Group #30624478 — subgroups

Subgroups — #30624478

Order 15616.

444 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 210 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	#21462236	C₂	1	8	yes	maximal, complemented
2	#21462185	C₂	1	4	yes	maximal
3	#21462180	C₂	1	2	yes	maximal
4	#21454749	C₂	1	1	yes	maximal, characteristic
5	#16840683	C₂²	1	8	yes	—
6	C₄⋊#11758187	C₂²	1	4	yes	—
7	C₂²⋊#11758187	C₂²	1	4	yes	—
8	#16840651	C₂²	1	4	yes	—
9	#16840650	C₂²	1	4	yes	—
10	#16840648	C₂²	1	4	yes	—
11	#16840319	C₄	1	1	yes	characteristic
12	#16839889	C₄	1	4	yes	—
13	#16839886	C₂²	1	4	yes	—
14	#16839790	C₄	1	2	yes	—
15	#16839720	C₂²	1	2	yes	—
16	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
17	C₆₁×#423	C₄	1	1	yes	characteristic
18	C₂×#11758187	C₂³	1	8	yes	—
19	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
20	C₂²×#92418	C₂³	1	1	yes	characteristic
21	#14519777	C₈	1	2	yes	—
22	#14519769	C₈	1	2	yes	—
23	#14519756	C₈	1	4	yes	—
24	C₄×Dic₁₂₂	C₂×C₄	1	4	yes	—
25	#14519736	C₂×C₄	1	2	yes	—
26	#14519706	C₂×C₄	1	8	yes	—
27	#14519687	C₂×C₄	1	4	yes	—
28	C₄×#92418	C₂×C₄	1	2	yes	—
29	C₄×#92418	C₂³	1	2	yes	—
30	C₈×Dic₆₁	C₂³	1	4	yes	—
31	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₂₂	C₂×C₈	1	4	yes	—
37	M₁₆×C₆₁	—	4	2	no	—
38	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
39	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
42	C₂²⋊Dic₆₁	C₂×C₈	1	4	yes	—
43	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
44	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
45	#11758196	C₂×C₈	1	8	yes	—
46	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
47	C₂×#92418	—	2	2	no	—
48	C₂×#92418	—	4	1	no	—
49	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
50	C₂×#92418	C₂²×C₄	1	2	yes	—
51	C₂×#92418	C₂⁴	1	1	yes	characteristic
52	#11758187	—	2	8	no	—
53	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₆₁	—	2	1	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	—	4	2	no	—
57	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
58	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
61	C₂×Dic₆₁	C₂²×C₈	1	8	yes	—
62	Dic₁₂₂	—	4	4	no	—
63	C₈×C₆₁	—	2	4	no	—
64	#92418	—	2	1	no	—
65	#92418	—	2	2	no	—
66	#92418	2₋¹⁺⁴	1	2	yes	—
67	#33501	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	—	2	2	no	—
69	C₂²×C₆₁	—	2	1	no	—
70	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	—	2	2	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	#519	1	1	yes	characteristic
75	C₄×C₆₁	#568	1	1	yes	characteristic
76	Dic₆₁	—	2	8	no	—
77	#2960	—	61	1	no	—
78	#2183	—	61	8	no	—
79	#2164	—	61	4	no	—
80	#2163	—	61	2	no	—
81	C₂×C₆₁	—	4	1	no	—
82	C₂×C₆₁	#2982	1	1	yes	characteristic
83	C₂×C₆₁	#3407	1	2	yes	—
84	#533	—	61	1	no	—
85	#446	—	61	4	no	—
86	C₈×Q₈	—	61	4	no	—
87	#424	—	61	2	no	—
88	#423	#6923	1	1	yes	characteristic
89	#405	—	61	2	no	—
90	C₂×C₄×C₈	—	61	1	no	—
91	C₄⋊C₁₆	—	61	4	no	—
92	C₄⋊C₁₆	—	61	8	no	—
93	C₂²⋊C₁₆	—	61	4	no	—
94	#346	—	61	4	no	—
95	#346	—	61	4	no	—
96	C₄×C₁₆	—	61	4	no	—
97	C₆₁	#33501	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	C₂×#6923	1	2	yes	—
99	C₂²×C₈	—	61	1	no	—
100	C₂²×C₈	—	61	1	no	—
101	C₄⋊Q₈	—	61	2	no	—
102	#125	—	61	2	no	—
103	C₂²⋊Q₈	—	61	4	no	—
104	C₄×Q₈	—	61	4	no	—
105	#117	—	61	2	no	—
106	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
107	C₂×C₁₆	—	61	8	no	—
108	C₄⋊C₈	—	61	8	no	—
109	C₄⋊C₈	C₂×#6923	1	4	yes	—
110	#97	—	61	4	no	—
111	C₄×C₈	—	61	4	no	—
112	C₄×C₈	—	61	2	no	—
113	C₄×C₈	—	61	2	no	—
114	C₂×Q₈	—	61	4	no	—
115	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
116	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	C₂×C₈	—	61	1	no	—
119	C₂×C₈	—	61	1	no	—
120	C₂×C₈	—	61	2	no	—
121	C₂×C₈	—	122	2	no	—
122	C₂×C₈	—	244	1	no	—
123	C₂×C₈	C₂²×#6923	1	4	yes	—
124	C₄⋊C₄	—	61	8	no	—
125	C₄⋊C₄	—	61	4	no	—
126	C₄⋊C₄	—	61	4	no	—
127	C₂²⋊C₄	—	61	4	no	—
128	C₄²	—	61	4	no	—
129	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
130	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
131	C₄²	C₂²×#6923	1	2	yes	—
132	C₁₆	—	122	8	no	—
133	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
134	Q₈	—	244	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	61	8	no	—
139	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
140	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
141	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
142	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₈	—	61	2	no	—
145	C₈	—	122	1	no	—
146	C₈	—	122	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	122	8	no	—
154	C₄	#16840830	1	1	yes	characteristic
155	C₄	#16840840	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	#21462714	1	1	yes	characteristic
158	C₂	#21462788	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21462236

