Group #20273351 — subgroups

Subgroups — #20273351

Order 6784.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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	#16811179	C₂	1	1	yes	maximal, characteristic
2	#16810678	C₂	1	2	yes	maximal
3	#16810365	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810364	C₂	1	2	yes	maximal, complemented
5	C₅₃×#330	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133216	C₄	1	2	yes	—
7	#13133209	C₂²	1	1	yes	characteristic
8	#13133195	—	2	2	no	—
9	#13133182	—	2	2	no	—
10	#13133175	C₄	1	2	yes	—
11	C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	#13133146	—	2	2	no	—
13	#13133129	—	2	2	no	—
14	#13133128	C₂²	1	2	yes	—
15	#125×C₅₃	C₂²	1	1	yes	characteristic
16	#97×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₅₃	D₄	1	2	yes	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#11719921	—	2	2	no	—
25	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
26	#11719919	—	2	2	no	—
27	#11719919	—	2	2	no	—
28	C₄×Dic₅₃	—	2	2	no	—
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅₃	D₄	1	2	yes	—
31	C₂×#88322	—	2	2	no	—
32	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
36	C₂×Dic₅₃	—	2	2	no	—
37	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×D₅₃	—	2	2	no	—
40	C₄×D₅₃	—	4	2	no	—
41	Dic₁₀₆	—	4	4	no	—
42	C₈×C₅₃	—	4	2	no	—
43	#88322	—	4	2	no	—
44	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×D₅₃	—	2	1	no	—
46	C₂×D₅₃	—	2	2	no	—
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	—	4	2	no	—
50	Dic₅₃	—	2	2	no	—
51	Dic₅₃	—	4	2	no	—
52	Dic₅₃	#101	1	2	yes	—
53	#1537	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
55	C₂×C₅₃	#421	1	2	yes	—
56	D₅₃	—	4	1	no	—
57	#533	—	53	1	no	—
58	#432	—	53	2	no	—
59	#330	—	53	1	no	—
60	#330	—	53	2	no	—
61	#330	D₅₃	1	1	yes	characteristic, complemented
62	C₅₃	#1537	1	1	yes	characteristic, complemented, Sylow
63	C₄⋊Q₈	—	53	2	no	—
64	#125	—	53	1	no	—
65	#125	C₂×D₅₃	1	1	yes	characteristic
66	C₂²⋊Q₈	—	106	2	no	—
67	C₄×Q₈	—	106	2	no	—
68	#117	—	53	2	no	—
69	C₂×C₄²	—	53	1	no	—
70	C₄⋊C₈	—	106	2	no	—
71	C₂²⋊C₈	—	106	2	no	—
72	#97	—	53	2	no	—
73	#97	C₂×D₅₃	1	2	yes	—
74	C₂×Q₈	—	106	2	no	—
75	C₂²×C₄	—	53	1	no	—
76	C₂²×C₄	—	53	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	106	2	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	53	2	no	—
81	C₄⋊C₄	—	106	2	no	—
82	C₄⋊C₄	—	106	2	no	—
83	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
84	C₂²⋊C₄	—	106	2	no	—
85	C₄²	—	53	1	no	—
86	C₄²	—	53	2	no	—
87	C₄²	—	106	2	no	—
88	C₄²	C₂²×D₅₃	1	1	yes	characteristic
89	C₂³	—	53	1	no	—
90	Q₈	—	212	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	53	1	no	—
93	C₂×C₄	—	53	2	no	—
94	C₂×C₄	—	106	2	no	—
95	C₂×C₄	—	106	2	no	—
96	C₂×C₄	—	212	2	no	—
97	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
98	C₂×C₄	D₄×D₅₃	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₈	—	212	2	no	—
101	C₂²	—	106	1	no	—
102	C₂²	—	106	2	no	—
103	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	—	4	2	no	—
107	C₄	—	53	2	no	—
108	C₄	—	106	2	no	—
109	C₄	—	212	2	no	—
110	C₂	—	212	1	no	—
111	C₂	#101×D₅₃	1	1	yes	characteristic
112	C₂	#16810638	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811179

C₂

yes

maximal, characteristic

#16810678

C₂

yes

maximal

#16810365

C₂

yes

maximal, characteristic, complemented

#16810364

C₂

yes

maximal, complemented

C₅₃×#330

C₂

yes

maximal, characteristic, complemented

#13133216

C₄

yes

—

#13133209

C₂²

yes

—

—

C₄

yes

—

C₄²×D₅₃

C₂²

yes

—

—

C₂²

yes

—

#125×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

C₂×#88322

—

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#101

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1537

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#125

—

#125

C₂×D₅₃

yes

characteristic

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

#97

C₂×D₅₃

yes

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₄×D₅₃

yes

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

—

C₈

—

100

C₈

—

212

—

101

C₂²

—

106

—

102

C₂²

—

106

—

103

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

106

—

109

C₄

—

212

—

110

C₂

—

212

—

111

C₂

#101×D₅₃

yes

characteristic

112

C₂

#16810638

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#330	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#330	D₅₃
hypocenter	C₅₃	#1537
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#330

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
3. #16810365	C₂	semidirect
4. #16810364	C₂	semidirect
5. C₅₃×#330	C₂	semidirect
61. #330	D₅₃	semidirect
62. C₅₃	#1537	semidirect

H ≅ G/N

type

3. #16810365

C₂

semidirect

4. #16810364

C₂

semidirect

5. C₅₃×#330

C₂

semidirect

61. #330

D₅₃

semidirect

62. C₅₃

#1537

semidirect

Subgroups — #20273351

Conjugacy classes of subgroups

Complements