Group #30603681 — subgroups

Subgroups — #30603681

Order 15616.

330 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 172 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	C₄×D₆₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#21461023	C₂	1	1	yes	maximal, characteristic
3	#21460730	C₂	1	2	yes	maximal, complemented
4	C₁₆×#92420	C₂	1	2	yes	maximal, complemented
5	#21447412	C₂	1	1	yes	maximal, characteristic
6	C₂×D₆₁⋊C₁₆	C₄	1	2	yes	—
7	#16840780	C₄	1	2	yes	—
8	#16840648	C₂²	1	1	yes	characteristic
9	#16840648	C₂²	1	1	yes	characteristic
10	C₄×#11758187	C₄	1	2	yes	—
11	#16840603	C₄	1	2	yes	—
12	C₈×#92420	C₂²	1	2	yes	—
13	C₂×#14519716	C₄	1	2	yes	—
14	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
15	#16839538	C₄	1	1	yes	characteristic
16	C₁₆×Dic₆₁	C₂²	1	2	yes	—
17	C₆₁×#346	C₄	1	1	yes	characteristic
18	D₆₁⋊C₁₆	C₈	1	8	yes	complemented
19	C₂×#11758187	C₂×C₄	1	2	yes	—
20	C₂×#11758187	C₂×C₄	1	2	yes	—
21	#14519844	C₈	1	4	yes	—
22	#14519834	C₈	1	4	yes	—
23	C₄×#92420	C₂×C₄	1	4	yes	—
24	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
27	#14519716	C₈	1	8	yes	complemented
28	C₂×#11758185	C₂×C₄	1	2	yes	—
29	C₄×#92418	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
32	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₆₁	C₄²	1	1	yes	characteristic
35	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
36	C₂×#92420	C₂×C₈	1	8	yes	—
37	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₆₁	C₄²	1	1	yes	characteristic
40	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
41	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×#92418	C₄²	1	1	yes	characteristic
43	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
44	C₈×D₆₁	C₂×C₈	1	4	yes	—
45	C₈×D₆₁	C₂×C₈	1	4	yes	—
46	#11758187	C₂×C₈	1	4	yes	—
47	#11758187	C₂×C₈	1	4	yes	—
48	C₁₆×C₆₁	C₂×C₈	1	4	yes	—
49	#11758185	C₂×C₈	1	4	yes	—
50	C₂²×D₆₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
53	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
54	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
55	C₄×D₆₁	—	2	4	no	—
56	C₄×D₆₁	C₄×C₈	1	2	yes	—
57	C₄×D₆₁	C₄×C₈	1	2	yes	—
58	#92420	—	2	8	no	—
59	C₈×C₆₁	C₄×C₈	1	2	yes	—
60	C₈×C₆₁	C₂²×C₈	1	2	yes	—
61	#92418	C₄×C₈	1	2	yes	—
62	#92418	C₂²×C₈	1	2	yes	—
