Group #29594711 — subgroups

Subgroups — #29594711

Order 15616.

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

normal subgroups only

This group has 268 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	#21448046	C₂	1	2	yes	maximal, complemented
2	#21448042	C₂	1	2	yes	maximal
3	#21445437	C₂	1	1	yes	maximal, characteristic, complemented
4	#21443736	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1709	C₂	1	1	yes	maximal, characteristic
6	#16840167	—	2	1	no	—
7	#16840167	C₂²	1	2	yes	—
8	#16840142	—	2	1	no	—
9	#16839566	—	2	4	no	—
10	#16839563	—	2	4	no	—
11	C₄×#11758185	C₂²	1	2	yes	—
12	C₆₁×#494	C₂²	1	1	yes	characteristic
13	C₆₁×#493	C₂²	1	1	yes	characteristic
14	C₆₁×#486	—	2	1	no	—
15	C₆₁×#486	—	2	1	no	—
16	C₈²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₄₄	—	4	2	no	—
18	#14519777	—	2	2	no	—
19	C₂×#11758185	D₄	1	4	yes	—
20	#14519711	—	2	2	no	—
21	#14519711	—	2	2	no	—
22	#14519711	—	4	2	no	—
23	C₂×Q₁₆×C₆₁	—	2	2	no	—
24	C₂×D₈×C₆₁	—	2	2	no	—
25	C₄⋊Q₈×C₆₁	—	2	1	no	—
26	#127×C₆₁	—	2	1	no	—
27	#102×C₆₁	—	4	2	no	—
28	#102×C₆₁	—	4	2	no	—
29	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₆₁	—	4	1	no	—
33	C₂×D₄×C₆₁	—	4	1	no	—
34	C₂×D₄×C₆₁	—	4	1	no	—
35	C₂×D₄×C₆₁	—	4	1	no	—
36	C₂×Dic₁₂₂	—	4	2	no	—
37	Q₁₆×C₆₁	—	4	4	no	—
38	D₈×C₆₁	—	4	4	no	—
39	C₂×C₈×C₆₁	D₈	1	2	yes	—
40	C₂×C₈×C₆₁	D₈	1	2	yes	—
41	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₆₁	—	4	1	no	—
43	C₄⋊C₄×C₆₁	—	4	1	no	—
44	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₆₁	—	4	2	no	—
46	C₄⋊Dic₆₁	—	4	2	no	—
47	Dic₂₄₄	—	8	4	no	—
48	#11758185	—	2	4	no	—
49	C₂³×C₆₁	—	8	1	no	—
50	Q₈×C₆₁	—	8	2	no	—
51	D₄×C₆₁	—	8	2	no	—
52	D₄×C₆₁	—	8	2	no	—
53	D₄×C₆₁	—	8	2	no	—
54	C₂×C₄×C₆₁	—	8	1	no	—
55	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₆₁	—	8	2	no	—
59	Dic₁₂₂	—	8	4	no	—
60	C₈×C₆₁	—	2	2	no	—
61	C₈×C₆₁	—	2	2	no	—
62	C₈×C₆₁	#135	1	4	yes	—
63	#12512	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	—	16	1	no	—
65	C₂²×C₆₁	—	16	1	no	—
66	C₂²×C₆₁	—	16	1	no	—
67	C₂²×C₆₁	#493	1	1	yes	characteristic
68	C₄×C₆₁	—	16	1	no	—
69	C₄×C₆₁	#495	1	2	yes	—
70	C₄×C₆₁	#509	1	2	yes	—
71	C₄×C₆₁	#510	1	2	yes	—
72	Dic₆₁	—	16	2	no	—
73	#2241	—	61	2	no	—
74	#2240	—	61	2	no	—
75	#1713	—	61	1	no	—
76	#1709	D₆₁	1	1	yes	characteristic
77	#1311	—	61	1	no	—
78	C₂×C₆₁	—	32	1	no	—
79	C₂×C₆₁	#1709	1	1	yes	characteristic
80	C₂×C₆₁	#2250	1	1	yes	characteristic
81	C₂×C₆₁	#2251	1	1	yes	characteristic
82	#500	—	61	2	no	—
83	#500	—	122	1	no	—
84	#494	—	122	1	no	—
85	#494	C₂×D₆₁	1	1	yes	characteristic
86	#493	C₂×D₆₁	1	1	yes	characteristic
87	#486	—	2	1	no	—
88	#486	—	2	1	no	—
89	#358	—	122	4	no	—
90	#357	—	122	4	no	—
91	C₄×C₁₆	—	61	2	no	—
92	C₈²	C₂×D₆₁	1	1	yes	characteristic
93	C₆₁	#12512	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	2	2	no	—
95	C₂×Q₁₆	—	244	2	no	—
96	C₂×D₈	—	2	2	no	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	122	2	no	—
99	#127	—	2	1	no	—
100	C₂×C₁₆	—	61	4	no	—
101	#107	—	122	2	no	—
102	#107	—	122	2	no	—
103	#107	—	244	2	no	—
104	#102	—	4	2	no	—
105	#102	—	4	2	no	—
106	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
109	C₂×Q₈	—	4	1	no	—
110	C₂×Q₈	—	244	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂×D₄	—	4	1	no	—
114	Q₁₆	—	4	4	no	—
115	Q₁₆	—	488	4	no	—
116	D₈	—	4	4	no	—
117	C₂×C₈	#11758199	1	2	yes	—
118	C₂×C₈	#11758199	1	2	yes	—
119	C₂×C₈	D₄×D₆₁	1	2	yes	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	244	2	no	—
123	C₄⋊C₄	—	244	2	no	—
124	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
125	C₁₆	—	122	4	no	—
126	C₂³	—	8	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	488	4	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	488	2	no	—
134	C₂×C₄	#14519773	1	1	yes	characteristic
135	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
136	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#14519796	1	4	yes	—
140	C₂²	—	16	1	no	—
141	C₂²	—	16	1	no	—
142	C₂²	—	16	1	no	—
143	C₂²	#16840138	1	1	yes	characteristic
144	C₄	—	16	1	no	—
145	C₄	—	976	2	no	—
146	C₄	#16840145	1	2	yes	—
147	C₄	#16840197	1	2	yes	—
148	C₄	#16840202	1	2	yes	—
149	C₂	—	32	1	no	—
150	C₂	#21445424	1	1	yes	characteristic
151	C₂	#21448086	1	1	yes	characteristic
152	C₂	#21448091	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448046