C₂

yes

maximal, complemented

#21462185

C₂

yes

maximal

#21462180

C₂

yes

maximal

#21454749

C₂

yes

maximal, characteristic

#16840683

C₂²

yes

—

C₄⋊#11758187

C₂²

yes

—

C₂²⋊#11758187

C₂²

yes

—

#16840651

C₂²

yes

—

#16840650

C₂²

yes

—

#16840648

C₂²

yes

—

#16840319

C₄

yes

characteristic

#16839889

C₄

yes

—

#16839886

C₂²

yes

—

#16839790

C₄

yes

—

#16839720

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₆₁×#423

C₄

yes

characteristic

C₂×#11758187

C₂³

yes

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂³

yes

characteristic

#14519777

C₈

yes

—

#14519769

C₈

yes

—

#14519756

C₈

yes

—

C₄×Dic₁₂₂

C₂×C₄

yes

—

#14519736

C₂×C₄

yes

—

#14519706

C₂×C₄

yes

—

#14519687

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

C₂³

yes

—

C₈×Dic₆₁

C₂³

yes

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₂₂

C₂×C₈

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

—

C₂²⋊Dic₆₁

C₂×C₈

yes

—

C₄⋊Dic₆₁

C₂×C₈

yes

—

C₄⋊Dic₆₁

C₂×C₈

yes

—

#11758196

C₂×C₈

yes

—

C₄×Dic₆₁

C₂²×C₄

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂²×C₄

yes

characteristic

C₂×#92418

C₂²×C₄

yes

—

C₂×#92418

C₂⁴

yes

characteristic

#11758187

—

C₂³×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂³×C₄

yes

characteristic

C₂×Dic₆₁

C₂²×C₈

yes

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

#92418

—

#92418

2₋¹⁺⁴

yes

—

#33501

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂³×C₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#2982

yes

characteristic

C₂×C₆₁

#3407

yes

—

#533

—

#446

—

C₈×Q₈

—

—

yes

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#346

—

C₄×C₁₆

—

C₆₁

#33501

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×#6923

yes

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₄⋊Q₈

—

102

#125

—

103

C₂²⋊Q₈

—

104

C₄×Q₈

—

105

#117

—

106

C₂×C₄²

C₂×#6923

yes

characteristic

107

C₂×C₁₆

—

108

C₄⋊C₈

—

109

C₄⋊C₈

C₂×#6923

yes

—

110

#97

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₂×Q₈

—

115

C₂²×C₄

D₆₁⋊C₈

yes

—

116

C₂²×C₄

C₂²×#6923

yes

characteristic

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

—

122

C₂×C₈

—

244

—

123

C₂×C₈

C₂²×#6923

yes

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

D₆₁⋊C₈

yes

characteristic

130

C₄²

D₆₁⋊C₈

yes

characteristic

131

C₄²

C₂²×#6923

yes

—

132

C₁₆

—

122

—

133

C₂³

C₂×D₆₁⋊C₈

yes

characteristic

134

Q₈

—

244

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

140

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

141

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

142

C₂×C₄

C₂³×#6923

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

122

—

146

C₈

—

122

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂²×D₆₁⋊C₈

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

122

—

154

C₄

#16840830

yes

characteristic

155

C₄

#16840840

yes

characteristic

156

C₂

—

157

C₂

#21462714

yes

characteristic

158

C₂

#21462788

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#423	C₄
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	#423	#6923
hypocenter	C₆₁	#33501
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#6923

commutator, socle

C₂²×C₆₁

C₂³×C₈

Fitting

C₆₁×#423

C₄

prime radical

C₂³×C₆₁

C₂²×C₈

C₂⁴

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. #21462236	C₂	semidirect
97. C₆₁	#33501	semidirect

H ≅ G/N

type

1. #21462236

C₂

semidirect

97. C₆₁

#33501

semidirect

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

subgroup	index
#21462236	2
#21462185	2
#21462180	2
#21454749	2
#33501	61

subgroup

index

Subgroups — #30624478

Conjugacy classes of subgroups

Complements

Maximal subgroups