Group #30599536 — subgroups

Subgroups — #30599536

Order 15616.

338 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 128 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	1	yes	maximal, characteristic
2	#21461758	C₂	1	1	yes	maximal, characteristic
3	#21461375	C₂	1	2	yes	maximal
4	#21461374	C₂	1	2	yes	maximal
5	#322×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519848	—	2	2	no	—
7	C₂×#14519847	—	2	2	no	—
8	#16840766	C₂²	1	1	yes	characteristic
9	#16840766	C₂²	1	1	yes	characteristic
10	#16840766	C₂²	1	2	yes	—
11	#16840624	—	2	4	no	—
12	#16840623	—	2	4	no	—
13	C₄×C₈×D₆₁	C₂²	1	2	yes	—
14	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
15	#16839484	C₄	1	2	yes	—
16	#16839483	C₄	1	1	yes	characteristic
17	C₆₁×#322	C₄	1	1	yes	characteristic
18	C₂×#11758213	—	2	2	no	—
19	C₂×#11758213	—	2	1	no	—
20	C₂×#11758213	—	2	1	no	—
21	#14519848	—	4	4	no	—
22	#14519847	—	4	4	no	—
23	C₄⋊#92420	—	2	2	no	—
24	C₄⋊#92420	—	2	2	no	—
25	C₄⋊#92420	—	2	2	no	—
26	C₄⋊#92420	—	2	2	no	—
27	C₂×C₈×D₆₁	D₄	1	2	yes	—
28	C₂×C₈×D₆₁	Q₈	1	4	yes	—
29	C₄²×D₆₁	C₂³	1	1	yes	characteristic
30	C₄×#92418	C₂×C₄	1	2	yes	—
31	C₄×#92418	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₆₁	D₄	1	4	yes	—
33	C₈×Dic₆₁	Q₈	1	2	yes	—
34	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×D₆₁	C₂×Q₈	1	2	yes	—
38	C₂×#92420	—	4	1	no	—
39	C₂×#92420	—	4	2	no	—
40	C₂×#92420	—	4	1	no	—
41	#11758213	—	4	4	no	—
42	#11758213	—	4	2	no	—
43	#11758213	—	4	2	no	—
44	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
45	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
46	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
48	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×#92418	C₄⋊C₄	1	4	yes	—
50	C₂×#92418	C₄⋊C₄	1	2	yes	—
51	C₈×D₆₁	—	4	4	no	—
52	C₈×D₆₁	SD₁₆	1	4	yes	—
53	C₈×D₆₁	SD₁₆	1	4	yes	—
54	C₂²×D₆₁	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×Dic₆₁	#127	1	1	yes	characteristic
58	C₂×Dic₆₁	C₄⋊Q₈	1	2	yes	—
59	C₄×D₆₁	—	2	4	no	—
60	C₄×D₆₁	C₂×SD₁₆	1	2	yes	—
61	C₄×D₆₁	C₂×SD₁₆	1	2	yes	—
62	#92420	—	8	2	no	—
63	#92420	—	8	1	no	—
64	#92420	—	8	1	no	—
65	C₈×C₆₁	—	2	4	no	—
66	C₈×C₆₁	#106	1	4	yes	—
67	#92418	—	2	4	no	—
68	#92418	#106	1	4	yes	—
69	#8825	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	#384	1	1	yes	characteristic
