Finite Group Databasebeta

← #27367675

Subgroups — #27367675

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286027C211yesmaximal, characteristic
2#20285856C211yesmaximal, characteristic, complemented
3#20285174C211yesmaximal, characteristic
4#20274209C212yesmaximal, complemented
5#20274187C212yesmaximal, complemented
6#20274085C212yesmaximal, complemented
7#20274063C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273365C211yesmaximal, characteristic
10C2×#16810370C211yesmaximal, characteristic, complemented
11C53×#2962C211yesmaximal, characteristic, complemented
12C2×#13133250—21no—
13C2×#13133250—21no—
14#16811173C2211yescharacteristic
15C2×#13133217C2211yescharacteristic
16#16811019—21no—
17#16811017—22no—
18#16811003C2212yes—
19#16810996C2212yes—
20#16810973—21no—
21#16810951—22no—
22#16810910—21no—
23#16810908C2212yes—
24#16810908C2212yes—
25#16810908C2212yes—
26#16810854—22no—
27#16810825—21no—
28#16810823C2212yes—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31#16810376C2212yes—
32#16810374C2212yes—
33#16810370C2212yes—
34#16810370C2211yescharacteristic
35#16810370C2212yes—
36#16810370C2211yescharacteristic
37#16810366C2211yescharacteristic
38#16810366C2211yescharacteristic
39C53×#533C2211yescharacteristic
40C53×#446—22no—
41C53×#446C2212yes—
42C8×Q8×C53C2212yes—
43C53×#424—21no—
44C2×C4⋊C8×C53C2211yescharacteristic
45C53×#405—21no—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C22×Dic106—21no—
49#13133250—44no—
50#13133250—44no—
51C2×Dic212—22no—
52C2×Dic212—22no—
53C2×#11719913—22no—
54C2×#11719913—22no—
55C2×#11719924—22no—
56C22×#88322—21no—
57#13133217—21no—
58#13133217—21no—
59#13133217—21no—
60#13133217C2312yes—
61#13133217C2311yescharacteristic
62#13133214C2311yescharacteristic
63#13133205—21no—
64#13133191—21no—
65#13133191—21no—
66C4×D212—21no—
67C4×D212—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70#13133148—22no—
71#13133148—22no—
72#13133148—22no—
73C4⋊#88322—21no—
74C4⋊#88322C2312yes—
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2311yescharacteristic
77#13133141—22no—
78#13133141—22no—
79#13133141—22no—
80#13133137—22no—
81C22⋊#88322—21no—
82C4×#88322—21no—
83C2×M16×C53D411yescharacteristic
84C2×M16×C53D411yescharacteristic
85C22×C8×C53—21no—
86C4⋊Q8×C53D411yescharacteristic
87C4⋊Q8×C53D411yescharacteristic
88#125×C53—21no—
89C22⋊Q8×C53—21no—
90C22⋊Q8×C53D412yes—
91C4×Q8×C53—21no—
92C4×Q8×C53C2312yes—
93#117×C53—21no—
94C2×C42×C53C2311yescharacteristic
95C4⋊C8×C53—22no—
96C4⋊C8×C53—22no—
97C4⋊C8×C53—21no—
98C4⋊C8×C53C2311yescharacteristic
99C4⋊C8×C53C2311yescharacteristic
100#97×C53—22no—
101#97×C53D412yes—
102C4×C8×C53—22no—
103C4×C8×C53C2312yes—
104C2×Q8×C53—21no—
105C2×Q8×C53C2×D412yes—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C2×C22⋊D53—41no—
110C22×Dic53—41no—
111#11719945—42no—
112#11719945—42no—
113C2×D212—21no—
114C2×D212—21no—
115C2×D212—21no—
116C2×C4×D53—41no—
117C2×Dic106—21no—
118C2×Dic106—22no—
119C2×Dic106—42no—
120C2×Dic106—42no—
121C2×Dic106—41no—
122M16×C53—22no—
123M16×C53SD1614yescomplemented
124C2×C8×C53—21no—
125C2×C8×C53—21no—
126C2×C8×C53—42no—
127C2×C8×C53C2×D412yes—
128C2×C8×C53C2×D412yes—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53C2×D412yes—
135C22⋊C4×C53—22no—
136C42×C53—22no—
137C42×C53C2×D411yescharacteristic
