Group #20289913 — subgroups

Subgroups — #20289913

Order 6784.

224 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 110 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	#16811641	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#16811464	C₂	1	2	yes	maximal, complemented
4	C₈×#88324	C₂	1	2	yes	maximal, complemented
5	#16810575	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₈	C₄	1	2	yes	—
7	#13133284	C₄	1	2	yes	—
8	#13133274	C₄	1	2	yes	—
9	C₄×#88324	C₄	1	2	yes	—
10	C₄×#88324	C₂²	1	1	yes	characteristic
11	C₄×#88324	C₂²	1	1	yes	characteristic
12	C₄×#88324	C₂²	1	2	yes	—
13	C₂×#11719912	C₄	1	2	yes	—
14	C₄²×D₅₃	C₂²	1	1	yes	characteristic
15	#13133128	C₄	1	1	yes	characteristic
16	C₈×Dic₅₃	C₂²	1	2	yes	—
17	#97×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#88324	C₂×C₄	1	2	yes	—
22	C₂×#88324	C₂×C₄	1	4	yes	—
23	C₂×#88324	C₂×C₄	1	2	yes	—
24	D₅₃⋊C₈	—	2	4	no	—
25	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
30	C₂×#88322	C₂×C₄	1	2	yes	—
31	#11719912	C₈	1	8	yes	complemented
32	C₂²×D₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
37	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₄×D₅₃	—	2	2	no	—
40	C₄×D₅₃	—	2	2	no	—
41	C₄×D₅₃	C₂×C₈	1	4	yes	—
42	#88324	—	2	2	no	—
43	#88324	—	2	4	no	—
44	#88324	M₁₆	1	4	yes	complemented
45	C₈×C₅₃	C₂×C₈	1	4	yes	complemented
46	#88322	C₂×C₈	1	4	yes	complemented
47	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×D₅₃	—	2	1	no	—
49	C₂×D₅₃	—	2	1	no	—
50	C₂×D₅₃	C₄×C₈	1	2	yes	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	C₂²×C₈	1	2	yes	—
54	Dic₅₃	C₄×C₈	1	2	yes	—
55	Dic₅₃	C₂×M₁₆	1	2	yes	—
56	Dic₅₃	#131	1	2	yes	—
57	C₈×M₁₆	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
59	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
60	C₂×C₅₃	#405	1	1	yes	characteristic
61	D₅₃	—	2	2	no	—
62	#431	—	53	1	no	—
63	C₄×M₁₆	—	53	1	no	—
64	C₂×C₄×C₈	—	53	1	no	—
65	#322	—	53	2	no	—
66	C₈²	—	53	2	no	—
67	C₅₃	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₁₆	—	53	2	no	—
69	C₂²×C₈	—	53	2	no	—
70	C₂×C₄²	—	53	1	no	—
71	C₄⋊C₈	—	53	2	no	—
72	C₂²⋊C₈	—	53	2	no	—
73	#97	—	53	1	no	—
74	#97	#6157	1	1	yes	characteristic
75	C₄×C₈	—	53	1	no	—
76	C₄×C₈	—	53	2	no	—
77	C₄×C₈	—	53	1	no	—
78	C₄×C₈	—	53	2	no	—
79	C₄×C₈	—	53	2	no	—
80	C₂²×C₄	—	53	1	no	—
81	C₂²×C₄	—	53	1	no	—
82	C₂²×C₄	—	53	1	no	—
83	M₁₆	—	53	8	no	—
84	C₂×C₈	—	53	2	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	53	2	no	—
87	C₂×C₈	—	53	4	no	—
88	C₂×C₈	—	106	4	no	—
89	C₂×C₈	C₂×#6157	1	2	yes	—
90	C₄²	—	53	1	no	—
91	C₄²	—	53	1	no	—
92	C₄²	—	53	1	no	—
93	C₄²	C₂×#6157	1	1	yes	characteristic
94	C₂³	—	53	1	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	53	4	no	—
99	C₂×C₄	—	106	2	no	—
100	C₂×C₄	—	106	2	no	—
101	C₂×C₄	C₄×#6157	1	1	yes	characteristic
102	C₂×C₄	C₄×#6157	1	1	yes	characteristic
103	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
104	C₈	—	53	4	no	—
105	C₈	—	53	4	no	—
106	C₈	—	106	4	no	—
107	C₈	—	106	2	no	—
108	C₈	D₅₃⋊C₈	1	4	yes	complemented
109	C₂²	—	53	2	no	—
110	C₂²	—	106	1	no	—
111	C₂²	—	106	1	no	—
112	C₂²	C₂×C₄×#6157	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	53	2	no	—
116	C₄	—	53	2	no	—
117	C₄	—	53	2	no	—
118	C₄	C₂×D₅₃⋊C₈	1	2	yes	—
119	C₂	—	106	2	no	—
120	C₂	C₄×D₅₃⋊C₈	1	1	yes	characteristic
121	C₂	M₁₆×#6157	1	1	yes	characteristic
122	C₂	#16811589	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811641