63	#11399	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂×D₆₁	—	2	2	no	—
66	C₂×D₆₁	C₈²	1	2	yes	—
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
70	Dic₆₁	C₈²	1	2	yes	—
71	Dic₆₁	#504	1	4	yes	—
72	C₄×M₃₂	—	61	1	no	—
73	C₂×C₄×C₁₆	—	61	1	no	—
74	#1450	—	61	1	no	—
75	#1311	—	61	2	no	—
76	C₈×C₁₆	—	61	2	no	—
77	C₂×C₆₁	C₂×C₈²	1	1	yes	characteristic
78	C₂×C₆₁	#2108	1	2	yes	—
79	D₆₁	—	2	2	no	—
80	C₂×M₃₂	—	61	2	no	—
81	C₂²×C₁₆	—	61	2	no	—
82	#431	—	61	2	no	—
83	C₂×C₄×C₈	—	61	1	no	—
84	#346	—	61	1	no	—
85	#346	#6923	1	1	yes	characteristic
86	C₄×C₁₆	—	61	1	no	—
87	C₄×C₁₆	—	61	1	no	—
88	C₄×C₁₆	—	61	2	no	—
89	C₄×C₁₆	—	61	2	no	—
90	#322	—	61	2	no	—
91	C₈²	—	61	2	no	—
92	C₆₁	#11399	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	61	1	no	—
94	C₂²×C₈	—	61	1	no	—
95	C₂×C₄²	—	61	1	no	—
96	M₃₂	—	61	8	no	—
97	C₂×C₁₆	—	61	2	no	—
98	C₂×C₁₆	—	61	2	no	—
99	C₂×C₁₆	—	61	8	no	—
100	C₂×C₁₆	—	61	2	no	—
101	C₂×C₁₆	C₂×#6923	1	2	yes	—
102	C₄⋊C₈	—	61	4	no	—
103	C₂²⋊C₈	—	61	4	no	—
104	C₄×C₈	—	61	4	no	—
105	C₄×C₈	—	61	1	no	—
106	C₄×C₈	—	61	1	no	—
107	C₄×C₈	—	61	1	no	—
108	C₄×C₈	C₂×#6923	1	1	yes	characteristic
109	C₂²×C₄	—	61	1	no	—
110	C₂²×C₄	—	61	2	no	—
111	C₂×C₈	—	61	1	no	—
112	C₂×C₈	—	61	1	no	—
113	C₂×C₈	—	61	4	no	—
114	C₂×C₈	—	61	4	no	—
115	C₂×C₈	—	61	8	no	—
116	C₂×C₈	C₄×#6923	1	1	yes	characteristic
117	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
118	C₄²	—	61	2	no	—
119	C₄²	—	61	1	no	—
120	C₄²	C₄×#6923	1	1	yes	characteristic
121	C₁₆	—	61	4	no	—
122	C₁₆	—	61	4	no	—
123	C₁₆	—	61	4	no	—
124	C₁₆	D₆₁⋊C₈	1	4	yes	—
125	C₂³	—	61	1	no	—
126	C₂×C₄	—	61	1	no	—
127	C₂×C₄	—	61	2	no	—
128	C₂×C₄	—	61	2	no	—
129	C₂×C₄	—	61	2	no	—
130	C₂×C₄	—	122	4	no	—
131	C₂×C₄	C₈×#6923	1	2	yes	—
132	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
133	C₈	—	61	2	no	—
134	C₈	—	61	2	no	—
135	C₈	—	122	8	no	—
136	C₈	C₈×#6923	1	2	yes	—
137	C₈	C₂×D₆₁⋊C₈	1	2	yes	—
138	C₂²	—	61	2	no	—
139	C₂²	—	122	2	no	—
140	C₂²	C₂×C₈×#6923	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	61	4	no	—
143	C₄	—	61	2	no	—
144	C₄	C₂×C₈×#6923	1	1	yes	characteristic
145	C₄	C₄×D₆₁⋊C₈	1	1	yes	characteristic
146	C₂	—	122	2	no	—
147	C₂	C₈×D₆₁⋊C₈	1	1	yes	characteristic
148	C₂	#21462031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₆₁⋊C₁₆

