Finite Group Databasebeta

← #28317476

Subgroups — #28317476

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291624C211yesmaximal, characteristic
2#20291618C211yesmaximal, characteristic
3#20290360C211yesmaximal, characteristic, complemented
4#20290355C211yesmaximal, characteristic, complemented
5#20290341C211yesmaximal, characteristic, complemented
6#20290340C211yesmaximal, characteristic, complemented
7#20290328C211yesmaximal, characteristic, complemented
8#20290326C211yesmaximal, characteristic, complemented
9#20290308C211yesmaximal, characteristic, complemented
10#20290302C211yesmaximal, characteristic, complemented
11#20290286C211yesmaximal, characteristic
12#20290077C211yesmaximal, characteristic
13#20290017C211yesmaximal, characteristic
14#20289936C211yesmaximal, characteristic
15#20284749C211yesmaximal, characteristic
16#16811675—21no—
17#16811674—21no—
18#16811674—21no—
19#16811674C2211yescharacteristic
20#16811674C2211yescharacteristic
21#16811674C2211yescharacteristic
22#16811673C2211yescharacteristic
23#16811672—21no—
24#16811672C2211yescharacteristic
25Q8×#88324C2211yescharacteristic
26Q8×#88324C2211yescharacteristic
27Q8×#88324C2211yescharacteristic
28#16811647—21no—
29#16811647—21no—
30#16811644—21no—
31#16811644C2211yescharacteristic
32#16811640—21no—
33#16811640C2211yescharacteristic
34#16811639C2211yescharacteristic
35#16811639C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46Q8⋊#88324C2211yescharacteristic
47Q8⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811175C2211yescharacteristic
53C2×Q8×Dic53C2211yescharacteristic
54#16810959C411yescharacteristic
55#16810956C411yescharacteristic
56#16810916C411yescharacteristic
57#16810916C411yescharacteristic
58#16810831C411yescharacteristic
59#16810831C411yescharacteristic
60#16810779C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62Q16×Dic53C2211yescharacteristic
63#16810715C2211yescharacteristic
64#16810715C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#451C411yescharacteristic
67#13133284—21no—
68#13133284—21no—
69#13133284—21no—
70#13133284—21no—
71#13133284D411yescharacteristic
72#13133284D411yescharacteristic
73C4⋊#88324—21no—
74C4⋊#88324—21no—
75C4⋊#88324—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324C2311yescharacteristic
79C4⋊#88324C2311yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C22⋊#88324—21no—
83C22⋊#88324—21no—
84C22⋊#88324—21no—
85C22⋊#88324D411yescharacteristic
86C22⋊#88324D411yescharacteristic
87#13133272—21no—
88#13133272—21no—
89#13133272—21no—
90#13133272—21no—
91#13133272—21no—
92C4×#88324—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719925C2×C411yescharacteristic
102C2×#11719925C2×C411yescharacteristic
103#13133216C2×C411yescharacteristic
104#13133215C2×C411yescharacteristic
105#13133207—21no—
106#13133196C2×C411yescharacteristic
107#13133194—21no—
108Q8×Dic53—21no—
109Q8×Dic53—21no—
110Q8×Dic53C2×C411yescharacteristic
111Q8×Dic53C2×C411yescharacteristic
112Q8×Dic53C2311yescharacteristic
113Q8×Dic53C2311yescharacteristic
114Q8×Dic53C2311yescharacteristic
115#13133182—21no—
116#13133182C2311yescharacteristic
117#13133173—21no—
118C2×C4⋊Dic53—21no—
119C2×C4⋊Dic53C2×C411yescharacteristic
120C2×C4×Dic53—21no—
121C2×C4×Dic53C2311yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133151C2×C411yescharacteristic
124#13133150C2×C411yescharacteristic
125#13133150C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133142C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133141C2×C411yescharacteristic
132#13133141C2×C411yescharacteristic
133#13133141C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C22×Q8×C53C2×C411yescharacteristic
138C2×Q16×C53C2×C411yescharacteristic
139C2×Q16×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144C2×Q8×C53—21no—
145C2×Q8×C53—22no—
146C2×Q8×C53—41no—
