Group #27215349 — subgroups

Subgroups — C₅₃×#7837 #27215349

Order 13568.

414 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₅₃×#2811	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2095	C₂	1	3	yes	maximal
3	C₅₃×#1272	C₂	1	3	yes	maximal
4	C₂×#107×C₅₃	—	2	3	no	—
5	C₂×#103×C₅₃	—	2	3	no	—
6	C₅₃×#395	—	2	3	no	—
7	C₅₃×#391	—	2	3	no	—
8	C₅₃×#391	C₂²	1	3	yes	—
9	C₅₃×#384	C₂²	1	1	yes	characteristic
10	C₅₃×#336	C₂²	1	3	yes	—
11	C₂²×Q₈×C₅₃	—	2	1	no	—
12	C₂²×Q₈×C₅₃	D₄	1	3	yes	—
13	C₂²×C₈×C₅₃	—	2	3	no	—
14	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
15	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
16	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
17	C₂×C₄²×C₅₃	—	2	1	no	—
18	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
19	#107×C₅₃	—	4	6	no	—
20	#103×C₅₃	—	4	6	no	—
21	#95×C₅₃	—	2	3	no	—
22	#95×C₅₃	—	2	6	no	—
23	C₂×Q₈×C₅₃	—	2	3	no	—
24	C₂×Q₈×C₅₃	—	4	3	no	—
25	C₂×Q₈×C₅₃	—	4	3	no	—
26	C₂×Q₈×C₅₃	—	8	3	no	—
27	C₂×Q₈×C₅₃	Q₁₆	1	6	yes	—
28	C₂²×C₄×C₅₃	—	2	3	no	—
29	C₂²×C₄×C₅₃	—	2	3	no	—
30	C₂²×C₄×C₅₃	—	4	1	no	—
31	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
32	C₂×C₈×C₅₃	—	2	6	no	—
33	C₂×C₈×C₅₃	—	4	3	no	—
34	C₂×C₈×C₅₃	—	4	3	no	—
35	C₄⋊C₄×C₅₃	—	4	3	no	—
36	C₄⋊C₄×C₅₃	—	4	3	no	—
37	C₄⋊C₄×C₅₃	—	4	3	no	—
38	C₄⋊C₄×C₅₃	—	8	3	no	—
39	C₄²×C₅₃	—	4	1	no	—
40	C₄²×C₅₃	—	8	1	no	—
41	C₂³×C₅₃	#120	1	1	yes	characteristic
42	Q₈×C₅₃	—	4	6	no	—
43	Q₈×C₅₃	—	8	3	no	—
44	Q₈×C₅₃	—	8	4	no	—
45	C₂×C₄×C₅₃	—	2	3	no	—
46	C₂×C₄×C₅₃	—	2	3	no	—
47	C₂×C₄×C₅₃	—	2	6	no	—
48	C₂×C₄×C₅₃	—	4	3	no	—
49	C₂×C₄×C₅₃	—	4	3	no	—
50	C₂×C₄×C₅₃	—	4	3	no	—
51	C₂×C₄×C₅₃	—	4	6	no	—
52	C₂×C₄×C₅₃	—	8	3	no	—
53	C₂×C₄×C₅₃	C₂×Q₁₆	1	3	yes	—
54	C₂×C₄×C₅₃	#137	1	3	yes	—
55	C₈×C₅₃	—	4	6	no	—
56	#7837	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₅₃	#451	1	3	yes	—
58	C₂²×C₅₃	#453	1	3	yes	—
59	C₂²×C₅₃	#458	1	1	yes	characteristic
60	C₄×C₅₃	—	2	3	no	—
61	C₄×C₅₃	—	2	3	no	—
62	C₄×C₅₃	—	4	6	no	—
63	C₄×C₅₃	—	8	3	no	—
64	C₄×C₅₃	—	16	1	no	—
65	#2811	C₂×C₅₃	1	1	yes	characteristic
66	#2095	C₂×C₅₃	1	3	yes	—
67	#1272	C₂×C₅₃	1	3	yes	—
68	C₂×C₅₃	#1614	1	3	yes	—
69	C₂×C₅₃	#1633	1	3	yes	—
70	C₂×C₅₃	#1655	1	1	yes	characteristic
71	C₂×#107	—	2	3	no	—
72	C₂×#103	—	2	3	no	—
73	#395	—	2	3	no	—
74	#391	—	2	3	no	—
75	#391	C₂²×C₅₃	1	3	yes	—
76	#384	C₂²×C₅₃	1	1	yes	characteristic
77	#336	C₂²×C₅₃	1	3	yes	—
78	C₅₃	#7837	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²×Q₈	—	2	1	no	—
80	C₂²×Q₈	D₄×C₅₃	1	3	yes	—
81	C₂²×C₈	—	2	3	no	—
82	C₂×C₄⋊C₄	—	2	3	no	—
83	C₂×C₄⋊C₄	—	2	3	no	—
84	C₂×C₄⋊C₄	D₄×C₅₃	1	3	yes	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
87	#107	—	4	6	no	—
88	#103	—	4	6	no	—
89	#95	—	2	3	no	—
90	#95	—	2	6	no	—
91	C₂×Q₈	—	2	3	no	—
92	C₂×Q₈	—	4	3	no	—
93	C₂×Q₈	—	4	3	no	—
94	C₂×Q₈	—	8	3	no	—
95	C₂×Q₈	Q₁₆×C₅₃	1	6	yes	—
96	C₂²×C₄	—	2	3	no	—
97	C₂²×C₄	—	2	3	no	—
98	C₂²×C₄	—	4	1	no	—
99	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
100	C₂×C₈	—	2	6	no	—
101	C₂×C₈	—	4	3	no	—
102	C₂×C₈	—	4	3	no	—
103	C₄⋊C₄	—	4	3	no	—
104	C₄⋊C₄	—	4	3	no	—
105	C₄⋊C₄	—	4	3	no	—
106	C₄⋊C₄	—	8	3	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	8	1	no	—
109	C₂³	#120×C₅₃	1	1	yes	characteristic
110	Q₈	—	4	6	no	—
111	Q₈	—	8	3	no	—
112	Q₈	—	8	4	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	4	3	no	—
117	C₂×C₄	—	4	3	no	—
118	C₂×C₄	—	4	3	no	—
119	C₂×C₄	—	4	6	no	—
120	C₂×C₄	—	8	3	no	—
121	C₂×C₄	C₂×Q₁₆×C₅₃	1	3	yes	—
122	C₂×C₄	#137×C₅₃	1	3	yes	—
123	C₈	—	4	6	no	—
124	C₂²	C₅₃×#451	1	3	yes	—
125	C₂²	C₅₃×#453	1	3	yes	—
126	C₂²	C₅₃×#458	1	1	yes	characteristic
127	C₄	—	2	3	no	—
128	C₄	—	2	3	no	—
129	C₄	—	4	6	no	—
130	C₄	—	8	3	no	—
131	C₄	—	16	1	no	—
132	C₂	C₅₃×#1614	1	3	yes	—
133	C₂	C₅₃×#1633	1	3	yes	—
134	C₂	C₅₃×#1655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2811

