Group #20277277 — subgroups

Subgroups — #20277277

Order 6784.

924 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 249 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	C₂×#13133179	C₂	1	6	yes	maximal, complemented
3	#16810524	C₂	1	6	yes	maximal, complemented
4	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	6	yes	—
7	C₂²×C₄×D₅₃	C₂²	1	4	yes	—
8	#13133214	C₄	1	8	yes	complemented
9	#13133179	—	2	24	no	—
10	C₂×#11719921	C₂²	1	12	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
12	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×D₅₃	C₂³	1	4	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
18	C₂²×Dic₅₃	C₂³	1	3	yes	—
19	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
20	C₂×D₂₁₂	—	2	12	no	—
21	C₂×D₂₁₂	—	2	12	no	—
22	C₂×D₂₁₂	C₂×C₄	1	24	yes	complemented
23	C₂×C₄×D₅₃	—	2	24	no	—
24	C₄⋊C₄×C₅₃	—	2	12	no	—
25	C₄²×C₅₃	C₂×C₄	1	4	yes	complemented
26	#11719921	—	2	24	no	—
27	C₄×Dic₅₃	—	2	6	no	—
28	C₄×Dic₅₃	D₄	1	12	yes	complemented
29	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂²×D₅₃	—	2	12	no	—
31	C₂²×D₅₃	—	2	12	no	—
32	C₂²×D₅₃	C₂²×C₄	1	8	yes	—
33	C₂×C₄×C₅₃	—	2	12	no	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₅₃	#41	1	6	yes	—
37	C₂×Dic₅₃	—	2	6	no	—
38	C₂×Dic₅₃	C₂×D₄	1	6	yes	—
39	C₂×Dic₅₃	C₂×D₄	1	6	yes	—
40	D₂₁₂	—	2	48	no	—
41	C₄×D₅₃	—	4	16	no	—
42	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
43	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
44	C₂²×C₅₃	C₂×#41	1	3	yes	—
45	C₂×D₅₃	—	2	24	no	—
46	C₂×D₅₃	—	4	4	no	—
47	C₂×D₅₃	—	4	12	no	—
48	C₄×C₅₃	—	4	4	no	—
49	C₄×C₅₃	C₄×D₄	1	12	yes	complemented
50	Dic₅₃	—	2	6	no	—
51	Dic₅₃	#127	1	4	yes	complemented
52	C₄×#127	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	C₂×C₄×D₄	1	3	yes	—
54	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
55	C₂×C₅₃	#532	1	3	yes	—
56	D₅₃	—	4	8	no	—
57	C₂×#127	—	53	1	no	—
58	C₂×C₄×D₄	—	53	6	no	—
59	#390	—	53	6	no	—
60	#384	D₅₃	1	1	yes	characteristic, complemented
61	C₄³	—	53	1	no	—
62	C₅₃	C₄×#127	1	1	yes	characteristic, complemented, Sylow
63	C₂²×D₄	—	53	6	no	—
64	C₂³×C₄	—	53	4	no	—
65	#127	—	53	8	no	—
66	C₄×D₄	—	106	24	no	—
67	C₂×C₄⋊C₄	C₂×D₅₃	1	6	yes	—
68	C₂×C₂²⋊C₄	—	53	12	no	—
69	C₂×C₄²	—	53	3	no	—
70	C₂×C₄²	—	53	3	no	—
71	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
72	C₂⁴	—	53	4	no	—
73	C₂×D₄	—	53	24	no	—
74	C₂×D₄	—	106	12	no	—
75	C₂×D₄	—	106	12	no	—
76	C₂²×C₄	—	53	1	no	—
77	C₂²×C₄	—	53	3	no	—
78	C₂²×C₄	—	106	24	no	—
79	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
80	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
81	C₄⋊C₄	—	2	12	no	—
82	C₂²⋊C₄	—	106	24	no	—
83	C₄²	—	53	12	no	—
84	C₄²	—	106	6	no	—
85	C₄²	C₄×D₅₃	1	4	yes	complemented
86	C₂³	—	53	8	no	—
87	C₂³	—	106	12	no	—
88	C₂³	—	106	12	no	—
89	C₂³	C₂³×D₅₃	1	1	yes	characteristic
90	D₄	—	106	48	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	53	6	no	—
93	C₂×C₄	—	53	6	no	—
94	C₂×C₄	—	106	6	no	—
95	C₂×C₄	—	212	16	no	—
96	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
97	C₂×C₄	D₄×D₅₃	1	6	yes	—
98	C₂×C₄	#11719949	1	6	yes	—
99	C₂²	—	106	24	no	—
100	C₂²	—	212	4	no	—
101	C₂²	—	212	12	no	—
102	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
103	C₂²	C₂×D₄×D₅₃	1	3	yes	—
104	C₂²	C₂×#11719949	1	3	yes	—
105	C₄	—	4	4	no	—
106	C₄	—	53	4	no	—
107	C₄	—	106	6	no	—
108	C₄	#13133179	1	12	yes	complemented
109	C₂	—	212	8	no	—
110	C₂	C₂×#13133179	1	3	yes	—
111	C₂	#127×D₅₃	1	1	yes	characteristic
112	C₂	#16811174	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133214

