Group #15753374 — subgroups

Subgroups — #15753374

Order 3328.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#13114587	C₂	1	1	yes	maximal, characteristic
2	#13113128	C₂	1	1	yes	maximal, characteristic
3	#13112666	C₂	1	2	yes	maximal
4	#13112664	C₂	1	2	yes	maximal
5	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#431	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718593	C₂²	1	1	yes	characteristic
9	#11718586	C₂²	1	2	yes	—
10	C₂×#88043	C₂²	1	1	yes	characteristic
11	C₁₃×#322	C₄	1	2	yes	complemented
12	C₈²×C₁₃	C₄	1	2	yes	complemented
13	#11718142	C₄	1	4	yes	—
14	#11718046	C₄	1	4	yes	complemented
15	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
18	C₂²×#1055	C₂×C₄	1	2	yes	—
19	C₁₃×C₄⋊C₈	—	2	2	no	—
20	C₁₃×C₂²⋊C₈	—	2	2	no	—
21	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	#88071	—	2	2	no	—
26	C₂²⋊#1055	—	2	2	no	—
27	C₄⋊#1055	—	2	2	no	—
28	#88043	—	2	2	no	—
29	C₄×#1055	C₂×C₄	1	4	yes	—
30	#8141	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	#41	1	2	yes	—
33	C₂²×Dic₁₃	—	2	2	no	—
34	C₂×C₈×C₁₃	—	2	4	no	—
35	C₂×C₈×C₁₃	—	2	4	no	—
36	C₂×C₈×C₁₃	C₄²	1	2	yes	—
37	C₂×C₈×C₁₃	C₄²	1	2	yes	—
38	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₁₃	#41	1	2	yes	—
41	C₄×Dic₁₃	—	2	4	no	—
42	C₂×#1055	—	2	4	no	—
43	C₂×#1055	—	2	2	no	—
44	C₂×#1055	—	2	4	no	—
45	#1746	—	13	1	no	—
46	#1452	—	13	1	no	—
47	#1450	D₁₃	1	1	yes	characteristic
48	#1275	—	13	2	no	—
49	#1274	—	13	2	no	—
50	C₂³×C₁₃	#117	1	1	yes	characteristic
51	C₂×C₄×C₁₃	—	4	2	no	—
52	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₁₃	#117	1	2	yes	—
54	C₂×C₄×C₁₃	#117	1	2	yes	—
55	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#131	1	4	yes	—
57	C₂×Dic₁₃	—	4	4	no	—
58	C₂×Dic₁₃	—	4	2	no	—
59	C₈×C₁₃	—	2	2	no	—
60	C₈×C₁₃	—	2	2	no	—
61	C₈×C₁₃	—	4	4	no	—
62	#1055	—	4	4	no	—
63	#1055	—	4	2	no	—
64	#431	—	13	2	no	—
65	#431	C₂×D₁₃	1	2	yes	—
66	C₂×#97	—	13	1	no	—
67	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
68	#376	—	13	1	no	—
69	#336	—	13	4	no	—
70	#322	—	13	4	no	—
71	#322	Dic₁₃	1	2	yes	complemented
72	C₈²	Dic₁₃	1	2	yes	complemented
73	C₂²×C₁₃	—	2	2	no	—
74	C₂²×C₁₃	#376	1	1	yes	characteristic
75	C₂²×C₁₃	#433	1	2	yes	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	#405	1	2	yes	—
79	C₄×C₁₃	#433	1	2	yes	—
80	Dic₁₃	—	8	2	no	—
81	C₂²×C₈	—	13	2	no	—
82	C₂²×C₈	C₄×D₁₃	1	2	yes	—
83	C₂×C₄²	—	13	2	no	—
84	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
85	C₄⋊C₈	—	2	2	no	—
86	C₄⋊C₈	—	26	2	no	—
87	C₂²⋊C₈	—	2	2	no	—
88	C₂²⋊C₈	—	26	2	no	—
89	#97	—	26	2	no	—
90	C₄×C₈	—	13	4	no	—
91	C₄×C₈	C₄×D₁₃	1	2	yes	—
92	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
93	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
94	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
95	#95	—	26	2	no	—
96	C₂×C₁₃	—	2	2	no	—
97	C₂×C₁₃	#1746	1	1	yes	characteristic
98	C₂×C₁₃	#1747	1	2	yes	—
99	C₂²×C₄	—	26	2	no	—
100	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
101	C₂²×C₄	#6127	1	2	yes	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	26	4	no	—
105	C₂×C₈	—	26	2	no	—
106	C₂×C₈	—	26	4	no	—
107	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
108	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
109	C₄²	—	26	4	no	—
110	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
111	C₄²	#6127	1	2	yes	—
112	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
113	C₁₃	#8141	1	1	yes	characteristic, complemented, Sylow
114	C₂³	#88126	1	1	yes	characteristic
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	52	4	no	—
117	C₂×C₄	—	52	2	no	—
118	C₂×C₄	#88126	1	2	yes	—
119	C₂×C₄	#88126	1	1	yes	characteristic
120	C₂×C₄	#88155	1	4	yes	—
121	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
122	C₂×C₄	#88180	1	2	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	4	no	—
126	C₈	—	52	4	no	—
127	C₈	—	52	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	#11718351	1	2	yes	—
130	C₂²	#11718593	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	104	2	no	—
134	C₄	#11718351	1	2	yes	—
135	C₄	#11718762	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#13114587	1	1	yes	characteristic
138	C₂	#13114591	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114587

