Finite Group Databasebeta

← #28314368

Subgroups — #28314368

Order 13568.

944 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291404C212yesmaximal, complemented
2#20289996C212yesmaximal, complemented
3#20289987C212yesmaximal, complemented
4#20289980C212yesmaximal
5#20289977C211yesmaximal, characteristic, complemented
6#20289966C211yesmaximal, characteristic, complemented
7#20289963C211yesmaximal, characteristic, complemented
8#20289957C211yesmaximal, characteristic, complemented
9#20289946C211yesmaximal, characteristic, complemented
10C2×#16811468C211yesmaximal, characteristic, complemented
11C2×#16811313C211yesmaximal, characteristic
12#16811647—22no—
13#16811647—22no—
14#16811644C2212yes—
15#16811643C2212yes—
16#16811640C2212yes—
17#16811639C2212yes—
18#16811629—22no—
19#16811593C2212yes—
20C2×C22⋊#88324C2212yes—
21#16811587—22no—
22#16811469C2212yes—
23#16811469C2212yes—
24#16811468C2211yescharacteristic
25#16811468C2211yescharacteristic
26#16811468C2212yes—
27#16811468C2211yescharacteristic
28#16811468C2211yescharacteristic
29#16811468C2212yes—
30#16811466C2212yes—
31#16811466C2212yes—
32#16811313—22no—
33#16811313—22no—
34#16811313—22no—
35#16811313—22no—
36C2×#13133210C411yescharacteristic
37C2×#13133210C2211yescharacteristic
38C2×#13133209C411yescharacteristic
39C2×#13133207C411yescharacteristic
40C2×#13133207C2211yescharacteristic
41C2×#13133198C411yescharacteristic
42C2×#13133198C2211yescharacteristic
43C2×#13133196C411yescharacteristic
44C2×#13133196C411yescharacteristic
45C2×C4×Dic106C2211yescharacteristic
46C2×#13133176C2211yescharacteristic
47C2×#13133173C411yescharacteristic
48C2×#13133173C2211yescharacteristic
49C22×C4⋊Dic53C2211yescharacteristic
50C2×C4×D4×C53C411yescharacteristic
51C2×#11719940—22no—
52C22×#88324—22no—
53#13133284—22no—
54#13133284—22no—
55C4⋊#88324—22no—
56C4⋊#88324—22no—
57C22⋊#88324—22no—
58C22⋊#88324—22no—
59C22⋊#88324C2312yes—
60C22⋊#88324C2312yes—
61C22⋊#88324C2312yes—
62C22⋊#88324C2312yes—
63#13133272—22no—
64#13133272—22no—
65C4×#88324—22no—
66C4×#88324—22no—
67C22×Dic106C2×C411yescharacteristic
68C23×Dic53C2×C411yescharacteristic
69C23×Dic53C2311yescharacteristic
70#13133210—22no—
71#13133210—22no—
72#13133210—22no—
73#13133210—22no—
74#13133209—22no—
75#13133209—22no—
76#13133207—22no—
77#13133207—22no—
78#13133207—22no—
79#13133207—22no—
80#13133198—22no—
81#13133198—22no—
82#13133198—22no—
83#13133198—22no—
84#13133196—22no—
85#13133196—22no—
86#13133196—22no—
87#13133196—22no—
88C4×Dic106—22no—
89C4×Dic106—22no—
90#13133176—22no—
91#13133176—22no—
92#13133173—21no—
93#13133173—21no—
94#13133173—21no—
95#13133173—21no—
96#13133173D412yes—
97#13133173D412yes—
98#13133173D412yes—
99#13133173D412yes—
100C2×C4⋊Dic53—22no—
101C2×C4⋊Dic53—22no—
102C2×C4⋊Dic53C2×C411yescharacteristic
103C2×C4⋊Dic53C2×C411yescharacteristic
104C2×C4⋊Dic53C2×C411yescharacteristic
105C2×C4⋊Dic53C2311yescharacteristic
106C2×C4⋊Dic53C2311yescharacteristic
107C2×#11719919C2×C411yescharacteristic
108C2×#11719919C2×C411yescharacteristic
109C2×#11719919C2×C411yescharacteristic
110C2×#11719919C2×C411yescharacteristic
111C2×#11719919C2×C411yescharacteristic
112C2×#11719919C2×C411yescharacteristic
113C2×#11719919C2×C411yescharacteristic
114C2×#11719919C2×C411yescharacteristic
115C2×#11719919C2×C411yescharacteristic
