Finite Group Databasebeta

← #28317496

Subgroups — #28317496

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291556C211yesmaximal, characteristic, complemented
2#20291552C211yesmaximal, characteristic, complemented
3#20290353C212yesmaximal, complemented
4#20290321C212yesmaximal, complemented
5#20290318C212yesmaximal, complemented
6#20290295C212yesmaximal, complemented
7#20290275C211yesmaximal, characteristic, complemented
8#20290048C211yesmaximal, characteristic, complemented
9#20290046C211yesmaximal, characteristic, complemented
10C8×#11719940C211yesmaximal, characteristic, complemented
11#20283483C211yesmaximal, characteristic
12#16811674C2212yes—
13#16811673C2212yes—
14#16811655—22no—
15#16811654—22no—
16#16811651C2212yes—
17D4×#88324C2212yes—
18#16811645—21no—
19#16811644C2211yescharacteristic
20#16811644C2211yescharacteristic
21#16811643—21no—
22#16811640C2211yescharacteristic
23C4×#11719940C2211yescharacteristic
24#16811582—21no—
25C2×#13133272—21no—
26#16811487C2212yes—
27#16811484C2212yes—
28#16811473C2212yes—
29Q8⋊#88324C2212yes—
30#16811471C2212yes—
31D4⋊#88324C2212yes—
32#88324⋊C8C2212yes—
33C8×#88324C2212yes—
34C2×#13133237C411yescharacteristic
35#41×Dic53C2211yescharacteristic
36#16811091C2211yescharacteristic
37#16810715C411yescharacteristic
38#16810715C2211yescharacteristic
39#16810714C411yescharacteristic
40#16810714C2211yescharacteristic
41#16810708C411yescharacteristic
42#16810708C2211yescharacteristic
43#16810703C411yescharacteristic
44#16810703C2211yescharacteristic
45#16810659C411yescharacteristic
46#16810620C411yescharacteristic
47C2×C8×Dic53C2211yescharacteristic
48C53×#416C411yescharacteristic
49C2×#11719940D412yes—
50C2×D53⋊C8—22no—
51C22×#88324—22no—
52#13133284—21no—
53#13133284—22no—
54C4⋊#88324—22no—
55C4⋊#88324C2312yes—
56C4⋊#88324C2312yes—
57#13133274—22no—
58C22⋊#88324—21no—
59C22⋊#88324—21no—
60C22⋊#88324—21no—
61C22⋊#88324—22no—
62#13133272—21no—
63#13133272—21no—
64#13133272—22no—
65#13133272—22no—
66#13133272D411yescharacteristic
67#13133272D411yescharacteristic
68C4×#88324C2312yes—
69C4×#88324C2312yes—
70C2×#11719945C2×C411yescharacteristic
71#13133237—24no—
72C2×#11719925C2×C411yescharacteristic
73C2×#11719923C2×C411yescharacteristic
74C2×#11719924C2×C411yescharacteristic
75C2×#11719922C2×C411yescharacteristic
76C22×#88322C2×C411yescharacteristic
77Q8×Dic53C2×C411yescharacteristic
78Q8×Dic53C2311yescharacteristic
79#13133182C2×C411yescharacteristic
80#13133182C2311yescharacteristic
81D4×Dic53—21no—
82D4×Dic53—21no—
83D4×Dic53C2×C411yescharacteristic
84D4×Dic53C2311yescharacteristic
85#13133179C2×C411yescharacteristic
86#13133179C2311yescharacteristic
87#13133177—21no—
88#13133177—21no—
89#13133176—21no—
90#13133176C2×C411yescharacteristic
91#13133174—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53C2311yescharacteristic
94C42×D53—21no—
95#13133150C2×C411yescharacteristic
96#13133150C2×C411yescharacteristic
