Group #20276854 — subgroups

Subgroups — #20276854

Order 6784.

130 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#16810450	C₂	1	2	yes	maximal
3	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
4	#16810426	C₂	1	2	yes	maximal
5	C₅₃×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₀₆	—	2	1	no	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	#13133151	C₂²	1	1	yes	characteristic
10	#13133142	—	2	1	no	—
11	C₄⋊Q₈×C₅₃	—	2	1	no	—
12	#107×C₅₃	—	2	1	no	—
13	#107×C₅₃	C₂²	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	4	1	no	—
16	C₂×Dic₁₀₆	—	2	1	no	—
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₄⋊C₄×C₅₃	—	4	1	no	—
21	C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₄⋊Dic₅₃	—	2	1	no	—
23	#11719919	—	4	1	no	—
24	C₄×Dic₅₃	—	4	1	no	—
25	Dic₂₁₂	—	2	1	no	—
26	Dic₂₁₂	Q₈	1	2	yes	complemented
27	#11719908	—	2	2	no	—
28	Q₈×C₅₃	—	8	2	no	—
29	C₂×C₄×C₅₃	—	4	2	no	—
30	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₅₃	—	4	1	no	—
34	C₂×Dic₅₃	—	4	1	no	—
35	Dic₁₀₆	—	2	2	no	—
36	Dic₁₀₆	—	4	1	no	—
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₅₃	#41	1	1	yes	characteristic
40	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
41	C₄×C₅₃	—	2	1	no	—
42	C₄×C₅₃	—	8	2	no	—
43	C₄×C₅₃	SD₃₂	1	2	yes	—
44	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
45	C₄×C₅₃	#137	1	1	yes	characteristic
46	Dic₅₃	—	4	2	no	—
47	Dic₅₃	—	8	1	no	—
48	#2225	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	#474	1	1	yes	characteristic
50	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
51	C₂×C₅₃	#510	1	1	yes	characteristic
52	#500	D₅₃	1	1	yes	characteristic
53	C₄×Q₁₆	—	53	1	no	—
54	#367	—	53	2	no	—
55	C₄⋊C₁₆	—	53	1	no	—
56	#358	—	53	2	no	—
57	C₅₃	#2225	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	—	53	1	no	—
59	C₄⋊Q₈	—	2	1	no	—
60	C₄×Q₈	—	106	1	no	—
61	C₂×C₁₆	—	53	2	no	—
62	#107	—	2	1	no	—
63	#107	—	53	1	no	—
64	#107	C₂×D₅₃	1	2	yes	—
65	#103	—	106	1	no	—
66	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
67	C₂×Q₈	—	4	1	no	—
68	C₂×Q₈	—	106	1	no	—
69	Q₁₆	—	53	2	no	—
70	Q₁₆	—	106	1	no	—
71	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	106	1	no	—
76	C₄⋊C₄	—	212	1	no	—
77	C₄²	—	212	1	no	—
78	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
79	C₁₆	—	106	2	no	—
80	Q₈	—	8	2	no	—
81	Q₈	—	106	2	no	—
82	Q₈	—	212	1	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	212	1	no	—
85	C₂×C₄	—	212	1	no	—
86	C₂×C₄	#11719922	1	1	yes	characteristic
87	C₂×C₄	#11719922	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	Q₈×D₅₃	1	1	yes	characteristic
91	C₈	#11719949	1	1	yes	characteristic
92	C₂²	C₂×#11719922	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	8	2	no	—
95	C₄	—	424	1	no	—
96	C₄	—	212	2	no	—
97	C₄	#13133161	1	2	yes	—
98	C₄	#13133193	1	1	yes	characteristic
99	C₄	#13133248	1	1	yes	characteristic
100	C₂	#16810900	1	1	yes	characteristic
101	C₂	C₂×#13133161	1	1	yes	characteristic
102	C₂	#16811064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic

#16810450

C₂

yes

maximal

C₄⋊#11719908

C₂

yes

maximal, characteristic

#16810426

C₂

yes

maximal

C₅₃×#500

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

C₂×#11719908

C₂²

yes

—

#13133151

C₂²

yes

characteristic

#13133142

—

C₄⋊Q₈×C₅₃

—

#107×C₅₃

—

#107×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

Q₈

yes

complemented

#11719908

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

SD₃₂

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#137

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×SD₃₂

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

C₄×Q₁₆

—

#367

—

C₄⋊C₁₆

—

#358

—

C₅₃

#2225

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

106

—

C₂×C₁₆

—

#107

—

#107

—

#107

C₂×D₅₃

yes

—

#103

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

Q₁₆

—

Q₁₆

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

212

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

Q₈

—

Q₈

—

106

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

Q₈×D₅₃

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

424

—

C₄

—

212

—

C₄

#13133161

yes

—

C₄

yes

C₄

yes

100

C₂

#16810900

yes

characteristic

101

C₂

C₂×#13133161

yes

characteristic

102

C₂

#16811064

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#500	D₅₃
hypocenter	C₅₃	#2225
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#500

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
26. Dic₂₁₂	Q₈	semidirect
57. C₅₃	#2225	semidirect

H ≅ G/N

type

26. Dic₂₁₂

Q₈

semidirect

57. C₅₃

#2225

semidirect

Subgroups — #20276854

Conjugacy classes of subgroups

Complements