Group #30597375 — subgroups

Subgroups — #30597375

Order 15616.

574 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 276 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	C₄²×#92420	C₂	1	2	yes	maximal
2	#21460641	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#92420	C₄	1	4	yes	—
5	C₂×C₄×#92420	C₄	1	4	yes	—
6	C₂×C₄×#92420	C₂²	1	4	yes	—
7	C₂×C₄×#92418	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
10	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂²×#92420	C₂×C₄	1	8	yes	—
13	C₄×#92420	—	2	8	no	—
14	C₄×#92420	—	2	8	no	—
15	C₄×#92420	C₈	1	16	yes	complemented
16	C₂²×#92418	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
21	C₄×#92418	—	2	2	no	—
22	C₈×Dic₆₁	—	2	4	no	—
23	C₈×Dic₆₁	C₈	1	8	yes	complemented
24	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	—	2	2	no	—
27	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
30	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
31	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
32	C₂×#92420	—	2	8	no	—
33	C₂×#92420	—	2	8	no	—
34	C₂×#92420	C₂×C₈	1	16	yes	—
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
38	C₄²×C₆₁	M₁₆	1	2	yes	—
39	C₄²×C₆₁	M₁₆	1	2	yes	—
40	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
41	C₄×Dic₆₁	C₂×C₈	1	8	yes	—
42	C₄×Dic₆₁	M₁₆	1	8	yes	—
43	C₄×Dic₆₁	#41	1	4	yes	—
44	C₂×#92418	—	2	2	no	—
45	C₂×#92418	—	2	2	no	—
46	C₂×#92418	C₂×C₈	1	4	yes	—
47	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₆₁	#97	1	4	yes	—
50	C₂×C₄×C₆₁	#97	1	4	yes	—
51	C₂×C₄×C₆₁	#117	1	2	yes	—
52	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
53	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
55	C₂×Dic₆₁	C₄×C₈	1	4	yes	—
56	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
57	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
58	C₂×Dic₆₁	#97	1	4	yes	—
59	C₂×Dic₆₁	#117	1	4	yes	—
60	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
61	C₂×Dic₆₁	C₂×M₁₆	1	4	yes	—
62	#92420	—	2	16	no	—
63	C₈×C₆₁	—	2	4	no	—
64	#92418	—	2	4	no	—
65	#8131	—	61	1	no	maximal, Sylow
66	C₂²×C₆₁	#376	1	1	yes	characteristic
67	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
68	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂²×C₆₁	C₂×#97	1	2	yes	—
70	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
71	C₄×C₆₁	#322	1	8	yes	—
72	C₄×C₆₁	#431	1	2	yes	—
73	C₄×C₆₁	#431	1	2	yes	—
74	Dic₆₁	C₈²	1	4	yes	—
75	Dic₆₁	#322	1	4	yes	—
76	Dic₆₁	#431	1	8	yes	—
