Group #20290897 — subgroups

Subgroups — #20290897

Order 6784.

252 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₄×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#16811567	C₂	1	1	yes	maximal, characteristic
3	#16811496	C₂	1	4	yes	maximal
4	#125×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#6157	C₂²	1	2	yes	—
7	C₄×#88324	—	2	2	no	—
8	(C₂×#6157)⋊C₄	—	2	2	no	—
9	#13133209	C₄	1	1	yes	characteristic, complemented
10	#13133208	C₄	1	2	yes	complemented
11	#13133207	—	2	2	no	—
12	C₄⋊C₄×D₅₃	—	2	2	no	—
13	C₄⋊C₄×D₅₃	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	—	2	2	no	—
18	C₂×C₄×D₅₃	D₄	1	2	yes	—
19	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
20	C₂×#88324	—	2	2	no	—
21	C₄⋊#6157	—	2	4	no	—
22	D₅₃⋊C₈	—	2	2	no	—
23	D₅₃⋊C₈	Q₈	1	4	yes	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
29	#11719919	—	2	2	no	—
30	#11719919	—	2	2	no	—
31	#11719919	—	2	2	no	—
32	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₅₃	D₄	1	2	yes	—
34	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×#6157	—	4	2	no	—
36	C₂×#6157	—	4	4	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₅₃	—	2	2	no	—
41	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
43	C₄×D₅₃	—	2	4	no	—
44	C₄×D₅₃	—	4	4	no	—
45	C₄×D₅₃	C₂×Q₈	1	2	yes	—
46	C₄×D₅₃	#41	1	2	yes	—
47	#88324	—	2	4	no	—
48	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₅₃	#124	1	1	yes	characteristic
50	C₂×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×D₅₃	#135	1	4	yes	—
52	#6157	—	8	2	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	—	4	2	no	—
55	C₄×C₅₃	C₄×Q₈	1	2	yes	—
56	Dic₅₃	—	2	2	no	—
57	Dic₅₃	—	4	2	no	—
58	Dic₅₃	C₂²⋊Q₈	1	2	yes	—
59	#1991	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	#391	1	1	yes	characteristic
61	C₂×C₅₃	#416	1	2	yes	—
62	D₅₃	#488	1	2	yes	—
63	D₅₃	#499	1	2	yes	—
64	C₂×#125	—	53	1	no	—
65	C₂×C₄×C₈	—	53	1	no	—
66	#382	—	53	1	no	—
67	#340	—	53	4	no	—
68	C₅₃	#1991	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	53	2	no	—
70	#125	—	53	2	no	—
71	#125	—	53	1	no	—
72	#125	—	106	2	no	—
73	#125	#6157	1	1	yes	characteristic, complemented
74	C₂×C₄⋊C₄	—	53	2	no	—
75	C₂×C₄⋊C₄	—	53	2	no	—
76	C₂×C₄⋊C₄	—	106	2	no	—
77	C₂×C₄²	—	53	1	no	—
78	C₄×C₈	—	106	2	no	—
79	#95	—	106	2	no	—
80	C₂²×C₄	—	53	1	no	—
81	C₂²×C₄	—	53	2	no	—
82	C₂²×C₄	—	106	2	no	—
83	C₂²×C₄	—	106	2	no	—
84	C₂×C₈	—	53	4	no	—
85	C₂×C₈	—	106	2	no	—
86	C₂×C₈	—	106	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	53	2	no	—
89	C₄⋊C₄	—	106	2	no	—
90	C₄⋊C₄	—	106	2	no	—
91	C₄⋊C₄	—	106	2	no	—
92	C₄⋊C₄	—	106	4	no	—
93	C₄⋊C₄	—	106	2	no	—
94	C₄⋊C₄	C₂×#6157	1	2	yes	—
95	C₄²	—	53	2	no	—
96	C₄²	—	53	1	no	—
97	C₄²	C₂×#6157	1	1	yes	characteristic
98	C₂³	—	53	1	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	53	2	no	—
102	C₂×C₄	—	53	2	no	—
103	C₂×C₄	—	53	2	no	—
104	C₂×C₄	—	106	4	no	—
105	C₂×C₄	—	106	2	no	—
106	C₂×C₄	—	212	4	no	—
107	C₂×C₄	—	212	2	no	—
108	C₂×C₄	—	212	4	no	—
109	C₂×C₄	C₂²⋊#6157	1	2	yes	—
110	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
111	C₈	—	106	4	no	—
112	C₂²	—	53	1	no	—
113	C₂²	—	53	1	no	—
114	C₂²	—	53	4	no	—
115	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	53	2	no	—
119	C₄	—	106	2	no	—
120	C₄	—	424	2	no	—
121	C₄	—	212	2	no	—
122	C₄	Q₈×#6157	1	2	yes	—
123	C₂	—	53	2	no	—
124	C₂	—	53	2	no	—
125	C₂	(C₂×#6157)⋊Q₈	1	1	yes	characteristic
126	C₂	#16811613	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₅₃⋊C₈

C₂

yes

maximal, characteristic

#16811567

C₂

yes

maximal, characteristic

#16811496

C₂

yes

maximal

#125×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₂²

yes

—

C₂×C₄⋊#6157

C₂²

yes

—

C₄×#88324

—

(C₂×#6157)⋊C₄

—

#13133209

C₄

yes

characteristic, complemented

#13133208

C₄

yes

complemented

#13133207

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#125×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×#88324

—

C₄⋊#6157

—

D₅₃⋊C₈

—

D₅₃⋊C₈

Q₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

#11719919

—

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

—

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

#41

yes

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

#124

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

characteristic

C₂×D₅₃

#135

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²⋊Q₈

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

#488

yes

—

D₅₃

#499

yes

—

C₂×#125

—

C₂×C₄×C₈

—

#382

—

#340

—

C₅₃

#1991

yes

characteristic, complemented, Sylow

C₂²×C₈

—

#125

—

#125

—

#125

—

106

—

#125

#6157

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₄×C₈

—

106

—

#95

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

212

—

107

C₂×C₄

—

212

—

108

C₂×C₄

—

212

—

109

C₂×C₄

C₂²⋊#6157

yes

—

110

C₂×C₄

C₂²×#6157

yes

characteristic

111

C₈

—

106

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₂²⋊#6157

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

106

—

120

C₄

—

424

—

121

C₄

—

212

—

122

C₄

Q₈×#6157

yes

—

123

C₂

—

124

C₂

—

125

C₂

(C₂×#6157)⋊Q₈

yes

characteristic

126

C₂

#16811613

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	#1991
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. #13133209	C₄	semidirect
10. #13133208	C₄	semidirect
15. #125×C₅₃	C₄	semidirect
68. C₅₃	#1991	semidirect
73. #125	#6157	semidirect

H ≅ G/N

type

9. #13133209

C₄

semidirect

10. #13133208

C₄

semidirect

15. #125×C₅₃

C₄

semidirect

68. C₅₃

#1991

semidirect

73. #125

#6157

semidirect

Subgroups — #20290897

Conjugacy classes of subgroups

Complements