Group #30603679 — subgroups

Subgroups — #30603679

Order 15616.

330 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 176 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×#14519852	C₂	1	1	yes	maximal, characteristic
2	C₄×D₆₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#21460731	C₂	1	2	yes	maximal, complemented
4	C₈×#11758187	C₂	1	2	yes	maximal, complemented
5	#21444104	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519852	C₄	1	2	yes	—
7	C₂×D₆₁⋊C₁₆	C₄	1	2	yes	—
8	#16840651	C₄	1	2	yes	—
9	#16840648	C₄	1	2	yes	—
10	C₄×#11758187	C₂²	1	1	yes	characteristic
11	C₄×#11758187	C₂²	1	1	yes	characteristic
12	C₄×#11758187	C₂²	1	2	yes	—
13	#16839813	C₄	1	2	yes	—
14	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
15	#16839483	C₄	1	1	yes	characteristic
16	C₈×#92418	C₂²	1	2	yes	—
17	C₆₁×#322	C₄	1	1	yes	characteristic
18	#14519852	C₈	1	8	yes	complemented
19	D₆₁⋊C₁₆	C₈	1	8	yes	complemented
20	C₂×#11758187	C₂×C₄	1	2	yes	—
21	C₂×#11758187	C₂×C₄	1	2	yes	—
22	C₂×#11758187	C₂×C₄	1	4	yes	—
23	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
26	#14519706	C₈	1	4	yes	—
27	#14519689	C₈	1	4	yes	—
28	C₄×#92418	C₂×C₄	1	2	yes	—
29	C₄×#92418	C₂³	1	1	yes	characteristic
30	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₆₁	C₄²	1	1	yes	characteristic
35	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
39	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
41	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
42	C₂×#92418	C₂×C₈	1	4	yes	—
43	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
44	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
45	C₈×D₆₁	C₂×C₈	1	4	yes	—
46	C₈×D₆₁	C₂×C₈	1	4	yes	—
47	#11758187	C₂×C₈	1	4	yes	—
48	#11758187	C₂×C₈	1	4	yes	—
49	#11758187	C₂×C₈	1	8	yes	—
50	C₂²×D₆₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
53	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
54	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
55	C₄×D₆₁	—	2	4	no	—
56	C₄×D₆₁	C₄×C₈	1	2	yes	—
57	C₄×D₆₁	C₄×C₈	1	2	yes	—
58	C₈×C₆₁	—	2	4	no	—
59	C₈×C₆₁	C₄×C₈	1	2	yes	—
60	C₈×C₆₁	C₄×C₈	1	2	yes	—
61	#92418	—	2	4	no	—
62	#92418	C₂²×C₈	1	2	yes	—
63	#92418	C₂²×C₈	1	2	yes	—
64	#11399	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
66	C₂×D₆₁	—	2	2	no	—
