Finite Group Databasebeta

← #27355488

Subgroups — #27355488

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#20283817C211yesmaximal, characteristic
2#20283804C211yesmaximal, characteristic
3#20273624C211yesmaximal, characteristic, complemented
4#20273612C211yesmaximal, characteristic, complemented
5#20273587C211yesmaximal, characteristic, complemented
6#20273587C211yesmaximal, characteristic, complemented
7#20273566C211yesmaximal, characteristic, complemented
8#20273566C211yesmaximal, characteristic, complemented
9#20273534C211yesmaximal, characteristic, complemented
10Q16×#88322C211yesmaximal, characteristic, complemented
11#20273495C211yesmaximal, characteristic
12#20273188C211yesmaximal, characteristic
13#20273110C211yesmaximal, characteristic
14#20273015C211yesmaximal, characteristic
15C53×#3104C211yesmaximal, characteristic
16#16810754—21no—
17#16810754—21no—
18#16810754—21no—
19#16810754C2211yescharacteristic
20#16810754C2211yescharacteristic
21#16810754C2211yescharacteristic
22#16810754C2211yescharacteristic
23Q8×#88322—21no—
24Q8×#88322C2211yescharacteristic
25Q8×#88322C2211yescharacteristic
26Q8×#88322C2211yescharacteristic
27Q8×#88322C2211yescharacteristic
28#16810684—21no—
29#16810684—21no—
30#16810680—21no—
31#16810680C2211yescharacteristic
32#16810676—21no—
33#16810676C2211yescharacteristic
34#16810676C2211yescharacteristic
35#16810676C2211yescharacteristic
36#16810376C2211yescharacteristic
37#16810376C2211yescharacteristic
38#16810374C2211yescharacteristic
39#16810374C2211yescharacteristic
40Q8⋊#88322C2211yescharacteristic
41Q8⋊#88322C2211yescharacteristic
42Q8⋊#88322C2211yescharacteristic
43Q8⋊#88322C2211yescharacteristic
44Q8⋊#88322C2211yescharacteristic
45Q8⋊#88322C2211yescharacteristic
46Q8⋊#88322C2211yescharacteristic
47Q8⋊#88322C2211yescharacteristic
48#16810345C2211yescharacteristic
49#16810345C2211yescharacteristic
50#16810344C2211yescharacteristic
51C8×#88322C2211yescharacteristic
52C53×#533C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C53×#484C411yescharacteristic
55C53×#484C411yescharacteristic
56C53×#477C411yescharacteristic
57C53×#477C411yescharacteristic
58C53×#462C411yescharacteristic
59C53×#462C411yescharacteristic
60C53×#451C411yescharacteristic
61C53×#451C411yescharacteristic
62C4×Q16×C53C2211yescharacteristic
63C4×Q16×C53C2211yescharacteristic
64C4×Q16×C53C2211yescharacteristic
65C4×Q16×C53C2211yescharacteristic
66C53×#431C2211yescharacteristic
67C4⋊#88322—21no—
68C4⋊#88322—21no—
69C4⋊#88322—21no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322—21no—
73C4⋊#88322—21no—
74C4⋊#88322—21no—
75C4⋊#88322—21no—
76C4⋊#88322D411yescharacteristic
77C4⋊#88322D411yescharacteristic
78C4⋊#88322C2311yescharacteristic
79C4⋊#88322C2311yescharacteristic
80C4⋊#88322C2311yescharacteristic
81C4⋊#88322C2311yescharacteristic
82C22⋊#88322—21no—
83C22⋊#88322—21no—
84C22⋊#88322—21no—
85C22⋊#88322D411yescharacteristic
86C22⋊#88322D411yescharacteristic
87#13133128—21no—
88#13133128—21no—
89#13133128—21no—
90#13133128—21no—
91#13133128—21no—
92C4×#88322—21no—
93C4×#88322—21no—
94C4×#88322—21no—
95C4×#88322—21no—
96C4×#88322—21no—
97C4×#88322C2311yescharacteristic
98C4×#88322C2311yescharacteristic
99C4×#88322C2311yescharacteristic
100C4×#88322C2311yescharacteristic
101C22×Q8×C53C2×C411yescharacteristic
102C2×Q16×C53C2×C411yescharacteristic
103C2×Q16×C53C2×C411yescharacteristic
104C2×Q16×C53C2×C411yescharacteristic
105C2×Q16×C53C2×C411yescharacteristic
106C4⋊Q8×C53C2×C411yescharacteristic
107C4⋊Q8×C53C2×C411yescharacteristic
108#125×C53—21no—
109C22⋊Q8×C53—21no—
110C22⋊Q8×C53C2×C411yescharacteristic
111C22⋊Q8×C53C2×C411yescharacteristic
112C4×Q8×C53—21no—
113C4×Q8×C53—21no—
114C4×Q8×C53—21no—
115C4×Q8×C53C2×C411yescharacteristic
116C4×Q8×C53C2×C411yescharacteristic
117C4×Q8×C53C2311yescharacteristic
118C4×Q8×C53C2311yescharacteristic
119C4×Q8×C53C2311yescharacteristic
120C4×Q8×C53C2311yescharacteristic
121#117×C53—21no—
122C2×C4⋊C4×C53—21no—
