Finite Group Databasebeta

← #27279321

Subgroups — #27279321

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285274C211yesmaximal, characteristic
2#20283662C211yesmaximal, characteristic, complemented
3#20275634C211yesmaximal, characteristic
4#20273215C211yesmaximal, characteristic, complemented
5#20273115C211yesmaximal, characteristic
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1278C211yesmaximal, characteristic, complemented
8#16811180C2211yescharacteristic
9#16811173C2211yescharacteristic
10#16811021C411yescharacteristic, complemented
11#16810996C411yescharacteristic, complemented
12#16810974C411yescharacteristic, complemented
13#16810970C411yescharacteristic, complemented
14#16810852—22no—
15#16810753—21no—
16#16810751—22no—
17C8×Dic106—21no—
18C4×#11719913—22no—
19#16810651—21no—
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133143—21no—
22C2×#13133143—21no—
23#16810581—21no—
24C4×C4⋊Dic53C2211yescharacteristic
25#16810361—22no—
26#16810352—21no—
27#16810352—21no—
28C53×#431C2211yescharacteristic
29C4×M16×C53C2211yescharacteristic
30C2×C4×C8×C53C2211yescharacteristic
31C2×#11719945—21no—
32C2×#11719913—22no—
33C22×#88322—21no—
34#13133217C2×C411yescharacteristic
35#13133217C2×C411yescharacteristic
36#13133217C2×C411yescharacteristic
37#13133214C2×C411yescharacteristic
38#13133209—21no—
39#13133205—21no—
40#13133196—21no—
41#13133196D412yes—
42#13133191—21no—
43#13133191—21no—
44C4×Dic106—21no—
45C4×Dic106D412yes—
46C4×D212—21no—
47C4×D212—21no—
48#13133176—21no—
49C2×C4⋊Dic53—21no—
50C2×C4×Dic53—21no—
51#13133149—22no—
52C4⋊#88322—21no—
53C4⋊#88322D411yescharacteristic
54C4⋊#88322D411yescharacteristic
55#13133146—22no—
56#13133146—22no—
57#13133143—42no—
58#13133143—42no—
59#13133142—22no—
60#13133139—22no—
61#13133127—22no—
62C8×Dic53—22no—
63#13133124—21no—
64C2×M16×C53D411yescharacteristic
65C2×M16×C53D411yescharacteristic
66C22×C8×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C22⋊C8×C53—21no—
70#97×C53—21no—
71C4×C8×C53—21no—
72C4×C8×C53—21no—
73C4×C8×C53—21no—
74C4×C8×C53C2×C411yescharacteristic
75C4×C8×C53C2×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C2×C22⋊D53—41no—
80C22×Dic53—41no—
81#11719945—41no—
82#11719945—41no—
83#11719945—41no—
84#11719945—41no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×C4×D53—41no—
89C2×Dic106—21no—
90C2×Dic106—21no—
91C2×Dic106C22⋊C412yes—
92M16×C53—21no—
93M16×C53—21no—
94M16×C53—21no—
95M16×C53SD1612yescomplemented
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—42no—
101C2×C8×C53—41no—
102C42×C53C22×C411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106C22⋊Dic53—22no—
107#11719921—41no—
108#11719921—41no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53—22no—
113C4⋊Dic53—21no—
114C4⋊Dic53—21no—
115C4⋊Dic53—41no—
116C4⋊Dic53C22⋊C412yes—
117#11719919—22no—
118#11719919—22no—
119C4×Dic53—22no—
120C4×Dic53—41no—
121C4×Dic53—41no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—41no—
125C2×#88322—41no—
126#11719913—44no—
127C23×C53#12111yescharacteristic
128C22×D53—41no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53C2×C22⋊C411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53#12011yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×C4×C53C2×SD1611yescharacteristic
139C2×C4×C53#13511yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C22⋊D53—81no—
143C22⋊D53—81no—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150D212—42no—
151D212—41no—
152D212—41no—
153D212—41no—
154D212—41no—
155C4×D53—81no—
156C4×D53—81no—
157Dic106—41no—
158Dic106—41no—
159Dic106—42no—
160Dic106—41no—
161Dic106—41no—
162C8×C53—22no—
163C8×C53—22no—
164C8×C53—41no—
165C8×C53—41no—
166C8×C53—81no—
167#88322—41no—
168#88322—41no—
169#9054—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53#38611yescharacteristic
173C22×C53#46011yescharacteristic
174C22×C53#46411yescharacteristic
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53C4×SD1611yescharacteristic
182C4×C53#44011yescharacteristic
183C4×C53#44311yescharacteristic
184C4×C53#44411yescharacteristic
185C4×C53#44811yescharacteristic
186C4×C53#44911yescharacteristic
187Dic53—42no—
188Dic53—42no—
189Dic53—82no—
190Dic53—81no—
191#3167—531no—
192#2962—531no—
193#1980—531no—
194#1503—531no—
195#1480—531no—
196#1279—531no—
197#1278D5311yescharacteristic, complemented
198C2×C53—21no—
199C2×C53—21no—
200C2×C53#187611yescharacteristic
