Group #30599528 — subgroups

Subgroups — #30599528

Order 15616.

354 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 140 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	#21461760	C₂	1	2	yes	maximal
2	#21461375	C₂	1	2	yes	maximal
3	#21461374	C₂	1	2	yes	maximal
4	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519848	—	2	4	no	—
6	#16840766	C₂²	1	4	yes	—
7	#16840624	—	2	4	no	—
8	#16840623	—	2	4	no	—
9	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
10	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
11	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
12	C₈×#92418	C₄	1	1	yes	characteristic
13	C₈×#92418	C₄	1	1	yes	characteristic
14	C₈×#92418	C₄	1	1	yes	characteristic
15	C₈²×C₆₁	C₄	1	1	yes	characteristic
16	C₂×#11758213	—	2	4	no	—
17	#14519848	—	4	8	no	—
18	C₄⋊#92420	—	2	4	no	—
19	C₄⋊#92420	—	2	4	no	—
20	C₂×C₈×D₆₁	D₄	1	2	yes	—
21	C₂×C₈×D₆₁	Q₈	1	2	yes	—
22	C₂×C₈×D₆₁	Q₈	1	2	yes	—
23	C₄²×D₆₁	C₂³	1	1	yes	characteristic
24	C₄×#92418	C₂×C₄	1	1	yes	characteristic
25	C₄×#92418	C₂×C₄	1	1	yes	characteristic
26	C₄×#92418	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₆₁	D₄	1	2	yes	—
28	C₈×Dic₆₁	D₄	1	2	yes	—
29	C₈×Dic₆₁	Q₈	1	2	yes	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×D₆₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×D₆₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×#92420	—	4	4	no	—
37	#11758213	—	4	8	no	—
38	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
41	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
43	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
44	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×#92418	C₄⋊C₄	1	2	yes	—
46	C₂×#92418	C₄⋊C₄	1	2	yes	—
47	C₂×#92418	C₄⋊C₄	1	2	yes	—
48	C₈×D₆₁	—	2	4	no	—
49	C₈×D₆₁	—	2	4	no	—
50	C₈×D₆₁	SD₁₆	1	8	yes	—
51	C₂²×D₆₁	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₆₁	#127	1	1	yes	characteristic
56	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×D₆₁	—	2	2	no	—
59	C₄×D₆₁	—	2	2	no	—
60	C₄×D₆₁	C₂×SD₁₆	1	4	yes	—
61	#92420	—	8	4	no	—
62	C₈×C₆₁	—	2	2	no	—
63	C₈×C₆₁	#106	1	4	yes	—
64	C₈×C₆₁	#108	1	4	yes	complemented
65	#92418	—	2	2	no	—
66	#92418	#106	1	4	yes	—
67	#92418	#108	1	4	yes	complemented
68	#8822	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	#384	1	1	yes	characteristic
70	C₂×D₆₁	—	2	1	no	—
71	C₂×D₆₁	—	2	1	no	—
72	C₂×D₆₁	#498	1	2	yes	—
