Group #27295931 — subgroups

Subgroups — #27295931

Order 13568.

212 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 117 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	Q₈×#11719908	C₂	1	2	yes	maximal
2	#20272779	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11719908	C₂	1	2	yes	maximal
4	#20272633	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	C₄×#11719908	—	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	2	no	—
17	C₂×#11719908	—	2	1	no	—
18	C₂×#11719908	D₄	1	2	yes	—
19	C₂×Q₁₆×C₅₃	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
21	#107×C₅₃	C₈	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	—	2	2	no	—
23	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
24	#103×C₅₃	C₈	1	2	yes	—
25	C₄×C₈×C₅₃	—	2	2	no	—
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
30	Q₁₆×C₅₃	C₁₆	1	4	yes	complemented
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
39	C₄²×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	#11719908	—	2	2	no	—
42	#11719908	—	4	1	no	—
43	#11719908	—	4	2	no	—
44	Q₈×C₅₃	—	2	2	no	—
45	Q₈×C₅₃	C₂×C₁₆	1	4	yes	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
51	C₈×C₅₃	—	2	1	no	—
52	C₈×C₅₃	—	2	1	no	—
53	C₈×C₅₃	—	4	1	no	—
54	C₈×C₅₃	—	4	2	no	—
55	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
56	C₈×C₅₃	#137	1	2	yes	—
57	#12059	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
59	C₄×C₅₃	—	2	4	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	4	2	no	—
62	C₄×C₅₃	#441	1	1	yes	characteristic
63	C₄×C₅₃	#444	1	1	yes	characteristic
64	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×C₅₃	#504	1	1	yes	characteristic
66	Q₈×C₁₆	—	53	2	no	—
67	C₈×Q₁₆	D₅₃	1	1	yes	characteristic
68	#1367	—	53	1	no	—
69	Q₈⋊C₁₆	—	53	2	no	—
70	#1312	—	53	1	no	—
71	C₂×C₅₃	#1579	1	1	yes	characteristic
72	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
73	C₂×C₅₃	#2171	1	1	yes	characteristic
74	C₈×Q₈	C₂×D₅₃	1	2	yes	—
75	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic
76	C₄⋊C₁₆	—	53	2	no	—
77	C₄⋊C₁₆	—	106	2	no	—
78	C₄×C₁₆	—	53	1	no	—
79	C₄×C₁₆	—	53	1	no	—
80	C₄×C₁₆	—	106	2	no	—
81	#335	Dic₅₃	1	1	yes	characteristic
82	Q₈⋊C₈	Dic₅₃	1	2	yes	—
83	C₈²	C₂×D₅₃	1	1	yes	characteristic
84	C₅₃	#12059	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	#88322	1	1	yes	characteristic
86	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
87	C₂×C₁₆	—	53	2	no	—
88	C₂×C₁₆	—	106	2	no	—
89	C₂×C₁₆	—	106	1	no	—
90	#107	#88322	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
93	#103	#88322	1	2	yes	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₂×Q₈	C₂×#88322	1	2	yes	—
99	Q₁₆	#11719908	1	4	yes	complemented
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	C₂×#88322	1	1	yes	characteristic
103	C₂×C₈	C₂×#88322	1	1	yes	characteristic
104	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
105	C₂×C₈	#11719947	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×#88322	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
110	C₁₆	—	106	2	no	—
111	C₁₆	—	212	1	no	—
112	C₁₆	—	212	2	no	—
113	Q₈	—	2	2	no	—
114	Q₈	C₂×#11719908	1	4	yes	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
118	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
119	C₂×C₄	#13133224	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	4	2	no	—
124	C₈	C₂×#11719908	1	2	yes	—
125	C₈	#13133249	1	2	yes	—
126	C₂²	D₄×#88322	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	2	no	—
130	C₄	#16810730	1	1	yes	characteristic
131	C₄	#16810745	1	1	yes	characteristic
132	C₄	C₂²×#11719908	1	1	yes	characteristic
133	C₄	#16811040	1	1	yes	characteristic
134	C₂	#20273534	1	1	yes	characteristic
135	C₂	D₄×#11719908	1	1	yes	characteristic
136	C₂	#20276557	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11719908

C₂

yes

maximal

#20272779

C₂

yes

maximal, characteristic

Q₈⋊#11719908

C₂

yes

maximal

#20272633

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#11719908

—

C₄⋊#11719908

C₂²

yes

—

C₄×#11719908

—

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

—

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

C₁₆

yes

complemented

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

—

—

—

Q₈×C₅₃

—

Q₈×C₅₃

C₂×C₁₆

yes

—

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

C₂×C₁₆

yes

—

C₈×C₅₃

#137

yes

—

#12059

—

maximal, Sylow

C₂²×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²×C₁₆

yes

characteristic

C₄×C₅₃

#504

yes

characteristic

Q₈×C₁₆

—

C₈×Q₁₆

D₅₃

yes

characteristic

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₂×C₅₃

#1579

yes

characteristic

C₂×C₅₃

D₄×C₁₆

yes

characteristic

C₂×C₅₃

#2171

yes

characteristic

C₈×Q₈

C₂×D₅₃

yes

—

C₄×Q₁₆

Dic₅₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

106

—

#335

Dic₅₃

yes

characteristic

Q₈⋊C₈

Dic₅₃

yes

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12059

yes

characteristic, complemented, Sylow

C₂×Q₁₆

#88322

yes

characteristic

C₄×Q₈

C₂×Dic₅₃

yes

—

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

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

C₂×#88322

yes

—

Q₁₆

#11719908

yes

complemented

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×#88322

yes

characteristic

103

C₂×C₈

C₂×#88322

yes

characteristic

104

C₂×C₈

D₄×D₅₃

yes

characteristic

105

C₂×C₈

#11719947

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×#88322

yes

—

108

C₄²

—

109

C₄²

C₂²×Dic₅₃

yes

characteristic

110

C₁₆

—

106

—

111

C₁₆

—

212

—

112

C₁₆

—

212

—

113

Q₈

—

114

Q₈

C₂×#11719908

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₄×Dic₅₃

yes

characteristic

118

C₂×C₄

C₂²×#88322

yes

characteristic

119

C₂×C₄

#13133224

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×#11719908

yes

—

125

C₈

#13133249

yes

—

126

C₂²

D₄×#88322

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#16810730

yes

characteristic

131

C₄

#16810745

yes

characteristic

132

C₄

C₂²×#11719908

yes

characteristic

133

C₄

#16811040

yes

characteristic

134

C₂

#20273534

yes

characteristic

135

C₂

D₄×#11719908

yes

characteristic

136

C₂

#20276557

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
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	#12059
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₈

D₄×D₅₃

commutator

C₄×C₅₃

C₂²×C₁₆

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

#12059

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

N	H ≅ G/N	type
30. Q₁₆×C₅₃	C₁₆	semidirect
84. C₅₃	#12059	semidirect
99. Q₁₆	#11719908	semidirect

H ≅ G/N

type

30. Q₁₆×C₅₃

C₁₆

semidirect

84. C₅₃

#12059

semidirect

99. Q₁₆

#11719908

semidirect

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

subgroup	index
Q₈×#11719908	2
#20272779	2
Q₈⋊#11719908	2
#20272633	2
C₈×Q₁₆×C₅₃	2
#12059	53

subgroup

index

C₈×Q₁₆×C₅₃

Subgroups — #27295931

Conjugacy classes of subgroups

Complements

Maximal subgroups