Group #16786653 — subgroups

Subgroups — #16786653

Order 3328.

306 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 152 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	#13131195	C₂	1	2	yes	maximal
2	#13129751	C₂	1	2	yes	maximal, complemented
3	C₈×#6092	C₂	1	2	yes	maximal, complemented
4	C₈²×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₄×C₈×D₁₃	C₄	1	1	yes	characteristic
6	C₄×C₈×D₁₃	C₄	1	1	yes	characteristic
7	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	#11718260	C₄	1	4	yes	—
9	#11718259	C₄	1	4	yes	—
10	C₄×#6092	C₂²	1	4	yes	—
11	C₈²×C₁₃	C₄	1	1	yes	characteristic
12	C₈×#1055	C₄	1	1	yes	characteristic
13	C₈×#1055	C₂²	1	1	yes	characteristic
14	C₈×#1055	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₁₃	C₈	1	2	yes	—
16	C₂×C₈×D₁₃	C₈	1	2	yes	—
17	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
18	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×#6092	C₂×C₄	1	8	yes	—
20	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₁₃	C₈	1	2	yes	—
24	C₈×Dic₁₃	C₈	1	2	yes	—
25	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
26	C₄×#1055	C₂×C₄	1	1	yes	characteristic
27	C₄×#1055	C₂×C₄	1	1	yes	characteristic
28	C₄×#1055	C₂³	1	1	yes	characteristic
29	#11410	—	13	1	no	maximal, Sylow
30	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₄²	1	2	yes	—
34	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
36	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
38	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
39	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂×#1055	C₂×C₈	1	2	yes	—
41	C₂×#1055	C₂×C₈	1	2	yes	—
42	C₂×#1055	C₂²×C₄	1	2	yes	—
43	C₈×D₁₃	—	2	4	no	—
44	C₈×D₁₃	—	2	4	no	—
45	C₈×D₁₃	C₁₆	1	8	yes	complemented
46	#6092	—	2	8	no	—
47	#2162	—	13	2	no	—
48	C₂×C₈²	—	13	1	no	—
49	#1312	—	13	2	no	—
50	C₈×C₁₆	—	13	2	no	—
51	C₂²×D₁₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
56	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
57	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
58	C₄×D₁₃	—	2	2	no	—
59	C₄×D₁₃	—	2	2	no	—
60	C₄×D₁₃	C₂×C₁₆	1	4	yes	—
61	C₈×C₁₃	—	2	2	no	—
62	C₈×C₁₃	—	2	2	no	—
63	C₈×C₁₃	C₂×C₁₆	1	4	yes	—
64	#1055	C₂×C₁₆	1	4	yes	—
65	#1055	M₃₂	1	4	yes	—
66	#1055	#131	1	4	yes	—
67	C₂×C₄×C₈	—	13	1	no	—
68	C₂×C₄×C₈	—	13	1	no	—
69	C₂×C₄×C₈	—	13	1	no	—
70	C₄⋊C₁₆	—	13	4	no	—
71	C₂²⋊C₁₆	—	13	4	no	—
