Group #27274533 — subgroups

Subgroups — #27274533

Order 13568.

354 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 195 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	#20272998	C₂	1	2	yes	maximal
2	#20272546	C₂	1	2	yes	maximal
3	#20272544	C₂	1	2	yes	maximal
4	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₂²	1	4	yes	—
6	#16810344	C₄	1	4	yes	complemented
7	C₈×#88322	C₄	1	4	yes	complemented
8	C₂×#97×C₅₃	C₄	1	2	yes	—
9	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#376	C₂²	1	1	yes	characteristic
13	C₄⋊#88322	—	2	4	no	—
14	C₂²⋊#88322	—	2	4	no	—
15	C₄×#88322	C₂×C₄	1	8	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	#97×C₅₃	—	2	4	no	—
22	#97×C₅₃	—	2	2	no	—
23	C₄×C₈×C₅₃	C₈	1	8	yes	complemented
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	#95×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	—	2	2	no	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	#41	1	2	yes	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₄²	1	2	yes	—
35	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
38	C₄²×C₅₃	—	2	2	no	—
39	C₄²×C₅₃	C₂×C₈	1	4	yes	—
40	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
41	C₄²×C₅₃	#41	1	2	yes	—
42	C₂×#88322	—	2	8	no	—
43	C₂³×C₅₃	#117	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#117	1	4	yes	—
49	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
52	C₈×C₅₃	—	2	4	no	—
53	C₈×C₅₃	C₄×C₈	1	2	yes	—
54	C₈×C₅₃	C₄×C₈	1	2	yes	—
55	C₈×C₅₃	#97	1	4	yes	—
56	#88322	—	4	8	no	—
57	#8137	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	—	2	2	no	—
59	C₂²×C₅₃	#376	1	1	yes	characteristic
60	C₂²×C₅₃	#431	1	2	yes	—
61	C₄×C₅₃	—	2	4	no	—
62	C₄×C₅₃	—	4	4	no	—
63	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
65	C₄×C₅₃	#431	1	2	yes	—
66	#1726	D₅₃	1	1	yes	characteristic
67	#1450	—	53	2	no	—
68	#1275	—	53	2	no	—
69	#1274	—	53	2	no	—
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	#1744	1	1	yes	characteristic
72	C₂×C₅₃	#1747	1	2	yes	—
73	#431	—	53	4	no	—
74	C₂×#97	Dic₅₃	1	2	yes	—
75	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
76	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
77	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
78	#376	C₂×D₅₃	1	1	yes	characteristic
79	#322	—	53	4	no	—
80	C₈²	—	53	4	no	—
81	C₅₃	#8137	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
83	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
85	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
86	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
87	C₄⋊C₈	—	106	4	no	—
88	C₂²⋊C₈	—	106	4	no	—
89	#97	—	2	4	no	—
90	#97	—	2	2	no	—
91	C₄×C₈	—	53	8	no	—
92	C₄×C₈	#88322	1	8	yes	complemented
93	C₄×C₈	C₄×D₅₃	1	2	yes	—
94	C₄×C₈	C₄×D₅₃	1	2	yes	—
95	#95	—	2	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	#11719945	1	2	yes	—
98	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	106	8	no	—
102	C₂×C₈	C₂×#88322	1	4	yes	—
103	C₂×C₈	C₂×#88322	1	4	yes	—
104	C₂×C₈	#11719917	1	4	yes	—
105	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
106	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
107	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
108	C₄²	—	2	2	no	—
109	C₄²	C₂×#88322	1	4	yes	—
110	C₄²	C₂×C₄×D₅₃	1	2	yes	—
111	C₄²	#11719945	1	2	yes	—
112	C₂³	#13133176	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
117	C₂×C₄	#13133174	1	4	yes	—
118	C₂×C₄	#13133176	1	1	yes	characteristic
119	C₂×C₄	C₂²×#88322	1	2	yes	—
120	C₂×C₄	C₂×#11719917	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	212	8	no	—
123	C₈	C₄×#88322	1	2	yes	—
124	C₈	C₄×#88322	1	2	yes	—
125	C₈	#13133128	1	4	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	#16810469	1	1	yes	characteristic
128	C₂²	#16810676	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	C₂×C₄×#88322	1	1	yes	characteristic
132	C₄	C₂×#13133128	1	1	yes	characteristic
133	C₄	#16810676	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	#20274461	1	1	yes	characteristic
136	C₂	#20274470	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20272998

