Group #20276818 — subgroups

Subgroups — #20276818

Order 6784.

170 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 53 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×D₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810859	C₂	1	2	yes	maximal, complemented
3	#16810451	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810425	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₂²	1	2	yes	—
7	#13133191	—	2	2	no	—
8	D₈₄₈	—	2	2	no	—
9	#13133151	C₂²	1	2	yes	—
10	#13133139	—	2	2	no	—
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	M₃₂×C₅₃	—	2	2	no	—
13	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₂²⋊D₅₃	—	4	2	no	—
17	C₂×D₂₁₂	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	1	no	—
19	C₂×C₈×C₅₃	—	2	1	no	—
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	#11719921	—	4	2	no	—
23	C₄⋊Dic₅₃	—	2	2	no	—
24	D₄₂₄	—	2	2	no	—
25	D₄₂₄	—	2	2	no	—
26	C₁₆×C₅₃	—	2	1	no	—
27	C₁₆×C₅₃	D₄	1	2	yes	complemented
28	C₂³×C₅₃	D₈	1	1	yes	characteristic
29	C₂²×D₅₃	—	4	2	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²⋊D₅₃	—	8	4	no	—
35	C₂×Dic₅₃	—	4	2	no	—
36	D₂₁₂	—	4	2	no	—
37	D₂₁₂	—	4	2	no	—
38	C₈×C₅₃	—	2	1	no	—
39	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₅₃	#41	1	1	yes	characteristic
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	—	2	2	no	—
43	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
44	C₂×D₅₃	—	8	2	no	—
45	C₂×D₅₃	—	8	2	no	—
46	C₂×D₅₃	—	8	2	no	—
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	#121	1	1	yes	characteristic
49	C₄×C₅₃	#135	1	1	yes	characteristic
50	Dic₅₃	—	8	2	no	—
51	#2218	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	4	1	no	—
53	C₂×C₅₃	#466	1	1	yes	characteristic
54	C₂×C₅₃	#509	1	2	yes	—
55	D₅₃	—	16	2	no	—
56	C₂×D₁₆	—	53	1	no	—
57	C₂×M₃₂	D₅₃	1	1	yes	characteristic, complemented
58	#466	—	53	2	no	—
59	#367	—	53	1	no	—
60	#357	—	53	2	no	—
61	C₅₃	#2218	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	—	53	2	no	—
63	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
64	#121	—	106	2	no	—
65	D₁₆	—	106	2	no	—
66	M₃₂	—	2	2	no	—
67	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
68	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
69	#107	—	53	2	no	—
70	#102	—	106	2	no	—
71	C₂×D₄	—	106	2	no	—
72	C₂×D₄	—	212	2	no	—
73	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
74	D₈	—	106	2	no	—
75	D₈	—	106	2	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	D₂₁₂	1	1	yes	characteristic
79	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
80	C₄⋊C₄	—	106	2	no	—
81	C₂²⋊C₄	—	212	2	no	—
82	C₁₆	—	2	1	no	—
83	C₁₆	C₂²⋊D₅₃	1	2	yes	complemented
84	C₂³	—	212	2	no	—
85	C₂³	D₄₂₄	1	1	yes	characteristic
86	D₄	—	424	4	no	—
87	D₄	—	212	2	no	—
88	D₄	—	212	2	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	212	2	no	—
92	C₂×C₄	D₄₂₄	1	1	yes	characteristic
93	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	#11719945	1	1	yes	characteristic
96	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	424	2	no	—
100	C₂²	—	424	2	no	—
101	C₂²	—	424	2	no	—
102	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	424	2	no	—
105	C₄	#13133191	1	1	yes	characteristic
106	C₄	#13133239	1	1	yes	characteristic
107	C₂	—	4	1	no	—
108	C₂	—	848	2	no	—
109	C₂	#16810859	1	1	yes	characteristic
110	C₂	#16811061	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₈₄₈

C₂

yes

maximal, characteristic, complemented

#16810859

C₂

yes

maximal, complemented

#16810451

C₂

yes

maximal, characteristic, complemented

#16810425

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₂²

yes

—

#13133191

—

D₈₄₈

—

#13133151

C₂²

yes

—

#13133139

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

D₄₂₄

—

D₄₂₄

—

C₁₆×C₅₃

—

C₁₆×C₅₃

D₄

yes

complemented

C₂³×C₅₃

D₈

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2218

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#466

yes

characteristic

C₂×C₅₃

#509

yes

—

D₅₃

—

C₂×D₁₆

—

C₂×M₃₂

D₅₃

yes

characteristic, complemented

#466

—

#367

—

#357

—

C₅₃

#2218

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#121

—

106

—

D₁₆

—

106

—

M₃₂

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

—

#102

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

D₂₁₂

yes

characteristic

D₈

—

106

—

D₈

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₁₆

—

C₁₆

C₂²⋊D₅₃

yes

complemented

C₂³

—

212

—

C₂³

D₄₂₄

yes

characteristic

D₄

—

424

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

D₄₂₄

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₈

—

C₈

#11719945

yes

characteristic

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

424

—

100

C₂²

—

424

—

101

C₂²

—

424

—

102

C₂²

C₂×D₄₂₄

yes

characteristic

103

C₄

—

104

C₄

—

424

—

105

C₄

#13133191

yes

characteristic

106

C₄

#13133239

yes

characteristic

107

C₂

—

108

C₂

—

848

—

109

C₂

#16810859

yes

characteristic

110

C₂

#16811061

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#2218
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×M₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₂×M₃₂

D₅₃

hypocenter

C₅₃

#2218

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
1. C₂×D₈₄₈	C₂	semidirect
2. #16810859	C₂	semidirect
3. #16810451	C₂	semidirect
4. #16810425	C₂	semidirect
5. C₂×M₃₂×C₅₃	C₂	semidirect
27. C₁₆×C₅₃	D₄	semidirect
57. C₂×M₃₂	D₅₃	semidirect
61. C₅₃	#2218	semidirect
83. C₁₆	C₂²⋊D₅₃	semidirect

H ≅ G/N

type

1. C₂×D₈₄₈

C₂

semidirect

2. #16810859

C₂

semidirect

3. #16810451

C₂

semidirect

4. #16810425

C₂

semidirect

5. C₂×M₃₂×C₅₃

C₂

semidirect

27. C₁₆×C₅₃

D₄

semidirect

57. C₂×M₃₂

D₅₃

semidirect

61. C₅₃

#2218

semidirect

83. C₁₆

C₂²⋊D₅₃

semidirect

Subgroups — #20276818

Conjugacy classes of subgroups

Complements