Group #20272864 — subgroups

Subgroups — #20272864

Order 6784.

112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 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	#16810800	C₂	1	1	yes	maximal, characteristic
2	#16810420	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#329	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133262	C₄	1	1	yes	characteristic
5	#13133248	—	2	1	no	—
6	#13133145	—	2	1	no	—
7	#13133136	—	2	1	no	—
8	#13133134	C₄	1	1	yes	characteristic
9	#124×C₅₃	C₂²	1	1	yes	characteristic
10	#97×C₅₃	—	2	1	no	—
11	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₂²⋊D₅₃	—	4	1	no	—
14	#11719947	—	4	1	no	—
15	D₄×D₅₃	—	4	1	no	—
16	#11719945	—	2	1	no	—
17	C₂×C₈×C₅₃	—	4	1	no	—
18	C₂²⋊C₄×C₅₃	—	4	1	no	—
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	#11719925	—	4	1	no	—
21	#11719924	—	4	1	no	—
22	#11719917	—	2	1	no	—
23	#11719917	—	4	1	no	—
24	C₂³×C₅₃	—	2	1	no	—
25	C₂²×D₅₃	—	4	1	no	—
26	Q₈×C₅₃	—	4	1	no	—
27	D₄×C₅₃	—	4	1	no	—
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²⋊D₅₃	—	4	2	no	—
32	C₂²⋊D₅₃	—	4	1	no	—
33	C₂×Dic₅₃	—	4	1	no	—
34	D₂₁₂	—	2	1	no	—
35	C₄×D₅₃	—	4	1	no	—
36	Dic₁₀₆	—	2	1	no	—
37	C₈×C₅₃	—	4	2	no	—
38	#88322	—	4	1	no	—
39	#88322	—	8	1	no	—
40	C₂²×C₅₃	—	4	1	no	—
41	C₂²×C₅₃	—	8	1	no	—
42	C₂²×C₅₃	#99	1	1	yes	characteristic
43	C₂×D₅₃	—	4	1	no	—
44	C₂×D₅₃	—	16	1	no	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	4	1	no	—
47	C₄×C₅₃	—	4	1	no	—
48	Dic₅₃	—	4	1	no	—
49	#1403	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	—	8	1	no	—
52	C₂×C₅₃	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
53	D₅₃	—	8	1	no	—
54	#455	—	53	1	no	—
55	#355	—	53	1	no	—
56	#329	D₅₃	1	1	yes	characteristic, complemented
57	C₅₃	#1403	1	1	yes	characteristic, complemented, Sylow
58	H(5,𝔽₂)	—	53	1	no	—
59	#137	—	106	1	no	—
60	#124	C₂×D₅₃	1	1	yes	characteristic
61	#104	—	106	1	no	—
62	#101	—	106	1	no	—
63	#100	—	53	1	no	—
64	#97	—	2	1	no	—
65	#41	—	106	1	no	—
66	#41	—	212	1	no	—
67	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
68	C₂×D₄	—	212	1	no	—
69	C₂×D₄	—	212	1	no	—
70	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
71	Q₁₆	—	212	1	no	—
72	SD₁₆	—	212	1	no	—
73	M₁₆	—	106	1	no	—
74	M₁₆	—	212	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂²⋊C₄	—	4	1	no	—
77	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
78	C₂³	—	2	1	no	—
79	C₂³	—	212	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	106	1	no	—
82	D₄	—	4	1	no	—
83	D₄	—	106	1	no	—
84	D₄	—	212	2	no	—
85	D₄	—	212	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	212	1	no	—
89	C₂×C₄	—	212	1	no	—
90	C₂×C₄	#11719921	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	—	424	1	no	—
93	C₈	—	212	1	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	8	1	no	—
96	C₂²	—	848	1	no	—
97	C₂²	—	212	1	no	—
98	C₂²	#13133131	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	212	1	no	—
103	C₂	—	2	1	no	—
104	C₂	—	8	1	no	—
105	C₂	—	424	1	no	—
106	C₂	#16810411	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810800

C₂

yes

maximal, characteristic

#16810420

C₂

yes

maximal, characteristic, complemented

C₅₃×#329

C₂

yes

maximal, characteristic, complemented

#13133262

C₄

yes

—

—

—

C₄

yes

characteristic

#124×C₅₃

C₂²

yes

characteristic

#97×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

#11719947

—

D₄×D₅₃

—

#11719945

—

C₂×C₈×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

—

—

—

C₂³×C₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#99

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1403

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

AΣL(1,𝔽₁₆)

yes

characteristic

D₅₃

—

#455

—

#355

—

#329

D₅₃

yes

characteristic, complemented

C₅₃

#1403

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

#137

—

106

—

#124

C₂×D₅₃

yes

characteristic

#104

—

106

—

#101

—

106

—

#100

—

#97

—

#41

—

106

—

#41

—

212

—

C₂×Q₈

D₂₁₂

yes

characteristic

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂×D₄

C₄×D₅₃

yes

characteristic

Q₁₆

—

212

—

SD₁₆

—

212

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

—

C₂³

—

212

—

Q₈

—

Q₈

—

106

—

D₄

—

D₄

—

106

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

#11719921

yes

characteristic

C₈

—

C₈

—

424

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

—

848

—

C₂²

—

212

—

C₂²

#13133131

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

212

—

103

C₂

—

104

C₂

—

105

C₂

—

424

—

106

C₂

#16810411

yes

characteristic

	subgroup	G/H
center	C₂	#16810411
commutator	C₂×D₄×C₅₃	C₂×C₄
Frattini	#124	C₂×D₅₃
Fitting	C₅₃×#329	C₂
socle	C₂×C₅₃	AΣL(1,𝔽₁₆)
hypercenter	#329	D₅₃
hypocenter	C₅₃	#1403
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #329

subgroup

G/H

center

C₂

#16810411

commutator

C₂×D₄×C₅₃

C₂×C₄

Frattini

#124

C₂×D₅₃

Fitting

C₅₃×#329

C₂

socle

C₂×C₅₃

AΣL(1,𝔽₁₆)

D₅₃

C₅₃

G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #329

N	H ≅ G/N	type
2. #16810420	C₂	semidirect
3. C₅₃×#329	C₂	semidirect
56. #329	D₅₃	semidirect
57. C₅₃	#1403	semidirect

H ≅ G/N

type

2. #16810420

C₂

semidirect

3. C₅₃×#329

C₂

semidirect

56. #329

D₅₃

semidirect

57. C₅₃

#1403

semidirect

Subgroups — #20272864

Conjugacy classes of subgroups

Complements