138C42×C53C2×D411yescharacteristic
139C42×C53#4111yescharacteristic
140C42×C53C2411yescharacteristic
141#11719924—82no—
142#11719921—41no—
143#11719921—41no—
144C4⋊Dic53—22no—
145C4⋊Dic53—21no—
146C4⋊Dic53—22no—
147C4⋊Dic53—21no—
148C4⋊Dic53—41no—
149C4⋊Dic53—41no—
150C2×#88322—21no—
151C2×#88322—21no—
152C2×#88322—22no—
153C2×#88322—81no—
154Dic212—44no—
155Dic212—44no—
156#11719913—44no—
157#11719913—44no—
158C23×C53#12111yescharacteristic
159C22×D53—41no—
160Q8×C53—22no—
161Q8×C53—22no—
162Q8×C53—81no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—41no—
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53C2×SD1612yes—
172C2×C4×C53AGL(1,ℤ/8)12yes—
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C2×#4111yescharacteristic
176C22⋊D53—82no—
177C2×Dic53—42no—
178C2×Dic53—41no—
179C2×Dic53—82no—
180D212—41no—
181D212—41no—
182D212—41no—
183D212—41no—
184D212—81no—
185C4×D53—82no—
186Dic106—42no—
187Dic106—41no—
188Dic106—42no—
189Dic106—41no—
190Dic106—82no—
191C8×C53—22no—
192C8×C53—41no—
193C8×C53—41no—
194C8×C53C2×SD1614yes—
195#88322—41no—
196#88322—42no—
197#88322—41no—
198#19239—531nomaximal, Sylow
199C22×C53—21no—
200C22×C53—21no—
201C22×C53#46012yes—
202C22×C53C2×#12111yescharacteristic
203C2×D53—81no—
204C2×D53—81no—
205C2×D53—81no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—42no—
210C4×C53—42no—
211C4×C53#46012yes—
212C4×C53C22×SD1611yescharacteristic
213C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
214C4×C53#57711yescharacteristic
215C4×C53#57811yescharacteristic
216Dic53—82no—
217Dic53—81no—
218#3246—531no—
219#3229—531no—
220#3152—531no—
221#2962D5311yescharacteristic, complemented
222#1690—532no—
223#1686—532no—
224#1670—532no—
225#1666—532no—
226#1565—531no—
227#1540—531no—
228C2×#332—531no—
229C2×C53—22no—
230C2×C53C2×#46011yescharacteristic
231C2×C53#306311yescharacteristic
232C2×C53#306911yescharacteristic
233D53—161no—
234C2×#137—1061no—
235C2×#137—1061no—
236#533C2×D5311yescharacteristic
237#532—531no—
238C2×C4⋊Q8—531no—
239#498—1061no—
240#497—1062no—
241#494—532no—
242#492—532no—
243#487—1061no—
244#483—1062no—
245#476—1061no—
246#475—532no—
247#475—532no—
248#475—532no—
249#465—1062no—
250#461—1061no—
251#460—532no—
252#446—22no—
253#446C2×D5312yes—
254C8×Q8C2×D5312yes—
255#431—531no—
256#424—21no—
257C2×C4⋊C8—531no—
258C2×C4⋊C8C2×D5311yescharacteristic
259#405—21no—
260C4×M16C2×D5311yescharacteristic
261#335—532no—
262#334—532no—
263#332—532no—
264#332—532no—
265#332—531no—
266#332—531no—
267#331—531no—
268#331—531no—
269C53#1923911yescharacteristic, complemented, Sylow
270C2×#41—1061no—
271C22×Q8—1061no—
272#137—2124no—
273#137—2124no—
274C2×Q16—1062no—
275C2×Q16—1062no—
276C2×SD16—1062no—
277C2×SD16—1062no—
278C2×SD16—1062no—
279C2×M16C22⋊D5311yescharacteristic
280C2×M16C22⋊D5311yescharacteristic
281C22×C8—21no—
282C22×C8—1061no—
283C4⋊Q8—532no—
284C4⋊Q8—531no—
285C4⋊Q8—1061no—
286C4⋊Q8—1061no—
287C4⋊Q8—1061no—
288C4⋊Q8D21211yescharacteristic
289C4⋊Q8D21211yescharacteristic
290#127—531no—
291#125—21no—
292#124—1061no—
293C22⋊Q8—21no—
294C22⋊Q8D21212yes—
295#121—1061no—
296#121—1061no—
297C4×Q8—21no—
298C4×Q8C22×D5312yes—
299C4×D4—1061no—
300C4×D4—1061no—
301#117—21no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4—1061no—
304C2×C42C22×D5311yescharacteristic
305#106—1062no—
