Group #20276823 — subgroups

Subgroups — #20276823

Order 6784.

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

normal subgroups only

This group has 150 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×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810864	C₂	1	2	yes	maximal
3	#16810451	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810428	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	2	yes	—
7	#13133196	—	2	2	no	—
8	Dic₄₂₄	—	2	2	no	—
9	#13133151	C₂²	1	2	yes	—
10	#13133142	—	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×Dic₁₀₆	—	2	2	no	—
17	C₂×C₈×C₅₃	—	2	1	no	—
18	C₂×C₈×C₅₃	—	2	1	no	—
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²⋊Dic₅₃	—	4	2	no	—
22	C₄⋊Dic₅₃	—	2	2	no	—
23	#11719919	—	4	2	no	—
24	Dic₂₁₂	—	2	2	no	—
25	Dic₂₁₂	—	2	2	no	—
26	C₁₆×C₅₃	—	2	1	no	—
27	C₁₆×C₅₃	D₄	1	2	yes	—
28	C₂³×C₅₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₅₃	—	4	2	no	—
34	C₂×Dic₅₃	—	4	2	no	—
35	Dic₁₀₆	—	4	2	no	—
36	Dic₁₀₆	—	4	2	no	—
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₅₃	#41	1	1	yes	characteristic
40	C₂²×C₅₃	—	2	1	no	—
41	C₂²×C₅₃	—	2	2	no	—
42	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
43	C₄×C₅₃	—	2	1	no	—
44	C₄×C₅₃	#121	1	1	yes	characteristic
45	C₄×C₅₃	#135	1	1	yes	characteristic
46	Dic₅₃	—	8	2	no	—
47	Dic₅₃	—	8	2	no	—
48	#2219	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	—	4	1	no	—
50	C₂×C₅₃	#466	1	1	yes	characteristic
51	C₂×C₅₃	#510	1	2	yes	—
52	C₂×Q₃₂	—	53	1	no	—
53	C₂×M₃₂	D₅₃	1	1	yes	characteristic
54	#467	—	53	2	no	—
55	#367	—	53	1	no	—
56	#358	—	53	2	no	—
57	C₅₃	#2219	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	—	53	2	no	—
59	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
60	C₂²⋊Q₈	—	106	2	no	—
61	Q₃₂	—	106	2	no	—
62	M₃₂	—	2	2	no	—
63	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
64	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
65	#107	—	53	2	no	—
66	#103	—	106	2	no	—
67	C₂×Q₈	—	106	2	no	—
68	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
69	Q₁₆	—	106	2	no	—
70	Q₁₆	—	106	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	D₂₁₂	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
75	C₄⋊C₄	—	106	2	no	—
76	C₄⋊C₄	—	212	2	no	—
77	C₂²⋊C₄	—	212	2	no	—
78	C₁₆	—	2	1	no	—
79	C₁₆	C₂²⋊D₅₃	1	2	yes	—
80	C₂³	D₄₂₄	1	1	yes	characteristic
81	Q₈	—	212	2	no	—
82	Q₈	—	212	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	212	2	no	—
86	C₂×C₄	—	212	2	no	—
87	C₂×C₄	D₄₂₄	1	1	yes	characteristic
88	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	#11719945	1	1	yes	characteristic
91	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	—	2	1	no	—
94	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	424	2	no	—
97	C₄	—	424	2	no	—
98	C₄	#13133191	1	1	yes	characteristic
99	C₄	#13133239	1	1	yes	characteristic
100	C₂	—	4	1	no	—
101	C₂	#16810859	1	1	yes	characteristic
102	C₂	#16811066	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₄₂₄

C₂

yes

maximal, characteristic, complemented

#16810864

C₂

yes

maximal

#16810451

C₂

yes

maximal, characteristic, complemented

#16810428

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

—

#13133196

—

Dic₄₂₄

—

#13133151

C₂²

yes

—

#13133142

—

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×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₁₆×C₅₃

D₄

yes

—

C₂³×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#2219

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#466

yes

characteristic

C₂×C₅₃

#510

yes

—

C₂×Q₃₂

—

C₂×M₃₂

D₅₃

yes

—

—

—

C₅₃

#2219

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²⋊Q₈

—

106

—

Q₃₂

—

106

—

M₃₂

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

—

#103

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

D₂₁₂

yes

characteristic

Q₁₆

—

106

—

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₁₆

—

C₁₆

C₂²⋊D₅₃

yes

—

C₂³

D₄₂₄

yes

characteristic

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

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

C₂×D₄₂₄

yes

characteristic

C₄

—

C₄

—

424

—

C₄

—

424

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#16810859

yes

characteristic

102

C₂

#16811066

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	#2219
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

#2219

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×Dic₄₂₄	C₂	semidirect
3. #16810451	C₂	semidirect
4. #16810428	C₂	semidirect
57. C₅₃	#2219	semidirect

H ≅ G/N

type

1. C₂×Dic₄₂₄

C₂

semidirect

3. #16810451

C₂

semidirect

4. #16810428

C₂

semidirect

57. C₅₃

#2219

semidirect

Subgroups — #20276823

Conjugacy classes of subgroups

Complements