Group #13123299 — subgroups

Subgroups — #13123299

Order 1664.

716 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 249 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×#88254	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719517	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718667	C₂	1	12	yes	maximal, complemented
5	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
6	#88254	—	2	4	no	—
7	#88254	—	2	4	no	—
8	C₂×#6131	C₄	1	4	yes	complemented
9	C₂×Q₈×D₁₃	C₄	1	4	yes	complemented
10	C₂×#6127	C₂²	1	6	yes	—
11	C₄×Q₈×C₁₃	—	2	4	no	—
12	C₁₃×#117	C₂²	1	6	yes	—
13	Q₈×Dic₁₃	—	2	4	no	—
14	#88180	C₂²	1	6	yes	—
15	#88046	C₂²	1	16	yes	—
16	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	—
17	C₂×Q₈×C₁₃	D₄	1	4	yes	—
18	C₂×Q₈×C₁₃	D₄	1	4	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
20	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
21	#6131	—	2	16	no	—
22	Q₈×D₁₃	—	2	16	no	—
23	#6127	—	2	12	no	—
24	#6127	—	2	12	no	—
25	C₂×D₅₂	C₂×C₄	1	6	yes	—
26	C₂×C₄×D₁₃	C₂×C₄	1	12	yes	—
27	C₂×Dic₂₆	C₂×C₄	1	6	yes	—
28	C₁₃×C₄⋊C₄	—	2	6	no	—
29	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
30	C₄²×C₁₃	—	2	6	no	—
31	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
32	C₄⋊Dic₁₃	—	2	6	no	—
33	C₄×Dic₁₃	—	2	6	no	—
34	#2884	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
36	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
37	Q₈×C₁₃	C₂²⋊C₄	1	16	yes	—
38	C₂×C₄×C₁₃	—	2	4	no	—
39	C₂×C₄×C₁₃	C₂²×C₄	1	6	yes	—
40	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
41	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
42	C₂²⋊D₁₃	—	2	4	no	—
43	C₂²⋊D₁₃	—	2	4	no	—
44	C₂×Dic₁₃	—	2	4	no	—
45	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
46	D₅₂	—	2	12	no	—
47	C₄×D₁₃	—	2	24	no	—
48	Dic₂₆	—	2	12	no	—
49	C₂×2₋¹⁺⁴	—	13	1	no	—
50	#519	—	13	1	no	—
51	#519	D₁₃	1	1	yes	characteristic, complemented
52	#410	—	13	12	no	—
53	C₂²×C₁₃	—	4	1	no	—
54	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
55	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
56	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
57	C₂×D₁₃	—	2	4	no	—
58	C₂×D₁₃	—	4	4	no	—
59	C₄×C₁₃	—	4	4	no	—
60	C₄×C₁₃	C₂×C₂²⋊C₄	1	12	yes	—
61	Dic₁₃	—	2	4	no	—
62	Dic₁₃	—	4	4	no	—
63	2₋¹⁺⁴	—	26	4	no	—
64	2₋¹⁺⁴	—	26	4	no	—
65	C₂×#41	—	13	4	no	—
66	C₂×#41	—	13	6	no	—
67	C₂²×Q₈	—	13	4	no	—
68	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
69	C₄×Q₈	—	2	4	no	—
70	C₄×Q₈	—	26	4	no	—
71	#117	—	13	6	no	—
72	#117	C₂×D₁₃	1	6	yes	—
73	#99	—	13	16	no	—
74	C₂×C₁₃	—	2	1	no	—
75	C₂×C₁₃	—	2	1	no	—
76	C₂×C₁₃	—	2	1	no	—
77	C₂×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
78	D₁₃	—	4	4	no	—
79	#41	—	26	16	no	—
80	#41	—	26	12	no	—
81	#41	—	26	12	no	—
82	C₂×Q₈	—	13	6	no	—
83	C₂×Q₈	—	26	16	no	—
84	C₂×Q₈	C₄×D₁₃	1	4	yes	—
85	C₂×Q₈	D₅₂	1	4	yes	—
86	C₂×Q₈	C₂²⋊D₁₃	1	4	yes	—
87	C₂×D₄	—	13	4	no	—
88	C₂×D₄	—	13	6	no	—
89	C₂²×C₄	—	13	12	no	—
90	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
91	C₄⋊C₄	—	2	6	no	—
92	C₄⋊C₄	—	26	6	no	—
93	C₂²⋊C₄	—	13	4	no	—
94	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
95	C₄²	—	2	6	no	—
96	C₄²	—	26	6	no	—
97	C₁₃	#2884	1	1	yes	characteristic, complemented, Sylow
98	C₂³	—	13	4	no	—
99	C₂³	C₂³×D₁₃	1	1	yes	characteristic
100	Q₈	—	26	12	no	—
101	Q₈	#6103	1	16	yes	—
102	D₄	—	26	4	no	—
103	D₄	—	26	4	no	—
104	D₄	—	26	12	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	13	4	no	—
107	C₂×C₄	—	26	24	no	—
108	C₂×C₄	—	26	4	no	—
109	C₂×C₄	C₂×C₄×D₁₃	1	6	yes	—
110	C₂×C₄	C₂×D₅₂	1	6	yes	—
111	C₂×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
112	C₂²	—	4	1	no	—
113	C₂²	—	26	4	no	—
114	C₂²	—	52	4	no	—
115	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
116	C₂²	C₂²×D₅₂	1	1	yes	characteristic
117	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	26	4	no	—
120	C₄	—	52	4	no	—
121	C₄	C₂×#6103	1	12	yes	—
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	52	4	no	—
126	C₂	C₂²×#6103	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88254

