Group #20290652 — subgroups

Subgroups — #20290652

Order 6784.

248 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 94 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×#13133286	C₂	1	2	yes	maximal
2	C₂×#13133285	C₂	1	2	yes	maximal
3	#16811567	C₂	1	2	yes	maximal
4	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#6157	C₂²	1	4	yes	—
6	#13133286	—	2	4	no	—
7	#13133285	—	2	4	no	—
8	(C₂×#6157)⋊C₄	—	2	4	no	—
9	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	C₄×#88322	C₄	1	1	yes	characteristic, complemented
13	C₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
14	C₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
15	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6157	—	2	4	no	—
17	C₂×C₄×D₅₃	D₄	1	2	yes	—
18	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊#6157	—	2	4	no	—
20	C₄⋊#6157	—	2	4	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₅₃	Q₈	1	2	yes	—
26	C₂×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂×#88322	C₂×C₄	1	1	yes	characteristic
28	C₈×D₅₃	D₄	1	2	yes	—
29	C₈×D₅₃	D₄	1	2	yes	—
30	C₈×D₅₃	Q₈	1	2	yes	—
31	C₈×D₅₃	Q₈	1	2	yes	—
32	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×#6157	—	4	4	no	—
34	C₂×#6157	—	4	8	no	—
35	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
38	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₄×D₅₃	—	2	4	no	—
40	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
41	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	#88322	C₄⋊C₄	1	2	yes	—
47	#88322	C₄⋊C₄	1	2	yes	—
48	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×D₅₃	#127	1	1	yes	characteristic
50	C₂×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×D₅₃	#135	1	4	yes	—
52	#6157	—	8	4	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	Dic₅₃	—	2	2	no	—
57	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
59	#1846	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	#384	1	1	yes	characteristic
61	C₂×C₅₃	#427	1	2	yes	—
62	D₅₃	#495	1	2	yes	—
63	D₅₃	#499	1	2	yes	—
64	C₂×#107	—	53	2	no	—
65	C₂×#106	—	53	2	no	—
66	C₂×C₄×C₈	—	53	1	no	—
67	#382	—	53	2	no	—
68	C₅₃	#1846	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	53	1	no	—
70	C₂²×C₈	—	53	1	no	—
71	C₂×C₄⋊C₄	—	53	4	no	—
72	C₂×C₄²	—	53	1	no	—
73	#107	—	106	4	no	—
74	#106	—	106	4	no	—
75	C₄×C₈	—	53	1	no	—
76	C₄×C₈	—	53	1	no	—
77	C₄×C₈	—	53	1	no	—
78	C₄×C₈	#6157	1	1	yes	characteristic, complemented
79	#95	—	106	4	no	—
80	C₂²×C₄	—	53	1	no	—
81	C₂²×C₄	—	53	2	no	—
82	C₂²×C₄	—	106	4	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	—	53	1	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	53	2	no	—
87	C₂×C₈	—	53	2	no	—
88	C₂×C₈	—	53	2	no	—
89	C₂×C₈	C₂×#6157	1	1	yes	characteristic
90	C₂×C₈	C₂×#6157	1	1	yes	characteristic
91	C₄⋊C₄	—	106	4	no	—
92	C₄⋊C₄	—	106	4	no	—
93	C₄²	—	53	2	no	—
94	C₄²	—	53	1	no	—
95	C₄²	C₂×#6157	1	1	yes	characteristic
96	C₂³	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	2	no	—
103	C₂×C₄	—	106	4	no	—
104	C₂×C₄	—	212	4	no	—
105	C₂×C₄	—	212	8	no	—
106	C₂×C₄	C₄⋊#6157	1	2	yes	—
107	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
108	C₈	—	53	2	no	—
109	C₈	—	53	2	no	—
110	C₈	C₄⋊#6157	1	2	yes	—
111	C₈	C₄⋊#6157	1	2	yes	—
112	C₂²	—	53	1	no	—
113	C₂²	—	53	1	no	—
114	C₂²	—	53	4	no	—
115	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	53	1	no	—
118	C₄	—	53	1	no	—
119	C₄	—	106	2	no	—
120	C₄	—	424	4	no	—
121	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
122	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
123	C₂	—	53	2	no	—
124	C₂	—	53	2	no	—
125	C₂	#16811569	1	1	yes	characteristic
126	C₂	#16811633	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133286

C₂

yes

maximal

C₂×#13133285

C₂

yes

maximal

#16811567

C₂

yes

maximal

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#6157

C₂²

yes

—

#13133286

—

#13133285

—

(C₂×#6157)⋊C₄

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₄²×D₅₃

C₂²

yes

characteristic

C₄×#88322

C₄

yes

characteristic, complemented

C₈×Dic₅₃

C₄

yes

characteristic, complemented

C₈×Dic₅₃

C₄

yes

characteristic, complemented

C₄×C₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₄⋊#6157

—

C₄⋊#6157

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₈×D₅₃

Q₈

yes

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

C₂×#6157

—

C₂×#6157

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

#127

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

characteristic

C₂×D₅₃

#135

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

Dic₅₃

—

Dic₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

#495

yes

—

D₅₃

#499

yes

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

—

#382

—

C₅₃

#1846

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

106

—

#106

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6157

yes

characteristic, complemented

#95

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

106

—

104

C₂×C₄

—

212

—

105

C₂×C₄

—

212

—

106

C₂×C₄

C₄⋊#6157

yes

—

107

C₂×C₄

C₂²×#6157

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

C₄⋊#6157

yes

—

111

C₈

C₄⋊#6157

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄⋊#6157

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

106

—

120

C₄

—

424

—

121

C₄

C₂×C₄⋊#6157

yes

characteristic

122

C₄

C₂×C₄⋊#6157

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#16811569

yes

characteristic

126

C₂

#16811633

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1846
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
12. C₄×#88322	C₄	semidirect
13. C₈×Dic₅₃	C₄	semidirect
14. C₈×Dic₅₃	C₄	semidirect
15. C₄×C₈×C₅₃	C₄	semidirect
68. C₅₃	#1846	semidirect
78. C₄×C₈	#6157	semidirect

H ≅ G/N

type

12. C₄×#88322

C₄

semidirect

13. C₈×Dic₅₃

C₄

semidirect

14. C₈×Dic₅₃

C₄

semidirect

15. C₄×C₈×C₅₃

C₄

semidirect

68. C₅₃

#1846

semidirect

78. C₄×C₈

#6157

semidirect

Subgroups — #20290652

Conjugacy classes of subgroups

Complements