Group #29585088 — subgroups

Subgroups — #29585088

Order 15616.

478 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 141 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	#21446625	C₂	1	3	yes	maximal
2	#21445947	C₂	1	3	yes	maximal, complemented
3	C₆₁×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14519774	C₂²	1	1	yes	characteristic
5	C₂×#14519699	—	2	6	no	—
6	#16839632	C₂²	1	3	yes	—
7	#16839525	—	2	6	no	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
9	C₂²×D₂₄₄	—	2	3	no	—
10	#14519774	—	2	3	no	—
11	#14519774	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₆₁	—	2	3	no	—
13	C₂×C₄⋊Dic₆₁	—	2	3	no	—
14	C₂×C₄⋊Dic₆₁	—	2	3	no	—
15	#14519699	—	4	12	no	—
16	C₂²×C₈×C₆₁	D₄	1	6	yes	—
17	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₆₁	—	2	6	no	—
19	C₂³×D₆₁	—	4	1	no	—
20	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
21	C₂²×Dic₆₁	—	4	3	no	—
22	C₂×D₂₄₄	—	2	6	no	—
23	C₂×D₂₄₄	—	4	6	no	—
24	C₂×D₂₄₄	—	4	6	no	—
25	C₂×D₂₄₄	—	4	3	no	—
26	C₂×C₈×C₆₁	—	2	6	no	—
27	C₂×C₈×C₆₁	—	2	6	no	—
28	C₂×C₈×C₆₁	—	2	6	no	—
29	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₆₁	#41	1	3	yes	—
31	C₄⋊Dic₆₁	—	4	6	no	—
32	C₄⋊Dic₆₁	—	4	6	no	—
33	C₄⋊Dic₆₁	—	4	6	no	—
34	C₂³×C₆₁	#127	1	1	yes	characteristic
35	C₂²×D₆₁	—	4	2	no	—
36	C₂²×D₆₁	—	8	3	no	—
37	C₂²×D₆₁	—	8	3	no	—
38	C₂×C₄×C₆₁	C₂²⋊Q₈	1	3	yes	—
39	C₂×C₄×C₆₁	#123	1	3	yes	—
40	C₂×C₄×C₆₁	#135	1	6	yes	—
41	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	6	yes	—
42	C₂×Dic₆₁	—	8	3	no	—
43	C₂×Dic₆₁	—	8	3	no	—
44	C₂×Dic₆₁	—	8	3	no	—
45	D₂₄₄	—	4	12	no	—
46	D₂₄₄	—	8	6	no	—
47	C₈×C₆₁	—	4	6	no	—
48	#10678	—	61	1	no	maximal, Sylow
49	C₂²×C₆₁	#399	1	1	yes	characteristic
50	C₂²×C₆₁	#495	1	3	yes	—
51	C₂²×C₆₁	#496	1	3	yes	—
52	C₂×D₆₁	—	8	6	no	—
53	C₂×D₆₁	—	16	1	no	—
54	C₂×D₆₁	—	16	3	no	—
55	C₄×C₆₁	#478	1	6	yes	—
56	C₄×C₆₁	#482	1	6	yes	—
57	Dic₆₁	—	16	3	no	—
58	#1957	—	61	3	no	—
59	#1834	—	61	3	no	—
60	#1273	D₆₁	1	1	yes	characteristic, complemented
61	C₂×C₆₁	#1711	1	1	yes	characteristic
62	C₂×C₆₁	#1717	1	3	yes	—
63	C₂×C₆₁	#2092	1	3	yes	—
64	D₆₁	—	16	2	no	—
65	C₂×#127	—	61	1	no	—
66	C₂×#102	—	122	6	no	—
67	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
68	#384	—	61	3	no	—
69	#340	—	122	6	no	—
70	C₆₁	#10678	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	122	3	no	—
72	C₂²×C₈	D₂₄₄	1	6	yes	—
73	#127	—	61	2	no	—
74	#127	—	122	3	no	—
75	C₂×C₄⋊C₄	—	122	3	no	—
76	C₂×C₄⋊C₄	—	122	3	no	—
77	C₂×C₄⋊C₄	—	122	3	no	—
78	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
79	#102	—	244	12	no	—
80	C₄×C₈	—	2	6	no	—
81	C₂⁴	—	244	1	no	—
82	C₂×D₄	—	122	6	no	—
83	C₂×D₄	—	244	3	no	—
84	C₂×D₄	—	244	6	no	—
85	C₂×D₄	—	244	6	no	—
86	C₂²×C₄	—	244	3	no	—
87	C₂²×C₄	C₂×D₂₄₄	1	3	yes	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	2	6	no	—
91	C₄⋊C₄	—	244	6	no	—
92	C₄⋊C₄	—	244	6	no	—
93	C₄⋊C₄	—	244	6	no	—
94	C₄²	#11758224	1	3	yes	—
95	C₄²	Q₈×D₆₁	1	1	yes	characteristic
96	C₂³	—	488	3	no	—
97	C₂³	—	488	3	no	—
98	C₂³	—	244	2	no	—
99	C₂³	#14519774	1	1	yes	characteristic
100	D₄	—	488	6	no	—
101	D₄	—	244	12	no	—
102	C₂×C₄	—	488	3	no	—
103	C₂×C₄	—	488	3	no	—
104	C₂×C₄	—	488	3	no	—
105	C₂×C₄	#14519757	1	3	yes	—
106	C₂×C₄	#14519761	1	3	yes	—
107	C₂×C₄	#14519799	1	6	yes	—
108	C₂×C₄	#14519805	1	6	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	976	1	no	—
111	C₂²	—	976	3	no	—
112	C₂²	—	488	6	no	—
113	C₂²	#16839699	1	1	yes	characteristic
114	C₂²	#16840146	1	3	yes	—
115	C₂²	#16840151	1	3	yes	—
116	C₄	—	976	3	no	—
117	C₄	#16840063	1	6	yes	—
118	C₄	#16840087	1	6	yes	—
119	C₂	—	976	2	no	—
120	C₂	#21445431	1	1	yes	characteristic
121	C₂	#21445458	1	3	yes	—
122	C₂	#21447339	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446625

