Group #27295898 — subgroups

Subgroups — #27295898

Order 13568.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20276624	C₂	1	2	yes	maximal
2	#20272785	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11719908	C₂	1	2	yes	maximal
4	C₈×#11719908	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11719908	—	2	2	no	—
7	C₄⋊#11719908	C₂²	1	2	yes	—
8	#16810399	—	2	2	no	—
9	C₄×#11719908	C₂²	1	1	yes	characteristic
10	C₄×#11719908	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#335	C₄	1	1	yes	characteristic
14	C₅₃×Q₈⋊C₈	C₄	1	2	yes	—
15	C₈²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719908	—	2	1	no	—
17	C₂×#11719908	—	2	2	no	—
18	C₂×#11719908	D₄	1	1	yes	characteristic
19	C₂×#11719908	D₄	1	1	yes	characteristic
20	C₂×Q₁₆×C₅₃	C₈	1	1	yes	characteristic
21	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
22	#107×C₅₃	C₈	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	—	2	2	no	—
24	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
25	#103×C₅₃	C₈	1	2	yes	—
26	C₄×C₈×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
31	Q₁₆×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	#11719908	—	2	2	no	—
43	#11719908	—	2	1	no	—
44	#11719908	—	4	2	no	—
45	#11719908	Q₁₆	1	2	yes	complemented
46	Q₈×C₅₃	—	2	2	no	—
47	Q₈×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
53	C₈×C₅₃	—	2	2	no	—
54	C₈×C₅₃	—	2	1	no	—
55	C₈×C₅₃	—	2	1	no	—
56	C₈×C₅₃	—	4	2	no	—
57	C₈×C₅₃	M₃₂	1	2	yes	—
58	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₅₃	#135	1	1	yes	characteristic
60	#12054	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	4	2	no	—
64	C₄×C₅₃	—	4	2	no	—
65	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
66	C₄×C₅₃	#443	1	1	yes	characteristic
67	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
68	C₄×C₅₃	#504	1	1	yes	characteristic
69	#2183	—	53	2	no	—
70	C₈×Q₁₆	D₅₃	1	1	yes	characteristic
71	#1370	—	53	1	no	—
72	Q₈⋊C₁₆	—	53	2	no	—
73	C₈×C₁₆	—	53	1	no	—
74	C₂×C₅₃	C₈×Q₁₆	1	1	yes	characteristic
75	C₂×C₅₃	#2169	1	1	yes	characteristic
76	C₂×C₅₃	#2170	1	1	yes	characteristic
77	C₈×Q₈	C₂×D₅₃	1	2	yes	—
78	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic
79	C₄⋊C₁₆	—	53	2	no	—
80	C₄⋊C₁₆	—	106	2	no	—
81	#346	—	106	2	no	—
82	C₄×C₁₆	—	53	1	no	—
83	C₄×C₁₆	—	53	1	no	—
84	#335	Dic₅₃	1	1	yes	characteristic
85	Q₈⋊C₈	Dic₅₃	1	2	yes	—
86	C₈²	C₂×D₅₃	1	1	yes	characteristic
87	C₅₃	#12054	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	#88322	1	1	yes	characteristic
89	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
90	C₂×C₁₆	—	53	1	no	—
91	C₂×C₁₆	—	53	1	no	—
92	C₂×C₁₆	—	106	1	no	—
93	C₂×C₁₆	—	106	2	no	—
94	#107	#88322	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
97	#103	#88322	1	2	yes	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
100	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
102	C₂×Q₈	C₂×#88322	1	2	yes	—
103	Q₁₆	—	2	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂×#88322	1	1	yes	characteristic
107	C₂×C₈	C₂×#88322	1	1	yes	characteristic
108	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
109	C₂×C₈	#11719947	1	1	yes	characteristic
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂×#88322	1	2	yes	—
112	C₄²	—	2	2	no	—
113	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
114	C₁₆	—	53	2	no	—
115	C₁₆	—	106	2	no	—
116	C₁₆	—	106	1	no	—
117	C₁₆	—	212	2	no	—
118	Q₈	—	2	2	no	—
119	Q₈	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
123	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
124	C₂×C₄	#13133224	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	4	2	no	—
129	C₈	#13133157	1	2	yes	—
130	C₈	Q₁₆×D₅₃	1	1	yes	characteristic
131	C₈	#13133240	1	1	yes	characteristic
132	C₂²	D₄×#88322	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	2	no	—
136	C₄	Q₁₆×Dic₅₃	1	1	yes	characteristic
137	C₄	#16810740	1	1	yes	characteristic
138	C₄	C₂×#13133157	1	1	yes	characteristic
139	C₄	#16811040	1	1	yes	characteristic
140	C₂	Q₁₆×#88322	1	1	yes	characteristic
141	C₂	#20276547	1	1	yes	characteristic
142	C₂	#20276552	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276624