72	C₄×C₁₆	—	13	4	no	—
73	C₈²	—	13	1	no	—
74	C₈²	—	13	1	no	—
75	C₈²	—	13	1	no	—
76	C₈²	#261	1	1	yes	characteristic
77	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
78	C₂×D₁₃	—	2	1	no	—
79	C₂×D₁₃	—	2	1	no	—
80	C₂×D₁₃	C₄×C₁₆	1	2	yes	—
81	C₄×C₁₃	#405	1	2	yes	—
82	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
83	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
84	Dic₁₃	—	2	1	no	—
85	Dic₁₃	—	2	1	no	—
86	Dic₁₃	C₄×C₁₆	1	2	yes	—
87	C₂²×C₈	—	13	2	no	—
88	C₂²×C₈	—	13	2	no	—
89	C₂²×C₈	—	13	2	no	—
90	C₂×C₄²	—	13	1	no	—
91	C₂×C₁₆	—	13	8	no	—
92	C₄×C₈	—	13	2	no	—
93	C₄×C₈	—	13	2	no	—
94	C₄×C₈	—	13	2	no	—
95	C₄×C₈	—	13	1	no	—
96	C₄×C₈	—	13	1	no	—
97	C₄×C₈	—	13	1	no	—
98	C₄×C₈	C₂×#261	1	1	yes	characteristic
99	C₄×C₈	C₂×#261	1	1	yes	characteristic
100	C₄×C₈	C₂×#261	1	1	yes	characteristic
101	C₂×C₁₃	#1450	1	1	yes	characteristic
102	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
103	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
104	D₁₃	—	2	2	no	—
105	C₂²×C₄	—	13	1	no	—
106	C₂²×C₄	—	13	1	no	—
107	C₂²×C₄	—	13	1	no	—
108	C₂×C₈	—	13	2	no	—
109	C₂×C₈	—	13	2	no	—
110	C₂×C₈	—	13	2	no	—
111	C₂×C₈	—	13	8	no	—
112	C₂×C₈	—	26	4	no	—
113	C₂×C₈	—	26	4	no	—
114	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
115	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
116	C₂×C₈	C₄×#261	1	2	yes	—
117	C₄²	—	13	1	no	—
118	C₄²	—	13	1	no	—
119	C₄²	—	13	1	no	—
120	C₄²	C₂²×#261	1	1	yes	characteristic
121	C₁₆	—	26	8	no	—
122	C₁₃	#11410	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	13	1	no	—
124	C₂×C₄	—	13	1	no	—
125	C₂×C₄	—	13	4	no	—
126	C₂×C₄	—	13	1	no	—
127	C₂×C₄	—	13	1	no	—
128	C₂×C₄	—	26	2	no	—
129	C₂×C₄	—	26	2	no	—
130	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
131	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	13	4	no	—
136	C₈	—	13	4	no	—
137	C₈	—	13	4	no	—
138	C₈	D₁₃⋊C₁₆	1	4	yes	—
139	C₂²	—	13	2	no	—
140	C₂²	—	26	1	no	—
141	C₂²	—	26	1	no	—
142	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
143	C₄	—	13	2	no	—
144	C₄	—	26	1	no	—
145	C₄	—	26	1	no	—
146	C₄	C₂×D₁₃⋊C₁₆	1	2	yes	—
147	C₄	D₁₃⋊M₃₂	1	2	yes	—
148	C₄	#11719089	1	2	yes	—
149	C₂	—	26	2	no	—
150	C₂	#13130040	1	1	yes	characteristic
151	C₂	C₄×D₁₃⋊C₁₆	1	1	yes	characteristic
152	C₂	#13131043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131195

