Group #30607316 — subgroups

Subgroups — #30607316

Order 15616.

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

normal subgroups only

This group has 234 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	#21462160	C₂	1	1	yes	maximal, characteristic
2	#21462156	C₂	1	1	yes	maximal, characteristic
3	#21460874	C₂	1	2	yes	maximal
4	#21460870	C₂	1	2	yes	maximal
5	#21445575	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₂²	1	1	yes	characteristic
7	C₂×#14519851	C₂²	1	2	yes	—
8	C₂²×#11758187	C₂²	1	1	yes	characteristic
9	#16840684	—	2	2	no	—
10	#16840683	—	2	2	no	—
11	#16839720	C₂²	1	2	yes	—
12	#16839720	C₂²	1	1	yes	characteristic
13	#16839519	C₄	1	2	yes	—
14	#16839519	C₄	1	1	yes	characteristic
15	C₆₁×#338	C₄	1	1	yes	characteristic
16	#14519851	—	2	2	no	—
17	#14519851	—	2	2	no	—
18	#14519851	—	2	2	no	—
19	C₂×#11758187	—	2	1	no	—
20	C₂×#11758187	—	2	1	no	—
21	C₂×#11758187	—	2	2	no	—
22	C₂×#11758187	—	2	2	no	—
23	C₂²×#92418	C₂³	1	1	yes	characteristic
24	#14519736	C₂×C₄	1	2	yes	—
25	#14519736	C₂×C₄	1	1	yes	characteristic
26	#14519687	—	2	2	no	—
27	#14519687	Q₈	1	2	yes	—
28	C₈×Dic₆₁	—	2	2	no	—
29	C₈×Dic₆₁	D₄	1	2	yes	—
30	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₆₁	—	2	2	no	—
34	M₁₆×C₆₁	—	2	2	no	—
35	M₁₆×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
38	C₂²⋊Dic₆₁	—	2	1	no	—
39	C₂²⋊Dic₆₁	—	2	2	no	—
40	C₄⋊Dic₆₁	—	2	1	no	—
41	C₄⋊Dic₆₁	—	2	2	no	—
42	C₄×Dic₆₁	—	2	2	no	—
43	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
44	C₂×#92418	C₂×D₄	1	1	yes	characteristic
45	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
46	C₂×#92418	#41	1	2	yes	—
47	C₂×#92418	#41	1	2	yes	—
48	#11758187	—	4	1	no	—
49	#11758187	—	4	1	no	—
50	#11758187	—	4	2	no	—
51	C₂³×C₆₁	#131	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#117	1	2	yes	—
54	C₂×C₄×C₆₁	#131	1	2	yes	—
55	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
56	C₂×Dic₆₁	—	2	2	no	—
57	C₂×Dic₆₁	—	4	2	no	—
58	C₈×C₆₁	—	2	2	no	—
59	C₈×C₆₁	—	4	2	no	—
60	#92418	#124	1	1	yes	characteristic
61	#92418	#124	1	1	yes	characteristic
62	#92418	#125	1	2	yes	—
63	#12885	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	—	4	1	no	—
65	C₂²×C₆₁	#424	1	1	yes	characteristic
66	C₂²×C₆₁	#433	1	2	yes	—
67	C₄×C₆₁	#382	1	1	yes	characteristic
68	C₄×C₆₁	#424	1	1	yes	characteristic
69	C₄×C₆₁	#433	1	2	yes	—
70	Dic₆₁	—	4	2	no	—
71	Dic₆₁	—	8	2	no	—
72	#2153	—	61	1	no	—
73	#2151	—	61	1	no	—
74	#1747	—	61	1	no	—
75	#1379	—	61	2	no	—
76	#1378	—	61	2	no	—
77	C₂×C₆₁	—	2	1	no	—
78	C₂×C₆₁	—	2	2	no	—
79	C₂×C₆₁	#1832	1	1	yes	characteristic
80	C₂×M₃₂	—	61	1	no	—
81	C₂×M₃₂	—	61	2	no	—
82	C₂²×C₁₆	—	61	1	no	—
83	#405	—	61	1	no	—
84	#405	—	61	2	no	—
85	#364	—	122	2	no	—
86	C₄⋊C₁₆	—	122	2	no	—
87	#338	#6923	1	1	yes	characteristic
88	H(3,ℤ/4)	—	61	2	no	—
89	H(3,ℤ/4)	—	61	1	no	—
90	C₆₁	#12885	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	C₂×#6923	1	2	yes	—
92	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
93	C₂²×C₈	—	61	1	no	—
94	#117	—	61	1	no	—
95	#117	—	61	2	no	—
96	M₃₂	—	122	2	no	—
97	M₃₂	—	122	2	no	—
98	M₃₂	—	122	2	no	—
99	C₂×C₁₆	—	122	1	no	—
100	C₂×C₁₆	—	122	1	no	—
101	C₂×C₁₆	—	122	2	no	—
102	C₂×C₁₆	—	122	2	no	—
103	#97	—	61	2	no	—
104	#97	—	122	2	no	—
105	C₄×C₈	—	61	2	no	—
106	C₄×C₈	—	122	2	no	—
107	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	61	1	no	—
113	C₂×C₈	—	61	1	no	—
114	C₂×C₈	—	61	2	no	—
115	C₂×C₈	—	61	2	no	—
116	C₂×C₈	C₄⋊#6923	1	2	yes	—
117	C₄⋊C₄	—	122	1	no	—
118	C₄⋊C₄	—	122	2	no	—
119	C₂²⋊C₄	—	122	1	no	—
120	C₂²⋊C₄	—	122	2	no	—
121	C₄²	—	61	2	no	—
122	C₄²	—	122	2	no	—
123	C₁₆	—	244	1	no	—
124	C₁₆	—	244	1	no	—
125	C₁₆	—	244	2	no	—
126	C₂³	#14519865	1	1	yes	characteristic
127	C₂×C₄	—	122	2	no	—
128	C₂×C₄	—	244	2	no	—
129	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
130	C₂×C₄	#14519856	1	2	yes	—
131	C₂×C₄	#14519864	1	2	yes	—
132	C₂×C₄	#14519864	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	—	61	1	no	—
136	C₈	—	61	1	no	—
137	C₈	—	61	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	#16840769	1	1	yes	characteristic
140	C₂²	#16840787	1	2	yes	—
141	C₄	—	488	2	no	—
142	C₄	—	244	2	no	—
143	C₄	#16840706	1	1	yes	characteristic
144	C₄	#16840768	1	1	yes	characteristic
145	C₄	#16840786	1	2	yes	—
146	C₂	—	2	2	no	—
147	C₂	—	2	1	no	—
148	C₂	#21461737	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462160

