Group #29594706 — subgroups

Subgroups — #29594706

Order 15616.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21448046	C₂	1	2	yes	maximal, complemented
2	#21448042	C₂	1	2	yes	maximal
3	#21445429	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1709	C₂	1	1	yes	maximal, characteristic
6	#16840167	C₂²	1	2	yes	—
7	#16840113	—	2	2	no	—
8	#16839566	—	2	4	no	—
9	#16839563	—	2	4	no	—
10	C₄×#11758185	C₂²	1	2	yes	—
11	C₆₁×#494	C₂²	1	1	yes	characteristic
12	C₆₁×#493	C₂²	1	1	yes	characteristic
13	C₆₁×#486	—	2	2	no	—
14	C₈²×C₆₁	C₂²	1	1	yes	characteristic
15	#14519777	—	2	2	no	—
16	C₂×#11758185	D₄	1	4	yes	—
17	#14519711	—	2	2	no	—
18	#14519711	—	2	2	no	—
19	#14519702	—	4	4	no	—
20	C₂×Q₁₆×C₆₁	—	2	2	no	—
21	C₂×D₈×C₆₁	—	2	2	no	—
22	C₄⋊Q₈×C₆₁	—	2	1	no	—
23	#127×C₆₁	—	2	1	no	—
24	#102×C₆₁	—	4	4	no	—
25	C₄×C₈×C₆₁	D₄	1	2	yes	—
26	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₆₁	—	4	1	no	—
28	C₂×D₄×C₆₁	—	4	1	no	—
29	C₂×D₄×C₆₁	—	4	2	no	—
30	C₂×Dic₁₂₂	—	4	2	no	—
31	Q₁₆×C₆₁	—	4	4	no	—
32	D₈×C₆₁	—	4	4	no	—
33	C₂×C₈×C₆₁	D₈	1	4	yes	—
34	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
35	C₄⋊C₄×C₆₁	—	4	2	no	—
36	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₆₁	—	4	2	no	—
38	C₄⋊Dic₆₁	—	4	2	no	—
39	#11758185	—	2	4	no	—
40	C₂³×C₆₁	—	8	1	no	—
41	Q₈×C₆₁	—	8	2	no	—
42	D₄×C₆₁	—	8	2	no	—
43	D₄×C₆₁	—	8	4	no	—
44	C₂×C₄×C₆₁	—	8	1	no	—
45	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
47	C₂×Dic₆₁	—	8	2	no	—
48	Dic₁₂₂	—	8	4	no	—
49	C₈×C₆₁	—	2	4	no	—
50	C₈×C₆₁	#135	1	4	yes	—
51	#12511	—	61	1	no	maximal, Sylow
52	C₂²×C₆₁	—	16	1	no	—
53	C₂²×C₆₁	—	16	2	no	—
54	C₂²×C₆₁	#493	1	1	yes	characteristic
55	C₄×C₆₁	—	16	1	no	—
56	C₄×C₆₁	#495	1	2	yes	—
57	C₄×C₆₁	#508	1	4	yes	—
58	Dic₆₁	—	16	2	no	—
59	#2241	—	61	2	no	—
60	#2240	—	61	2	no	—
61	#1710	—	61	1	no	—
62	#1709	D₆₁	1	1	yes	characteristic
63	C₈×C₁₆	—	61	1	no	—
64	C₂×C₆₁	—	32	1	no	—
65	C₂×C₆₁	#1709	1	1	yes	characteristic
66	C₂×C₆₁	#2249	1	2	yes	—
67	#500	—	61	2	no	—
68	#494	C₂×D₆₁	1	1	yes	characteristic
69	#493	C₂×D₆₁	1	1	yes	characteristic
70	#487	—	122	2	no	—
71	#486	—	2	2	no	—
72	#358	—	122	4	no	—
73	#357	—	122	4	no	—
74	C₄×C₁₆	—	61	2	no	—
75	C₈²	C₂×D₆₁	1	1	yes	characteristic
76	C₆₁	#12511	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×D₈	—	2	2	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	—	122	2	no	—
81	#127	—	2	1	no	—
82	C₂×C₁₆	—	61	4	no	—
83	#107	—	122	2	no	—
84	#107	—	122	2	no	—
85	#103	—	244	4	no	—
86	#102	—	4	4	no	—
87	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
88	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	244	2	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	D₈	—	4	4	no	—
95	C₂×C₈	#11758199	1	4	yes	—
96	C₂×C₈	D₄×D₆₁	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	244	2	no	—
99	C₄⋊C₄	—	244	2	no	—
100	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
101	C₁₆	—	122	4	no	—
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	488	4	no	—
105	D₄	—	8	4	no	—
106	D₄	—	8	2	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	488	2	no	—
109	C₂×C₄	#14519773	1	1	yes	characteristic
110	C₂×C₄	C₂×#11758199	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#14519796	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	#16840138	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	976	2	no	—
118	C₄	#16840145	1	2	yes	—
119	C₄	#16840192	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	#21445424	1	1	yes	characteristic
122	C₂	#21448078	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448046

