Group #30594162 — subgroups

Subgroups — #30594162

Order 15616.

206 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 62 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	#21460860	C₂	1	2	yes	maximal
2	#21446789	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519851	—	2	2	no	—
4	C₂×#14519688	C₂²	1	1	yes	characteristic
5	#16839632	C₄	1	1	yes	characteristic
6	#16839524	—	2	2	no	—
7	C₂×#125×C₆₁	C₄	1	1	yes	characteristic
8	#14519851	—	4	2	no	—
9	#14519851	—	4	2	no	—
10	C₂×#11758187	—	2	4	no	—
11	C₂²×#92418	D₄	1	2	yes	—
12	C₂×C₄⋊Dic₆₁	—	2	1	no	—
13	C₂×C₄⋊Dic₆₁	—	4	1	no	—
14	#14519688	—	4	1	no	—
15	#125×C₆₁	—	2	1	no	—
16	#125×C₆₁	—	4	1	no	—
17	#125×C₆₁	C₈	1	2	yes	—
18	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
19	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
20	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	—	2	1	no	—
22	C₂²×C₄×C₆₁	—	4	1	no	—
23	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×Dic₆₁	—	4	1	no	—
25	C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₄⋊C₄×C₆₁	—	4	2	no	—
27	C₄⋊C₄×C₆₁	—	4	2	no	—
28	C₄⋊C₄×C₆₁	—	4	1	no	—
29	C₄²×C₆₁	—	2	1	no	—
30	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
31	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
32	C₄⋊Dic₆₁	—	4	2	no	—
33	C₄⋊Dic₆₁	—	8	2	no	—
34	C₂×#92418	—	2	2	no	—
35	C₂×#92418	—	2	2	no	—
36	C₂×#92418	SD₁₆	1	2	yes	—
37	C₂×#92418	Q₁₆	1	2	yes	—
38	#11758187	—	4	4	no	—
39	C₂³×C₆₁	#100	1	1	yes	characteristic
40	C₂×C₄×C₆₁	—	2	1	no	—
41	C₂×C₄×C₆₁	—	2	1	no	—
42	C₂×C₄×C₆₁	—	4	1	no	—
43	C₂×C₄×C₆₁	—	4	1	no	—
44	C₂×C₄×C₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	—	8	1	no	—
46	C₂×C₄×C₆₁	—	8	1	no	—
47	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
49	C₂×C₄×C₆₁	#103	1	2	yes	—
50	C₂×C₄×C₆₁	#104	1	2	yes	—
51	C₂×Dic₆₁	—	8	2	no	—
52	C₂×Dic₆₁	—	8	1	no	—
53	#92418	—	2	2	no	—
54	#92418	—	2	2	no	—
55	#7130	—	61	1	no	maximal, Sylow
56	C₂²×C₆₁	—	2	1	no	—
57	C₂²×C₆₁	—	2	1	no	—
58	C₂²×C₆₁	#323	1	1	yes	characteristic
59	C₂²×C₆₁	#329	1	1	yes	characteristic
60	C₂²×C₆₁	#333	1	1	yes	characteristic
61	C₄×C₆₁	—	4	1	no	—
62	C₄×C₆₁	—	4	1	no	—
63	C₄×C₆₁	—	8	2	no	—
64	C₄×C₆₁	Q₈⋊C₈	1	2	yes	—
65	C₄×C₆₁	#328	1	2	yes	—
66	Dic₆₁	—	16	1	no	—
67	#1992	—	61	1	no	—
68	#1375	—	61	2	no	—
69	C₂×C₆₁	—	2	1	no	—
70	C₂×C₆₁	—	2	1	no	—
71	C₂×C₆₁	#1272	1	1	yes	characteristic
72	C₂×C₆₁	#1355	1	1	yes	characteristic
73	C₂×C₆₁	#1356	1	1	yes	characteristic
74	C₂×#125	#6923	1	1	yes	characteristic
75	C₂×M₃₂	—	122	2	no	—
76	C₂×#97	—	61	1	no	—
77	#384	—	61	1	no	—
78	#340	—	122	2	no	—
79	C₆₁	#7130	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	61	2	no	—
81	#125	—	2	1	no	—
82	#125	—	4	1	no	—
83	#125	#92420	1	2	yes	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	4	1	no	—
86	C₂×C₄⋊C₄	—	122	1	no	—
87	C₂×C₄⋊C₄	—	244	1	no	—
88	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
89	M₃₂	—	244	2	no	—
90	M₃₂	—	244	2	no	—
91	C₂×C₁₆	—	122	4	no	—
92	#97	—	244	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	—	244	1	no	—
96	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
97	C₂×C₈	—	61	2	no	—
98	C₂×C₈	—	61	2	no	—
99	C₂×C₈	—	122	2	no	—
100	C₂×C₈	—	122	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	488	2	no	—
106	C₄⋊C₄	—	244	2	no	—
107	C₄²	—	2	1	no	—
108	C₄²	C₂×#92420	1	1	yes	characteristic
109	C₄²	#11758217	1	1	yes	characteristic
110	C₁₆	—	244	4	no	—
111	C₂³	#14519837	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	488	2	no	—
120	C₂×C₄	—	488	1	no	—
121	C₂×C₄	C₂²⋊#92420	1	1	yes	characteristic
122	C₂×C₄	#14519835	1	1	yes	characteristic
123	C₂×C₄	#14519840	1	2	yes	—
124	C₂×C₄	#14519842	1	2	yes	—
125	C₈	—	122	2	no	—
126	C₈	—	122	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#16840605	1	1	yes	characteristic
130	C₂²	#16840618	1	1	yes	characteristic
131	C₂²	#16840622	1	1	yes	characteristic
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	8	2	no	—
135	C₄	—	976	1	no	—
136	C₄	Q₈⋊#92420	1	2	yes	—
137	C₄	#16840616	1	2	yes	—
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	#21460639	1	1	yes	characteristic
141	C₂	#21460823	1	1	yes	characteristic
142	C₂	#21460825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21460860

