Group #27304884 — subgroups

Subgroups — #27304884

Order 13568.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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×#16810463	C₂	1	2	yes	maximal
2	C₄×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13133123	C₂	1	1	yes	maximal, characteristic
4	#20272896	C₂	1	2	yes	maximal
5	C₅₃×#2247	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
7	C₄×Dic₂₁₂	—	2	1	no	—
8	#16810463	—	2	4	no	—
9	C₂×#13133123	C₂²	1	2	yes	—
10	#16810449	C₂²	1	1	yes	characteristic
11	#16810428	—	2	1	no	—
12	C₂×Q₃₂×C₅₃	—	2	1	no	—
13	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#494	—	2	1	no	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₂	—	2	1	no	—
17	C₄×Dic₁₀₆	—	4	1	no	—
18	Dic₄₂₄	—	2	1	no	—
19	Dic₄₂₄	D₄	1	2	yes	—
20	#13133151	—	2	1	no	—
21	#13133142	—	4	1	no	—
22	#13133123	—	2	2	no	—
23	C₂×Q₁₆×C₅₃	—	2	2	no	—
24	C₂×Q₁₆×C₅₃	—	4	1	no	—
25	C₄⋊Q₈×C₅₃	—	4	1	no	—
26	Q₃₂×C₅₃	—	2	2	no	—
27	Q₃₂×C₅₃	—	2	2	no	—
28	Q₃₂×C₅₃	—	4	2	no	—
29	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×Q₈×C₅₃	—	4	2	no	—
33	C₂×Dic₁₀₆	—	4	1	no	—
34	Q₁₆×C₅₃	—	4	2	no	—
35	Q₁₆×C₅₃	—	4	2	no	—
36	Q₁₆×C₅₃	—	8	2	no	—
37	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₅₃	—	8	1	no	—
40	C₄⋊C₄×C₅₃	—	8	1	no	—
41	C₄²×C₅₃	D₈	1	1	yes	characteristic
42	C₄⋊Dic₅₃	—	4	1	no	—
43	#11719919	—	8	1	no	—
44	C₄×Dic₅₃	—	8	1	no	—
45	Dic₂₁₂	—	2	2	no	—
46	Dic₂₁₂	—	4	1	no	—
47	C₁₆×C₅₃	—	2	1	no	—
48	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₅₃	#41	1	1	yes	characteristic
50	Q₈×C₅₃	—	8	2	no	—
51	Q₈×C₅₃	—	8	2	no	—
52	C₂×C₄×C₅₃	—	8	2	no	—
53	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₅₃	—	8	1	no	—
57	C₂×Dic₅₃	—	8	1	no	—
58	Dic₁₀₆	—	4	2	no	—
59	Dic₁₀₆	—	8	1	no	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	#121	1	1	yes	characteristic
62	C₈×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	#13688	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	16	2	no	—
67	C₄×C₅₃	Q₆₄	1	2	yes	complemented
68	C₄×C₅₃	#459	1	1	yes	characteristic
69	C₄×C₅₃	#509	1	1	yes	characteristic
70	Dic₅₃	—	8	2	no	—
71	Dic₅₃	—	16	1	no	—
72	C₂×Q₆₄	—	53	2	no	—
73	#2247	D₅₃	1	1	yes	characteristic
74	C₄×Q₃₂	—	53	1	no	—
75	C₄⋊C₃₂	—	53	1	no	—
76	#1416	—	53	2	no	—
77	C₂×C₅₃	#2206	1	1	yes	characteristic
78	C₂×C₅₃	C₂×Q₆₄	1	1	yes	characteristic
79	C₂×C₅₃	#2264	1	1	yes	characteristic
80	C₂×Q₃₂	—	2	1	no	—
81	C₂×Q₃₂	—	53	1	no	—
82	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
83	#494	—	2	1	no	—
84	C₄×Q₁₆	—	106	1	no	—
85	Q₆₄	—	106	4	no	—
86	C₂×C₃₂	—	53	2	no	—
87	#366	—	53	1	no	—
88	#358	—	106	1	no	—
89	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
90	C₅₃	#13688	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	4	1	no	—
93	C₂×Q₁₆	—	106	1	no	—
94	C₄⋊Q₈	—	4	1	no	—
95	C₄×Q₈	—	212	1	no	—
96	Q₃₂	—	2	2	no	—
97	Q₃₂	—	2	2	no	—
98	Q₃₂	—	4	2	no	—
99	Q₃₂	—	53	2	no	—
100	Q₃₂	—	106	1	no	—
101	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
103	#107	—	106	1	no	—
104	#103	—	212	1	no	—
105	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₃₂	—	106	2	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	—	212	1	no	—
109	Q₁₆	—	4	2	no	—
110	Q₁₆	—	4	2	no	—
111	Q₁₆	—	8	2	no	—
112	Q₁₆	—	106	2	no	—
113	Q₁₆	—	212	1	no	—
114	C₂×C₈	#11719922	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
116	C₄⋊C₄	—	8	1	no	—
117	C₄⋊C₄	—	8	1	no	—
118	C₄⋊C₄	—	424	1	no	—
119	C₄⋊C₄	—	212	1	no	—
120	C₄²	—	424	1	no	—
121	C₄²	#11719922	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	D₄×D₅₃	1	1	yes	characteristic
124	C₁₆	#11719947	1	1	yes	characteristic
125	Q₈	—	8	2	no	—
126	Q₈	—	8	2	no	—
127	Q₈	—	424	1	no	—
128	Q₈	—	212	2	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	424	1	no	—
131	C₂×C₄	—	424	1	no	—
132	C₂×C₄	#13133158	1	1	yes	characteristic
133	C₂×C₄	#13133158	1	1	yes	characteristic
134	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#13133188	1	1	yes	characteristic
137	C₈	#13133243	1	1	yes	characteristic
138	C₂²	C₂×#13133158	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	16	2	no	—
141	C₄	—	848	1	no	—
142	C₄	—	424	2	no	—
143	C₄	#16810463	1	2	yes	complemented
144	C₄	#16810815	1	1	yes	characteristic
145	C₄	#16811059	1	1	yes	characteristic
146	C₂	#20276747	1	1	yes	characteristic
147	C₂	C₂×#16810463	1	1	yes	characteristic
148	C₂	#20277033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810463