97#13133148C2×C411yescharacteristic
98#13133148C2×C411yescharacteristic
99#13133142C2×C411yescharacteristic
100#13133142C2×C411yescharacteristic
101#13133139C2×C411yescharacteristic
102#13133139C2×C411yescharacteristic
103C8×Dic53C2×C411yescharacteristic
104C8×Dic53C2×C411yescharacteristic
105C8×Dic53C2311yescharacteristic
106C8×Dic53C2311yescharacteristic
107C2×#41×C53C2×C411yescharacteristic
108C22×C8×C53C2×C411yescharacteristic
109#117×C53C2×C411yescharacteristic
110#103×C53C2×C411yescharacteristic
111#103×C53C2×C411yescharacteristic
112#102×C53C2×C411yescharacteristic
113#102×C53C2×C411yescharacteristic
114#41×C53—22no—
115#41×C53—41no—
116C2×Q8×C53C22×C411yescharacteristic
117C2×D4×C53—21no—
118C2×D4×C53C22×C411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53C22×C411yescharacteristic
121C2×C22⋊D53—21no—
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53C2×D411yescharacteristic
125C22×Dic53#4111yescharacteristic
126#11719945—22no—
127#11719945—41no—
128C2×D212C22×C411yescharacteristic
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×Dic106C22×C411yescharacteristic
133#11719940—24no—
134C2×#88324—22no—
135C2×#88324—22no—
136C2×#88324—22no—
137C2×#88324—44no—
138C2×#88324C2×D412yes—
139C2×#88324C2×D412yes—
140D53⋊C8—44no—
141C2×C8×C53—41no—
142C2×C8×C53C22×C411yescharacteristic
143C2×C8×C53C22×C411yescharacteristic
144C4⋊C4×C53C22×C411yescharacteristic
145C4⋊C4×C53C22×C411yescharacteristic
146C22⋊C4×C53—21no—
147C42×C53—21no—
148C22⋊Dic53—21no—
149C22⋊Dic53—21no—
150C22⋊Dic53—21no—
151#11719925—22no—
152#11719924—22no—
153#11719923—22no—
154#11719922M1614yescomplemented
155#11719921—21no—
156#11719921—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53C22×C411yescharacteristic
160C4⋊Dic53C22×C411yescharacteristic
161#11719919—21no—
162#11719919—21no—
163C4×Dic53—21no—
164C4×Dic53—21no—
165C4×Dic53—21no—
166C4×Dic53—21no—
167C4×Dic53—21no—
168C4×Dic53C22×C411yescharacteristic
169C4×Dic53C2×D411yescharacteristic
170C4×Dic53#4111yescharacteristic
171C4×Dic53C2411yescharacteristic
172C2×#88322—22no—
173C2×#88322C22×C411yescharacteristic
174C2×#88322C22×C411yescharacteristic
175C23×C53—21no—
176C23×C53C4×D411yescharacteristic
177C22×D53—21no—
178Q8×C53—21no—
179Q8×C53—21no—
180D4×C53—21no—
181D4×C53—42no—
182D4×C53C2×M1612yes—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—42no—
189C2×C4×C53C4×D411yescharacteristic
190C2×C4×C53C23×C411yescharacteristic
191C22⋊D53—42no—
192C2×Dic53—22no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—41no—
197C2×Dic53—42no—
198C2×Dic53—81no—
199C2×Dic53C4×D411yescharacteristic
200C2×Dic53C4×D411yescharacteristic
201C2×Dic53C22×D411yescharacteristic
202C2×Dic53C2×#4111yescharacteristic
203D212—21no—
204D212C2×M1612yes—
205C4×D53—42no—
206C4×D53—42no—
207C4×D53—81no—
208Dic106—21no—
209Dic106—21no—
210#88324—22no—
211#88324—42no—
212#88324—42no—
213#88324—42no—
214#88324#13514yescomplemented
215C8×C53—21no—
