Group #13131804 — subgroups

Subgroups — #13131804

Order 1664.

360 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 198 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	D₁₃×#131	C₂	1	1	yes	maximal, characteristic
2	#11719162	C₂	1	2	yes	maximal, complemented
3	#11719096	C₂	1	6	yes	maximal, complemented
4	D₁₃⋊M₃₂	C₂	1	3	yes	maximal, complemented
5	C₂×#88098	C₂	1	3	yes	maximal, complemented
6	#41×D₁₃	C₄	1	1	yes	characteristic
7	C₁₃×#131	C₄	1	1	yes	characteristic
8	#88202	C₂²	1	1	yes	characteristic
9	#88161	C₂²	1	3	yes	—
10	M₁₆×D₁₃	C₄	1	3	yes	—
11	#88155	C₄	1	3	yes	—
12	C₂×C₈×D₁₃	C₂²	1	3	yes	—
13	#88106	C₂²	1	6	yes	complemented
14	C₂×#6092	C₂²	1	6	yes	complemented
15	#88098	C₂²	1	1	yes	characteristic
16	#88098	C₂²	1	3	yes	—
17	#88098	C₂²	1	6	yes	complemented
18	D₁₃⋊C₁₆	C₂²	1	6	yes	complemented
19	C₁₃×#41	C₂×C₄	1	1	yes	characteristic
20	#6131	C₈	1	1	yes	characteristic
21	Q₈×D₁₃	C₈	1	1	yes	characteristic
22	#6129	C₈	1	3	yes	—
23	D₄×D₁₃	C₈	1	3	yes	—
24	#6127	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₁₃	C₂×C₄	1	3	yes	—
26	C₁₃×M₁₆	C₂×C₄	1	3	yes	—
27	C₂×C₈×C₁₃	C₂×C₄	1	3	yes	—
28	#6099	C₂×C₄	1	3	yes	—
29	C₂×#1055	C₂³	1	3	yes	—
30	#6094	C₂×C₄	1	6	yes	—
31	C₈×D₁₃	C₂×C₄	1	3	yes	—
32	C₈×D₁₃	C₂×C₄	1	3	yes	—
33	C₈×D₁₃	C₂³	1	3	yes	—
34	C₈×D₁₃	C₂³	1	1	yes	characteristic
35	#6092	C₂³	1	6	yes	—
36	#6092	C₂³	1	2	yes	—
37	#3407	—	13	1	no	maximal, Sylow
38	C₂²×D₁₃	C₂×C₈	1	3	yes	—
39	Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
40	D₄×C₁₃	C₂×C₈	1	3	yes	—
41	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
42	C₂²⋊D₁₃	C₂×C₈	1	6	yes	—
43	C₂×Dic₁₃	C₂×C₈	1	3	yes	—
44	D₅₂	C₂×C₈	1	3	yes	—
45	C₄×D₁₃	C₂×C₈	1	3	yes	—
46	C₄×D₁₃	C₂×C₈	1	3	yes	—
47	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₄×D₁₃	C₂²×C₄	1	3	yes	—
49	Dic₂₆	C₂×C₈	1	3	yes	—
50	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₁₃	C₂²×C₄	1	3	yes	—
52	#1055	C₂²×C₄	1	3	yes	—
53	#1055	C₂⁴	1	1	yes	characteristic
54	C₂×#131	—	13	1	no	—
