristic

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×#6099

yes

—

108

C₂²

C₂³×Dic₁₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₂×#6099

yes

—

114

C₄

#88254

yes

—

115

C₂

—

116

C₂

C₂²×#6099

yes

characteristic

117

C₂

#11719517

yes

characteristic

118

C₂

#11719524

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₁₃	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	#2983
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₁₃

C₂

prime radical

C₂³×C₁₃

C₂²×C₄

cosocle

C₂×C₄×C₁₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
2. #11718685	C₂	semidirect
90. C₁₃	#2983	semidirect

H ≅ G/N

type

2. #11718685

C₂

semidirect

90. C₁₃

#2983

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₁₃	2
#11718685	2
#11718587	2
#11718586	2
#2983	13

subgroup

index

C₂×C₄×Q₈×C₁₃

Subgroups — #13123921

Conjugacy classes of subgroups

Complements

Maximal subgroups