Finite Group Databasebeta

← #27469873

Subgroups — #27469873

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280472C211yesmaximal, characteristic
2#20277338C211yesmaximal, characteristic, complemented
3#20276135C211yesmaximal, characteristic, complemented
4#20276130C211yesmaximal, characteristic, complemented
5#20276125C211yesmaximal, characteristic, complemented
6#20276120C211yesmaximal, characteristic, complemented
7#20275531C211yesmaximal, characteristic, complemented
8#20275526C211yesmaximal, characteristic, complemented
9#20275035C211yesmaximal, characteristic
10#20275030C211yesmaximal, characteristic
11#20275025C211yesmaximal, characteristic, complemented
12#20275020C211yesmaximal, characteristic, complemented
13#20274849C211yesmaximal, characteristic, complemented
14#20274843C211yesmaximal, characteristic, complemented
15C53×#1750C211yesmaximal, characteristic, complemented
16C2×#13133205C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C2×#13133142C2211yescharacteristic
22C2×#13133142C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29#16810545—21no—
30#16810545—21no—
31#16810539—21no—
32#16810539—21no—
33#16810529C2211yescharacteristic
34#16810525C2211yescharacteristic
35#16810522—21no—
36#16810522—21no—
37#16810498—21no—
38#16810498—21no—
39#16810493C2211yescharacteristic
40#16810493C2211yescharacteristic
41#16810493C2211yescharacteristic
42#16810489C2211yescharacteristic
43#16810489C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C23×Dic53—21no—
63#13133205—22no—
64#13133205—22no—
65#13133196—22no—
66#13133196—22no—
67#13133191—22no—
68#13133191—22no—
69#13133176—41no—
70#13133176—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—41no—
79C2×C4⋊Dic53—41no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×#11719919—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×C4×Dic53—21no—
95#13133142—41no—
96#13133142—41no—
97#13133142—41no—
98#13133142—41no—
99#13133139—41no—
100#13133139—41no—
101#13133139—41no—
102#13133139—41no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114#117×C53—22no—
115#117×C53—22no—
116C2×C4⋊C4×C53D411yescharacteristic
117C2×C4⋊C4×C53D411yescharacteristic
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122C22⋊C8×C53—22no—
123C22⋊C8×C53—22no—
124#97×C53—22no—
125C4×C8×C53—22no—
126C24×C53C2×D411yescharacteristic
127C23×D53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53#4112yes—
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C2×C22⋊D53—42no—
141C2×C22⋊D53—81no—
142C2×C22⋊D53—81no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—42no—
151C22×Dic53—81no—
152C22×Dic53—81no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53#4112yes—
184C42×C53#4112yes—
185C22⋊Dic53—42no—
186C22⋊Dic53—42no—
187#11719921—42no—
188#11719921—42no—
189#11719921—42no—
190#11719921—81no—
191#11719921—81no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—42no—
205C4⋊Dic53—81no—
206C4⋊Dic53—81no—
207C4⋊Dic53—81no—
208C4⋊Dic53—81no—
209C4⋊Dic53—81no—
210C4⋊Dic53—81no—
211#11719919—42no—
212C4×Dic53—42no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53#12412yes—
220C23×C53C22×D411yescharacteristic
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
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#12312yes—
245C2×C4×C53#12312yes—
246C2×C4×C53#12412yes—
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C22⋊D53—84no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—84no—
276D212—41no—
277D212—41no—
278D212—41no—
279D212—41no—
280Dic106—41no—
281Dic106—41no—
282Dic106—41no—
283Dic106—41no—
284C8×C53—41no—
285C8×C53—41no—
286C8×C53—41no—
287C8×C53—41no—
288#27984—531nomaximal, Sylow
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53C2×#12311yescharacteristic
297C22×C53C2×#12411yescharacteristic
298C22×C53#53511yescharacteristic
299C22×C53#57611yescharacteristic
300C22×C53#57711yescharacteristic
301C22×C53#57711yescharacteristic
302C22×C53#57811yescharacteristic
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C2×D53—81no—
308C2×D53—81no—
309C2×D53—81no—
310C4×C53—22no—
311C4×C53—22no—
312C4×C53—22no—
313C4×C53—41no—
314C4×C53—41no—
315C4×C53#55311yescharacteristic
316C4×C53#55311yescharacteristic
317C4×C53#55311yescharacteristic
318C4×C53#55311yescharacteristic
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322Dic53—81no—
323Dic53—81no—
324Dic53—81no—
325Dic53—81no—
326#2620—531no—
327#2317—531no—
328#2074—531no—
329#2073—531no—
330#2072—531no—
331#2071—531no—
332#1959—531no—
333#1958—531no—
334#1875—531no—
335#1874—531no—
336#1873—531no—
337#1872—531no—
338#1836—531no—
339#1834—531no—
340#1750D5311yescharacteristic, complemented
341C2×C53—42no—
342C2×C53#250411yescharacteristic
343C2×C53#309811yescharacteristic
344C2×C53#309911yescharacteristic
345C2×C53#313611yescharacteristic
346C2×C53#313711yescharacteristic
347C2×C53#322411yescharacteristic
348C2×C53#322511yescharacteristic
349D53—161no—
350C2×#124—531no—
351C2×C22⋊Q8—531no—
352C2×#121—531no—
353C2×#117—531no—
354C2×#117C2×D5311yescharacteristic
355C22×C4⋊C4—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×C22⋊C8C2×D5311yescharacteristic