C₂

yes

maximal, characteristic

C₂×#13133179

C₂

yes

maximal, complemented

#16810524

C₂

yes

maximal, complemented

C₄²×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₅₃×#384

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂²×C₄×D₅₃

C₂²

yes

—

#13133214

C₄

yes

complemented

#13133179

—

C₂×#11719921

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

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×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

complemented

C₂×C₄×D₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

complemented

#11719921

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

complemented

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

complemented

Dic₅₃

—

Dic₅₃

yes

—

C₂×C₅₃

C₂×C₄×D₄

yes

—

C₂×C₅₃

C₂×#127

yes

characteristic

C₂×C₅₃

#532

yes

—

D₅₃

—

C₂×#127

—

C₂×C₄×D₄

—

#390

—

#384

D₅₃

yes

characteristic, complemented

C₄³

—

C₅₃

C₄×#127

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

#127

—

C₄×D₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

—

106

—

C₄²

C₄×D₅₃

yes

complemented

C₂³

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂³×D₅₃

yes

characteristic

D₄

—

106

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

D₄×D₅₃

yes

—

C₂×C₄

#11719949

yes

—

C₂²

—

106

—

100

C₂²

—

212

—

101

C₂²

—

212

—

102

C₂²

C₂²×C₄×D₅₃

yes

characteristic

103

C₂²

C₂×D₄×D₅₃

yes

—

104

C₂²

C₂×#11719949

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

106

—

108

C₄

#13133179

yes

complemented

109

C₂

—

212

—

110

C₂

C₂×#13133179

yes

—

111

C₂

#127×D₅₃

yes

characteristic

112

C₂

#16811174

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#384	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	#384	D₅₃
hypocenter	C₅₃	C₄×#127
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#384

C₂

socle

C₂³×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
2. C₂×#13133179	C₂	semidirect
3. #16810524	C₂	semidirect
4. C₄²×Dic₅₃	C₂	semidirect
5. C₅₃×#384	C₂	semidirect
8. #13133214	C₄	semidirect
22. C₂×D₂₁₂	C₂×C₄	semidirect
25. C₄²×C₅₃	C₂×C₄	semidirect
28. C₄×Dic₅₃	D₄	semidirect
49. C₄×C₅₃	C₄×D₄	semidirect
51. Dic₅₃	#127	semidirect
60. #384	D₅₃	semidirect
62. C₅₃	C₄×#127	semidirect
85. C₄²	C₄×D₅₃	semidirect
108. C₄	#13133179	semidirect

H ≅ G/N

type

2. C₂×#13133179

C₂

semidirect

3. #16810524

C₂

semidirect

4. C₄²×Dic₅₃

C₂

semidirect

5. C₅₃×#384

C₂

semidirect

8. #13133214

C₄

semidirect

22. C₂×D₂₁₂

C₂×C₄

semidirect

25. C₄²×C₅₃

C₂×C₄

semidirect

28. C₄×Dic₅₃

D₄

semidirect

49. C₄×C₅₃

C₄×D₄

semidirect

51. Dic₅₃

#127

semidirect

60. #384

D₅₃

semidirect

62. C₅₃

C₄×#127

semidirect

85. C₄²

C₄×D₅₃

semidirect

108. C₄

#13133179

semidirect

Subgroups — #20277277

Conjugacy classes of subgroups

Complements