Finite Group Databasebeta

← #28324147

Subgroups — #28324147

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290889C211yesmaximal, characteristic, complemented
2#20290889C211yesmaximal, characteristic, complemented
3#20290791C211yesmaximal, characteristic, complemented
4#20290789C211yesmaximal, characteristic, complemented
5#20290786C211yesmaximal, characteristic, complemented
6#20290785C211yesmaximal, characteristic, complemented
7#20290690C211yesmaximal, characteristic
8#20290689C211yesmaximal, characteristic
9#20290660C211yesmaximal, characteristic, complemented
10#20290659C211yesmaximal, characteristic, complemented
11#20290644C211yesmaximal, characteristic, complemented
12#20290503C211yesmaximal, characteristic, complemented
13#20290453C211yesmaximal, characteristic, complemented
14#20290452C211yesmaximal, characteristic, complemented
15#20277997C211yesmaximal, characteristic
16C22×D53⋊C8C2211yescharacteristic
17C23×#88324C2211yescharacteristic
18C2×#13133284C2211yescharacteristic
19C2×#13133284C2211yescharacteristic
20C2×#13133284C2211yescharacteristic
21C2×C4⋊#88324C2211yescharacteristic
22C2×#13133274C2211yescharacteristic
23C2×#13133274C2211yescharacteristic
24C2×#13133274C2211yescharacteristic
25C2×C22⋊#88324C2211yescharacteristic
26C2×C22⋊#88324C2211yescharacteristic
27C2×C22⋊#88324C2211yescharacteristic
28C2×#13133272C2211yescharacteristic
29C2×#13133272C2211yescharacteristic
30C2×#13133272C2211yescharacteristic
31C2×C4×#88324C2211yescharacteristic
32#16811489C2211yescharacteristic
33#16811489C2211yescharacteristic
34#16811489C2211yescharacteristic
35#16811489C2211yescharacteristic
36#16811489C2211yescharacteristic
37#16811489C2211yescharacteristic
38#16811489C2211yescharacteristic
39#16811489C2211yescharacteristic
40#16811489C2211yescharacteristic
41#16811489C2211yescharacteristic
42#16811489C2211yescharacteristic
43#16811489C2211yescharacteristic
44C2×Q8×Dic53C2211yescharacteristic
45C2×#13133177C2211yescharacteristic
46#16810565C411yescharacteristic
47#16810565C411yescharacteristic
48#16810550C411yescharacteristic
49#16810548C411yescharacteristic
50#16810539C411yescharacteristic
51#16810539C411yescharacteristic
52#16810534C411yescharacteristic
53C22⋊C4×Dic53C2211yescharacteristic
54C22⋊C4×Dic53C2211yescharacteristic
55#16810470C2211yescharacteristic
56#16810470C2211yescharacteristic
57#16810468C2211yescharacteristic
58C53×#393C411yescharacteristic
59C2×D53⋊C8—22no—
60C2×D53⋊C8—41no—
61C2×D53⋊C8—41no—
62C22×#88324—22no—
63C22×#88324—41no—
64C22×#88324—41no—
65C22×#88324C2311yescharacteristic
66C22×#88324C2311yescharacteristic
67C22×#88324C2311yescharacteristic
68C22×#88324C2311yescharacteristic
69C22×#88324C2311yescharacteristic
70C22×#88324C2311yescharacteristic
71C22×#88324C2311yescharacteristic
72C22×#88324C2311yescharacteristic
73#13133284—22no—
74#13133284—41no—
75#13133284—41no—
76C4⋊#88324—22no—
77#13133274—22no—
78#13133274—41no—
79#13133274—41no—
80C22⋊#88324—22no—
81C22⋊#88324—41no—
82C22⋊#88324—41no—
83#13133272—41no—
84#13133272—41no—
85#13133272—41no—
86C4×#88324—41no—
87C22×C22⋊D53C2×C411yescharacteristic
88C23×Dic53C2311yescharacteristic
89C22×C4×D53C2311yescharacteristic
90Q8×Dic53—41no—
91Q8×Dic53—41no—
92#13133177—41no—
93#13133177—41no—
94C2×C4⋊Dic53C2×C411yescharacteristic
