Finite Group Databasebeta

← #27294932

Subgroups — #27294932

Order 13568.

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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#20285445C211yesmaximal, characteristic
2#20276524C211yesmaximal, characteristic, complemented
3#20274862C211yesmaximal, characteristic, complemented
4#20272709C211yesmaximal, characteristic, complemented
5#20272707C211yesmaximal, characteristic
6#20272699C211yesmaximal, characteristic
7C53×#1340C211yesmaximal, characteristic, complemented
8#16811159—21no—
9#16811023C411yescharacteristic, complemented
10#16811008C411yescharacteristic, complemented
11#16810979—21no—
12#16810876C2211yescharacteristic
13#16810871—21no—
14#16810734C411yescharacteristic, complemented
15#16810730C411yescharacteristic, complemented
16#16810726—21no—
17M16×Dic53C2211yescharacteristic
18#16810443—21no—
19#16810405C2211yescharacteristic
20#16810400—21no—
21#16810393—21no—
22#16810382—21no—
23#16810381C2211yescharacteristic
24C53×#470C2211yescharacteristic
25C53×#349C2211yescharacteristic
26C53×H(3,ℤ/4)C2211yescharacteristic
27C2×#11719924—21no—
28C2×D424C2×C411yescharacteristic
29C2×#11719917—21no—
30C2×#11719917D411yescharacteristic
31#13133212—41no—
32#13133208—21no—
33#13133206—21no—
34#13133201—41no—
35#13133191—21no—
36Q8×Dic53—21no—
37#13133179—21no—
38#13133177—41no—
39#13133175—41no—
40C2×C4×Dic53D411yescharacteristic
41#13133157—21no—
42#13133150—21no—
43#13133149C2×C411yescharacteristic
44#13133141—21no—
45#13133139—21no—
46#13133137—21no—
47#13133127C2×C411yescharacteristic
48#13133127D411yescharacteristic
49C8×Dic53C2×C411yescharacteristic
50C8×Dic53D411yescharacteristic
51C2×Q16×C53C2×C411yescharacteristic
52C2×M16×C53C2311yescharacteristic
53C22⋊Q8×C53—21no—
54#117×C53—21no—
55M32×C53—21no—
56#106×C53C2×C411yescharacteristic
57#103×C53—21no—
58C2×Q8×C53—21no—
59C22×C4×C53C2×D411yescharacteristic
60C2×C22⋊D53—41no—
61C22×Dic53—21no—
62C2×D212—21no—
63C2×C4×D53—41no—
64Q16×C53—41no—
65M16×C53—41no—
66C2×C8×C53C22×C411yescharacteristic
67C2×C8×C53C2×D411yescharacteristic
68C4⋊C4×C53—21no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—41no—
73C42×C53—41no—
74#11719924—42no—
75#11719921—41no—
76#11719921—41no—
77#11719921—41no—
78#11719921—41no—
79C4⋊Dic53—21no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82#11719919—41no—
83C4×Dic53—21no—
84C4×Dic53—41no—
85C4×Dic53C22⋊C411yescharacteristic
86C4×Dic53C22⋊C411yescharacteristic
87#11719917—41no—
88#11719917—41no—
89#11719917—41no—
90C2×#88322—41no—
91C2×#88322C22⋊C411yescharacteristic
92C2×#88322C22⋊C411yescharacteristic
93D424—41no—
94C16×C53—41no—
95#11719908—41no—
96C23×C53AGL(1,ℤ/8)11yescharacteristic
97C22×D53—41no—
98Q8×C53—41no—
99Q8×C53—41no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—81no—
105C2×C4×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
107C22⋊D53—82no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—41no—
111C2×Dic53—81no—
112D212—41no—
113D212—41no—
114C4×D53—82no—
115C8×C53—21no—
116C8×C53—21no—
117#88322—21no—
118#88322—21no—
119#88322—81no—
120#11836—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—41no—
123C22×C53#41811yescharacteristic
124C2×D53—81no—
125C2×D53—81no—
126C2×D53—81no—
127C4×C53—21no—
128C4×C53—81no—
129C4×C53—81no—
130C4×C53—161no—
131C4×C53#41011yescharacteristic
132C4×C53#42111yescharacteristic
