Group #13119865 — subgroups

Subgroups — #13119865

Order 1664.

556 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 197 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×#88142	C₂	1	4	yes	maximal
2	C₁₃×#384	C₂	1	1	yes	maximal, characteristic
3	#11718640	C₂	1	4	yes	maximal
4	#11718634	C₂	1	4	yes	maximal
5	C₄×C₄⋊Dic₁₃	C₂	1	2	yes	maximal
6	C₂²×Dic₂₆	C₂²	1	4	yes	—
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×#6101	C₂²	1	8	yes	—
13	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
14	#88142	—	2	16	no	—
15	#88071	C₂²	1	4	yes	—
16	#88071	C₂²	1	4	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
20	C₂²×Dic₁₃	C₂³	1	4	yes	—
21	C₂²×Dic₁₃	C₂³	1	4	yes	—
22	C₂×Dic₂₆	—	2	16	no	—
23	C₂×Dic₂₆	—	4	4	no	—
24	C₁₃×C₄⋊C₄	—	2	8	no	—
25	C₁₃×C₄⋊C₄	—	4	2	no	—
26	C₄²×C₁₃	Q₈	1	4	yes	complemented
27	C₄⋊Dic₁₃	D₄	1	8	yes	—
28	C₄⋊Dic₁₃	Q₈	1	8	yes	complemented
29	#6101	—	2	16	no	—
30	C₄×Dic₁₃	—	2	8	no	—
31	#2551	—	13	1	no	maximal, Sylow
32	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
33	C₂×C₄×C₁₃	—	2	1	no	—
34	C₂×C₄×C₁₃	—	2	1	no	—
35	C₂×C₄×C₁₃	—	2	8	no	—
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
40	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
41	C₂×Dic₁₃	—	2	4	no	—
42	C₂×Dic₁₃	—	2	4	no	—
43	C₂×Dic₁₃	—	2	4	no	—
44	C₂×Dic₁₃	—	2	4	no	—
45	C₂×Dic₁₃	C₂×D₄	1	8	yes	—
46	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
47	Dic₂₆	—	4	16	no	—
48	C₂×C₄⋊Q₈	—	13	4	no	—
49	#395	—	13	4	no	—
50	#389	—	13	4	no	—
51	#384	D₁₃	1	1	yes	characteristic
52	C₄×C₄⋊C₄	—	13	2	no	—
53	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
54	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
55	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
56	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
58	C₄×C₁₃	—	2	2	no	—
59	C₄×C₁₃	—	4	4	no	—
60	C₄×C₁₃	C₄⋊Q₈	1	8	yes	complemented
61	Dic₁₃	—	2	8	no	—
62	Dic₁₃	—	2	8	no	—
63	C₂²×Q₈	—	13	4	no	—
64	C₄⋊Q₈	—	26	16	no	—
65	C₂×C₄⋊C₄	—	13	2	no	—
66	C₂×C₄⋊C₄	—	13	2	no	—
67	C₂×C₄⋊C₄	—	13	8	no	—
68	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
69	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
70	C₂×C₄²	—	13	4	no	—
71	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
72	#95	—	13	4	no	—
73	#95	—	13	4	no	—
74	C₂×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
75	C₂×C₁₃	#544	1	1	yes	characteristic
76	C₂×C₁₃	D₄²	1	1	yes	characteristic
77	C₂×C₁₃	D₄×Q₈	1	2	yes	—
78	C₂×C₁₃	Q₈²	1	1	yes	characteristic
79	C₂×Q₈	—	26	16	no	—
80	C₂×Q₈	—	52	4	no	—
81	C₂²×C₄	—	13	4	no	—
82	C₂²×C₄	—	13	4	no	—
83	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
84	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
85	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
86	C₄⋊C₄	—	2	8	no	—
87	C₄⋊C₄	—	4	2	no	—
88	C₄⋊C₄	—	13	8	no	—
89	C₄⋊C₄	—	13	8	no	—
90	C₄⋊C₄	—	26	16	no	—
91	C₄²	—	26	8	no	—
92	C₄²	Dic₂₆	1	4	yes	complemented
93	C₁₃	#2551	1	1	yes	characteristic, complemented, Sylow
94	C₂³	C₂³×D₁₃	1	1	yes	characteristic
95	Q₈	—	52	16	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	13	8	no	—
101	C₂×C₄	—	13	8	no	—
102	C₂×C₄	—	26	4	no	—
103	C₂×C₄	—	26	4	no	—
104	C₂×C₄	—	26	4	no	—
105	C₂×C₄	—	26	4	no	—
106	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
107	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
108	C₂×C₄	D₄×D₁₃	1	4	yes	—
109	C₂×C₄	Q₈×D₁₃	1	4	yes	—
110	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
111	C₂²	C₂×D₄×D₁₃	1	2	yes	—
112	C₂²	#88251	1	1	yes	characteristic
113	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
114	C₂²	#88254	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	4	no	—
117	C₄	—	26	8	no	—
118	C₄	—	26	8	no	—
119	C₄	#88142	1	8	yes	complemented
120	C₂	C₂×#88142	1	2	yes	—
121	C₂	#11719205	1	1	yes	characteristic
122	C₂	#11719422	1	1	yes	characteristic
123	C₂	#11719427	1	1	yes	characteristic
124	C₂	#11719432	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88142

