Group #13130727 — subgroups

Subgroups — #13130727

Order 1664.

380 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₂²×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#1057	C₂	1	2	yes	maximal
3	#11718290	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×Dic₁₃	C₄	1	1	yes	characteristic
6	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
7	C₂²×#1057	C₂²	1	4	yes	—
8	C₂×C₄×Dic₁₃	—	2	4	no	—
9	C₂×C₄×Dic₁₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
11	C₂²⋊#1057	—	2	4	no	—
12	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₁₃	—	2	2	no	—
14	C₂²×C₄×C₁₃	—	2	4	no	—
15	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
16	C₂²×Dic₁₃	—	2	2	no	—
17	C₂²×Dic₁₃	—	2	2	no	—
18	C₂²×Dic₁₃	—	2	4	no	—
19	C₂²×Dic₁₃	C₈	1	4	yes	complemented
20	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
22	C₂²×Dic₁₃	D₄	1	4	yes	—
23	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
24	C₂×#1057	—	2	4	no	—
25	C₂×#1057	—	2	8	no	—
26	C₄×Dic₁₃	—	2	4	no	—
27	C₄×Dic₁₃	—	2	4	no	—
28	#1817	—	13	1	no	maximal, Sylow
29	C₂³×C₁₃	—	2	2	no	—
30	C₂³×C₁₃	—	2	2	no	—
31	C₂³×C₁₃	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₁₃	C₂×C₈	1	2	yes	—
33	C₂³×C₁₃	M₁₆	1	2	yes	—
34	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃	—	2	8	no	—
36	C₂×C₄×C₁₃	—	2	2	no	—
37	C₂×C₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	M₁₆	1	4	yes	—
39	C₂×Dic₁₃	—	2	8	no	—
40	C₂×Dic₁₃	—	2	2	no	—
41	C₂×Dic₁₃	—	2	2	no	—
42	C₂×Dic₁₃	—	2	2	no	—
43	C₂×Dic₁₃	—	2	8	no	—
44	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₁₃	C₂×C₈	1	4	yes	—
46	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
47	C₂×Dic₁₃	#41	1	4	yes	—
48	#1057	—	4	4	no	—
49	C₂²×C₄²	—	13	1	no	—
50	C₂×C₂²⋊C₈	—	13	2	no	—
51	#336	—	13	4	no	—
52	C₂²×C₁₃	—	2	2	no	—
53	C₂²×C₁₃	—	2	2	no	—
54	C₂²×C₁₃	—	2	4	no	—
55	C₂²×C₁₃	—	2	4	no	—
56	C₂²×C₁₃	C₂²⋊C₈	1	4	yes	—
57	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₃	#117	1	2	yes	—
59	C₂²×C₁₃	C₂²×C₈	1	1	yes	characteristic
60	C₂²×C₁₃	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
62	C₄×C₁₃	—	2	4	no	—
63	Dic₁₃	—	2	2	no	—
64	Dic₁₃	—	2	4	no	—
65	Dic₁₃	#123	1	4	yes	—
66	C₂³×C₄	—	13	1	no	—
67	C₂³×C₄	—	13	1	no	—
68	C₂³×C₄	#261	1	1	yes	characteristic
69	C₂²×C₈	—	13	4	no	—
70	C₂×C₄²	—	13	2	no	—
71	C₂×C₄²	—	13	2	no	—
72	C₂×C₄²	—	26	4	no	—
73	C₂²⋊C₈	—	26	4	no	—
74	C₂×C₁₃	—	2	4	no	—
75	C₂×C₁₃	#381	1	1	yes	characteristic
76	C₂×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₁₃	#407	1	1	yes	characteristic
78	C₂×C₁₃	#431	1	2	yes	—
79	C₂×C₁₃	#432	1	2	yes	—
80	C₂⁴	C₂×#261	1	1	yes	characteristic
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	2	4	no	—
83	C₂²×C₄	—	13	1	no	—
84	C₂²×C₄	—	13	4	no	—
85	C₂²×C₄	—	13	4	no	—
86	C₂²×C₄	—	13	2	no	—
87	C₂²×C₄	—	13	1	no	—
88	C₂²×C₄	—	26	4	no	—
89	C₂²×C₄	—	26	2	no	—
90	C₂²×C₄	—	26	2	no	—
91	C₂²×C₄	C₂×#261	1	2	yes	—
92	C₂×C₈	—	26	4	no	—
93	C₂×C₈	—	26	8	no	—
94	C₄²	—	26	4	no	—
95	C₄²	—	26	4	no	—
96	C₁₃	#1817	1	1	yes	characteristic, complemented, Sylow
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	D₁₃⋊C₈	1	2	yes	—
100	C₂³	#6118	1	2	yes	—
101	C₂³	C₂²⋊#261	1	2	yes	—
102	C₂³	C₂²×#261	1	1	yes	characteristic
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	13	4	no	—
107	C₂×C₄	—	13	2	no	—
108	C₂×C₄	—	13	4	no	—
109	C₂×C₄	—	13	2	no	—
110	C₂×C₄	—	26	8	no	—
111	C₂×C₄	—	26	8	no	—
112	C₂×C₄	—	26	2	no	—
113	C₂×C₄	—	26	2	no	—
114	C₂×C₄	—	26	2	no	—
115	C₂×C₄	#6122	1	4	yes	—
116	C₈	—	52	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	4	no	—
120	C₂²	—	2	2	no	—
121	C₂²	#88111	1	4	yes	—
122	C₂²	C₂×D₁₃⋊C₈	1	1	yes	characteristic
123	C₂²	C₂×#6118	1	1	yes	characteristic
124	C₂²	#88237	1	2	yes	—
125	C₂²	C₂×#6122	1	2	yes	—
126	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	13	4	no	—
129	C₄	—	26	4	no	—
130	C₄	—	26	2	no	—
131	C₂	—	2	4	no	—
132	C₂	C₂×#88111	1	1	yes	characteristic
133	C₂	#11719125	1	2	yes	—
134	C₂	#11719130	1	2	yes	—
135	C₂	#11719135	1	1	yes	characteristic
136	C₂	#11719169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×Dic₁₃

