Group #29567244 — subgroups

Subgroups — #29567244

Order 15616.

194 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21447618	C₂	1	1	yes	maximal, characteristic
2	#21444582	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic
4	#16839819	—	2	1	no	—
5	C₄⋊#11758185	—	2	1	no	—
6	C₂²⋊#11758185	C₄	1	2	yes	—
7	C₄×#11758185	—	2	1	no	—
8	#16839515	C₄	1	2	yes	—
9	#16839513	—	2	1	no	—
10	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#334	—	2	1	no	—
13	C₆₁×#334	C₄	1	2	yes	—
14	C₂×#11758185	—	2	2	no	—
15	C₄⋊#92418	—	2	2	no	—
16	C₂²⋊#92418	—	4	1	no	—
17	#14519688	—	4	1	no	—
18	C₂²×C₈×C₆₁	—	4	1	no	—
19	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	—	2	2	no	—
23	C₄⋊C₈×C₆₁	—	4	1	no	—
24	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
31	C₂×C₈×C₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	—	4	1	no	—
33	C₂×C₈×C₆₁	—	4	2	no	—
34	C₂×C₈×C₆₁	—	8	2	no	—
35	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
42	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
43	C₂×#92418	—	4	2	no	—
44	#11758185	—	4	2	no	—
45	C₂³×C₆₁	#107	1	1	yes	characteristic
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	1	no	—
48	C₂×C₄×C₆₁	—	2	1	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
56	C₈×C₆₁	—	2	1	no	—
57	C₈×C₆₁	—	2	1	no	—
58	C₈×C₆₁	—	2	1	no	—
59	C₈×C₆₁	—	8	2	no	—
60	C₈×C₆₁	#108	1	2	yes	—
61	#92418	—	8	2	no	—
62	#7290	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	—	2	1	no	—
64	C₂²×C₆₁	—	2	1	no	—
65	C₂²×C₆₁	#340	1	1	yes	characteristic
66	C₂²×C₆₁	#368	1	2	yes	—
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	2	1	no	—
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	#341	1	1	yes	characteristic
72	C₄×C₆₁	#344	1	1	yes	characteristic
73	C₄×C₆₁	#360	1	2	yes	—
74	#2162	—	61	1	no	—
75	#1568	—	61	1	no	—
76	C₂×#334	D₆₁	1	1	yes	characteristic
77	C₂×C₆₁	—	2	2	no	—
78	C₂×C₆₁	#1301	1	1	yes	characteristic
79	C₂×C₆₁	#1381	1	1	yes	characteristic
80	C₂×C₆₁	#1386	1	1	yes	characteristic
81	#432	—	122	1	no	—
82	C₂×C₄⋊C₈	—	2	1	no	—
83	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
84	C₄⋊C₁₆	—	122	1	no	—
85	C₂²⋊C₁₆	—	61	2	no	—
86	C₄×C₁₆	—	122	1	no	—
87	#335	—	61	2	no	—
88	#334	—	2	1	no	—
89	#334	—	122	1	no	—
90	#334	Dic₆₁	1	2	yes	—
91	C₆₁	#7290	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	4	1	no	—
93	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
96	C₂×C₁₆	—	122	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	4	1	no	—
99	C₄⋊C₈	—	122	2	no	—
100	C₂²⋊C₈	—	244	1	no	—
101	#97	—	244	1	no	—
102	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
103	C₄×C₈	D₂₄₄	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
106	C₂²×C₄	#11758196	1	1	yes	characteristic
107	C₂²×C₄	#11758199	1	1	yes	characteristic
108	C₂²×C₄	#11758202	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	8	2	no	—
113	C₂×C₈	—	244	2	no	—
114	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
115	C₂×C₈	#11758196	1	1	yes	characteristic
116	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
117	C₂×C₈	#11758198	1	1	yes	characteristic
118	C₄²	#11758198	1	1	yes	characteristic
119	C₄²	#11758200	1	1	yes	characteristic
120	C₄²	#11758201	1	1	yes	characteristic
121	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
122	C₁₆	—	244	2	no	—
123	C₂³	#14519710	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	#14519684	1	1	yes	characteristic
129	C₂×C₄	#14519697	1	1	yes	characteristic
130	C₂×C₄	#14519698	1	1	yes	characteristic
131	C₂×C₄	#14519700	1	1	yes	characteristic
132	C₂×C₄	#14519701	1	1	yes	characteristic
133	C₂×C₄	#14519708	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	8	2	no	—
138	C₈	—	488	2	no	—
139	C₈	#14519713	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#16839524	1	1	yes	characteristic
143	C₂²	#16839591	1	2	yes	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	#16839527	1	1	yes	characteristic
149	C₄	#16839534	1	1	yes	characteristic
150	C₄	#16839571	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	#21443716	1	1	yes	characteristic
153	C₂	#21443912	1	1	yes	characteristic
154	C₂	#21443926	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447618