C₂

yes

maximal

C₁₃×#384

C₂

yes

maximal, characteristic

#11718640

C₂

yes

maximal

#11718634

C₂

yes

maximal

C₄×C₄⋊Dic₁₃

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88142

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

Q₈

yes

complemented

C₄⋊Dic₁₃

D₄

yes

—

C₄⋊Dic₁₃

Q₈

yes

complemented

#6101

—

C₄×Dic₁₃

—

#2551

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

#395

—

#389

—

#384

D₁₃

yes

characteristic

C₄×C₄⋊C₄

—

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄⋊Q₈

yes

complemented

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

#95

—

C₂×C₁₃

C₂×C₄⋊Q₈

yes

—

C₂×C₁₃

#544

yes

characteristic

C₂×C₁₃

D₄²

yes

characteristic

C₂×C₁₃

D₄×Q₈

yes

—

C₂×C₁₃

Q₈²

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

Dic₂₆

yes

complemented

C₁₃

#2551

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₂₆

yes

—

107

C₂×C₄

C₂×Dic₂₆

yes

—

108

C₂×C₄

D₄×D₁₃

yes

—

109

C₂×C₄

Q₈×D₁₃

yes

—

110

C₂²

C₂²×Dic₂₆

yes

characteristic

111

C₂²

C₂×D₄×D₁₃

yes

—

112

C₂²

#88251

yes

characteristic

113

C₂²

C₂×Q₈×D₁₃

yes

—

114

C₂²

#88254

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#88142

yes

complemented

120

C₂

C₂×#88142

yes

—

121

C₂

#11719205

yes

characteristic

122

C₂

#11719422

yes

characteristic

123

C₂

#11719427

yes

characteristic

124

C₂

#11719432

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#384	C₂
hypercenter	#384	D₁₃
hypocenter	C₁₃	#2551
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#384

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₁₃	Q₈	semidirect
28. C₄⋊Dic₁₃	Q₈	semidirect
60. C₄×C₁₃	C₄⋊Q₈	semidirect
92. C₄²	Dic₂₆	semidirect
93. C₁₃	#2551	semidirect
119. C₄	#88142	semidirect

H ≅ G/N

type

26. C₄²×C₁₃

Q₈

semidirect

28. C₄⋊Dic₁₃

Q₈

semidirect

60. C₄×C₁₃

C₄⋊Q₈

semidirect

92. C₄²

Dic₂₆

semidirect

93. C₁₃

#2551

semidirect

119. C₄

#88142

semidirect

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

subgroup	index
C₂×#88142	2
C₁₃×#384	2
#11718640	2
#11718634	2
C₄×C₄⋊Dic₁₃	2
#2551	13

subgroup

index

C₁₃×#384

C₄×C₄⋊Dic₁₃

Subgroups — #13119865

Conjugacy classes of subgroups

Complements

Maximal subgroups