Finite Group Databasebeta

← #27270768

Subgroups — #27270768

Order 13568.

218 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20272639C211yesmaximal, characteristic
2#20272583C211yesmaximal, characteristic
3C53×#2076C211yesmaximal, characteristic
4#16810483—21no—
5C22⋊#11719908—21no—
6C2×#107×C53C411yescharacteristic, complemented
7C2×C22⋊C8×C53C2211yescharacteristic
8C53×#380—21no—
9C53×#340—21no—
10C53×#340C411yescharacteristic, complemented
11C2×C4⋊Dic53—41no—
12C2×C22⋊Dic53—41no—
13C2×#11719908—41no—
14#13133124—22no—
15C23×C4×C53D411yescharacteristic
16C22×C8×C53C2×C411yescharacteristic
17C22×C8×C53D411yescharacteristic
18C2×C4⋊C4×C53—21no—
19C2×C4⋊C4×C53—21no—
20C2×C22⋊C4×C53—41no—
21#107×C53—41no—
22C22⋊C8×C53—41no—
23#95×C53—41no—
24C24×C53D811yescharacteristic
25C22×C4×C53—21no—
26C22×C4×C53—21no—
27C22×C4×C53—41no—
28C22×C4×C53—41no—
29C22×C4×C53—41no—
30C22×C4×C53—41no—
31C22×C4×C53C22⋊C411yescharacteristic
32C22×C4×C53SD1611yescharacteristic
33C22×Dic53—42no—
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C2×C8×C53—41no—
37C2×C8×C53—41no—
38C4⋊C4×C53—41no—
39C4⋊C4×C53—41no—
40C4⋊C4×C53—41no—
41C4⋊C4×C53—41no—
42C22⋊C4×C53—82no—
43C22⋊Dic53—81no—
44C22⋊Dic53—81no—
45C4⋊Dic53—81no—
46C4⋊Dic53—81no—
47#11719908—81no—
48C23×C53—21no—
49C23×C53—21no—
50C23×C53—41no—
51C23×C53—41no—
52C23×C53#10211yescharacteristic
53C23×C53SD3211yescharacteristic
54C23×C53Q3211yescharacteristic
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—81no—
62C2×C4×C53—81no—
63C2×C4×C53—81no—
64C2×C4×C53—81no—
65C2×C4×C53—81no—
66C2×C4×C53—81no—
67C2×C4×C53—81no—
68C2×C4×C53#9911yescharacteristic
69C2×C4×C53#10411yescharacteristic
70C2×Dic53—82no—
71C2×Dic53—82no—
72C2×Dic53—82no—
73C8×C53—41no—
74C8×C53—41no—
75#7077—531nomaximal, Sylow
76C22×C53—21no—
77C22×C53—21no—
78C22×C53—21no—
79C22×C53—21no—
80C22×C53—41no—
81C22×C53—41no—
82C22×C53—41no—
83C22×C53—41no—
84C22×C53—81no—
85C22×C53#32711yescharacteristic
86C22×C53#35811yescharacteristic
87C22×C53#36011yescharacteristic
88C4×C53—21no—
89C4×C53—41no—
90C4×C53—161no—
91C4×C53—161no—
92C4×C53#35211yescharacteristic
93C4×C53#35411yescharacteristic
94Dic53—162no—
95#2076D5311yescharacteristic
96#1314—531no—
97#1289—531no—
98C2×C53—21no—
99C2×C53—21no—
100C2×C53—41no—
101C2×C53—41no—
102C2×C53#133911yescharacteristic
103C2×C53#134411yescharacteristic
104C2×C53#134811yescharacteristic
105C2×#107Dic5311yescharacteristic, complemented
106C2×C22⋊C8C2×D5311yescharacteristic
107#380—21no—
108#380—1061no—
109C22⋊C16—1061no—
110#340—21no—
111#340Dic5311yescharacteristic, complemented
112C53#707711yescharacteristic, complemented, Sylow
113C23×C4C22⋊D5311yescharacteristic
114C22×C8C2×Dic5311yescharacteristic
115C22×C8C22⋊D5311yescharacteristic
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—2121no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—2121no—
121C2×C16—2121no—
122#107—41no—
123C22⋊C8—41no—
124#95—41no—
125#95—1062no—
126C24#1171992211yescharacteristic
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—2122no—
134C22×C4#1171992311yescharacteristic
135C22×C4C22⋊Dic5311yescharacteristic
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—41no—
139C2×C8—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—4241no—
145C4⋊C4—4241no—
146C22⋊C4—82no—
147C22⋊C4—4241no—
148C22⋊C4—4241no—
149C16—4241no—
150C23—21no—
151C23—21no—
152C23—41no—
153C23—41no—
154C23#1313313811yescharacteristic
155C23#1313316011yescharacteristic
156C23#1313316311yescharacteristic
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—4242no—
171C2×C4—4242no—
172C2×C4—4242no—
173C2×C4#1313313211yescharacteristic
174C2×C4#1313314411yescharacteristic
175C8—41no—
176C8—41no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22—81no—
186C22#1681035711yescharacteristic
187C22#1681042711yescharacteristic
188C22#1681043311yescharacteristic
189C4—21no—
190C4—41no—
191C4—161no—
192C4—161no—
193C4—8482no—
194C4#1681041311yescharacteristic
195C4#1681041811yescharacteristic
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2#2027270011yescharacteristic
201C2#2027271511yescharacteristic
202C2#2027272711yescharacteristic
subgroupG/H
centerC22#16810357
commutatorC22×C8×C53C2×C4
FrattiniC2×C22⋊C8C2×D53
FittingC53×#2076C2
socleC22×C53#327
prime radicalC24×C53D8
cosocleC2×C22⋊C8×C53C22
hypercenter#2076D53
hypocenterC53#7077
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2076

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×#107×C53C4semidirect
10. C53×#340C4semidirect
105. C2×#107Dic53semidirect
111. #340Dic53semidirect
112. C53#7077semidirect

Maximal subgroups

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

subgroupindex
#202726392
#202725832
C53×#20762
#707753