Finite Group Databasebeta

← #28317708

Subgroups — #28317708

Order 13568.

652 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291642C211yesmaximal, characteristic, complemented
2#20291641C211yesmaximal, characteristic, complemented
3#20290360C212yesmaximal, complemented
4#20290321C212yesmaximal, complemented
5#20290318C212yesmaximal, complemented
6#20290312C212yesmaximal, complemented
7#20290269C211yesmaximal, characteristic, complemented
8#20290052C211yesmaximal, characteristic, complemented
9#20290050C211yesmaximal, characteristic, complemented
10#20289924C211yesmaximal, characteristic, complemented
11#20283483C211yesmaximal, characteristic
12#16811674C2212yes—
13#16811673C2212yes—
14#16811655—22no—
15#16811654—22no—
16#16811651C2212yes—
17D4×#88324C2212yes—
18#16811646—21no—
19#16811643—21no—
20#16811640C2211yescharacteristic
21#16811626C2211yescharacteristic
22#16811626C2211yescharacteristic
23C2×#13133284—21no—
24D53⋊(C4⋊C8)—21no—
25C4×#11719940C2211yescharacteristic
26#16811485C2212yes—
27#16811484C2212yes—
28#16811473C2212yes—
29Q8⋊#88324C2212yes—
30#16811471C2212yes—
31D4⋊#88324C2212yes—
32#88324⋊C8C2212yes—
33#16811464C2212yes—
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×#11719940—21no—
50C2×#11719940—21no—
51C2×#11719940—21no—
52C2×D53⋊C8—22no—
53C22×#88324—22no—
54#13133284—21no—
55#13133284—21no—
56#13133284—22no—
57#13133284D411yescharacteristic
58#13133284D411yescharacteristic
59C4⋊#88324—22no—
60C4⋊#88324C2312yes—
61C4⋊#88324C2312yes—
62#13133274—22no—
63C22⋊#88324—22no—
64C22⋊#88324D412yes—
65#13133272—21no—
66#13133272—22no—
67#13133272—22no—
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—42no—
134#11719940—81no—
135#11719940—81no—
136C2×#88324—22no—
137C2×#88324—22no—
138C2×#88324—22no—
139C2×#88324—82no—
140C2×#88324C2×D412yes—
141C2×#88324C2×D412yes—
142D53⋊C8—82no—
143C2×C8×C53—41no—
144C2×C8×C53C22×C411yescharacteristic
145C2×C8×C53C22×C411yescharacteristic
146C4⋊C4×C53C22×C411yescharacteristic
147C4⋊C4×C53C22×C411yescharacteristic
148C22⋊C4×C53—21no—
149C42×C53—21no—
150C22⋊Dic53—21no—
151C22⋊Dic53—21no—
152C22⋊Dic53—21no—
153#11719925—22no—
154#11719924—22no—
155#11719923—22no—
156#11719922M1614yescomplemented
157#11719921—21no—
158#11719921—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53C22×C411yescharacteristic
162C4⋊Dic53C22×C411yescharacteristic
163#11719919—21no—
164#11719919—21no—
165C4×Dic53—21no—
166C4×Dic53—21no—
167C4×Dic53—21no—
168C4×Dic53—21no—
169C4×Dic53—21no—
170C4×Dic53C22×C411yescharacteristic
171C4×Dic53C2×D411yescharacteristic
172C4×Dic53#4111yescharacteristic
173C4×Dic53C2411yescharacteristic
174C2×#88322—22no—
175C2×#88322C22×C411yescharacteristic
176C2×#88322C22×C411yescharacteristic
177C23×C53—21no—
178C23×C53C4×D411yescharacteristic
179C22×D53—21no—
180Q8×C53—21no—
181Q8×C53—21no—
182D4×C53—21no—
183D4×C53—42no—
184D4×C53C2×M1612yes—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—42no—
191C2×C4×C53C4×D411yescharacteristic
192C2×C4×C53C23×C411yescharacteristic
193C22⋊D53—42no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—22no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—42no—