306#106—1062no—
307#106—1062no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—21no—
311C4⋊C8—531no—
312C4⋊C8—531no—
313C4⋊C8—532no—
314C4⋊C8—1061no—
315C4⋊C8C22×D5311yescharacteristic
316C4⋊C8C22×D5311yescharacteristic
317#103—1062no—
318#103—1062no—
319#103—1062no—
320#102—1062no—
321C22⋊C8—1061no—
322#97—22no—
323#97C22⋊D5312yes—
324C4×C8—22no—
325C4×C8—1061no—
326C4×C8C22×D5312yes—
327#41—2122no—
328#41—2122no—
329C2×Q8—21no—
330C2×Q8—1061no—
331C2×Q8—1062no—
332C2×Q8—2122no—
333C2×Q8—2122no—
334C2×Q8—2121no—
335C2×Q8C2×D21212yes—
336C2×D4—1061no—
337C2×D4—1061no—
338C2×D4—1061no—
339C2×D4—2121no—
340C22×C4—2121no—
341C22×C4—2121no—
342C22×C4C2×D21211yescharacteristic
343C22×C4#1171994511yescharacteristic
344C22×C4C2×C22⋊D5311yescharacteristic
345Q16—2124no—
346Q16—2124no—
347SD16—4242no—
348SD16—2124no—
349SD16—2124no—
350M16—22no—
351M16#1171992414yescomplemented
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—42no—
355C2×C8—1061no—
356C2×C8—1061no—
357C2×C8—1062no—
358C2×C8—4241no—
359C2×C8C2×C22⋊D5312yes—
360C2×C8C2×C22⋊D5312yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—1062no—
367C4⋊C4—1061no—
368C4⋊C4—1062no—
369C4⋊C4—1061no—
370C4⋊C4—2121no—
371C4⋊C4—2121no—
372C4⋊C4C2×D21212yes—
373C22⋊C4—22no—
374C22⋊C4—2121no—
375C22⋊C4—2121no—
376C42—22no—
377C42C2×D21211yescharacteristic
378C42#1171994511yescharacteristic
379C42C2×C22⋊D5311yescharacteristic
380C42C23×D5311yescharacteristic
381C23—2121no—
382C23#1313319111yescharacteristic
383Q8—22no—
384Q8—22no—
385Q8—81no—
386Q8—4242no—
387Q8—2122no—
388Q8—2121no—
389Q8—2122no—
390Q8—2121no—
391D4—4241no—
392D4—4242no—
393D4—2121no—
394D4—2121no—
395D4—2121no—
396D4—2121no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—4242no—
403C2×C4—4242no—
404C2×C4—2121no—
405C2×C4—2122no—
406C2×C4#1313319111yescharacteristic
407C2×C4#1313319111yescharacteristic
408C2×C4#1313319111yescharacteristic
409C2×C4C2×#1171992412yes—
410C2×C4#1313324212yes—
411C2×C4C22×C22⋊D5311yescharacteristic
412C2×C4C22×D21211yescharacteristic
413C2×C4C2×#1171994511yescharacteristic
414C8—22no—
415C8—41no—
416C8—41no—
417C8—2121no—
418C8—2122no—
419C8—2121no—
420C8C2×#1171992414yes—
421C22—21no—
422C22—21no—
423C22—4241no—
424C22—4241no—
425C22—4241no—
426C22#1681082312yes—
427C22C2×#1313319111yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—42no—
433C4—4241no—
434C4—4242no—
435C4#1681082312yes—
436C4C22×#1171992411yescharacteristic
437C4C2×#1313324211yescharacteristic
438C4#1681143511yescharacteristic
439C4#1681144011yescharacteristic
440C2—22no—
441C2—8481no—
442C2C2×#1681082311yescharacteristic
443C2#2028440611yescharacteristic
444C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2962C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#2962D53
hypocenterC53#19239
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
2. #20285856C2semidirect
4. #20274209C2semidirect
5. #20274187C2semidirect
6. #20274085C2semidirect
7. #20274063C2semidirect
8. #20273467C2semidirect
10. C2×#16810370C2semidirect
11. C53×#2962C2semidirect
123. M16×C53SD16semidirect
221. #2962D53semidirect
269. C53#19239semidirect
351. M16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202860272
#202858562
#202851742
#202742092
#202741872
#202740852
#202740632
#202734672
#202733652
C2×#168103702
C53×#29622
#1923953