Group #30605965 — subgroups

Subgroups — #30605965

Order 15616.

252 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 86 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	#21462198	C₂	1	2	yes	maximal, complemented
2	#21460841	C₂	1	1	yes	maximal, characteristic, complemented
3	#21460770	C₂	1	2	yes	maximal, complemented
4	#21460732	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444613	C₂	1	1	yes	maximal, characteristic
6	C₂×D₆₁⋊C₁₆	—	2	2	no	—
7	C₄⋊#11758187	C₂²	1	2	yes	—
8	#16840657	—	2	2	no	—
9	C₄×#11758187	C₂²	1	1	yes	characteristic
10	C₄×#11758187	C₂²	1	1	yes	characteristic
11	C₄×D₄₈₈	C₄	1	1	yes	characteristic
12	#16839827	C₂²	1	2	yes	—
13	#16839489	C₄	1	2	yes	—
14	C₈×#92418	C₂²	1	1	yes	characteristic
15	C₆₁×#335	C₄	1	1	yes	characteristic
16	D₆₁⋊C₁₆	—	4	4	no	—
17	C₂×#11758187	—	2	2	no	—
18	C₂×#11758187	—	2	1	no	—
19	C₂×#11758187	D₄	1	2	yes	—
20	C₂×D₄₈₈	C₈	1	1	yes	characteristic
21	C₂×C₈×D₆₁	—	2	2	no	—
22	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
23	#14519711	C₈	1	1	yes	characteristic
24	#14519699	C₈	1	2	yes	—
25	#14519689	—	2	2	no	—
26	C₄×#92418	C₂×C₄	1	1	yes	characteristic
27	C₄×#92418	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
31	C₂×C₄×D₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	#11758198	—	2	2	no	—
37	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
38	C₂×#92418	—	2	1	no	—
39	C₂×#92418	C₂×D₄	1	1	yes	characteristic
40	C₂×#92418	#41	1	1	yes	characteristic
41	D₄₈₈	C₁₆	1	4	yes	complemented
42	C₈×D₆₁	—	4	2	no	—
43	C₈×D₆₁	—	4	2	no	—
44	#11758187	—	2	2	no	—
45	#11758187	—	4	1	no	—
46	#11758187	—	4	2	no	—
47	C₂²×D₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	2	no	—
52	D₂₄₄	—	2	2	no	—
53	D₂₄₄	C₂×C₁₆	1	4	yes	complemented
54	C₄×D₆₁	—	4	2	no	—
55	C₄×D₆₁	—	4	2	no	—
56	C₈×C₆₁	—	2	1	no	—
57	C₈×C₆₁	—	4	2	no	—
58	C₈×C₆₁	C₂×C₁₆	1	2	yes	complemented
59	#92418	—	2	1	no	—
60	#92418	—	4	1	no	—
61	#92418	AGL(1,ℤ/8)	1	2	yes	—
62	#12062	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
64	C₂×D₆₁	—	2	4	no	—
65	C₂×D₆₁	—	4	2	no	—
66	C₂×D₆₁	—	4	2	no	—
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	#442	1	1	yes	characteristic
69	C₄×C₆₁	#444	1	1	yes	characteristic
70	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
71	C₄×C₆₁	#504	1	1	yes	characteristic
72	Dic₆₁	—	4	2	no	—