C₂

yes

maximal, characteristic

C₄×D₅₃⋊C₈

C₂

yes

maximal, characteristic

#16811464

C₂

yes

maximal, complemented

C₈×#88324

C₂

yes

maximal, complemented

#16810575

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₄

yes

—

#13133284

C₄

yes

—

#13133274

C₄

yes

—

C₄×#88324

C₄

yes

—

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

—

C₂×#11719912

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133128

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

—

#97×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×#88324

C₂×C₄

yes

—

C₂×#88324

C₂×C₄

yes

—

C₂×#88324

C₂×C₄

yes

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

#11719912

C₈

yes

complemented

C₂²×D₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×C₈

yes

—

#88324

—

#88324

—

#88324

M₁₆

yes

complemented

C₈×C₅₃

C₂×C₈

yes

complemented

#88322

C₂×C₈

yes

complemented

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₄×C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

—

Dic₅₃

C₄×C₈

yes

—

Dic₅₃

C₂×M₁₆

yes

—

Dic₅₃

#131

yes

—

C₈×M₁₆

—

maximal, Sylow

C₂×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×C₅₃

C₄×M₁₆

yes

characteristic

C₂×C₅₃

#405

yes

characteristic

D₅₃

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

C₈²

—

C₅₃

C₈×M₁₆

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

—

yes

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

100

C₂×C₄

—

106

—

101

C₂×C₄

C₄×#6157

yes

characteristic

102

C₂×C₄

C₄×#6157

yes

characteristic

103

C₂×C₄

C₂²×#6157

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

106

—

107

C₈

—

106

—

108

C₈

D₅₃⋊C₈

yes

complemented

109

C₂²

—

110

C₂²

—

106

—

111

C₂²

—

106

—

112

C₂²

C₂×C₄×#6157

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂×D₅₃⋊C₈

yes

—

119

C₂

—

106

—

120

C₂

C₄×D₅₃⋊C₈

yes

characteristic

121

C₂

M₁₆×#6157

yes

characteristic

122

C₂

#16811589

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂×C₄×C₈
Fitting	#97×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄²
hypercenter	#97	#6157
hypocenter	C₅₃	C₈×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator

C₂×C₅₃

C₂×C₄×C₈

Fitting

#97×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄²

C₅₃

C₈×M₁₆

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
3. #16811464	C₂	semidirect
4. C₈×#88324	C₂	semidirect
31. #11719912	C₈	semidirect
44. #88324	M₁₆	semidirect
45. C₈×C₅₃	C₂×C₈	semidirect
46. #88322	C₂×C₈	semidirect
67. C₅₃	C₈×M₁₆	semidirect
108. C₈	D₅₃⋊C₈	semidirect

H ≅ G/N

type

3. #16811464

C₂

semidirect

4. C₈×#88324

C₂

semidirect

31. #11719912

C₈

semidirect

44. #88324

M₁₆

semidirect

45. C₈×C₅₃

C₂×C₈

semidirect

46. #88322

C₂×C₈

semidirect

67. C₅₃

C₈×M₁₆

semidirect

108. C₈

D₅₃⋊C₈

semidirect

Subgroups — #20289913

Conjugacy classes of subgroups

Complements