C₂

yes

maximal

C₄×Dic₄₂₄

C₂

yes

maximal, characteristic

C₄⋊#13133123

C₂

yes

maximal, characteristic

#20272896

C₂

yes

maximal

C₅₃×#2247

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

C₂²

yes

characteristic

C₄×Dic₂₁₂

—

#16810463

—

C₂×#13133123

C₂²

yes

—

#16810449

C₂²

yes

characteristic

#16810428

—

C₂×Q₃₂×C₅₃

—

C₂×Q₃₂×C₅₃

C₂²

yes

—

C₅₃×#494

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₄×Dic₁₀₆

—

Dic₄₂₄

—

Dic₄₂₄

D₄

yes

—

#13133151

—

#13133142

—

#13133123

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

Q₃₂×C₅₃

—

Q₃₂×C₅₃

—

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₈

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

#41

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₁₆

yes

characteristic

C₂×C₄×C₅₃

D₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

C₂×D₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

Q₆₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

C₂×Q₆₄

—

#2247

D₅₃

yes

characteristic

C₄×Q₃₂

—

C₄⋊C₃₂

—

#1416

—

C₂×C₅₃

#2206

yes

characteristic

C₂×C₅₃

C₂×Q₆₄

yes

characteristic

C₂×C₅₃

#2264

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₅₃

yes

—

#494

—

C₄×Q₁₆

—

106

—

Q₆₄

—

106

—

C₂×C₃₂

—

#366

—

#358

—

106

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₅₃

#13688

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄×Q₈

—

212

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

100

Q₃₂

—

106

—

101

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

102

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

103

#107

—

106

—

104

#103

—

212

—

105

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

106

C₃₂

—

106

—

107

C₂×Q₈

—

108

C₂×Q₈

—

212

—

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

—

112

Q₁₆

—

106

—

113

Q₁₆

—

212

—

114

C₂×C₈

#11719922

yes

characteristic

115

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

424

—

119

C₄⋊C₄

—

212

—

120

C₄²

—

424

—

121

C₄²

#11719922

yes

characteristic

122

C₁₆

—

123

C₁₆

D₄×D₅₃

yes

characteristic

124

C₁₆

#11719947

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

424

—

128

Q₈

—

212

—

129

C₂×C₄

—

130

C₂×C₄

—

424

—

131

C₂×C₄

—

424

—

132

C₂×C₄

#13133158

yes

characteristic

133

C₂×C₄

#13133158

yes

characteristic

134

C₂×C₄

C₂×#11719922

yes

characteristic

135

C₈

—

136

C₈

#13133188

yes

characteristic

137

C₈

#13133243

yes

characteristic

138

C₂²

C₂×#13133158

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

848

—

142

C₄

—

424

—

143

C₄

#16810463

yes

complemented

144

C₄

#16810815

yes

characteristic

145

C₄

#16811059

yes

characteristic

146

C₂

#20276747

yes

characteristic

147

C₂

C₂×#16810463

yes

characteristic

148

C₂

#20277033

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13133158
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#2247	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
hypercenter	#2247	D₅₃
hypocenter	C₅₃	#13688
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

subgroup

G/H

center

C₂²

C₂×#13133158

commutator, cosocle

C₂×C₁₆×C₅₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₅₃

Fitting

C₅₃×#2247

C₂

socle, prime radical

C₂²×C₅₃

C₂×D₁₆

D₅₃

C₅₃

G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

N	H ≅ G/N	type
67. C₄×C₅₃	Q₆₄	semidirect
90. C₅₃	#13688	semidirect
143. C₄	#16810463	semidirect

H ≅ G/N

type

67. C₄×C₅₃

Q₆₄

semidirect

90. C₅₃

#13688

semidirect

143. C₄

#16810463

semidirect

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

subgroup	index
C₂×#16810463	2
C₄×Dic₄₂₄	2
C₄⋊#13133123	2
#20272896	2
C₅₃×#2247	2
#13688	53

subgroup

index

C₄×Dic₄₂₄

C₅₃×#2247

Subgroups — #27304884

Conjugacy classes of subgroups

Complements

Maximal subgroups