203C2×Dic53C4×D411yescharacteristic
204C2×Dic53C4×D411yescharacteristic
205C2×Dic53C22×D411yescharacteristic
206C2×Dic53C2×#4111yescharacteristic
207D212—21no—
208D212C2×M1612yes—
209C4×D53—42no—
210C4×D53—41no—
211C4×D53—41no—
212C4×D53—42no—
213Dic106—21no—
214Dic106—21no—
215#88324—22no—
216#88324—22no—
217#88324—42no—
218#88324—42no—
219#88324—42no—
220C8×C53—21no—
221C8×C53—21no—
222#88322—21no—
223#88322C2×M1612yes—
224#18065—531nomaximal, Sylow
225C22×C53—21no—
226C22×C53—22no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—41no—
230C22×C53—41no—
231C22×C53C2×C4×D411yescharacteristic
232C2×D53—22no—
233C2×D53—41no—
234C2×D53—41no—
235C4×C53—21no—
236C4×C53—41no—
237C4×C53—41no—
238C4×C53—41no—
239C4×C53C22×M1611yescharacteristic
240C4×C53#56811yescharacteristic
241Dic53—21no—
242Dic53—21no—
243Dic53—41no—
244Dic53—41no—
245Dic53—41no—
246Dic53#44412yes—
247Dic53#57711yescharacteristic
248Dic53#57711yescharacteristic
249#2990—531no—
250#2990—531no—
251C4×#135—531no—
252#1593—532no—
253#1582—532no—
254#1582—532no—
255#1580—532no—
256#1566—531no—
257#1489—531no—
258#1489—531no—
259#1451—531no—
260C2×C53—41no—
261C2×C53—42no—
262C2×C53D4×M1611yescharacteristic
263C2×C53#295211yescharacteristic
264C2×C53C2×#44411yescharacteristic
265D53—42no—
266C2×#135—531no—
267C4×#41—531no—
268C4×#41—531no—
269#446—532no—
270#446—532no—
271C4×Q16—531no—
272C4×Q16—531no—
273C4×SD16—531no—
274C4×SD16—531no—
275C4×SD16—531no—
276C4×SD16—531no—
277C4×D8—531no—
278C4×D8—531no—
279#435—1062no—
280#435—1062no—
281C8×D4—532no—
282C8×D4—532no—
283#432—1061no—
284#432—1061no—
285#431—531no—
286#427—531no—
287#423—531no—
288#423—531no—
289C2×C4⋊C8—1061no—
290C2×C4⋊C8—1061no—
291#416—531no—
292#416#615711yescharacteristic
293C4×M16—531no—
294C2×C4×C8—531no—
295#334—532no—
296#334—532no—
297Q8⋊C8—532no—
298Q8⋊C8—532no—
299D4⋊C8—532no—
300D4⋊C8—532no—
301#322—532no—
302#322—532no—
303C53#1806511yescharacteristic, complemented, Sylow
304C2×#41—531no—
305C2×#41C2×#615711yescharacteristic
306#135—1064no—
307C2×Q16—531no—
308C2×SD16—531no—
309C2×SD16—531no—
310C2×D8—531no—
311C2×M16—1061no—
312C2×M16—1061no—
313C2×M16—1061no—
314C22×C8—531no—
315C22×C8—1062no—
316C22×C8—1062no—
317C22×C8C2×#615711yescharacteristic
318C4×Q8—531no—
319C4×Q8—531no—
320C4×Q8—531no—
321C4×Q8—531no—
322C4×D4—531no—
323C4×D4—531no—
324C4×D4—531no—
325C4×D4—531no—
326C4×D4—1061no—
327C4×D4—1061no—
328C4×D4—1061no—
329C4×D4—1061no—
330#117—531no—
331#117—1061no—
332#117—1061no—
333#117C2×#615711yescharacteristic
334C2×C42—531no—
335C2×C42—1061no—
336C2×C42—1061no—
337#107—531no—
338#107—531no—
339#106—531no—
340#106—531no—
341C4⋊C8—531no—
342C4⋊C8—531no—
343C4⋊C8—532no—
344C4⋊C8—532no—
345C4⋊C8—1061no—
346C4⋊C8—1061no—
347C4⋊C8—1062no—
348C4⋊C8—1062no—
349#103—531no—
350#103—531no—
351#103C2×#615711yescharacteristic
352#103C2×#615711yescharacteristic
353#102—531no—
354#102—531no—
355#102C2×#615711yescharacteristic
356#102C2×#615711yescharacteristic
357C22⋊C8—532no—
358C22⋊C8—1062no—
359C22⋊C8—1062no—
360#97—1061no—
361#97—1062no—
362#97—1062no—
363C4×C8—532no—
364C4×C8—531no—
365C4×C8—531no—
366C4×C8—531no—
