Group #30605755 — subgroups

Subgroups — #30605755

Order 15616.

234 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 100 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	#21461367	C₂	1	2	yes	maximal
2	#21460845	C₂	1	2	yes	maximal, complemented
3	#21460844	C₂	1	1	yes	maximal, characteristic, complemented
4	#21460843	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×D₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840781	—	2	2	no	—
7	#16840626	—	2	2	no	—
8	#16840623	C₂²	1	2	yes	—
9	#16840623	C₂²	1	2	yes	—
10	C₂×C₁₆×D₆₁	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₆₁	C₄	1	1	yes	characteristic
12	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
13	#16839538	C₄	1	1	yes	characteristic
14	#16839537	C₂²	1	1	yes	characteristic
15	#16839537	C₂²	1	1	yes	characteristic
16	C₆₁×#346	C₄	1	1	yes	characteristic
17	C₄⋊#92420	—	2	2	no	—
18	C₄⋊#92420	—	2	2	no	—
19	#14519834	—	4	2	no	—
20	C₄×#92420	—	4	2	no	—
21	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
23	C₄²×D₆₁	D₄	1	1	yes	characteristic
24	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
25	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
26	C₁₆×D₆₁	—	2	2	no	—
27	C₁₆×D₆₁	C₈	1	4	yes	complemented
28	C₄×#92418	Q₈	1	1	yes	characteristic
29	C₈×Dic₆₁	D₄	1	1	yes	characteristic
30	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₆₁	SD₁₆	1	2	yes	—
36	C₂×#92420	—	4	2	no	—
37	C₂×#92420	—	4	2	no	—
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₆₁	SD₁₆	1	2	yes	—
42	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
45	C₈×D₆₁	—	2	2	no	—
46	C₈×D₆₁	C₂×C₈	1	2	yes	—
47	C₈×D₆₁	M₁₆	1	2	yes	—
48	C₁₆×C₆₁	—	2	1	no	—
49	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
50	#11758185	—	2	1	no	—
51	#11758185	C₂×C₈	1	2	yes	—
52	C₂²×D₆₁	#106	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#106	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₆₁	#106	1	1	yes	characteristic
56	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
57	C₄×D₆₁	—	4	2	no	—
58	C₄×D₆₁	C₄⋊C₈	1	2	yes	—
59	C₄×D₆₁	#108	1	2	yes	—
60	#92420	—	8	2	no	—
61	#92420	—	8	2	no	—
62	C₈×C₆₁	—	2	1	no	—
63	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
64	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
65	#92418	—	2	1	no	—
66	#92418	C₄⋊C₈	1	2	yes	—
67	#11999	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
69	C₂×D₆₁	—	2	2	no	—
70	C₂×D₆₁	#334	1	2	yes	—
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
74	Dic₆₁	—	2	2	no	—
75	Dic₆₁	#334	1	2	yes	—
76	C₂×#346	—	61	1	no	—
77	#1564	—	61	2	no	—
78	#1368	—	61	1	no	—
79	#1368	—	61	1	no	—
80	#1368	—	61	2	no	—
81	C₂×C₆₁	C₂×#334	1	1	yes	characteristic
82	C₂×C₆₁	#2155	1	2	yes	—
83	D₆₁	—	2	2	no	—
84	C₂²×C₁₆	—	61	1	no	—
85	C₂²×C₁₆	—	61	1	no	—
86	#431	—	122	2	no	—
87	C₂×C₄×C₈	—	61	1	no	—
88	#346	—	61	1	no	—
89	#346	—	61	1	no	—
90	#346	—	61	1	no	—
91	#346	#6923	1	1	yes	characteristic
92	#335	—	122	2	no	—
93	#334	—	61	2	no	—
94	#334	—	61	2	no	—
95	C₆₁	#11999	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	61	1	no	—
97	C₂²×C₈	—	61	1	no	—
98	C₂×C₄²	—	61	1	no	—
99	C₂×C₁₆	—	61	1	no	—
100	C₂×C₁₆	—	61	4	no	—
101	C₂×C₁₆	—	61	1	no	—
102	C₂×C₁₆	—	122	2	no	—
103	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
104	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
105	C₄⋊C₈	—	122	2	no	—
106	C₄⋊C₈	—	122	2	no	—
107	C₂²⋊C₈	—	244	2	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	—	61	1	no	—
110	C₄×C₈	—	61	1	no	—
111	C₄×C₈	—	244	2	no	—
112	C₄×C₈	C₂×#6923	1	1	yes	characteristic
113	C₂²×C₄	—	61	1	no	—
114	C₂²×C₄	—	61	2	no	—
115	C₂×C₈	—	61	1	no	—
116	C₂×C₈	—	61	2	no	—
117	C₂×C₈	—	61	2	no	—
118	C₂×C₈	—	61	1	no	—
119	C₂×C₈	—	122	2	no	—
120	C₂×C₈	—	244	2	no	—
121	C₂×C₈	—	244	2	no	—
122	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
123	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
124	C₄²	—	61	2	no	—
125	C₄²	—	61	1	no	—
126	C₄²	C₄⋊#6923	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	61	2	no	—
129	C₁₆	—	122	1	no	—
130	C₁₆	D₆₁⋊C₈	1	2	yes	—
131	C₂³	—	61	1	no	—
132	C₂×C₄	—	61	1	no	—
133	C₂×C₄	—	61	2	no	—
134	C₂×C₄	—	61	2	no	—
135	C₂×C₄	—	61	2	no	—
136	C₂×C₄	—	244	2	no	—
137	C₂×C₄	#14519845	1	2	yes	—
138	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	61	2	no	—
141	C₈	—	122	1	no	—
142	C₈	—	488	2	no	—
143	C₈	—	488	2	no	—
144	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
145	C₈	D₆₁⋊M₁₆	1	1	yes	characteristic
146	C₂²	—	61	2	no	—
147	C₂²	—	122	2	no	—
148	C₂²	C₂×#14519845	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	61	2	no	—
151	C₄	—	122	2	no	—
152	C₄	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
153	C₄	D₆₁⋊#108	1	1	yes	characteristic
154	C₂	—	122	2	no	—
155	C₂	D₆₁⋊#334	1	1	yes	characteristic
156	C₂	#21462167	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461367

