Group #13116254 — subgroups

Subgroups — #13116254

Order 1664.

222 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 81 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×#391	C₂	1	1	yes	maximal, characteristic
2	C₂×#88075	C₂	1	2	yes	maximal
3	#11718597	C₂	1	1	yes	maximal, characteristic
4	#11718121	C₂	1	2	yes	maximal
5	#11718116	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	C₂²×#1055	C₂²	1	2	yes	—
11	C₁₃×#95	—	2	2	no	—
12	#88075	—	2	4	no	—
13	#88071	—	2	2	no	—
14	C₂×Q₈×C₁₃	—	2	1	no	—
15	C₂×Q₈×C₁₃	—	2	2	no	—
16	C₂×Q₈×C₁₃	—	2	2	no	—
17	C₂×Q₈×C₁₃	D₄	1	2	yes	—
18	C₂²×C₄×C₁₃	—	2	1	no	—
19	C₂²×C₄×C₁₃	—	2	1	no	—
20	C₂²×C₄×C₁₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₃	—	2	2	no	—
23	C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₄²×C₁₃	—	2	2	no	—
25	C₄⋊Dic₁₃	—	2	2	no	—
26	C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×#1055	—	2	2	no	—
28	C₂×#1055	—	2	2	no	—
29	C₂×#1055	—	2	2	no	—
30	#2074	—	13	1	no	maximal, Sylow
31	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₁₃	—	2	4	no	—
33	Q₈×C₁₃	—	4	2	no	—
34	C₂×C₄×C₁₃	—	2	2	no	—
35	C₂×C₄×C₁₃	—	2	2	no	—
36	C₂×C₄×C₁₃	—	2	2	no	—
37	C₂×C₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	—	4	1	no	—
39	C₂×C₄×C₁₃	—	4	1	no	—
40	C₂×C₄×C₁₃	—	4	1	no	—
41	C₂×C₄×C₁₃	—	4	2	no	—
42	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	#41	1	2	yes	—
44	C₂×C₄×C₁₃	#41	1	2	yes	—
45	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
46	C₂×Dic₁₃	—	4	2	no	—
47	C₂×Dic₁₃	—	4	2	no	—
48	C₂×Dic₁₃	—	4	2	no	—
49	#1055	—	4	2	no	—
50	C₂×#103	—	13	2	no	—
51	#391	D₁₃	1	1	yes	characteristic
52	#382	—	13	1	no	—
53	#340	—	13	2	no	—
54	#336	—	13	1	no	—
55	C₂²×C₁₃	#121	1	1	yes	characteristic
56	C₂²×C₁₃	#123	1	1	yes	characteristic
57	C₂²×C₁₃	#124	1	1	yes	characteristic
58	C₂²×C₁₃	#135	1	2	yes	—
59	C₂²×C₁₃	#137	1	2	yes	—
60	C₄×C₁₃	—	4	2	no	—
61	C₄×C₁₃	—	4	2	no	—
62	C₄×C₁₃	—	8	1	no	—
63	C₄×C₁₃	#123	1	2	yes	—
64	C₄×C₁₃	#126	1	2	yes	—
65	Dic₁₃	—	8	2	no	—
66	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
67	C₂²×C₈	—	13	2	no	—
68	C₂×C₄⋊C₄	—	13	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
71	#103	—	26	4	no	—
72	#95	—	2	2	no	—
73	#95	—	26	2	no	—
74	C₂×C₁₃	#397	1	1	yes	characteristic
75	C₂×C₁₃	#464	1	2	yes	—
76	C₂×C₁₃	#485	1	2	yes	—
77	C₂×C₁₃	#488	1	1	yes	characteristic
78	C₂×C₁₃	#491	1	1	yes	characteristic
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	D₅₂	1	2	yes	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	26	2	no	—
86	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
88	C₂×C₈	—	26	2	no	—
89	C₂×C₈	—	26	2	no	—
90	C₂×C₈	—	26	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	26	2	no	—
93	C₄⋊C₄	—	26	2	no	—
94	C₄²	—	2	2	no	—
95	C₁₃	#2074	1	1	yes	characteristic, complemented, Sylow
96	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
97	Q₈	—	2	4	no	—
98	Q₈	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	52	2	no	—
108	C₂×C₄	—	52	2	no	—
109	C₂×C₄	—	52	2	no	—
110	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
111	C₂×C₄	#6127	1	1	yes	characteristic
112	C₂×C₄	#6129	1	2	yes	—
113	C₂×C₄	#6129	1	2	yes	—
114	C₈	—	52	2	no	—
115	C₂²	#88184	1	1	yes	characteristic
116	C₂²	#88189	1	1	yes	characteristic
117	C₂²	#88190	1	1	yes	characteristic
118	C₂²	#88196	1	2	yes	—
119	C₂²	#88204	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	—	8	1	no	—
123	C₄	—	104	2	no	—
124	C₄	#88133	1	2	yes	—
125	C₄	#88140	1	2	yes	—
126	C₂	#11718614	1	1	yes	characteristic
127	C₂	#11718689	1	2	yes	—
128	C₂	#11718706	1	2	yes	—
129	C₂	#11718717	1	1	yes	characteristic
130	C₂	#11718743	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#391

C₂

yes

maximal, characteristic

C₂×#88075

C₂

yes

maximal

#11718597

C₂

yes

maximal, characteristic

#11718121

C₂

yes

maximal

#11718116

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂²×#1055

C₂²

yes

—

C₁₃×#95

—

#88075

—

#88071

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

C₂×#103

—

#391

D₁₃

yes

—

—

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#137

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#123

yes

—

C₄×C₁₃

#126

yes

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

#397

yes

characteristic

C₂×C₁₃

#464

yes

—

C₂×C₁₃

#485

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₅₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₁₃

#2074

yes

characteristic, complemented, Sylow

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

Q₈

—

Q₈

—

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×D₅₂

yes

characteristic

111

C₂×C₄

#6127

yes

characteristic

112

C₂×C₄

#6129

yes

—

113

C₂×C₄

#6129

yes

—

114

C₈

—

115

C₂²

#88184

yes

characteristic

116

C₂²

#88189

yes

characteristic

117

C₂²

#88190

yes

characteristic

118

C₂²

#88196

yes

—

119

C₂²

#88204

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

104

—

124

C₄

#88133

yes

—

125

C₄

#88140

yes

—

126

C₂

#11718614

yes

characteristic

127

C₂

#11718689

yes

—

128

C₂

#11718706

yes

—

129

C₂

#11718717

yes

characteristic

130

C₂

#11718743

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#391	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#391	D₁₃
hypocenter	C₁₃	#2074
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#391

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
95. C₁₃	#2074	semidirect

H ≅ G/N

type

95. C₁₃

#2074

semidirect

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

subgroup	index
C₁₃×#391	2
C₂×#88075	2
#11718597	2
#11718121	2
#11718116	2
#2074	13

subgroup

index

C₁₃×#391

Subgroups — #13116254

Conjugacy classes of subgroups

Complements

Maximal subgroups