Group #30603757 — subgroups

Subgroups — #30603757

Order 15616.

290 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 132 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	#21462178	C₂	1	2	yes	maximal
2	#21460734	C₂	1	2	yes	maximal, complemented
3	#21460731	C₂	1	2	yes	maximal, complemented
4	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
5	#16840683	C₄	1	4	yes	—
6	#16840657	C₄	1	4	yes	—
7	C₄×#11758187	C₂²	1	4	yes	—
8	C₄×C₈×D₆₁	C₄	1	1	yes	characteristic
9	C₄×C₈×D₆₁	C₄	1	1	yes	characteristic
10	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
11	C₈×#92418	C₄	1	1	yes	characteristic
12	C₈×#92418	C₂²	1	1	yes	characteristic
13	C₈×#92418	C₂²	1	1	yes	characteristic
14	C₈²×C₆₁	C₄	1	1	yes	characteristic
15	C₂×#11758187	C₂×C₄	1	8	yes	—
16	C₂×C₈×D₆₁	C₈	1	2	yes	—
17	C₂×C₈×D₆₁	C₈	1	2	yes	—
18	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
19	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
20	C₄×#92418	C₂×C₄	1	1	yes	characteristic
21	C₄×#92418	C₂×C₄	1	1	yes	characteristic
22	C₄×#92418	C₂³	1	1	yes	characteristic
23	C₈×Dic₆₁	C₈	1	2	yes	—
24	C₈×Dic₆₁	C₈	1	2	yes	—
25	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₆₁	C₄²	1	1	yes	characteristic
30	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₄²	1	2	yes	—
33	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
35	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
37	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
38	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
39	C₂×#92418	C₂×C₈	1	2	yes	—
40	C₂×#92418	C₂×C₈	1	2	yes	—
41	C₂×#92418	C₂²×C₄	1	2	yes	—
42	C₈×D₆₁	—	2	4	no	—
43	C₈×D₆₁	—	2	4	no	—
44	C₈×D₆₁	—	2	4	no	—
45	#11758187	—	2	8	no	—
46	C₂²×D₆₁	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
51	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
52	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
53	C₄×D₆₁	—	2	2	no	—
54	C₄×D₆₁	—	2	2	no	—
55	C₄×D₆₁	M₃₂	1	4	yes	—
56	C₈×C₆₁	—	2	2	no	—
57	C₈×C₆₁	—	2	2	no	—
58	C₈×C₆₁	—	2	2	no	—
59	#92418	—	2	2	no	—
60	#92418	M₃₂	1	4	yes	—
61	#92418	#131	1	4	yes	—
62	#11411	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂×D₆₁	—	2	1	no	—
65	C₂×D₆₁	—	2	1	no	—
66	C₂×D₆₁	#346	1	2	yes	—
67	C₄×C₆₁	#405	1	2	yes	—
68	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