365C2×C22⋊C8C2×D5311yescharacteristic
366C2×#97C2×D5311yescharacteristic
367C2×C4×C8C2×D5311yescharacteristic
368#396—1061no—
369#396—1061no—
370#394—1061no—
371#394—1061no—
372#391—531no—
373#390—531no—
374#389—1061no—
375#389—1061no—
376#385—1061no—
377#385—1061no—
378#384—531no—
379#384—531no—
380#384—531no—
381#382—531no—
382#382—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#336C2×D5311yescharacteristic
392#336C2×D5311yescharacteristic
393C53#2798411yescharacteristic, complemented, Sylow
394C22×Q8—531no—
395C22×D4—531no—
396C22×D4—1061no—
397C23×C4—1061no—
398C23×C4C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C22×C8C22×D5311yescharacteristic
401C22×C8C22×D5311yescharacteristic
402C22×C8C22×D5311yescharacteristic
403#124—1062no—
404#124—1062no—
405C22⋊Q8—1062no—
406C22⋊Q8—1062no—
407#121—1062no—
408#121—1062no—
409#117—22no—
410#117—22no—
411#117—2121no—
412#117—2121no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—2121no—
428C2×C4⋊C4—2121no—
429C2×C4⋊C4D21211yescharacteristic
430C2×C4⋊C4D21211yescharacteristic
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4D21211yescharacteristic
439C2×C22⋊C4D21211yescharacteristic
440C2×C42—1061no—
441C2×C42C22×D5311yescharacteristic
442C2×C42C22×D5311yescharacteristic
443#103—2121no—
444#103—2121no—
445#103—2121no—
446#103—2121no—
447#102—2121no—
448#102—2121no—
449#102—2121no—
450#102—2121no—
451C22⋊C8—22no—
452C22⋊C8—22no—
453#97—22no—
454C4×C8—22no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461C24—1061no—
462C24C2×D21211yescharacteristic
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8—1061no—
469C2×D4—1061no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C2×D4—1061no—
474C2×D4—1061no—
475C2×D4—4241no—
476C2×D4—4241no—
477C2×D4—2122no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—4241no—
490C22×C4—4241no—
491C22×C4—2122no—
492C22×C4C2×D21211yescharacteristic
493C22×C4C2×D21211yescharacteristic
494C22×C4C2×D21211yescharacteristic
495C22×C4C2×D21211yescharacteristic
496C22×C4C2×D21211yescharacteristic
497C22×C4#1171994512yes—
498C22×C4#1171994512yes—
499C22×C4C23×D5311yescharacteristic
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—4241no—
527C4⋊C4—4241no—
528C4⋊C4—4241no—
529C4⋊C4—4241no—
530C4⋊C4—4241no—
531C4⋊C4—4241no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—4241no—
537C22⋊C4—4241no—
538C22⋊C4—2122no—
539C22⋊C4—2122no—
540C22⋊C4—2122no—
541C22⋊C4—2122no—
542C22⋊C4—2122no—
543C42—21no—
544C42—21no—
545C42—2122no—
546C42#1171994712yes—
547C42#1171994712yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23—2121no—
559C23—2121no—
560C23—2121no—
561C23#1313320512yes—
562C23C22×D21211yescharacteristic
563Q8—2121no—
564Q8—2121no—
565Q8—2121no—
566Q8—2121no—
567D4—4244no—
568D4—2121no—
569D4—2121no—
570D4—2121no—
571D4—2121no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—4244no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4#1313320112yes—
611C2×C4#1313320112yes—
612C2×C4#1313320512yes—
613C2×C4C2×#1171994711yescharacteristic
614C2×C4C2×#1171994711yescharacteristic
615C2×C4C2×#1171994511yescharacteristic
616C2×C4C2×#1171994511yescharacteristic
617C2×C4#1313326311yescharacteristic
618C2×C4#1313326311yescharacteristic
619C8—41no—
620C8—41no—
621C8—41no—
622C8—41no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—4241no—
631C22—4241no—
632C22—4241no—
633C22—4241no—
634C22—4241no—
635C22—4241no—
636C22—4241no—
637C22C2×#1313320111yescharacteristic
638C22C2×#1313320511yescharacteristic
639C22#1681119211yescharacteristic
640C22#1681142811yescharacteristic
641C22#1681143511yescharacteristic
642C22#1681143511yescharacteristic
643C22#1681144011yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—41no—
648C4—41no—
649C4—4241no—
650C4—4241no—
651C4—4241no—
652C4—4241no—
653C4—4241no—
654C4—4241no—
655C4—4241no—
656C4#1681131911yescharacteristic
657C4#1681131911yescharacteristic
658C4#1681131911yescharacteristic
659C4#1681131911yescharacteristic
660C2—42no—
661C2—8481no—
662C2#2027920711yescharacteristic
663C2#2028469511yescharacteristic
664C2#2028470611yescharacteristic
665C2#2028507111yescharacteristic
666C2#2028508211yescharacteristic
667C2#2028581111yescharacteristic
668C2#2028582611yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1750C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1750D53
hypocenterC53#27984
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. #20277338C2semidirect
3. #20276135C2semidirect
4. #20276130C2semidirect
5. #20276125C2semidirect
6. #20276120C2semidirect
7. #20275531C2semidirect
8. #20275526C2semidirect
11. #20275025C2semidirect
12. #20275020C2semidirect
13. #20274849C2semidirect
14. #20274843C2semidirect
15. C53×#1750C2semidirect
340. #1750D53semidirect
393. C53#27984semidirect

Maximal subgroups

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

subgroupindex
#202804722
#202773382
#202761352
#202761302
#202761252
#202761202
#202755312
#202755262
#202750352
#202750302
#202750252
#202750202
#202748492
#202748432
C53×#17502
#2798453