Group #20275534 — subgroups

Subgroups — #20275534

Order 6784.

308 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 105 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133140	C₂	1	4	yes	maximal
3	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
6	C₂²×#88322	C₂²	1	2	yes	—
7	#13133217	—	2	2	no	—
8	#13133217	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₅₃	—	2	2	no	—
10	#13133140	—	2	8	no	—
11	#13133128	—	2	2	no	—
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	#95×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	D₄	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	2	no	—
19	C₂×Dic₁₀₆	—	2	4	no	—
20	C₂×Dic₁₀₆	—	2	4	no	—
21	C₂×Dic₁₀₆	—	2	2	no	—
22	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
23	C₄⋊C₄×C₅₃	—	2	4	no	—
24	C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	C₄²×C₅₃	D₄	1	2	yes	—
26	C₄⋊Dic₅₃	—	2	8	no	—
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂×#88322	D₄	1	4	yes	—
30	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	4	no	—
32	C₂×C₄×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	—	4	4	no	—
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₅₃	#41	1	2	yes	—
39	C₂×Dic₅₃	—	2	4	no	—
40	C₂×Dic₅₃	—	4	2	no	—
41	C₂×Dic₅₃	—	4	2	no	—
42	Dic₁₀₆	—	2	8	no	—
43	Dic₁₀₆	—	4	4	no	—
44	#88322	—	2	4	no	—
45	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₅₃	#124	1	1	yes	characteristic
47	C₂²×C₅₃	#127	1	1	yes	characteristic
48	C₂²×C₅₃	#137	1	4	yes	—
49	C₄×C₅₃	—	2	4	no	—
50	C₄×C₅₃	—	8	2	no	—
51	C₄×C₅₃	C₄×D₄	1	2	yes	—
52	C₄×C₅₃	#121	1	2	yes	—
53	Dic₅₃	—	4	4	no	—
54	#1960	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	#390	1	1	yes	characteristic
56	C₂×C₅₃	#419	1	2	yes	—
57	C₂×C₅₃	#491	1	2	yes	—
58	C₂×C₅₃	#497	1	2	yes	—
59	C₂×C₄⋊Q₈	—	53	1	no	—
60	C₂×#103	—	53	4	no	—
61	C₂×#97	—	53	1	no	—
62	#382	D₅₃	1	1	yes	characteristic
63	C₅₃	#1960	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	—	53	2	no	—
65	C₂²×C₈	—	53	2	no	—
66	C₄⋊Q₈	—	53	4	no	—
67	C₄⋊Q₈	—	106	2	no	—
68	C₂×C₄⋊C₄	—	106	2	no	—
69	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
70	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
71	#103	—	106	8	no	—
72	#97	—	106	2	no	—
73	#95	—	2	2	no	—
74	C₂×Q₈	—	53	4	no	—
75	C₂×Q₈	—	106	2	no	—
76	C₂×Q₈	—	106	4	no	—
77	C₂×Q₈	—	106	4	no	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	106	2	no	—
80	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
82	C₂×C₈	—	53	4	no	—
83	C₂×C₈	—	106	2	no	—
84	C₂×C₈	—	106	2	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₄⋊C₄	—	106	8	no	—
87	C₄²	C₄×D₅₃	1	2	yes	—
88	C₄²	D₂₁₂	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
90	Q₈	—	106	8	no	—
91	Q₈	—	212	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	4	no	—
95	C₂×C₄	—	106	4	no	—
96	C₂×C₄	—	212	2	no	—
97	C₂×C₄	—	212	2	no	—
98	C₂×C₄	#11719921	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
100	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
101	C₂×C₄	D₄×D₅₃	1	2	yes	—
102	C₂×C₄	#11719949	1	2	yes	—
103	C₈	—	106	4	no	—
104	C₂²	C₂×#11719921	1	1	yes	characteristic
105	C₂²	#13133206	1	1	yes	characteristic
106	C₂²	#13133213	1	1	yes	characteristic
107	C₂²	#13133247	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	8	2	no	—
110	C₄	—	212	4	no	—
111	C₄	#13133179	1	2	yes	—
112	C₄	#13133192	1	2	yes	—
113	C₂	#16810524	1	1	yes	characteristic
114	C₂	#16810631	1	2	yes	—
115	C₂	#16810993	1	2	yes	—
116	C₂	#16811018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

C₂×#13133140

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal, characteristic

C₅₃×#382

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

#13133217

—

#13133217

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

—

#13133140

—

#13133128

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

#121

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#491

yes

—

C₂×C₅₃

#497

yes

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#97

—

#382

D₅₃

yes

characteristic

C₅₃

#1960

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

106

—

#97

—

106

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

C₄×D₅₃

yes

—

C₄²

D₂₁₂

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

Q₈

—

106

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

#11719921

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

100

C₂×C₄

C₂×D₂₁₂

yes

characteristic

101

C₂×C₄

D₄×D₅₃

yes

—

102

C₂×C₄

#11719949

yes

—

103

C₈

—

106

—

104

C₂²

C₂×#11719921

yes

characteristic

105

C₂²

#13133206

yes

characteristic

106

C₂²

#13133213

yes

characteristic

107

C₂²

#13133247

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

212

—

111

C₄

#13133179

yes

—

112

C₄

#13133192

yes

—

113

C₂

#16810524

yes

characteristic

114

C₂

#16810631

yes

—

115

C₂

#16810993

yes

—

116

C₂

#16811018

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#382	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#382	D₅₃
hypocenter	C₅₃	#1960
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#382

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #13133217	C₄	semidirect
63. C₅₃	#1960	semidirect

H ≅ G/N

type

8. #13133217

C₄

semidirect

63. C₅₃

#1960

semidirect

Subgroups — #20275534

Conjugacy classes of subgroups

Complements