Group #20290901 — subgroups

Subgroups — #20290901

Order 6784.

284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 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	#16811628	C₂	1	1	yes	maximal, characteristic
2	C₄²×#6157	C₂	1	1	yes	maximal, characteristic
3	#16811493	C₂	1	4	yes	maximal
4	#16811179	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719936	C₂²	1	2	yes	—
6	C₂×C₄×#6157	—	2	2	no	—
7	C₂×C₄×#6157	C₂²	1	2	yes	—
8	#13133284	—	2	2	no	—
9	#13133216	C₄	1	2	yes	complemented
10	#13133209	C₄	1	1	yes	characteristic, complemented
11	#13133195	—	2	2	no	—
12	#13133182	—	2	2	no	—
13	#13133175	C₂²	1	2	yes	—
14	C₄²×D₅₃	C₂²	1	1	yes	characteristic
15	#125×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6157	—	2	2	no	—
17	C₂×C₄×D₅₃	D₄	1	2	yes	—
18	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₁₀₆	—	2	2	no	—
20	C₂×#88324	—	2	2	no	—
21	C₄×#6157	—	2	8	no	—
22	C₄×#6157	—	2	2	no	—
23	C₄×#6157	Q₈	1	4	yes	complemented
24	#11719936	—	2	4	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#11719921	—	2	2	no	—
29	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	#11719919	—	2	2	no	—
31	#11719919	—	2	2	no	—
32	C₄×Dic₅₃	—	2	2	no	—
33	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₅₃	D₄	1	2	yes	—
35	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×#6157	—	2	8	no	—
37	C₂×#6157	—	2	4	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₅₃	—	2	2	no	—
42	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
44	C₄×D₅₃	—	2	4	no	—
45	C₄×D₅₃	C₂×Q₈	1	2	yes	—
46	C₄×D₅₃	#41	1	2	yes	—
47	Dic₁₀₆	—	4	4	no	—
48	#88324	—	4	2	no	—
49	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₅₃	—	2	2	no	—
51	C₂×D₅₃	C₂²⋊Q₈	1	2	yes	—
52	#6157	—	2	8	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	—	4	2	no	—
55	C₄×C₅₃	C₄×Q₈	1	2	yes	—
56	Dic₅₃	—	4	2	no	—
57	Dic₅₃	#104	1	4	yes	—
58	Dic₅₃	#124	1	1	yes	characteristic
59	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
60	#1994	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	#391	1	1	yes	characteristic
62	C₂×C₅₃	C₂×#104	1	2	yes	—
63	D₅₃	—	2	2	no	—
64	#533	—	53	1	no	—
65	#423	—	53	1	no	—
66	C₄³	—	53	1	no	—
67	#339	—	53	4	no	—
68	C₅₃	#1994	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	53	2	no	—
70	C₄⋊Q₈	—	53	2	no	—
71	#125	—	53	1	no	—
72	#125	#6157	1	1	yes	characteristic, complemented
73	C₂²⋊Q₈	—	106	2	no	—
74	C₄×Q₈	—	106	2	no	—
75	#117	—	53	2	no	—
76	C₂×C₄²	—	53	1	no	—
77	C₂×C₄²	—	53	2	no	—
78	C₂×C₄²	—	106	2	no	—
79	C₄⋊C₈	—	106	2	no	—
80	C₂×Q₈	—	106	2	no	—
81	C₂²×C₄	—	53	1	no	—
82	C₂²×C₄	—	53	2	no	—
83	C₂²×C₄	—	106	2	no	—
84	M₁₆	—	106	4	no	—
85	C₂×C₈	—	106	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	53	2	no	—
88	C₄⋊C₄	—	106	2	no	—
89	C₄⋊C₄	—	106	2	no	—
90	C₄⋊C₄	C₂×#6157	1	2	yes	—
91	C₂²⋊C₄	—	106	2	no	—
92	C₄²	—	53	2	no	—
93	C₄²	—	53	1	no	—
94	C₄²	—	53	4	no	—
95	C₄²	—	106	2	no	—
96	C₄²	—	106	2	no	—
97	C₄²	—	106	8	no	—
98	C₄²	C₂×#6157	1	1	yes	characteristic
99	C₂³	—	53	1	no	—
100	Q₈	—	212	4	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	53	2	no	—
103	C₂×C₄	—	53	1	no	—
104	C₂×C₄	—	53	2	no	—
105	C₂×C₄	—	53	2	no	—
106	C₂×C₄	—	106	4	no	—
107	C₂×C₄	—	106	4	no	—
108	C₂×C₄	—	106	8	no	—
109	C₂×C₄	—	106	2	no	—
110	C₂×C₄	C₂²⋊#6157	1	2	yes	—
111	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
112	C₈	—	212	2	no	—
113	C₂²	—	53	2	no	—
114	C₂²	—	106	2	no	—
115	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	53	1	no	—
119	C₄	—	53	1	no	—
120	C₄	—	53	4	no	—
121	C₄	—	106	8	no	—
122	C₄	—	212	2	no	—
123	C₄	Q₈×#6157	1	2	yes	—
124	C₂	—	106	2	no	—
125	C₂	(C₂×#6157)⋊Q₈	1	1	yes	characteristic
126	C₂	D₅₃⋊#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811628

C₂

yes

maximal, characteristic

C₄²×#6157

C₂

yes

maximal, characteristic

#16811493

C₂

yes

maximal

#16811179

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₂×C₄×#6157

—

C₂×C₄×#6157

C₂²

yes

—

#13133284

—

#13133216

C₄

yes

complemented

#13133209

C₄

yes

characteristic, complemented

#13133195

—

#13133182

—

#13133175

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#125×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

—

—

—

Q₈

yes

complemented

#11719936

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

C₂×#6157

—

C₂×#6157

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊Q₈

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

Dic₅₃

#104

yes

—

Dic₅₃

#124

yes

characteristic

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

#533

—

#423

—

C₄³

—

#339

—

C₅₃

#1994

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

#6157

yes

characteristic, complemented

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

106

—

C₄⋊C₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂×#6157

yes

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

100

Q₈

—

212

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

106

—

110

C₂×C₄

C₂²⋊#6157

yes

—

111

C₂×C₄

C₂²×#6157

yes

characteristic

112

C₈

—

212

—

113

C₂²

—

114

C₂²

—

106

—

115

C₂²

C₂×C₂²⋊#6157

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

106

—

122

C₄

—

212

—

123

C₄

Q₈×#6157

yes

—

124

C₂

—

106

—

125

C₂

(C₂×#6157)⋊Q₈

yes

characteristic

126

C₂

D₅₃⋊#104

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#125×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#125	#6157
hypocenter	C₅₃	#1994
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#125×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #13133216	C₄	semidirect
10. #13133209	C₄	semidirect
15. #125×C₅₃	C₄	semidirect
23. C₄×#6157	Q₈	semidirect
68. C₅₃	#1994	semidirect
72. #125	#6157	semidirect

H ≅ G/N

type

9. #13133216

C₄

semidirect

10. #13133209

C₄

semidirect

15. #125×C₅₃

C₄

semidirect

23. C₄×#6157

Q₈

semidirect

68. C₅₃

#1994

semidirect

72. #125

#6157

semidirect

Subgroups — #20290901

Conjugacy classes of subgroups

Complements