95C2×C4⋊Dic53C2×C411yescharacteristic
96C2×C4⋊Dic53C2×C411yescharacteristic
97C2×#11719919C2×C411yescharacteristic
98C2×C22⋊Dic53C2×C411yescharacteristic
99C2×C22⋊Dic53C2×C411yescharacteristic
100C2×#11719921C2×C411yescharacteristic
101C2×#11719921C2×C411yescharacteristic
102C2×#11719921C2×C411yescharacteristic
103C2×#11719921C2×C411yescharacteristic
104C2×#11719921C2×C411yescharacteristic
105C2×C4×Dic53C2311yescharacteristic
106C2×C4×Dic53C2311yescharacteristic
107C2×C4×Dic53C2311yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109C2×C4×Dic53C2311yescharacteristic
110#13133124C2×C411yescharacteristic
111#13133124C2×C411yescharacteristic
112#13133124C2×C411yescharacteristic
113#13133124C2×C411yescharacteristic
114#13133124C2×C411yescharacteristic
115#13133124C2×C411yescharacteristic
116#13133124C2×C411yescharacteristic
117#13133124C2×C411yescharacteristic
118#13133124C2×C411yescharacteristic
119C22×Q8×C53C2×C411yescharacteristic
120C2×C22⋊C4×C53C2×C411yescharacteristic
121C2×C22⋊C4×C53C2×C411yescharacteristic
122C2×C22⋊C4×C53C2×C411yescharacteristic
123#95×C53C2×C411yescharacteristic
124#95×C53C2×C411yescharacteristic
125#95×C53C2×C411yescharacteristic
126C24×C53C22×C411yescharacteristic
127C23×D53C22×C411yescharacteristic
128C2×Q8×C53—41no—
129C2×Q8×C53—41no—
130C2×Q8×C53—41no—
131C22×C4×C53C22×C411yescharacteristic
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C22×C411yescharacteristic
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C22×C411yescharacteristic
137C2×C22⋊D53—22no—
138C2×C22⋊D53—22no—
139C2×C22⋊D53—41no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53C22×C411yescharacteristic
147C22×Dic53C22×C411yescharacteristic
148C22×Dic53C22×C411yescharacteristic
149C22×Dic53C22×C411yescharacteristic
150C22×Dic53C22×C411yescharacteristic
151C22×Dic53C22×C411yescharacteristic
152C22×Dic53C2411yescharacteristic
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324—21no—
174C2×#88324—21no—
175C2×#88324—21no—
176C2×#88324—21no—
177C2×#88324—21no—
178C2×#88324—21no—
179C2×#88324—21no—
180C2×#88324—21no—
181C2×#88324—44no—
182C2×#88324#4112yes—
183C2×#88324#4112yes—
184D53⋊C8—44no—
185C22⋊C4×C53—41no—
186C22⋊C4×C53—41no—
187C22⋊C4×C53—41no—
188C22⋊Dic53—22no—
189C22⋊Dic53—22no—
190#11719921—22no—
191#11719921—22no—
192#11719921—41no—
193#11719921—41no—
194#11719921—41no—
195C4⋊Dic53—41no—
196C4⋊Dic53—41no—
197C4⋊Dic53—41no—
198#11719919—41no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—21no—
208C4×Dic53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53#13112yes—
216C23×C53C23×C411yescharacteristic
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53—21no—
223C22×D53#13112yes—
224Q8×C53—44no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
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—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C22⋊D53—44no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53C2×#4111yescharacteristic
267C2×Dic53C2×#4111yescharacteristic
268C2×Dic53H(5,𝔽2)11yescharacteristic
269C2×Dic53H(5,𝔽2)11yescharacteristic
270C2×Dic53H(5,𝔽2)11yescharacteristic
271C2×Dic532−1+411yescharacteristic
272C4×D53—41no—
273C4×D53—41no—
274C4×D53—41no—
275C4×D53—41no—
276#88324—22no—
277#88324—22no—
278#88324—41no—
279#88324—41no—
280#88324—41no—
281#88324—41no—
