Group #27270916 — subgroups

Subgroups — #27270916

Order 13568.

230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20272548	C₂	1	2	yes	maximal
2	C₅₃×#1994	C₂	1	1	yes	maximal, characteristic
3	#16810676	—	2	2	no	—
4	C₅₃×#533	C₄	1	1	yes	characteristic
5	C₅₃×#423	C₄	1	1	yes	characteristic
6	C₄³×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#339	—	2	2	no	—
8	C₄⋊#88322	—	2	4	no	—
9	C₂²⋊#88322	—	4	2	no	—
10	C₄×#88322	—	4	2	no	—
11	C₂×M₁₆×C₅₃	—	2	1	no	—
12	C₄⋊Q₈×C₅₃	—	2	1	no	—
13	#125×C₅₃	C₈	1	2	yes	complemented
14	C₂²⋊Q₈×C₅₃	—	4	1	no	—
15	C₄×Q₈×C₅₃	—	4	1	no	—
16	#117×C₅₃	—	2	1	no	—
17	C₂×C₄²×C₅₃	—	4	1	no	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	D₄	1	2	yes	—
20	C₄⋊C₈×C₅₃	—	4	1	no	—
21	C₂×Q₈×C₅₃	—	4	1	no	—
22	C₂²×C₄×C₅₃	—	2	1	no	—
23	C₂²×C₄×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	M₁₆×C₅₃	—	4	2	no	—
26	C₂×C₈×C₅₃	—	4	1	no	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	—	4	1	no	—
29	C₄⋊C₄×C₅₃	—	4	2	no	—
30	C₂²⋊C₄×C₅₃	—	4	1	no	—
31	C₄²×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	4	4	no	—
34	C₄²×C₅₃	—	4	1	no	—
35	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
37	C₄²×C₅₃	SD₁₆	1	2	yes	—
38	C₄²×C₅₃	Q₁₆	1	2	yes	—
39	C₂×#88322	—	4	4	no	—
40	C₂³×C₅₃	#99	1	1	yes	characteristic
41	Q₈×C₅₃	—	8	2	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#103	1	2	yes	—
52	C₂×C₄×C₅₃	#104	1	2	yes	—
53	C₈×C₅₃	—	8	1	no	—
54	#88322	—	8	4	no	—
55	#7134	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	—	4	1	no	—
57	C₂²×C₅₃	#323	1	1	yes	characteristic
58	C₂²×C₅₃	#328	1	2	yes	—
59	C₄×C₅₃	—	2	4	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	4	1	no	—
62	C₄×C₅₃	—	4	2	no	—
63	C₄×C₅₃	—	8	2	no	—
64	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
65	C₄×C₅₃	#329	1	1	yes	characteristic
66	C₄×C₅₃	#333	1	1	yes	characteristic
67	#1994	D₅₃	1	1	yes	characteristic
68	#1276	—	53	2	no	—
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	#1272	1	1	yes	characteristic
72	#533	Dic₅₃	1	1	yes	characteristic
73	#431	—	106	2	no	—
74	#423	Dic₅₃	1	1	yes	characteristic
75	C₄³	C₂×D₅₃	1	1	yes	characteristic
76	#339	—	2	2	no	—
77	C₅₃	#7134	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₁₆	—	2	1	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	#125	#88322	1	2	yes	complemented
81	C₂²⋊Q₈	—	4	1	no	—
82	C₄×Q₈	—	4	1	no	—
83	#117	—	2	1	no	—
84	C₂×C₄²	—	4	1	no	—
85	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
86	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
87	C₄⋊C₈	—	4	1	no	—
88	C₄⋊C₈	—	106	4	no	—
89	C₂²⋊C₈	—	212	2	no	—
90	C₄×C₈	—	212	2	no	—
91	C₂×Q₈	—	4	1	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
95	M₁₆	—	4	2	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	212	4	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₂²⋊C₄	—	4	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	4	4	no	—
105	C₄²	—	4	1	no	—
106	C₄²	C₂×#88322	1	1	yes	characteristic
107	C₄²	#11719917	1	1	yes	characteristic
108	C₄²	#11719924	1	2	yes	—
109	C₄²	#11719925	1	2	yes	—
110	C₂³	#13133132	1	1	yes	characteristic
111	Q₈	—	8	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
120	C₂×C₄	#13133134	1	1	yes	characteristic
121	C₂×C₄	#13133141	1	2	yes	—
122	C₂×C₄	#13133144	1	2	yes	—
123	C₈	—	8	1	no	—
124	C₈	—	424	4	no	—
125	C₂²	—	4	1	no	—
126	C₂²	#16810347	1	1	yes	characteristic
127	C₂²	#16810360	1	2	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	2	no	—
132	C₄	—	8	2	no	—
133	C₄	Q₈⋊#88322	1	2	yes	—
134	C₄	#16810363	1	1	yes	characteristic
135	C₄	#16810372	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	2	2	no	—
138	C₂	#20272541	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272548

