Group #30594269 — subgroups

Subgroups — #30594269

Order 15616.

216 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 84 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	#21462176	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519851	C₂	1	1	yes	maximal, characteristic
3	#21444109	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519851	—	2	1	no	—
5	C₄⋊#11758187	C₄	1	2	yes	—
6	C₂²⋊#11758187	—	2	1	no	—
7	#16840650	C₄	1	2	yes	—
8	C₄×#11758187	—	2	1	no	—
9	C₄×#11758187	—	2	1	no	—
10	C₄×#11758194	C₄	1	1	yes	characteristic
11	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
12	#16839484	—	2	1	no	—
13	#16839483	—	2	1	no	—
14	C₆₁×#431	C₄	1	1	yes	characteristic
15	#14519851	—	2	4	no	—
16	C₂×#11758187	—	2	2	no	—
17	C₂×#11758187	—	2	2	no	—
18	C₂×#11758194	C₈	1	2	yes	—
19	C₂²×#92418	D₄	1	1	yes	characteristic
20	C₂²×#92418	Q₈	1	1	yes	characteristic
21	#14519688	—	2	1	no	—
22	C₄×#92418	—	2	1	no	—
23	C₄×#92418	C₂×C₄	1	1	yes	characteristic
24	C₄×#92418	C₂×C₄	1	1	yes	characteristic
25	C₄×#92418	D₄	1	1	yes	characteristic
26	C₄×#92418	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	C₈	1	2	yes	—
29	C₂²⋊C₈×C₆₁	—	2	1	no	—
30	C₄×C₈×C₆₁	—	2	1	no	—
31	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
34	C₂×C₈×C₆₁	—	2	2	no	—
35	C₄²×C₆₁	C₄²	1	1	yes	characteristic
36	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
39	#11758194	—	4	2	no	—
40	C₂×#92418	—	2	2	no	—
41	C₂×#92418	—	2	2	no	—
42	C₂×#92418	—	2	2	no	—
43	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
46	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
47	#11758187	—	2	4	no	—
48	#11758187	—	4	2	no	—
49	C₂³×C₆₁	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	—	2	2	no	—
54	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₆₁	—	4	2	no	—
61	#92418	—	2	1	no	—
62	#92418	—	2	1	no	—
63	#92418	—	4	2	no	—
64	#92418	#104	1	2	yes	—
65	#92418	#108	1	2	yes	—
66	#7166	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	—	2	1	no	—
68	C₂²×C₆₁	—	2	1	no	—
69	C₂²×C₆₁	—	2	1	no	—
70	C₂²×C₆₁	#336	1	1	yes	characteristic
71	C₄×C₆₁	—	2	1	no	—
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	#339	1	1	yes	characteristic
75	C₄×C₆₁	#341	1	1	yes	characteristic
76	C₄×C₆₁	#349	1	2	yes	—
77	C₄×C₆₁	#350	1	2	yes	—
78	#2162	—	61	1	no	—
79	C₄×M₃₂	—	61	1	no	—
80	#1451	—	61	1	no	—
81	C₂×C₆₁	—	4	1	no	—
82	C₂×C₆₁	#1278	1	1	yes	characteristic
83	C₂×C₆₁	#1377	1	1	yes	characteristic
84	C₂×C₆₁	#1379	1	1	yes	characteristic
85	C₂×M₃₂	—	122	1	no	—
86	#431	#6923	1	1	yes	characteristic
87	C₄×M₁₆	—	61	1	no	—
88	C₂×C₄×C₈	—	61	1	no	—
89	C₄⋊C₁₆	—	61	2	no	—
90	C₂²⋊C₁₆	—	122	1	no	—
91	#346	—	61	2	no	—
92	C₄×C₁₆	—	122	1	no	—
93	C₄×C₁₆	—	122	1	no	—
94	#322	—	122	1	no	—
95	#322	—	122	1	no	—
96	C₆₁	#7166	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	61	2	no	—
98	C₂²×C₈	—	61	1	no	—
99	C₂²×C₈	—	61	1	no	—
100	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
101	M₃₂	—	122	4	no	—
102	C₂×C₁₆	—	122	2	no	—
103	C₂×C₁₆	—	122	2	no	—
104	C₄⋊C₈	#92420	1	2	yes	—
105	C₂²⋊C₈	—	2	1	no	—
106	#97	—	122	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	—	61	1	no	—
110	C₄×C₈	—	61	1	no	—
111	C₄×C₈	—	61	1	no	—
112	C₄×C₈	—	122	1	no	—
113	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
115	C₂²×C₄	#11758217	1	1	yes	characteristic
116	M₁₆	—	244	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	61	1	no	—
119	C₂×C₈	—	61	1	no	—
120	C₂×C₈	—	61	1	no	—
121	C₂×C₈	—	61	1	no	—
122	C₂×C₈	—	122	2	no	—
123	C₂×C₈	—	122	2	no	—
124	C₂×C₈	—	122	2	no	—
125	C₄²	#11758213	1	1	yes	characteristic
126	C₄²	C₄×#6923	1	1	yes	characteristic
127	C₄²	C₂×#92420	1	1	yes	characteristic
128	C₄²	C₂²⋊#6923	1	1	yes	characteristic
129	C₁₆	—	122	4	no	—
130	C₁₆	—	244	2	no	—
131	C₂³	#14519844	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₄×#92420	1	1	yes	characteristic
138	C₂×C₄	#14519832	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#92420	1	1	yes	characteristic
140	C₂×C₄	#14519834	1	1	yes	characteristic
141	C₂×C₄	C₄⋊#92420	1	1	yes	characteristic
142	C₈	—	4	2	no	—
143	C₈	—	61	2	no	—
144	C₈	—	61	2	no	—
145	C₈	—	122	1	no	—
146	C₈	—	122	1	no	—
147	C₈	—	244	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#16840628	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#16840634	1	1	yes	characteristic
156	C₄	#16840637	1	1	yes	characteristic
157	C₄	#16840659	1	2	yes	—
158	C₄	#16840660	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#21460652	1	1	yes	characteristic
161	C₂	#21460867	1	1	yes	characteristic
162	C₂	#21460873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462176

