Group #20274381 — subgroups

Subgroups — #20274381

Order 6784.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16811000	C₂	1	2	yes	maximal, complemented
2	#16811000	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810996	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810970	C₂	1	2	yes	maximal, complemented
5	C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719913	—	2	2	no	—
7	C₂×D₄₂₄	—	2	4	no	—
8	C₂×D₄₂₄	—	2	2	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133214	C₂²	1	2	yes	—
11	#13133214	C₂²	1	1	yes	characteristic
12	#13133139	—	2	4	no	—
13	C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₂₁₂	—	2	2	no	—
16	C₂×D₂₁₂	—	2	2	no	—
17	C₂×D₂₁₂	—	2	2	no	—
18	C₂×D₂₁₂	—	2	2	no	—
19	C₂×D₂₁₂	—	2	1	no	—
20	C₂×Dic₁₀₆	—	2	1	no	—
21	C₂×C₈×C₅₃	D₄	1	4	yes	—
22	C₂×C₈×C₅₃	D₄	1	2	yes	—
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₅₃	—	2	2	no	—
25	D₄₂₄	—	4	8	no	—
26	D₄₂₄	—	4	4	no	—
27	#11719913	—	4	4	no	—
28	C₂²×D₅₃	—	4	2	no	—
29	C₂²×D₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
32	C₂×Dic₅₃	—	4	1	no	—
33	D₂₁₂	—	4	2	no	—
34	D₂₁₂	—	4	4	no	—
35	D₂₁₂	—	4	4	no	—
36	D₂₁₂	—	4	4	no	—
37	D₂₁₂	—	4	4	no	—
38	Dic₁₀₆	—	4	2	no	—
39	C₈×C₅₃	—	2	4	no	—
40	C₈×C₅₃	D₈	1	4	yes	complemented
41	C₂²×C₅₃	#127	1	1	yes	characteristic
42	C₂×D₅₃	—	8	1	no	—
43	C₂×D₅₃	—	8	2	no	—
44	C₂×D₅₃	—	8	2	no	—
45	C₂×D₅₃	—	8	2	no	—
46	C₂×D₅₃	—	8	2	no	—
47	C₄×C₅₃	C₂×D₈	1	2	yes	—
48	C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₅₃	—	8	1	no	—
50	#1721	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	#493	1	1	yes	characteristic
52	C₂×C₅₃	#496	1	2	yes	—
53	D₅₃	—	16	1	no	—
54	D₅₃	—	16	2	no	—
55	#493	—	53	2	no	—
56	#493	—	53	1	no	—
57	#492	—	53	1	no	—
58	#486	—	53	2	no	—
59	#322	D₅₃	1	1	yes	characteristic, complemented
60	C₅₃	#1721	1	1	yes	characteristic, complemented, Sylow
61	C₂×SD₁₆	—	106	2	no	—
62	C₂×D₈	—	106	4	no	—
63	C₂×D₈	—	106	2	no	—
64	C₄⋊Q₈	—	53	1	no	—
65	#127	—	53	2	no	—
66	#127	—	53	1	no	—
67	#102	—	106	4	no	—
68	C₄×C₈	C₂×D₅₃	1	2	yes	—
69	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
70	C₂×Q₈	—	106	1	no	—
71	C₂×D₄	—	106	2	no	—
72	C₂×D₄	—	106	2	no	—
73	C₂×D₄	—	106	2	no	—
74	C₂×D₄	—	106	2	no	—
75	C₂×D₄	—	106	1	no	—
76	SD₁₆	—	212	4	no	—
77	D₈	—	212	8	no	—
78	D₈	—	212	4	no	—
79	C₂×C₈	D₂₁₂	1	4	yes	—
80	C₂×C₈	D₂₁₂	1	2	yes	—
81	C₄⋊C₄	—	106	2	no	—
82	C₄²	C₂²×D₅₃	1	1	yes	characteristic
83	C₂³	—	212	2	no	—
84	C₂³	—	212	1	no	—
85	Q₈	—	212	2	no	—
86	D₄	—	212	4	no	—
87	D₄	—	212	2	no	—
88	D₄	—	212	4	no	—
89	D₄	—	212	4	no	—
90	D₄	—	212	4	no	—
91	C₂×C₄	—	212	1	no	—
92	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
93	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₄₂₄	1	4	yes	complemented
96	C₂²	—	424	2	no	—
97	C₂²	—	424	2	no	—
98	C₂²	—	424	2	no	—
99	C₂²	—	424	2	no	—
100	C₂²	—	424	1	no	—
101	C₂²	#13133214	1	1	yes	characteristic
102	C₄	—	424	1	no	—
103	C₄	C₂×D₄₂₄	1	2	yes	—
104	C₄	#13133245	1	4	yes	—
105	C₂	—	848	1	no	—
106	C₂	—	848	2	no	—
107	C₂	#16811000	1	1	yes	characteristic
108	C₂	#16811012	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811000

C₂

yes

maximal, complemented

#16811000

C₂

yes

maximal, characteristic, complemented

#16810996

C₂

yes

maximal, characteristic, complemented

#16810970

C₂

yes

maximal, complemented

C₅₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×#11719913

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

#13133217

C₂²

yes

characteristic

#13133214

C₂²

yes

—

#13133214

C₂²

yes

characteristic

#13133139

—

C₄×C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

D₄₂₄

—

D₄₂₄

—

#11719913

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

complemented

C₂²×C₅₃

#127

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1721

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₄⋊Q₈

—

#127

—

#127

—

#102

—

106

—

C₄×C₈

C₂×D₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

SD₁₆

—

212

—

D₈

—

212

—

D₈

—

212

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

D₂₁₂

yes

—

C₄⋊C₄

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

212

—

C₂³

—

212

—

Q₈

—

212

—

D₄

—

212

—

D₄

—

212

—

D₄

—

212

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

—

C₈

—

C₈

D₄₂₄

yes

complemented

C₂²

—

424

—

C₂²

—

424

—

C₂²

—

424

—

C₂²

—

424

—

100

C₂²

—

424

—

101

C₂²

#13133214

yes

characteristic

102

C₄

—

424

—

103

C₄

C₂×D₄₂₄

yes

—

104

C₄

#13133245

yes

—

105

C₂

—

848

—

106

C₂

—

848

—

107

C₂

#16811000

yes

characteristic

108

C₂

#16811012

yes

—

	subgroup	G/H
center	C₂²	#13133214
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#322	C₂
socle	C₂²×C₅₃	#127
hypercenter	#322	D₅₃
hypocenter	C₅₃	#1721
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#13133214

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#322

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #16811000	C₂	semidirect
2. #16811000	C₂	semidirect
3. #16810996	C₂	semidirect
4. #16810970	C₂	semidirect
5. C₅₃×#322	C₂	semidirect
40. C₈×C₅₃	D₈	semidirect
59. #322	D₅₃	semidirect
60. C₅₃	#1721	semidirect
95. C₈	D₄₂₄	semidirect

H ≅ G/N

type

1. #16811000

C₂

semidirect

2. #16811000

C₂

semidirect

3. #16810996

C₂

semidirect

4. #16810970

C₂

semidirect

5. C₅₃×#322

C₂

semidirect

40. C₈×C₅₃

D₈

semidirect

59. #322

D₅₃

semidirect

60. C₅₃

#1721

semidirect

95. C₈

D₄₂₄

semidirect

Subgroups — #20274381

Conjugacy classes of subgroups

Complements