Group #20275696 — subgroups

Subgroups — #20275696

Order 6784.

244 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16810647	C₂	1	1	yes	maximal, characteristic
3	#16810383	C₂	1	4	yes	maximal
4	C₅₃×#376	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	#13133217	C₄	1	2	yes	complemented
7	#13133209	C₄	1	2	yes	complemented
8	#13133196	—	2	2	no	—
9	C₄×Dic₁₀₆	—	2	2	no	—
10	#13133176	C₂²	1	2	yes	—
11	C₄⋊#88322	—	2	2	no	—
12	C₂×C₄²×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	C₂×Dic₁₀₆	—	2	2	no	—
19	C₄²×C₅₃	—	2	2	no	—
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	2	yes	—
23	C₄²×C₅₃	Q₈	1	4	yes	complemented
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	2	no	—
29	#11719917	—	2	4	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	4	no	—
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	—	4	4	no	—
35	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₅₃	#41	1	2	yes	—
41	C₂×Dic₅₃	—	2	4	no	—
42	Dic₁₀₆	—	4	4	no	—
43	#88322	—	4	2	no	—
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	—	4	4	no	—
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	—	4	4	no	—
54	#1995	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	2	no	—
56	C₂×C₅₃	#391	1	1	yes	characteristic
57	C₂×C₅₃	#421	1	2	yes	—
58	#533	—	53	1	no	—
59	#423	—	53	1	no	—
60	#376	D₅₃	1	1	yes	characteristic
61	#339	—	53	4	no	—
62	C₅₃	#1995	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₁₆	—	53	2	no	—
64	C₄⋊Q₈	—	53	2	no	—
65	#125	—	53	2	no	—
66	C₂²⋊Q₈	—	106	2	no	—
67	C₄×Q₈	—	106	2	no	—
68	#117	—	53	2	no	—
69	C₂×C₄²	C₂×D₅₃	1	2	yes	—
70	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
71	C₄⋊C₈	—	106	2	no	—
72	#95	—	2	2	no	—
73	C₂×Q₈	—	106	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
76	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
77	M₁₆	—	106	4	no	—
78	C₂×C₈	—	106	2	no	—
79	C₄⋊C₄	—	53	4	no	—
80	C₄⋊C₄	—	106	4	no	—
81	C₄⋊C₄	—	106	2	no	—
82	C₂²⋊C₄	—	106	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	106	2	no	—
85	C₄²	Dic₁₀₆	1	4	yes	complemented
86	C₄²	C₄×D₅₃	1	1	yes	characteristic
87	C₄²	C₄×D₅₃	1	1	yes	characteristic
88	C₄²	D₂₁₂	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
90	Q₈	—	212	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	—	106	4	no	—
95	C₂×C₄	#11719921	1	2	yes	—
96	C₂×C₄	#11719921	1	2	yes	—
97	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
98	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
99	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
100	C₂×C₄	#11719945	1	2	yes	—
101	C₈	—	212	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×#11719921	1	1	yes	characteristic
104	C₂²	#13133196	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	212	4	no	—
109	C₄	C₄×Dic₁₀₆	1	2	yes	—
110	C₄	#13133205	1	1	yes	characteristic
111	C₄	#13133217	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#16810529	1	1	yes	characteristic
114	C₂	#16810641	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

#16810647

C₂

yes

maximal, characteristic

#16810383

C₂

yes

maximal

C₅₃×#376

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

#13133217

C₄

yes

complemented

#13133209

C₄

yes

complemented

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂²

yes

—

C₄⋊#88322

—

C₂×C₄²×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

C₂×Dic₁₀₆

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

complemented

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

—

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

—

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

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

#1995

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#391

yes

characteristic

C₂×C₅₃

#421

yes

—

#533

—

#423

—

#376

D₅₃

yes

characteristic

#339

—

C₅₃

#1995

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#95

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

—

106

—

C₄²

Dic₁₀₆

yes

complemented

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

D₂₁₂

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

#11719921

yes

—

C₂×C₄

#11719921

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

100

C₂×C₄

#11719945

yes

—

101

C₈

—

212

—

102

C₂²

—

103

C₂²

C₂×#11719921

yes

characteristic

104

C₂²

#13133196

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

212

—

109

C₄

C₄×Dic₁₀₆

yes

—

110

C₄

#13133205

yes

characteristic

111

C₄

#13133217

yes

characteristic

112

C₂

—

113

C₂

#16810529

yes

characteristic

114

C₂

#16810641

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#376	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#376	D₅₃
hypocenter	C₅₃	#1995
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#376

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
6. #13133217	C₄	semidirect
7. #13133209	C₄	semidirect
23. C₄²×C₅₃	Q₈	semidirect
62. C₅₃	#1995	semidirect
85. C₄²	Dic₁₀₆	semidirect

H ≅ G/N

type

6. #13133217

C₄

semidirect

7. #13133209

C₄

semidirect

23. C₄²×C₅₃

Q₈

semidirect

62. C₅₃

#1995

semidirect

85. C₄²

Dic₁₀₆

semidirect

Subgroups — #20275696

Conjugacy classes of subgroups

Complements