Group #29566624 — subgroups

Subgroups — #29566624

Order 15616.

304 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 115 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	#21445672	C₂	1	2	yes	maximal
2	C₆₁×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#92418	—	2	2	no	—
4	C₂×#14519688	—	2	2	no	—
5	#16839517	C₄	1	4	yes	—
6	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
7	C₄³×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#336	—	2	2	no	—
9	C₂²×#92418	—	2	4	no	—
10	C₄⋊#92418	—	2	8	no	—
11	#14519688	—	4	2	no	—
12	#14519688	—	4	2	no	—
13	C₂²×C₈×C₆₁	—	2	2	no	—
14	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₆₁	D₄	1	2	yes	—
18	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
19	C₄⋊C₈×C₆₁	—	4	2	no	—
20	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
24	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₁	—	4	2	no	—
26	C₂×C₈×C₆₁	—	4	4	no	—
27	C₄²×C₆₁	—	2	1	no	—
28	C₄²×C₆₁	—	2	1	no	—
29	C₄²×C₆₁	—	2	4	no	—
30	C₄²×C₆₁	—	4	2	no	—
31	C₄²×C₆₁	—	4	1	no	—
32	C₄²×C₆₁	D₈	1	1	yes	characteristic
33	C₄²×C₆₁	SD₁₆	1	2	yes	—
34	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
35	C₂×#92418	—	2	8	no	—
36	C₂×#92418	—	4	4	no	—
37	C₂×#92418	—	4	4	no	—
38	C₂³×C₆₁	#95	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	—	2	2	no	—
41	C₂×C₄×C₆₁	—	2	1	no	—
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	—	2	2	no	—
45	C₂×C₄×C₆₁	—	2	1	no	—
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
49	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#102	1	2	yes	—
51	C₂×C₄×C₆₁	#103	1	2	yes	—
52	C₂×C₄×C₆₁	#104	1	4	yes	—
53	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
55	C₈×C₆₁	—	8	2	no	—
56	#92418	—	4	8	no	—
57	#7049	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	#338	1	1	yes	characteristic
59	C₂²×C₆₁	#339	1	2	yes	—
60	C₂²×C₆₁	#340	1	1	yes	characteristic
61	C₂²×C₆₁	#343	1	1	yes	characteristic
62	C₂²×C₆₁	#344	1	2	yes	—
63	C₄×C₆₁	—	2	1	no	—
64	C₄×C₆₁	—	2	1	no	—
65	C₄×C₆₁	—	2	2	no	—
66	C₄×C₆₁	—	2	2	no	—
67	C₄×C₆₁	—	4	2	no	—
68	C₄×C₆₁	—	4	1	no	—
69	C₄×C₆₁	#329	1	2	yes	—
70	C₄×C₆₁	#330	1	2	yes	—
71	#1831	D₆₁	1	1	yes	characteristic
72	#1767	—	61	2	no	—
73	C₂×C₆₁	#1291	1	2	yes	—
74	C₂×C₆₁	#1292	1	1	yes	characteristic
75	C₂×C₆₁	#1295	1	1	yes	characteristic
76	C₂×C₆₁	#1303	1	2	yes	—
77	C₂×C₆₁	#1307	1	1	yes	characteristic
78	C₂×C₄⋊C₈	—	122	2	no	—
79	C₂×C₄⋊C₈	Dic₆₁	1	2	yes	—
80	C₂×#97	—	122	2	no	—
81	C₄³	C₂×D₆₁	1	1	yes	characteristic
82	#336	—	2	2	no	—
83	#336	—	61	4	no	—
84	C₆₁	#7049	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	122	4	no	—
87	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
88	C₂×C₄²	C₄×D₆₁	1	2	yes	—
89	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
90	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	122	8	no	—
94	#97	—	244	2	no	—
95	#97	—	244	2	no	—
96	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
97	C₂²×C₄	#11758196	1	2	yes	—
98	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
99	C₂²×C₄	#11758198	1	2	yes	—
100	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	122	8	no	—
104	C₂×C₈	—	244	4	no	—
105	C₂×C₈	—	244	4	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	4	no	—
109	C₄²	—	4	1	no	—
110	C₄²	—	4	2	no	—
111	C₄²	#11758190	1	2	yes	—
112	C₄²	D₄₈₈	1	1	yes	characteristic
113	C₄²	Dic₂₄₄	1	1	yes	characteristic
114	C₂³	#14519684	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	#14519694	1	1	yes	characteristic
125	C₂×C₄	#14519696	1	1	yes	characteristic
126	C₂×C₄	#14519699	1	2	yes	—
127	C₂×C₄	#14519702	1	2	yes	—
128	C₂×C₄	#14519704	1	4	yes	—
129	C₂×C₄	#14519709	1	1	yes	characteristic
130	C₂×C₄	#14519711	1	1	yes	characteristic
131	C₈	—	8	2	no	—
132	C₈	—	244	8	no	—
133	C₂²	#16839520	1	1	yes	characteristic
134	C₂²	#16839521	1	2	yes	—
135	C₂²	#16839525	1	1	yes	characteristic
136	C₂²	#16839531	1	1	yes	characteristic
137	C₂²	#16839534	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	#16839502	1	2	yes	—
145	C₄	#16839504	1	2	yes	—
146	C₂	#21443690	1	2	yes	—
147	C₂	#21443694	1	1	yes	characteristic
148	C₂	#21443700	1	1	yes	characteristic
149	C₂	#21443721	1	2	yes	—
150	C₂	#21443730	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445672