C₂

yes

maximal, characteristic

C₂×C₂²⋊#1057

C₂

yes

maximal

#11718290

C₂

yes

maximal, complemented

C₂³×C₄×C₁₃

C₄

yes

characteristic

C₂³×Dic₁₃

C₄

yes

characteristic

C₂³×Dic₁₃

C₂²

yes

characteristic

C₂²×#1057

C₂²

yes

—

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

C₄

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂²⋊#1057

—

C₂⁴×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₈

yes

complemented

C₂²×Dic₁₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

D₄

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂×#1057

—

C₂×#1057

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

#1817

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₄

yes

—

C₂³×C₁₃

C₂×C₈

yes

—

C₂³×C₁₃

M₁₆

yes

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

M₁₆

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₃

C₂×C₈

yes

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

#41

yes

—

#1057

—

C₂²×C₄²

—

C₂×C₂²⋊C₈

—

#336

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²⋊C₈

yes

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

#117

yes

—

C₂²×C₁₃

C₂²×C₈

yes

characteristic

C₂²×C₁₃

C₂×M₁₆

yes

characteristic

C₂²×C₁₃

C₂×M₁₆

yes

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#123

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

#261

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂×C₁₃

—

C₂×C₁₃

#381

yes

characteristic

C₂×C₁₃

C₂×C₂²⋊C₈

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×C₁₃

#432

yes

—

C₂⁴

C₂×#261

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#261

yes

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₁₃

#1817

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

D₁₃⋊C₈

yes

—

100

C₂³

#6118

yes

—

101

C₂³

C₂²⋊#261

yes

—

102

C₂³

C₂²×#261

yes

characteristic

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

—

114

C₂×C₄

—

115

C₂×C₄

#6122

yes

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#88111

yes

—

122

C₂²

C₂×D₁₃⋊C₈

yes

characteristic

123

C₂²

C₂×#6118

yes

characteristic

124

C₂²

#88237

yes

—

125

C₂²

C₂×#6122

yes

—

126

C₂²

C₂×C₂²⋊#261

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₂

—

132

C₂

C₂×#88111

yes

characteristic

133

C₂

#11719125

yes

—

134

C₂

#11719130

yes

—

135

C₂

#11719135

yes

characteristic

136

C₂

#11719169

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₈
Fitting	C₂³×C₄×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₄
prime radical	C₂⁴×C₁₃	C₂×C₄
cosocle	C₂²×Dic₁₃	C₂³
hypercenter	C₂³×C₄	#261
hypocenter	C₁₃	#1817
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#261

commutator

C₂²×C₁₃

C₂²×C₈

Fitting

C₂³×C₄×C₁₃

C₄

socle

C₂³×C₁₃

C₂²×C₄

prime radical

C₂⁴×C₁₃

C₂×C₄

cosocle

C₂²×Dic₁₃

C₂³

C₂³×C₄

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂³×C₄

N	H ≅ G/N	type
3. #11718290	C₂	semidirect
19. C₂²×Dic₁₃	C₈	semidirect
96. C₁₃	#1817	semidirect

H ≅ G/N

type

3. #11718290

C₂

semidirect

19. C₂²×Dic₁₃

C₈

semidirect

96. C₁₃

#1817

semidirect

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

subgroup	index
C₂²×C₄×Dic₁₃	2
C₂×C₂²⋊#1057	2
#11718290	2
#1817	13

subgroup

index

C₂²×C₄×Dic₁₃

C₂×C₂²⋊#1057

#11718290

#1817

Subgroups — #13130727

Conjugacy classes of subgroups

Complements

Maximal subgroups