Group #13113915 — subgroups

Subgroups — #13113915

Order 1664.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#11719427	C₂	1	1	yes	maximal, characteristic, complemented
2	#11718746	C₂	1	1	yes	maximal, characteristic
3	#11718509	C₂	1	2	yes	maximal
4	C₁₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718076	C₂	1	2	yes	maximal, complemented
6	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	#88198	—	2	2	no	—
8	C₂×#6106	—	2	2	no	—
9	C₂×#6095	—	2	2	no	—
10	#88142	—	2	2	no	—
11	#88141	—	2	2	no	—
12	#88128	C₂²	1	1	yes	characteristic
13	C₄×Dic₂₆	C₂²	1	2	yes	—
14	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	#88061	—	2	2	no	—
16	C₄×#1055	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₃	D₄	1	2	yes	—
18	C₂×D₅₂	—	2	2	no	—
19	C₂×D₅₂	—	2	1	no	—
20	C₂×Dic₂₆	—	2	2	no	—
21	C₂×Dic₂₆	D₄	1	2	yes	—
22	C₂×C₈×C₁₃	—	2	2	no	—
23	C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	#6106	—	4	4	no	—
26	C₄⋊Dic₁₃	D₄	1	2	yes	—
27	#6101	—	2	4	no	—
28	#6101	—	2	2	no	—
29	C₄×Dic₁₃	—	2	2	no	—
30	C₄×Dic₁₃	—	2	2	no	—
31	C₂×#1055	—	2	1	no	—
32	#6095	—	4	4	no	—
33	#1623	—	13	1	no	maximal, Sylow
34	C₂²×D₁₃	—	4	1	no	—
35	Q₈×C₁₃	—	4	2	no	—
36	C₂×C₄×C₁₃	—	2	2	no	—
37	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
39	C₂×Dic₁₃	—	2	2	no	—
40	C₂×Dic₁₃	—	2	2	no	—
41	D₅₂	—	4	2	no	—
42	D₅₂	—	4	2	no	—
43	D₅₂	—	4	2	no	—
44	Dic₂₆	—	2	2	no	—
45	Dic₂₆	—	8	2	no	—
46	Dic₂₆	SD₁₆	1	4	yes	complemented
47	C₈×C₁₃	—	4	2	no	—
48	#1055	—	2	2	no	—
49	Q₈²	—	13	1	no	—
50	#492	—	13	1	no	—
51	#460	—	13	2	no	—
52	#333	D₁₃	1	1	yes	characteristic, complemented
53	Q₈⋊C₈	—	13	2	no	—
54	C₂²×C₁₃	#120	1	1	yes	characteristic
55	C₂×D₁₃	—	8	2	no	—
56	C₂×D₁₃	—	8	1	no	—
57	C₄×C₁₃	—	2	1	no	—
58	C₄×C₁₃	—	4	2	no	—
59	C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
60	C₄×C₁₃	#137	1	2	yes	—
61	Dic₁₃	—	2	4	no	—
62	Dic₁₃	—	4	2	no	—
63	C₂×SD₁₆	—	26	2	no	—
64	C₂×SD₁₆	—	26	2	no	—
65	C₄⋊Q₈	—	26	2	no	—
66	C₄⋊Q₈	—	26	2	no	—
67	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
68	#127	—	13	1	no	—
69	C₄×Q₈	—	13	2	no	—
70	C₄×Q₈	—	26	2	no	—
71	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
72	#102	—	26	2	no	—
73	C₄×C₈	—	13	1	no	—
74	C₂×C₁₃	#448	1	2	yes	—
75	C₂×C₁₃	#453	1	1	yes	characteristic
76	D₁₃	—	16	1	no	—
77	C₂×Q₈	—	13	2	no	—
78	C₂×Q₈	—	26	2	no	—
79	C₂×Q₈	D₅₂	1	2	yes	—
80	C₂×D₄	—	26	1	no	—
81	C₂×D₄	—	26	2	no	—
82	SD₁₆	—	52	4	no	—
83	SD₁₆	—	52	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	26	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	13	2	no	—
88	C₄⋊C₄	—	26	4	no	—
89	C₄⋊C₄	—	26	2	no	—
90	C₄²	—	26	2	no	—
91	C₄²	—	26	2	no	—
92	C₄²	C₂²×D₁₃	1	1	yes	characteristic
93	C₁₃	#1623	1	1	yes	characteristic, complemented, Sylow
94	C₂³	—	52	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	13	4	no	—
97	Q₈	—	26	2	no	—
98	Q₈	—	104	2	no	—
99	D₄	—	52	2	no	—
100	D₄	—	52	2	no	—
101	D₄	—	52	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	26	2	no	—
104	C₂×C₄	—	26	2	no	—
105	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
106	C₂×C₄	D₄×D₁₃	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	26	2	no	—
109	C₂²	—	104	1	no	—
110	C₂²	—	104	2	no	—
111	C₂²	#88136	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	26	4	no	—
115	C₄	—	52	2	no	—
116	C₄	SD₁₆×D₁₃	1	2	yes	—
117	C₄	#88172	1	2	yes	—
118	C₂	—	208	1	no	—
119	C₂	#11718408	1	1	yes	characteristic
120	C₂	#11718467	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719427