C₂

yes

maximal, characteristic

#13113128

C₂

yes

maximal, characteristic

#13112666

C₂

yes

maximal

#13112664

C₂

yes

maximal

C₁₃×#1450

C₂

yes

maximal, characteristic

C₁₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718593

C₂²

yes

characteristic

#11718586

C₂²

yes

—

C₂×#88043

C₂²

yes

characteristic

C₁₃×#322

C₄

yes

complemented

C₈²×C₁₃

C₄

yes

complemented

#11718142

C₄

yes

—

#11718046

C₄

yes

complemented

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂²×#1055

C₂×C₄

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88071

—

C₂²⋊#1055

—

C₄⋊#1055

—

#88043

—

C₄×#1055

C₂×C₄

yes

—

#8141

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

—

C₂×C₈×C₁₃

C₄²

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄×Dic₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic

#1275

—

#1274

—

C₂³×C₁₃

#117

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#117

yes

characteristic

C₂×C₄×C₁₃

#131

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

C₂×D₁₃

yes

—

C₂×#97

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

Dic₁₃

yes

complemented

C₈²

Dic₁₃

yes

complemented

C₂²×C₁₃

—

C₂²×C₁₃

#376

yes

characteristic

C₂²×C₁₃

#433

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#405

yes

—

C₄×C₁₃

#433

yes

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

C₄×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₄×D₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

#95

—

C₂×C₁₃

—

C₂×C₁₃

#1746

yes

characteristic

C₂×C₁₃

#1747

yes

—

C₂²×C₄

—

100

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

101

C₂²×C₄

#6127

yes

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₄×Dic₁₃

yes

—

108

C₂×C₈

C₄×Dic₁₃

yes

—

109

C₄²

—

110

C₄²

C₂×C₄×D₁₃

yes

characteristic

111

C₄²

#6127

yes

—

112

C₄²

C₂²×Dic₁₃

yes

characteristic

113

C₁₃

#8141

yes

characteristic, complemented, Sylow

114

C₂³

#88126

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#88126

yes

—

119

C₂×C₄

#88126

yes

characteristic

120

C₂×C₄

#88155

yes

—

121

C₂×C₄

C₂×C₄×Dic₁₃

yes

characteristic

122

C₂×C₄

#88180

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

#11718351

yes

—

130

C₂²

#11718593

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

104

—

134

C₄

#11718351

yes

—

135

C₄

#11718762

yes

—

136

C₂

—

137

C₂

#13114587

yes

characteristic

138

C₂

#13114591

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1450	C₂
socle	C₂²×C₁₃	#376
prime radical	C₂³×C₁₃	#117
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1450	D₁₃
hypocenter	C₁₃	#8141
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #1450

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₁₃

commutator

C₂×C₄×C₁₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1450

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #1450

N	H ≅ G/N	type
11. C₁₃×#322	C₄	semidirect
12. C₈²×C₁₃	C₄	semidirect
14. #11718046	C₄	semidirect
71. #322	Dic₁₃	semidirect
72. C₈²	Dic₁₃	semidirect
113. C₁₃	#8141	semidirect

H ≅ G/N

type

11. C₁₃×#322

C₄

semidirect

12. C₈²×C₁₃

C₄

semidirect

14. #11718046

C₄

semidirect

71. #322

Dic₁₃

semidirect

72. C₈²

Dic₁₃

semidirect

113. C₁₃

#8141

semidirect

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

subgroup	index
#13114587	2
#13113128	2
#13112666	2
#13112664	2
C₁₃×#1450	2
#8141	13

subgroup

index

C₁₃×#1450

Subgroups — #15753374

Conjugacy classes of subgroups

Complements

Maximal subgroups