Group #29591210 — subgroups

Subgroups — #29591210

Order 15616.

274 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21447631	C₂	1	2	yes	maximal
2	#21443854	C₂	1	4	yes	maximal, complemented
3	C₆₁×#3139	C₂	1	1	yes	maximal, characteristic
4	#16839583	—	2	2	no	—
5	#16839544	—	2	2	no	—
6	#16839540	C₂²	1	4	yes	—
7	C₆₁×#533	C₄	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
9	C₆₁×#491	C₄	1	2	yes	—
10	C₆₁×#491	C₂²	1	2	yes	—
11	C₆₁×#487	—	2	2	no	—
12	C₆₁×#419	—	2	2	no	—
13	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	#14519717	—	2	4	no	—
15	C₂²×Q₈×C₆₁	—	2	1	no	—
16	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
17	C₄⋊Q₈×C₆₁	—	2	1	no	—
18	C₄⋊Q₈×C₆₁	—	2	1	no	—
19	C₄⋊Q₈×C₆₁	—	4	1	no	—
20	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
21	#125×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²⋊Q₈×C₆₁	—	4	1	no	—
23	C₄×Q₈×C₆₁	—	4	1	no	—
24	#117×C₆₁	—	2	1	no	—
25	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	#103×C₆₁	—	2	4	no	—
28	#103×C₆₁	—	2	4	no	—
29	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
30	#97×C₆₁	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₆₁	D₄	1	2	yes	—
32	C₂×Q₈×C₆₁	—	2	2	no	—
33	C₂×Q₈×C₆₁	—	4	1	no	—
34	C₂×Q₈×C₆₁	—	4	4	no	—
35	C₂×Q₈×C₆₁	—	4	1	no	—
36	C₂²×C₄×C₆₁	—	2	1	no	—
37	C₂²×C₄×C₆₁	—	4	1	no	—
38	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	M₁₆×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₄⋊C₄×C₆₁	—	4	2	no	—
44	C₄⋊C₄×C₆₁	—	4	1	no	—
45	C₄⋊C₄×C₆₁	—	4	2	no	—
46	C₄⋊C₄×C₆₁	—	4	2	no	—
47	C₂²⋊C₄×C₆₁	—	4	1	no	—
48	C₄²×C₆₁	—	2	1	no	—
49	C₄²×C₆₁	—	4	1	no	—
50	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	#11758185	—	4	4	no	—
53	C₂³×C₆₁	#100	1	1	yes	characteristic
54	Q₈×C₆₁	—	4	4	no	—
55	Q₈×C₆₁	—	8	2	no	—
56	Q₈×C₆₁	—	8	2	no	—
57	C₂×C₄×C₆₁	—	2	1	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	1	no	—
60	C₂×C₄×C₆₁	—	4	2	no	—
61	C₂×C₄×C₆₁	—	4	2	no	—
62	C₂×C₄×C₆₁	—	4	2	no	—
63	C₂×C₄×C₆₁	—	8	2	no	—
64	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₆₁	—	2	2	no	—
67	C₈×C₆₁	—	2	2	no	—
68	#11916	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	—	2	1	no	—
70	C₂²×C₆₁	—	4	1	no	—
71	C₂²×C₆₁	C₂×#100	1	1	yes	characteristic
72	C₄×C₆₁	—	2	2	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	—	4	1	no	—
75	C₄×C₆₁	—	8	2	no	—
76	C₄×C₆₁	—	8	2	no	—
77	C₄×C₆₁	#421	1	2	yes	—
78	#3139	D₆₁	1	1	yes	characteristic
79	#2165	—	61	2	no	—
80	#1356	—	61	4	no	—
81	C₂×C₆₁	—	2	1	no	—
82	C₂×C₆₁	—	4	1	no	—
83	C₂×C₆₁	#1533	1	1	yes	characteristic
84	#533	Dic₆₁	1	1	yes	characteristic
85	C₂×C₄⋊Q₈	Dic₆₁	1	1	yes	characteristic
86	#491	Dic₆₁	1	2	yes	—
87	#491	C₂×D₆₁	1	2	yes	—
88	#487	—	2	2	no	—
89	#419	—	2	2	no	—
90	C₄×M₁₆	C₂×D₆₁	1	1	yes	characteristic
91	#364	—	122	2	no	—
92	#349	—	122	2	no	—
93	#347	—	61	4	no	—
94	C₆₁	#11916	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	—	2	1	no	—
96	C₂×M₁₆	C₂²⋊D₆₁	1	2	yes	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	2	1	no	—
99	C₄⋊Q₈	—	4	1	no	—
100	C₄⋊Q₈	C₂×Dic₆₁	1	2	yes	—
101	#125	C₂×Dic₆₁	1	2	yes	—
102	C₂²⋊Q₈	—	4	1	no	—
103	C₄×Q₈	—	4	1	no	—
104	#117	—	2	1	no	—
105	C₂×C₄⋊C₄	—	4	1	no	—
106	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
107	M₃₂	—	122	4	no	—
108	#103	—	2	4	no	—
109	#103	—	2	4	no	—
110	#97	C₂×Dic₆₁	1	1	yes	characteristic
111	#97	C₂²×D₆₁	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
113	C₂×Q₈	—	2	2	no	—
114	C₂×Q₈	—	4	1	no	—
115	C₂×Q₈	—	4	4	no	—
116	C₂×Q₈	—	4	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	4	1	no	—
119	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
122	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₂²⋊C₄	—	4	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	4	1	no	—
131	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
132	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
133	C₁₆	—	244	4	no	—
134	C₂³	#14519694	1	1	yes	characteristic
135	Q₈	—	4	4	no	—
136	Q₈	—	8	2	no	—
137	Q₈	—	8	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#14519694	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	C₂×#14519694	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	2	no	—
156	C₄	—	8	2	no	—
157	C₄	#16839779	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	4	1	no	—
160	C₂	#21444443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447631

