Group #21453893 — subgroups

Subgroups — #21453893

Order 7808.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×#14519771	C₂	1	2	yes	maximal, complemented
2	#16839692	C₂	1	3	yes	maximal, complemented
3	#16839684	C₂	1	6	yes	maximal, complemented
4	#16839679	C₂	1	3	yes	maximal
5	C₆₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₄₄	C₂²	1	1	yes	characteristic
7	#14519771	—	2	8	no	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	3	yes	—
9	C₂×#11758196	C₂²	1	6	yes	—
10	C₂×#11758198	C₂²	1	6	yes	—
11	C₂×#11758198	C₂²	1	6	yes	—
12	#14519684	C₂²	1	3	yes	—
13	#14519684	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂³×D₆₁	C₂³	1	2	yes	—
18	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
20	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
21	C₂²×Dic₆₁	C₂³	1	6	yes	—
22	C₂×D₂₄₄	—	4	3	no	—
23	C₄⋊C₄×C₆₁	—	4	3	no	—
24	C₄⋊C₄×C₆₁	—	4	3	no	—
25	C₄²×C₆₁	D₄	1	4	yes	complemented
26	#11758198	—	2	12	no	—
27	#11758198	—	4	6	no	—
28	C₄⋊Dic₆₁	—	4	3	no	—
29	#11758196	—	2	12	no	—
30	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
31	C₂²×D₆₁	—	2	2	no	—
32	C₂²×D₆₁	—	2	6	no	—
33	C₂²×D₆₁	—	2	6	no	—
34	C₂×C₄×C₆₁	—	2	3	no	—
35	C₂×C₄×C₆₁	—	2	3	no	—
36	C₂×C₄×C₆₁	—	2	3	no	—
37	C₂×C₄×C₆₁	—	2	6	no	—
38	C₂×C₄×C₆₁	—	2	3	no	—
39	C₂×C₄×C₆₁	C₂×D₄	1	6	yes	—
40	C₂×Dic₆₁	—	2	6	no	—
41	C₂×Dic₆₁	—	2	6	no	—
42	C₂×Dic₆₁	—	2	6	no	—
43	D₂₄₄	—	4	4	no	—
44	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
45	C₂²×C₆₁	H(5,𝔽₂)	1	3	yes	—
46	C₂²×C₆₁	2₋¹⁺⁴	1	3	yes	—
47	C₂×D₆₁	—	4	6	no	—
48	C₂×D₆₁	—	4	2	no	—
49	C₂×D₆₁	—	4	6	no	—
50	C₄×C₆₁	—	2	6	no	—
51	C₄×C₆₁	—	4	1	no	—
52	C₄×C₆₁	—	4	3	no	—
53	Dic₆₁	—	4	6	no	—
54	#2823	—	61	1	no	maximal, Sylow
55	C₂×C₆₁	#537	1	3	yes	—
56	C₂×C₆₁	#560	1	1	yes	characteristic
57	C₂×C₆₁	#563	1	3	yes	—
58	D₆₁	—	8	2	no	—
59	C₂×#126	—	61	2	no	—
60	#398	—	61	3	no	—
61	#396	—	61	6	no	—
62	#394	—	61	3	no	—
63	#384	D₆₁	1	1	yes	characteristic, complemented
64	C₆₁	#2823	1	1	yes	characteristic, complemented, Sylow
65	C₂²×D₄	—	61	1	no	—
66	#126	—	122	8	no	—
67	C₂×C₄⋊C₄	—	61	6	no	—
68	C₂×C₄⋊C₄	—	61	3	no	—
69	C₂×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
70	C₂×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
71	C₂×C₂²⋊C₄	—	61	6	no	—
72	C₂×C₂²⋊C₄	—	61	6	no	—
73	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
74	#95	—	61	3	no	—
75	#95	—	61	3	no	—
76	C₂⁴	—	61	2	no	—
77	C₂×D₄	—	244	3	no	—
78	C₂²×C₄	—	61	6	no	—
79	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
80	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
81	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
82	C₄⋊C₄	—	4	3	no	—
83	C₄⋊C₄	—	4	3	no	—
84	C₄⋊C₄	—	122	12	no	—
85	C₄⋊C₄	—	244	3	no	—
86	C₂²⋊C₄	—	122	12	no	—
87	C₂²⋊C₄	—	244	6	no	—
88	C₄²	D₂₄₄	1	4	yes	complemented
89	C₂³	—	122	2	no	—
90	C₂³	—	122	6	no	—
91	C₂³	—	122	6	no	—
92	C₂³	C₂³×D₆₁	1	1	yes	characteristic
93	D₄	—	244	4	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	122	6	no	—
100	C₂×C₄	—	122	6	no	—
101	C₂×C₄	—	122	6	no	—
102	C₂×C₄	C₂×D₂₄₄	1	6	yes	—
103	C₂²	—	244	2	no	—
104	C₂²	—	244	6	no	—
105	C₂²	—	244	6	no	—
106	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
107	C₂²	#14519822	1	3	yes	—
108	C₂²	#14519825	1	3	yes	—
109	C₄	—	2	6	no	—
110	C₄	—	4	3	no	—
111	C₄	—	4	1	no	—
112	C₄	—	244	6	no	—
113	C₂	—	488	2	no	—
114	C₂	#16840343	1	3	yes	—
115	C₂	#16840499	1	1	yes	characteristic
116	C₂	#16840511	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519771

