Group #27410647 — subgroups

Subgroups — #27410647

Order 13568.

912 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 255 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	#20275013	C₂	1	12	yes	maximal, complemented
2	#20274858	C₂	1	2	yes	maximal, complemented
3	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₂²	1	4	yes	—
5	C₂×C₂²⋊#88322	C₂²	1	12	yes	—
6	#16810378	C₂²	1	12	yes	—
7	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₅₃	C₂²	1	6	yes	—
9	C₅₃×#390	C₄	1	6	yes	—
10	C₅₃×#384	C₄	1	1	yes	characteristic
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#11719917	—	4	12	no	—
13	C₂²×#88322	C₂³	1	8	yes	—
14	C₂²⋊#88322	—	4	12	no	—
15	C₂²×D₄×C₅₃	C₂×C₄	1	6	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
17	#127×C₅₃	—	2	4	no	—
18	C₄×D₄×C₅₃	—	4	6	no	—
19	C₄×D₄×C₅₃	—	4	6	no	—
20	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
21	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	12	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
25	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
26	C₂×D₄×C₅₃	—	2	12	no	—
27	C₂×D₄×C₅₃	—	4	6	no	—
28	C₂×D₄×C₅₃	—	4	6	no	—
29	C₂²×C₄×C₅₃	—	2	12	no	—
30	C₂²×C₄×C₅₃	—	2	12	no	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
34	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	4	6	no	—
36	C₂²⋊C₄×C₅₃	—	4	12	no	—
37	C₄²×C₅₃	—	2	6	no	—
38	C₄²×C₅₃	—	4	3	no	—
39	C₄²×C₅₃	M₁₆	1	4	yes	complemented
40	#11719917	—	4	16	no	—
41	C₂×#88322	—	2	24	no	—
42	C₂³×C₅₃	—	2	12	no	—
43	C₂³×C₅₃	—	2	4	no	—
44	C₂³×C₅₃	—	2	12	no	—
45	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
46	D₄×C₅₃	—	4	24	no	—
47	C₂×C₄×C₅₃	—	2	3	no	—
48	C₂×C₄×C₅₃	—	2	3	no	—
49	C₂×C₄×C₅₃	—	2	12	no	—
50	C₂×C₄×C₅₃	—	2	3	no	—
51	C₂×C₄×C₅₃	—	2	6	no	—
52	C₂×C₄×C₅₃	—	4	12	no	—
53	C₂×C₄×C₅₃	—	4	4	no	—
54	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
55	C₂×C₄×C₅₃	H(5,𝔽₂)	1	6	yes	—
56	#88322	—	4	8	no	—
57	#23357	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	—	4	4	no	—
59	C₂²×C₅₃	—	4	12	no	—
60	C₂²×C₅₃	—	4	12	no	—
61	C₂²×C₅₃	#518	1	3	yes	—
62	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
63	C₂²×C₅₃	#568	1	3	yes	—
64	C₄×C₅₃	—	2	6	no	—
65	C₄×C₅₃	—	4	4	no	—
66	C₄×C₅₃	—	4	3	no	—
67	C₄×C₅₃	#560	1	4	yes	—
68	C₄×#127	D₅₃	1	1	yes	characteristic
69	#1870	—	53	12	no	—
70	#1839	—	53	2	no	—
71	C₂×C₅₃	—	8	4	no	—
72	C₂×C₅₃	#2380	1	1	yes	characteristic
73	C₂×C₅₃	#2975	1	3	yes	—
74	C₂×C₅₃	#2980	1	3	yes	—
75	C₂²×M₁₆	—	53	4	no	—
76	C₂×#127	Dic₅₃	1	1	yes	characteristic
77	C₂×C₄×D₄	C₂×D₅₃	1	6	yes	—
78	C₂×C₂²⋊C₈	—	53	12	no	—
79	#390	Dic₅₃	1	6	yes	—
80	#384	Dic₅₃	1	1	yes	characteristic
81	C₄³	C₂×D₅₃	1	1	yes	characteristic
82	#336	—	53	12	no	—
83	C₅₃	#23357	1	1	yes	characteristic, complemented, Sylow
84	C₂²×D₄	C₂×Dic₅₃	1	6	yes	—
85	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
86	C₂×M₁₆	—	212	12	no	—
87	C₂²×C₈	—	53	8	no	—
88	#127	—	2	4	no	—
89	C₄×D₄	—	4	6	no	—
90	C₄×D₄	—	4	6	no	—
91	C₂×C₄⋊C₄	C₂×Dic₅₃	1	6	yes	—
92	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	12	yes	—
93	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
94	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
96	C₂²⋊C₈	—	212	12	no	—
97	C₂⁴	C₂²×Dic₅₃	1	4	yes	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	—	4	6	no	—
100	C₂×D₄	—	4	6	no	—
101	C₂²×C₄	—	2	12	no	—
102	C₂²×C₄	—	2	12	no	—
103	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
104	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
105	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
106	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
107	M₁₆	—	212	16	no	—
108	C₂×C₈	—	106	24	no	—
109	C₄⋊C₄	—	4	6	no	—
110	C₂²⋊C₄	—	4	12	no	—
111	C₄²	—	2	6	no	—
112	C₄²	—	4	3	no	—
113	C₄²	#11719917	1	4	yes	complemented
114	C₂³	—	2	12	no	—
115	C₂³	—	2	4	no	—
116	C₂³	—	2	12	no	—
117	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
118	D₄	—	4	24	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	2	3	no	—
121	C₂×C₄	—	2	3	no	—
122	C₂×C₄	—	2	12	no	—
123	C₂×C₄	—	2	6	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	4	12	no	—
126	C₂×C₄	C₂×#11719917	1	6	yes	—
127	C₂×C₄	#13133262	1	6	yes	—
128	C₈	—	212	8	no	—
129	C₂²	—	4	12	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	4	12	no	—
132	C₂²	#16811098	1	3	yes	—
133	C₂²	C₂²×#11719917	1	1	yes	characteristic
134	C₂²	#16811390	1	3	yes	—
135	C₄	—	2	6	no	—
136	C₄	—	4	4	no	—
137	C₄	—	4	3	no	—
138	C₄	#16811360	1	4	yes	—
139	C₂	—	8	4	no	—
140	C₂	#20277892	1	1	yes	characteristic
141	C₂	#20283756	1	3	yes	—
142	C₂	#20283786	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275013