147C2×Q8×C53C22×C411yescharacteristic
148C2×Q8×C53C22×C411yescharacteristic
149C2×Q8×C53C22×C411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53#4111yescharacteristic
155C2×Dic106—21no—
156C2×Dic106C22×C411yescharacteristic
157C2×#88324—21no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324C2×D411yescharacteristic
164C2×#88324C2×D411yescharacteristic
165C2×#88324C2×D411yescharacteristic
166C2×#88324C2×D411yescharacteristic
167Q16×C53—22no—
168Q16×C53—41no—
169C2×C8×C53C22×C411yescharacteristic
170C2×C8×C53C22×C411yescharacteristic
171C4⋊C4×C53—21no—
172C4⋊C4×C53C22×C411yescharacteristic
173C22⋊C4×C53—21no—
174C22⋊Dic53—21no—
175#11719925—22no—
176#11719925—41no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53C22×C411yescharacteristic
184C4⋊Dic53C22×C411yescharacteristic
185C4⋊Dic53C22×C411yescharacteristic
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53C22×C411yescharacteristic
198C4×Dic53C2×D411yescharacteristic
199C4×Dic53#4111yescharacteristic
200C4×Dic53C2411yescharacteristic
201C2×#88322C22×C411yescharacteristic
202C2×#88322C22×C411yescharacteristic
203C23×C53C4×D411yescharacteristic
204Q8×C53—21no—
205Q8×C53—21no—
206Q8×C53—21no—
207Q8×C53—21no—
208Q8×C53—21no—
209Q8×C53—42no—
210Q8×C53#13112yes—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—42no—
218C2×C4×C53C4×D411yescharacteristic
219C2×C4×C53C23×C411yescharacteristic
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—41no—
227C2×Dic53—42no—
228C2×Dic53C4×D411yescharacteristic
229C2×Dic53C4×D411yescharacteristic
230C2×Dic53C22×D411yescharacteristic
231C2×Dic53C2×#4111yescharacteristic
232Dic106—21no—
233Dic106—21no—
234Dic106—42no—
235#88324—21no—
236#88324—21no—
237#88324—21no—
238#88324—21no—
239#88324—41no—
240#88324—41no—
241#88324—41no—
242#88324—41no—
243#88324—41no—
244#88324—41no—
245C8×C53—21no—
246C8×C53—21no—
247#88322—21no—
248#88322—21no—
249#18032—531nomaximal, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53C2×C4×D411yescharacteristic
254C4×C53—21no—
255C4×C53—22no—
256C4×C53—41no—
257C4×C53—41no—
258C4×C53—41no—
259C4×C53C2×#13111yescharacteristic
260C4×C53#56811yescharacteristic
261Dic53—21no—
262Dic53—21no—
263Dic53—21no—
264Dic53—41no—
265Dic53—41no—
266Dic53—41no—
267Dic53—41no—
268Dic53#57811yescharacteristic
269Dic53#57811yescharacteristic
270#3104—531no—
271#2986—531no—
272#2984—531no—
273#1593—531no—
274#1591—531no—
275#1587—531no—
276#1587—531no—
277#1584—531no—
278#1584—531no—
279#1579—531no—
280C8×Q16—531no—
281#1572—531no—
282#1497—531no—
283#1479—531no—
284#1454—531no—
285C2×C53—41no—
286C2×C53#293311yescharacteristic
287C2×C53#295011yescharacteristic
288C2×C53#295511yescharacteristic
289#533—531no—
290C2×C4×Q8—531no—
291#484—531no—
292#484—531no—
293#477—531no—
294#477—531no—
295#462—531no—
296#462—531no—
297#451—531no—
298#451#615711yescharacteristic
299#446—531no—
300#446—531no—
301#446—531no—
302#446—531no—
303#446—1061no—
304#446—1061no—
305#446—1061no—
306C8×Q8—531no—
307C8×Q8—531no—
308C8×Q8—531no—
309C8×Q8—531no—
310C8×Q8—1061no—
311C4×Q16—531no—
312C4×Q16—531no—
313C4×Q16—531no—
314C4×Q16—531no—
315#433—1061no—
316#433—1061no—
317#432—531no—
318#432—1061no—
319#431—531no—
320#431—531no—
321#431—531no—
322#431—531no—
323#431—1061no—
324#335—531no—
325#335—531no—
326#334—531no—
327#334—531no—
328Q8⋊C8—531no—
329Q8⋊C8—531no—
330Q8⋊C8—531no—
331Q8⋊C8—531no—
332Q8⋊C8—531no—
333Q8⋊C8—531no—
334Q8⋊C8—531no—
335Q8⋊C8—531no—
336#322—531no—
337#322—531no—
338#322—531no—
339C82—531no—
340C53#1803211yescharacteristic, complemented, Sylow
341C22×Q8C2×#615711yescharacteristic
342C2×Q16—531no—
343C2×Q16—531no—
344C2×Q16C2×#615711yescharacteristic
345C2×Q16C2×#615711yescharacteristic
346C4⋊Q8—531no—
347C4⋊Q8—531no—
348#125—1061no—
349C22⋊Q8—531no—
350C22⋊Q8—1061no—
351C22⋊Q8C2×#615711yescharacteristic
352C4×Q8—531no—
