Group #29591566 — subgroups

Subgroups — #29591566

Order 15616.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#346×D₆₁	C₂	1	1	yes	maximal, characteristic
2	#21444564	C₂	1	2	yes	maximal
3	#21443885	C₂	1	1	yes	maximal, characteristic, complemented
4	#21443884	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₆₁	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₆₁	C₄	1	1	yes	characteristic
8	#16839814	—	2	2	no	—
9	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
10	#16839538	C₂²	1	1	yes	characteristic
11	#16839537	C₄	1	1	yes	characteristic
12	#16839537	C₄	1	1	yes	characteristic
13	#16839514	—	2	2	no	—
14	#16839513	C₂²	1	2	yes	—
15	C₆₁×#346	C₂²	1	1	yes	characteristic
16	C₆₁×#334	C₂²	1	2	yes	—
17	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
19	C₄²×D₆₁	D₄	1	1	yes	characteristic
20	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
21	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
22	C₁₆×D₆₁	—	2	2	no	—
23	C₁₆×D₆₁	C₈	1	4	yes	complemented
24	C₄⋊#92418	—	2	2	no	—
25	#14519689	—	4	2	no	—
26	C₄×#92418	D₄	1	1	yes	characteristic
27	C₈×Dic₆₁	—	4	2	no	—
28	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₆₁	—	2	2	no	—
33	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₆₁	SD₁₆	1	2	yes	—
36	C₂×C₈×C₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₆₁	SD₁₆	1	2	yes	—
42	C₂×#92418	—	4	2	no	—
43	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
45	C₈×D₆₁	—	2	2	no	—
46	C₈×D₆₁	C₂×C₈	1	2	yes	—
47	C₈×D₆₁	M₁₆	1	2	yes	—
48	C₁₆×C₆₁	—	2	1	no	—
49	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
50	#11758185	—	2	1	no	—
51	#11758185	C₂×C₈	1	2	yes	—
52	C₂²×D₆₁	#106	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
55	C₂×Dic₆₁	#106	1	2	yes	—
56	C₂×Dic₆₁	#106	1	1	yes	characteristic
57	C₄×D₆₁	—	4	2	no	—
58	C₄×D₆₁	C₄⋊C₈	1	2	yes	—
59	C₄×D₆₁	#108	1	2	yes	—
60	C₈×C₆₁	—	2	1	no	—
61	C₈×C₆₁	—	8	2	no	—
62	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
64	#92418	—	2	1	no	—
65	#92418	—	8	2	no	—
66	#92418	C₄⋊C₈	1	2	yes	—
67	#11999	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
69	C₂×D₆₁	—	2	2	no	—
70	C₂×D₆₁	#334	1	2	yes	—
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
74	Dic₆₁	—	2	2	no	—
75	Dic₆₁	#334	1	2	yes	—
76	C₂×#346	—	61	1	no	—
77	#1564	—	61	2	no	—
78	#1368	—	61	1	no	—
79	#1368	—	61	2	no	—
80	#1368	D₆₁	1	1	yes	characteristic, complemented
81	C₂×C₆₁	C₂×#334	1	1	yes	characteristic
82	C₂×C₆₁	#2155	1	2	yes	—
83	D₆₁	—	2	2	no	—
84	C₂²×C₁₆	—	61	1	no	—
85	C₂²×C₁₆	—	61	1	no	—
86	#431	—	122	2	no	—
87	C₂×C₄×C₈	—	61	1	no	—
88	#346	—	61	1	no	—
89	#346	—	61	1	no	—
90	#346	—	61	1	no	—
91	#346	C₂×D₆₁	1	1	yes	characteristic
92	#335	—	122	2	no	—
93	#334	—	61	2	no	—
94	#334	C₂×D₆₁	1	2	yes	—
95	C₆₁	#11999	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	61	1	no	—
97	C₂²×C₈	—	61	1	no	—
98	C₂×C₄²	—	61	1	no	—
99	C₂×C₁₆	—	61	1	no	—
100	C₂×C₁₆	—	61	4	no	—
101	C₂×C₁₆	—	61	1	no	—
102	C₂×C₁₆	—	122	2	no	—
103	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
104	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	122	2	no	—
107	C₂²⋊C₈	—	244	2	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	—	61	1	no	—
110	C₄×C₈	—	61	1	no	—
111	C₄×C₈	—	244	2	no	—
112	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
113	C₂²×C₄	—	61	1	no	—
114	C₂²×C₄	—	61	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	61	1	no	—
117	C₂×C₈	—	61	2	no	—
118	C₂×C₈	—	61	2	no	—
119	C₂×C₈	—	61	1	no	—
120	C₂×C₈	—	122	2	no	—
121	C₂×C₈	—	244	2	no	—
122	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
123	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
124	C₄²	—	61	1	no	—
125	C₄²	—	61	2	no	—
126	C₄²	D₄×D₆₁	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	61	2	no	—
129	C₁₆	—	122	1	no	—
130	C₁₆	C₈×D₆₁	1	2	yes	—
131	C₂³	—	61	1	no	—
132	C₂×C₄	—	61	1	no	—
133	C₂×C₄	—	61	2	no	—
134	C₂×C₄	—	61	2	no	—
135	C₂×C₄	—	61	2	no	—
136	C₂×C₄	—	244	2	no	—
137	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄	SD₁₆×D₆₁	1	2	yes	—
139	C₈	—	2	1	no	—
140	C₈	—	8	2	no	—
141	C₈	—	61	2	no	—
142	C₈	—	122	1	no	—
143	C₈	—	488	2	no	—
144	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
145	C₈	M₁₆×D₆₁	1	1	yes	characteristic
146	C₂²	—	61	2	no	—
147	C₂²	—	122	2	no	—
148	C₂²	#106×D₆₁	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	61	2	no	—
151	C₄	—	122	2	no	—
152	C₄	C₄⋊C₈×D₆₁	1	1	yes	characteristic
153	C₄	#108×D₆₁	1	1	yes	characteristic
154	C₂	—	122	2	no	—
155	C₂	#334×D₆₁	1	1	yes	characteristic
156	C₂	#21447590	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#346×D₆₁