69	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
70	Dic₆₁	—	2	1	no	—
71	Dic₆₁	—	2	1	no	—
72	Dic₆₁	#346	1	2	yes	—
73	#2162	—	61	2	no	—
74	C₂×C₈²	—	61	1	no	—
75	#1312	—	61	2	no	—
76	#1311	—	61	2	no	—
77	C₂×C₆₁	#1450	1	1	yes	characteristic
78	C₂×C₆₁	C₂×#346	1	1	yes	characteristic
79	C₂×C₆₁	C₄×M₃₂	1	1	yes	characteristic
80	D₆₁	—	2	2	no	—
81	C₂×C₄×C₈	—	61	1	no	—
82	C₂×C₄×C₈	—	61	1	no	—
83	C₂×C₄×C₈	—	61	1	no	—
84	C₄⋊C₁₆	—	61	4	no	—
85	C₂²⋊C₁₆	—	61	4	no	—
86	C₄×C₁₆	—	61	4	no	—
87	C₈²	—	61	1	no	—
88	C₈²	—	61	1	no	—
89	C₈²	—	61	1	no	—
90	C₈²	#6923	1	1	yes	characteristic
91	C₆₁	#11411	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	61	2	no	—
93	C₂²×C₈	—	61	2	no	—
94	C₂²×C₈	—	61	2	no	—
95	C₂×C₄²	—	61	1	no	—
96	C₂×C₁₆	—	61	8	no	—
97	C₄×C₈	—	61	2	no	—
98	C₄×C₈	—	61	2	no	—
99	C₄×C₈	—	61	2	no	—
100	C₄×C₈	—	61	1	no	—
101	C₄×C₈	—	61	1	no	—
102	C₄×C₈	—	61	1	no	—
103	C₄×C₈	C₂×#6923	1	1	yes	characteristic
104	C₄×C₈	C₂×#6923	1	1	yes	characteristic
105	C₄×C₈	C₂×#6923	1	1	yes	characteristic
106	C₂²×C₄	—	61	1	no	—
107	C₂²×C₄	—	61	1	no	—
108	C₂²×C₄	—	61	1	no	—
109	C₂×C₈	—	61	2	no	—
110	C₂×C₈	—	61	2	no	—
111	C₂×C₈	—	61	2	no	—
112	C₂×C₈	—	122	4	no	—
113	C₂×C₈	—	122	4	no	—
114	C₂×C₈	—	122	4	no	—
115	C₂×C₈	D₆₁⋊C₈	1	2	yes	—
116	C₂×C₈	D₆₁⋊C₈	1	2	yes	—
117	C₂×C₈	C₄×#6923	1	2	yes	—
118	C₄²	—	61	1	no	—
119	C₄²	—	61	1	no	—
120	C₄²	—	61	1	no	—
121	C₄²	C₂²×#6923	1	1	yes	characteristic
122	C₁₆	—	122	8	no	—
123	C₂³	—	61	1	no	—
124	C₂×C₄	—	61	1	no	—
125	C₂×C₄	—	61	4	no	—
126	C₂×C₄	—	61	1	no	—
127	C₂×C₄	—	61	1	no	—
128	C₂×C₄	—	122	2	no	—
129	C₂×C₄	—	122	2	no	—
130	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
131	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
132	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	2	2	no	—
136	C₈	—	61	4	no	—
137	C₈	—	61	4	no	—
138	C₈	—	122	2	no	—
139	C₂²	—	61	2	no	—
140	C₂²	—	122	1	no	—
141	C₂²	—	122	1	no	—
142	C₂²	C₄×D₆₁⋊C₈	1	1	yes	characteristic
143	C₄	—	61	2	no	—
144	C₄	—	122	1	no	—
145	C₄	—	122	1	no	—
146	C₄	#16840725	1	2	yes	—
147	C₄	D₆₁⋊M₃₂	1	2	yes	—
148	C₄	D₆₁⋊M₃₂	1	2	yes	—
149	C₂	—	122	2	no	—
150	C₂	#21461023	1	1	yes	characteristic
151	C₂	#21462020	1	1	yes	characteristic
152	C₂	#21462026	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462178