C₂

yes

maximal, characteristic

C₄×#14519851

C₂

yes

maximal, characteristic

#21444109

C₂

yes

maximal, characteristic

C₂×#14519851

—

C₄⋊#11758187

C₄

yes

—

C₂²⋊#11758187

—

#16840650

C₄

yes

—

C₄×#11758187

—

C₄×#11758187

—

C₄×#11758194

C₄

yes

characteristic

C₂×C₄×#92418

C₂²

yes

characteristic

#16839484

—

#16839483

—

C₆₁×#431

C₄

yes

—

—

—

C₈

yes

—

C₂²×#92418

D₄

yes

characteristic

C₂²×#92418

Q₈

yes

—

—

C₂×C₄

yes

characteristic

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

D₄

yes

characteristic

C₄×#92418

D₄

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

C₈

yes

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

M₁₆

yes

characteristic

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₄²

yes

characteristic

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

#11758187

—

#11758187

—

C₂³×C₆₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#95

yes

characteristic

C₂×C₄×C₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

#97

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₈

yes

characteristic

C₈×C₆₁

—

—

—

—

yes

—

#92418

#108

yes

—

#7166

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#336

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

#350

yes

—

#2162

—

C₄×M₃₂

—

#1451

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×M₃₂

—

122

—

#431

#6923

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

122

—

#346

—

C₄×C₁₆

—

122

—

C₄×C₁₆

—

122

—

#322

—

122

—

#322

—

122

—

C₆₁

#7166

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

C₂×#6923

yes

characteristic

101

M₃₂

—

122

—

102

C₂×C₁₆

—

122

—

103

C₂×C₁₆

—

122

—

104

C₄⋊C₈

#92420

yes

—

105

C₂²⋊C₈

—

106

#97

—

122

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

122

—

113

C₂²×C₄

D₆₁⋊C₈

yes

characteristic

114

C₂²×C₄

C₄⋊#6923

yes

characteristic

115

C₂²×C₄

#11758217

yes

characteristic

116

M₁₆

—

244

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

122

—

123

C₂×C₈

—

122

—

124

C₂×C₈

—

122

—

125

C₄²

#11758213

yes

characteristic

126

C₄²

C₄×#6923

yes

characteristic

127

C₄²

C₂×#92420

yes

characteristic

128

C₄²

C₂²⋊#6923

yes

characteristic

129

C₁₆

—

122

—

130

C₁₆

—

244

—

131

C₂³

#14519844

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

137

C₂×C₄

C₄×#92420

yes

characteristic

138

C₂×C₄

#14519832

yes

characteristic

139

C₂×C₄

C₂²⋊#92420

yes

characteristic

140

C₂×C₄

#14519834

yes

characteristic

141

C₂×C₄

C₄⋊#92420

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

122

—

146

C₈

—

122

—

147

C₈

—

244

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#16840628

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#16840634

yes

characteristic

156

C₄

#16840637

yes

characteristic

157

C₄

#16840659

yes

—

158

C₄

#16840660

yes

—

159

C₂

—

160

C₂

#21460652

yes

characteristic

161

C₂

#21460867

yes

characteristic

162

C₂

#21460873

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₄×C₈
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₆₁×#431	C₄
socle	C₂²×C₆₁	#336
prime radical	C₂³×C₆₁	C₄⋊C₈
cosocle	C₂×C₄×#92418	C₂²
hypercenter	#431	#6923
hypocenter	C₆₁	#7166
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₄×C₈

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₆₁×#431

C₄

socle

C₂²×C₆₁

#336

prime radical

C₂³×C₆₁

C₄⋊C₈

cosocle

C₂×C₄×#92418

C₂²

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
96. C₆₁	#7166	semidirect

H ≅ G/N

type

96. C₆₁

#7166

semidirect

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

subgroup	index
#21462176	2
C₄×#14519851	2
#21444109	2
#7166	61

subgroup

index

Subgroups — #30594269

Conjugacy classes of subgroups

Complements

Maximal subgroups