71	C₂×D₆₁	—	2	2	no	—
72	C₂×D₆₁	#498	1	2	yes	—
73	C₄×C₆₁	C₂×#106	1	2	yes	—
74	C₄×C₆₁	#430	1	4	yes	—
75	Dic₆₁	—	2	2	no	—
76	Dic₆₁	#492	1	1	yes	characteristic
77	Dic₆₁	#492	1	1	yes	characteristic
78	#1848	—	61	1	no	—
79	#1848	—	61	1	no	—
80	#1566	—	61	2	no	—
81	#1566	—	61	2	no	—
82	C₂×#322	—	61	1	no	—
83	C₂×C₆₁	#1844	1	1	yes	characteristic
84	C₂×C₆₁	#1849	1	2	yes	—
85	D₆₁	—	2	2	no	—
86	C₂×#108	—	122	2	no	—
87	C₂×#108	—	122	2	no	—
88	#423	—	61	1	no	—
89	#423	—	61	2	no	—
90	#423	—	61	1	no	—
91	C₂×C₄×C₈	—	61	2	no	—
92	C₂×C₄×C₈	—	61	1	no	—
93	#334	—	122	4	no	—
94	#334	—	122	4	no	—
95	#322	—	61	1	no	—
96	#322	—	61	2	no	—
97	#322	#6923	1	1	yes	characteristic
98	C₆₁	#8825	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	122	2	no	—
100	C₂×M₁₆	—	122	1	no	—
101	C₂×M₁₆	—	122	1	no	—
102	C₂²×C₈	—	61	2	no	—
103	C₂²×C₈	—	61	4	no	—
104	C₂×C₄²	—	61	1	no	—
105	#108	—	244	4	no	—
106	#108	—	244	4	no	—
107	C₄⋊C₈	—	122	2	no	—
108	C₄⋊C₈	—	122	2	no	—
109	C₄⋊C₈	—	122	2	no	—
110	C₄⋊C₈	—	122	2	no	—
111	C₄×C₈	—	61	4	no	—
112	C₄×C₈	—	61	2	no	—
113	C₄×C₈	—	61	1	no	—
114	C₄×C₈	—	61	2	no	—
115	C₄×C₈	C₂×#6923	1	2	yes	—
116	C₄×C₈	C₂×#6923	1	1	yes	characteristic
117	C₂²×C₄	—	61	1	no	—
118	C₂²×C₄	—	61	2	no	—
119	M₁₆	—	244	4	no	—
120	M₁₆	—	244	2	no	—
121	M₁₆	—	244	2	no	—
122	C₂×C₈	—	61	2	no	—
123	C₂×C₈	—	61	4	no	—
124	C₂×C₈	—	61	4	no	—
125	C₂×C₈	—	61	4	no	—
126	C₂×C₈	—	244	4	no	—
127	C₂×C₈	—	244	1	no	—
128	C₂×C₈	—	244	2	no	—
129	C₂×C₈	—	244	1	no	—
130	C₂×C₈	C₄⋊#6923	1	4	yes	—
131	C₂×C₈	C₄⋊#6923	1	2	yes	—
132	C₄²	—	61	2	no	—
133	C₄²	—	61	1	no	—
134	C₄²	C₂²×#6923	1	1	yes	characteristic
135	C₂³	—	61	1	no	—
136	C₂×C₄	—	61	1	no	—
137	C₂×C₄	—	61	2	no	—
138	C₂×C₄	—	61	2	no	—
139	C₂×C₄	—	61	2	no	—
140	C₂×C₄	—	122	4	no	—
141	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	61	4	no	—
145	C₈	—	122	4	no	—
146	C₈	—	488	2	no	—
147	C₈	—	488	1	no	—
148	C₈	—	488	1	no	—
149	C₈	#14519845	1	4	yes	—
150	C₂²	—	61	2	no	—
151	C₂²	—	122	2	no	—
152	C₂²	#16840708	1	1	yes	characteristic
153	C₄	—	61	1	no	—
154	C₄	—	61	1	no	—
155	C₄	—	122	2	no	—
156	C₄	C₂×#14519845	1	2	yes	—
157	C₄	#16840777	1	4	yes	—
158	C₂	—	122	2	no	—
159	C₂	#21461754	1	1	yes	characteristic
160	C₂	#21461761	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461760