C₂

yes

maximal, characteristic

#21462156

C₂

yes

maximal, characteristic

#21460874

C₂

yes

maximal

#21460870

C₂

yes

maximal

#21445575

C₂

yes

maximal, characteristic

C₂×#14519851

C₂²

yes

characteristic

C₂×#14519851

C₂²

yes

—

C₂²×#11758187

C₂²

yes

—

—

C₂²

yes

—

#16839720

C₂²

yes

characteristic

#16839519

C₄

yes

—

#16839519

C₄

yes

characteristic

C₆₁×#338

C₄

yes

—

—

—

—

—

—

—

C₂²×#92418

C₂³

yes

characteristic

#14519736

C₂×C₄

yes

—

#14519736

C₂×C₄

yes

characteristic

#14519687

—

#14519687

Q₈

yes

—

C₈×Dic₆₁

—

C₈×Dic₆₁

D₄

yes

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₄⋊C₄

yes

—

C₂×#92418

C₂×D₄

yes

characteristic

C₂×#92418

C₂×Q₈

yes

characteristic

C₂×#92418

#41

yes

—

C₂×#92418

#41

yes

—

#11758187

—

#11758187

—

#11758187

—

C₂³×C₆₁

#131

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

yes

yes

yes

—

#12885

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#424

yes

characteristic

C₂²×C₆₁

#433

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1832

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#405

—

#364

—

122

—

C₄⋊C₁₆

—

122

—

yes

—

—

C₆₁

#12885

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×#6923

yes

—

C₂×M₁₆

C₂×#6923

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

M₃₂

—

122

—

M₃₂

—

122

—

M₃₂

—

122

—

C₂×C₁₆

—

122

—

100

C₂×C₁₆

—

122

—

101

C₂×C₁₆

—

122

—

102

C₂×C₁₆

—

122

—

103

#97

—

104

#97

—

122

—

105

C₄×C₈

—

106

C₄×C₈

—

122

—

107

C₂²×C₄

C₂²×#6923

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄⋊#6923

yes

—

117

C₄⋊C₄

—

122

—

118

C₄⋊C₄

—

122

—

119

C₂²⋊C₄

—

122

—

120

C₂²⋊C₄

—

122

—

121

C₄²

—

122

C₄²

—

122

—

123

C₁₆

—

244

—

124

C₁₆

—

244

—

125

C₁₆

—

244

—

126

C₂³

#14519865

yes

characteristic

127

C₂×C₄

—

122

—

128

C₂×C₄

—

244

—

129

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

130

C₂×C₄

#14519856

yes

—

131

C₂×C₄

#14519864

yes

—

132

C₂×C₄

#14519864

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

#16840769

yes

characteristic

140

C₂²

#16840787

yes

—

141

C₄

—

488

—

142

C₄

—

244

—

143

C₄

#16840706

yes

characteristic

144

C₄

#16840768

yes

characteristic

145

C₄

#16840786

yes

—

146

C₂

—

147

C₂

—

148

C₂

#21461737

yes

characteristic

	subgroup	G/H
center	C₄	#16840706
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#338	C₄
socle	C₂×C₆₁	#1832
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×#92418	C₂³
hypercenter	#338	#6923
hypocenter	C₆₁	#12885
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

subgroup

G/H

center

C₄

#16840706

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₆₁×#338

C₄

C₂×C₆₁

C₂³×C₆₁

C₂²×#92418

C₂³

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

N	H ≅ G/N	type
90. C₆₁	#12885	semidirect

H ≅ G/N

type

90. C₆₁

#12885

semidirect

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

subgroup	index
#21462160	2
#21462156	2
#21460874	2
#21460870	2
#21445575	2
#12885	61

subgroup

index

Subgroups — #30607316

Conjugacy classes of subgroups

Complements

Maximal subgroups