C₂

yes

maximal

#21460734

C₂

yes

maximal, complemented

#21460731

C₂

yes

maximal, complemented

C₈²×D₆₁

C₂

yes

maximal, characteristic

#16840683

C₄

yes

—

#16840657

C₄

yes

—

C₄×#11758187

C₂²

yes

—

C₄×C₈×D₆₁

C₄

yes

characteristic

C₄×C₈×D₆₁

C₄

yes

characteristic

C₄×C₈×D₆₁

C₂²

yes

characteristic

C₈×#92418

C₄

yes

characteristic

C₈×#92418

C₂²

yes

characteristic

C₈×#92418

C₂²

yes

characteristic

C₈²×C₆₁

C₄

yes

characteristic

C₂×#11758187

C₂×C₄

yes

—

C₂×C₈×D₆₁

C₈

yes

—

C₂×C₈×D₆₁

C₈

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₄²×D₆₁

C₂×C₄

yes

characteristic

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂³

yes

characteristic

C₈×Dic₆₁

C₈

yes

—

C₈×Dic₆₁

C₈

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₄²

yes

characteristic

C₂×C₄×D₆₁

C₂×C₈

yes

characteristic

C₂×C₄×D₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₄²

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₄²

yes

characteristic

C₄×Dic₆₁

C₂×C₈

yes

characteristic

C₄×Dic₆₁

C₂×C₈

yes

characteristic

C₂×#92418

C₂×C₈

yes

—

C₂×#92418

C₂×C₈

yes

—

C₂×#92418

C₂²×C₄

yes

—

C₈×D₆₁

—

C₈×D₆₁

—

C₈×D₆₁

—

#11758187

—

C₂²×D₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×Dic₆₁

C₄×C₈

yes

characteristic

C₂×Dic₆₁

C₄×C₈

yes

characteristic

C₂×Dic₆₁

C₄×C₈

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

M₃₂

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

M₃₂

yes

—

#92418

#131

yes

—

#11411

—

maximal, Sylow

C₂²×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#346

yes

—

C₄×C₆₁

#405

yes

—

C₄×C₆₁

C₂×M₃₂

yes

—

C₄×C₆₁

C₂×M₃₂

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#346

yes

—

#2162

—

C₂×C₈²

—

#1312

—

#1311

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

C₄×M₃₂

yes

characteristic

D₆₁

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

—

C₈²

#6923

yes

characteristic

C₆₁

#11411

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×#6923

yes

characteristic

104

C₄×C₈

C₂×#6923

yes

characteristic

105

C₄×C₈

C₂×#6923

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

122

—

113

C₂×C₈

—

122

—

114

C₂×C₈

—

122

—

115

C₂×C₈

D₆₁⋊C₈

yes

—

116

C₂×C₈

D₆₁⋊C₈

yes

—

117

C₂×C₈

C₄×#6923

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₂²×#6923

yes

characteristic

122

C₁₆

—

122

—

123

C₂³

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

122

—

129

C₂×C₄

—

122

—

130

C₂×C₄

C₂×C₄×#6923

yes

characteristic

131

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

132

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

122

—

139

C₂²

—

140

C₂²

—

122

—

141

C₂²

—

122

—

142

C₂²

C₄×D₆₁⋊C₈

yes

characteristic

143

C₄

—

144

C₄

—

122

—

145

C₄

—

122

—

146

C₄

#16840725

yes

—

147

C₄

D₆₁⋊M₃₂

yes

—

148

C₄

D₆₁⋊M₃₂

yes

—

149

C₂

—

122

—

150

C₂

#21461023

yes

characteristic

151

C₂

#21462020

yes

characteristic

152

C₂

#21462026

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6923
commutator, socle	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₈²×C₆₁	C₄
prime radical	C₂²×D₆₁	C₄×C₈
cosocle	C₄×#92418	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#11411
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₈²

subgroup

G/H

center, Frattini

C₄²

C₂²×#6923

commutator, socle

C₂²×C₆₁

C₂×C₄×C₈

Fitting

C₈²×C₆₁

C₄

prime radical

C₂²×D₆₁

C₄×C₈

C₂³

C₈²

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
2. #21460734	C₂	semidirect
3. #21460731	C₂	semidirect
91. C₆₁	#11411	semidirect

H ≅ G/N

type

2. #21460734

C₂

semidirect

3. #21460731

C₂

semidirect

91. C₆₁

#11411

semidirect

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

subgroup	index
#21462178	2
#21460734	2
#21460731	2
C₈²×D₆₁	2
#11411	61

subgroup

index

C₈²×D₆₁

Subgroups — #30603757

Conjugacy classes of subgroups

Complements

Maximal subgroups