73	D₄×C₁₆	—	61	2	no	—
74	C₈×D₈	—	61	1	no	—
75	#1367	—	61	1	no	—
76	D₄⋊C₁₆	—	61	2	no	—
77	#1312	—	61	1	no	—
78	C₂×C₆₁	#1580	1	1	yes	characteristic
79	C₂×C₆₁	D₄×C₁₆	1	1	yes	characteristic
80	C₂×C₆₁	#2171	1	1	yes	characteristic
81	D₆₁	—	4	4	no	—
82	C₂²×C₁₆	—	122	2	no	—
83	C₄×D₈	—	61	1	no	—
84	C₈×D₄	—	61	2	no	—
85	C₄⋊C₁₆	—	61	2	no	—
86	C₂²⋊C₁₆	—	122	2	no	—
87	C₄×C₁₆	—	61	1	no	—
88	C₄×C₁₆	—	61	1	no	—
89	#335	#6923	1	1	yes	characteristic
90	D₄⋊C₈	—	61	2	no	—
91	C₈²	—	61	1	no	—
92	C₆₁	#12062	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₈	—	61	1	no	—
94	C₂²×C₈	—	122	2	no	—
95	C₄×D₄	—	61	2	no	—
96	C₂×C₁₆	—	61	2	no	—
97	C₂×C₁₆	—	122	1	no	—
98	C₂×C₁₆	—	122	2	no	—
99	C₂×C₁₆	—	244	4	no	—
100	#107	—	61	1	no	—
101	C₄⋊C₈	C₂×#6923	1	2	yes	—
102	#102	—	61	2	no	—
103	C₂²⋊C₈	—	122	2	no	—
104	C₄×C₈	—	61	1	no	—
105	C₄×C₈	—	61	1	no	—
106	C₄×C₈	C₂×#6923	1	1	yes	characteristic
107	C₂×D₄	—	61	2	no	—
108	C₂²×C₄	—	122	2	no	—
109	D₈	—	61	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	61	1	no	—
112	C₂×C₈	—	61	1	no	—
113	C₂×C₈	—	122	1	no	—
114	C₂×C₈	—	244	2	no	—
115	C₂×C₈	—	244	2	no	—
116	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
117	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
118	C₄⋊C₄	—	61	2	no	—
119	C₂²⋊C₄	—	122	2	no	—
120	C₄²	C₂²×#6923	1	1	yes	characteristic
121	C₁₆	—	122	2	no	—
122	C₁₆	—	244	2	no	—
123	C₁₆	—	244	1	no	—
124	C₂³	—	122	2	no	—
125	D₄	—	61	4	no	—
126	D₄	—	122	2	no	—
127	C₂×C₄	—	122	2	no	—
128	C₂×C₄	—	244	2	no	—
129	C₂×C₄	—	244	2	no	—
130	C₂×C₄	D₄×#6923	1	1	yes	characteristic
131	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
132	C₂×C₄	#14519865	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	4	2	no	—
135	C₈	—	61	2	no	—
136	C₈	—	122	1	no	—
137	C₈	—	244	1	no	—
138	C₈	D₆₁⋊C₁₆	1	2	yes	complemented
139	C₂²	—	122	4	no	—
140	C₂²	—	244	2	no	—
141	C₂²	—	244	2	no	—
142	C₂²	#16840790	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	244	2	no	—
145	C₄	#16840803	1	1	yes	characteristic
146	C₄	#16840808	1	1	yes	characteristic
147	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
148	C₄	#16840821	1	1	yes	characteristic
149	C₂	—	244	4	no	—
150	C₂	#21461417	1	1	yes	characteristic
151	C₂	#21462198	1	1	yes	characteristic
152	C₂	#21462211	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462198

