Group #30605765 — subgroups

Subgroups — #30605765

Order 15616.

220 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 96 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	D₆₁⋊#364	C₂	1	2	yes	maximal
2	#21460847	C₂	1	2	yes	maximal, complemented
3	#21460846	C₂	1	2	yes	maximal, complemented
4	C₄×#14519716	C₂	1	1	yes	maximal, characteristic
5	D₆₁⋊M₃₂	—	2	2	no	—
6	#16840685	—	2	2	no	—
7	#16840684	C₂²	1	4	yes	—
8	C₂×#14519716	C₄	1	1	yes	characteristic
9	C₂×#14519716	C₄	1	1	yes	characteristic
10	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
11	#16839537	C₂²	1	1	yes	characteristic
12	#16839537	C₂²	1	1	yes	characteristic
13	C₄×#11758185	C₄	1	1	yes	characteristic
14	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
15	#14519852	—	4	2	no	—
16	#14519851	—	2	4	no	—
17	D₆₁⋊C₁₆	—	4	2	no	—
18	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
20	C₄²×D₆₁	Q₈	1	1	yes	characteristic
21	#14519716	—	2	2	no	—
22	#14519716	C₈	1	4	yes	—
23	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
24	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
25	C₄×#92418	D₄	1	1	yes	characteristic
26	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
27	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₆₁	—	2	1	no	—
32	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₄×Dic₆₁	—	2	1	no	—
37	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#92418	C₂×D₄	1	1	yes	characteristic
40	C₈×D₆₁	C₂×C₈	1	2	yes	—
41	C₈×D₆₁	M₁₆	1	2	yes	—
42	C₈×D₆₁	SD₁₆	1	2	yes	—
43	C₈×D₆₁	SD₁₆	1	2	yes	—
44	#11758187	—	4	4	no	—
45	C₁₆×C₆₁	—	2	1	no	—
46	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
47	#11758185	—	2	1	no	—
48	#11758185	C₂×C₈	1	2	yes	—
49	C₂²×D₆₁	#108	1	1	yes	characteristic
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×Dic₆₁	—	2	1	no	—
53	C₂×Dic₆₁	#108	1	1	yes	characteristic
54	C₄×D₆₁	—	4	2	no	—
55	C₄×D₆₁	C₄⋊C₈	1	2	yes	—
56	C₄×D₆₁	#106	1	2	yes	—
57	C₈×C₆₁	#106	1	1	yes	characteristic
58	C₈×C₆₁	#106	1	1	yes	characteristic
59	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
61	#92418	C₄⋊C₈	1	2	yes	—
62	#92418	C₂×SD₁₆	1	2	yes	—
63	#12002	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	C₂×#108	1	1	yes	characteristic
65	C₂×D₆₁	—	4	1	no	—
66	C₂×D₆₁	#334	1	2	yes	—
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
70	Dic₆₁	—	4	1	no	—
71	Dic₆₁	#334	1	2	yes	—
72	C₂×#364	—	61	2	no	—
73	C₄×M₃₂	—	61	1	no	—
74	#1369	—	61	2	no	—
75	#1369	—	61	2	no	—
76	C₂×C₆₁	—	2	1	no	—
77	C₂×C₆₁	C₂×#334	1	1	yes	characteristic
78	D₆₁	—	2	2	no	—
79	C₂×M₃₂	—	61	1	no	—
80	C₂×M₃₂	—	61	1	no	—
81	C₂×M₃₂	—	122	2	no	—
82	C₂×C₄×C₈	—	61	1	no	—
83	#364	—	61	4	no	—
84	#364	—	122	2	no	—
85	#346	—	61	1	no	—
86	#346	—	61	1	no	—
87	C₄×C₁₆	—	61	1	no	—
88	C₄×C₁₆	#6923	1	1	yes	characteristic
89	C₆₁	#12002	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	61	1	no	—
91	C₂²×C₈	—	61	1	no	—
92	C₂×C₄²	—	61	1	no	—
93	M₃₂	—	61	4	no	—
94	M₃₂	—	122	2	no	—
95	M₃₂	—	122	4	no	—
96	M₃₂	—	244	2	no	—
97	C₂×C₁₆	—	61	1	no	—
98	C₂×C₁₆	—	61	1	no	—
99	C₂×C₁₆	—	244	2	no	—
100	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
101	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
102	C₄×C₈	—	61	1	no	—
103	C₄×C₈	—	61	1	no	—
104	C₄×C₈	—	61	1	no	—
105	C₄×C₈	C₂×#6923	1	1	yes	characteristic
106	C₂²×C₄	—	61	1	no	—
107	C₂²×C₄	—	122	1	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	61	1	no	—
110	C₂×C₈	—	61	2	no	—
111	C₂×C₈	—	61	2	no	—
112	C₂×C₈	—	61	2	no	—
113	C₂×C₈	—	61	2	no	—
114	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
115	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
116	C₄²	—	61	1	no	—
117	C₄²	—	122	1	no	—
118	C₄²	C₄⋊#6923	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	61	2	no	—
121	C₁₆	—	122	1	no	—
122	C₁₆	—	244	4	no	—
123	C₁₆	D₆₁⋊C₈	1	2	yes	—
124	C₂³	—	61	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	61	1	no	—
127	C₂×C₄	—	61	2	no	—
128	C₂×C₄	—	61	2	no	—
129	C₂×C₄	—	122	1	no	—
130	C₂×C₄	—	244	2	no	—
131	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
132	C₈	—	61	2	no	—
133	C₈	—	61	2	no	—
134	C₈	#14519845	1	1	yes	characteristic
135	C₈	#14519845	1	1	yes	characteristic
136	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
137	C₈	D₆₁⋊M₁₆	1	1	yes	characteristic
138	C₂²	—	61	2	no	—
139	C₂²	—	244	1	no	—
140	C₂²	D₆₁⋊#108	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	61	2	no	—
143	C₄	—	244	1	no	—
144	C₄	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
145	C₄	C₂×#14519845	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	122	2	no	—
148	C₂	D₆₁⋊#334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₆₁⋊#364

