Group #13116182 — subgroups

Subgroups — #13116182

Order 1664.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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	#11719205	C₂	1	1	yes	maximal, characteristic
2	#11718782	C₂	1	2	yes	maximal
3	#11718637	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#343	C₂	1	1	yes	maximal, characteristic
5	#11718147	C₂	1	2	yes	maximal
6	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	#88180	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
10	#88178	—	2	2	no	—
11	C₂×#6101	—	2	2	no	—
12	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
13	#88144	—	2	2	no	—
14	#88142	—	2	2	no	—
15	#88058	—	2	2	no	—
16	#88057	—	2	2	no	—
17	C₂²×C₄×C₁₃	D₄	1	2	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₃	—	2	1	no	—
20	C₂²×Dic₁₃	—	2	1	no	—
21	C₂×Dic₂₆	—	2	2	no	—
22	C₁₃×M₁₆	—	2	2	no	—
23	C₁₃×M₁₆	Q₈	1	4	yes	complemented
24	C₂×C₈×C₁₃	—	2	2	no	—
25	C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂²⋊Dic₁₃	—	2	2	no	—
27	C₄⋊Dic₁₃	—	2	2	no	—
28	C₄⋊Dic₁₃	—	2	2	no	—
29	C₄⋊Dic₁₃	D₄	1	2	yes	—
30	C₄⋊Dic₁₃	D₄	1	2	yes	—
31	#6101	—	2	4	no	—
32	#6101	—	4	4	no	—
33	C₄×Dic₁₃	—	2	2	no	—
34	C₄×Dic₁₃	—	2	1	no	—
35	C₄×Dic₁₃	Q₈	1	2	yes	complemented
36	#2060	—	13	1	no	maximal, Sylow
37	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	4	2	no	—
40	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
44	C₂×Dic₁₃	—	2	2	no	—
45	C₂×Dic₁₃	—	2	2	no	—
46	C₂×Dic₁₃	—	2	2	no	—
47	C₂×Dic₁₃	—	2	2	no	—
48	C₂×Dic₁₃	—	2	2	no	—
49	C₂×Dic₁₃	—	4	2	no	—
50	C₂×Dic₁₃	—	4	1	no	—
51	Dic₂₆	—	8	2	no	—
52	C₈×C₁₃	—	2	4	no	—
53	#544	—	13	1	no	—
54	#428	—	13	2	no	—
55	#384	—	13	1	no	—
56	#343	D₁₃	1	1	yes	characteristic
57	#339	—	13	2	no	—
58	C₂²×C₁₃	—	2	1	no	—
59	C₂²×C₁₃	—	2	1	no	—
60	C₂²×C₁₃	#120	1	1	yes	characteristic
61	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₄×C₁₃	—	4	2	no	—
64	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
66	Dic₁₃	—	2	4	no	—
67	Dic₁₃	—	4	2	no	—
68	Dic₁₃	—	4	2	no	—
69	Dic₁₃	—	8	1	no	—
70	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
71	C₄⋊Q₈	—	26	2	no	—
72	#125	—	26	2	no	—
73	C₂²⋊Q₈	—	26	2	no	—
74	#117	—	13	2	no	—
75	C₂×C₄⋊C₄	—	13	1	no	—
76	C₂×C₄⋊C₄	—	26	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
78	C₂×C₄²	—	13	1	no	—
79	#107	—	26	2	no	—
80	#106	—	26	2	no	—
81	C₂×C₁₃	—	2	1	no	—
82	C₂×C₁₃	—	2	1	no	—
83	C₂×C₁₃	#395	1	1	yes	characteristic
84	C₂×C₁₃	#453	1	1	yes	characteristic
85	C₂×C₁₃	#456	1	1	yes	characteristic
86	C₂×Q₈	—	26	2	no	—
87	C₂²×C₄	—	26	1	no	—
88	C₂²×C₄	—	26	1	no	—
89	C₂²×C₄	D₅₂	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
91	M₁₆	—	2	2	no	—
92	M₁₆	Dic₂₆	1	4	yes	complemented
93	C₂×C₈	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	13	2	no	—
96	C₄⋊C₄	—	13	2	no	—
97	C₄⋊C₄	—	26	2	no	—
98	C₄⋊C₄	—	26	4	no	—
99	C₄⋊C₄	—	26	2	no	—
100	C₄⋊C₄	—	52	4	no	—
101	C₂²⋊C₄	—	26	2	no	—
102	C₄²	—	13	2	no	—
103	C₄²	—	26	2	no	—
104	C₄²	—	26	1	no	—
105	C₁₃	#2060	1	1	yes	characteristic, complemented, Sylow
106	C₂³	C₂×D₅₂	1	1	yes	characteristic
107	Q₈	—	104	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	26	2	no	—
111	C₂×C₄	—	26	2	no	—
112	C₂×C₄	—	26	2	no	—
113	C₂×C₄	—	26	2	no	—
114	C₂×C₄	—	26	2	no	—
115	C₂×C₄	—	52	2	no	—
116	C₂×C₄	—	52	1	no	—
117	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
118	C₂×C₄	D₄×D₁₃	1	2	yes	—
119	C₂×C₄	#6129	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₁₃	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#88123	1	1	yes	characteristic
125	C₂²	#88136	1	1	yes	characteristic
126	C₂²	#88151	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	26	4	no	—
129	C₄	—	52	2	no	—
130	C₄	—	52	2	no	—
131	C₄	—	104	1	no	—
132	C₄	#88132	1	2	yes	—
133	C₄	#88142	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#11718317	1	1	yes	characteristic
137	C₂	#11718408	1	1	yes	characteristic
138	C₂	#11718413	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719205