77	C₄²×C₈	—	61	2	no	—
78	C₄²×C₈	—	61	1	no	—
79	#1273	—	61	4	no	—
80	C₂×C₆₁	C₂×C₈²	1	1	yes	characteristic
81	C₂×C₆₁	C₂×#322	1	1	yes	characteristic
82	C₂×C₆₁	C₂×#322	1	2	yes	—
83	C₂×C₆₁	#1744	1	2	yes	—
84	C₂×C₆₁	#1744	1	1	yes	characteristic
85	C₂×C₄×C₈	—	61	4	no	—
86	C₂×C₄×C₈	—	61	2	no	—
87	C₂×C₄×C₈	—	61	4	no	—
88	C₂×C₄×C₈	—	61	4	no	—
89	C₂×C₄×C₈	—	61	2	no	—
90	C₂×C₄×C₈	—	61	1	no	—
91	C₂×C₄×C₈	#6923	1	1	yes	characteristic
92	C₄³	—	61	1	no	—
93	C₆₁	#8131	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	61	8	no	—
95	C₂²×C₈	—	61	2	no	—
96	C₂²×C₈	C₂×#6923	1	2	yes	—
97	C₂×C₄²	—	61	2	no	—
98	C₂×C₄²	—	61	2	no	—
99	C₂×C₄²	—	61	1	no	—
100	C₂×C₄²	—	61	1	no	—
101	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	—	61	16	no	—
104	C₄×C₈	—	61	8	no	—
105	C₄×C₈	—	122	4	no	—
106	C₄×C₈	—	122	8	no	—
107	C₄×C₈	—	122	8	no	—
108	C₄×C₈	—	122	2	no	—
109	C₂²×C₄	—	61	2	no	—
110	C₂²×C₄	—	61	1	no	—
111	C₂²×C₄	—	61	1	no	—
112	C₂²×C₄	C₄×#6923	1	2	yes	—
113	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	61	16	no	—
117	C₂×C₈	—	61	4	no	—
118	C₂×C₈	—	122	8	no	—
119	C₂×C₈	—	122	8	no	—
120	C₂×C₈	—	122	2	no	—
121	C₂×C₈	—	122	2	no	—
122	C₂×C₈	D₆₁⋊C₈	1	4	yes	—
123	C₄²	—	61	8	no	—
124	C₄²	—	61	8	no	—
125	C₄²	—	61	4	no	—
126	C₄²	—	61	4	no	—
127	C₄²	#11758213	1	2	yes	—
128	C₄²	#11758217	1	2	yes	—
129	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
130	C₂×C₄	—	61	4	no	—
131	C₂×C₄	—	61	4	no	—
132	C₂×C₄	—	61	2	no	—
133	C₂×C₄	—	61	2	no	—
134	C₂×C₄	—	61	4	no	—
135	C₂×C₄	—	61	4	no	—
136	C₂×C₄	—	61	4	no	—
137	C₂×C₄	C₈×#6923	1	4	yes	—
138	C₂×C₄	#14519831	1	4	yes	—
139	C₂×C₄	#14519832	1	4	yes	—
140	C₂×C₄	#14519856	1	2	yes	—
141	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
142	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
143	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	122	16	no	—
146	C₈	—	122	4	no	—
147	C₂²	#16840698	1	1	yes	characteristic
148	C₂²	C₂×C₈×#6923	1	2	yes	—
149	C₂²	C₄×D₆₁⋊C₈	1	1	yes	characteristic
150	C₂²	C₂×#14519831	1	2	yes	—
151	C₂²	C₂×#14519832	1	1	yes	characteristic
152	C₄	—	61	4	no	—
153	C₄	—	61	4	no	—
154	C₄	—	61	8	no	—
155	C₄	#92420⋊C₈	1	8	yes	—
156	C₄	#16840779	1	2	yes	—
157	C₄	#16840781	1	2	yes	—
158	C₂	C₈×D₆₁⋊C₈	1	1	yes	characteristic
159	C₂	#21461013	1	1	yes	characteristic
160	C₂	C₂×#92420⋊C₈	1	2	yes	—
161	C₂	#21461532	1	2	yes	—
162	C₂	#21461534	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×#92420