C₂

yes

maximal, complemented

#21460841

C₂

yes

maximal, characteristic, complemented

#21460770

C₂

yes

maximal, complemented

#21460732

C₂

yes

maximal, characteristic, complemented

#21444613

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₁₆

—

C₄⋊#11758187

C₂²

yes

—

#16840657

—

C₄×#11758187

C₂²

yes

characteristic

C₄×#11758187

C₂²

yes

characteristic

C₄×D₄₈₈

C₄

yes

characteristic

#16839827

C₂²

yes

—

#16839489

C₄

yes

—

C₈×#92418

C₂²

yes

characteristic

C₆₁×#335

C₄

yes

characteristic

D₆₁⋊C₁₆

—

C₂×#11758187

—

C₂×#11758187

—

C₂×#11758187

D₄

yes

—

C₂×D₄₈₈

C₈

yes

characteristic

C₂×C₈×D₆₁

—

C₄×D₂₄₄

C₂×C₄

yes

—

#14519711

C₈

yes

characteristic

#14519699

C₈

yes

—

#14519689

—

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×D₂₄₄

C₂×C₈

yes

—

C₂×C₄×D₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

C₂×C₈

yes

—

C₂×#92418

—

C₂×#92418

C₂×D₄

yes

yes

D₄₈₈

C₁₆

yes

complemented

C₈×D₆₁

—

C₈×D₆₁

—

#11758187

—

#11758187

—

#11758187

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

#131

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

C₂×C₁₆

yes

complemented

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₁₆

yes

—

—

yes

—

#12062

—

maximal, Sylow

C₂²×C₆₁

C₈×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂²×C₁₆

yes

characteristic

C₄×C₆₁

#504

yes

characteristic

Dic₆₁

—

D₄×C₁₆

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

—

C₂×C₆₁

#1580

yes

characteristic

C₂×C₆₁

D₄×C₁₆

yes

characteristic

C₂×C₆₁

#2171

yes

characteristic

D₆₁

—

C₂²×C₁₆

—

122

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

122

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

#6923

yes

characteristic

D₄⋊C₈

—

C₈²

—

C₆₁

#12062

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

122

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

—

122

—

C₂×C₁₆

—

244

—

100

#107

—

101

C₄⋊C₈

C₂×#6923

yes

—

102

#102

—

103

C₂²⋊C₈

—

122

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₂×#6923

yes

characteristic

107

C₂×D₄

—

108

C₂²×C₄

—

122

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

122

—

114

C₂×C₈

—

244

—

115

C₂×C₈

—

244

—

116

C₂×C₈

D₆₁⋊C₈

yes

characteristic

117

C₂×C₈

D₆₁⋊C₈

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

122

—

120

C₄²

C₂²×#6923

yes

characteristic

121

C₁₆

—

122

—

122

C₁₆

—

244

—

123

C₁₆

—

244

—

124

C₂³

—

122

—

125

D₄

—

126

D₄

—

122

—

127

C₂×C₄

—

122

—

128

C₂×C₄

—

244

—

129

C₂×C₄

—

244

—

130

C₂×C₄

D₄×#6923

yes

characteristic

131

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

132

C₂×C₄

#14519865

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

122

—

137

C₈

—

244

—

138

C₈

D₆₁⋊C₁₆

yes

complemented

139

C₂²

—

122

—

140

C₂²

—

244

—

141

C₂²

—

244

—

142

C₂²

#16840790

yes

characteristic

143

C₄

—

144

C₄

—

244

—

145

C₄

#16840803

yes

characteristic

146

C₄

#16840808

yes

characteristic

147

C₄

C₂×D₆₁⋊C₁₆

yes

characteristic

148

C₄

#16840821

yes

characteristic

149

C₂

—

244

—

150

C₂

#21461417

yes

characteristic

151

C₂

#21462198

yes

characteristic

152

C₂

#21462211

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×#335	C₄
socle	C₂²×C₆₁	C₈×D₄
prime radical	C₂×D₄₈₈	C₈
cosocle	C₄×#92418	C₂³
hypercenter	#335	#6923
hypocenter	C₆₁	#12062
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#6923

commutator

C₄×C₆₁

C₂²×C₁₆

Frattini

C₄²

C₂²×#6923

Fitting

C₆₁×#335

C₄

socle

C₂²×C₆₁

C₈×D₄

prime radical

C₂×D₄₈₈

C₈

C₂³

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #21462198	C₂	semidirect
2. #21460841	C₂	semidirect
3. #21460770	C₂	semidirect
4. #21460732	C₂	semidirect
41. D₄₈₈	C₁₆	semidirect
53. D₂₄₄	C₂×C₁₆	semidirect
58. C₈×C₆₁	C₂×C₁₆	semidirect
92. C₆₁	#12062	semidirect
138. C₈	D₆₁⋊C₁₆	semidirect

H ≅ G/N

type

1. #21462198

C₂

semidirect

2. #21460841

C₂

semidirect

3. #21460770

C₂

semidirect

4. #21460732

C₂

semidirect

41. D₄₈₈

C₁₆

semidirect

53. D₂₄₄

C₂×C₁₆

semidirect

58. C₈×C₆₁

C₂×C₁₆

semidirect

92. C₆₁

#12062

semidirect

138. C₈

D₆₁⋊C₁₆

semidirect

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

subgroup	index
#21462198	2
#21460841	2
#21460770	2
#21460732	2
#21444613	2
#12062	61

subgroup

index

Subgroups — #30605965

Conjugacy classes of subgroups

Complements

Maximal subgroups