123C2×C4⋊C4×C53C2×C411yescharacteristic
124C2×C42×C53—21no—
125C2×C42×C53C2311yescharacteristic
126#107×C53C2×C411yescharacteristic
127#107×C53C2×C411yescharacteristic
128#107×C53C2×C411yescharacteristic
129#107×C53C2×C411yescharacteristic
130C4⋊C8×C53C2×C411yescharacteristic
131C4⋊C8×C53C2×C411yescharacteristic
132#103×C53C2×C411yescharacteristic
133#103×C53C2×C411yescharacteristic
134#103×C53C2×C411yescharacteristic
135#103×C53C2×C411yescharacteristic
136#103×C53C2×C411yescharacteristic
137#103×C53C2×C411yescharacteristic
138#103×C53C2×C411yescharacteristic
139#103×C53C2×C411yescharacteristic
140C22⋊C8×C53C2×C411yescharacteristic
141C22⋊C8×C53C2×C411yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—22no—
147C2×Q8×C53—41no—
148C2×Q8×C53C22×C411yescharacteristic
149C2×Q8×C53C22×C411yescharacteristic
150C2×Q8×C53C22×C411yescharacteristic
151C2×Q8×C53C22×C411yescharacteristic
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53C22×C411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53#4111yescharacteristic
157Q16×C53—22no—
158Q16×C53—22no—
159Q16×C53—41no—
160Q16×C53—41no—
161C2×C8×C53C22×C411yescharacteristic
162C2×C8×C53C22×C411yescharacteristic
163C2×C8×C53C22×C411yescharacteristic
164C2×C8×C53C22×C411yescharacteristic
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53C22×C411yescharacteristic
178C4⋊C4×C53C22×C411yescharacteristic
179C4⋊C4×C53C22×C411yescharacteristic
180C4⋊C4×C53C22×C411yescharacteristic
181C22⋊C4×C53—21no—
182C22⋊C4×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53C22×C411yescharacteristic
190C42×C53C2×D411yescharacteristic
191C42×C53#4111yescharacteristic
192C42×C53C2411yescharacteristic
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C2×D411yescharacteristic
200C2×#88322C2×D411yescharacteristic
201C2×#88322C2×D411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C23×C53C4×D411yescharacteristic
204Q8×C53—21no—
205Q8×C53—21no—
206Q8×C53—21no—
207Q8×C53—21no—
208Q8×C53—21no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—42no—
212Q8×C53—42no—
213Q8×C53#13112yes—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—41no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C23×C411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C8×C53—21no—
236C8×C53—21no—
237C8×C53—21no—
238C8×C53—21no—
239#88322—21no—
240#88322—21no—
241#88322—21no—
242#88322—21no—
243#88322—41no—
244#88322—41no—
245#88322—41no—
246#88322—41no—
247#88322—41no—
248#88322—41no—
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—21no—
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—41no—
260C4×C53—41no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53C2×#13111yescharacteristic
267C4×C53#56811yescharacteristic
268C4×C53#57811yescharacteristic
269C4×C53#57811yescharacteristic
270#3104D5311yescharacteristic
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#533C2×D5311yescharacteristic
290C2×C4×Q8C2×D5311yescharacteristic
291#484Dic5311yescharacteristic
292#484Dic5311yescharacteristic
293#477Dic5311yescharacteristic
294#477Dic5311yescharacteristic
295#462Dic5311yescharacteristic
296#462Dic5311yescharacteristic
297#451Dic5311yescharacteristic
298#451Dic5311yescharacteristic
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×Q16C2×D5311yescharacteristic
312C4×Q16C2×D5311yescharacteristic
313C4×Q16C2×D5311yescharacteristic
314C4×Q16C2×D5311yescharacteristic
315#433—1061no—
316#433—1061no—
317#432—531no—
318#432—1061no—
319#431—531no—
320#431—531no—
321#431—531no—
322#431—1061no—
323#431C2×D5311yescharacteristic
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×Dic5311yescharacteristic
342C2×Q16C2×Dic5311yescharacteristic
343C2×Q16C2×Dic5311yescharacteristic
344C2×Q16C2×Dic5311yescharacteristic
345C2×Q16C2×Dic5311yescharacteristic
346C4⋊Q8C2×Dic5311yescharacteristic
347C4⋊Q8C2×Dic5311yescharacteristic
348#125—21no—
349C22⋊Q8—21no—
350C22⋊Q8C2×Dic5311yescharacteristic
351C22⋊Q8C2×Dic5311yescharacteristic
