Group #29570471 — subgroups

Subgroups — #29570471

Order 15616.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#21445570	C₂	1	1	yes	maximal, characteristic
2	#21444102	C₂	1	1	yes	maximal, characteristic
3	#21443649	C₂	1	2	yes	maximal
4	#21443647	C₂	1	2	yes	maximal
5	C₆₁×#1452	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×#14519688	C₂²	1	1	yes	characteristic
8	#16839608	C₂²	1	1	yes	characteristic
9	#16839516	C₄	1	4	yes	—
10	#16839483	C₄	1	2	yes	complemented
11	C₈×#92418	C₄	1	2	yes	complemented
12	C₆₁×#431	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#322	C₄	1	4	yes	complemented
15	C₂²×#92418	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
17	C₄⋊#92418	—	2	2	no	—
18	C₂²⋊#92418	—	2	2	no	—
19	#14519688	—	2	2	no	—
20	C₄×#92418	C₂×C₄	1	4	yes	—
21	#14519684	—	2	2	no	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₆₁	—	2	2	no	—
25	C₂²⋊C₈×C₆₁	—	2	2	no	—
26	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	#41	1	2	yes	—
32	C₂²×Dic₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	—	2	4	no	—
34	C₂×C₈×C₆₁	—	2	4	no	—
35	C₂×C₈×C₆₁	C₄²	1	2	yes	—
36	C₂×C₈×C₆₁	C₄²	1	2	yes	—
37	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₆₁	#41	1	2	yes	—
40	C₄×Dic₆₁	—	2	4	no	—
41	C₂×#92418	—	2	4	no	—
42	C₂×#92418	—	2	2	no	—
43	C₂×#92418	—	2	4	no	—
44	C₂³×C₆₁	#117	1	1	yes	characteristic
45	C₂×C₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₆₁	#117	1	2	yes	—
48	C₂×C₄×C₆₁	#117	1	2	yes	—
49	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#131	1	4	yes	—
51	C₂×Dic₆₁	—	4	4	no	—
52	C₂×Dic₆₁	—	4	2	no	—
53	C₈×C₆₁	—	2	2	no	—
54	C₈×C₆₁	—	2	2	no	—
55	C₈×C₆₁	—	4	4	no	—
56	#92418	—	4	4	no	—
57	#92418	—	4	2	no	—
58	#8141	—	61	1	no	maximal, Sylow
59	C₂²×C₆₁	—	2	2	no	—
60	C₂²×C₆₁	#376	1	1	yes	characteristic
61	C₂²×C₆₁	#433	1	2	yes	—
62	C₄×C₆₁	—	2	2	no	—
63	C₄×C₆₁	—	2	2	no	—
64	C₄×C₆₁	#405	1	2	yes	—
65	C₄×C₆₁	#433	1	2	yes	—
66	Dic₆₁	—	8	2	no	—
67	#1746	—	61	1	no	—
68	#1452	D₆₁	1	1	yes	characteristic
69	#1450	—	61	1	no	—
70	#1275	—	61	2	no	—
71	#1274	—	61	2	no	—
72	C₂×C₆₁	—	2	2	no	—
73	C₂×C₆₁	#1746	1	1	yes	characteristic
74	C₂×C₆₁	#1747	1	2	yes	—
75	#431	—	61	2	no	—
76	#431	C₂×D₆₁	1	2	yes	—
77	C₂×#97	—	61	1	no	—
78	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
79	#376	—	61	1	no	—
80	#336	—	61	4	no	—
81	#322	—	61	2	no	—
82	#322	Dic₆₁	1	4	yes	complemented
83	C₈²	—	61	2	no	—
84	C₆₁	#8141	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	61	2	no	—
86	C₂²×C₈	C₄×D₆₁	1	2	yes	—
87	C₂×C₄²	—	61	2	no	—
88	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	122	2	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₂²⋊C₈	—	122	2	no	—
93	#97	—	122	2	no	—
94	C₄×C₈	—	61	4	no	—
95	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
96	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
97	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
98	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
99	#95	—	122	2	no	—
100	C₂²×C₄	—	122	2	no	—
101	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
102	C₂²×C₄	#11758222	1	2	yes	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	122	4	no	—
106	C₂×C₈	—	122	2	no	—
107	C₂×C₈	—	122	4	no	—
108	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
109	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
110	C₄²	—	122	4	no	—
111	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
112	C₄²	#11758222	1	2	yes	—
113	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
114	C₂³	#14519734	1	1	yes	characteristic
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	244	4	no	—
117	C₂×C₄	—	244	2	no	—
118	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
119	C₂×C₄	#14519734	1	2	yes	—
120	C₂×C₄	#14519734	1	1	yes	characteristic
121	C₂×C₄	#14519736	1	2	yes	—
122	C₂×C₄	#14519785	1	4	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	4	no	—
126	C₈	—	244	4	no	—
127	C₈	—	244	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	#16839608	1	1	yes	characteristic
130	C₂²	#16839824	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	488	2	no	—
134	C₄	#16839720	1	2	yes	—
135	C₄	#16839824	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#21445572	1	1	yes	characteristic
138	C₂	#21445574	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445570

