Group #13116400 — subgroups

Subgroups — #13116400

Order 1664.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#11718596	C₂	1	1	yes	maximal, characteristic
2	#11718589	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#336	C₂	1	1	yes	maximal, characteristic
4	#11718143	C₂	1	2	yes	maximal
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	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6101	—	2	1	no	—
14	#88071	—	2	1	no	—
15	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₃	—	2	1	no	—
19	C₂²×Dic₁₃	—	2	1	no	—
20	C₂²×Dic₁₃	—	2	2	no	—
21	C₂×C₈×C₁₃	—	2	2	no	—
22	C₂×C₈×C₁₃	—	2	2	no	—
23	C₂×C₈×C₁₃	—	2	2	no	—
24	C₄²×C₁₃	—	2	2	no	—
25	C₄⋊Dic₁₃	—	2	1	no	—
26	C₄⋊Dic₁₃	—	2	4	no	—
27	C₄⋊Dic₁₃	—	2	2	no	—
28	C₄⋊Dic₁₃	—	4	2	no	—
29	C₄⋊Dic₁₃	—	4	2	no	—
30	C₄⋊Dic₁₃	Q₈	1	2	yes	—
31	#6101	—	4	2	no	—
32	#2100	—	13	1	no	maximal, Sylow
33	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃	—	2	1	no	—
35	C₂×C₄×C₁₃	—	2	1	no	—
36	C₂×C₄×C₁₃	—	2	1	no	—
37	C₂×C₄×C₁₃	—	2	1	no	—
38	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
39	C₂×C₄×C₁₃	Q₁₆	1	2	yes	—
40	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₃	#41	1	2	yes	—
42	C₂×C₄×C₁₃	#41	1	2	yes	—
43	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
44	C₂×Dic₁₃	—	2	2	no	—
45	C₂×Dic₁₃	—	4	1	no	—
46	C₂×Dic₁₃	—	4	1	no	—
47	C₂×Dic₁₃	—	4	1	no	—
48	C₂×Dic₁₃	—	4	2	no	—
49	C₂×Dic₁₃	—	4	1	no	—
50	C₂×Dic₁₃	—	4	2	no	—
51	C₂×Dic₁₃	—	4	2	no	—
52	C₂×Dic₁₃	—	4	1	no	—
53	C₈×C₁₃	—	4	2	no	—
54	#389	—	13	1	no	—
55	#384	—	13	1	no	—
56	#340	—	13	2	no	—
57	#340	—	13	2	no	—
58	#336	D₁₃	1	1	yes	characteristic
59	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₁₃	#123	1	1	yes	characteristic
61	C₂²×C₁₃	#124	1	1	yes	characteristic
62	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂²×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₄×C₁₃	—	2	2	no	—
67	C₄×C₁₃	—	2	2	no	—
68	C₄×C₁₃	#125	1	2	yes	—
69	C₄×C₁₃	#126	1	2	yes	—
70	Dic₁₃	—	4	2	no	—
71	Dic₁₃	—	8	1	no	—
72	Dic₁₃	—	8	2	no	—
73	C₂²×C₈	C₂×D₁₃	1	2	yes	—
74	C₂×C₄⋊C₄	—	13	1	no	—
75	C₂×C₄⋊C₄	—	13	2	no	—
76	C₂×C₄⋊C₄	—	13	1	no	—
77	C₂×C₄⋊C₄	—	26	1	no	—
78	C₂×C₄⋊C₄	—	26	1	no	—
79	C₂×C₄⋊C₄	—	26	1	no	—
80	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
81	#95	—	26	1	no	—
82	C₂×C₁₃	#400	1	1	yes	characteristic
83	C₂×C₁₃	#476	1	1	yes	characteristic
84	C₂×C₁₃	#477	1	1	yes	characteristic
85	C₂×C₁₃	#481	1	1	yes	characteristic
86	C₂×C₁₃	#484	1	1	yes	characteristic
87	C₂×C₁₃	#487	1	1	yes	characteristic
88	C₂×C₁₃	#490	1	1	yes	characteristic
89	C₂²×C₄	—	26	1	no	—
90	C₂²×C₄	—	26	1	no	—
91	C₂²×C₄	—	26	2	no	—
92	C₂²×C₄	D₅₂	1	1	yes	characteristic
93	C₂²×C₄	D₅₂	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₄⋊C₄	—	13	2	no	—
99	C₄⋊C₄	—	26	1	no	—
100	C₄⋊C₄	—	26	4	no	—
101	C₄⋊C₄	—	26	2	no	—
102	C₄⋊C₄	—	52	2	no	—
103	C₄⋊C₄	—	52	2	no	—
104	C₄⋊C₄	—	52	2	no	—
105	C₄²	—	2	2	no	—
106	C₁₃	#2100	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₂×D₅₂	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	26	2	no	—
113	C₂×C₄	—	52	1	no	—
114	C₂×C₄	—	52	1	no	—
115	C₂×C₄	—	52	2	no	—
116	C₂×C₄	—	52	1	no	—
117	C₂×C₄	—	52	1	no	—
118	C₂×C₄	—	52	2	no	—
119	C₂×C₄	—	52	2	no	—
120	C₂×C₄	—	52	1	no	—
121	C₂×C₄	#6095	1	2	yes	—
122	C₂×C₄	Dic₅₂	1	2	yes	—
123	C₂×C₄	#6127	1	2	yes	—
124	C₂×C₄	#6129	1	1	yes	characteristic
125	C₂×C₄	#6129	1	2	yes	—
126	C₂×C₄	Q₈×D₁₃	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	#88129	1	1	yes	characteristic
129	C₂²	#88140	1	1	yes	characteristic
130	C₂²	#88151	1	1	yes	characteristic
131	C₂²	C₂×#6095	1	1	yes	characteristic
132	C₂²	C₂×Dic₅₂	1	1	yes	characteristic
133	C₂²	#88162	1	1	yes	characteristic
134	C₂²	#88162	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	52	2	no	—
138	C₄	—	104	2	no	—
139	C₄	—	104	1	no	—
140	C₄	#88133	1	2	yes	—
141	C₄	#88143	1	2	yes	—
142	C₂	#11718320	1	1	yes	characteristic
143	C₂	#11718342	1	1	yes	characteristic
144	C₂	#11718374	1	1	yes	characteristic
145	C₂	#11718383	1	1	yes	characteristic
146	C₂	#11718396	1	1	yes	characteristic
147	C₂	#11718503	1	1	yes	characteristic
148	C₂	#11718511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11718596