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8C2×Dic5311yescharacteristic
356C4×Q8C2×Dic5311yescharacteristic
357C4×Q8C22×D5311yescharacteristic
358C4×Q8C22×D5311yescharacteristic
359C4×Q8C22×D5311yescharacteristic
360C4×Q8C22×D5311yescharacteristic
361#117—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C2×Dic5311yescharacteristic
364C2×C42—21no—
365C2×C42C22×D5311yescharacteristic
366#107C2×Dic5311yescharacteristic
367#107C2×Dic5311yescharacteristic
368#107C2×Dic5311yescharacteristic
369#107C2×Dic5311yescharacteristic
370C4⋊C8—531no—
371C4⋊C8—531no—
372C4⋊C8—531no—
373C4⋊C8—531no—
374C4⋊C8—531no—
375C4⋊C8—531no—
376C4⋊C8—1061no—
377C4⋊C8—1061no—
378C4⋊C8—1061no—
379C4⋊C8—1061no—
380C4⋊C8—1061no—
381C4⋊C8—1061no—
382C4⋊C8—1061no—
383C4⋊C8—1061no—
384C4⋊C8—1061no—
385C4⋊C8C2×Dic5311yescharacteristic
386C4⋊C8C2×Dic5311yescharacteristic
387#103C2×Dic5311yescharacteristic
388#103C2×Dic5311yescharacteristic
389#103C2×Dic5311yescharacteristic
390#103C2×Dic5311yescharacteristic
391#103C2×Dic5311yescharacteristic
392#103C2×Dic5311yescharacteristic
393#103C2×Dic5311yescharacteristic
394#103C2×Dic5311yescharacteristic
395C22⋊C8—531no—
396C22⋊C8—531no—
397C22⋊C8—1061no—
398C22⋊C8—1061no—
399C22⋊C8—1061no—
400C22⋊C8C2×Dic5311yescharacteristic
401C22⋊C8C2×Dic5311yescharacteristic
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—1061no—
412C4×C8—1061no—
413C4×C8—1061no—
414C4×C8—1061no—
415C4×C8—1061no—
416C4×C8C22×D5311yescharacteristic
417C4×C8C22×D5311yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—41no—
422C2×Q8C22×Dic5311yescharacteristic
423C2×Q8C22×Dic5311yescharacteristic
424C2×Q8C22×Dic5311yescharacteristic
425C2×Q8C22×Dic5311yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4D4×D5311yescharacteristic
429C22×C4#1171994711yescharacteristic
430C22×C4C22×Dic5311yescharacteristic
431Q16—22no—
432Q16—22no—
433Q16—41no—
434Q16—41no—
435C2×C8—531no—
436C2×C8—531no—
437C2×C8—531no—
438C2×C8—531no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8C22×Dic5311yescharacteristic
446C2×C8C22×Dic5311yescharacteristic
447C2×C8C22×Dic5311yescharacteristic
448C2×C8C22×Dic5311yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4C22×Dic5311yescharacteristic
462C4⋊C4C22×Dic5311yescharacteristic
463C4⋊C4C22×Dic5311yescharacteristic
464C4⋊C4C22×Dic5311yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42D4×D5311yescharacteristic
474C42#1171994711yescharacteristic
475C42C22×Dic5311yescharacteristic
476C42C23×D5311yescharacteristic
477C23D4×Dic5311yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—42no—
486Q8—42no—
487Q8#1313322412yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4D4×Dic5311yescharacteristic
504C2×C4D4×Dic5311yescharacteristic
505C2×C4D4×Dic5311yescharacteristic
506C2×C4C23×Dic5311yescharacteristic
507C2×C4C2×D4×D5311yescharacteristic
508C2×C4C2×#1171994711yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
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×Dic5311yescharacteristic
527C4—21no—
528C4—22no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4C2×#1313322411yescharacteristic
540C4#1681139011yescharacteristic
541C4#1681143611yescharacteristic
542C4#1681143611yescharacteristic
543C2—41no—
544C2#2028344211yescharacteristic
545C2#2028358311yescharacteristic
546C2#2028361811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#3104C2
socleC22×C53C2×C4×D4
prime radicalC23×C53C4×D4
cosocleC42×C53C24
hypercenter#3104D53
hypocenterC53#18032
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. #20273624C2semidirect
4. #20273612C2semidirect
5. #20273587C2semidirect
6. #20273587C2semidirect
7. #20273566C2semidirect
8. #20273566C2semidirect
9. #20273534C2semidirect
10. Q16×#88322C2semidirect
340. C53#18032semidirect

Maximal subgroups

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

subgroupindex
#202838172
#202838042
#202736242
#202736122
#202735872
#202735662
#202735342
Q16×#883222
#202734952
#202731882
#202731102
#202730152
C53×#31042
#1803253