C₂

yes

maximal, characteristic

#21444102

C₂

yes

maximal, characteristic

#21443649

C₂

yes

maximal

#21443647

C₂

yes

maximal

C₆₁×#1452

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×#14519688

C₂²

yes

characteristic

#16839608

C₂²

yes

characteristic

#16839516

C₄

yes

—

#16839483

C₄

yes

complemented

C₈×#92418

C₄

yes

complemented

C₆₁×#431

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#322

C₄

yes

complemented

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

#14519684

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

—

C₂×C₈×C₆₁

C₄²

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#117

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#117

yes

characteristic

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#376

yes

characteristic

C₂²×C₆₁

#433

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#405

yes

—

C₄×C₆₁

#433

yes

—

Dic₆₁

—

#1746

—

#1452

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#1746

yes

characteristic

C₂×C₆₁

#1747

yes

—

#431

—

#431

C₂×D₆₁

yes

—

C₂×#97

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

—

—

Dic₆₁

yes

complemented

C₈²

—

C₆₁

#8141

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

C₂²⋊C₈

—

122

—

#97

—

122

—

C₄×C₈

—

C₄×C₈

C₄×D₆₁

yes

characteristic

C₄×C₈

C₄×D₆₁

yes

characteristic

C₄×C₈

C₂×Dic₆₁

yes

—

C₄×C₈

C₂×Dic₆₁

yes

—

#95

—

122

—

100

C₂²×C₄

—

122

—

101

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

102

C₂²×C₄

#11758222

yes

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

122

—

106

C₂×C₈

—

122

—

107

C₂×C₈

—

122

—

108

C₂×C₈

C₄×Dic₆₁

yes

—

109

C₂×C₈

C₄×Dic₆₁

yes

—

110

C₄²

—

122

—

111

C₄²

C₂×C₄×D₆₁

yes

characteristic

112

C₄²

#11758222

yes

—

113

C₄²

C₂²×Dic₆₁

yes

characteristic

114

C₂³

#14519734

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

244

—

117

C₂×C₄

—

244

—

118

C₂×C₄

C₂×C₄×Dic₆₁

yes

characteristic

119

C₂×C₄

#14519734

yes

—

120

C₂×C₄

#14519734

yes

characteristic

121

C₂×C₄

#14519736

yes

—

122

C₂×C₄

#14519785

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

244

—

127

C₈

—

244

—

128

C₂²

—

129

C₂²

#16839608

yes

characteristic

130

C₂²

#16839824

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

488

—

134

C₄

#16839720

yes

—

135

C₄

#16839824

yes

—

136

C₂

—

137

C₂

#21445572

yes

characteristic

138

C₂

#21445574

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1452	C₂
socle	C₂²×C₆₁	#376
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1452	D₆₁
hypocenter	C₆₁	#8141
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #1452

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₆₁

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1452

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #1452

N	H ≅ G/N	type
10. #16839483	C₄	semidirect
11. C₈×#92418	C₄	semidirect
14. C₆₁×#322	C₄	semidirect
82. #322	Dic₆₁	semidirect
84. C₆₁	#8141	semidirect

H ≅ G/N

type

10. #16839483

C₄

semidirect

11. C₈×#92418

C₄

semidirect

14. C₆₁×#322

C₄

semidirect

82. #322

Dic₆₁

semidirect

84. C₆₁

#8141

semidirect

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

subgroup	index
#21445570	2
#21444102	2
#21443649	2
#21443647	2
C₆₁×#1452	2
#8141	61

subgroup

index

C₆₁×#1452

Subgroups — #29570471

Conjugacy classes of subgroups

Complements

Maximal subgroups