C₂

yes

maximal, characteristic

#11718589

C₂

yes

maximal, characteristic

C₁₃×#336

C₂

yes

maximal, characteristic

#11718143

C₂

yes

maximal

#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

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×#6101

—

#88071

—

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

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

Q₈

yes

—

#6101

—

#2100

—

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

Q₁₆

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

#41

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

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃

C₂×Q₁₆

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#125

yes

—

C₄×C₁₃

#126

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

#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

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

—

106

C₁₃

#2100

yes

characteristic, complemented, Sylow

107

C₂³

C₂×D₅₂

yes

characteristic

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

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#6095

yes

—

122

C₂×C₄

Dic₅₂

yes

—

123

C₂×C₄

#6127

yes

—

124

C₂×C₄

#6129

yes

characteristic

125

C₂×C₄

#6129

yes

—

126

C₂×C₄

Q₈×D₁₃

yes

characteristic

127

C₈

—

128

C₂²

#88129

yes

characteristic

129

C₂²

#88140

yes

characteristic

130

C₂²

#88151

yes

characteristic

131

C₂²

C₂×#6095

yes

characteristic

132

C₂²

C₂×Dic₅₂

yes

characteristic

133

C₂²

#88162

yes

characteristic

134

C₂²

#88162

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

104

—

139

C₄

—

104

—

140

C₄

#88133

yes

—

141

C₄

#88143

yes

—

142

C₂

#11718320

yes

characteristic

143

C₂

#11718342

yes

characteristic

144

C₂

#11718374

yes

characteristic

145

C₂

#11718383

yes

characteristic

146

C₂

#11718396

yes

characteristic

147

C₂

#11718503

yes

characteristic

148

C₂

#11718511

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	#2100
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
106. C₁₃	#2100	semidirect

H ≅ G/N

type

106. C₁₃

#2100

semidirect

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

subgroup	index
#11718596	2
#11718589	2
C₁₃×#336	2
#11718143	2
#2100	13

subgroup

index

C₁₃×#336

Subgroups — #13116400

Conjugacy classes of subgroups

Complements

Maximal subgroups