Group #20274827 — subgroups

Subgroups — #20274827

Order 6784.

236 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16811101	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133136	C₂	1	1	yes	maximal, characteristic
3	#16810395	C₂	1	2	yes	maximal
4	#16810394	C₂	1	2	yes	maximal
5	C₂×#101×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	Q₈×Dic₅₃	—	2	2	no	—
8	Q₈×Dic₅₃	—	2	2	no	—
9	#13133176	C₄	1	4	yes	—
10	#13133176	C₂²	1	1	yes	characteristic
11	#13133176	C₂²	1	1	yes	characteristic
12	#13133136	—	2	2	no	—
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
15	#101×C₅₃	—	2	2	no	—
16	C₂×Q₈×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	D₄	1	2	yes	—
19	C₂×Q₈×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₅₃	—	2	2	no	—
23	M₁₆×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₅₃	—	2	4	no	—
28	C₄⋊Dic₅₃	—	2	1	no	—
29	C₄⋊Dic₅₃	—	2	1	no	—
30	C₄×Dic₅₃	—	2	4	no	—
31	C₄×Dic₅₃	—	2	1	no	—
32	C₄×Dic₅₃	—	2	1	no	—
33	#11719917	—	2	2	no	—
34	#11719917	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₅₃	—	2	4	no	—
38	Q₈×C₅₃	—	2	4	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	2	yes	—
45	C₂×C₄×C₅₃	#41	1	2	yes	—
46	C₂×Dic₅₃	—	2	2	no	—
47	C₂×Dic₅₃	—	2	2	no	—
48	C₈×C₅₃	—	4	2	no	—
49	#88322	—	4	2	no	—
50	C₂²×C₅₃	—	4	1	no	—
51	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₅₃	#117	1	1	yes	characteristic
53	C₂²×C₅₃	#117	1	1	yes	characteristic
54	C₄×C₅₃	—	2	4	no	—
55	C₄×C₅₃	#123	1	2	yes	—
56	C₄×C₅₃	#123	1	2	yes	—
57	Dic₅₃	—	4	2	no	—
58	Dic₅₃	—	4	2	no	—
59	#1830	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	—	2	1	no	—
61	C₂×C₅₃	—	2	1	no	—
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	#381	1	1	yes	characteristic
64	#519	—	53	1	no	—
65	C₂×#101	—	53	1	no	—
66	C₂×#101	D₅₃	1	1	yes	characteristic
67	#344	—	53	2	no	—
68	#344	—	53	2	no	—
69	C₅₃	#1830	1	1	yes	characteristic, complemented, Sylow
70	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
71	C₂×M₁₆	—	53	2	no	—
72	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
73	C₄×Q₈	—	106	2	no	—
74	C₄×Q₈	—	106	2	no	—
75	#117	—	53	4	no	—
76	#117	—	53	1	no	—
77	#117	—	53	1	no	—
78	#101	—	2	2	no	—
79	#101	—	106	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	D₂₁₂	1	2	yes	—
83	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
84	C₂²×C₄	C₄×D₅₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	106	2	no	—
89	M₁₆	—	106	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	106	2	no	—
92	C₄⋊C₄	—	106	1	no	—
93	C₄⋊C₄	—	106	1	no	—
94	C₄⋊C₄	—	106	4	no	—
95	C₂²⋊C₄	—	53	2	no	—
96	C₂²⋊C₄	—	53	2	no	—
97	C₄²	—	106	1	no	—
98	C₄²	—	106	1	no	—
99	C₄²	—	106	4	no	—
100	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	106	2	no	—
106	C₂×C₄	—	106	2	no	—
107	C₂×C₄	#11719921	1	4	yes	—
108	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
109	C₂×C₄	#11719947	1	2	yes	—
110	C₂×C₄	#11719947	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	212	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₂×#11719921	1	1	yes	characteristic
116	C₂²	#13133173	1	1	yes	characteristic
117	C₂²	#13133173	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	212	2	no	—
120	C₄	—	212	2	no	—
121	C₄	#13133198	1	2	yes	—
122	C₄	#13133201	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#16810484	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811101

C₂

yes

maximal, characteristic

C₂×#13133136

C₂

yes

maximal, characteristic

#16810395

C₂

yes

maximal

#16810394

C₂

yes

maximal

C₂×#101×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

Q₈×Dic₅₃

—

Q₈×Dic₅₃

—

#13133176

C₄

yes

—

#13133176

C₂²

yes

characteristic

#13133176

C₂²

yes

characteristic

#13133136

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

#101×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

#123

yes

—

C₄×C₅₃

#123

yes

—

Dic₅₃

—

Dic₅₃

—

#1830

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#344

—

#344

—

C₅₃

#1830

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

—

—

—

—

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₂₁₂

yes

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

—

106

—

100

C₂³

C₂×C₄×D₅₃

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

106

—

107

C₂×C₄

#11719921

yes

—

108

C₂×C₄

C₂×D₂₁₂

yes

characteristic

109

C₂×C₄

#11719947

yes

—

110

C₂×C₄

#11719947

yes

—

111

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

112

C₈

—

113

C₈

—

212

—

114

C₂²

—

115

C₂²

C₂×#11719921

yes

characteristic

116

C₂²

#13133173

yes

characteristic

117

C₂²

#13133173

yes

characteristic

118

C₄

—

119

C₄

—

212

—

120

C₄

—

212

—

121

C₄

#13133198

yes

—

122

C₄

#13133201

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#16810484

yes

characteristic

	subgroup	G/H
center	C₂	#16810484
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#101×C₅₃	C₂
socle	C₂×C₅₃	#381
hypercenter	C₂×#101	D₅₃
hypocenter	C₅₃	#1830
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#16810484

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#101×C₅₃

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
69. C₅₃	#1830	semidirect

H ≅ G/N

type

69. C₅₃

#1830

semidirect

Subgroups — #20274827

Conjugacy classes of subgroups

Complements