C₂

yes

maximal

#20272546

C₂

yes

maximal

#20272544

C₂

yes

maximal

C₅₃×#1726

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

#16810344

C₄

yes

complemented

C₈×#88322

C₄

yes

complemented

C₂×#97×C₅₃

C₄

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#376

C₂²

yes

—

—

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#97×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

C₈

yes

complemented

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

—

C₈×C₅₃

C₄×C₈

yes

—

C₈×C₅₃

#97

yes

—

#88322

—

#8137

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#376

yes

characteristic

C₂²×C₅₃

#431

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

C₂×#97

yes

characteristic

C₄×C₅₃

#431

yes

—

#1726

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1744

yes

characteristic

C₂×C₅₃

#1747

yes

—

#431

—

C₂×#97

Dic₅₃

yes

—

C₂×#97

C₂×D₅₃

yes

characteristic

C₂×C₄×C₈

Dic₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#376

C₂×D₅₃

yes

characteristic

#322

—

C₈²

—

C₅₃

#8137

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

#88322

yes

complemented

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

C₄×D₅₃

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11719945

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

106

—

102

C₂×C₈

C₂×#88322

yes

—

103

C₂×C₈

C₂×#88322

yes

—

104

C₂×C₈

#11719917

yes

—

105

C₂×C₈

C₄×Dic₅₃

yes

characteristic

106

C₂×C₈

C₄×Dic₅₃

yes

—

107

C₂×C₈

C₄×Dic₅₃

yes

characteristic

108

C₄²

—

109

C₄²

C₂×#88322

yes

—

110

C₄²

C₂×C₄×D₅₃

yes

—

111

C₄²

#11719945

yes

—

112

C₂³

#13133176

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×C₄×Dic₅₃

yes

characteristic

117

C₂×C₄

#13133174

yes

—

118

C₂×C₄

#13133176

yes

characteristic

119

C₂×C₄

C₂²×#88322

yes

—

120

C₂×C₄

C₂×#11719917

yes

—

121

C₈

—

122

C₈

—

212

—

123

C₈

C₄×#88322

yes

—

124

C₈

C₄×#88322

yes

—

125

C₈

#13133128

yes

—

126

C₂²

—

127

C₂²

#16810469

yes

characteristic

128

C₂²

#16810676

yes

—

129

C₄

—

130

C₄

—

131

C₄

C₂×C₄×#88322

yes

characteristic

132

C₄

C₂×#13133128

yes

characteristic

133

C₄

#16810676

yes

—

134

C₂

—

135

C₂

#20274461

yes

characteristic

136

C₂

#20274470

yes

—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1726	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#8137
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₄×Dic₅₃

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1726

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ #1726

N	H ≅ G/N	type
6. #16810344	C₄	semidirect
7. C₈×#88322	C₄	semidirect
23. C₄×C₈×C₅₃	C₈	semidirect
81. C₅₃	#8137	semidirect
92. C₄×C₈	#88322	semidirect

H ≅ G/N

type

6. #16810344

C₄

semidirect

7. C₈×#88322

C₄

semidirect

23. C₄×C₈×C₅₃

C₈

semidirect

81. C₅₃

#8137

semidirect

92. C₄×C₈

#88322

semidirect

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

subgroup	index
#20272998	2
#20272546	2
#20272544	2
C₅₃×#1726	2
#8137	53

subgroup

index

C₅₃×#1726

Subgroups — #27274533

Conjugacy classes of subgroups

Complements

Maximal subgroups