C₂

yes

maximal, complemented

#21448042

C₂

yes

maximal

#21445429

C₂

yes

maximal, characteristic, complemented

C₈×#11758185

C₂

yes

maximal, characteristic, complemented

C₆₁×#1709

C₂

yes

maximal, characteristic

#16840167

C₂²

yes

—

—

—

—

C₂²

yes

—

C₆₁×#494

C₂²

yes

characteristic

C₆₁×#493

C₂²

yes

characteristic

C₆₁×#486

—

C₈²×C₆₁

C₂²

yes

characteristic

#14519777

—

C₂×#11758185

D₄

yes

—

#14519711

—

#14519711

—

#14519702

—

C₂×Q₁₆×C₆₁

—

C₂×D₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

#127×C₆₁

—

#102×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758185

—

C₂³×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

#135

yes

—

#12511

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#493

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

#508

yes

—

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic

C₈×C₁₆

—

C₂×C₆₁

—

C₂×C₆₁

#1709

yes

characteristic

C₂×C₆₁

#2249

yes

—

#500

—

#494

C₂×D₆₁

yes

characteristic

#493

C₂×D₆₁

yes

characteristic

#487

—

122

—

#486

—

#358

—

122

—

#357

—

122

—

C₄×C₁₆

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12511

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

#127

—

C₂×C₁₆

—

#107

—

122

—

#107

—

122

—

#103

—

244

—

#102

—

C₄×C₈

C₂²⋊D₆₁

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

244

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#11758199

yes

—

C₂×C₈

D₄×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

100

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

101

C₁₆

—

122

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

488

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

488

—

109

C₂×C₄

#14519773

yes

characteristic

110

C₂×C₄

C₂×#11758199

yes

—

111

C₈

—

112

C₈

#14519796

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

#16840138

yes

characteristic

116

C₄

—

117

C₄

—

976

—

118

C₄

#16840145

yes

—

119

C₄

#16840192

yes

—

120

C₂

—

121

C₂

#21445424

yes

characteristic

122

C₂

#21448078

yes

—

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1709	C₂
socle	C₂²×C₆₁	#493
prime radical	C₆₁×#493	C₂²
hypercenter	#1709	D₆₁
hypocenter	C₆₁	#12511
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16840138

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1709

C₂

socle

C₂²×C₆₁

#493

prime radical

C₆₁×#493

C₂²

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #21448046	C₂	semidirect
3. #21445429	C₂	semidirect
4. C₈×#11758185	C₂	semidirect
76. C₆₁	#12511	semidirect

H ≅ G/N

type

1. #21448046

C₂

semidirect

3. #21445429

C₂

semidirect

4. C₈×#11758185

C₂

semidirect

76. C₆₁

#12511

semidirect

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

subgroup	index
#21448046	2
#21448042	2
#21445429	2
C₈×#11758185	2
C₆₁×#1709	2
#12511	61

subgroup

index

C₆₁×#1709

Subgroups — #29594706

Conjugacy classes of subgroups

Complements

Maximal subgroups