C₂

yes

maximal, characteristic

#21444564

C₂

yes

maximal

#21443885

C₂

yes

maximal, characteristic, complemented

#21443884

C₂

yes

maximal, complemented

C₆₁×#1368

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₆₁

C₄

yes

characteristic

C₂×C₁₆×D₆₁

C₄

yes

characteristic

#16839814

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839538

C₂²

yes

characteristic

#16839537

C₄

yes

characteristic

#16839537

C₄

yes

characteristic

#16839514

—

#16839513

C₂²

yes

—

C₆₁×#346

C₂²

yes

characteristic

C₆₁×#334

C₂²

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

Q₈

yes

characteristic

C₄²×D₆₁

D₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

characteristic

C₁₆×D₆₁

—

C₁₆×D₆₁

C₈

yes

—

—

D₄

yes

characteristic

C₈×Dic₆₁

—

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄×D₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×D₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

—

C₂×#92418

—

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

C₂×C₈

yes

—

C₈×D₆₁

M₁₆

yes

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂×C₈

yes

—

#11758185

—

#11758185

C₂×C₈

yes

—

C₂²×D₆₁

#106

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×Dic₆₁

#106

yes

—

C₂×Dic₆₁

#106

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₄⋊C₈

yes

—

C₄×D₆₁

#108

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₈

yes

characteristic

C₈×C₆₁

C₂×M₁₆

yes

—

—

C₄⋊C₈

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

#334

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₂×#108

yes

characteristic

Dic₆₁

—

Dic₆₁

#334

yes

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₂×C₆₁

C₂×#334

yes

characteristic

C₂×C₆₁

#2155

yes

—

D₆₁

—

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

—

122

—

C₂×C₄×C₈

—

—

—

—

C₂×D₆₁

yes

characteristic

#335

—

122

—

#334

—

#334

C₂×D₆₁

yes

—

C₆₁

#11999

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

122

—

103

C₂×C₁₆

C₄×D₆₁

yes

characteristic

104

C₂×C₁₆

C₄×D₆₁

yes

characteristic

105

C₄⋊C₈

—

106

C₄⋊C₈

—

122

—

107

C₂²⋊C₈

—

244

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

244

—

112

C₄×C₈

C₂²×D₆₁

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

122

—

121

C₂×C₈

—

244

—

122

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

123

C₂×C₈

Q₈×D₆₁

yes

characteristic

124

C₄²

—

125

C₄²

—

126

C₄²

D₄×D₆₁

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

122

—

130

C₁₆

C₈×D₆₁

yes

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

244

—

137

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

138

C₂×C₄

SD₁₆×D₆₁

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

122

—

143

C₈

—

488

—

144

C₈

C₂×C₈×D₆₁

yes

characteristic

145

C₈

M₁₆×D₆₁

yes

characteristic

146

C₂²

—

147

C₂²

—

122

—

148

C₂²

#106×D₆₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

122

—

152

C₄

C₄⋊C₈×D₆₁

yes

characteristic

153

C₄

#108×D₆₁

yes

characteristic

154

C₂

—

122

—

155

C₂

#334×D₆₁

yes

characteristic

156

C₂

#21447590

yes

—

	subgroup	G/H
center	C₂²	#106×D₆₁
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1368	C₂
socle	C₂²×C₆₁	C₂×#106
prime radical	C₂²×D₆₁	#106
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1368	D₆₁
hypocenter	C₆₁	#11999
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#106×D₆₁

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1368

C₂

C₂²×C₆₁

C₂²×D₆₁

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
3. #21443885	C₂	semidirect
4. #21443884	C₂	semidirect
5. C₆₁×#1368	C₂	semidirect
23. C₁₆×D₆₁	C₈	semidirect
80. #1368	D₆₁	semidirect
95. C₆₁	#11999	semidirect

H ≅ G/N

type

3. #21443885

C₂

semidirect

4. #21443884

C₂

semidirect

5. C₆₁×#1368

C₂

semidirect

23. C₁₆×D₆₁

C₈

semidirect

80. #1368

D₆₁

semidirect

95. C₆₁

#11999

semidirect

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

subgroup	index
#346×D₆₁	2
#21444564	2
#21443885	2
#21443884	2
C₆₁×#1368	2
#11999	61

subgroup

index

#346×D₆₁

C₆₁×#1368

Subgroups — #29591566

Conjugacy classes of subgroups

Complements

Maximal subgroups