67	C₂×D₆₁	C₈²	1	2	yes	—
68	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
70	C₄×C₆₁	#504	1	4	yes	—
71	Dic₆₁	—	2	2	no	—
72	Dic₆₁	C₈²	1	2	yes	—
73	C₄×M₃₂	—	61	1	no	—
74	C₂×C₄×C₁₆	—	61	1	no	—
75	#1450	—	61	1	no	—
76	#1311	—	61	2	no	—
77	C₈×C₁₆	—	61	2	no	—
78	C₂×C₆₁	C₂×C₈²	1	1	yes	characteristic
79	C₂×C₆₁	#2108	1	2	yes	—
80	D₆₁	—	2	2	no	—
81	C₂×M₃₂	—	61	2	no	—
82	C₂²×C₁₆	—	61	2	no	—
83	#431	—	61	2	no	—
84	C₂×C₄×C₈	—	61	1	no	—
85	#346	—	61	2	no	—
86	C₄×C₁₆	—	61	1	no	—
87	C₄×C₁₆	—	61	2	no	—
88	C₄×C₁₆	—	61	1	no	—
89	C₄×C₁₆	—	61	2	no	—
90	#322	—	61	1	no	—
91	#322	#6923	1	1	yes	characteristic
92	C₈²	—	61	2	no	—
93	C₆₁	#11399	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	61	1	no	—
95	C₂²×C₈	—	61	1	no	—
96	C₂×C₄²	—	61	1	no	—
97	M₃₂	—	61	8	no	—
98	C₂×C₁₆	—	61	2	no	—
99	C₂×C₁₆	—	61	2	no	—
100	C₂×C₁₆	—	61	4	no	—
101	C₂×C₁₆	—	61	8	no	—
102	C₄⋊C₈	—	61	4	no	—
103	C₂²⋊C₈	—	61	4	no	—
104	C₄×C₈	—	61	1	no	—
105	C₄×C₈	—	61	1	no	—
106	C₄×C₈	—	61	1	no	—
107	C₄×C₈	—	61	2	no	—
108	C₄×C₈	C₂×#6923	1	2	yes	—
109	C₄×C₈	C₂×#6923	1	1	yes	characteristic
110	C₂²×C₄	—	61	1	no	—
111	C₂²×C₄	—	61	2	no	—
112	C₂×C₈	—	61	1	no	—
113	C₂×C₈	—	61	1	no	—
114	C₂×C₈	—	61	4	no	—
115	C₂×C₈	—	61	4	no	—
116	C₂×C₈	—	61	4	no	—
117	C₂×C₈	D₆₁⋊C₈	1	4	yes	—
118	C₂×C₈	C₄×#6923	1	1	yes	characteristic
119	C₂×C₈	C₄×#6923	1	1	yes	characteristic
120	C₄²	—	61	1	no	—
121	C₄²	—	61	2	no	—
122	C₄²	C₂²×#6923	1	1	yes	characteristic
123	C₁₆	—	61	4	no	—
124	C₁₆	—	61	4	no	—
125	C₁₆	—	61	8	no	—
126	C₂³	—	61	1	no	—
127	C₂×C₄	—	61	1	no	—
128	C₂×C₄	—	61	2	no	—
129	C₂×C₄	—	61	2	no	—
130	C₂×C₄	—	61	2	no	—
131	C₂×C₄	—	122	4	no	—
132	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
133	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
134	C₈	—	2	4	no	—
135	C₈	—	61	2	no	—
136	C₈	—	61	2	no	—
137	C₈	—	122	4	no	—
138	C₈	C₈×#6923	1	2	yes	—
139	C₈	C₈×#6923	1	2	yes	—
140	C₂²	—	61	2	no	—
141	C₂²	—	122	2	no	—
142	C₂²	C₄×D₆₁⋊C₈	1	1	yes	characteristic
143	C₄	—	61	2	no	—
144	C₄	—	122	2	no	—
145	C₄	C₂×C₈×#6923	1	1	yes	characteristic
146	C₄	C₂×C₈×#6923	1	1	yes	characteristic
147	C₄	#16840821	1	4	yes	—
148	C₂	—	122	2	no	—
149	C₂	C₈×D₆₁⋊C₈	1	1	yes	characteristic
150	C₂	#21462029	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519852