282#88324—41no—
283#88324—41no—
284#22396—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53C4×#4111yescharacteristic
293C22×C53#51811yescharacteristic
294C22×C53#52011yescharacteristic
295C22×C53C2×#13111yescharacteristic
296C22×C53C2×#13111yescharacteristic
297C22×C53#56811yescharacteristic
298C22×C53#56811yescharacteristic
299C2×D53—22no—
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—41no—
303C2×D53—41no—
304C2×D53—41no—
305C2×D53—41no—
306C4×C53—41no—
307C4×C53—41no—
308C4×C53—41no—
309C4×C53—41no—
310C4×C53—41no—
311C4×C53—41no—
312Dic53—41no—
313Dic53—41no—
314Dic53—41no—
315Dic53—41no—
316Dic53—41no—
317Dic53—41no—
318Dic53#53811yescharacteristic
319Dic53#53811yescharacteristic
320Dic53#54211yescharacteristic
321Dic53#54211yescharacteristic
322#2398—531no—
323#1985—531no—
324#1985—531no—
325#1924—531no—
326#1924—531no—
327#1923—531no—
328#1923—531no—
329#1869—531no—
330#1869—531no—
331#1850—531no—
332#1850—531no—
333#1840—531no—
334#1768—531no—
335#1752—531no—
336#1752—531no—
337C2×C53—42no—
338C2×C53#233711yescharacteristic
339C2×C53#296811yescharacteristic
340C2×C53#296811yescharacteristic
341C2×C53#297711yescharacteristic
342C2×C53#297811yescharacteristic
343C2×C53#299211yescharacteristic
344C2×C53#299211yescharacteristic
345D53—42no—
346C23×C8—531no—
347C23×C8—531no—
348C2×C4×Q8—531no—
349C2×C4×D4—531no—
350C2×C4⋊C8—531no—
351C2×C4⋊C8—531no—
352C2×C4⋊C8—531no—
353C2×C4⋊C8—531no—
354C2×C22⋊C8—531no—
355C2×C22⋊C8—531no—
356C2×C22⋊C8—531no—
357C2×C22⋊C8—531no—
358C2×C22⋊C8—531no—
359C2×C22⋊C8—531no—
360C2×#97—531no—
361C2×#97—531no—
362C2×#97—531no—
363C2×C4×C8—531no—
364#400—531no—
365#400—531no—
366#397—531no—
367#397—531no—
368#394—531no—
369#394—531no—
370#393—531no—
371#393#615711yescharacteristic
372C4×C22⋊C4—531no—
373C4×C22⋊C4—531no—
374C4×C22⋊C4—531no—
375C4×C22⋊C4—531no—
376#376—531no—
377#336—531no—
378#336—531no—
379#336—531no—
380#336—531no—
381#336—531no—
382#336—531no—
383#336—531no—
384#336—531no—
385#336—531no—
386#336—531no—
387#336—531no—
388#336—531no—
389C53#2239611yescharacteristic, complemented, Sylow
390C22×Q8C2×#615711yescharacteristic
391C22×D4—531no—
392C23×C4—531no—
393C23×C4—531no—
394C22×C8—531no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8—531no—
401C22×C8—531no—
402C22×C8—1062no—
403C22×C8—1062no—
404C22×C8—2121no—
405C22×C8—2121no—
406C22×C8—2121no—
407C22×C8—2121no—
408C4×Q8—2121no—
409C4×Q8—2121no—
410C4×D4—2121no—
411C4×D4—2121no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C22⋊C4—531no—
417C2×C22⋊C4—531no—
418C2×C22⋊C4—531no—
419C2×C22⋊C4—531no—
420C2×C22⋊C4—531no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—531no—
423C2×C22⋊C4C2×#615711yescharacteristic
424C2×C22⋊C4C2×#615711yescharacteristic
425C2×C22⋊C4C2×#615711yescharacteristic
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—531no—
431C4⋊C8—1062no—
432C4⋊C8—1062no—
433C4⋊C8—2121no—
434C4⋊C8—2121no—
435C22⋊C8—1062no—
436C22⋊C8—1062no—
437C22⋊C8—2121no—
438C22⋊C8—2121no—
439C22⋊C8—2121no—
440C22⋊C8—2121no—
441#97—2121no—
442#97—2121no—
443#97—2121no—
444C4×C8—2121no—
445#95—531no—
446#95—531no—
447#95—531no—
448#95—531no—
449#95—531no—
450#95—531no—
451#95—531no—
452#95—531no—
453#95—531no—
454#95C2×#615711yescharacteristic
455#95C2×#615711yescharacteristic