116C2×#11719919C2311yescharacteristic
117C2×C22⋊Dic53C2×C411yescharacteristic
118C2×C22⋊Dic53C2×C411yescharacteristic
119C2×C22⋊Dic53C2×C411yescharacteristic
120C2×C22⋊Dic53C2×C411yescharacteristic
121C2×C22⋊Dic53C2×C411yescharacteristic
122C2×C22⋊Dic53C2311yescharacteristic
123C2×C4×Dic53C2×C411yescharacteristic
124C2×C4×Dic53C2×C411yescharacteristic
125C2×C4×Dic53C2311yescharacteristic
126C2×C4×Dic53C2311yescharacteristic
127C22×D4×C53C2×C411yescharacteristic
128C23×C4×C53C2×C411yescharacteristic
129C23×C4×C53C2×C411yescharacteristic
130C4×D4×C53—21no—
131C4×D4×C53—21no—
132C4×D4×C53—21no—
133C4×D4×C53—21no—
134C2×C4⋊C4×C53C2×C411yescharacteristic
135C2×C22⋊C4×C53C2×C411yescharacteristic
136C2×C22⋊C4×C53C2×C411yescharacteristic
137C2×C42×C53C2×C411yescharacteristic
138C24×C53C22×C411yescharacteristic
139C24×C53C22×C411yescharacteristic
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—42no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—41no—
150C22×C4×C53—41no—
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C22×C411yescharacteristic
154C22×C4×C53C22×C411yescharacteristic
155C22×C4×C53C22×C411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—42no—
159C22×Dic53—42no—
160C22×Dic53C22×C411yescharacteristic
161C22×Dic53C22×C411yescharacteristic
162C22×Dic53C22×C411yescharacteristic
163C22×Dic53C22×C411yescharacteristic
164C22×Dic53C22×C411yescharacteristic
165C22×Dic53C22×C411yescharacteristic
166C22×Dic53C22×C411yescharacteristic
167C22×Dic53C2×D412yes—
168C22×Dic53C2×D412yes—
169C22×Dic53C2411yescharacteristic
170C2×Dic106—22no—
171C2×Dic106—22no—
172C2×Dic106—41no—
173#11719940—44no—
174C2×#88324—22no—
175C2×#88324—22no—
176C2×#88324—22no—
177C2×#88324—22no—
178C2×#88324—44no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—21no—
183C22⋊C4×C53C22⋊C411yescharacteristic
184C22⋊C4×C53C22⋊C411yescharacteristic
185C22⋊C4×C53C22⋊C411yescharacteristic
186C22⋊C4×C53C22⋊C411yescharacteristic
187C42×C53—21no—
188C42×C53—21no—
189C22⋊Dic53—21no—
190C22⋊Dic53—21no—
191C22⋊Dic53—22no—
192C22⋊Dic53—22no—
193C22⋊Dic53—22no—
194C22⋊Dic53—22no—
195C22⋊Dic53C22⋊C412yes—
196C22⋊Dic53C22⋊C412yes—
197C4⋊Dic53—22no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4⋊Dic53—22no—
201C4⋊Dic53—22no—
202#11719919—21no—
203#11719919—21no—
204#11719919—21no—
205#11719919—21no—
206#11719919—22no—
207#11719919—22no—
208#11719919—22no—
209#11719919—22no—
210#11719919—22no—
211#11719919—22no—
212#11719919C22⋊C412yes—
213#11719919C22⋊C412yes—
214#11719919C22⋊C412yes—
215#11719919C22⋊C412yes—
216C4×Dic53—21no—
217C4×Dic53—21no—
218C4×Dic53—21no—
219C4×Dic53—21no—
220C4×Dic53C2×D412yes—
221C4×Dic53C2×D412yes—
222C4×Dic53C2×D412yes—
223C4×Dic53C2×D412yes—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—22no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53C2×C22⋊C411yescharacteristic
234C23×C53C2×C22⋊C411yescharacteristic
235C23×C53C23×C411yescharacteristic
236D4×C53—42no—
237D4×C53—42no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—41no—
252C2×C4×C53C2×C22⋊C411yescharacteristic
253C2×C4×C53C2×C22⋊C411yescharacteristic
254C2×C4×C53C2×C22⋊C411yescharacteristic
255C2×C4×C53C2×C22⋊C411yescharacteristic
256C2×C4×C53#13112yes—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—24no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—41no—
