Group #16788661 — subgroups

Subgroups — #16788661

Order 3328.

234 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 100 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	#13130384	C₂	1	2	yes	maximal
2	#13129862	C₂	1	2	yes	maximal, complemented
3	#13129861	C₂	1	1	yes	maximal, characteristic, complemented
4	#13129860	C₂	1	1	yes	maximal, characteristic, complemented
5	D₁₃×#346	C₂	1	1	yes	maximal, characteristic
6	#11719051	—	2	2	no	—
7	C₂×C₁₆×D₁₃	C₄	1	1	yes	characteristic
8	C₂×C₁₆×D₁₃	C₄	1	1	yes	characteristic
9	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	#11718256	—	2	2	no	—
11	#11718253	C₂²	1	2	yes	—
12	#11718253	C₂²	1	2	yes	—
13	C₁₃×#346	C₄	1	1	yes	characteristic
14	#11718094	C₂²	1	1	yes	characteristic
15	#11718094	C₂²	1	1	yes	characteristic
16	#11718053	C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
19	C₄²×D₁₃	D₄	1	1	yes	characteristic
20	#88111	—	4	2	no	—
21	C₄⋊#1057	—	2	2	no	—
22	C₄⋊#1057	—	2	2	no	—
23	C₄×#1057	—	4	2	no	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₃	D₄	1	1	yes	characteristic
28	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
29	C₂×#6090	C₂×C₄	1	1	yes	characteristic
30	C₂×#6090	C₂×C₄	1	1	yes	characteristic
31	C₄×#1055	Q₈	1	1	yes	characteristic
32	C₁₆×D₁₃	—	2	2	no	—
33	C₁₆×D₁₃	C₈	1	4	yes	complemented
34	#11999	—	13	1	no	maximal, Sylow
35	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
37	C₂×#1057	—	4	2	no	—
38	C₂×#1057	—	4	2	no	—
39	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₃	SD₁₆	1	2	yes	—
43	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
46	C₈×D₁₃	—	2	2	no	—
47	C₈×D₁₃	C₂×C₈	1	2	yes	—
48	C₈×D₁₃	M₁₆	1	2	yes	—
49	C₁₃×C₁₆	—	2	1	no	—
50	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
51	#6090	—	2	1	no	—
52	#6090	C₂×C₈	1	2	yes	—
53	C₂×#346	—	13	1	no	—
54	#1564	—	13	2	no	—
55	#1368	—	13	1	no	—
56	#1368	—	13	1	no	—
57	#1368	—	13	2	no	—
58	C₂²×D₁₃	#106	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#106	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×Dic₁₃	#106	1	1	yes	characteristic
62	C₂×Dic₁₃	C₂×SD₁₆	1	2	yes	—
63	C₄×D₁₃	—	4	2	no	—
64	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
65	C₄×D₁₃	#108	1	2	yes	—
66	#1057	—	8	2	no	—
67	#1057	—	8	2	no	—
68	C₈×C₁₃	—	2	1	no	—
69	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
70	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
71	#1055	—	2	1	no	—
72	#1055	C₄⋊C₈	1	2	yes	—
73	C₂²×C₁₆	—	13	1	no	—
74	C₂²×C₁₆	—	13	1	no	—
75	#431	—	26	2	no	—
76	C₂×C₄×C₈	—	13	1	no	—
77	#346	—	13	1	no	—
78	#346	—	13	1	no	—
79	#346	—	13	1	no	—
80	#346	#261	1	1	yes	characteristic
81	#335	—	26	2	no	—
82	#334	—	13	2	no	—
83	#334	—	13	2	no	—
84	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
85	C₂×D₁₃	—	2	2	no	—
86	C₂×D₁₃	#334	1	2	yes	—
87	C₄×C₁₃	—	2	2	no	—
88	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
90	Dic₁₃	—	2	2	no	—
91	Dic₁₃	#334	1	2	yes	—
92	C₂²×C₈	—	13	1	no	—
93	C₂²×C₈	—	13	1	no	—
94	C₂×C₄²	—	13	1	no	—
95	C₂×C₁₆	—	13	1	no	—
96	C₂×C₁₆	—	13	4	no	—
97	C₂×C₁₆	—	13	1	no	—
98	C₂×C₁₆	—	26	2	no	—
99	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
100	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
101	C₄⋊C₈	—	26	2	no	—
102	C₄⋊C₈	—	26	2	no	—
103	C₂²⋊C₈	—	52	2	no	—
104	C₄×C₈	—	13	1	no	—
105	C₄×C₈	—	13	1	no	—
106	C₄×C₈	—	13	1	no	—
107	C₄×C₈	—	52	2	no	—
108	C₄×C₈	C₂×#261	1	1	yes	characteristic
109	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
110	C₂×C₁₃	#2155	1	2	yes	—
111	D₁₃	—	2	2	no	—
112	C₂²×C₄	—	13	1	no	—
113	C₂²×C₄	—	13	2	no	—
114	C₂×C₈	—	13	1	no	—
115	C₂×C₈	—	13	2	no	—
116	C₂×C₈	—	13	2	no	—
117	C₂×C₈	—	13	1	no	—
118	C₂×C₈	—	26	2	no	—
119	C₂×C₈	—	52	2	no	—
120	C₂×C₈	—	52	2	no	—
121	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
122	C₂×C₈	C₂²×#261	1	1	yes	characteristic
123	C₄²	—	13	2	no	—
124	C₄²	—	13	1	no	—
125	C₄²	C₄⋊#261	1	1	yes	characteristic
126	C₁₆	—	2	1	no	—
127	C₁₆	—	13	2	no	—
128	C₁₆	—	26	1	no	—
129	C₁₆	D₁₃⋊C₈	1	2	yes	—
130	C₁₃	#11999	1	1	yes	characteristic, complemented, Sylow
131	C₂³	—	13	1	no	—
132	C₂×C₄	—	13	1	no	—
133	C₂×C₄	—	13	2	no	—
134	C₂×C₄	—	13	2	no	—
135	C₂×C₄	—	13	2	no	—
136	C₂×C₄	—	52	2	no	—
137	C₂×C₄	#88101	1	2	yes	—
138	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	13	2	no	—
141	C₈	—	26	1	no	—
142	C₈	—	104	2	no	—
143	C₈	—	104	2	no	—
144	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
145	C₈	D₁₃⋊M₁₆	1	1	yes	characteristic
146	C₂²	—	13	2	no	—
147	C₂²	—	26	2	no	—
148	C₂²	C₂×#88101	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	13	2	no	—
151	C₄	—	26	2	no	—
152	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
153	C₄	D₁₃⋊#108	1	1	yes	characteristic
154	C₂	—	26	2	no	—
155	C₂	D₁₃⋊#334	1	1	yes	characteristic
156	C₂	#13131184	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13130384

