Group #13123103 — subgroups

Subgroups — #13123103

Order 1664.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₁₃×#383	C₂	1	1	yes	maximal, characteristic
2	#11718598	C₂	1	1	yes	maximal, characteristic
3	#11718596	C₂	1	1	yes	maximal, characteristic
4	#11718322	C₂	1	3	yes	maximal
5	#11718321	C₂	1	3	yes	maximal
6	#11718318	C₂	1	3	yes	maximal
7	#11718317	C₂	1	3	yes	maximal
8	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
12	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
13	C₂×#6101	C₂²	1	3	yes	—
14	C₂×#6101	C₂²	1	3	yes	—
15	C₁₃×#95	C₂²	1	3	yes	—
16	C₁₃×#95	C₂²	1	3	yes	—
17	#88071	C₂²	1	3	yes	—
18	#88071	C₂²	1	3	yes	—
19	#88071	C₂²	1	3	yes	—
20	#88071	C₂²	1	3	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
24	C₂²×Dic₁₃	C₂³	1	3	yes	—
25	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₃	C₂³	1	3	yes	—
28	C₂×Dic₂₆	—	4	3	no	—
29	C₄²×C₁₃	Q₈	1	4	yes	complemented
30	C₄⋊Dic₁₃	—	4	3	no	—
31	C₄⋊Dic₁₃	—	4	3	no	—
32	C₄⋊Dic₁₃	—	4	3	no	—
33	#6101	—	4	3	no	—
34	#6101	—	4	3	no	—
35	#2843	—	13	1	no	maximal, Sylow
36	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₁₃	—	2	3	no	—
38	C₂×C₄×C₁₃	—	2	3	no	—
39	C₂×C₄×C₁₃	—	2	3	no	—
40	C₂×C₄×C₁₃	—	2	3	no	—
41	C₂×C₄×C₁₃	—	2	6	no	—
42	C₂×C₄×C₁₃	C₂×Q₈	1	6	yes	—
43	C₂×Dic₁₃	—	2	3	no	—
44	C₂×Dic₁₃	—	2	3	no	—
45	C₂×Dic₁₃	—	2	3	no	—
46	C₂×Dic₁₃	—	2	3	no	—
47	C₂×Dic₁₃	—	2	3	no	—
48	C₂×Dic₁₃	—	2	3	no	—
49	C₂×Dic₁₃	—	2	6	no	—
50	Dic₂₆	—	4	4	no	—
51	#400	—	13	3	no	—
52	#398	—	13	3	no	—
53	#398	—	13	3	no	—
54	#395	—	13	3	no	—
55	#384	—	13	1	no	—
56	#383	—	13	1	no	—
57	#383	D₁₃	1	1	yes	characteristic
58	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
59	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
60	C₂²×C₁₃	2₋¹⁺⁴	1	3	yes	—
61	C₄×C₁₃	—	2	6	no	—
62	C₄×C₁₃	—	4	1	no	—
63	C₄×C₁₃	—	4	3	no	—
64	Dic₁₃	—	4	1	no	—
65	Dic₁₃	—	4	1	no	—
66	Dic₁₃	—	4	3	no	—
67	Dic₁₃	—	4	3	no	—
68	C₂²×Q₈	—	13	1	no	—
69	C₂×C₄⋊C₄	—	13	3	no	—
70	C₂×C₄⋊C₄	—	13	3	no	—
71	C₂×C₄⋊C₄	—	13	3	no	—
72	C₂×C₄⋊C₄	—	13	3	no	—
73	C₂×C₄⋊C₄	—	13	3	no	—
74	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
75	#95	—	13	3	no	—
76	#95	—	13	3	no	—
77	#95	—	13	3	no	—
78	#95	—	13	3	no	—
79	#95	C₂×D₁₃	1	3	yes	—
80	#95	C₂×D₁₃	1	3	yes	—
81	C₂×C₁₃	#544	1	3	yes	—
82	C₂×C₁₃	#560	1	1	yes	characteristic
83	C₂×C₁₃	#563	1	3	yes	—
84	C₂×Q₈	—	52	3	no	—
85	C₂²×C₄	—	13	3	no	—
86	C₂²×C₄	—	13	1	no	—
87	C₂²×C₄	—	13	1	no	—
88	C₂²×C₄	—	13	3	no	—
89	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
90	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
91	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
92	C₄⋊C₄	—	52	3	no	—
93	C₄⋊C₄	—	52	3	no	—
94	C₄⋊C₄	—	52	3	no	—
95	C₄⋊C₄	—	52	3	no	—
96	C₄⋊C₄	—	52	3	no	—
97	C₄²	Dic₂₆	1	4	yes	complemented
98	C₁₃	#2843	1	1	yes	characteristic, complemented, Sylow
99	C₂³	C₂³×D₁₃	1	1	yes	characteristic
100	Q₈	—	52	4	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	26	3	no	—
107	C₂×C₄	—	26	3	no	—
108	C₂×C₄	—	26	3	no	—
109	C₂×C₄	—	26	3	no	—
110	C₂×C₄	—	26	3	no	—
111	C₂×C₄	—	26	3	no	—
112	C₂×C₄	—	26	6	no	—
113	C₂×C₄	C₂×Dic₂₆	1	6	yes	—
114	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
115	C₂²	#88256	1	3	yes	—
116	C₂²	#88257	1	3	yes	—
117	C₄	—	2	6	no	—
118	C₄	—	4	1	no	—
119	C₄	—	4	3	no	—
120	C₄	—	52	3	no	—
121	C₄	—	52	1	no	—
122	C₄	—	52	1	no	—
123	C₄	—	52	3	no	—
124	C₂	#11719224	1	3	yes	—
125	C₂	#11719416	1	1	yes	characteristic
126	C₂	#11719417	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#383

C₂

yes

maximal, characteristic

#11718598

C₂

yes

maximal, characteristic

#11718596

C₂

yes

maximal, characteristic

#11718322

C₂

yes

maximal

#11718321

C₂

yes

maximal

#11718318

C₂

yes

maximal

#11718317

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#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

characteristic

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₄²×C₁₃

Q₈

yes

complemented

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

—

C₂×D₁₃

yes

—

#95

C₂×D₁₃

yes

—

C₂×C₁₃

#544

yes

—

C₂×C₁₃

#560

yes

characteristic

C₂×C₁₃

#563

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

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

Dic₂₆

yes

complemented

C₁₃

#2843

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×Dic₂₆

yes

—

114

C₂²

C₂²×Dic₂₆

yes

characteristic

115

C₂²

#88256

yes

—

116

C₂²

#88257

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₂

#11719224

yes

—

125

C₂

#11719416

yes

characteristic

126

C₂

#11719417

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#383	C₂
hypercenter	#383	D₁₃
hypocenter	C₁₃	#2843
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#383

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
29. C₄²×C₁₃	Q₈	semidirect
97. C₄²	Dic₂₆	semidirect
98. C₁₃	#2843	semidirect

H ≅ G/N

type

29. C₄²×C₁₃

Q₈

semidirect

97. C₄²

Dic₂₆

semidirect

98. C₁₃

#2843

semidirect

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

subgroup	index
C₁₃×#383	2
#11718598	2
#11718596	2
#11718322	2
#11718321	2
#11718318	2
#11718317	2
#2843	13

subgroup

index

C₁₃×#383

Subgroups — #13123103

Conjugacy classes of subgroups

Complements

Maximal subgroups