Group #13116314 — subgroups

Subgroups — #13116314

Order 1664.

230 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 95 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	C₂×#88057	C₂	1	2	yes	maximal
2	#11718596	C₂	1	1	yes	maximal, characteristic
3	#11718589	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#336	C₂	1	1	yes	maximal, characteristic
5	#11718143	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃	—	2	1	no	—
9	C₂×C₄⋊Dic₁₃	—	2	1	no	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6101	—	2	1	no	—
14	#88071	—	2	1	no	—
15	#88057	—	2	4	no	—
16	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₃	—	2	1	no	—
20	C₂²×Dic₁₃	—	2	1	no	—
21	C₂²×Dic₁₃	—	2	2	no	—
22	C₂×C₈×C₁₃	—	2	2	no	—
23	C₂×C₈×C₁₃	—	2	2	no	—
24	C₂×C₈×C₁₃	Q₈	1	4	yes	—
25	C₄²×C₁₃	—	2	2	no	—
26	C₄⋊Dic₁₃	—	2	1	no	—
27	C₄⋊Dic₁₃	—	2	2	no	—
28	C₄⋊Dic₁₃	—	2	2	no	—
29	C₄⋊Dic₁₃	—	2	2	no	—
30	C₄⋊Dic₁₃	—	4	2	no	—
31	C₄⋊Dic₁₃	—	4	2	no	—
32	C₄⋊Dic₁₃	D₄	1	2	yes	—
33	#6101	—	4	2	no	—
34	#2084	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃	—	2	1	no	—
37	C₂×C₄×C₁₃	—	2	1	no	—
38	C₂×C₄×C₁₃	—	2	1	no	—
39	C₂×C₄×C₁₃	—	2	1	no	—
40	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
41	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₃	#41	1	2	yes	—
45	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
46	C₂×Dic₁₃	—	2	2	no	—
47	C₂×Dic₁₃	—	4	1	no	—
48	C₂×Dic₁₃	—	4	1	no	—
49	C₂×Dic₁₃	—	4	2	no	—
50	C₂×Dic₁₃	—	4	1	no	—
51	C₂×Dic₁₃	—	4	1	no	—
52	C₂×Dic₁₃	—	4	2	no	—
53	C₂×Dic₁₃	—	4	2	no	—
54	C₂×Dic₁₃	—	4	1	no	—
55	C₈×C₁₃	—	2	4	no	—
56	C₂×#106	—	13	2	no	—
57	#389	—	13	1	no	—
58	#384	—	13	1	no	—
59	#340	—	13	2	no	—
60	#336	D₁₃	1	1	yes	characteristic
61	C₂²×C₁₃	#121	1	1	yes	characteristic
62	C₂²×C₁₃	#123	1	1	yes	characteristic
63	C₂²×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂²×C₁₃	#137	1	1	yes	characteristic
68	C₄×C₁₃	—	2	2	no	—
69	C₄×C₁₃	—	2	2	no	—
70	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
71	C₄×C₁₃	#125	1	2	yes	—
72	Dic₁₃	—	4	2	no	—
73	Dic₁₃	—	8	2	no	—
74	Dic₁₃	—	8	1	no	—
75	C₂²×C₈	C₂×D₁₃	1	2	yes	—
76	C₂×C₄⋊C₄	—	13	1	no	—
77	C₂×C₄⋊C₄	—	13	1	no	—
78	C₂×C₄⋊C₄	—	13	2	no	—
79	C₂×C₄⋊C₄	—	26	1	no	—
80	C₂×C₄⋊C₄	—	26	1	no	—
81	C₂×C₄⋊C₄	—	26	1	no	—
82	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
83	#106	—	26	4	no	—
84	#95	—	26	1	no	—
85	C₂×C₁₃	#398	1	1	yes	characteristic
86	C₂×C₁₃	#460	1	1	yes	characteristic
87	C₂×C₁₃	#461	1	1	yes	characteristic
88	C₂×C₁₃	#481	1	1	yes	characteristic
89	C₂×C₁₃	#483	1	1	yes	characteristic
90	C₂×C₁₃	#498	1	1	yes	characteristic
91	C₂×C₁₃	#501	1	1	yes	characteristic
92	C₂²×C₄	—	26	1	no	—
93	C₂²×C₄	—	26	1	no	—
94	C₂²×C₄	—	26	2	no	—
95	C₂²×C₄	D₅₂	1	1	yes	characteristic
96	C₂²×C₄	D₅₂	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	Dic₂₆	1	4	yes	—
101	C₄⋊C₄	—	13	2	no	—
102	C₄⋊C₄	—	26	1	no	—
103	C₄⋊C₄	—	26	2	no	—
104	C₄⋊C₄	—	26	2	no	—
105	C₄⋊C₄	—	26	2	no	—
106	C₄⋊C₄	—	52	2	no	—
107	C₄⋊C₄	—	52	2	no	—
108	C₄⋊C₄	—	52	2	no	—
109	C₄²	—	2	2	no	—
110	C₁₃	#2084	1	1	yes	characteristic, complemented, Sylow
111	C₂³	C₂×D₅₂	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	26	2	no	—
117	C₂×C₄	—	52	1	no	—
118	C₂×C₄	—	52	1	no	—
119	C₂×C₄	—	52	1	no	—
120	C₂×C₄	—	52	2	no	—
121	C₂×C₄	—	52	2	no	—
122	C₂×C₄	—	52	1	no	—
123	C₂×C₄	—	52	2	no	—
124	C₂×C₄	—	52	1	no	—
125	C₂×C₄	#6095	1	2	yes	—
126	C₂×C₄	#6095	1	2	yes	—
127	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
128	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
129	C₂×C₄	#6129	1	2	yes	—
130	C₂×C₄	#6131	1	1	yes	characteristic
131	C₈	—	2	4	no	—
132	C₂²	#88123	1	1	yes	characteristic
133	C₂²	#88140	1	1	yes	characteristic
134	C₂²	#88149	1	1	yes	characteristic
135	C₂²	C₂×#6095	1	1	yes	characteristic
136	C₂²	C₂×#6095	1	1	yes	characteristic
137	C₂²	#88162	1	1	yes	characteristic
138	C₂²	#88163	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	52	2	no	—
142	C₄	—	104	2	no	—
143	C₄	—	104	1	no	—
144	C₄	#88132	1	2	yes	—
145	C₄	#88144	1	2	yes	—
146	C₂	#11718322	1	1	yes	characteristic
147	C₂	#11718341	1	1	yes	characteristic
148	C₂	#11718363	1	1	yes	characteristic
149	C₂	#11718382	1	1	yes	characteristic
150	C₂	#11718396	1	1	yes	characteristic
151	C₂	#11718502	1	1	yes	characteristic
152	C₂	#11718509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88057