267C2×Dic53—42no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53C2×C22⋊C412yes—
276C2×Dic53C2×C22⋊C412yes—
277C2×Dic53C2×C22⋊C412yes—
278C2×Dic53C22×D412yes—
279Dic106—44no—
280#88324—42no—
281#88324—42no—
282#88324—42no—
283#88324—42no—
284#17417—531nomaximal, Sylow
285C22×C53—21no—
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—21no—
289C22×C53—21no—
290C22×C53—21no—
291C22×C53—21no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53#41012yes—
300C22×C53C22×C22⋊C411yescharacteristic
301C22×C53C2×#13111yescharacteristic
302C22×C53#56811yescharacteristic
303C4×C53—22no—
304C4×C53—21no—
305C4×C53—21no—
306C4×C53—21no—
307C4×C53—21no—
308C4×C53—41no—
309C4×C53—41no—
310Dic53—22no—
311Dic53—22no—
312Dic53—22no—
313Dic53—22no—
314Dic53—41no—
315Dic53—41no—
316Dic53—41no—
317Dic53—41no—
318C2×#552—531no—
319#2924—532no—
320#1473—532no—
321#1470—532no—
322#1468—532no—
323#1467—531no—
324#1464—531no—
325#1463—531no—
326#1461—531no—
327#1458—531no—
328C2×#324—531no—
329C2×C53—21no—
330C2×C53—21no—
331C2×C53—22no—
332C2×C53—41no—
333C2×C53—81no—
334C2×C53#288011yescharacteristic
335C2×C53C2×#41011yescharacteristic
336C2×C53#288911yescharacteristic
337#552—1062no—
338#552—1062no—
339#552—1062no—
340#552—1062no—
341C2×#126—531no—
342C2×#126—531no—
343C2×#125—531no—
344C2×#125—531no—
345C2×#125—531no—
346C2×#123—531no—
347C2×#123—531no—
348C2×C22⋊Q8—531no—
349C2×C22⋊Q8—531no—
350C2×C4×Q8—531no—
351C2×C4×D4#615711yescharacteristic
352C2×#117—531no—
353C2×#117—531no—
354C2×#117—531no—
355C22×C4⋊C4—531no—
356#433—1062no—
357#433—1062no—
358#432—532no—
359#432—532no—
360#431—532no—
361#431—532no—
362#424—1062no—
363#407—532no—
364C2×C22⋊C8—532no—
365#405—1062no—
366#324—531no—
367#324—531no—
368#324—532no—
369#324—531no—
370#324—531no—
371#324—532no—
372#324—532no—
373#324—532no—
374#323—532no—
375#323—532no—
376C53#1741711yescharacteristic, complemented, Sylow
377C22×Q8—531no—
378C22×D4C2×#615711yescharacteristic
379C23×C4—531no—
380C23×C4—531no—
381C23×C4C2×#615711yescharacteristic
382C23×C4C2×#615711yescharacteristic
383C2×M16—1062no—
384C22×C8—1062no—
385#126—1062no—
386#126—1062no—
387#126—1062no—
388#126—1062no—
389#125—1062no—
390#125—1062no—
391#125—1062no—
392#125—1062no—
393#125—1062no—
394#125—1062no—
395#123—1062no—
396#123—1062no—
397#123—1062no—
398#123—1062no—
399C22⋊Q8—1062no—
400C22⋊Q8—1062no—
401C22⋊Q8—1062no—
402C22⋊Q8—1062no—
403C4×Q8—1062no—
404C4×Q8—1062no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409#117—532no—
410#117—532no—
411#117—532no—
412#117—532no—
413#117—1062no—
414#117—1062no—
415#117—1061no—
416#117—1061no—
417#117—1061no—
418#117—1061no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—1062no—
435C2×C4⋊C4—1062no—
436C2×C4⋊C4C2×#615711yescharacteristic
437C2×C22⋊C4—531no—
438C2×C22⋊C4—531no—
439C2×C22⋊C4—531no—
440C2×C22⋊C4—531no—
441C2×C22⋊C4—531no—
442C2×C22⋊C4—531no—
443C2×C22⋊C4C2×#615711yescharacteristic
444C2×C22⋊C4C2×#615711yescharacteristic
445C2×C42—531no—
446C2×C42—531no—
447C2×C42—531no—
448C2×C42—531no—
449C2×C42C2×#615711yescharacteristic
450C4⋊C8—1062no—
451C4⋊C8—1062no—
452C4⋊C8—1062no—
453C4⋊C8—1062no—
454C22⋊C8—532no—
455C22⋊C8—532no—
456C22⋊C8—532no—
457C22⋊C8—532no—
458C22⋊C8—1062no—
459C22⋊C8—1062no—
460#97—1062no—
461#97—1062no—
462C4×C8—1062no—
463C4×C8—1062no—