C₂

yes

maximal

#21443854

C₂

yes

maximal, complemented

C₆₁×#3139

C₂

yes

maximal, characteristic

#16839583

—

#16839544

—

#16839540

C₂²

yes

—

C₆₁×#533

C₄

yes

characteristic

C₂×C₄⋊Q₈×C₆₁

C₄

yes

characteristic

C₆₁×#491

C₄

yes

—

C₆₁×#491

C₂²

yes

—

C₆₁×#487

—

C₆₁×#419

—

C₄×M₁₆×C₆₁

C₂²

yes

characteristic

#14519717

—

C₂²×Q₈×C₆₁

—

C₂×M₁₆×C₆₁

D₄

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

#125×C₆₁

C₂×C₄

yes

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

#103×C₆₁

—

#103×C₆₁

—

#97×C₆₁

C₂×C₄

yes

characteristic

#97×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

#11758185

—

C₂³×C₆₁

#100

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#100

yes

characteristic

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

—

#11916

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#100

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#421

yes

—

#3139

D₆₁

yes

characteristic

#2165

—

#1356

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1533

yes

characteristic

#533

Dic₆₁

yes

characteristic

C₂×C₄⋊Q₈

Dic₆₁

yes

characteristic

#491

Dic₆₁

yes

—

#491

C₂×D₆₁

yes

—

#487

—

#419

—

C₄×M₁₆

C₂×D₆₁

yes

characteristic

#364

—

122

—

#349

—

122

—

#347

—

C₆₁

#11916

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

C₂²⋊D₆₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

C₂×Dic₆₁

yes

—

101

#125

C₂×Dic₆₁

yes

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

#117

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

107

M₃₂

—

122

—

108

#103

—

109

#103

—

110

#97

C₂×Dic₆₁

yes

characteristic

111

#97

C₂²×D₆₁

yes

characteristic

112

C₄×C₈

C₂²⋊D₆₁

yes

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

120

M₁₆

—

121

C₂×C₈

C₂²⋊Dic₆₁

yes

—

122

C₂×C₈

C₂×C₂²⋊D₆₁

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²⋊Dic₆₁

yes

characteristic

132

C₄²

C₂²×Dic₆₁

yes

characteristic

133

C₁₆

—

244

—

134

C₂³

#14519694

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#14519694

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

147

C₈

—

148

C₈

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂×#14519694

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#16839779

yes

—

158

C₂

—

159

C₂

—

160

C₂

#21444443

yes

characteristic

	subgroup	G/H
center	C₂	#21444443
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	#97	C₂²×D₆₁
Fitting	C₆₁×#3139	C₂
socle	C₂×C₆₁	#1533
prime radical	C₂³×C₆₁	#100
cosocle	#97×C₆₁	C₂³
hypercenter	#3139	D₆₁
hypocenter	C₆₁	#11916
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

subgroup

G/H

center

C₂

#21444443

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

#97

C₂²×D₆₁

Fitting

C₆₁×#3139

C₂

C₂×C₆₁

C₂³×C₆₁

#97×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

N	H ≅ G/N	type
2. #21443854	C₂	semidirect
94. C₆₁	#11916	semidirect

H ≅ G/N

type

2. #21443854

C₂

semidirect

94. C₆₁

#11916

semidirect

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

subgroup	index
#21447631	2
#21443854	2
C₆₁×#3139	2
#11916	61

subgroup

index

#21447631

#21443854

C₆₁×#3139

#11916

Subgroups — #29591210

Conjugacy classes of subgroups

Complements

Maximal subgroups