C₂

yes

maximal

#11718596

C₂

yes

maximal, characteristic

#11718589

C₂

yes

maximal, characteristic

C₁₃×#336

C₂

yes

maximal, characteristic

#11718143

C₂

yes

maximal

C₂²×C₈×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

—

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

Q₈

yes

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

#6101

—

#2084

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

SD₁₆

yes

—

C₂×C₄×C₁₃

SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

—

—

—

—

D₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃

C₂×SD₁₆

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

#125

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#106

—

#95

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₅₂

yes

characteristic

C₂²×C₄

D₅₂

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₂₆

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₁₃

#2084

yes

characteristic, complemented, Sylow

111

C₂³

C₂×D₅₂

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#6095

yes

—

126

C₂×C₄

#6095

yes

—

127

C₂×C₄

C₂×Dic₂₆

yes

—

128

C₂×C₄

D₄×D₁₃

yes

characteristic

129

C₂×C₄

#6129

yes

—

130

C₂×C₄

#6131

yes

characteristic

131

C₈

—

132

C₂²

#88123

yes

characteristic

133

C₂²

#88140

yes

characteristic

134

C₂²

#88149

yes

characteristic

135

C₂²

C₂×#6095

yes

characteristic

136

C₂²

C₂×#6095

yes

characteristic

137

C₂²

#88162

yes

characteristic

138

C₂²

#88163

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

104

—

143

C₄

—

104

—

144

C₄

#88132

yes

—

145

C₄

#88144

yes

—

146

C₂

#11718322

yes

characteristic

147

C₂

#11718341

yes

characteristic

148

C₂

#11718363

yes

characteristic

149

C₂

#11718382

yes

characteristic

150

C₂

#11718396

yes

characteristic

151

C₂

#11718502

yes

characteristic

152

C₂

#11718509

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#336	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#336	D₁₃
hypocenter	C₁₃	#2084
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#336

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
110. C₁₃	#2084	semidirect

H ≅ G/N

type

110. C₁₃

#2084

semidirect

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

subgroup	index
C₂×#88057	2
#11718596	2
#11718589	2
C₁₃×#336	2
#11718143	2
#2084	13

subgroup

index

C₁₃×#336

Subgroups — #13116314

Conjugacy classes of subgroups

Complements

Maximal subgroups