464C24C22×#615711yescharacteristic
465C24C22×#615711yescharacteristic
466C2×Q8—1062no—
467C2×Q8—1062no—
468C2×Q8—2121no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—42no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—532no—
481C22×C4—531no—
482C22×C4—532no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—531no—
489C22×C4—531no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4C22×#615711yescharacteristic
495C22×C4C22×#615711yescharacteristic
496C22×C4C22×#615711yescharacteristic
497C22×C4C22×#615711yescharacteristic
498C22×C4C22×#615711yescharacteristic
499M16—2124no—
500C2×C8—1062no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—2124no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—532no—
508C4⋊C4—532no—
509C4⋊C4—532no—
510C4⋊C4—532no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C4⋊C4—1062no—
518C4⋊C4—1062no—
519C4⋊C4—1062no—
520C4⋊C4—1062no—
521C4⋊C4—1062no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—532no—
529C22⋊C4—532no—
530C22⋊C4—1062no—
531C22⋊C4—1062no—
532C22⋊C4—1062no—
533C22⋊C4—1062no—
534C22⋊C4—1061no—
535C22⋊C4—1061no—
536C22⋊C4C22⋊#615711yescharacteristic
537C22⋊C4C22⋊#615711yescharacteristic
538C22⋊C4C22⋊#615711yescharacteristic
539C22⋊C4C22⋊#615711yescharacteristic
540C42—21no—
541C42—21no—
542C42—532no—
543C42—532no—
544C42—532no—
545C42—532no—
546C42—1061no—
547C42—1061no—
548C42—1061no—
549C42—1061no—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×C22⋊#615711yescharacteristic
560C23C2×C22⋊#615711yescharacteristic
561C23C23×#615711yescharacteristic
562Q8—2124no—
563D4—42no—
564D4—42no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—532no—
580C2×C4—532no—
581C2×C4—532no—
582C2×C4—532no—
583C2×C4—1064no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2122no—
595C2×C4—2121no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4C2×C22⋊#615711yescharacteristic
602C2×C4C2×C22⋊#615711yescharacteristic
603C2×C4C2×C22⋊#615711yescharacteristic
604C2×C4C2×C22⋊#615711yescharacteristic
605C2×C4#1313330412yes—
606C8—2122no—
607C8—2122no—
608C8—2122no—
609C8—2122no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—42no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—81no—
623C22—81no—
624C22#1681160212yes—
625C22C22×C22⋊#615711yescharacteristic
626C22C2×#1313330411yescharacteristic
627C22#1681170111yescharacteristic
628C4—22no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—41no—
634C4—41no—
635C4—1062no—
636C4—1062no—
637C4—1062no—
638C4—1062no—
639C4—2121no—
640C4—2121no—
641C4—2121no—
642C4—2121no—
643C2—21no—
644C2—21no—
645C2—22no—
646C2—41no—
647C2—81no—
648C2#2029130311yescharacteristic
649C2C2×#1681160211yescharacteristic
650C2#2029132711yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC22×Dic53C24
hypercenterC2×C4×D4#6157
hypocenterC53#17417
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20291404C2semidirect
2. #20289996C2semidirect
3. #20289987C2semidirect
5. #20289977C2semidirect
6. #20289966C2semidirect
7. #20289963C2semidirect
8. #20289957C2semidirect
9. #20289946C2semidirect
10. C2×#16811468C2semidirect
376. C53#17417semidirect

Maximal subgroups

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

subgroupindex
#202914042
#202899962
#202899872
#202899802
#202899772
#202899662
#202899632
#202899572
#202899462
C2×#168114682
C2×#168113132
#1741753