C₂

yes

maximal, complemented

#20274858

C₂

yes

maximal, complemented

C₄×#127×C₅₃

C₂

yes

maximal, characteristic

C₂²×#11719917

C₂²

yes

—

C₂×C₂²⋊#88322

C₂²

yes

—

#16810378

C₂²

yes

—

C₂×#127×C₅₃

C₄

yes

characteristic

C₂×C₄×D₄×C₅₃

C₂²

yes

—

C₅₃×#390

C₄

yes

—

C₅₃×#384

C₄

yes

characteristic

C₄³×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

C₂²×#88322

C₂³

yes

—

C₂²⋊#88322

—

C₂²×D₄×C₅₃

C₂×C₄

yes

—

C₂³×C₄×C₅₃

C₂³

yes

—

#127×C₅₃

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂²×C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

M₁₆

yes

complemented

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

D₄×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×M₁₆

yes

—

C₂×C₄×C₅₃

H(5,𝔽₂)

yes

—

#88322

—

#23357

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#518

yes

—

C₂²×C₅₃

C₂²×M₁₆

yes

characteristic

C₂²×C₅₃

#568

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#560

yes

—

C₄×#127

D₅₃

yes

characteristic

#1870

—

#1839

—

C₂×C₅₃

—

C₂×C₅₃

#2380

yes

characteristic

C₂×C₅₃

#2975

yes

—

C₂×C₅₃

#2980

yes

—

C₂²×M₁₆

—

C₂×#127

Dic₅₃

yes

characteristic

C₂×C₄×D₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₈

—

#390

Dic₅₃

yes

—

#384

Dic₅₃

yes

characteristic

C₄³

C₂×D₅₃

yes

characteristic

#336

—

C₅₃

#23357

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×Dic₅₃

yes

—

C₂³×C₄

C₂²×D₅₃

yes

—

C₂×M₁₆

—

212

—

C₂²×C₈

—

#127

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂×C₂²⋊C₄

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂²×D₅₃

yes

—

C₂²⋊C₈

—

212

—

C₂⁴

C₂²×Dic₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×Dic₅₃

yes

—

104

C₂²×C₄

C₂²×Dic₅₃

yes

—

105

C₂²×C₄

C₂²×Dic₅₃

yes

—

106

C₂²×C₄

C₂³×D₅₃

yes

characteristic

107

M₁₆

—

212

—

108

C₂×C₈

—

106

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

#11719917

yes

complemented

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂³×Dic₅₃

yes

characteristic

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×#11719917

yes

—

127

C₂×C₄

#13133262

yes

—

128

C₈

—

212

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#16811098

yes

—

133

C₂²

C₂²×#11719917

yes

characteristic

134

C₂²

#16811390

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#16811360

yes

—

139

C₂

—

140

C₂

#20277892

yes

characteristic

141

C₂

#20283756

yes

—

142

C₂

#20283786

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₄×#127×C₅₃	C₂
prime radical	C₂×#127×C₅₃	C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#127	D₅₃
hypocenter	C₅₃	#23357
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#127

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂³×C₄

Fitting

C₄×#127×C₅₃

C₂

prime radical

C₂×#127×C₅₃

C₄

cosocle

C₂²×C₄×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₄×#127

N	H ≅ G/N	type
1. #20275013	C₂	semidirect
2. #20274858	C₂	semidirect
39. C₄²×C₅₃	M₁₆	semidirect
83. C₅₃	#23357	semidirect
113. C₄²	#11719917	semidirect

H ≅ G/N

type

1. #20275013

C₂

semidirect

2. #20274858

C₂

semidirect

39. C₄²×C₅₃

M₁₆

semidirect

83. C₅₃

#23357

semidirect

113. C₄²

#11719917

semidirect

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

subgroup	index
#20275013	2
#20274858	2
C₄×#127×C₅₃	2
#23357	53

subgroup

index

#20275013

#20274858

C₄×#127×C₅₃

#23357

Subgroups — #27410647

Conjugacy classes of subgroups

Complements

Maximal subgroups