C₂

yes

maximal

C₅₃×#1994

C₂

yes

maximal, characteristic

#16810676

—

C₅₃×#533

C₄

yes

characteristic

C₅₃×#423

C₄

yes

characteristic

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#339

—

C₄⋊#88322

—

C₂²⋊#88322

—

C₄×#88322

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

#125×C₅₃

C₈

yes

complemented

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₄⋊C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₂×#88322

—

C₂³×C₅₃

#99

yes

characteristic

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

C₂²⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₈×C₅₃

—

#88322

—

#7134

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#323

yes

characteristic

C₂²×C₅₃

#328

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

Q₈⋊C₈

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1276

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1272

yes

characteristic

#533

Dic₅₃

yes

characteristic

#431

—

106

—

#423

Dic₅₃

yes

characteristic

C₄³

C₂×D₅₃

yes

characteristic

#339

—

C₅₃

#7134

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

#88322

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

212

—

C₄×C₈

—

212

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×#88322

yes

characteristic

107

C₄²

#11719917

yes

characteristic

108

C₄²

#11719924

yes

—

109

C₄²

#11719925

yes

—

110

C₂³

#13133132

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊#88322

yes

characteristic

120

C₂×C₄

#13133134

yes

characteristic

121

C₂×C₄

#13133141

yes

—

122

C₂×C₄

#13133144

yes

—

123

C₈

—

124

C₈

—

424

—

125

C₂²

—

126

C₂²

#16810347

yes

characteristic

127

C₂²

#16810360

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

Q₈⋊#88322

yes

—

134

C₄

#16810363

yes

characteristic

135

C₄

#16810372

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#20272541

yes

characteristic

	subgroup	G/H
center	C₄	#16810372
commutator	C₄²×C₅₃	C₂×C₈
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1994	C₂
socle	C₂×C₅₃	#1272
prime radical	C₂³×C₅₃	#99
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1994	D₅₃
hypocenter	C₅₃	#7134
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄³ ◁ #1994

subgroup

G/H

center

C₄

#16810372

commutator

C₄²×C₅₃

C₂×C₈

Frattini

C₄³

C₂×D₅₃

Fitting

C₅₃×#1994

C₂

C₂×C₅₃

C₂³×C₅₃

C₄³×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄³ ◁ #1994

N	H ≅ G/N	type
13. #125×C₅₃	C₈	semidirect
77. C₅₃	#7134	semidirect
80. #125	#88322	semidirect

H ≅ G/N

type

13. #125×C₅₃

C₈

semidirect

77. C₅₃

#7134

semidirect

80. #125

#88322

semidirect

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

subgroup	index
#20272548	2
C₅₃×#1994	2
#7134	53

subgroup

index

#20272548

C₅₃×#1994

#7134

Subgroups — #27270916

Conjugacy classes of subgroups

Complements

Maximal subgroups