C₂

yes

maximal, characteristic

C₄×D₆₁⋊C₁₆

C₂

yes

maximal, characteristic

#21460731

C₂

yes

maximal, complemented

C₈×#11758187

C₂

yes

maximal, complemented

#21444104

C₂

yes

maximal, characteristic

C₂×#14519852

C₄

yes

—

C₂×D₆₁⋊C₁₆

C₄

yes

—

#16840651

C₄

yes

—

#16840648

C₄

yes

—

C₄×#11758187

C₂²

yes

characteristic

C₄×#11758187

C₂²

yes

characteristic

C₄×#11758187

C₂²

yes

—

#16839813

C₄

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839483

C₄

yes

characteristic

C₈×#92418

C₂²

yes

—

C₆₁×#322

C₄

yes

characteristic

#14519852

C₈

yes

complemented

D₆₁⋊C₁₆

C₈

yes

complemented

C₂×#11758187

C₂×C₄

yes

—

C₂×#11758187

C₂×C₄

yes

—

C₂×#11758187

C₂×C₄

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₄²×D₆₁

C₂×C₄

yes

characteristic

#14519706

C₈

yes

—

#14519689

C₈

yes

—

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

C₂³

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₄²

yes

characteristic

C₂×C₄×D₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₄²

yes

characteristic

C₄×Dic₆₁

C₂×C₈

yes

—

C₂×#92418

C₂×C₈

yes

—

C₂×#92418

C₂²×C₄

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

—

#11758187

C₂×C₈

yes

—

C₂²×D₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×Dic₆₁

C₄×C₈

yes

—

C₂×Dic₆₁

C₄×C₈

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₄×C₈

yes

—

C₄×D₆₁

C₄×C₈

yes

—

C₈×C₆₁

—

C₈×C₆₁

C₄×C₈

yes

—

C₈×C₆₁

C₄×C₈

yes

—

#92418

—

#92418

C₂²×C₈

yes

—

#92418

C₂²×C₈

yes

—

#11399

—

maximal, Sylow

C₂²×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

C₈²

yes

—

C₄×C₆₁

C₂×C₄×C₈

yes

characteristic

C₄×C₆₁

C₂×C₄×C₈

yes

characteristic

C₄×C₆₁

#504

yes

—

Dic₆₁

—

Dic₆₁

C₈²

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

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

—

yes

C₈²

—

C₆₁

#11399

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₂×#6923

yes

—

109

C₄×C₈

C₂×#6923

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

D₆₁⋊C₈

yes

—

118

C₂×C₈

C₄×#6923

yes

characteristic

119

C₂×C₈

C₄×#6923

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₄²

C₂²×#6923

yes

characteristic

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

122

—

132

C₂×C₄

C₂×C₄×#6923

yes

characteristic

133

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

122

—

138

C₈

C₈×#6923

yes

—

139

C₈

C₈×#6923

yes

—

140

C₂²

—

141

C₂²

—

122

—

142

C₂²

C₄×D₆₁⋊C₈

yes

characteristic

143

C₄

—

144

C₄

—

122

—

145

C₄

C₂×C₈×#6923

yes

characteristic

146

C₄

C₂×C₈×#6923

yes

characteristic

147

C₄

#16840821

yes

—

148

C₂

—

122

—

149

C₂

C₈×D₆₁⋊C₈

yes

characteristic

150

C₂

#21462029

yes

—

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6923
commutator	C₂×C₆₁	C₂×C₈²
Fitting	C₆₁×#322	C₄
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂²×D₆₁	C₄×C₈
cosocle	C₄×#92418	C₂³
hypercenter	#322	#6923
hypocenter	C₆₁	#11399
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×#6923

commutator

C₂×C₆₁

C₂×C₈²

Fitting

C₆₁×#322

C₄

socle

C₂²×C₆₁

C₂×C₄×C₈

prime radical

C₂²×D₆₁

C₄×C₈

C₂³

C₆₁

G ▷ C₂×C₆₁ ▷ C₆₁

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
3. #21460731	C₂	semidirect
4. C₈×#11758187	C₂	semidirect
18. #14519852	C₈	semidirect
19. D₆₁⋊C₁₆	C₈	semidirect
93. C₆₁	#11399	semidirect

H ≅ G/N

type

3. #21460731

C₂

semidirect

4. C₈×#11758187

C₂

semidirect

18. #14519852

C₈

semidirect

19. D₆₁⋊C₁₆

C₈

semidirect

93. C₆₁

#11399

semidirect

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

subgroup	index
C₄×#14519852	2
C₄×D₆₁⋊C₁₆	2
#21460731	2
C₈×#11758187	2
#21444104	2
#11399	61

subgroup

index

C₄×D₆₁⋊C₁₆

Subgroups — #30603679

Conjugacy classes of subgroups

Complements

Maximal subgroups