73	C₄×C₆₁	C₂×#106	1	2	yes	—
74	C₄×C₆₁	C₂×#108	1	2	yes	—
75	C₄×C₆₁	C₂×#108	1	2	yes	—
76	Dic₆₁	—	2	1	no	—
77	Dic₆₁	—	2	1	no	—
78	Dic₆₁	#492	1	2	yes	—
79	#1848	—	61	2	no	—
80	#1566	—	61	2	no	—
81	#1566	—	61	2	no	—
82	C₂×C₈²	—	61	1	no	—
83	C₂×C₆₁	#1844	1	1	yes	characteristic
84	C₂×C₆₁	#1848	1	1	yes	characteristic
85	C₂×C₆₁	#1848	1	1	yes	characteristic
86	D₆₁	—	2	2	no	—
87	C₂×#108	—	122	4	no	—
88	#423	—	61	4	no	—
89	C₂×C₄×C₈	—	61	1	no	—
90	C₂×C₄×C₈	—	61	1	no	—
91	C₂×C₄×C₈	—	61	1	no	—
92	#334	—	122	4	no	—
93	#334	—	122	4	no	—
94	C₈²	—	61	1	no	—
95	C₈²	—	61	1	no	—
96	C₈²	—	61	1	no	—
97	C₈²	#6923	1	1	yes	characteristic
98	C₆₁	#8822	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	122	4	no	—
100	C₂²×C₈	—	61	2	no	—
101	C₂²×C₈	—	61	2	no	—
102	C₂²×C₈	—	61	2	no	—
103	C₂×C₄²	—	61	1	no	—
104	#108	—	244	8	no	—
105	C₄⋊C₈	—	122	4	no	—
106	C₄⋊C₈	—	122	4	no	—
107	C₄×C₈	—	61	2	no	—
108	C₄×C₈	—	61	2	no	—
109	C₄×C₈	—	61	2	no	—
110	C₄×C₈	—	61	1	no	—
111	C₄×C₈	—	61	1	no	—
112	C₄×C₈	—	61	1	no	—
113	C₄×C₈	C₂×#6923	1	1	yes	characteristic
114	C₄×C₈	C₂×#6923	1	1	yes	characteristic
115	C₄×C₈	C₂×#6923	1	1	yes	characteristic
116	C₂²×C₄	—	61	1	no	—
117	C₂²×C₄	—	61	1	no	—
118	C₂²×C₄	—	61	1	no	—
119	M₁₆	—	244	8	no	—
120	C₂×C₈	—	61	2	no	—
121	C₂×C₈	—	61	2	no	—
122	C₂×C₈	—	61	8	no	—
123	C₂×C₈	—	61	2	no	—
124	C₂×C₈	—	122	4	no	—
125	C₂×C₈	—	122	4	no	—
126	C₂×C₈	—	244	4	no	—
127	C₂×C₈	C₄⋊#6923	1	2	yes	—
128	C₂×C₈	C₄⋊#6923	1	2	yes	—
129	C₂×C₈	C₄⋊#6923	1	2	yes	—
130	C₄²	—	61	1	no	—
131	C₄²	—	61	1	no	—
132	C₄²	—	61	1	no	—
133	C₄²	C₂²×#6923	1	1	yes	characteristic
134	C₂³	—	61	1	no	—
135	C₂×C₄	—	61	1	no	—
136	C₂×C₄	—	61	4	no	—
137	C₂×C₄	—	61	1	no	—
138	C₂×C₄	—	61	1	no	—
139	C₂×C₄	—	122	2	no	—
140	C₂×C₄	—	122	2	no	—
141	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	61	4	no	—
146	C₈	—	61	4	no	—
147	C₈	—	122	2	no	—
148	C₈	—	488	4	no	—
149	C₈	#14519845	1	4	yes	—
150	C₈	#14519848	1	4	yes	complemented
151	C₂²	—	61	2	no	—
152	C₂²	—	122	1	no	—
153	C₂²	—	122	1	no	—
154	C₂²	#16840708	1	1	yes	characteristic
155	C₄	—	61	2	no	—
156	C₄	—	122	1	no	—
157	C₄	—	122	1	no	—
158	C₄	C₂×#14519845	1	2	yes	—
159	C₄	D₆₁⋊#108	1	2	yes	—
160	C₄	C₂×#14519848	1	2	yes	—
161	C₂	—	122	2	no	—
162	C₂	#21461754	1	1	yes	characteristic
163	C₂	#21461759	1	1	yes	characteristic
164	C₂	#21461760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461760

