Group #29566610 — subgroups

Subgroups — #29566610

Order 15616.

304 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₄	1	4	yes	—
5	#16839517	—	2	2	no	—
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	4	no	—
12	C₂²×C₈×C₆₁	—	2	2	no	—
13	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
16	C₂×C₄²×C₆₁	D₄	1	2	yes	—
17	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
18	C₄⋊C₈×C₆₁	—	4	2	no	—
19	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
21	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₁	—	4	2	no	—
25	C₂×C₈×C₆₁	—	4	4	no	—
26	C₄²×C₆₁	—	2	1	no	—
27	C₄²×C₆₁	—	2	2	no	—
28	C₄²×C₆₁	—	2	1	no	—
29	C₄²×C₆₁	—	2	2	no	—
30	C₄²×C₆₁	—	2	4	no	—
31	C₄²×C₆₁	—	2	2	no	—
32	C₄²×C₆₁	—	4	1	no	—
33	C₂×#92418	—	2	8	no	—
34	C₂×#92418	—	4	4	no	—
35	C₂×#92418	—	4	4	no	—
36	C₂³×C₆₁	#95	1	1	yes	characteristic
37	C₂×C₄×C₆₁	—	2	2	no	—
38	C₂×C₄×C₆₁	—	2	1	no	—
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	2	1	no	—
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	—	2	1	no	—
45	C₂×C₄×C₆₁	—	2	2	no	—
46	C₂×C₄×C₆₁	#100	1	2	yes	—
47	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#101	1	2	yes	—
49	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#104	1	4	yes	—
51	C₂×C₄×C₆₁	#108	1	2	yes	—
52	C₈×C₆₁	—	8	2	no	—
53	#92418	—	4	8	no	—
54	#7042	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	H(3,ℤ/4)	1	1	yes	characteristic
56	C₂²×C₆₁	#339	1	2	yes	—
57	C₂²×C₆₁	#341	1	1	yes	characteristic
58	C₂²×C₆₁	#343	1	2	yes	—
59	C₂²×C₆₁	#343	1	1	yes	characteristic
60	C₄×C₆₁	—	2	2	no	—
61	C₄×C₆₁	—	2	2	no	—
62	C₄×C₆₁	—	2	2	no	—
63	C₄×C₆₁	—	4	2	no	—
64	C₄×C₆₁	—	4	1	no	—
65	C₄×C₆₁	#329	1	2	yes	—
66	C₄×C₆₁	#330	1	2	yes	—
67	#1831	D₆₁	1	1	yes	characteristic
68	#1767	—	61	2	no	—
69	C₂×C₆₁	#1285	1	2	yes	—
70	C₂×C₆₁	#1290	1	1	yes	characteristic
71	C₂×C₆₁	#1295	1	1	yes	characteristic
72	C₂×C₆₁	#1299	1	2	yes	—
73	C₂×C₆₁	#1309	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	122	2	no	—
75	C₂×C₄⋊C₈	Dic₆₁	1	2	yes	—
76	C₂×#97	—	61	4	no	—
77	C₄³	C₂×D₆₁	1	1	yes	characteristic
78	#336	—	2	2	no	—
79	#336	—	122	2	no	—
80	C₆₁	#7042	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	122	4	no	—
83	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
84	C₂×C₄²	C₄×D₆₁	1	2	yes	—
85	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
86	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
87	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	122	8	no	—
90	#97	—	244	4	no	—
91	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
92	C₂²×C₄	#11758196	1	2	yes	—
93	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
94	C₂²×C₄	#11758198	1	2	yes	—
95	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	—	122	8	no	—
99	C₂×C₈	—	244	4	no	—
100	C₂×C₈	—	244	4	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	4	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	4	1	no	—
108	C₂³	#14519684	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	#14519693	1	2	yes	—
119	C₂×C₄	#14519694	1	1	yes	characteristic
120	C₂×C₄	#14519695	1	2	yes	—
121	C₂×C₄	#14519696	1	1	yes	characteristic
122	C₂×C₄	#14519704	1	4	yes	—
123	C₂×C₄	#14519713	1	2	yes	—
124	C₈	—	8	2	no	—
125	C₈	—	244	8	no	—
126	C₂²	#16839519	1	1	yes	characteristic
127	C₂²	#16839521	1	2	yes	—
128	C₂²	#16839527	1	1	yes	characteristic
129	C₂²	#16839530	1	2	yes	—
130	C₂²	#16839531	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	1	no	—
136	C₄	#16839502	1	2	yes	—
137	C₄	#16839504	1	2	yes	—
138	C₂	#21443676	1	2	yes	—
139	C₂	#21443689	1	1	yes	characteristic
140	C₂	#21443700	1	1	yes	characteristic
141	C₂	#21443710	1	2	yes	—
142	C₂	#21443732	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

C₄

yes

—

#16839517

—

C₂×C₄⋊C₈×C₆₁

C₄

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₆₁×#336

—

C₂²×#92418

—

C₄⋊#92418

—

#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

—

C₄²×C₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

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

#100

yes

—

C₂×C₄×C₆₁

#100

yes

characteristic

C₂×C₄×C₆₁

#101

yes

—

C₂×C₄×C₆₁

#101

yes

characteristic

C₂×C₄×C₆₁

#104

yes

—

C₂×C₄×C₆₁

#108

yes

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#341

yes

characteristic

C₂²×C₆₁

#343

yes

—

C₂²×C₆₁

#343

yes

characteristic

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

#1285

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#1309

yes

characteristic

C₂×C₄⋊C₈

—

122

—

C₂×C₄⋊C₈

Dic₆₁

yes

—

C₂×#97

—

C₄³

C₂×D₆₁

yes

characteristic

#336

—

#336

—

122

—

C₆₁

#7042

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

—

C₂²×C₄

C₄×Dic₆₁

yes

characteristic

C₂²×C₄

#11758196

yes

—

C₂²×C₄

C₄⋊Dic₆₁

yes

characteristic

C₂²×C₄

#11758198

yes

—

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

100

C₂×C₈

—

244

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂³

#14519684

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#14519693

yes

—

119

C₂×C₄

#14519694

yes

characteristic

120

C₂×C₄

#14519695

yes

—

121

C₂×C₄

#14519696

yes

characteristic

122

C₂×C₄

#14519704

yes

—

123

C₂×C₄

#14519713

yes

—

124

C₈

—

125

C₈

—

244

—

126

C₂²

#16839519

yes

characteristic

127

C₂²

#16839521

yes

—

128

C₂²

#16839527

yes

characteristic

129

C₂²

#16839530

yes

—

130

C₂²

#16839531

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#16839502

yes

—

137

C₄

#16839504

yes

—

138

C₂

#21443676

yes

—

139

C₂

#21443689

yes

characteristic

140

C₂

#21443700

yes

characteristic

141

C₂

#21443710

yes

—

142

C₂

#21443732

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	#7042
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
80. C₆₁	#7042	semidirect

H ≅ G/N

type

80. C₆₁

#7042

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
#7042	61

subgroup

index

#21445672

C₆₁×#1831

#7042

Subgroups — #29566610

Conjugacy classes of subgroups

Complements

Maximal subgroups