Group #30606374 — subgroups

Subgroups — #30606374

Order 15616.

258 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 100 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	#21462042	C₂	1	1	yes	maximal, characteristic
2	#21462030	C₂	1	1	yes	maximal, characteristic
3	#21460873	C₂	1	2	yes	maximal
4	#21460869	C₂	1	2	yes	maximal
5	#21445573	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₂²	1	2	yes	—
7	C₂×#14519851	C₂²	1	1	yes	characteristic
8	C₂²×#11758187	C₂²	1	1	yes	characteristic
9	#16840655	C₄	1	4	yes	complemented
10	#16840651	C₄	1	2	yes	complemented
11	#16840648	C₄	1	2	yes	complemented
12	#16839720	C₂²	1	1	yes	characteristic
13	#16839718	C₂²	1	2	yes	—
14	#16839520	C₄	1	1	yes	characteristic
15	#16839518	C₄	1	2	yes	—
16	C₆₁×H(3,ℤ/4)	C₄	1	1	yes	characteristic
17	#14519851	—	2	2	no	—
18	#14519851	—	2	2	no	—
19	#14519851	C₂×C₄	1	4	yes	—
20	C₂×#11758187	—	2	2	no	—
21	C₂×#11758187	—	2	2	no	—
22	C₂×#11758187	C₂×C₄	1	1	yes	characteristic
23	C₂×#11758187	C₂×C₄	1	1	yes	characteristic
24	C₂×#11758187	C₂×C₄	1	2	yes	—
25	C₂×#11758194	C₂×C₄	1	2	yes	—
26	C₂²×#92418	C₂³	1	1	yes	characteristic
27	#14519736	C₂×C₄	1	1	yes	characteristic
28	#14519688	—	2	2	no	—
29	#14519687	C₂×C₄	1	2	yes	—
30	C₄×#92418	—	2	2	no	—
31	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
33	#117×C₆₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	C₄²	1	2	yes	—
37	C₄⋊C₄×C₆₁	—	2	2	no	—
38	C₂²⋊C₄×C₆₁	—	2	2	no	—
39	C₄²×C₆₁	—	2	2	no	—
40	C₂²⋊Dic₆₁	—	2	1	no	—
41	C₄⋊Dic₆₁	—	2	1	no	—
42	C₄×Dic₆₁	C₄²	1	2	yes	—
43	#11758194	—	2	2	no	—
44	#11758194	—	2	2	no	—
45	C₂×#92418	—	2	2	no	—
46	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
47	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
48	C₂×#92418	#41	1	2	yes	—
49	C₂×#92418	#41	1	2	yes	—
50	#11758187	—	2	1	no	—
51	#11758187	—	2	1	no	—
52	#11758187	—	2	4	no	—
53	#11758187	—	2	2	no	—
54	C₂³×C₆₁	#131	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	4	2	no	—
56	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#117	1	2	yes	—
58	C₂×C₄×C₆₁	#131	1	2	yes	—
59	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
60	C₂×Dic₆₁	—	2	2	no	—
61	C₈×C₆₁	—	2	2	no	—
62	#92418	—	4	2	no	—
63	#92418	#117	1	1	yes	characteristic
64	#92418	#117	1	2	yes	—
65	#92418	#117	1	1	yes	characteristic
66	#12597	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	—	4	1	no	—
68	C₂²×C₆₁	#405	1	1	yes	characteristic
69	C₂²×C₆₁	#433	1	2	yes	—
70	C₄×C₆₁	—	8	2	no	—
71	C₄×C₆₁	#376	1	1	yes	characteristic
72	C₄×C₆₁	#405	1	1	yes	characteristic
73	C₄×C₆₁	#433	1	2	yes	—
74	Dic₆₁	—	4	2	no	—
75	#2110	—	61	1	no	—
76	#2108	—	61	1	no	—
77	#1747	—	61	1	no	—
78	#1379	—	61	2	no	—
79	#1378	—	61	2	no	—
80	C₂×C₆₁	—	2	1	no	—
81	C₂×C₆₁	—	2	2	no	—
82	C₂×C₆₁	#1746	1	1	yes	characteristic
83	C₂×M₃₂	—	61	1	no	—
84	C₂×M₃₂	—	61	2	no	—
85	C₂²×C₁₆	—	61	1	no	—
86	#405	—	61	1	no	—
87	#405	—	61	2	no	—
88	#347	—	61	4	no	—
89	#346	—	61	2	no	—
90	C₄×C₁₆	—	61	2	no	—
91	#338	—	61	1	no	—
92	H(3,ℤ/4)	—	61	2	no	—
93	H(3,ℤ/4)	#6923	1	1	yes	characteristic
94	C₆₁	#12597	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	61	2	no	—
96	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
97	C₂²×C₈	—	61	1	no	—
98	#117	—	61	1	no	—
99	#117	C₂×#6923	1	2	yes	—
100	M₃₂	—	61	4	no	—
101	M₃₂	—	122	2	no	—
102	M₃₂	—	122	2	no	—
103	C₂×C₁₆	—	61	1	no	—
104	C₂×C₁₆	—	61	2	no	—
105	C₂×C₁₆	—	61	1	no	—
106	C₂×C₁₆	—	122	2	no	—
107	C₂×C₁₆	—	122	2	no	—
108	#97	—	61	2	no	—
109	#97	—	122	2	no	—
110	C₄×C₈	—	61	2	no	—
111	C₄×C₈	—	122	2	no	—
112	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	122	2	no	—
115	M₁₆	—	122	2	no	—
116	C₂×C₈	—	61	1	no	—
117	C₂×C₈	—	61	1	no	—
118	C₂×C₈	—	61	2	no	—
119	C₂×C₈	—	61	2	no	—
120	C₂×C₈	—	122	2	no	—
121	C₂×C₈	C₄×#6923	1	2	yes	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	122	1	no	—
124	C₂²⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	122	1	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	61	2	no	—
128	C₁₆	—	122	1	no	—
129	C₁₆	—	122	1	no	—
130	C₁₆	—	122	2	no	—
131	C₁₆	—	122	4	no	—
132	C₂³	#14519864	1	1	yes	characteristic
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	122	2	no	—
135	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
136	C₂×C₄	#14519856	1	2	yes	—
137	C₂×C₄	#14519864	1	2	yes	—
138	C₂×C₄	#14519865	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	61	1	no	—
141	C₈	—	61	1	no	—
142	C₈	—	61	2	no	—
143	C₈	—	244	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	#16840727	1	1	yes	characteristic
146	C₂²	#16840786	1	2	yes	—
147	C₄	—	8	2	no	—
148	C₄	—	244	2	no	—
149	C₄	#16840698	1	1	yes	characteristic
150	C₄	#16840726	1	1	yes	characteristic
151	C₄	#16840787	1	2	yes	—
152	C₂	—	2	2	no	—
153	C₂	—	2	1	no	—
154	C₂	#21461538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462042