C₂

yes

maximal

C₆₁×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

—

C₂×#14519688

—

#16839517

C₄

yes

—

C₂×C₄⋊C₈×C₆₁

C₄

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₆₁×#336

—

C₂²×#92418

—

C₄⋊#92418

—

#14519688

—

#14519688

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

Q₈

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

C₄²

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₈

yes

characteristic

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

Q₁₆

yes

—

—

—

C₂³×C₆₁

#95

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

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

#104

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#329

yes

—

C₄×C₆₁

#330

yes

—

#1831

D₆₁

yes

characteristic

#1767

—

C₂×C₆₁

#1291

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#1307

yes

characteristic

C₂×C₄⋊C₈

—

122

—

C₂×C₄⋊C₈

Dic₆₁

yes

—

C₂×#97

—

122

—

C₄³

C₂×D₆₁

yes

characteristic

#336

—

#336

—

C₆₁

#7049

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

122

—

C₂×C₄²

Dic₁₂₂

yes

characteristic

C₂×C₄²

C₄×D₆₁

yes

—

C₂×C₄²

D₂₄₄

yes

characteristic

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

C₂×C₄²

C₂²⋊D₆₁

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

#97

—

244

—

#97

—

244

—

C₂²×C₄

C₄×Dic₆₁

yes

characteristic

C₂²×C₄

#11758196

yes

—

C₂²×C₄

C₄⋊Dic₆₁

yes

characteristic

C₂²×C₄

#11758198

yes

—

100

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

122

—

104

C₂×C₈

—

244

—

105

C₂×C₈

—

244

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

#11758190

yes

—

112

C₄²

D₄₈₈

yes

characteristic

113

C₄²

Dic₂₄₄

yes

characteristic

114

C₂³

#14519684

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#14519694

yes

characteristic

125

C₂×C₄

#14519696

yes

characteristic

126

C₂×C₄

#14519699

yes

—

127

C₂×C₄

#14519702

yes

—

128

C₂×C₄

#14519704

yes

—

129

C₂×C₄

#14519709

yes

characteristic

130

C₂×C₄

#14519711

yes

characteristic

131

C₈

—

132

C₈

—

244

—

133

C₂²

#16839520

yes

characteristic

134

C₂²

#16839521

yes

—

135

C₂²

#16839525

yes

characteristic

136

C₂²

#16839531

yes

characteristic

137

C₂²

#16839534

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#16839502

yes

—

145

C₄

#16839504

yes

—

146

C₂

#21443690

yes

—

147

C₂

#21443694

yes

characteristic

148

C₂

#21443700

yes

characteristic

149

C₂

#21443721

yes

—

150

C₂

#21443730

yes

characteristic

	subgroup	G/H
center	C₂³	#14519684
commutator	C₂²×C₄×C₆₁	C₄²
Frattini	C₄³	C₂×D₆₁
Fitting	C₆₁×#1831	C₂
socle, prime radical	C₂³×C₆₁	#95
cosocle	C₄³×C₆₁	C₂²
hypercenter	#1831	D₆₁
hypocenter	C₆₁	#7049
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

#14519684

commutator

C₂²×C₄×C₆₁

C₄²

Frattini

C₄³

C₂×D₆₁

Fitting

C₆₁×#1831

C₂

socle, prime radical

C₂³×C₆₁

#95

cosocle

C₄³×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
84. C₆₁	#7049	semidirect

H ≅ G/N

type

84. C₆₁

#7049

semidirect

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

subgroup	index
#21445672	2
C₆₁×#1831	2
#7049	61

subgroup

index

#21445672

C₆₁×#1831

#7049

Subgroups — #29566624

Conjugacy classes of subgroups

Complements

Maximal subgroups