C₂

yes

maximal, characteristic

#21461023

C₂

yes

maximal, characteristic

#21460730

C₂

yes

maximal, complemented

C₁₆×#92420

C₂

yes

maximal, complemented

#21447412

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₁₆

C₄

yes

—

#16840780

C₄

yes

—

#16840648

C₂²

yes

characteristic

#16840648

C₂²

yes

characteristic

C₄×#11758187

C₄

yes

—

#16840603

C₄

yes

—

C₈×#92420

C₂²

yes

—

C₂×#14519716

C₄

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839538

C₄

yes

characteristic

C₁₆×Dic₆₁

C₂²

yes

—

C₆₁×#346

C₄

yes

characteristic

D₆₁⋊C₁₆

C₈

yes

complemented

C₂×#11758187

C₂×C₄

yes

—

C₂×#11758187

C₂×C₄

yes

—

#14519844

C₈

yes

—

#14519834

C₈

yes

—

C₄×#92420

C₂×C₄

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₄²×D₆₁

C₂×C₄

yes

characteristic

#14519716

C₈

yes

complemented

C₂×#11758185

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₄²

yes

characteristic

C₂×C₄×D₆₁

C₂×C₈

yes

—

C₂×#92420

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₄²

yes

characteristic

C₄×Dic₆₁

C₂×C₈

yes

—

C₄×Dic₆₁

C₂²×C₄

yes

characteristic

C₂×#92418

C₄²

yes

characteristic

C₂×#92418

C₂²×C₄

yes

characteristic

C₈×D₆₁

C₂×C₈

yes

—

C₈×D₆₁

C₂×C₈

yes

—

#11758187

C₂×C₈

yes

—

#11758187

C₂×C₈

yes

—

C₁₆×C₆₁

C₂×C₈

yes

—

#11758185

C₂×C₈

yes

—

C₂²×D₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

C₄×C₈

yes

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×Dic₆₁

C₄×C₈

yes

characteristic

C₂×Dic₆₁

C₂²×C₈

yes

—

C₄×D₆₁

—

C₄×D₆₁

C₄×C₈

yes

—

C₄×D₆₁

C₄×C₈

yes

—

#92420

—

C₈×C₆₁

C₄×C₈

yes

—

C₈×C₆₁

C₂²×C₈

yes

—

#92418

C₄×C₈

yes

—

#92418

C₂²×C₈

yes

—

#11399

—

maximal, Sylow

C₂²×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

C₈²

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄×C₈

yes

characteristic

C₄×C₆₁

C₂×C₄×C₈

yes

characteristic

Dic₆₁

C₈²

yes

—

Dic₆₁

#504

yes

—

C₄×M₃₂

—

C₂×C₄×C₁₆

—

#1450

—

#1311

—

C₈×C₁₆

—

C₂×C₆₁

C₂×C₈²

yes

characteristic

C₂×C₆₁

#2108

yes

—

D₆₁

—

C₂×M₃₂

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

—

C₈²

—

C₆₁

#11399

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂×#6923

yes

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₂×#6923

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄×#6923

yes

characteristic

117

C₂×C₈

C₂²×#6923

yes

characteristic

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×#6923

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

D₆₁⋊C₈

yes

—

125

C₂³

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

122

—

131

C₂×C₄

C₈×#6923

yes

—

132

C₂×C₄

C₂×C₄×#6923

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

122

—

136

C₈

C₈×#6923

yes

—

137

C₈

C₂×D₆₁⋊C₈

yes

—

138

C₂²

—

139

C₂²

—

122

—

140

C₂²

C₂×C₈×#6923

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

C₂×C₈×#6923

yes

characteristic

145

C₄

C₄×D₆₁⋊C₈

yes

characteristic

146

C₂

—

122

—

147

C₂

C₈×D₆₁⋊C₈

yes

characteristic

148

C₂

#21462031

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6923
commutator	C₂×C₆₁	C₂×C₈²
Fitting	C₆₁×#346	C₄
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂²×D₆₁	C₄×C₈
cosocle	C₈×Dic₆₁	C₂³
hypercenter	#346	#6923
hypocenter	C₆₁	#11399
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#6923

commutator

C₂×C₆₁

C₂×C₈²

Fitting

C₆₁×#346

C₄

socle

C₂²×C₆₁

C₂×C₄×C₈

prime radical

C₂²×D₆₁

C₄×C₈

cosocle

C₈×Dic₆₁

C₂³

C₆₁

G ▷ C₂×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
3. #21460730	C₂	semidirect
4. C₁₆×#92420	C₂	semidirect
18. D₆₁⋊C₁₆	C₈	semidirect
27. #14519716	C₈	semidirect
92. C₆₁	#11399	semidirect

H ≅ G/N

type

3. #21460730

C₂

semidirect

4. C₁₆×#92420

C₂

semidirect

18. D₆₁⋊C₁₆

C₈

semidirect

27. #14519716

C₈

semidirect

92. C₆₁

#11399

semidirect

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

subgroup	index
C₄×D₆₁⋊C₁₆	2
#21461023	2
#21460730	2
C₁₆×#92420	2
#21447412	2
#11399	61

subgroup

index

C₄×D₆₁⋊C₁₆

C₁₆×#92420

Subgroups — #30603681

Conjugacy classes of subgroups

Complements

Maximal subgroups