201C2×C53#188011yescharacteristic
202C2×C53#188411yescharacteristic
203D53—161no—
204#533—531no—
205#532—531no—
206#498—531no—
207#492—531no—
208#487—531no—
209#486—531no—
210#465—1062no—
211#446—1061no—
212#446—1062no—
213C8×Q8—1061no—
214C4×SD16—1062no—
215#431C2×D5311yescharacteristic
216#424—1061no—
217C2×C4⋊C8—531no—
218C2×#104—1061no—
219C2×#104—1061no—
220#405—1061no—
221C4×M16C2×D5311yescharacteristic
222C2×C4×C8C2×D5311yescharacteristic
223C4×C4⋊C4—531no—
224#328—1062no—
225D4⋊C8—1061no—
226D4⋊C8—1061no—
227C53#905411yescharacteristic, complemented, Sylow
228C2×#41—1061no—
229C2×SD16—1062no—
230C2×M16D21211yescharacteristic
231C2×M16D21211yescharacteristic
232C22×C8—21no—
233C22×C8—1061no—
234C4⋊Q8—531no—
235C4⋊Q8—531no—
236C4⋊Q8—531no—
237#127—531no—
238#125—1061no—
239#124—1061no—
240C22⋊Q8—532no—
241C22⋊Q8—1061no—
242#121—1061no—
243#121—1061no—
244C4×Q8—532no—
245C4×Q8—1061no—
246C4×D4—1061no—
247C4×D4—1061no—
248#117—1061no—
249C2×C4⋊C4—1061no—
250C2×C42—1061no—
251C2×C42C22×D5311yescharacteristic
252#106—1062no—
253C4⋊C8—21no—
254C4⋊C8—531no—
255C4⋊C8—531no—
256C4⋊C8—1062no—
257C4⋊C8—1062no—
258C4⋊C8—1061no—
259#104—2122no—
260#104—2122no—
261#103—1062no—
262#102—1062no—
263C22⋊C8—21no—
264#97—21no—
265#97—1062no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—1062no—
270C4×C8C4×D5311yescharacteristic
271C4×C8C4×D5311yescharacteristic
272#95—1061no—
273#41—2121no—
274#41—2121no—
275#41—2121no—
276#41—2121no—
277C2×Q8—532no—
278C2×Q8—1061no—
279C2×Q8—1061no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—2121no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4C2×D21211yescharacteristic
287C22×C4D4×D5311yescharacteristic
288C22×C4#1171994911yescharacteristic
289SD16—2124no—
290M16—21no—
291M16—21no—
292M16—21no—
293M16#1171991312yescomplemented
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—2121no—
303C2×C8—2121no—
304C4⋊C4—532no—
305C4⋊C4—1062no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C4⋊C4—1062no—
310C4⋊C4—1062no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—2121no—
314C22⋊C4—1062no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C42—1062no—
318C42—2121no—
319C42—2121no—
320C42C2×C4×D5311yescharacteristic
321C42#1171994511yescharacteristic
322C42D4×D5311yescharacteristic
323C42D4×D5311yescharacteristic
324C23—2121no—
325C23#1313319211yescharacteristic
326Q8—2122no—
327Q8—2121no—
328Q8—2121no—
329Q8—2121no—
330Q8—2121no—
331D4—4241no—
332D4—4241no—
333D4—2122no—
334D4—2121no—
335D4—2121no—
336D4—2121no—
337D4—2121no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4—4241no—
344C2×C4—4241no—
345C2×C4—4241no—
346C2×C4—4241no—
347C2×C4—2122no—
348C2×C4—2121no—
349C2×C4C22⋊C4×D5311yescharacteristic
350C2×C4#1313317911yescharacteristic
351C2×C4C4×D21211yescharacteristic
352C2×C4#1313318711yescharacteristic
353C2×C4#1313319011yescharacteristic
354C2×C4#1313320211yescharacteristic
355C2×C4C2×#1171991311yescharacteristic
356C2×C4#1313323911yescharacteristic
357C2×C4#1313324511yescharacteristic
358C2×C4#1313325011yescharacteristic
359C8—22no—
360C8—22no—
361C8—41no—
362C8—41no—
363C8—81no—
364C8—2121no—
365C8—2121no—
366C22—21no—
367C22—21no—
368C22—4241no—
369C22—4241no—
370C22—4241no—
371C22#1681050111yescharacteristic
372C22#1681082411yescharacteristic
373C22#1681084611yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—4242no—
378C4—4241no—
379C4—2122no—
380C4—2122no—
381C4C4×#1171991311yescharacteristic
382C4#1681072511yescharacteristic
383C4#1681073911yescharacteristic
384C4#1681074411yescharacteristic
385C4#1681076411yescharacteristic
386C4#1681077011yescharacteristic
387C2—21no—
388C2—21no—
389C2—8481no—
390C2#2027504311yescharacteristic
391C2#2027506911yescharacteristic
392C2#2027509711yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1278C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#1278D53
hypocenterC53#9054
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
2. #20283662C2semidirect
4. #20273215C2semidirect
6. #20272555C2semidirect
7. C53×#1278C2semidirect
10. #16811021C4semidirect
11. #16810996C4semidirect
12. #16810974C4semidirect
13. #16810970C4semidirect
95. M16×C53SD16semidirect
197. #1278D53semidirect
227. C53#9054semidirect
293. M16#11719913semidirect

Maximal subgroups

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

subgroupindex
#202852742
#202836622
#202756342
#202732152
#202731152
#202725552
C53×#12782
#905453