456#95C2×#615711yescharacteristic
457C24—531no—
458C24C22×#615711yescharacteristic
459C2×Q8—41no—
460C2×Q8—41no—
461C2×Q8—41no—
462C2×D4—1062no—
463C2×D4—1062no—
464C2×D4—2121no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4C22×#615711yescharacteristic
485C22×C4C22×#615711yescharacteristic
486C22×C4C22×#615711yescharacteristic
487C22×C4C22×#615711yescharacteristic
488C22×C4C22×#615711yescharacteristic
489C22×C4C22×#615711yescharacteristic
490C2×C8—532no—
491C2×C8—532no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—2124no—
515C2×C8—2124no—
516C4⋊C4—2121no—
517C4⋊C4—2121no—
518C4⋊C4—2121no—
519C4⋊C4—2121no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—1062no—
524C22⋊C4—1062no—
525C22⋊C4—1062no—
526C22⋊C4—1062no—
527C22⋊C4—2121no—
528C22⋊C4—2121no—
529C22⋊C4—2121no—
530C42—1061no—
531C42—1061no—
532C42—1061no—
533C42—1061no—
534C42—1061no—
535C42—1061no—
536C42—1061no—
537C42—1061no—
538C42—1061no—
539C42—1061no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—532no—
547C23—1061no—
548C23—1061no—
549C23—1061no—
550C23—1061no—
551C23—1061no—
552C23—1061no—
553C23#1313330512yes—
554C23C23×#615711yescharacteristic
555Q8—44no—
556D4—2124no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
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—21no—
574C2×C4—21no—
575C2×C4—531no—
576C2×C4—531no—
577C2×C4—531no—
578C2×C4—531no—
579C2×C4—531no—
580C2×C4—531no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
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—1061no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C8—1062no—
608C8—1062no—
609C8—2121no—
610C8—2121no—
611C8—2121no—
612C8—2121no—
613C8—2121no—
614C8—2121no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—1062no—
623C22—1062no—
624C22—1062no—
625C22—2121no—
626C22—2121no—
627C22—2121no—
628C22—2121no—
629C22#41×#615711yescharacteristic
630C22#1681169011yescharacteristic
631C22#1681169211yescharacteristic
632C22D53⋊#13111yescharacteristic
633C22C2×#1313330511yescharacteristic
634C22#1681170111yescharacteristic
635C22#1681170111yescharacteristic
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—531no—
643C4—531no—
644C4—531no—
645C4—531no—
646C4—2121no—
647C4—2121no—
648C4—2121no—
649C4—2121no—
650C4—2121no—
651C4—2121no—
652C2—42no—
653C2—2122no—
654C2#2029121611yescharacteristic
655C2#2029154111yescharacteristic
656C2#2029154311yescharacteristic
657C2#2029159011yescharacteristic
658C2#2029159111yescharacteristic
659C2#2029165211yescharacteristic
660C2#2029165411yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC53×#393C4
prime radicalC22×C22⋊D53C2×C4
cosocleC22×Dic53C24
hypercenter#393#6157
hypocenterC53#22396
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #20290889C2semidirect
2. #20290889C2semidirect
3. #20290791C2semidirect
4. #20290789C2semidirect
5. #20290786C2semidirect
6. #20290785C2semidirect
9. #20290660C2semidirect
10. #20290659C2semidirect
11. #20290644C2semidirect
12. #20290503C2semidirect
13. #20290453C2semidirect
14. #20290452C2semidirect
389. C53#22396semidirect

Maximal subgroups

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

subgroupindex
#202908892
#202907912
#202907892
#202907862
#202907852
#202906902
#202906892
#202906602
#202906592
#202906442
#202905032
#202904532
#202904522
#202779972
#2239653