C₂

yes

maximal, characteristic

#21461758

C₂

yes

maximal, characteristic

#21461375

C₂

yes

maximal

#21461374

C₂

yes

maximal

#322×D₆₁

C₂

yes

maximal, characteristic

C₂×#14519848

—

C₂×#14519847

—

#16840766

C₂²

yes

characteristic

#16840766

C₂²

yes

characteristic

#16840766

C₂²

yes

—

#16840624

—

#16840623

—

C₄×C₈×D₆₁

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839484

C₄

yes

—

#16839483

C₄

yes

characteristic

C₆₁×#322

C₄

yes

—

—

—

—

—

—

—

—

—

C₂×C₈×D₆₁

D₄

yes

—

C₂×C₈×D₆₁

Q₈

yes

—

C₄²×D₆₁

C₂³

yes

characteristic

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

Q₈

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×D₄

yes

characteristic

C₂×C₄×D₆₁

C₂×Q₈

yes

—

—

—

—

—

—

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂×D₄

yes

—

C₄×Dic₆₁

C₂×Q₈

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

—

C₂×#92418

C₄⋊C₄

yes

—

C₈×D₆₁

—

C₈×D₆₁

SD₁₆

yes

—

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

—

C₂×Dic₆₁

#127

yes

characteristic

C₂×Dic₆₁

C₄⋊Q₈

yes

—

C₄×D₆₁

—

C₄×D₆₁

C₂×SD₁₆

yes

—

C₄×D₆₁

C₂×SD₁₆

yes

—

#92420

—

#92420

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

#106

yes

—

#92418

—

#92418

#106

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

#498

yes

—

C₄×C₆₁

C₂×#106

yes

—

C₄×C₆₁

#430

yes

—

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

—

—

—

—

—

C₂×C₆₁

#1844

yes

characteristic

C₂×C₆₁

#1849

yes

—

D₆₁

—

C₂×#108

—

122

—

C₂×#108

—

122

—

#423

—

#423

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

122

—

#334

—

122

—

—

—

yes

C₆₁

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

100

C₂×M₁₆

—

122

—

101

C₂×M₁₆

—

122

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

#108

—

244

—

106

#108

—

244

—

107

C₄⋊C₈

—

122

—

108

C₄⋊C₈

—

122

—

109

C₄⋊C₈

—

122

—

110

C₄⋊C₈

—

122

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

C₂×#6923

yes

—

116

C₄×C₈

C₂×#6923

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

244

—

120

M₁₆

—

244

—

121

M₁₆

—

244

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

244

—

127

C₂×C₈

—

244

—

128

C₂×C₈

—

244

—

129

C₂×C₈

—

244

—

130

C₂×C₈

C₄⋊#6923

yes

—

131

C₂×C₈

C₄⋊#6923

yes

—

132

C₄²

—

133

C₄²

—

134

C₄²

C₂²×#6923

yes

characteristic

135

C₂³

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

122

—

141

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

142

C₂×C₄

C₂×C₄⋊#6923

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

122

—

146

C₈

—

488

—

147

C₈

—

488

—

148

C₈

—

488

—

149

C₈

#14519845

yes

—

150

C₂²

—

151

C₂²

—

122

—

152

C₂²

#16840708

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

122

—

156

C₄

C₂×#14519845

yes

—

157

C₄

#16840777

yes

—

158

C₂

—

122

—

159

C₂

#21461754

yes

characteristic

160

C₂

#21461761

yes

—

	subgroup	G/H
center	C₂²	#16840708
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×#322	C₄
socle	C₂²×C₆₁	#384
prime radical	C₂²×D₆₁	C₄⋊Q₈
cosocle	C₄²×D₆₁	C₂³
hypercenter	#322	#6923
hypocenter	C₆₁	#8825
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#16840708

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×#6923

Fitting

C₆₁×#322

C₄

socle

C₂²×C₆₁

#384

prime radical

C₂²×D₆₁

C₄⋊Q₈

cosocle

C₄²×D₆₁

C₂³

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
98. C₆₁	#8825	semidirect

H ≅ G/N

type

98. C₆₁

#8825

semidirect

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

subgroup	index
#21461760	2
#21461758	2
#21461375	2
#21461374	2
#322×D₆₁	2
#8825	61

subgroup

index

#322×D₆₁

Subgroups — #30599536

Conjugacy classes of subgroups

Complements

Maximal subgroups