C₂

yes

maximal

#21461375

C₂

yes

maximal

#21461374

C₂

yes

maximal

C₈²×D₆₁

C₂

yes

maximal, characteristic

C₂×#14519848

—

#16840766

C₂²

yes

—

#16840624

—

#16840623

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

C₄×C₈×D₆₁

C₂²

yes

characteristic

C₄×C₈×D₆₁

C₂²

yes

characteristic

C₈×#92418

C₄

yes

characteristic

C₈×#92418

C₄

yes

characteristic

C₈×#92418

C₄

yes

characteristic

C₈²×C₆₁

C₄

yes

—

—

—

—

C₂×C₈×D₆₁

D₄

yes

—

C₂×C₈×D₆₁

Q₈

yes

—

C₂×C₈×D₆₁

Q₈

yes

—

C₄²×D₆₁

C₂³

yes

characteristic

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

Q₈

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×D₄

yes

characteristic

C₂×C₄×D₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×D₆₁

C₂×Q₈

yes

characteristic

C₂×#92420

—

#11758213

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂×D₄

yes

characteristic

C₄×Dic₆₁

C₂×D₄

yes

characteristic

C₄×Dic₆₁

C₂×Q₈

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

—

C₂×#92418

C₄⋊C₄

yes

—

C₂×#92418

C₄⋊C₄

yes

—

C₈×D₆₁

—

C₈×D₆₁

—

C₈×D₆₁

SD₁₆

yes

—

C₂²×D₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

#127

yes

characteristic

C₂×Dic₆₁

C₄⋊Q₈

yes

characteristic

C₂×Dic₆₁

C₄⋊Q₈

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×SD₁₆

yes

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

#106

yes

—

C₈×C₆₁

yes

—

yes

—

yes

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#498

yes

—

C₄×C₆₁

C₂×#106

yes

—

C₄×C₆₁

C₂×#108

yes

—

C₄×C₆₁

C₂×#108

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#492

yes

—

#1848

—

#1566

—

#1566

—

C₂×C₈²

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂×#108

—

122

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

122

—

#334

—

122

—

C₈²

—

C₈²

—

C₈²

—

C₈²

#6923

yes

characteristic

C₆₁

#8822

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄²

—

104

#108

—

244

—

105

C₄⋊C₈

—

122

—

106

C₄⋊C₈

—

122

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂×#6923

yes

characteristic

114

C₄×C₈

C₂×#6923

yes

characteristic

115

C₄×C₈

C₂×#6923

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

244

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

122

—

125

C₂×C₈

—

122

—

126

C₂×C₈

—

244

—

127

C₂×C₈

C₄⋊#6923

yes

—

128

C₂×C₈

C₄⋊#6923

yes

—

129

C₂×C₈

C₄⋊#6923

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²×#6923

yes

characteristic

134

C₂³

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

122

—

140

C₂×C₄

—

122

—

141

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

142

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

143

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

122

—

148

C₈

—

488

—

149

C₈

#14519845

yes

—

150

C₈

#14519848

yes

complemented

151

C₂²

—

152

C₂²

—

122

—

153

C₂²

—

122

—

154

C₂²

#16840708

yes

characteristic

155

C₄

—

156

C₄

—

122

—

157

C₄

—

122

—

158

C₄

C₂×#14519845

yes

—

159

C₄

D₆₁⋊#108

yes

—

160

C₄

C₂×#14519848

yes

—

161

C₂

—

122

—

162

C₂

#21461754

yes

characteristic

163

C₂

#21461759

yes

characteristic

164

C₂

#21461760

yes

characteristic

	subgroup	G/H
center	C₂²	#16840708
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×#6923
Fitting	C₈²×C₆₁	C₄
socle	C₂²×C₆₁	#384
prime radical	C₂²×D₆₁	C₄⋊Q₈
cosocle	C₄²×D₆₁	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#8822
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#16840708

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×#6923

Fitting

C₈²×C₆₁

C₄

socle

C₂²×C₆₁

#384

prime radical

C₂²×D₆₁

C₄⋊Q₈

cosocle

C₄²×D₆₁

C₂³

C₈²

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
64. C₈×C₆₁	#108	semidirect
67. #92418	#108	semidirect
98. C₆₁	#8822	semidirect
150. C₈	#14519848	semidirect

H ≅ G/N

type

64. C₈×C₆₁

#108

semidirect

67. #92418

#108

semidirect

98. C₆₁

#8822

semidirect

150. C₈

#14519848

semidirect

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

subgroup	index
#21461760	2
#21461375	2
#21461374	2
C₈²×D₆₁	2
#8822	61

subgroup

index

C₈²×D₆₁

Subgroups — #30599528

Conjugacy classes of subgroups

Complements

Maximal subgroups