367C4×C8—531no—
368C4×C8—532no—
369#41—22no—
370#41—41no—
371#41—1062no—
372#41—2121no—
373C2×Q8—531no—
374C2×Q8C22×#615711yescharacteristic
375C2×D4—21no—
376C2×D4—531no—
377C2×D4—1061no—
378C2×D4C22×#615711yescharacteristic
379C22×C4—21no—
380C22×C4—531no—
381C22×C4—531no—
382C22×C4—1061no—
383C22×C4—1061no—
384C22×C4—1061no—
385C22×C4—1061no—
386C22×C4—1061no—
387C22×C4C22×#615711yescharacteristic
388Q16—1062no—
389SD16—1062no—
390SD16—1062no—
391D8—534no—
392M16—4241no—
393M16—4241no—
394M16—2122no—
395C2×C8—41no—
396C2×C8—531no—
397C2×C8—532no—
398C2×C8—532no—
399C2×C8—531no—
400C2×C8—1062no—
401C2×C8—1062no—
402C2×C8—1062no—
403C2×C8—1062no—
404C2×C8—4242no—
405C2×C8—4242no—
406C2×C8C22×#615711yescharacteristic
407C2×C8C22×#615711yescharacteristic
408C4⋊C4—531no—
409C4⋊C4—531no—
410C4⋊C4—1061no—
411C4⋊C4—1061no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4C22×#615711yescharacteristic
415C4⋊C4C22×#615711yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—1061no—
418C22⋊C4—1061no—
419C22⋊C4—1061no—
420C22⋊C4—1061no—
421C22⋊C4—1061no—
422C42—21no—
423C42—531no—
424C42—531no—
425C42—531no—
426C42—531no—
427C42—1061no—
428C42—1061no—
429C42—1061no—
430C42—1061no—
431C42—1061no—
432C23—21no—
433C23—1061no—
434C23D4×#615711yescharacteristic
435Q8—21no—
436Q8—21no—
437Q8—1061no—
438Q8—1061no—
439D4—21no—
440D4—42no—
441D4—532no—
442D4—1061no—
443D4—2122no—
444D4D53⋊M1612yes—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—42no—
451C2×C4—531no—
452C2×C4—531no—
453C2×C4—531no—
454C2×C4—531no—
455C2×C4—1061no—
456C2×C4—1061no—
457C2×C4—1062no—
458C2×C4—1061no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—2122no—
462C2×C4—2121no—
463C2×C4—2121no—
464C2×C4—2122no—
465C2×C4—2121no—
466C2×C4—2121no—
467C2×C4—2122no—
468C2×C4D4×#615711yescharacteristic
469C2×C4C23×#615711yescharacteristic
470C8—21no—
471C8—21no—
472C8—532no—
473C8—1062no—
474C8—1062no—
475C8—1061no—
476C8—2122no—
477C8—2122no—
478C8—2122no—
479C22—21no—
480C22—22no—
481C22—21no—
482C22—21no—
483C22—41no—
484C22—41no—
485C22—1062no—
486C22—2121no—
487C22—2121no—
488C22C2×D4×#615711yescharacteristic
489C4—21no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—531no—
494C4—531no—
495C4—532no—
496C4—1061no—
497C4—1061no—
498C4—2121no—
499C4—2121no—
500C4—2121no—
501C4C2×D53⋊M1611yescharacteristic
502C4#1681170111yescharacteristic
503C2—41no—
504C2—42no—
505C2—2122no—
506C2#2029145311yescharacteristic
507C2#2029147811yescharacteristic
508C2D53⋊#44411yescharacteristic
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#18065
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. #20291642C2semidirect
2. #20291641C2semidirect
3. #20290360C2semidirect
4. #20290321C2semidirect
5. #20290318C2semidirect
6. #20290312C2semidirect
7. #20290269C2semidirect
8. #20290052C2semidirect
9. #20290050C2semidirect
10. #20289924C2semidirect
156. #11719922M16semidirect
303. C53#18065semidirect

Maximal subgroups

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

subgroupindex
#202916422
#202916412
#202903602
#202903212
#202903182
#202903122
#202902692
#202900522
#202900502
#202899242
#202834832
#1806553