Group #20275691 — subgroups

Subgroups — #20275691

Order 6784.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810384	C₂	1	4	yes	maximal
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	#13133217	C₄	1	2	yes	complemented
6	#13133209	C₄	1	2	yes	complemented
7	#13133196	—	2	2	no	—
8	C₄×Dic₁₀₆	—	2	2	no	—
9	#13133176	C₂²	1	2	yes	—
10	C₂×C₄×Dic₅₃	—	2	2	no	—
11	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	2	no	—
18	C₂×Dic₁₀₆	—	2	2	no	—
19	M₁₆×C₅₃	—	2	4	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	D₄	1	2	yes	—
24	C₂²⋊Dic₅₃	—	2	2	no	—
25	C₄⋊Dic₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
27	#11719919	—	2	4	no	—
28	C₄×Dic₅₃	—	2	8	no	—
29	C₄×Dic₅₃	—	2	2	no	—
30	C₄×Dic₅₃	—	2	2	no	—
31	C₄×Dic₅₃	Q₈	1	4	yes	complemented
32	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₅₃	#41	1	2	yes	—
40	C₂×Dic₅₃	—	2	8	no	—
41	C₂×Dic₅₃	—	2	4	no	—
42	C₂×Dic₅₃	—	2	4	no	—
43	Dic₁₀₆	—	4	4	no	—
44	C₈×C₅₃	—	4	2	no	—
45	C₂²×C₅₃	—	2	2	no	—
46	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	#104	1	4	yes	—
50	C₄×C₅₃	C₄×Q₈	1	2	yes	—
51	C₄×C₅₃	#124	1	1	yes	characteristic
52	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₅₃	—	2	8	no	—
54	Dic₅₃	—	4	4	no	—
55	#1994	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	2	2	no	—
57	C₂×C₅₃	#391	1	1	yes	characteristic
58	C₂×C₅₃	C₂×#104	1	2	yes	—
59	#533	—	53	1	no	—
60	#423	D₅₃	1	1	yes	characteristic
61	C₄³	—	53	1	no	—
62	#339	—	53	4	no	—
63	C₅₃	#1994	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
65	C₄⋊Q₈	—	53	2	no	—
66	#125	—	53	2	no	—
67	C₂²⋊Q₈	—	106	2	no	—
68	C₄×Q₈	—	106	2	no	—
69	#117	—	53	2	no	—
70	C₂×C₄²	—	53	2	no	—
71	C₂×C₄²	—	106	2	no	—
72	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂×Q₈	—	106	2	no	—
75	C₂²×C₄	—	106	2	no	—
76	C₂²×C₄	D₂₁₂	1	2	yes	—
77	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	53	4	no	—
81	C₄⋊C₄	—	106	4	no	—
82	C₄⋊C₄	—	106	2	no	—
83	C₂²⋊C₄	—	106	2	no	—
84	C₄²	—	53	4	no	—
85	C₄²	—	106	8	no	—
86	C₄²	—	106	2	no	—
87	C₄²	—	106	2	no	—
88	C₄²	C₄×D₅₃	1	1	yes	characteristic
89	C₄²	C₄×D₅₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₅₃	1	2	yes	—
91	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
92	Q₈	—	212	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	106	8	no	—
95	C₂×C₄	—	106	4	no	—
96	C₂×C₄	—	106	4	no	—
97	C₂×C₄	#11719921	1	2	yes	—
98	C₂×C₄	#11719921	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
100	C₂×C₄	#11719947	1	2	yes	—
101	C₂×C₄	Q₈×D₅₃	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#11719921	1	1	yes	characteristic
106	C₂²	#13133197	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	106	8	no	—
109	C₄	—	212	4	no	—
110	C₄	#13133143	1	4	yes	—
111	C₄	#13133182	1	2	yes	—
112	C₄	#13133204	1	1	yes	characteristic
113	C₄	#13133215	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#16810528	1	1	yes	characteristic
116	C₂	C₂×#13133143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

C₄²×Dic₅₃

C₂

yes

maximal, characteristic

#16810384

C₂

yes

maximal

C₅₃×#423

C₂

yes

maximal, characteristic

#13133217

C₄

yes

complemented

#13133209

C₄

yes

complemented

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂²

yes

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

Q₈

yes

complemented

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

#104

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#1994

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#391

yes

characteristic

C₂×C₅₃

C₂×#104

yes

—

#533

—

#423

D₅₃

yes

characteristic

C₄³

—

#339

—

C₅₃

#1994

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₅₃

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂×D₂₁₂

yes

characteristic

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

#11719921

yes

—

C₂×C₄

#11719921

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

100

C₂×C₄

#11719947

yes

—

101

C₂×C₄

Q₈×D₅₃

yes

—

102

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

C₂×#11719921

yes

characteristic

106

C₂²

#13133197

yes

—

107

C₄

—

108

C₄

—

106

—

109

C₄

—

212

—

110

C₄

#13133143

yes

—

111

C₄

#13133182

yes

—

112

C₄

#13133204

yes

characteristic

113

C₄

#13133215

yes

characteristic

114

C₂

—

115

C₂

#16810528

yes

characteristic

116

C₂

C₂×#13133143

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1994
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#423

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #13133217	C₄	semidirect
6. #13133209	C₄	semidirect
31. C₄×Dic₅₃	Q₈	semidirect
63. C₅₃	#1994	semidirect

H ≅ G/N

type

5. #13133217

C₄

semidirect

6. #13133209

C₄

semidirect

31. C₄×Dic₅₃

Q₈

semidirect

63. C₅₃

#1994

semidirect

Subgroups — #20275691

Conjugacy classes of subgroups

Complements