Group #20272999 — subgroups

Subgroups — #20272999

Order 6784.

212 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#16810345	C₂	1	2	yes	maximal, complemented
4	C₈×#88322	C₂	1	2	yes	maximal, complemented
5	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₄	1	2	yes	—
7	C₄⋊#88322	C₄	1	2	yes	—
8	C₂²⋊#88322	C₄	1	2	yes	—
9	C₄×#88322	C₄	1	2	yes	—
10	C₄×#88322	C₂²	1	1	yes	characteristic
11	C₄×#88322	C₂²	1	1	yes	characteristic
12	C₄×#88322	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	C₄	1	2	yes	—
15	C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×#88322	C₈	1	8	yes	complemented
26	C₂×#88322	C₂×C₄	1	4	yes	—
27	C₂×#88322	C₂×C₄	1	2	yes	—
28	C₂×#88322	C₂×C₄	1	2	yes	—
29	C₂³×C₅₃	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	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	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₈×C₅₃	—	2	4	no	—
40	#88322	—	2	4	no	—
41	#88322	C₂×C₈	1	4	yes	—
42	#88322	C₂×C₈	1	4	yes	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	—	2	1	no	—
45	C₂²×C₅₃	C₄×C₈	1	2	yes	—
46	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	C₄×C₈	1	2	yes	—
50	C₄×C₅₃	C₂²×C₈	1	2	yes	—
51	C₄×C₅₃	#131	1	2	yes	—
52	C₄×C₅₃	#131	1	2	yes	—
53	#1450	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	—	2	2	no	—
55	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
56	C₂×C₅₃	#405	1	1	yes	characteristic
57	C₂×C₅₃	#405	1	1	yes	characteristic
58	#431	—	53	1	no	—
59	#431	D₅₃	1	1	yes	characteristic
60	C₂×C₄×C₈	—	53	1	no	—
61	#322	—	53	2	no	—
62	C₈²	—	53	2	no	—
63	C₅₃	#1450	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₈	—	53	2	no	—
65	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
66	C₄⋊C₈	—	53	2	no	—
67	C₄⋊C₈	Dic₅₃	1	2	yes	—
68	C₂²⋊C₈	—	53	2	no	—
69	C₂²⋊C₈	Dic₅₃	1	2	yes	—
70	C₄×C₈	—	53	2	no	—
71	C₄×C₈	—	53	1	no	—
72	C₄×C₈	—	53	2	no	—
73	C₄×C₈	—	53	1	no	—
74	C₄×C₈	C₂×D₅₃	1	2	yes	—
75	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
77	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
78	C₂×C₈	—	53	2	no	—
79	C₂×C₈	—	53	2	no	—
80	C₂×C₈	—	53	8	no	—
81	C₂×C₈	—	53	4	no	—
82	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
83	C₄²	C₄×D₅₃	1	1	yes	characteristic
84	C₄²	C₄×D₅₃	1	1	yes	characteristic
85	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
86	C₄²	C₂²×D₅₃	1	1	yes	characteristic
87	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	C₈×D₅₃	1	4	yes	—
91	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
92	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
94	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
95	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
96	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
97	C₈	—	2	4	no	—
98	C₈	—	53	4	no	—
99	C₈	—	53	4	no	—
100	C₈	—	106	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₈×Dic₅₃	1	2	yes	—
104	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	C₈×Dic₅₃	1	2	yes	—
108	C₄	C₂×C₈×D₅₃	1	2	yes	—
109	C₄	#13133224	1	2	yes	—
110	C₄	#13133225	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	C₂×C₈×Dic₅₃	1	1	yes	characteristic
113	C₂	#16810579	1	1	yes	characteristic
114	C₂	#16810581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, complemented

C₈×#88322

C₂

yes

maximal, complemented

C₅₃×#431

C₂

yes

maximal, characteristic

C₂²×#88322

C₄

yes

—

C₄⋊#88322

C₄

yes

—

C₂²⋊#88322

C₄

yes

—

C₄×#88322

C₄

yes

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₄

yes

—

C₂²⋊C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂×#88322

C₈

yes

complemented

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

—

#88322

—

#88322

C₂×C₈

yes

—

#88322

C₂×C₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×C₈

yes

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×C₈

yes

—

C₄×C₅₃

C₂²×C₈

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

#131

yes

—

#1450

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₂×C₄×C₈

—

#322

—

C₈²

—

C₅₃

#1450

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₅₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₅₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

C₄×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₅₃

yes

—

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

106

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₈×Dic₅₃

yes

—

104

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

C₈×Dic₅₃

yes

—

108

C₄

C₂×C₈×D₅₃

yes

—

109

C₄

#13133224

yes

—

110

C₄

#13133225

yes

—

111

C₂

—

112

C₂

C₂×C₈×Dic₅₃

yes

characteristic

113

C₂

#16810579

yes

characteristic

114

C₂

#16810581

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×C₄²
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1450
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₅₃

commutator

C₂×C₅₃

C₂×C₄×C₈

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂×C₄²

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #16810345	C₂	semidirect
4. C₈×#88322	C₂	semidirect
25. C₂×#88322	C₈	semidirect
63. C₅₃	#1450	semidirect

H ≅ G/N

type

3. #16810345

C₂

semidirect

4. C₈×#88322

C₂

semidirect

25. C₂×#88322

C₈

semidirect

63. C₅₃

#1450

semidirect

Subgroups — #20272999

Conjugacy classes of subgroups

Complements