Group #20283134 — subgroups

Subgroups — #20283134

Order 6784.

564 conjugacy classes of subgroups; only the 373 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 373 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	C₂²×#11719917	C₂	1	2	yes	maximal
2	#16810579	C₂	1	8	yes	maximal, complemented
3	C₄×#11719917	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₄	1	16	yes	complemented
6	C₂×#11719917	C₂²	1	4	yes	—
7	C₂×#11719917	C₂²	1	4	yes	—
8	C₂²×#88322	C₂²	1	4	yes	—
9	#13133128	C₂²	1	8	yes	complemented
10	C₄×#88322	C₂²	1	8	yes	complemented
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	#117×C₅₃	C₄	1	4	yes	—
13	#117×C₅₃	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
15	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
17	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
24	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
25	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
26	C₄²×C₅₃	C₂×C₄	1	4	yes	—
27	C₄²×C₅₃	C₂³	1	4	yes	—
28	#11719917	C₂×C₄	1	16	yes	—
29	#11719917	C₂×C₄	1	16	yes	complemented
30	C₂×#88322	C₂×C₄	1	16	yes	complemented
31	C₂×#88322	C₂³	1	8	yes	—
32	C₂³×C₅₃	—	4	2	no	—
33	C₂³×C₅₃	C₄²	1	4	yes	—
34	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₅₃	—	4	4	no	—
37	C₂×C₄×C₅₃	C₄²	1	4	yes	—
38	C₂×C₄×C₅₃	C₄²	1	8	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	#88322	C₂²×C₄	1	16	yes	—
47	C₂²×C₅₃	—	2	2	no	—
48	C₂²×C₅₃	—	2	4	no	—
49	C₂²×C₅₃	—	4	4	no	—
50	C₂²×C₅₃	C₂×C₄²	1	4	yes	—
51	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
52	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₅₃	—	2	8	no	—
54	C₄×C₅₃	C₂×C₄²	1	4	yes	—
55	C₄×C₅₃	C₂×C₄²	1	2	yes	—
56	C₄×C₅₃	C₂³×C₄	1	2	yes	—
57	#2873	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	2	2	no	—
59	C₂×C₅₃	—	2	4	no	—
60	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
61	C₂×C₅₃	#568	1	2	yes	—
62	C₂²×M₁₆	—	53	2	no	—
63	C₂×#117	D₅₃	1	1	yes	characteristic
64	#405	—	53	8	no	—
65	C₄×M₁₆	—	53	4	no	—
66	C₅₃	#2873	1	1	yes	characteristic, complemented, Sylow
67	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
68	C₂×M₁₆	—	53	16	no	—
69	C₂×M₁₆	—	53	4	no	—
70	C₂×M₁₆	—	53	4	no	—
71	C₂²×C₈	—	53	4	no	—
72	#117	Dic₅₃	1	4	yes	—
73	#117	C₂×D₅₃	1	4	yes	—
74	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	—
75	C₂×C₂²⋊C₄	Dic₅₃	1	2	yes	—
76	C₂×C₄²	C₂×D₅₃	1	2	yes	—
77	#97	—	53	8	no	—
78	C₄×C₈	—	53	8	no	—
79	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
80	C₂²×C₄	C₄×D₅₃	1	8	yes	—
81	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
82	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
83	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
84	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
85	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
86	M₁₆	—	53	16	no	—
87	M₁₆	—	53	16	no	—
88	C₂×C₈	—	53	16	no	—
89	C₂×C₈	—	53	8	no	—
90	C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
91	C₂²⋊C₄	C₂×Dic₅₃	1	8	yes	—
92	C₄²	C₂×Dic₅₃	1	4	yes	—
93	C₄²	C₂²×D₅₃	1	4	yes	—
94	C₂³	—	4	2	no	—
95	C₂³	C₄×Dic₅₃	1	4	yes	—
96	C₂³	C₂²×Dic₅₃	1	2	yes	—
97	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
100	C₂×C₄	C₄×Dic₅₃	1	8	yes	—
101	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
102	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
103	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
104	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
105	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
107	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
108	C₈	—	53	16	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	4	no	—
111	C₂²	—	4	4	no	—
112	C₂²	C₂×C₄×Dic₅₃	1	4	yes	—
113	C₂²	C₂×C₄×Dic₅₃	1	2	yes	—
114	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
115	C₄	—	2	8	no	—
116	C₄	C₂×C₄×Dic₅₃	1	4	yes	—
117	C₄	C₂×C₄×Dic₅₃	1	2	yes	—
118	C₄	C₂²×C₄×D₅₃	1	2	yes	—
119	C₂	—	2	2	no	—
120	C₂	—	2	4	no	—
121	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
122	C₂	#16811390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719917