C₂

yes

maximal

#21445947

C₂

yes

maximal, complemented

C₆₁×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14519774

C₂²

yes

characteristic

C₂×#14519699

—

#16839632

C₂²

yes

—

#16839525

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂²×D₂₄₄

—

#14519774

—

#14519774

Q₈

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519699

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

—

C₂³×D₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

#127

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#496

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#478

yes

—

C₄×C₆₁

#482

yes

—

Dic₆₁

—

#1957

—

#1834

—

#1273

D₆₁

yes

characteristic, complemented

C₂×C₆₁

#1711

yes

characteristic

C₂×C₆₁

#1717

yes

—

C₂×C₆₁

#2092

yes

—

D₆₁

—

C₂×#127

—

C₂×#102

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

#384

—

#340

—

122

—

C₆₁

#10678

yes

characteristic, complemented, Sylow

C₂²×D₄

—

122

—

C₂²×C₈

D₂₄₄

yes

—

#127

—

#127

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#102

—

244

—

C₄×C₈

—

C₂⁴

—

244

—

C₂×D₄

—

122

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂×D₂₄₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

#11758224

yes

—

C₄²

Q₈×D₆₁

yes

characteristic

C₂³

—

488

—

C₂³

—

488

—

C₂³

—

244

—

C₂³

#14519774

yes

characteristic

100

D₄

—

488

—

101

D₄

—

244

—

102

C₂×C₄

—

488

—

103

C₂×C₄

—

488

—

104

C₂×C₄

—

488

—

105

C₂×C₄

#14519757

yes

—

106

C₂×C₄

#14519761

yes

—

107

C₂×C₄

#14519799

yes

—

108

C₂×C₄

#14519805

yes

—

109

C₈

—

110

C₂²

—

976

—

111

C₂²

—

976

—

112

C₂²

—

488

—

113

C₂²

#16839699

yes

characteristic

114

C₂²

#16840146

yes

—

115

C₂²

#16840151

yes

—

116

C₄

—

976

—

117

C₄

#16840063

yes

—

118

C₄

#16840087

yes

—

119

C₂

—

976

—

120

C₂

#21445431

yes

characteristic

121

C₂

#21445458

yes

—

122

C₂

#21447339

yes

—

	subgroup	G/H
center	C₂³	#14519774
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1273	C₂
socle	C₂³×C₆₁	#127
prime radical	C₂×#14519774	C₂²
hypercenter	#1273	D₆₁
hypocenter	C₆₁	#10678
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14519774

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1273

C₂

C₂³×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #21445947	C₂	semidirect
3. C₆₁×#1273	C₂	semidirect
60. #1273	D₆₁	semidirect
70. C₆₁	#10678	semidirect

H ≅ G/N

type

2. #21445947

C₂

semidirect

3. C₆₁×#1273

C₂

semidirect

60. #1273

D₆₁

semidirect

70. C₆₁

#10678

semidirect

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

subgroup	index
#21446625	2
#21445947	2
C₆₁×#1273	2
#10678	61

subgroup

index

#21446625

#21445947

C₆₁×#1273

#10678

Subgroups — #29585088

Conjugacy classes of subgroups

Complements

Maximal subgroups