C₂

yes

maximal

#20272785

C₂

yes

maximal, characteristic

Q₈⋊#11719908

C₂

yes

maximal

C₈×#11719908

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#11719908

—

C₄⋊#11719908

C₂²

yes

—

#16810399

—

C₄×#11719908

C₂²

yes

characteristic

C₄×#11719908

C₂²

yes

characteristic

C₈×Q₈×C₅₃

C₂²

yes

—

C₄×Q₁₆×C₅₃

C₄

yes

characteristic

C₅₃×#335

C₄

yes

characteristic

C₅₃×Q₈⋊C₈

C₄

yes

—

C₈²×C₅₃

C₂²

yes

—

—

D₄

yes

characteristic

C₂×#11719908

D₄

yes

characteristic

C₂×Q₁₆×C₅₃

C₈

yes

characteristic

C₄×Q₈×C₅₃

C₂×C₄

yes

—

#107×C₅₃

C₈

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

#103×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₈

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

—

—

—

Q₁₆

yes

complemented

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

M₃₂

yes

—

C₈×C₅₃

C₂×Q₁₆

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₁₆

yes

characteristic

C₄×C₅₃

#443

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

C₄×C₅₃

yes

—

C₈×Q₁₆

D₅₃

yes

characteristic

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂×C₅₃

C₈×Q₁₆

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₈×Q₈

C₂×D₅₃

yes

—

C₄×Q₁₆

Dic₅₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

#346

—

106

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

Dic₅₃

yes

characteristic

Q₈⋊C₈

Dic₅₃

yes

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12054

yes

characteristic, complemented, Sylow

C₂×Q₁₆

#88322

yes

characteristic

C₄×Q₈

C₂×Dic₅₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

#107

#88322

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×Dic₅₃

yes

—

#103

#88322

yes

—

C₄×C₈

—

C₄×C₈

C₂×Dic₅₃

yes

characteristic

100

C₄×C₈

C₂×Dic₅₃

yes

characteristic

101

C₄×C₈

C₂²×D₅₃

yes

characteristic

102

C₂×Q₈

C₂×#88322

yes

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×#88322

yes

characteristic

107

C₂×C₈

C₂×#88322

yes

characteristic

108

C₂×C₈

D₄×D₅₃

yes

characteristic

109

C₂×C₈

#11719947

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×#88322

yes

—

112

C₄²

—

113

C₄²

C₂²×Dic₅₃

yes

characteristic

114

C₁₆

—

115

C₁₆

—

106

—

116

C₁₆

—

106

—

117

C₁₆

—

212

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

D₄×Dic₅₃

yes

characteristic

123

C₂×C₄

C₂²×#88322

yes

characteristic

124

C₂×C₄

#13133224

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#13133157

yes

—

130

C₈

Q₁₆×D₅₃

yes

characteristic

131

C₈

#13133240

yes

characteristic

132

C₂²

D₄×#88322

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

Q₁₆×Dic₅₃

yes

characteristic

137

C₄

#16810740

yes

characteristic

138

C₄

C₂×#13133157

yes

characteristic

139

C₄

#16811040

yes

characteristic

140

C₂

Q₁₆×#88322

yes

characteristic

141

C₂

#20276547

yes

characteristic

142

C₂

#20276552

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×Q₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₈×D₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₈×Q₁₆	D₅₃
hypocenter	C₅₃	#12054
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₈

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₈×Q₁₆×C₅₃

C₂

socle, prime radical

C₂²×C₅₃

C₈×D₄

cosocle

C₄×C₈×C₅₃

C₂³

hypercenter

C₈×Q₁₆

D₅₃

hypocenter

C₅₃

#12054

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

N	H ≅ G/N	type
45. #11719908	Q₁₆	semidirect
87. C₅₃	#12054	semidirect

H ≅ G/N

type

45. #11719908

Q₁₆

semidirect

87. C₅₃

#12054

semidirect

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

subgroup	index
#20276624	2
#20272785	2
Q₈⋊#11719908	2
C₈×#11719908	2
C₈×Q₁₆×C₅₃	2
#12054	53

subgroup

index

C₈×Q₁₆×C₅₃

Subgroups — #27295898

Conjugacy classes of subgroups

Complements

Maximal subgroups