C₂

yes

maximal, characteristic

#21444582

C₂

yes

maximal, characteristic

C₂×C₆₁×#334

C₂

yes

maximal, characteristic

#16839819

—

C₄⋊#11758185

—

C₂²⋊#11758185

C₄

yes

—

C₄×#11758185

—

#16839515

C₄

yes

—

#16839513

—

C₂×C₄⋊C₈×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#334

—

C₆₁×#334

C₄

yes

—

—

—

—

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

Q₈

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

D₈

yes

characteristic

C₂²×C₄×C₆₁

Q₁₆

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄²×C₆₁

SD₁₆

yes

characteristic

C₄²×C₆₁

SD₁₆

yes

characteristic

C₂×#92418

—

#11758185

—

C₂³×C₆₁

#107

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#108

yes

—

#92418

—

#7290

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#340

yes

characteristic

C₂²×C₆₁

#368

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#2162

—

#1568

—

C₂×#334

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

122

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

122

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

122

—

#335

—

#334

—

#334

—

122

—

#334

Dic₆₁

yes

—

C₆₁

#7290

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

Dic₁₂₂

yes

characteristic

C₂²×C₈

C₄×D₆₁

yes

characteristic

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

C₂×C₁₆

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

100

C₂²⋊C₈

—

244

—

101

#97

—

244

—

102

C₄×C₈

C₄×D₆₁

yes

characteristic

103

C₄×C₈

D₂₄₄

yes

characteristic

104

C₄×C₈

C₂×Dic₆₁

yes

characteristic

105

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

106

C₂²×C₄

#11758196

yes

characteristic

107

C₂²×C₄

#11758199

yes

characteristic

108

C₂²×C₄

#11758202

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

244

—

114

C₂×C₈

C₄×Dic₆₁

yes

characteristic

115

C₂×C₈

#11758196

yes

characteristic

116

C₂×C₈

C₄⋊Dic₆₁

yes

characteristic

117

C₂×C₈

#11758198

yes

characteristic

118

C₄²

#11758198

yes

characteristic

119

C₄²

#11758200

yes

characteristic

120

C₄²

#11758201

yes

characteristic

121

C₄²

C₂²⋊Dic₆₁

yes

characteristic

122

C₁₆

—

244

—

123

C₂³

#14519710

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#14519684

yes

characteristic

129

C₂×C₄

#14519697

yes

characteristic

130

C₂×C₄

#14519698

yes

characteristic

131

C₂×C₄

#14519700

yes

characteristic

132

C₂×C₄

#14519701

yes

characteristic

133

C₂×C₄

#14519708

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

488

—

139

C₈

#14519713

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

#16839524

yes

characteristic

143

C₂²

#16839591

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#16839527

yes

characteristic

149

C₄

#16839534

yes

characteristic

150

C₄

#16839571

yes

—

151

C₂

—

152

C₂

#21443716

yes

characteristic

153

C₂

#21443912

yes

characteristic

154

C₂

#21443926

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14519697
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₂×C₆₁×#334	C₂
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#107
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	C₂×#334	D₆₁
hypocenter	C₆₁	#7290
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

subgroup

G/H

center

C₂×C₄

#14519697

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₁

Fitting

C₂×C₆₁×#334

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄×C₈×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

N	H ≅ G/N	type
91. C₆₁	#7290	semidirect

H ≅ G/N

type

91. C₆₁

#7290

semidirect

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

subgroup	index
#21447618	2
#21444582	2
C₂×C₆₁×#334	2
#7290	61

subgroup

index

#21447618

#21444582

C₂×C₆₁×#334

#7290

Subgroups — #29567244

Conjugacy classes of subgroups

Complements

Maximal subgroups