55	#504	—	13	6	no	—
56	#504	—	13	2	no	—
57	C₂×M₃₂	—	13	3	no	—
58	C₂×M₃₂	—	13	3	no	—
59	C₂²×C₁₃	C₂²×C₈	1	3	yes	—
60	C₂×D₁₃	—	4	3	no	—
61	C₂×D₁₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×D₁₃	C₂²×C₈	1	3	yes	—
63	C₄×C₁₃	C₂²×C₈	1	3	yes	—
64	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
65	Dic₁₃	C₂²×C₈	1	3	yes	—
66	Dic₁₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×#41	—	13	1	no	—
68	#131	—	13	1	no	—
69	#131	—	13	3	no	—
70	#131	—	13	3	no	—
71	#131	#261	1	1	yes	characteristic
72	C₂×M₁₆	—	13	3	no	—
73	C₂²×C₈	—	13	3	no	—
74	M₃₂	—	13	1	no	—
75	M₃₂	—	13	3	no	—
76	M₃₂	—	13	6	no	—
77	M₃₂	—	13	6	no	—
78	C₂×C₁₆	—	13	6	no	—
79	C₂×C₁₆	—	13	6	no	—
80	C₂×C₁₃	—	2	3	no	—
81	C₂×C₁₃	C₂³×C₈	1	1	yes	characteristic
82	D₁₃	—	2	1	no	—
83	D₁₃	—	2	3	no	—
84	#41	—	13	1	no	—
85	#41	—	13	3	no	—
86	#41	—	13	3	no	—
87	#41	C₂×#261	1	1	yes	characteristic
88	C₂×Q₈	—	13	1	no	—
89	C₂×D₄	—	13	3	no	—
90	C₂²×C₄	—	13	3	no	—
91	M₁₆	—	13	3	no	—
92	M₁₆	—	13	6	no	—
93	M₁₆	C₂×#261	1	3	yes	—
94	C₂×C₈	—	13	3	no	—
95	C₂×C₈	—	13	3	no	—
96	C₂×C₈	—	13	1	no	—
97	C₂×C₈	—	13	3	no	—
98	C₂×C₈	—	13	3	no	—
99	C₂×C₈	C₂×#261	1	3	yes	—
100	C₁₆	—	13	6	no	—
101	C₁₆	—	13	2	no	—
102	C₁₃	#3407	1	1	yes	characteristic, complemented, Sylow
103	C₂³	—	13	3	no	—
104	Q₈	—	13	3	no	—
105	Q₈	D₁₃⋊C₈	1	1	yes	characteristic
106	D₄	—	13	6	no	—
107	D₄	—	13	3	no	—
108	D₄	D₁₃⋊C₈	1	3	yes	—
109	C₂×C₄	—	13	3	no	—
110	C₂×C₄	—	13	3	no	—
111	C₂×C₄	—	13	3	no	—
112	C₂×C₄	—	13	1	no	—
113	C₂×C₄	—	13	3	no	—
114	C₂×C₄	C₂²×#261	1	3	yes	—
115	C₈	—	13	1	no	—
116	C₈	—	13	3	no	—
117	C₈	C₂²×#261	1	1	yes	characteristic
118	C₈	C₂²×#261	1	3	yes	—
119	C₂²	—	13	1	no	—
120	C₂²	—	13	3	no	—
121	C₂²	—	52	3	no	—
122	C₂²	C₂×D₁₃⋊C₈	1	3	yes	—
123	C₄	—	13	1	no	—
124	C₄	—	13	3	no	—
125	C₄	C₂×D₁₃⋊C₈	1	3	yes	—
126	C₄	C₂³×#261	1	1	yes	characteristic
127	C₂	—	2	3	no	—
128	C₂	—	26	1	no	—
129	C₂	—	26	3	no	—
130	C₂	C₂²×D₁₃⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₃×#131