C₂

yes

maximal, characteristic

C₅₃×#2095

C₂

yes

maximal

C₅₃×#1272

C₂

yes

maximal

C₂×#107×C₅₃

—

C₂×#103×C₅₃

—

C₅₃×#395

—

C₅₃×#391

—

C₅₃×#391

C₂²

yes

—

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#336

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

#103×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

#7837

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

#451

yes

—

C₂²×C₅₃

#453

yes

—

C₂²×C₅₃

#458

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#2811

C₂×C₅₃

yes

characteristic

#2095

C₂×C₅₃

yes

—

#1272

C₂×C₅₃

yes

—

C₂×C₅₃

#1614

yes

—

C₂×C₅₃

#1633

yes

—

C₂×C₅₃

yes

—

—

—

—

C₂²×C₅₃

yes

—

#384

C₂²×C₅₃

yes

characteristic

#336

C₂²×C₅₃

yes

—

C₅₃

#7837

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

—

C₂²×Q₈

D₄×C₅₃

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³×C₅₃

yes

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₁₆×C₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₅₃

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₂³

#120×C₅₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂×Q₁₆×C₅₃

yes

—

122

C₂×C₄

#137×C₅₃

yes

—

123

C₈

—

124

C₂²

C₅₃×#451

yes

—

125

C₂²

C₅₃×#453

yes

—

126

C₂²

C₅₃×#458

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₂

C₅₃×#1614

yes

—

133

C₂

C₅₃×#1633

yes

—

134

C₂

C₅₃×#1655

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅₃

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

LCS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
56. #7837	C₅₃	direct (N × H)
78. C₅₃	#7837	direct (N × H)

H ≅ G/N

type

56. #7837

C₅₃

direct (N × H)

78. C₅₃

#7837

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#2811	2
C₅₃×#2095	2
C₅₃×#1272	2
#7837	53

subgroup

index

C₅₃×#2811

C₅₃×#2095

C₅₃×#1272

#7837

Subgroups — C₅₃×#7837 #27215349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7837 #27215349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7837 #27215349