216C8×C53—21no—
217#88322—21no—
218#88322C2×M1612yes—
219#18035—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—22no—
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—41no—
225C22×C53—41no—
226C22×C53C2×C4×D411yescharacteristic
227C2×D53—22no—
228C2×D53—41no—
229C2×D53—41no—
230C4×C53—21no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—41no—
234C4×C53C22×M1611yescharacteristic
235C4×C53#56811yescharacteristic
236Dic53—21no—
237Dic53—21no—
238Dic53—21no—
239Dic53—41no—
240Dic53—41no—
241Dic53—41no—
242Dic53C2×#13512yes—
243#2970—531no—
244#2970—531no—
245C4×#135—531no—
246#1591—532no—
247#1582—532no—
248#1582—532no—
249C8×D8—532no—
250#1568—531no—
251#1488—531no—
252#1488—531no—
253C8×M16—531no—
254C2×C53—41no—
255C2×C53—42no—
256C2×C53D4×M1611yescharacteristic
257C2×C53C4×#13511yescharacteristic
258C2×C53#295711yescharacteristic
259D53—42no—
260C2×#135—531no—
261C4×#41—531no—
262C4×#41—531no—
263#446—532no—
264#446—532no—
265C4×Q16—531no—
266C4×Q16—531no—
267C4×SD16—531no—
268C4×SD16—531no—
269C4×SD16—531no—
270C4×SD16—531no—
271C4×D8—531no—
272C4×D8—531no—
273#435—1062no—
274#435—1062no—
275C8×D4—532no—
276C8×D4—532no—
277#432—531no—
278#432—531no—
279#432—1061no—
280#432—1061no—
281#431—531no—
282#427—531no—
283#416—531no—
284#416#615711yescharacteristic
285C4×M16—531no—
286C2×#97—1061no—
287C2×#97—1061no—
288C2×C4×C8—531no—
289#335—532no—
290#334—532no—
291Q8⋊C8—532no—
292Q8⋊C8—532no—
293D4⋊C8—532no—
294D4⋊C8—532no—
295#322—532no—
296C82—532no—
297C53#1803511yescharacteristic, complemented, Sylow
298C2×#41—531no—
299C2×#41C2×#615711yescharacteristic
300#135—1064no—
301C2×Q16—531no—
302C2×SD16—531no—
303C2×SD16—531no—
304C2×D8—531no—
305C2×M16—532no—
306C22×C8—531no—
307C22×C8—1062no—
308C22×C8—1062no—
309C22×C8C2×#615711yescharacteristic
310C4×Q8—531no—
311C4×Q8—531no—
312C4×Q8—531no—
313C4×Q8—531no—
314C4×D4—531no—
315C4×D4—531no—
316C4×D4—531no—
317C4×D4—531no—
318C4×D4—1061no—
319C4×D4—1061no—
320C4×D4—1061no—
321C4×D4—1061no—
322#117—531no—
323#117—1061no—
324#117—1061no—
325#117C2×#615711yescharacteristic
326C2×C42—531no—
327C2×C42—1061no—
328C2×C42—1061no—
329#107—531no—
330#107—531no—
331#106—531no—
332#106—531no—
333C4⋊C8—532no—
334C4⋊C8—532no—
335C4⋊C8—1061no—
336C4⋊C8—1062no—
337C4⋊C8—1062no—
338#103—531no—
339#103—531no—
340#103C2×#615711yescharacteristic
341#103C2×#615711yescharacteristic
342#102—531no—
343#102—531no—
344#102C2×#615711yescharacteristic
345#102C2×#615711yescharacteristic
346C22⋊C8—1062no—
347C22⋊C8—1062no—
348C22⋊C8—1061no—
349C22⋊C8—1061no—
350C22⋊C8—1061no—
351#97—531no—
352#97—531no—
353#97—1061no—
354#97—1061no—
355#97—1062no—
356#97—1062no—
357C4×C8—532no—
358C4×C8—531no—
359C4×C8—531no—
360C4×C8—531no—
361C4×C8—531no—
362C4×C8—532no—
363#41—22no—
364#41—41no—
365#41—1062no—
366#41—2121no—
367C2×Q8—531no—
368C2×Q8C22×#615711yescharacteristic
369C2×D4—21no—
370C2×D4—531no—
371C2×D4—1061no—
372C2×D4C22×#615711yescharacteristic