C₂

yes

maximal

#16810579

C₂

yes

maximal, complemented

C₄×#11719917

C₂

yes

maximal, complemented

C₂×#117×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

complemented

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

#13133128

C₂²

yes

complemented

C₄×#88322

C₂²

yes

complemented

C₂³×C₄×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₄

yes

—

#117×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₂²⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719917

C₂×C₄

yes

—

#11719917

C₂×C₄

yes

complemented

C₂×#88322

C₂×C₄

yes

complemented

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄²

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂³×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

C₂²×C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

—

C₂²×C₅₃

C₂×C₄²

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

C₂³×C₄

yes

—

#2873

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

#568

yes

—

C₂²×M₁₆

—

C₂×#117

D₅₃

yes

characteristic

#405

—

C₄×M₁₆

—

C₅₃

#2873

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

Dic₅₃

yes

—

#117

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₂²⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

#97

—

C₄×C₈

—

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

—

C₂³

C₄×Dic₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×Dic₅₃

yes

—

100

C₂×C₄

C₄×Dic₅₃

yes

—

101

C₂×C₄

C₂×C₄×D₅₃

yes

—

102

C₂×C₄

C₂×C₄×D₅₃

yes

—

103

C₂×C₄

C₂×C₄×D₅₃

yes

—

104

C₂×C₄

C₂²×Dic₅₃

yes

—

105

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

106

C₂×C₄

C₂²×Dic₅₃

yes

—

107

C₂×C₄

C₂³×D₅₃

yes

characteristic

108

C₈

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂×C₄×Dic₅₃

yes

—

113

C₂²

C₂×C₄×Dic₅₃

yes

—

114

C₂²

C₂³×Dic₅₃

yes

characteristic

115

C₄

—

116

C₄

C₂×C₄×Dic₅₃

yes

—

117

C₄

C₂×C₄×Dic₅₃

yes

—

118

C₄

C₂²×C₄×D₅₃

yes

—

119

C₂

—

120

C₂

—

121

C₂

C₂²×C₄×Dic₅₃

yes

characteristic

122

C₂

#16811390

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Fitting	C₂×#117×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	#2873
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Fitting

C₂×#117×C₅₃

C₂

socle

C₂²×C₅₃

C₂³×C₄

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #16810579	C₂	semidirect
3. C₄×#11719917	C₂	semidirect
5. C₂×#11719917	C₄	semidirect
9. #13133128	C₂²	semidirect
10. C₄×#88322	C₂²	semidirect
29. #11719917	C₂×C₄	semidirect
30. C₂×#88322	C₂×C₄	semidirect
66. C₅₃	#2873	semidirect

H ≅ G/N

type

2. #16810579

C₂

semidirect

3. C₄×#11719917

C₂

semidirect

5. C₂×#11719917

C₄

semidirect

9. #13133128

C₂²

semidirect

10. C₄×#88322

C₂²

semidirect

29. #11719917

C₂×C₄

semidirect

30. C₂×#88322

C₂×C₄

semidirect

66. C₅₃

#2873

semidirect

Subgroups — #20283134

Conjugacy classes of subgroups

Complements