C₂

yes

maximal

#21460641

C₂

yes

maximal

C₄×C₈×Dic₆₁

C₂

yes

maximal, characteristic

C₂×C₄×#92420

C₄

yes

—

C₂×C₄×#92420

C₄

yes

—

C₂×C₄×#92420

C₂²

yes

—

C₂×C₄×#92418

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₄

yes

—

C₂×C₈×Dic₆₁

C₂²

yes

—

C₄²×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₁

C₄

yes

characteristic

C₂²×#92420

C₂×C₄

yes

—

C₄×#92420

—

C₄×#92420

—

C₄×#92420

C₈

yes

complemented

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂³

yes

characteristic

C₄×#92418

—

C₈×Dic₆₁

—

C₈×Dic₆₁

C₈

yes

complemented

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₄²

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₄²

yes

characteristic

C₂²×Dic₆₁

C₄²

yes

characteristic

C₂²×Dic₆₁

C₂²×C₄

yes

—

C₂×#92420

—

C₂×#92420

—

C₂×#92420

C₂×C₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

M₁₆

yes

—

C₄²×C₆₁

M₁₆

yes

—

C₄×Dic₆₁

C₂×C₈

yes

—

C₄×Dic₆₁

C₂×C₈

yes

—

C₄×Dic₆₁

M₁₆

yes

—

C₄×Dic₆₁

#41

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂×C₈

yes

—

C₂³×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₄×C₈

yes

—

C₂×C₄×C₆₁

#97

yes

—

C₂×C₄×C₆₁

#97

yes

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×Dic₆₁

C₄×C₈

yes

—

C₂×Dic₆₁

C₄×C₈

yes

—

C₂×Dic₆₁

C₄×C₈

yes

—

C₂×Dic₆₁

#97

yes

—

C₂×Dic₆₁

#117

yes

—

C₂×Dic₆₁

C₂²×C₈

yes

—

C₂×Dic₆₁

C₂×M₁₆

yes

—

#92420

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×C₄×C₈

yes

—

C₂²×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂²×C₆₁

C₂×#97

yes

—

C₂²×C₆₁

C₂×#97

yes

characteristic

C₄×C₆₁

#322

yes

—

C₄×C₆₁

#431

yes

—

C₄×C₆₁

#431

yes

—

Dic₆₁

C₈²

yes

—

Dic₆₁

#322

yes

—

Dic₆₁

#431

yes

—

C₄²×C₈

—

C₄²×C₈

—

#1273

—

C₂×C₆₁

C₂×C₈²

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#1744

yes

—

C₂×C₆₁

#1744

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6923

yes

characteristic

C₄³

—

C₆₁

#8131

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×#6923

yes

characteristic

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

122

—

106

C₄×C₈

—

122

—

107

C₄×C₈

—

122

—

108

C₄×C₈

—

122

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₄×#6923

yes

—

113

C₂²×C₄

C₂²×#6923

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

122

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

122

—

121

C₂×C₈

—

122

—

122

C₂×C₈

D₆₁⋊C₈

yes

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11758213

yes

—

128

C₄²

#11758217

yes

—

129

C₂³

C₂×C₄×#6923

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₈×#6923

yes

—

138

C₂×C₄

#14519831

yes

—

139

C₂×C₄

#14519832

yes

—

140

C₂×C₄

#14519856

yes

—

141

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

142

C₂×C₄

C₂×#11758217

yes

characteristic

143

C₂×C₄

C₂×#11758213

yes

characteristic

144

C₈

—

145

C₈

—

122

—

146

C₈

—

122

—

147

C₂²

#16840698

yes

characteristic

148

C₂²

C₂×C₈×#6923

yes

—

149

C₂²

C₄×D₆₁⋊C₈

yes

characteristic

150

C₂²

C₂×#14519831

yes

—

151

C₂²

C₂×#14519832

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#92420⋊C₈

yes

—

156

C₄

#16840779

yes

—

157

C₄

#16840781

yes

—

158

C₂

C₈×D₆₁⋊C₈

yes

characteristic

159

C₂

#21461013

yes

characteristic

160

C₂

C₂×#92420⋊C₈

yes

—

161

C₂

#21461532

yes

—

162

C₂

#21461534

yes

characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#6923
commutator	C₂×C₆₁	C₂×C₈²
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#8131
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄²

C₂×#6923

commutator

C₂×C₆₁

C₂×C₈²

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₂×C₄×C₈×C₆₁

C₄

socle, prime radical

C₂³×C₆₁

C₂×C₄²

cosocle

C₂×C₄×Dic₆₁

C₂³

C₂×C₄×C₈

C₆₁

G ▷ C₂×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
15. C₄×#92420	C₈	semidirect
23. C₈×Dic₆₁	C₈	semidirect
93. C₆₁	#8131	semidirect

H ≅ G/N

type

15. C₄×#92420

C₈

semidirect

23. C₈×Dic₆₁

C₈

semidirect

93. C₆₁

#8131

semidirect

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

subgroup	index
C₄²×#92420	2
#21460641	2
C₄×C₈×Dic₆₁	2
#8131	61

subgroup

index

C₄²×#92420

#21460641

C₄×C₈×Dic₆₁

#8131

Subgroups — #30597375

Conjugacy classes of subgroups

Complements

Maximal subgroups