Finite Group Databasebeta

← #28309603

Subgroups — #28309603

Order 13568.

296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 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#20291090C211yesmaximal, characteristic, complemented
2#20290937C211yesmaximal, characteristic, complemented
3#20289719C212yesmaximal, complemented
4#20289718C212yesmaximal, complemented
5#20285366C211yesmaximal, characteristic
6D53⋊M32—21no—
7#16811545—21no—
8#16811520—21no—
9#16811516—21no—
10#16811515C2212yes—
11#16811515C2211yescharacteristic
12#16811515C2211yescharacteristic
13#16811229C412yes—
14#16811000C411yescharacteristic
15#16810989C2212yes—
16#16810877—21no—
17#16810860—21no—
18#16810734—21no—
19#16810705—21no—
20#16810577C2211yescharacteristic
21C53×#499C411yescharacteristic
22#13133292—22no—
23#13133291—21no—
24#13133291—21no—
25#13133291—22no—
26D53⋊C16—22no—
27C2×D424—21no—
28C2×D424—21no—
29C2×#11719912D411yescharacteristic
30C2×C8×D53D411yescharacteristic
31#13133214C2×C412yes—
32#13133202—42no—
33#13133192—22no—
34#13133192—42no—
35#13133179—22no—
36#13133174C2×C411yescharacteristic
37#13133137—22no—
38#13133137—22no—
39#13133128C2311yescharacteristic
40#13133127D411yescharacteristic
41C8×Dic53D411yescharacteristic
42#125×C53C2×C412yes—
43#107×C53—21no—
44#106×C53—21no—
45C4×C8×C53C2×C411yescharacteristic
46C2×D212—22no—
47C2×D212—42no—
48C2×D212—42no—
49C2×C4×D53—42no—
50C2×C4×D53C22⋊C411yescharacteristic
51C2×C8×C53C22⋊C411yescharacteristic
52C2×C8×C53C22⋊C411yescharacteristic
53C4⋊C4×C53—22no—
54C4⋊C4×C53—42no—
55C42×C53C22×C411yescharacteristic
56#11719921—21no—
57#11719921—42no—
58#11719921—42no—
59#11719919—21no—
60C4×Dic53C22⋊C411yescharacteristic
61C2×#88322C2×D411yescharacteristic
62C2×#88322C2×D411yescharacteristic
63D424—42no—
64D424—41no—
65D424—41no—
66#11719912—41no—
67C8×D53—21no—
68C8×D53—21no—
69#11719910—22no—
70#11719910—22no—
71#11719910—42no—
72C22×D53—42no—
73C22×D53#10011yescharacteristic
74C2×C4×C53—21no—
75C2×C4×C53—42no—
76C2×C4×C53C2×C22⋊C411yescharacteristic
77C2×Dic53—21no—
78C2×Dic53#10011yescharacteristic
79D212—42no—
80D212—42no—
81D212—162no—
82D212—162no—
83C4×D53—21no—
84C4×D53—21no—
85C4×D53—162no—
86C8×C53—21no—
87C8×C53#10411yescharacteristic
88C8×C53#10411yescharacteristic
89#88322—21no—
90#88322—21no—
91#11913—531nomaximal, Sylow
92C22×C53C2×#10011yescharacteristic
93C2×D53—21no—
94C2×D53—41no—
95C2×D53—82no—
96C2×D53—82no—
97C2×D53—82no—
98C4×C53—41no—
99C4×C53—82no—
100C4×C53C2×#10411yescharacteristic
101C4×C53#42111yescharacteristic
102Dic53—21no—
103Dic53—41no—
104#3176—531no—
105#2166—531no—
106#2110—531no—
107#1355—532no—
108#1355—532no—
109C2×C53—21no—
110C2×C53#153011yescharacteristic
111D53—41no—
112D53—162no—
113#540—532no—
114C2×M32—1061no—
115#499#615711yescharacteristic
116#493—531no—
117#490—532no—
118#469—1061no—
119#466—1061no—
120#442—1061no—
121C4×D8—1061no—
122#405—531no—
123#364—1061no—
124#349—1061no—
125#347—531no—
126#347—531no—
127#347—532no—
128#347—1061no—
129C53#1191311yescharacteristic, complemented, Sylow
130C2×D8—1061no—
131C2×D8—1061no—
132C2×M16—531no—
133C22×C8—531no—
134#127—532no—
135#125C2×#615712yes—
136#123—2122no—
137#121—1062no—
138#121—2122no—
139C4×D4—1062no—
140#117—531no—
141M32—1061no—
142M32—1061no—
143M32—1062no—
144M32—1062no—
145C2×C16—1062no—
146#107—21no—
147#106—21no—
148#102—1062no—
149#102—1062no—
150#97—531no—
151#97—531no—
152C4×C8—531no—
153C4×C8C2×#615711yescharacteristic
154C2×D4—1062no—
155C2×D4—2122no—
156C2×D4—2122no—
157C22×C4—531no—
158C22×C4—2122no—
159D8—2122no—
160D8—2121no—
161D8—2121no—
162M16—2121no—
163C2×C8—531no—
164C2×C8—531no—
165C2×C8—1061no—
166C2×C8—1061no—
167C2×C8C22⋊#615711yescharacteristic
168C2×C8C22⋊#615711yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—1061no—
172C22⋊C4—1061no—
173C22⋊C4—2122no—
174C22⋊C4—2122no—
175C42—531no—
176C42C22×#615711yescharacteristic
177C16—1062no—
178C16—1062no—
179C16—2122no—
180C23—531no—
181C23—2122no—
182D4—8482no—
183D4—8482no—
184D4—2122no—
185D4—2122no—
186C2×C4—21no—
187C2×C4—42no—
188C2×C4—531no—
189C2×C4—8482no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4C2×C22⋊#615711yescharacteristic
194C8—21no—
195C8—1061no—
196C8—1061no—
197C8#1313328211yescharacteristic
198C8#1313328211yescharacteristic
199C22—1061no—
200C22—4242no—
201C22—4242no—
202C22—4242no—
203C22—2121no—
204C22D53⋊#10011yescharacteristic
205C4—41no—
206C4—82no—
207C4—1061no—
208C4—2121no—
209C4C2×#1313328211yescharacteristic
210C4#1681162211yescharacteristic
211C2—21no—
212C2—8482no—
213C2—2121no—
214C2#2029018611yescharacteristic
subgroupG/H
centerC2#20290186
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#499C4
socleC2×C53#1530
prime radical#20285366C2
cosocle#13133128C23
hypercenter#499#6157
hypocenterC53#11913
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #499

Complements

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

NH ≅ G/Ntype
1. #20291090C2semidirect
2. #20290937C2semidirect
3. #20289719C2semidirect
4. #20289718C2semidirect
129. C53#11913semidirect

Maximal subgroups

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

subgroupindex
#202910902
#202909372
#202897192
#202897182
#202853662
#1191353