C₂

yes

maximal

#21460847

C₂

yes

maximal, complemented

#21460846

C₂

yes

maximal, complemented

C₄×#14519716

C₂

yes

maximal, characteristic

D₆₁⋊M₃₂

—

#16840685

—

#16840684

C₂²

yes

—

C₂×#14519716

C₄

yes

characteristic

C₂×#14519716

C₄

yes

characteristic

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839537

C₂²

yes

characteristic

#16839537

C₂²

yes

characteristic

C₄×#11758185

C₄

yes

characteristic

C₄×C₁₆×C₆₁

C₄

yes

characteristic

#14519852

—

#14519851

—

D₆₁⋊C₁₆

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

D₄

yes

characteristic

C₄²×D₆₁

Q₈

yes

characteristic

#14519716

—

#14519716

C₈

yes

—

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

characteristic

C₄×#92418

D₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₈×Dic₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×Q₈

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₂×D₄

yes

characteristic

C₈×D₆₁

C₂×C₈

yes

—

C₈×D₆₁

M₁₆

yes

—

C₈×D₆₁

SD₁₆

yes

—

C₈×D₆₁

SD₁₆

yes

—

#11758187

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂×C₈

yes

—

#11758185

—

#11758185

C₂×C₈

yes

—

C₂²×D₆₁

#108

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

#108

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₄⋊C₈

yes

—

C₄×D₆₁

#106

yes

—

C₈×C₆₁

yes

C₈×C₆₁

yes

C₈×C₆₁

C₂²×C₈

yes

characteristic

C₈×C₆₁

C₂×M₁₆

yes

characteristic

#92418

C₄⋊C₈

yes

—

#92418

C₂×SD₁₆

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

#334

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₂×#106

yes

characteristic

Dic₆₁

—

Dic₆₁

#334

yes

—

C₂×#364

—

C₄×M₃₂

—

#1369

—

#1369

—

C₂×C₆₁

—

C₂×C₆₁

C₂×#334

yes

characteristic

D₆₁

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

122

—

C₂×C₄×C₈

—

#364

—

#364

—

122

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

#6923

yes

characteristic

C₆₁

#12002

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

122

—

M₃₂

—

122

—

M₃₂

—

244

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

244

—

100

C₂×C₁₆

C₂×#6923

yes

characteristic

101

C₂×C₁₆

C₂×#6923

yes

characteristic

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₂×#6923

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

122

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄⋊#6923

yes

characteristic

115

C₂×C₈

C₂²×#6923

yes

characteristic

116

C₄²

—

117

C₄²

—

122

—

118

C₄²

C₄⋊#6923

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

122

—

122

C₁₆

—

244

—

123

C₁₆

D₆₁⋊C₈

yes

—

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

122

—

130

C₂×C₄

—

244

—

131

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

#14519845

yes

characteristic

135

C₈

#14519845

yes

characteristic

136

C₈

C₂×D₆₁⋊C₈

yes

characteristic

137

C₈

D₆₁⋊M₁₆

yes

characteristic

138

C₂²

—

139

C₂²

—

244

—

140

C₂²

D₆₁⋊#108

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

244

—

144

C₄

D₆₁⋊(C₄⋊C₈)

yes

characteristic

145

C₄

C₂×#14519845

yes

characteristic

146

C₂

—

147

C₂

—

122

—

148

C₂

D₆₁⋊#334

yes

characteristic

	subgroup	G/H
center	C₈	#14519845
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle	C₂×C₆₁	C₂×#334
prime radical	C₂²×D₆₁	#108
cosocle	C₈×Dic₆₁	C₂³
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#12002
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₈

#14519845

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₄×C₁₆×C₆₁

C₄

C₂×C₆₁

C₂²×D₆₁

C₈×Dic₆₁

C₂³

C₄×C₁₆

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #21460847	C₂	semidirect
3. #21460846	C₂	semidirect
89. C₆₁	#12002	semidirect

H ≅ G/N

type

2. #21460847

C₂

semidirect

3. #21460846

C₂

semidirect

89. C₆₁

#12002

semidirect

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

subgroup	index
D₆₁⋊#364	2
#21460847	2
#21460846	2
C₄×#14519716	2
#12002	61

subgroup

index

D₆₁⋊#364

Subgroups — #30605765

Conjugacy classes of subgroups

Complements

Maximal subgroups