Group #20280661 — subgroups

Subgroups — #20280661

Order 6784.

584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133214	C₂	1	1	yes	maximal, characteristic
2	#16810540	C₂	1	8	yes	maximal, complemented
3	#16810524	C₂	1	4	yes	maximal, complemented
4	#16810469	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	2	yes	—
7	C₂²×D₂₁₂	C₂²	1	4	yes	—
8	#13133214	—	2	4	no	—
9	C₂×#11719921	C₂²	1	16	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
11	#13133124	C₂²	1	4	yes	—
12	#125×C₅₃	—	2	4	no	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂³×D₅₃	C₂³	1	4	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	C₂³	1	4	yes	—
21	C₂×D₂₁₂	—	2	8	no	—
22	C₂×D₂₁₂	—	2	4	no	—
23	C₂×D₂₁₂	—	2	8	no	—
24	C₂×D₂₁₂	—	4	8	no	—
25	C₄⋊C₄×C₅₃	—	2	8	no	—
26	C₄⋊C₄×C₅₃	—	2	4	no	—
27	C₄²×C₅₃	D₄	1	4	yes	—
28	#11719921	—	4	16	no	—
29	C₄×Dic₅₃	—	2	4	no	—
30	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
31	C₂²×D₅₃	—	2	4	no	—
32	C₂²×D₅₃	—	2	8	no	—
33	C₂²×D₅₃	—	2	16	no	—
34	C₂×C₄×C₅₃	—	2	4	no	—
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	—	2	8	no	—
37	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₅₃	#41	1	4	yes	—
40	C₂×Dic₅₃	—	2	4	no	—
41	C₂×Dic₅₃	—	2	8	no	—
42	D₂₁₂	—	4	16	no	—
43	D₂₁₂	—	4	8	no	—
44	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
45	C₂²×C₅₃	C₂×#41	1	2	yes	—
46	C₂²×C₅₃	H(5,𝔽₂)	1	4	yes	—
47	C₂×D₅₃	—	4	4	no	—
48	C₂×D₅₃	—	4	8	no	—
49	C₂×D₅₃	—	4	16	no	—
50	C₄×C₅₃	—	2	4	no	—
51	C₄×C₅₃	—	4	4	no	—
52	C₄×C₅₃	#124	1	4	yes	—
53	Dic₅₃	—	4	4	no	—
54	#2641	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
56	C₂×C₅₃	#535	1	2	yes	—
57	C₂×C₅₃	#540	1	4	yes	—
58	D₅₃	—	8	4	no	—
59	C₂×#127	—	53	1	no	—
60	C₂×#125	D₅₃	1	1	yes	characteristic, complemented
61	#394	—	53	8	no	—
62	#390	—	53	4	no	—
63	#376	—	53	1	no	—
64	C₅₃	#2641	1	1	yes	characteristic, complemented, Sylow
65	C₂²×D₄	—	53	2	no	—
66	C₂²×D₄	—	53	4	no	—
67	#127	—	106	4	no	—
68	#125	—	2	4	no	—
69	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
70	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
71	C₂×C₂²⋊C₄	—	53	16	no	—
72	C₂×C₄²	—	53	2	no	—
73	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
74	#95	—	53	4	no	—
75	C₂⁴	—	53	4	no	—
76	C₂×D₄	—	106	8	no	—
77	C₂×D₄	—	106	8	no	—
78	C₂×D₄	—	106	4	no	—
79	C₂×D₄	—	212	8	no	—
80	C₂²×C₄	—	53	4	no	—
81	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
82	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	C₄⋊C₄	—	2	8	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₂²⋊C₄	—	212	16	no	—
87	C₄²	—	106	4	no	—
88	C₄²	C₂²⋊D₅₃	1	4	yes	—
89	C₂³	—	106	4	no	—
90	C₂³	—	106	8	no	—
91	C₂³	—	106	16	no	—
92	C₂³	C₂³×D₅₃	1	1	yes	characteristic
93	D₄	—	212	16	no	—
94	D₄	—	212	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	106	4	no	—
99	C₂×C₄	—	106	8	no	—
100	C₂×C₄	#11719949	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
102	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
103	C₂²	—	212	4	no	—
104	C₂²	—	212	8	no	—
105	C₂²	—	212	16	no	—
106	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
107	C₂²	C₂×#11719949	1	2	yes	—
108	C₂²	#13133263	1	4	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	4	4	no	—
111	C₄	—	212	4	no	—
112	C₄	#13133206	1	4	yes	—
113	C₂	—	424	4	no	—
114	C₂	C₂×#13133206	1	1	yes	characteristic
115	C₂	#16811187	1	2	yes	—
116	C₂	#16811229	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133214

C₂

yes

maximal, characteristic

#16810540

C₂

yes

maximal, complemented

#16810524

C₂

yes

maximal, complemented

#16810469

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂²×D₂₁₂

C₂²

yes

—

#13133214

—

C₂×#11719921

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

#11719921

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#540

yes

—

D₅₃

—

C₂×#127

—

C₂×#125

D₅₃

yes

characteristic, complemented

#394

—

#390

—

#376

—

C₅₃

#2641

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

#127

—

106

—

#125

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

C₂⁴

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

212

—

C₄²

—

106

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂³×D₅₃

yes

characteristic

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

#11719949

yes

—

101

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

102

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

103

C₂²

—

212

—

104

C₂²

—

212

—

105

C₂²

—

212

—

106

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

107

C₂²

C₂×#11719949

yes

—

108

C₂²

#13133263

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

212

—

112

C₄

#13133206

yes

—

113

C₂

—

424

—

114

C₂

C₂×#13133206

yes

characteristic

115

C₂

#16811187

yes

—

116

C₂

#16811229

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂×#125×C₅₃	C₂
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	#2641
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂×#125×C₅₃

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
2. #16810540	C₂	semidirect
3. #16810524	C₂	semidirect
4. #16810469	C₂	semidirect
5. C₂×#125×C₅₃	C₂	semidirect
60. C₂×#125	D₅₃	semidirect
64. C₅₃	#2641	semidirect

H ≅ G/N

type

2. #16810540

C₂

semidirect

3. #16810524

C₂

semidirect

4. #16810469

C₂

semidirect

5. C₂×#125×C₅₃

C₂

semidirect

60. C₂×#125

D₅₃

semidirect

64. C₅₃

#2641

semidirect

Subgroups — #20280661

Conjugacy classes of subgroups

Complements