Group #20291107 — subgroups

Subgroups — #20291107

Order 6784.

144 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 58 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	#16811682	C₂	1	2	yes	maximal, complemented
2	#16811545	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811522	C₂	1	2	yes	maximal, complemented
4	#16811512	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810739	C₂	1	1	yes	maximal, characteristic
6	#13133292	—	2	2	no	—
7	#13133291	C₂²	1	2	yes	—
8	D₅₃⋊C₁₆	—	2	2	no	—
9	C₂×#11719910	C₂²	1	1	yes	characteristic
10	C₂×#11719910	C₂²	1	1	yes	characteristic
11	#13133239	C₄	1	1	yes	characteristic
12	#13133225	C₂²	1	2	yes	—
13	#13133143	C₄	1	2	yes	—
14	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	#108×C₅₃	C₄	1	1	yes	characteristic
16	#11719945	C₂×C₄	1	2	yes	—
17	M₁₆×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×#88322	C₂³	1	1	yes	characteristic
21	Dic₂₁₂	C₈	1	1	yes	characteristic
22	D₄₂₄	C₈	1	1	yes	characteristic
23	#11719913	C₈	1	2	yes	—
24	#11719912	—	2	2	no	—
25	C₈×D₅₃	—	2	2	no	—
26	#11719910	—	2	2	no	—
27	#11719910	—	2	1	no	—
28	#11719910	D₄	1	2	yes	—
29	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²⋊D₅₃	—	2	2	no	—
31	C₂×Dic₅₃	—	2	1	no	—
32	D₂₁₂	C₂×C₈	1	2	yes	—
33	C₄×D₅₃	—	2	2	no	—
34	Dic₁₀₆	C₂×C₈	1	2	yes	—
35	C₈×C₅₃	—	2	2	no	—
36	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
38	#88322	C₂×D₄	1	1	yes	characteristic
39	#88322	#41	1	1	yes	characteristic
40	C₂²×C₅₃	#131	1	1	yes	characteristic
41	C₂×D₅₃	—	2	2	no	—
42	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
43	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
44	Dic₅₃	—	2	2	no	—
45	Dic₅₃	—	4	1	no	—
46	#2171	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	—	2	1	no	—
48	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
49	D₅₃	—	4	2	no	—
50	#504	—	53	2	no	—
51	#443	—	53	1	no	—
52	#364	—	53	1	no	—
53	#350	—	53	2	no	—
54	#346	—	53	1	no	—
55	C₅₃	#2171	1	1	yes	characteristic, complemented, Sylow
56	#135	—	53	1	no	—
57	#131	—	53	2	no	—
58	M₃₂	—	53	2	no	—
59	M₃₂	—	106	2	no	—
60	C₂×C₁₆	—	53	1	no	—
61	C₂×C₁₆	—	53	1	no	—
62	C₂×C₁₆	—	106	2	no	—
63	#108	#6157	1	1	yes	characteristic
64	#104	—	53	2	no	—
65	C₄×C₈	—	53	1	no	—
66	#41	—	53	2	no	—
67	Q₁₆	—	53	1	no	—
68	SD₁₆	—	53	2	no	—
69	D₈	—	53	1	no	—
70	M₁₆	—	106	2	no	—
71	M₁₆	C₂×#6157	1	2	yes	—
72	C₂×C₈	—	53	1	no	—
73	C₂×C₈	—	106	2	no	—
74	C₂×C₈	C₂×#6157	1	1	yes	characteristic
75	C₄²	—	53	1	no	—
76	C₁₆	—	53	2	no	—
77	C₁₆	—	106	1	no	—
78	C₁₆	—	106	2	no	—
79	Q₈	—	53	2	no	—
80	D₄	—	53	2	no	—
81	D₄	—	106	2	no	—
82	C₂×C₄	—	106	2	no	—
83	C₂×C₄	—	106	1	no	—
84	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	53	1	no	—
87	C₈	—	53	1	no	—
88	C₈	D₅₃⋊C₈	1	1	yes	characteristic
89	C₈	D₅₃⋊C₈	1	1	yes	characteristic
90	C₂²	—	106	2	no	—
91	C₂²	#13133305	1	1	yes	characteristic
92	C₄	—	106	2	no	—
93	C₄	—	212	1	no	—
94	C₄	D₄×#6157	1	1	yes	characteristic
95	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	212	2	no	—
98	C₂	#16811651	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811682

C₂

yes

maximal, complemented

#16811545

C₂

yes

maximal, characteristic, complemented

#16811522

C₂

yes

maximal, complemented

#16811512

C₂

yes

maximal, characteristic, complemented

#16810739

C₂

yes

maximal, characteristic

#13133292

—

#13133291

C₂²

yes

—

D₅₃⋊C₁₆

—

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

#13133239

C₄

yes

characteristic

#13133225

C₂²

yes

—

#13133143

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

characteristic

#108×C₅₃

C₄

yes

characteristic

#11719945

C₂×C₄

yes

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

Dic₂₁₂

C₈

yes

characteristic

D₄₂₄

C₈

yes

characteristic

#11719913

C₈

yes

—

#11719912

—

C₈×D₅₃

—

#11719910

—

#11719910

—

#11719910

D₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

C₂×C₈

yes

—

C₄×D₅₃

—

Dic₁₀₆

C₂×C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₈

yes

characteristic

C₈×C₅₃

C₂×C₈

yes

characteristic

#88322

C₂×D₄

yes

yes

C₂²×C₅₃

yes

C₂×D₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2171

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₈×D₄

yes

characteristic

D₅₃

—

—

—

—

—

—

C₅₃

#2171

yes

characteristic, complemented, Sylow

#135

—

#131

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

yes

—

C₄×C₈

—

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

106

—

M₁₆

C₂×#6157

yes

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄²

—

C₁₆

—

C₁₆

—

106

—

C₁₆

—

106

—

Q₈

—

D₄

—

D₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

D₅₃⋊C₈

yes

characteristic

C₈

D₅₃⋊C₈

yes

characteristic

C₂²

—

106

—

C₂²

#13133305

yes

characteristic

C₄

—

106

—

C₄

—

212

—

C₄

D₄×#6157

yes

characteristic

C₄

C₂×D₅₃⋊C₈

yes

characteristic

C₂

—

C₂

—

212

—

C₂

#16811651

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	#108×C₅₃	C₄
socle	C₂×C₅₃	C₈×D₄
hypercenter	#108	#6157
hypocenter	C₅₃	#2171
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₄

D₄×#6157

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

#108×C₅₃

C₄

socle

C₂×C₅₃

C₈×D₄

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
1. #16811682	C₂	semidirect
2. #16811545	C₂	semidirect
3. #16811522	C₂	semidirect
4. #16811512	C₂	semidirect
55. C₅₃	#2171	semidirect

H ≅ G/N

type

1. #16811682

C₂

semidirect

2. #16811545

C₂

semidirect

3. #16811522

C₂

semidirect

4. #16811512

C₂

semidirect

55. C₅₃

#2171

semidirect

Subgroups — #20291107

Conjugacy classes of subgroups

Complements