C₂

yes

maximal, complemented

#21448042

C₂

yes

maximal

#21445437

C₂

yes

maximal, characteristic, complemented

#21443736

C₂

yes

maximal, characteristic, complemented

C₆₁×#1709

C₂

yes

maximal, characteristic

#16840167

—

#16840167

C₂²

yes

—

—

—

—

C₂²

yes

—

C₆₁×#494

C₂²

yes

characteristic

C₆₁×#493

C₂²

yes

characteristic

C₆₁×#486

—

C₆₁×#486

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

#14519777

—

C₂×#11758185

D₄

yes

—

#14519711

—

#14519711

—

#14519711

—

C₂×Q₁₆×C₆₁

—

C₂×D₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

#127×C₆₁

—

#102×C₆₁

—

#102×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

#11758185

—

C₂³×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#135

yes

—

#12512

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#493

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

#509

yes

—

C₄×C₆₁

#510

yes

—

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic

#1311

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

122

—

#494

—

122

—

#494

C₂×D₆₁

yes

characteristic

#493

C₂×D₆₁

yes

—

—

—

122

—

#357

—

122

—

C₄×C₁₆

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12512

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

244

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

#127

—

100

C₂×C₁₆

—

101

#107

—

122

—

102

#107

—

122

—

103

#107

—

244

—

104

#102

—

105

#102

—

106

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

107

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

108

C₄×C₈

C₂²×D₆₁

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

244

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

Q₁₆

—

488

—

116

D₈

—

117

C₂×C₈

#11758199

yes

—

118

C₂×C₈

#11758199

yes

—

119

C₂×C₈

D₄×D₆₁

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

244

—

123

C₄⋊C₄

—

244

—

124

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

125

C₁₆

—

122

—

126

C₂³

—

127

Q₈

—

128

Q₈

—

488

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

488

—

134

C₂×C₄

#14519773

yes

characteristic

135

C₂×C₄

C₂×#11758199

yes

characteristic

136

C₂×C₄

C₂×#11758199

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#14519796

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#16840138

yes

characteristic

144

C₄

—

145

C₄

—

976

—

146

C₄

#16840145

yes

—

147

C₄

#16840197

yes

—

148

C₄

#16840202

yes

—

149

C₂

—

150

C₂

#21445424

yes

characteristic

151

C₂

#21448086

yes

characteristic

152

C₂

#21448091

yes

characteristic

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1709	C₂
socle	C₂²×C₆₁	#493
prime radical	C₆₁×#493	C₂²
hypercenter	#1709	D₆₁
hypocenter	C₆₁	#12512
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16840138

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1709

C₂

socle

C₂²×C₆₁

#493

prime radical

C₆₁×#493

C₂²

D₆₁

C₆₁

G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #21448046	C₂	semidirect
3. #21445437	C₂	semidirect
4. #21443736	C₂	semidirect
93. C₆₁	#12512	semidirect

H ≅ G/N

type

1. #21448046

C₂

semidirect

3. #21445437

C₂

semidirect

4. #21443736

C₂

semidirect

93. C₆₁

#12512

semidirect

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

subgroup	index
#21448046	2
#21448042	2
#21445437	2
#21443736	2
C₆₁×#1709	2
#12512	61

subgroup

index

C₆₁×#1709

Subgroups — #29594711

Conjugacy classes of subgroups

Complements

Maximal subgroups