Group #13120506 — subgroups

Subgroups — #13120506

Order 1664.

428 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 141 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×#381	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Dic₁₃	C₂	1	2	yes	maximal
3	#11718609	C₂	1	4	yes	maximal
4	#11718315	C₂	1	4	yes	maximal, complemented
5	#11718308	C₂	1	4	yes	maximal, complemented
6	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
10	C₂×#6101	C₂²	1	4	yes	—
11	C₂×#6101	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
13	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
14	C₁₃×#95	C₂²	1	4	yes	—
15	#88071	C₂²	1	4	yes	—
16	#88071	C₂²	1	4	yes	—
17	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	—	4	2	no	—
19	C₂²×C₄×C₁₃	Q₈	1	4	yes	—
20	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
22	C₂²×Dic₁₃	C₂³	1	4	yes	—
23	C₂²×Dic₁₃	C₂³	1	4	yes	—
24	C₁₃×C₂²⋊C₄	—	4	2	no	—
25	C₂²⋊Dic₁₃	—	2	8	no	—
26	C₄⋊Dic₁₃	—	2	8	no	—
27	#6101	—	2	8	no	—
28	#6101	—	4	4	no	—
29	C₄×Dic₁₃	—	4	2	no	—
30	C₄×Dic₁₃	—	4	2	no	—
31	#2608	—	13	1	no	maximal, Sylow
32	C₂³×C₁₃	—	2	4	no	—
33	C₂³×C₁₃	—	2	2	no	—
34	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
35	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₁₃	—	2	2	no	—
37	C₂×C₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	—	2	4	no	—
39	C₂×C₄×C₁₃	—	2	4	no	—
40	C₂×C₄×C₁₃	—	2	4	no	—
41	C₂×C₄×C₁₃	—	4	4	no	—
42	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
43	C₂×Dic₁₃	—	2	4	no	—
44	C₂×Dic₁₃	—	2	4	no	—
45	C₂×Dic₁₃	—	2	4	no	—
46	C₂×Dic₁₃	—	2	4	no	—
47	C₂×Dic₁₃	—	2	4	no	—
48	C₂×Dic₁₃	#41	1	8	yes	—
49	#396	—	13	4	no	—
50	#389	—	13	4	no	—
51	#387	—	13	4	no	—
52	#381	D₁₃	1	1	yes	characteristic
53	C₄×C₂²⋊C₄	—	13	2	no	—
54	C₂²×C₁₃	—	2	4	no	—
55	C₂²×C₁₃	—	2	2	no	—
56	C₂²×C₁₃	—	4	4	no	—
57	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
58	C₂²×C₁₃	C₂×#41	1	4	yes	—
59	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
60	C₄×C₁₃	—	2	4	no	—
61	C₄×C₁₃	—	4	4	no	—
62	Dic₁₃	—	2	8	no	—
63	Dic₁₃	—	4	4	no	—
64	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	—	13	4	no	—
66	C₂×C₄⋊C₄	—	13	4	no	—
67	C₂×C₄⋊C₄	—	13	4	no	—
68	C₂×C₂²⋊C₄	—	13	4	no	—
69	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
70	C₂×C₄²	—	13	2	no	—
71	C₂×C₄²	—	13	2	no	—
72	#95	—	13	4	no	—
73	#95	—	13	4	no	—
74	#95	C₂×D₁₃	1	4	yes	—
75	C₂×C₁₃	—	4	2	no	—
76	C₂×C₁₃	#533	1	2	yes	—
77	C₂×C₁₃	#543	1	1	yes	characteristic
78	C₂×C₁₃	#553	1	4	yes	—
79	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
80	C₂²×C₄	—	4	2	no	—
81	C₂²×C₄	—	13	4	no	—
82	C₂²×C₄	—	13	4	no	—
83	C₂²×C₄	Dic₂₆	1	4	yes	—
84	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
85	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
86	C₄⋊C₄	—	26	8	no	—
87	C₄⋊C₄	—	26	8	no	—
88	C₄⋊C₄	—	52	4	no	—
89	C₂²⋊C₄	—	4	2	no	—
90	C₂²⋊C₄	—	26	8	no	—
91	C₄²	—	52	2	no	—
92	C₄²	—	52	2	no	—
93	C₁₃	#2608	1	1	yes	characteristic, complemented, Sylow
94	C₂³	—	2	2	no	—
95	C₂³	—	2	4	no	—
96	C₂³	C₂×Dic₂₆	1	2	yes	—
97	C₂³	C₂³×D₁₃	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	13	8	no	—
105	C₂×C₄	—	26	4	no	—
106	C₂×C₄	—	26	4	no	—
107	C₂×C₄	—	26	4	no	—
108	C₂×C₄	—	26	4	no	—
109	C₂×C₄	—	26	4	no	—
110	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
115	C₂²	#88251	1	2	yes	—
116	C₂²	#41×D₁₃	1	4	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	4	4	no	—
119	C₄	—	26	8	no	—
120	C₄	—	52	4	no	—
121	C₂	—	4	2	no	—
122	C₂	#11719188	1	1	yes	characteristic
123	C₂	#11719222	1	2	yes	—
124	C₂	#11719366	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#381

C₂

yes

maximal, characteristic

C₄×C₂²⋊Dic₁₃

C₂

yes

maximal

#11718609

C₂

yes

maximal

#11718315

C₂

yes

maximal, complemented

#11718308

C₂

yes

maximal, complemented

C₂³×C₄×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

Q₈

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

#2608

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×Q₈

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

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

#41

yes

—

—

—

—

D₁₃

yes

characteristic

C₄×C₂²⋊C₄

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂³×C₄

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

#533

yes

—

C₂×C₁₃

#543

yes

characteristic

C₂×C₁₃

#553

yes

—

C₂⁴

C₂²×D₁₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₂₆

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₁₃

#2608

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

C₂×Dic₂₆

yes

—

C₂³

C₂³×D₁₃

yes

characteristic

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

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×Dic₂₆

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂²×Dic₂₆

yes

characteristic

115

C₂²

#88251

yes

—

116

C₂²

#41×D₁₃

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

#11719188

yes

characteristic

123

C₂

#11719222

yes

—

124

C₂

#11719366

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#381	C₂
prime radical	C₂⁴×C₁₃	C₂³
hypercenter	#381	D₁₃
hypocenter	C₁₃	#2608
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #381

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#381

C₂

prime radical

C₂⁴×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #381

N	H ≅ G/N	type
4. #11718315	C₂	semidirect
5. #11718308	C₂	semidirect
93. C₁₃	#2608	semidirect

H ≅ G/N

type

4. #11718315

C₂

semidirect

5. #11718308

C₂

semidirect

93. C₁₃

#2608

semidirect

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

subgroup	index
C₁₃×#381	2
C₄×C₂²⋊Dic₁₃	2
#11718609	2
#11718315	2
#11718308	2
#2608	13

subgroup

index

C₁₃×#381

C₄×C₂²⋊Dic₁₃

Subgroups — #13120506

Conjugacy classes of subgroups

Complements

Maximal subgroups