C₂

yes

maximal, characteristic

C₁₃×#519

C₂

yes

maximal, characteristic, complemented

#11719517

C₂

yes

maximal, characteristic, complemented

#11718667

C₂

yes

maximal, complemented

C₂²×Q₈×C₁₃

C₂²

yes

—

—

C₄

yes

complemented

C₂×Q₈×D₁₃

C₄

yes

complemented

C₂×#6127

C₂²

yes

—

C₄×Q₈×C₁₃

—

C₁₃×#117

C₂²

yes

—

Q₈×Dic₁₃

—

#88180

C₂²

yes

—

#88046

C₂²

yes

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×Q₈×C₁₃

D₄

yes

—

C₂×Q₈×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂×C₂²⋊D₁₃

C₂³

yes

—

#6131

—

Q₈×D₁₃

—

#6127

—

#6127

—

C₂×D₅₂

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂×C₄

yes

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

C₂³

yes

—

C₄²×C₁₃

—

C₂²⋊Dic₁₃

C₂³

yes

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#2884

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

C₂²×C₄

yes

—

Q₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂²⋊D₁₃

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²×C₄

yes

—

D₅₂

—

C₄×D₁₃

—

Dic₂₆

—

C₂×2₋¹⁺⁴

—

#519

—

#519

D₁₃

yes

characteristic, complemented

#410

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₂²⋊C₄

yes

—

Dic₁₃

—

Dic₁₃

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₁₃

yes

—

#99

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂²×C₂²⋊C₄

yes

characteristic

D₁₃

—

#41

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₁₃

yes

—

C₂×Q₈

D₅₂

yes

—

C₂×Q₈

C₂²⋊D₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂²×D₁₃

yes

—

C₄²

—

C₄²

—

C₁₃

#2884

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

C₂³×D₁₃

yes

characteristic

100

Q₈

—

101

Q₈

#6103

yes

—

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×C₄×D₁₃

yes

—

110

C₂×C₄

C₂×D₅₂

yes

—

111

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂²×C₄×D₁₃

yes

characteristic

116

C₂²

C₂²×D₅₂

yes

characteristic

117

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×#6103

yes

—

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂²×#6103

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#6103
commutator	C₂²×C₁₃	C₂³×C₄
Frattini	C₂³	C₂³×D₁₃
Fitting	C₁₃×#519	C₂
socle	C₂×C₁₃	C₂²×C₂²⋊C₄
prime radical	C₂×#88254	C₂
cosocle	C₂³×C₁₃	C₂⁴
hypercenter	#519	D₁₃
hypocenter	C₁₃	#2884
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

subgroup

G/H

center

C₂

C₂²×#6103

commutator

C₂²×C₁₃

C₂³×C₄

Frattini

C₂³

C₂³×D₁₃

Fitting

C₁₃×#519

C₂

socle

C₂×C₁₃

C₂²×C₂²⋊C₄

prime radical

C₂×#88254

C₂

cosocle

C₂³×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

N	H ≅ G/N	type
2. C₁₃×#519	C₂	semidirect
3. #11719517	C₂	semidirect
4. #11718667	C₂	semidirect
8. C₂×#6131	C₄	semidirect
9. C₂×Q₈×D₁₃	C₄	semidirect
51. #519	D₁₃	semidirect
97. C₁₃	#2884	semidirect

H ≅ G/N

type

2. C₁₃×#519

C₂

semidirect

3. #11719517

C₂

semidirect

4. #11718667

C₂

semidirect

8. C₂×#6131

C₄

semidirect

9. C₂×Q₈×D₁₃

C₄

semidirect

51. #519

D₁₃

semidirect

97. C₁₃

#2884

semidirect

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

subgroup	index
C₂×#88254	2
C₁₃×#519	2
#11719517	2
#11718667	2
#2884	13

subgroup

index

C₁₃×#519

Subgroups — #13123299

Conjugacy classes of subgroups

Complements

Maximal subgroups