Group #20277435 — subgroups

Subgroups — #20277435

Order 6784.

516 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q₈×Dic₅₃	C₂	1	2	yes	maximal
2	#16810530	C₂	1	2	yes	maximal
3	#16810509	C₂	1	8	yes	maximal
4	#16810489	C₂	1	1	yes	maximal, characteristic
5	#16810469	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	Q₈×Dic₅₃	—	2	8	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
13	#13133124	C₂²	1	8	yes	—
14	#13133124	C₂²	1	4	yes	—
15	#13133124	C₂²	1	4	yes	—
16	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₅₃	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	2	4	no	—
21	C₂×Q₈×C₅₃	—	2	4	no	—
22	C₂×Q₈×C₅₃	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₃	C₂³	1	2	yes	—
27	C₂²×Dic₅₃	C₂³	1	4	yes	—
28	C₂²×Dic₅₃	C₂³	1	2	yes	—
29	C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
30	C₄²×C₅₃	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₅₃	—	2	4	no	—
32	C₄⋊Dic₅₃	—	2	8	no	—
33	C₄×Dic₅₃	—	2	2	no	—
34	C₄×Dic₅₃	—	2	8	no	—
35	C₄×Dic₅₃	D₄	1	4	yes	—
36	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
37	Q₈×C₅₃	—	2	16	no	—
38	C₂×C₄×C₅₃	—	2	4	no	—
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₃	#41	1	2	yes	—
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	—	2	4	no	—
48	C₂×Dic₅₃	—	2	2	no	—
49	C₂×Dic₅₃	—	2	8	no	—
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
52	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
55	C₂²×C₅₃	2₋¹⁺⁴	1	4	yes	—
56	C₄×C₅₃	—	2	8	no	—
57	C₄×C₅₃	—	2	4	no	—
58	C₄×C₅₃	C₄×D₄	1	4	yes	—
59	Dic₅₃	—	2	4	no	—
60	Dic₅₃	—	4	4	no	—
61	Dic₅₃	—	4	2	no	—
62	#2327	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
64	C₂×C₅₃	#519	1	2	yes	—
65	C₂×C₅₃	#536	1	2	yes	—
66	C₂×C₅₃	#541	1	2	yes	—
67	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
68	C₂×C₄×Q₈	—	53	2	no	—
69	#391	—	53	2	no	—
70	#387	—	53	8	no	—
71	#382	—	53	1	no	—
72	#376	—	53	1	no	—
73	C₅₃	#2327	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
75	C₄⋊Q₈	Dic₅₃	1	8	yes	complemented
76	C₄×Q₈	—	106	8	no	—
77	C₂×C₄⋊C₄	—	53	2	no	—
78	C₂×C₄⋊C₄	—	53	4	no	—
79	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
80	C₂×C₄²	—	53	1	no	—
81	C₂×C₄²	—	53	4	no	—
82	C₂×C₄²	—	53	1	no	—
83	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
84	#95	—	53	4	no	—
85	#95	—	53	4	no	—
86	#95	—	53	8	no	—
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	4	no	—
89	C₂×Q₈	C₂×Dic₅₃	1	8	yes	—
90	C₂²×C₄	—	53	2	no	—
91	C₂²×C₄	—	53	2	no	—
92	C₂²×C₄	—	53	4	no	—
93	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
94	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
95	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
96	C₄⋊C₄	—	106	8	no	—
97	C₄⋊C₄	—	106	4	no	—
98	C₄⋊C₄	C₂×Dic₅₃	1	16	yes	—
99	C₄²	—	53	4	no	—
100	C₄²	—	106	8	no	—
101	C₄²	—	106	2	no	—
102	C₄²	C₂×Dic₅₃	1	4	yes	—
103	C₂³	C₂³×D₅₃	1	1	yes	characteristic
104	Q₈	—	2	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	53	4	no	—
109	C₂×C₄	—	106	4	no	—
110	C₂×C₄	—	106	4	no	—
111	C₂×C₄	—	106	2	no	—
112	C₂×C₄	—	106	8	no	—
113	C₂×C₄	—	106	4	no	—
114	C₂×C₄	D₄×D₅₃	1	2	yes	—
115	C₂×C₄	#11719947	1	2	yes	—
116	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
117	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
118	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
119	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
120	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
121	C₂²	C₂×#11719947	1	1	yes	characteristic
122	C₂²	#13133265	1	4	yes	—
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	106	4	no	—
126	C₄	—	212	4	no	—
127	C₄	—	212	2	no	—
128	C₄	D₄×Dic₅₃	1	4	yes	—
129	C₂	C₂×D₄×Dic₅₃	1	1	yes	characteristic
130	C₂	#16811101	1	2	yes	—
131	C₂	#16811198	1	2	yes	—
132	C₂	#16811234	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅₃

C₂

yes

maximal

#16810530

C₂

yes

maximal

#16810509

C₂

yes

maximal

#16810489

C₂

yes

maximal, characteristic

#16810469

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

Q₈×Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×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

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2327

—

maximal, Sylow

C₂×C₅₃

C₂×C₄×D₄

yes

characteristic

C₂×C₅₃

#519

yes

—

C₂×C₅₃

#536

yes

—

C₂×C₅₃

#541

yes

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

C₅₃

#2327

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₄⋊Q₈

Dic₅₃

yes

complemented

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

—

100

C₄²

—

106

—

101

C₄²

—

106

—

102

C₄²

C₂×Dic₅₃

yes

—

103

C₂³

C₂³×D₅₃

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

106

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

106

—

113

C₂×C₄

—

106

—

114

C₂×C₄

D₄×D₅₃

yes

—

115

C₂×C₄

#11719947

yes

—

116

C₂×C₄

C₂²×Dic₅₃

yes

—

117

C₂×C₄

C₂²×Dic₅₃

yes

—

118

C₂×C₄

C₂²×Dic₅₃

yes

—

119

C₂²

C₂³×Dic₅₃

yes

characteristic

120

C₂²

C₂×D₄×D₅₃

yes

characteristic

121

C₂²

C₂×#11719947

yes

characteristic

122

C₂²

#13133265

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

106

—

126

C₄

—

212

—

127

C₄

—

212

—

128

C₄

D₄×Dic₅₃

yes

—

129

C₂

C₂×D₄×Dic₅₃

yes

characteristic

130

C₂

#16811101

yes

—

131

C₂

#16811198

yes

—

132

C₂

#16811234

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#2327
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
17. C₄⋊Q₈×C₅₃	C₄	semidirect
73. C₅₃	#2327	semidirect
75. C₄⋊Q₈	Dic₅₃	semidirect

H ≅ G/N

type

17. C₄⋊Q₈×C₅₃

C₄

semidirect

73. C₅₃

#2327

semidirect

75. C₄⋊Q₈

Dic₅₃

semidirect

Subgroups — #20277435

Conjugacy classes of subgroups

Complements