C₂

yes

maximal

#21446789

C₂

yes

maximal, characteristic

C₂×#14519851

—

C₂×#14519688

C₂²

yes

characteristic

#16839632

C₄

yes

characteristic

#16839524

—

C₂×#125×C₆₁

C₄

yes

—

—

—

C₂²×#92418

D₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519688

—

#125×C₆₁

—

#125×C₆₁

—

#125×C₆₁

C₈

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

SD₁₆

yes

—

C₂×#92418

Q₁₆

yes

—

#11758187

—

C₂³×C₆₁

#100

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#104

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

Q₈⋊C₈

yes

—

C₄×C₆₁

#328

yes

—

Dic₆₁

—

#1992

—

#1375

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

yes

C₂×M₃₂

—

122

—

C₂×#97

—

#384

—

#340

—

122

—

C₆₁

#7130

yes

characteristic, complemented, Sylow

C₂²×C₈

—

—

—

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

244

—

C₂×C₄²

C₂×#6923

yes

characteristic

M₃₂

—

244

—

M₃₂

—

244

—

C₂×C₁₆

—

122

—

#97

—

244

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂²⋊#6923

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

100

C₂×C₈

—

122

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

488

—

106

C₄⋊C₄

—

244

—

107

C₄²

—

108

C₄²

C₂×#92420

yes

characteristic

109

C₄²

#11758217

yes

characteristic

110

C₁₆

—

244

—

111

C₂³

#14519837

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

488

—

120

C₂×C₄

—

488

—

121

C₂×C₄

C₂²⋊#92420

yes

characteristic

122

C₂×C₄

#14519835

yes

characteristic

123

C₂×C₄

#14519840

yes

—

124

C₂×C₄

#14519842

yes

—

125

C₈

—

122

—

126

C₈

—

122

—

127

C₂²

—

128

C₂²

—

129

C₂²

#16840605

yes

characteristic

130

C₂²

#16840618

yes

characteristic

131

C₂²

#16840622

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

976

—

136

C₄

Q₈⋊#92420

yes

—

137

C₄

#16840616

yes

—

138

C₂

—

139

C₂

—

140

C₂

#21460639

yes

characteristic

141

C₂

#21460823

yes

characteristic

142

C₂

#21460825

yes

characteristic

	subgroup	G/H
center	C₂²	#16840618
commutator	C₄²×C₆₁	C₂×C₈
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₂×#125×C₆₁	C₄
socle	C₂²×C₆₁	#329
prime radical	C₂³×C₆₁	#100
cosocle	C₂×#14519688	C₂²
hypercenter	C₂×#125	#6923
hypocenter	C₆₁	#7130
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

#16840618

commutator

C₄²×C₆₁

C₂×C₈

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₂×#125×C₆₁

C₄

C₂²×C₆₁

C₂³×C₆₁

C₂²

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
79. C₆₁	#7130	semidirect

H ≅ G/N

type

79. C₆₁

#7130

semidirect

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

subgroup	index
#21460860	2
#21446789	2
#7130	61

subgroup

index

#21460860

#21446789

#7130

Subgroups — #30594162

Conjugacy classes of subgroups

Complements

Maximal subgroups