353C4×Q8—531no—
354C4×Q8—531no—
355C4×Q8—531no—
356C4×Q8—531no—
357C4×Q8—531no—
358C4×Q8—1061no—
359C4×Q8—1061no—
360C4×Q8—1061no—
361#117—1061no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—1061no—
364C2×C42—531no—
365C2×C42—1061no—
366#107—531no—
367#107—531no—
368#107—531no—
369#107—531no—
370C4⋊C8—531no—
371C4⋊C8—531no—
372C4⋊C8—531no—
373C4⋊C8—531no—
374C4⋊C8—531no—
375C4⋊C8—531no—
376C4⋊C8—531no—
377C4⋊C8—531no—
378C4⋊C8—1061no—
379C4⋊C8—1061no—
380C4⋊C8—1061no—
381C4⋊C8—1061no—
382C4⋊C8—1061no—
383C4⋊C8—1061no—
384C4⋊C8—1061no—
385C4⋊C8—1061no—
386C4⋊C8—1061no—
387#103—531no—
388#103—531no—
389#103—531no—
390#103—531no—
391#103—531no—
392#103—531no—
393#103C2×#615711yescharacteristic
394#103C2×#615711yescharacteristic
395C22⋊C8—531no—
396C22⋊C8—531no—
397C22⋊C8—531no—
398C22⋊C8—1061no—
399C22⋊C8—1061no—
400C22⋊C8—1061no—
401C22⋊C8C2×#615711yescharacteristic
402#97—1061no—
403#97—1061no—
404#97—1061no—
405#97—1061no—
406#97—1061no—
407C4×C8—531no—
408C4×C8—531no—
409C4×C8—531no—
410C4×C8—531no—
411C4×C8—531no—
412C4×C8—531no—
413C4×C8—1061no—
414C4×C8—1061no—
415C4×C8—1061no—
416C4×C8—1061no—
417C4×C8—1061no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8—531no—
422C2×Q8—1061no—
423C2×Q8C22×#615711yescharacteristic
424C2×Q8C22×#615711yescharacteristic
425C2×Q8C22×#615711yescharacteristic
426C22×C4—21no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—1061no—
430C22×C4C22×#615711yescharacteristic
431Q16—22no—
432Q16—41no—
433Q16—1062no—
434Q16—2121no—
435C2×C8—531no—
436C2×C8—531no—
437C2×C8—531no—
438C2×C8—531no—
439C2×C8—531no—
440C2×C8—531no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8C22×#615711yescharacteristic
448C2×C8C22×#615711yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—531no—
451C4⋊C4—531no—
452C4⋊C4—531no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4C22×#615711yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—1061no—
467C42—531no—
468C42—531no—
469C42—531no—
470C42—531no—
471C42—1061no—
472C42—1061no—
473C42—1061no—
474C42—1061no—
475C42—1061no—
476C42—1061no—
477C23D4×#615711yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—42no—
484Q8—1061no—
485Q8—1061no—
486Q8—2122no—
487Q8#1313330412yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—42no—
495C2×C4—531no—
496C2×C4—531no—
497C2×C4—531no—
498C2×C4—531no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—2121no—
506C2×C4—2122no—
507C2×C4D4×#615711yescharacteristic
508C2×C4C23×#615711yescharacteristic
509C8—21no—
510C8—21no—
511C8—1061no—
512C8—1061no—
513C8—1061no—
514C8—1061no—
515C8—1061no—
516C8—1061no—
517C8—2121no—
518C8—2121no—
519C8—2121no—
520C8—2121no—
521C8—2121no—
522C8—2121no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×D4×#615711yescharacteristic
527C4—21no—
528C4—22no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—531no—
533C4—531no—
534C4—1061no—
535C4—1061no—
536C4—1061no—
537C4—2121no—
538C4—2121no—
539C4—2121no—
540C4—2121no—
541C4C2×#1313330411yescharacteristic
542C4#1681170111yescharacteristic
543C2—41no—
544C2#2029144711yescharacteristic
545C2#2029147611yescharacteristic
546C2#2029148811yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#451C4
socleC22×C53C2×C4×D4
prime radicalC23×C53C4×D4
cosocleC4×Dic53C24
hypercenter#451#6157
hypocenterC53#18032
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #20290360C2semidirect
4. #20290355C2semidirect
5. #20290341C2semidirect
6. #20290340C2semidirect
7. #20290328C2semidirect
8. #20290326C2semidirect
9. #20290308C2semidirect
10. #20290302C2semidirect
340. C53#18032semidirect

Maximal subgroups

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

subgroupindex
#202916242
#202916182
#202903602
#202903552
#202903412
#202903402
#202903282
#202903262
#202903082
#202903022
#202902862
#202900772
#202900172
#202899362
#202847492
#1803253