Group #20283656 — subgroups

Subgroups — #20283656

Order 6784.

348 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 221 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	#16810754	C₂	1	8	yes	maximal, complemented
2	#16810650	C₂	1	2	yes	maximal
3	#16810647	C₂	1	1	yes	maximal, characteristic
4	#16810579	C₂	1	2	yes	maximal
5	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
7	C₂×#11719917	C₂²	1	2	yes	—
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₄⋊#88322	C₂²	1	4	yes	—
11	C₄⋊#88322	C₂²	1	8	yes	—
12	#13133128	C₂²	1	4	yes	—
13	#13133128	C₂²	1	4	yes	—
14	C₄×#88322	C₂²	1	4	yes	—
15	C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
16	#125×C₅₃	C₄	1	2	yes	—
17	C₂²⋊Q₈×C₅₃	C₄	1	4	yes	—
18	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
19	#117×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₂³	1	2	yes	—
31	C₄²×C₅₃	C₂³	1	4	yes	—
32	#11719917	—	2	4	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	Q₈	1	4	yes	complemented
35	C₂×#88322	C₂³	1	4	yes	—
36	C₂×#88322	C₂³	1	2	yes	—
37	C₂×#88322	C₂³	1	2	yes	—
38	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₅₃	—	4	4	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	#88322	—	2	2	no	—
49	#88322	—	2	4	no	—
50	#88322	C₂×Q₈	1	4	yes	—
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
53	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	8	no	—
57	C₄×C₅₃	C₄×Q₈	1	2	yes	—
58	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
60	#2961	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	2	no	—
62	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
63	C₂×C₅₃	#568	1	2	yes	—
64	#533	D₅₃	1	1	yes	characteristic
65	#446	—	53	8	no	—
66	#424	—	53	2	no	—
67	#423	—	53	1	no	—
68	#405	—	53	2	no	—
69	C₂×#97	—	53	1	no	—
70	C₅₃	#2961	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	53	2	no	—
72	C₂²×C₈	—	53	1	no	—
73	C₂²×C₈	—	53	1	no	—
74	C₄⋊Q₈	Dic₅₃	1	2	yes	—
75	#125	Dic₅₃	1	2	yes	—
76	C₂²⋊Q₈	Dic₅₃	1	4	yes	—
77	C₄×Q₈	C₂×D₅₃	1	4	yes	—
78	#117	C₂×D₅₃	1	2	yes	—
79	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
80	C₄⋊C₈	—	53	4	no	—
81	C₄⋊C₈	—	53	8	no	—
82	#97	—	53	4	no	—
83	#97	—	53	4	no	—
84	C₄×C₈	—	53	4	no	—
85	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
86	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
88	M₁₆	—	106	4	no	—
89	C₂×C₈	—	53	4	no	—
90	C₂×C₈	—	53	2	no	—
91	C₂×C₈	—	53	4	no	—
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	106	2	no	—
94	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
95	C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
96	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
98	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
99	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
100	C₄²	C₂²×D₅₃	1	4	yes	—
101	C₄²	C₂²×D₅₃	1	2	yes	—
102	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	Q₈×D₅₃	1	2	yes	—
106	C₂×C₄	#11719949	1	2	yes	—
107	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
108	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
109	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
110	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
112	C₈	—	53	4	no	—
113	C₈	—	106	2	no	—
114	C₈	—	106	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	Q₈×Dic₅₃	1	2	yes	—
117	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	Q₈×Dic₅₃	1	2	yes	—
122	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
123	C₄	C₂×#11719949	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
126	C₂	#16811390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810754

C₂

yes

maximal, complemented

#16810650

C₂

yes

maximal

#16810647

C₂

yes

maximal, characteristic

#16810579

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal, characteristic

C₅₃×#533

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133128

C₂²

yes

—

#13133128

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

—

#125×C₅₃

C₄

yes

—

C₂²⋊Q₈×C₅₃

C₄

yes

—

C₄×Q₈×C₅₃

C₂²

yes

—

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

Q₈

yes

complemented

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

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

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

—

C₂×Q₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₈

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

#2961

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₅₃

#568

yes

—

#533

D₅₃

yes

—

—

—

—

C₂×#97

—

C₅₃

#2961

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₅₃

yes

—

#125

Dic₅₃

yes

—

C₂²⋊Q₈

Dic₅₃

yes

—

C₄×Q₈

C₂×D₅₃

yes

—

#117

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

100

C₄²

C₂²×D₅₃

yes

—

101

C₄²

C₂²×D₅₃

yes

—

102

C₂³

C₂²×Dic₅₃

yes

characteristic

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

Q₈×D₅₃

yes

—

106

C₂×C₄

#11719949

yes

—

107

C₂×C₄

C₂²×Dic₅₃

yes

—

108

C₂×C₄

C₂²×Dic₅₃

yes

—

109

C₂×C₄

C₂²×Dic₅₃

yes

—

110

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

111

C₂×C₄

C₂³×D₅₃

yes

characteristic

112

C₈

—

113

C₈

—

106

—

114

C₈

—

106

—

115

C₂²

—

116

C₂²

Q₈×Dic₅₃

yes

—

117

C₂²

C₂³×Dic₅₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×Dic₅₃

yes

—

122

C₄

C₂×Q₈×D₅₃

yes

characteristic

123

C₄

C₂×#11719949

yes

characteristic

124

C₂

—

125

C₂

C₂×Q₈×Dic₅₃

yes

characteristic

126

C₂

#16811390

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#533	C₂
hypercenter	#533	D₅₃
hypocenter	C₅₃	#2961
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#533

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
34. C₂×#88322	Q₈	semidirect
70. C₅₃	#2961	semidirect

H ≅ G/N

type

1. #16810754

C₂

semidirect

34. C₂×#88322

Q₈

semidirect

70. C₅₃

#2961

semidirect

Subgroups — #20283656

Conjugacy classes of subgroups

Complements