C₂

yes

maximal, characteristic, complemented

#11718746

C₂

yes

maximal, characteristic

#11718509

C₂

yes

maximal

C₁₃×#333

C₂

yes

maximal, characteristic, complemented

#11718076

C₂

yes

maximal, complemented

C₁₃×C₄⋊Q₈

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₂₆

C₂²

yes

—

C₁₃×C₄⋊C₈

C₂²

yes

—

#88061

—

C₄×#1055

C₂²

yes

characteristic

C₂×Q₈×C₁₃

D₄

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₄

yes

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂³

yes

characteristic

#6106

—

C₄⋊Dic₁₃

D₄

yes

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

C₂²×D₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

SD₁₆

yes

complemented

C₈×C₁₃

—

#1055

—

Q₈²

—

#492

—

#460

—

#333

D₁₃

yes

characteristic, complemented

Q₈⋊C₈

—

C₂²×C₁₃

#120

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×SD₁₆

yes

—

C₄×C₁₃

#137

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₃

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

—

C₄⋊C₈

C₂×D₁₃

yes

—

#102

—

C₄×C₈

—

C₂×C₁₃

#448

yes

—

C₂×C₁₃

#453

yes

characteristic

D₁₃

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₅₂

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₃

yes

characteristic

C₁₃

#1623

yes

characteristic, complemented, Sylow

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

104

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×D₅₂

yes

characteristic

106

C₂×C₄

D₄×D₁₃

yes

—

107

C₈

—

108

C₈

—

109

C₂²

—

104

—

110

C₂²

—

104

—

111

C₂²

#88136

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

SD₁₆×D₁₃

yes

—

117

C₄

#88172

yes

—

118

C₂

—

208

—

119

C₂

#11718408

yes

characteristic

120

C₂

#11718467

yes

—

	subgroup	G/H
center	C₂²	#88136
commutator, cosocle	C₄²×C₁₃	C₂³
Frattini	C₄²	C₂²×D₁₃
Fitting	C₁₃×#333	C₂
socle	C₂²×C₁₃	#120
prime radical	#88128	C₂²
hypercenter	#333	D₁₃
hypocenter	C₁₃	#1623
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#88136

commutator, cosocle

C₄²×C₁₃

C₂³

Frattini

C₄²

C₂²×D₁₃

Fitting

C₁₃×#333

C₂

C₂²×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #11719427	C₂	semidirect
4. C₁₃×#333	C₂	semidirect
5. #11718076	C₂	semidirect
46. Dic₂₆	SD₁₆	semidirect
52. #333	D₁₃	semidirect
93. C₁₃	#1623	semidirect

H ≅ G/N

type

1. #11719427

C₂

semidirect

4. C₁₃×#333

C₂

semidirect

5. #11718076

C₂

semidirect

46. Dic₂₆

SD₁₆

semidirect

52. #333

D₁₃

semidirect

93. C₁₃

#1623

semidirect

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

subgroup	index
#11719427	2
#11718746	2
#11718509	2
C₁₃×#333	2
#11718076	2
#1623	13

subgroup

index

C₁₃×#333

Subgroups — #13113915

Conjugacy classes of subgroups

Complements

Maximal subgroups