133Dic53—41no—
134Dic53—41no—
135Dic53—81no—
136#3183—531no—
137#2165—531no—
138#1841—531no—
139#1342—531no—
140#1341—531no—
141#1340D5311yescharacteristic, complemented
142#1339—531no—
143C2×C53—21no—
144C2×C53—41no—
145C2×C53#151411yescharacteristic
146D53—161no—
147#529—1061no—
148#499—531no—
149#495—531no—
150#488—1061no—
151#470C2×D5311yescharacteristic
152#469—531no—
153#468—1061no—
154#442—531no—
155#441—531no—
156#440—1061no—
157C4×M16—531no—
158#364—1061no—
159#349—531no—
160#349C2×D5311yescharacteristic
161#347—1061no—
162#344—1061no—
163#339—1061no—
164#338—531no—
165H(3,ℤ/4)C2×D5311yescharacteristic
166C53#1183611yescharacteristic, complemented, Sylow
167C2×Q16C4×D5311yescharacteristic
168C2×SD16—1061no—
169C2×D8—531no—
170C2×M16—531no—
171C2×M16—1061no—
172C2×M16C22×D5311yescharacteristic
173#126—2121no—
174#125—1061no—
175#124—1061no—
176#123—2121no—
177C22⋊Q8—21no—
178#121—1061no—
179C4×Q8—1061no—
180C4×D4—1061no—
181C4×D4—2121no—
182#117—21no—
183#117—2121no—
184C2×C42—531no—
185M32—21no—
186M32—1061no—
187#107—1061no—
188#106—531no—
189#106C4×D5311yescharacteristic
190#103—21no—
191#103—1061no—
192#102—1061no—
193#102—1061no—
194#97—531no—
195#97—531no—
196C4×C8—531no—
197C4×C8—531no—
198C2×Q8—21no—
199C2×D4—1061no—
200C2×D4—2121no—
201C22×C4—1061no—
202C22×C4—2121no—
203C22×C4D4×D5311yescharacteristic
204Q16—41no—
205SD16—2122no—
206D8—2121no—
207M16—41no—
208M16—2121no—
209M16—2121no—
210M16—2121no—
211C2×C8—531no—
212C2×C8—531no—
213C2×C8—2121no—
214C2×C8C2×C4×D5311yescharacteristic
215C2×C8D4×D5311yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—1061no—
220C4⋊C4—2121no—
221C4⋊C4—2121no—
222C4⋊C4—2121no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—2121no—
226C22⋊C4—2121no—
227C22⋊C4—2121no—
228C22⋊C4—2121no—
229C42—41no—
230C42—531no—
231C42—531no—
232C42—1061no—
233C42—2121no—
234C16—41no—
235C16—2121no—
236C23—2121no—
237C23#1313324111yescharacteristic
238Q8—41no—
239Q8—41no—
240D4—4242no—
241D4—2121no—
242D4—2121no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—1061no—
249C2×C4—1061no—
250C2×C4—4242no—
251C2×C4—4241no—
252C2×C4—2121no—
253C2×C4C22⋊C4×D5311yescharacteristic
254C2×C4#1313324411yescharacteristic
255C8—21no—
256C8—21no—
257C8—1061no—
258C8—1061no—
259C8—4241no—
260C22—21no—
261C22—41no—
262C22—4241no—
263C22—4241no—
264C22—4241no—
265C22#1681062611yescharacteristic
266C4—21no—
267C4—81no—
268C4—81no—
269C4—161no—
270C4—4241no—
271C4—2121no—
272C4—2121no—
273C4#1681059511yescharacteristic
274C4#1681063811yescharacteristic
275C2—21no—
276C2—41no—
277C2—8481no—
278C2#2027326011yescharacteristic
subgroupG/H
centerC2#20273260
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1340C2
socleC2×C53#1514
prime radical#16810876C22
cosocleC2×M16×C53C23
hypercenter#1340D53
hypocenterC53#11836
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1340

Complements

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

NH ≅ G/Ntype
2. #20276524C2semidirect
3. #20274862C2semidirect
4. #20272709C2semidirect
7. C53×#1340C2semidirect
9. #16811023C4semidirect
10. #16811008C4semidirect
14. #16810734C4semidirect
15. #16810730C4semidirect
141. #1340D53semidirect
166. C53#11836semidirect

Maximal subgroups

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

subgroupindex
#202854452
#202765242
#202748622
#202727092
#202727072
#202726992
C53×#13402
#1183653