Group #20274868 — subgroups

Subgroups — #20274868

Order 6784.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133146	C₂	1	4	yes	maximal
3	#16810469	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
7	C₄⋊#88322	—	2	2	no	—
8	#13133146	—	2	8	no	—
9	#13133124	C₄	1	2	yes	—
10	#13133124	C₄	1	2	yes	—
11	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₃	—	2	2	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	—	2	4	no	—
20	C₄²×C₅₃	D₄	1	2	yes	—
21	C₄²×C₅₃	Q₈	1	2	yes	—
22	C₄×Dic₅₃	D₄	1	4	yes	—
23	C₄×Dic₅₃	Q₈	1	4	yes	—
24	C₂×#88322	—	2	4	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	2	2	no	—
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
35	C₂×Dic₅₃	—	2	4	no	—
36	C₂×Dic₅₃	—	2	4	no	—
37	C₂×Dic₅₃	C₄⋊C₄	1	8	yes	—
38	C₈×C₅₃	—	4	2	no	—
39	#88322	—	4	2	no	—
40	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₅₃	#131	1	2	yes	—
43	C₂²×C₅₃	#131	1	2	yes	—
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	—	2	2	no	—
46	C₄×C₅₃	#127	1	1	yes	characteristic
47	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
49	Dic₅₃	—	2	8	no	—
50	#1843	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	#384	1	1	yes	characteristic
52	C₂×C₅₃	#424	1	4	yes	—
53	C₂×C₅₃	#424	1	1	yes	characteristic
54	C₂×C₅₃	#424	1	1	yes	characteristic
55	C₂×C₄⋊C₈	—	53	4	no	—
56	C₂×C₄⋊C₈	—	53	1	no	—
57	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
58	#376	—	53	1	no	—
59	C₅₃	#1843	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	53	2	no	—
61	C₂²×C₈	C₂×D₅₃	1	2	yes	—
62	C₂×C₄²	—	53	2	no	—
63	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	106	8	no	—
66	C₄⋊C₈	—	106	2	no	—
67	#95	—	53	2	no	—
68	#95	—	53	2	no	—
69	C₂²×C₄	—	53	4	no	—
70	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	106	4	no	—
76	C₂×C₈	—	106	2	no	—
77	C₄²	—	53	4	no	—
78	C₄²	—	53	4	no	—
79	C₄²	Dic₁₀₆	1	2	yes	—
80	C₄²	C₂²⋊D₅₃	1	2	yes	—
81	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	53	8	no	—
85	C₂×C₄	—	106	4	no	—
86	C₂×C₄	—	106	4	no	—
87	C₂×C₄	#11719919	1	2	yes	—
88	C₂×C₄	#11719919	1	2	yes	—
89	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
90	C₂×C₄	D₄×D₅₃	1	2	yes	—
91	C₂×C₄	Q₈×D₅₃	1	2	yes	—
92	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	212	2	no	—
95	C₂²	C₄⋊C₄×D₅₃	1	2	yes	—
96	C₂²	C₂×#11719919	1	1	yes	characteristic
97	C₂²	#13133223	1	2	yes	—
98	C₂²	#13133225	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	106	8	no	—
102	C₄	#13133213	1	1	yes	characteristic
103	C₄	#13133215	1	1	yes	characteristic
104	C₄	#13133216	1	2	yes	—
105	C₂	#16810492	1	1	yes	characteristic
106	C₂	#16810648	1	4	yes	—
107	C₂	#16810649	1	1	yes	characteristic
108	C₂	#16810651	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal

#16810469

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄⋊#88322

—

#13133146

—

#13133124

C₄

yes

—

#13133124

C₄

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

Q₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#131

yes

—

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#127

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₅₃

yes

characteristic

#376

—

C₅₃

#1843

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₄²

Dic₁₀₆

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

#11719919

yes

—

C₂×C₄

#11719919

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

—

C₂×C₄

Q₈×D₅₃

yes

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₂²

C₄⋊C₄×D₅₃

yes

—

C₂²

C₂×#11719919

yes

characteristic

C₂²

#13133223

yes

—

C₂²

#13133225

yes

—

C₄

—

100

C₄

—

101

C₄

—

106

—

102

C₄

#13133213

yes

characteristic

103

C₄

#13133215

yes

characteristic

104

C₄

#13133216

yes

—

105

C₂

#16810492

yes

characteristic

106

C₂

#16810648

yes

—

107

C₂

#16810649

yes

characteristic

108

C₂

#16810651

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1843
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

hypercenter

C₂×C₄⋊C₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
59. C₅₃	#1843	semidirect

H ≅ G/N

type

59. C₅₃

#1843

semidirect

Subgroups — #20274868

Conjugacy classes of subgroups

Complements