373C22×C4—21no—
374C22×C4—531no—
375C22×C4—531no—
376C22×C4—1061no—
377C22×C4—1061no—
378C22×C4—1061no—
379C22×C4—1061no—
380C22×C4—1061no—
381C22×C4C22×#615711yescharacteristic
382Q16—1062no—
383SD16—1062no—
384SD16—1062no—
385D8—534no—
386M16—1064no—
387C2×C8—41no—
388C2×C8—532no—
389C2×C8—531no—
390C2×C8—532no—
391C2×C8—531no—
392C2×C8—1062no—
393C2×C8—1062no—
394C2×C8—1062no—
395C2×C8—1062no—
396C2×C8—2124no—
397C2×C8—2124no—
398C2×C8C22×#615711yescharacteristic
399C2×C8C22×#615711yescharacteristic
400C4⋊C4—531no—
401C4⋊C4—531no—
402C4⋊C4—1061no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4—1061no—
406C4⋊C4C22×#615711yescharacteristic
407C4⋊C4C22×#615711yescharacteristic
408C22⋊C4—21no—
409C22⋊C4—1061no—
410C22⋊C4—1061no—
411C22⋊C4—1061no—
412C22⋊C4—1061no—
413C22⋊C4—1061no—
414C42—21no—
415C42—531no—
416C42—531no—
417C42—531no—
418C42—531no—
419C42—1061no—
420C42—1061no—
421C42—1061no—
422C42—1061no—
423C42—1061no—
424C23—21no—
425C23—1061no—
426C23D4×#615711yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—1061no—
430Q8—1061no—
431D4—21no—
432D4—42no—
433D4—532no—
434D4—1061no—
435D4—2122no—
436D4D53⋊M1612yes—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—42no—
443C2×C4—531no—
444C2×C4—531no—
445C2×C4—531no—
446C2×C4—531no—
447C2×C4—1062no—
448C2×C4—1061no—
449C2×C4—1061no—
450C2×C4—1061no—
451C2×C4—4241no—
452C2×C4—4241no—
453C2×C4—2122no—
454C2×C4—2122no—
455C2×C4—2122no—
456C2×C4—2121no—
457C2×C4D4×#615711yescharacteristic
458C2×C4C23×#615711yescharacteristic
459C8—21no—
460C8—21no—
461C8—534no—
462C8—532no—
463C8—1062no—
464C8—1061no—
465C8—2122no—
466C8—2122no—
467C8—2122no—
468C22—21no—
469C22—22no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—1062no—
475C22—2121no—
476C22—2121no—
477C22C2×D4×#615711yescharacteristic
478C4—21no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—532no—
483C4—1061no—
484C4—1061no—
485C4—1061no—
486C4—2121no—
487C4—2121no—
488C4—2121no—
489C4C2×D53⋊M1611yescharacteristic
490C4#1681170111yescharacteristic
491C2—41no—
492C2—42no—
493C2—2122no—
494C2#2029145311yescharacteristic
495C2#135×#615711yescharacteristic
496C2#2029149411yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#416C4
socleC22×C53C2×C4×D4
prime radicalC2×#13133237C4
cosocleC4×Dic53C24
hypercenter#416#6157
hypocenterC53#18035
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20291556C2semidirect
2. #20291552C2semidirect
3. #20290353C2semidirect
4. #20290321C2semidirect
5. #20290318C2semidirect
6. #20290295C2semidirect
7. #20290275C2semidirect
8. #20290048C2semidirect
9. #20290046C2semidirect
10. C8×#11719940C2semidirect
154. #11719922M16semidirect
214. #88324#135semidirect
297. C53#18035semidirect

Maximal subgroups

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

subgroupindex
#202915562
#202915522
#202903532
#202903212
#202903182
#202902952
#202902752
#202900482
#202900462
C8×#117199402
#202834832
#1803553