C₂

yes

maximal

#13129751

C₂

yes

maximal, complemented

C₈×#6092

C₂

yes

maximal, complemented

C₈²×D₁₃

C₂

yes

maximal, characteristic

C₄×C₈×D₁₃

C₄

yes

characteristic

C₄×C₈×D₁₃

C₄

yes

characteristic

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718260

C₄

yes

—

#11718259

C₄

yes

—

C₄×#6092

C₂²

yes

—

C₈²×C₁₃

C₄

yes

characteristic

C₈×#1055

C₄

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₂×C₈×D₁₃

C₈

yes

—

C₂×C₈×D₁₃

C₈

yes

—

C₂×C₈×D₁₃

C₂×C₄

yes

—

C₄²×D₁₃

C₂×C₄

yes

characteristic

C₂×#6092

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

C₈

yes

—

C₈×Dic₁₃

C₈

yes

—

C₈×Dic₁₃

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

C₄×#1055

C₂×C₄

yes

characteristic

C₄×#1055

C₂³

yes

characteristic

#11410

—

maximal, Sylow

C₂×C₄×D₁₃

C₄²

yes

characteristic

C₂×C₄×D₁₃

C₂×C₈

yes

characteristic

C₂×C₄×D₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₄²

yes

—

C₂×C₈×C₁₃

C₂×C₈

yes

—

C₂×C₈×C₁₃

C₂×C₈

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄×Dic₁₃

C₄²

yes

characteristic

C₄×Dic₁₃

C₂×C₈

yes

characteristic

C₄×Dic₁₃

C₂×C₈

yes

characteristic

C₂×#1055

C₂×C₈

yes

—

C₂×#1055

C₂×C₈

yes

—

C₂×#1055

C₂²×C₄

yes

—

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

C₁₆

yes

complemented

#6092

—

#2162

—

C₂×C₈²

—

#1312

—

C₈×C₁₆

—

C₂²×D₁₃

C₄×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×Dic₁₃

C₄×C₈

yes

characteristic

C₂×Dic₁₃

C₄×C₈

yes

characteristic

C₂×Dic₁₃

C₄×C₈

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×C₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×C₁₆

yes

—

#1055

C₂×C₁₆

yes

—

#1055

M₃₂

yes

—

#1055

#131

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

—

C₈²

#261

yes

characteristic

C₂²×C₁₃

C₂×C₄×C₈

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

C₄×C₁₆

yes

—

C₄×C₁₃

#405

yes

—

C₄×C₁₃

C₂²×C₁₆

yes

—

C₄×C₁₃

C₂×M₃₂

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

C₄×C₁₆

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

C₄×C₈

yes

100

C₄×C₈

C₂×#261

yes

characteristic

101

C₂×C₁₃

#1450

yes

characteristic

102

C₂×C₁₃

C₂×C₄×C₁₆

yes

characteristic

103

C₂×C₁₃

C₄×M₃₂

yes

characteristic

104

D₁₃

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

D₁₃⋊C₈

yes

—

115

C₂×C₈

D₁₃⋊C₈

yes

—

116

C₂×C₈

C₄×#261

yes

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂²×#261

yes

characteristic

121

C₁₆

—

122

C₁₃

#11410

yes

characteristic, complemented, Sylow

123

C₂³

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

131

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

132

C₂×C₄

C₂×C₄×#261

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

D₁₃⋊C₁₆

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₄×D₁₃⋊C₈

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

C₂×D₁₃⋊C₁₆

yes

—

147

C₄

D₁₃⋊M₃₂

yes

—

148

C₄

#11719089

yes

—

149

C₂

—

150

C₂

#13130040

yes

characteristic

151

C₂

C₄×D₁₃⋊C₁₆

yes

characteristic

152

C₂

#13131043

yes

characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Frattini	C₄²	C₂²×#261
Fitting	C₈²×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂²×D₁₃	C₄×C₈
cosocle	C₄×#1055	C₂³
hypercenter	C₈²	#261
hypocenter	C₁₃	#11410
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₄×C₈

C₂×#261

commutator

C₂×C₁₃

C₂×C₄×C₁₆

Frattini

C₄²

C₂²×#261

Fitting

C₈²×C₁₃

C₄

socle

C₂²×C₁₃

C₂×C₄×C₈

prime radical

C₂²×D₁₃

C₄×C₈

C₂³

C₈²

C₁₃

G ▷ C₂×C₁₃ ▷ C₁₃

C₁ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
2. #13129751	C₂	semidirect
3. C₈×#6092	C₂	semidirect
45. C₈×D₁₃	C₁₆	semidirect
122. C₁₃	#11410	semidirect

H ≅ G/N

type

2. #13129751

C₂

semidirect

3. C₈×#6092

C₂

semidirect

45. C₈×D₁₃

C₁₆

semidirect

122. C₁₃

#11410

semidirect

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

subgroup	index
#13131195	2
#13129751	2
C₈×#6092	2
C₈²×D₁₃	2
#11410	13

subgroup

index

C₈²×D₁₃

Subgroups — #16786653

Conjugacy classes of subgroups

Complements

Maximal subgroups