Group #27300512 — subgroups

Subgroups — #27300512

Order 13568.

342 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16810444	C₂	1	4	yes	maximal
2	#20276471	C₂	1	2	yes	maximal
3	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	C₂²	1	4	yes	—
5	#16810444	—	2	8	no	—
6	C₄⋊#11719908	—	2	4	no	—
7	C₂×#97×C₅₃	C₄	1	2	yes	—
8	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#376	C₄	1	1	yes	characteristic
12	#13133157	—	2	8	no	—
13	C₂×#11719908	—	2	4	no	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#97×C₅₃	—	2	4	no	—
20	#97×C₅₃	C₈	1	4	yes	—
21	C₄×C₈×C₅₃	D₄	1	2	yes	—
22	C₄×C₈×C₅₃	D₄	1	4	yes	—
23	C₄×C₈×C₅₃	Q₈	1	4	yes	—
24	C₄×C₈×C₅₃	Q₈	1	2	yes	—
25	#95×C₅₃	—	2	2	no	—
26	C₂²×C₄×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
34	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
35	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₅₃	—	2	2	no	—
39	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₄²×C₅₃	C₂×C₈	1	2	yes	—
42	#11719908	—	4	4	no	—
43	C₂³×C₅₃	C₂×M₁₆	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	C₄⋊C₈	1	4	yes	—
48	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
52	C₈×C₅₃	—	2	2	no	—
53	C₈×C₅₃	C₄⋊C₈	1	8	yes	—
54	C₈×C₅₃	#127	1	1	yes	characteristic
55	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
56	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
57	#12901	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	—	2	2	no	—
59	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₂²×C₅₃	#423	1	2	yes	—
61	C₄×C₅₃	—	2	4	no	—
62	C₄×C₅₃	—	4	4	no	—
63	C₄×C₅₃	#384	1	1	yes	characteristic
64	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
65	C₄×C₅₃	#423	1	1	yes	characteristic
66	C₂×#364	—	53	4	no	—
67	#2150	—	53	2	no	—
68	#1726	D₅₃	1	1	yes	characteristic
69	C₂×C₅₃	—	2	2	no	—
70	C₂×C₅₃	#1842	1	1	yes	characteristic
71	C₂×C₅₃	#2153	1	2	yes	—
72	C₂×M₃₂	—	53	4	no	—
73	C₂×#97	Dic₅₃	1	2	yes	—
74	C₂×#97	Dic₅₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
76	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
77	#376	Dic₅₃	1	1	yes	characteristic
78	#364	—	106	8	no	—
79	C₄⋊C₁₆	—	106	4	no	—
80	C₅₃	#12901	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
83	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
84	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
85	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
86	M₃₂	—	106	8	no	—
87	C₂×C₁₆	—	106	4	no	—
88	#97	—	2	4	no	—
89	#97	#88322	1	4	yes	—
90	C₄×C₈	Dic₁₀₆	1	4	yes	—
91	C₄×C₈	Dic₁₀₆	1	2	yes	—
92	C₄×C₈	D₂₁₂	1	4	yes	—
93	C₄×C₈	D₂₁₂	1	2	yes	—
94	#95	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#11719917	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₂×#88322	1	4	yes	—
100	C₂×C₈	#11719917	1	4	yes	—
101	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
102	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
103	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
104	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
105	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
106	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
107	C₄²	—	2	2	no	—
108	C₄²	C₂×#88322	1	2	yes	—
109	C₄²	C₄⋊Dic₅₃	1	4	yes	—
110	C₄²	C₄⋊Dic₅₃	1	2	yes	—
111	C₁₆	—	212	4	no	—
112	C₂³	C₂×#11719917	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	C₄⋊#88322	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
119	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
120	C₂×C₄	C₂×#11719917	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	C₄⋊#88322	1	8	yes	—
123	C₈	#13133214	1	1	yes	characteristic
124	C₈	#13133217	1	2	yes	—
125	C₈	#13133217	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
128	C₂²	#16810647	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#16810493	1	1	yes	characteristic
132	C₄	C₂×C₄⋊#88322	1	2	yes	—
133	C₄	#16810647	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#20274867	1	1	yes	characteristic
136	C₂	#20276481	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810444

C₂

yes

maximal

#20276471

C₂

yes

maximal

C₅₃×#1726

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

#16810444

—

C₄⋊#11719908

—

C₂×#97×C₅₃

C₄

yes

—

C₂×#97×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#376

C₄

yes

characteristic

#13133157

—

C₂×#11719908

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#97×C₅₃

—

#97×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

#11719908

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

#127

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

—

C₈×C₅₃

C₄⋊Q₈

yes

characteristic

#12901

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₃

#423

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#384

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₈

yes

—

C₄×C₅₃

yes

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1842

yes

characteristic

C₂×C₅₃

#2153

yes

—

C₂×M₃₂

—

C₂×#97

Dic₅₃

yes

—

C₂×#97

Dic₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#376

Dic₅₃

yes

characteristic

#364

—

106

—

C₄⋊C₁₆

—

106

—

C₅₃

#12901

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

M₃₂

—

106

—

C₂×C₁₆

—

106

—

#97

—

#97

#88322

yes

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

D₂₁₂

yes

—

C₄×C₈

D₂₁₂

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11719917

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×#88322

yes

—

100

C₂×C₈

#11719917

yes

—

101

C₂×C₈

C₄⋊Dic₅₃

yes

—

102

C₂×C₈

C₄⋊Dic₅₃

yes

—

103

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

104

C₂×C₈

C₂×Dic₁₀₆

yes

—

105

C₂×C₈

C₂×D₂₁₂

yes

—

106

C₂×C₈

C₂×D₂₁₂

yes

characteristic

107

C₄²

—

108

C₄²

C₂×#88322

yes

—

109

C₄²

C₄⋊Dic₅₃

yes

—

110

C₄²

C₄⋊Dic₅₃

yes

—

111

C₁₆

—

212

—

112

C₂³

C₂×#11719917

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄⋊#88322

yes

—

117

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

—

118

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

119

C₂×C₄

C₂²×#88322

yes

characteristic

120

C₂×C₄

C₂×#11719917

yes

—

121

C₈

—

122

C₈

C₄⋊#88322

yes

—

123

C₈

#13133214

yes

characteristic

124

C₈

#13133217

yes

—

125

C₈

#13133217

yes

characteristic

126

C₂²

—

127

C₂²

C₂×C₄⋊#88322

yes

characteristic

128

C₂²

#16810647

yes

—

129

C₄

—

130

C₄

—

131

C₄

#16810493

yes

characteristic

132

C₄

C₂×C₄⋊#88322

yes

—

133

C₄

#16810647

yes

characteristic

134

C₂

—

135

C₂

#20274867

yes

characteristic

136

C₂

#20276481

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1726	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#12901
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1726

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

N	H ≅ G/N	type
80. C₅₃	#12901	semidirect

H ≅ G/N

type

80. C₅₃

#12901

semidirect

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

subgroup	index
C₂×#16810444	2
#20276471	2
C₅₃×#1726	2
#12901	53

subgroup

index

C₂×#16810444

#20276471

C₅₃×#1726

#12901

Subgroups — #27300512

Conjugacy classes of subgroups

Complements

Maximal subgroups