C₂

yes

maximal, characteristic

#11718782

C₂

yes

maximal

#11718637

C₂

yes

maximal, characteristic

C₁₃×#343

C₂

yes

maximal, characteristic

#11718147

C₂

yes

maximal

C₂×C₁₃×M₁₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

#88180

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

#88178

—

C₂×#6101

—

C₂×C₄×Dic₁₃

C₂²

yes

—

—

—

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

Q₈

yes

complemented

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

C₄⋊Dic₁₃

D₄

yes

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

Q₈

yes

complemented

#2060

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

—

—

—

D₁₃

yes

characteristic

#339

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#120

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃

C₄⋊Q₈

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×M₁₆

C₂×D₁₃

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄²

—

#107

—

#106

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₅₂

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

M₁₆

Dic₂₆

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₁₃

#2060

yes

characteristic, complemented, Sylow

106

C₂³

C₂×D₅₂

yes

characteristic

107

Q₈

—

104

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×Dic₂₆

yes

—

118

C₂×C₄

D₄×D₁₃

yes

—

119

C₂×C₄

#6129

yes

characteristic

120

C₂×C₄

Q₈×D₁₃

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#88123

yes

characteristic

125

C₂²

#88136

yes

characteristic

126

C₂²

#88151

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

104

—

132

C₄

#88132

yes

—

133

C₄

#88142

yes

—

134

C₂

—

135

C₂

—

136

C₂

#11718317

yes

characteristic

137

C₂

#11718408

yes

characteristic

138

C₂

#11718413

yes

characteristic

	subgroup	G/H
center	C₂²	#88136
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#343	C₂
socle	C₂²×C₁₃	#120
prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#343	D₁₃
hypocenter	C₁₃	#2060
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#88136

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#343

C₂

socle

C₂²×C₁₃

#120

prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
23. C₁₃×M₁₆	Q₈	semidirect
35. C₄×Dic₁₃	Q₈	semidirect
92. M₁₆	Dic₂₆	semidirect
105. C₁₃	#2060	semidirect

H ≅ G/N

type

23. C₁₃×M₁₆

Q₈

semidirect

35. C₄×Dic₁₃

Q₈

semidirect

92. M₁₆

Dic₂₆

semidirect

105. C₁₃

#2060

semidirect

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

subgroup	index
#11719205	2
#11718782	2
#11718637	2
C₁₃×#343	2
#11718147	2
#2060	13

subgroup

index

C₁₃×#343

Subgroups — #13116182

Conjugacy classes of subgroups

Complements

Maximal subgroups