C₂

yes

maximal

#21460845

C₂

yes

maximal, complemented

#21460844

C₂

yes

maximal, characteristic, complemented

#21460843

C₂

yes

maximal, characteristic, complemented

#346×D₆₁

C₂

yes

maximal, characteristic

#16840781

—

#16840626

—

#16840623

C₂²

yes

—

#16840623

C₂²

yes

—

C₂×C₁₆×D₆₁

C₄

yes

characteristic

C₂×C₁₆×D₆₁

C₄

yes

characteristic

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839538

C₄

yes

characteristic

#16839537

C₂²

yes

characteristic

#16839537

C₂²

yes

characteristic

C₆₁×#346

C₄

yes

—

—

—

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

Q₈

yes

characteristic

C₄²×D₆₁

D₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

characteristic

C₁₆×D₆₁

—

C₁₆×D₆₁

C₈

yes

complemented

C₄×#92418

Q₈

yes

characteristic

C₈×Dic₆₁

D₄

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

C₄⋊C₄

yes

characteristic

C₂×C₄×D₆₁

SD₁₆

yes

—

C₂×#92420

—

C₂×#92420

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

C₂×D₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₂×Q₈

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

C₂×C₈

yes

—

C₈×D₆₁

M₁₆

yes

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂×C₈

yes

—

#11758185

—

#11758185

C₂×C₈

yes

—

C₂²×D₆₁

#106

yes

characteristic

C₂×C₄×C₆₁

#106

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

#106

yes

characteristic

C₂×Dic₆₁

C₂×SD₁₆

yes

—

C₄×D₆₁

—

C₄×D₆₁

C₄⋊C₈

yes

—

C₄×D₆₁

#108

yes

—

#92420

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₈

yes

characteristic

C₈×C₆₁

C₂×M₁₆

yes

characteristic

#92418

—

#92418

C₄⋊C₈

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

#334

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₂×#108

yes

characteristic

Dic₆₁

—

Dic₆₁

#334

yes

—

—

—

—

—

—

C₂×C₆₁

C₂×#334

yes

characteristic

C₂×C₆₁

#2155

yes

—

D₆₁

—

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

—

122

—

C₂×C₄×C₈

—

—

—

—

yes

—

122

—

#334

—

#334

—

C₆₁

#11999

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

122

—

103

C₂×C₁₆

C₂×#6923

yes

characteristic

104

C₂×C₁₆

C₂×#6923

yes

characteristic

105

C₄⋊C₈

—

122

—

106

C₄⋊C₈

—

122

—

107

C₂²⋊C₈

—

244

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

244

—

112

C₄×C₈

C₂×#6923

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

244

—

121

C₂×C₈

—

244

—

122

C₂×C₈

C₄⋊#6923

yes

characteristic

123

C₂×C₈

C₂²×#6923

yes

characteristic

124

C₄²

—

125

C₄²

—

126

C₄²

C₄⋊#6923

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

122

—

130

C₁₆

D₆₁⋊C₈

yes

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

244

—

137

C₂×C₄

#14519845

yes

—

138

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

122

—

142

C₈

—

488

—

143

C₈

—

488

—

144

C₈

C₂×D₆₁⋊C₈

yes

characteristic

145

C₈

D₆₁⋊M₁₆

yes

characteristic

146

C₂²

—

147

C₂²

—

122

—

148

C₂²

C₂×#14519845

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

122

—

152

C₄

D₆₁⋊(C₄⋊C₈)

yes

characteristic

153

C₄

D₆₁⋊#108

yes

characteristic

154

C₂

—

122

—

155

C₂

D₆₁⋊#334

yes

characteristic

156

C₂

#21462167

yes

—

	subgroup	G/H
center	C₂²	C₂×#14519845
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#346	C₄
socle	C₂²×C₆₁	C₂×#106
prime radical	C₂²×D₆₁	#106
cosocle	C₈×Dic₆₁	C₂³
hypercenter	#346	#6923
hypocenter	C₆₁	#11999
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#14519845

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₆₁×#346

C₄

C₂²×C₆₁

C₂²×D₆₁

C₈×Dic₆₁

C₂³

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #21460845	C₂	semidirect
3. #21460844	C₂	semidirect
4. #21460843	C₂	semidirect
27. C₁₆×D₆₁	C₈	semidirect
95. C₆₁	#11999	semidirect

H ≅ G/N

type

2. #21460845

C₂

semidirect

3. #21460844

C₂

semidirect

4. #21460843

C₂

semidirect

27. C₁₆×D₆₁

C₈

semidirect

95. C₆₁

#11999

semidirect

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

subgroup	index
#21461367	2
#21460845	2
#21460844	2
#21460843	2
#346×D₆₁	2
#11999	61

subgroup

index

#346×D₆₁

Subgroups — #30605755

Conjugacy classes of subgroups

Complements

Maximal subgroups