C₂

yes

maximal, characteristic

#21462030

C₂

yes

maximal, characteristic

#21460873

C₂

yes

maximal

#21460869

C₂

yes

maximal

#21445573

C₂

yes

maximal, characteristic

C₂×#14519851

C₂²

yes

—

C₂×#14519851

C₂²

yes

characteristic

C₂²×#11758187

C₂²

yes

characteristic

#16840655

C₄

yes

complemented

#16840651

C₄

yes

complemented

#16840648

C₄

yes

complemented

#16839720

C₂²

yes

characteristic

#16839718

C₂²

yes

—

#16839520

C₄

yes

characteristic

#16839518

C₄

yes

—

C₆₁×H(3,ℤ/4)

C₄

yes

—

—

C₂×C₄

yes

—

C₂×#11758187

—

C₂×#11758187

—

C₂×#11758187

C₂×C₄

yes

characteristic

C₂×#11758187

C₂×C₄

yes

characteristic

C₂×#11758187

C₂×C₄

yes

—

C₂×#11758194

C₂×C₄

yes

—

C₂²×#92418

C₂³

yes

characteristic

#14519736

C₂×C₄

yes

characteristic

#14519688

—

#14519687

C₂×C₄

yes

—

C₄×#92418

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

characteristic

#117×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

C₄²

yes

—

—

—

—

C₂²×C₄

yes

characteristic

C₂×#92418

C₂²×C₄

yes

characteristic

C₂×#92418

#41

yes

—

C₂×#92418

#41

yes

—

—

—

—

—

C₂³×C₆₁

#131

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

characteristic

C₂×Dic₆₁

—

C₈×C₆₁

—

—

yes

yes

—

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

#405

yes

characteristic

C₂²×C₆₁

#433

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

—

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1746

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

—

—

—

—

C₄×C₁₆

—

—

—

yes

C₆₁

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×#6923

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×#6923

yes

—

100

M₃₂

—

101

M₃₂

—

122

—

102

M₃₂

—

122

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

122

—

107

C₂×C₁₆

—

122

—

108

#97

—

109

#97

—

122

—

110

C₄×C₈

—

111

C₄×C₈

—

122

—

112

C₂²×C₄

C₂²×#6923

yes

characteristic

113

M₁₆

—

114

M₁₆

—

122

—

115

M₁₆

—

122

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

122

—

121

C₂×C₈

C₄×#6923

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

122

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

122

—

126

C₄²

—

127

C₄²

—

128

C₁₆

—

122

—

129

C₁₆

—

122

—

130

C₁₆

—

122

—

131

C₁₆

—

122

—

132

C₂³

#14519864

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

122

—

135

C₂×C₄

C₂×C₄×#6923

yes

characteristic

136

C₂×C₄

#14519856

yes

—

137

C₂×C₄

#14519864

yes

—

138

C₂×C₄

#14519865

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

244

—

144

C₂²

—

145

C₂²

#16840727

yes

characteristic

146

C₂²

#16840786

yes

—

147

C₄

—

148

C₄

—

244

—

149

C₄

#16840698

yes

characteristic

150

C₄

#16840726

yes

characteristic

151

C₄

#16840787

yes

—

152

C₂

—

153

C₂

—

154

C₂

#21461538

yes

characteristic

	subgroup	G/H
center	C₄	#16840698
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×H(3,ℤ/4)	C₄
socle	C₂×C₆₁	#1746
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×#92418	C₂³
hypercenter	H(3,ℤ/4)	#6923
hypocenter	C₆₁	#12597
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16840698

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₆₁×H(3,ℤ/4)

C₄

C₂×C₆₁

C₂³×C₆₁

C₂²×#92418

C₂³

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #16840655	C₄	semidirect
10. #16840651	C₄	semidirect
11. #16840648	C₄	semidirect
94. C₆₁	#12597	semidirect

H ≅ G/N

type

9. #16840655

C₄

semidirect

10. #16840651

C₄

semidirect

11. #16840648

C₄

semidirect

94. C₆₁

#12597

semidirect

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

subgroup	index
#21462042	2
#21462030	2
#21460873	2
#21460869	2
#21445573	2
#12597	61

subgroup

index

Subgroups — #30606374

Conjugacy classes of subgroups

Complements

Maximal subgroups