Group #20285219 — subgroups

Subgroups — C₂×#16811025 #20285219

Order 6784.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133209	C₂	1	2	yes	maximal
2	#16811025	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133151	C₂	1	2	yes	maximal
4	C₂×#13133149	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133209	—	2	4	no	—
7	#13133209	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂×#11719919	—	2	4	no	—
10	#13133151	C₂²	1	8	yes	—
11	#13133149	C₂²	1	8	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂²	1	4	yes	—
16	C₂²×C₄×C₅₃	D₄	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	4	no	—
19	C₂×C₈×C₅₃	Q₈	1	4	yes	—
20	C₂×C₈×C₅₃	Q₈	1	4	yes	—
21	C₂×C₈×C₅₃	C₂³	1	2	yes	—
22	C₂×C₈×C₅₃	C₂³	1	2	yes	—
23	C₄²×C₅₃	D₄	1	2	yes	—
24	C₄²×C₅₃	C₂³	1	2	yes	—
25	C₄⋊Dic₅₃	—	2	4	no	—
26	C₄⋊Dic₅₃	C₂³	1	8	yes	—
27	#11719919	—	2	16	no	—
28	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	4	no	—
30	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₅₃	—	2	8	no	—
36	C₂×Dic₅₃	—	4	8	no	—
37	C₈×C₅₃	C₂×Q₈	1	4	yes	—
38	C₈×C₅₃	C₂×Q₈	1	4	yes	—
39	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₅₃	#135	1	4	yes	—
41	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₅₃	—	2	4	no	—
43	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
44	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
46	Dic₅₃	—	4	8	no	—
47	C₂×#499	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	#499	1	4	yes	complemented
49	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₅₃	C₂×#135	1	2	yes	—
51	C₂×#125	—	53	2	no	—
52	#499	—	53	8	no	—
53	C₂×#107	—	53	2	no	—
54	C₂×#106	—	53	2	no	—
55	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
56	C₅₃	C₂×#499	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
59	#125	—	53	8	no	—
60	#125	—	106	4	no	—
61	C₂×C₄⋊C₄	—	53	4	no	—
62	C₂×C₄⋊C₄	—	106	4	no	—
63	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
64	#107	—	53	8	no	—
65	#106	—	53	8	no	—
66	C₄×C₈	C₂×D₅₃	1	4	yes	—
67	C₂²×C₄	—	106	4	no	—
68	C₂²×C₄	D₂₁₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
70	C₂×C₈	Dic₁₀₆	1	4	yes	—
71	C₂×C₈	Dic₁₀₆	1	4	yes	—
72	C₂×C₈	C₂²×D₅₃	1	2	yes	—
73	C₂×C₈	C₂²×D₅₃	1	2	yes	—
74	C₄⋊C₄	—	53	8	no	—
75	C₄⋊C₄	—	106	4	no	—
76	C₄⋊C₄	—	106	16	no	—
77	C₄²	D₂₁₂	1	2	yes	—
78	C₄²	C₂²×D₅₃	1	2	yes	—
79	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	106	8	no	—
82	C₂×C₄	—	212	8	no	—
83	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
84	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
85	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
86	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
87	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
88	C₈	C₂×Dic₁₀₆	1	4	yes	—
89	C₈	C₂×Dic₁₀₆	1	4	yes	—
90	C₂²	#13133217	1	2	yes	—
91	C₂²	#13133239	1	4	yes	—
92	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	212	8	no	—
95	C₄	#13133217	1	2	yes	—
96	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
97	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
98	C₂	#16811025	1	4	yes	direct product factor
99	C₂	C₂×#13133217	1	1	yes	characteristic
100	C₂	C₂×#13133239	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133209

C₂

yes

maximal

#16811025

C₂

yes

maximal, direct product factor

C₂×#13133151

C₂

yes

maximal

C₂×#13133149

C₂

yes

maximal

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

#13133209

—

#13133209

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

—

#13133151

C₂²

yes

—

#13133149

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

C₂×Q₈

yes

—

C₈×C₅₃

C₂×Q₈

yes

—

C₂²×C₅₃

C₄⋊Q₈

yes

—

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₃

C₂×#135

yes

—

—

—

—

—

C₂×C₄×C₈

D₅₃

yes

characteristic

C₅₃

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#125

—

#125

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₅₃

yes

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

C₂²×D₅₃

yes

—

C₂×C₈

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

D₂₁₂

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

C₂×D₂₁₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂³×D₅₃

yes

characteristic

C₈

C₂×Dic₁₀₆

yes

—

C₈

C₂×Dic₁₀₆

yes

—

C₂²

#13133217

yes

—

C₂²

#13133239

yes

—

C₂²

C₂²×D₂₁₂

yes

characteristic

C₄

—

C₄

—

212

—

C₄

#13133217

yes

—

C₄

C₂²×Dic₁₀₆

yes

characteristic

C₄

C₂²×Dic₁₀₆

yes

characteristic

C₂

#16811025

yes

direct product factor

C₂

C₂×#13133217

yes

characteristic

100

C₂

C₂×#13133239

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#499
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

C₂×#499

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16811025	C₂	direct (N × H)
48. C₂×C₅₃	#499	semidirect
56. C₅₃	C₂×#499	semidirect
98. C₂	#16811025	direct (N × H)

H ≅ G/N

type

2. #16811025

C₂

direct (N × H)

48. C₂×C₅₃

#499

semidirect

56. C₅₃

C₂×#499

semidirect

98. C₂

#16811025

direct (N × H)

Subgroups — C₂×#16811025 #20285219

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811025 #20285219

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811025 #20285219