C₂

yes

maximal

#13129862

C₂

yes

maximal, complemented

#13129861

C₂

yes

maximal, characteristic, complemented

#13129860

C₂

yes

maximal, characteristic, complemented

D₁₃×#346

C₂

yes

maximal, characteristic

#11719051

—

C₂×C₁₆×D₁₃

C₄

yes

characteristic

C₂×C₁₆×D₁₃

C₄

yes

characteristic

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718256

—

#11718253

C₂²

yes

—

#11718253

C₂²

yes

—

C₁₃×#346

C₄

yes

characteristic

#11718094

C₂²

yes

characteristic

#11718094

C₂²

yes

characteristic

#11718053

C₄

yes

characteristic

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

Q₈

yes

characteristic

C₄²×D₁₃

D₄

yes

—

—

—

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

characteristic

C₈×Dic₁₃

C₂³

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

C₄×#1055

Q₈

yes

characteristic

C₁₆×D₁₃

—

C₁₆×D₁₃

C₈

yes

complemented

#11999

—

maximal, Sylow

C₂×C₄×D₁₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₃

SD₁₆

yes

—

C₂×#1057

—

C₂×#1057

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

SD₁₆

yes

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

C₂×Q₈

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

C₂×C₈

yes

—

C₈×D₁₃

M₁₆

yes

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×C₈

yes

—

#6090

—

#6090

C₂×C₈

yes

—

—

—

—

—

—

C₂²×D₁₃

#106

yes

characteristic

C₂×C₄×C₁₃

#106

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₃

#106

yes

characteristic

C₂×Dic₁₃

C₂×SD₁₆

yes

—

C₄×D₁₃

—

C₄×D₁₃

C₄⋊C₈

yes

—

C₄×D₁₃

#108

yes

—

#1057

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

C₂²×C₈

yes

characteristic

C₈×C₁₃

C₂×M₁₆

yes

characteristic

#1055

—

#1055

C₄⋊C₈

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

—

—

—

yes

—

—

—

C₂²×C₁₃

C₂×#106

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

#334

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₃

C₂×#108

yes

characteristic

Dic₁₃

—

Dic₁₃

#334

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#261

yes

characteristic

100

C₂×C₁₆

C₂×#261

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₂×#261

yes

characteristic

109

C₂×C₁₃

C₂×#334

yes

characteristic

110

C₂×C₁₃

#2155

yes

—

111

D₁₃

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₄⋊#261

yes

characteristic

122

C₂×C₈

C₂²×#261

yes

characteristic

123

C₄²

—

124

C₄²

—

125

C₄²

C₄⋊#261

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

D₁₃⋊C₈

yes

—

130

C₁₃

#11999

yes

characteristic, complemented, Sylow

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#88101

yes

—

138

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

104

—

143

C₈

—

104

—

144

C₈

C₂×D₁₃⋊C₈

yes

characteristic

145

C₈

D₁₃⋊M₁₆

yes

characteristic

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×#88101

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

D₁₃⋊(C₄⋊C₈)

yes

characteristic

153

C₄

D₁₃⋊#108

yes

characteristic

154

C₂

—

155

C₂

D₁₃⋊#334

yes

characteristic

156

C₂

#13131184

yes

—

	subgroup	G/H
center	C₂²	C₂×#88101
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#346	C₄
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂²×D₁₃	#106
cosocle	C₈×Dic₁₃	C₂³
hypercenter	#346	#261
hypocenter	C₁₃	#11999
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#88101

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#261

Fitting

C₁₃×#346

C₄

C₂²×C₁₃

C₂²×D₁₃

C₈×Dic₁₃

C₂³

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #13129862	C₂	semidirect
3. #13129861	C₂	semidirect
4. #13129860	C₂	semidirect
33. C₁₆×D₁₃	C₈	semidirect
130. C₁₃	#11999	semidirect

H ≅ G/N

type

2. #13129862

C₂

semidirect

3. #13129861

C₂

semidirect

4. #13129860

C₂

semidirect

33. C₁₆×D₁₃

C₈

semidirect

130. C₁₃

#11999

semidirect

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

subgroup	index
#13130384	2
#13129862	2
#13129861	2
#13129860	2
D₁₃×#346	2
#11999	13

subgroup

index

D₁₃×#346

Subgroups — #16788661

Conjugacy classes of subgroups

Complements

Maximal subgroups