C₂

yes

maximal, characteristic

#11719162

C₂

yes

maximal, complemented

#11719096

C₂

yes

maximal, complemented

D₁₃⋊M₃₂

C₂

yes

maximal, complemented

C₂×#88098

C₂

yes

maximal, complemented

#41×D₁₃

C₄

yes

characteristic

C₁₃×#131

C₄

yes

characteristic

#88202

C₂²

yes

characteristic

#88161

C₂²

yes

—

M₁₆×D₁₃

C₄

yes

—

#88155

C₄

yes

—

C₂×C₈×D₁₃

C₂²

yes

—

#88106

C₂²

yes

complemented

C₂×#6092

C₂²

yes

complemented

#88098

C₂²

yes

characteristic

#88098

C₂²

yes

—

#88098

C₂²

yes

complemented

D₁₃⋊C₁₆

C₂²

yes

complemented

C₁₃×#41

C₂×C₄

yes

characteristic

#6131

C₈

yes

characteristic

Q₈×D₁₃

C₈

yes

characteristic

#6129

C₈

yes

—

D₄×D₁₃

C₈

yes

—

#6127

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂×C₄

yes

—

C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

#6099

C₂×C₄

yes

—

C₂×#1055

C₂³

yes

—

#6094

C₂×C₄

yes

—

C₈×D₁₃

C₂×C₄

yes

—

C₈×D₁₃

C₂×C₄

yes

—

C₈×D₁₃

C₂³

yes

—

C₈×D₁₃

C₂³

yes

characteristic

#6092

C₂³

yes

—

#6092

C₂³

yes

—

#3407

—

maximal, Sylow

C₂²×D₁₃

C₂×C₈

yes

—

Q₈×C₁₃

C₂×C₈

yes

characteristic

D₄×C₁₃

C₂×C₈

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂²⋊D₁₃

C₂×C₈

yes

—

C₂×Dic₁₃

C₂×C₈

yes

—

D₅₂

C₂×C₈

yes

—

C₄×D₁₃

C₂×C₈

yes

—

C₄×D₁₃

C₂×C₈

yes

—

C₄×D₁₃

C₂²×C₄

yes

characteristic

C₄×D₁₃

C₂²×C₄

yes

—

Dic₂₆

C₂×C₈

yes

—

C₈×C₁₃

C₂²×C₄

yes

characteristic

C₈×C₁₃

C₂²×C₄

yes

—

#1055

C₂²×C₄

yes

—

#1055

C₂⁴

yes

—

—

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₃

C₂²×C₈

yes

—

C₂×D₁₃

—

C₂×D₁₃

C₂²×C₈

yes

characteristic

C₂×D₁₃

C₂²×C₈

yes

—

C₄×C₁₃

C₂²×C₈

yes

—

C₄×C₁₃

C₂³×C₄

yes

characteristic

Dic₁₃

C₂²×C₈

yes

—

Dic₁₃

C₂²×C₈

yes

characteristic

C₂×#41

—

—

—

—

yes

C₂×M₁₆

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₃

—

C₂×C₁₃

C₂³×C₈

yes

characteristic

D₁₃

—

D₁₃

—

—

—

—

yes

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

C₂×#261

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#261

yes

—

100

C₁₆

—

101

C₁₆

—

102

C₁₃

#3407

yes

characteristic, complemented, Sylow

103

C₂³

—

104

Q₈

—

105

Q₈

D₁₃⋊C₈

yes

characteristic

106

D₄

—

107

D₄

—

108

D₄

D₁₃⋊C₈

yes

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×#261

yes

—

115

C₈

—

116

C₈

—

117

C₈

C₂²×#261

yes

characteristic

118

C₈

C₂²×#261

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×D₁₃⋊C₈

yes

—

123

C₄

—

124

C₄

—

125

C₄

C₂×D₁₃⋊C₈

yes

—

126

C₄

C₂³×#261

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

C₂²×D₁₃⋊C₈

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#261
commutator, socle	C₂×C₁₃	C₂³×C₈
Fitting	C₁₃×#131	C₄
prime radical	#41×D₁₃	C₄
cosocle	#1055	C₂⁴
hypercenter	#131	#261
hypocenter	C₁₃	#3407
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #131

subgroup

G/H

center, Frattini

C₄

C₂³×#261

commutator, socle

C₂×C₁₃

C₂³×C₈

Fitting

C₁₃×#131

C₄

prime radical

#41×D₁₃

C₄

C₂⁴

C₁₃

G ▷ C₂×C₁₃ ▷ C₁₃

C₁ ◁ C₄ ◁ #131

N	H ≅ G/N	type
2. #11719162	C₂	semidirect
3. #11719096	C₂	semidirect
4. D₁₃⋊M₃₂	C₂	semidirect
5. C₂×#88098	C₂	semidirect
13. #88106	C₂²	semidirect
14. C₂×#6092	C₂²	semidirect
17. #88098	C₂²	semidirect
18. D₁₃⋊C₁₆	C₂²	semidirect
102. C₁₃	#3407	semidirect

H ≅ G/N

type

2. #11719162

C₂

semidirect

3. #11719096

C₂

semidirect

4. D₁₃⋊M₃₂

C₂

semidirect

5. C₂×#88098

C₂

semidirect

13. #88106

C₂²

semidirect

14. C₂×#6092

C₂²

semidirect

17. #88098

C₂²

semidirect

18. D₁₃⋊C₁₆

C₂²

semidirect

102. C₁₃

#3407

semidirect

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

subgroup	index
D₁₃×#131	2
#11719162	2
#11719096	2
D₁₃⋊M₃₂	2
C₂×#88098	2
#3407	13

subgroup

index

D₁₃×#131

D₁₃⋊M₃₂

Subgroups — #13131804

Conjugacy classes of subgroups

Complements

Maximal subgroups