C₂

yes

maximal, complemented

#16839692

C₂

yes

maximal, complemented

#16839684

C₂

yes

maximal, complemented

#16839679

C₂

yes

maximal

C₆₁×#384

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₄₄

C₂²

yes

characteristic

#14519771

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×#11758198

C₂²

yes

—

C₂×#11758198

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂³×D₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×D₂₄₄

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₄

yes

complemented

#11758198

—

#11758198

—

C₄⋊Dic₆₁

—

#11758196

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂²×C₆₁

H(5,𝔽₂)

yes

—

C₂²×C₆₁

2₋¹⁺⁴

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

#2823

—

maximal, Sylow

C₂×C₆₁

#537

yes

—

C₂×C₆₁

#560

yes

characteristic

C₂×C₆₁

#563

yes

—

D₆₁

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₆₁

#2823

yes

characteristic, complemented, Sylow

C₂²×D₄

—

#126

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#95

—

#95

—

C₂⁴

—

C₂×D₄

—

244

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₂²⋊C₄

—

122

—

C₂²⋊C₄

—

244

—

C₄²

D₂₄₄

yes

complemented

C₂³

—

122

—

C₂³

—

122

—

C₂³

—

122

—

C₂³

C₂³×D₆₁

yes

characteristic

D₄

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

100

C₂×C₄

—

122

—

101

C₂×C₄

—

122

—

102

C₂×C₄

C₂×D₂₄₄

yes

—

103

C₂²

—

244

—

104

C₂²

—

244

—

105

C₂²

—

244

—

106

C₂²

C₂²×D₂₄₄

yes

characteristic

107

C₂²

#14519822

yes

—

108

C₂²

#14519825

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

244

—

113

C₂

—

488

—

114

C₂

#16840343

yes

—

115

C₂

#16840499

yes

characteristic

116

C₂

#16840511

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₆₁×#384	C₂
hypercenter	#384	D₆₁
hypocenter	C₆₁	#2823
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₆₁×#384

C₂

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#14519771	C₂	semidirect
2. #16839692	C₂	semidirect
3. #16839684	C₂	semidirect
5. C₆₁×#384	C₂	semidirect
25. C₄²×C₆₁	D₄	semidirect
63. #384	D₆₁	semidirect
64. C₆₁	#2823	semidirect
88. C₄²	D₂₄₄	semidirect

H ≅ G/N

type

1. C₂×#14519771

C₂

semidirect

2. #16839692

C₂

semidirect

3. #16839684

C₂

semidirect

5. C₆₁×#384

C₂

semidirect

25. C₄²×C₆₁

D₄

semidirect

63. #384

D₆₁

semidirect

64. C₆₁

#2823

semidirect

88. C₄²

D₂₄₄

semidirect

Subgroups — #21453893

Conjugacy classes of subgroups

Complements