Finite Group Databasebeta

← #27446696

Subgroups — #27446696

Order 13568.

858 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
1C2×#16810924C211yesmaximal, characteristic, complemented
2C2×#16810661C211yesmaximal, characteristic, complemented
3#20278428C211yesmaximal, characteristic, complemented
4#20277751C211yesmaximal, characteristic
5#20276247C211yesmaximal, characteristic, complemented
6#20276213C211yesmaximal, characteristic, complemented
7#20276102C211yesmaximal, characteristic, complemented
8#20276054C211yesmaximal, characteristic, complemented
9#20275474C211yesmaximal, characteristic, complemented
10#20275294C211yesmaximal, characteristic, complemented
11#20275143C211yesmaximal, characteristic
12#20275035C211yesmaximal, characteristic
13#20274965C211yesmaximal, characteristic, complemented
14#20274916C211yesmaximal, characteristic, complemented
15C53×#1850C211yesmaximal, characteristic, complemented
16C2×#13133209C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21#16810924—24no—
22#16810661—24no—
23C2×#13133151C2211yescharacteristic
24C2×#13133151C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133139C2211yescharacteristic
30C2×#13133139C2211yescharacteristic
31C2×#13133139C2211yescharacteristic
32C2×#13133139C2211yescharacteristic
33#16810559—21no—
34#16810546—21no—
35#16810525C2211yescharacteristic
36#16810523—21no—
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810520—21no—
41#16810512—21no—
42#16810506—21no—
43#16810499—21no—
44#16810493C2211yescharacteristic
45#16810483C2211yescharacteristic
46C4×C4⋊Dic53C2211yescharacteristic
47#16810470—21no—
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C22⋊D53—21no—
63C22×C4×D53—21no—
64#13133209—22no—
65#13133209—41no—
66#13133191—22no—
67#13133191Q814yes—
68C4×D212—22no—
69C4×D212—41no—
70#13133176—22no—
71#13133176—22no—
72#13133176—41no—
73#13133176—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98#13133151—22no—
99#13133151—22no—
100#13133151—22no—
101#13133149—22no—
102#13133149—41no—
103#13133149D414yescomplemented
104#13133139—22no—
105#13133139—22no—
106#13133139—22no—
107#13133139—41no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113C23×C4×C53C2311yescharacteristic
114C22×C8×C53—22no—
115C22×C8×C53—22no—
116C22×C8×C53—22no—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C42×C53C2311yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C4⋊C8×C53—22no—
126C4⋊C8×C53—22no—
127C22⋊C8×C53—22no—
128#95×C53D411yescharacteristic
129#95×C53D411yescharacteristic
130C24×C53C2×D411yescharacteristic
131C23×D53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×Q812yes—
142C22×C4×C53#4112yes—
143C22×C4×C53C2411yescharacteristic
144C2×C22⋊D53—24no—
145C2×C22⋊D53—81no—
146C2×C22⋊D53—81no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212C2×Q812yes—
160C2×C4×D53—42no—
161C2×C4×D53—42no—
162C2×C4×D53—81no—
163C2×C8×C53—28no—
164C2×C8×C53—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×C53C2×D412yes—
175C2×C8×C53#4112yes—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—41no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C22⋊Dic53—42no—
183C22⋊Dic53—42no—
184C22⋊Dic53—81no—
185#11719921—24no—
186#11719921—42no—
187#11719921—42no—
188#11719921—81no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
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—42no—
201C4⋊Dic53—81no—
202C4⋊Dic53C2×D412yes—
203C4⋊Dic53C2×D412yes—
204C4⋊Dic53C2×Q812yes—
205#11719919—42no—
206#11719919—42no—
207#11719919—42no—
208#11719919—81no—
209#11719919—81no—
210C4×Dic53—42no—
211C4×Dic53—42no—
212C4×Dic53—81no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53C22⋊Q812yes—
220C23×C53C22×D411yescharacteristic
221C22×D53—22no—
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—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
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—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53C22⋊Q812yes—
246C2×C4×C53#13512yes—
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×Q811yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C22⋊D53—84no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
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—
275D212—22no—
276D212—22no—
277D212—41no—
278D212—41no—
279C4×D53—84no—
280C8×C53—22no—
281C8×C53—22no—
282C8×C53—41no—
283C8×C53—41no—
284#26345—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53C2×C22⋊Q811yescharacteristic
293C22×C53D4211yescharacteristic
294C22×C53#54711yescharacteristic
295C22×C53C2×#13511yescharacteristic
296C22×C53#57511yescharacteristic
297C22×C53#57611yescharacteristic
298C22×C53#57711yescharacteristic
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—81no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—41no—
309C4×C53—41no—
310C4×C53—41no—
311C4×C53#52511yescharacteristic
312C4×C53#53911yescharacteristic
313C4×C53D4×Q811yescharacteristic
314C4×C53#55411yescharacteristic
315Dic53—42no—
316Dic53—42no—
317Dic53—42no—
318Dic53—81no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322C2×#478—531no—
323C2×#427—531no—
324#2441—531no—
325#2363—531no—
326#2094—531no—
327#2088—531no—
328#2067—531no—
329#2059—531no—
330#1947—531no—
331#1919—531no—
332#1893—531no—
333#1875—531no—
334#1862—531no—
335#1854—531no—
336#1850D5311yescharacteristic, complemented
337C2×C53—42no—
338C2×C53#244911yescharacteristic
339C2×C53#307611yescharacteristic
340C2×C53#308111yescharacteristic
341C2×C53#326811yescharacteristic
342C2×C53#328411yescharacteristic
343C2×C53#333911yescharacteristic
344C2×C53#334811yescharacteristic
345D53—82no—
346C23×C8C2×D5311yescharacteristic
347C2×#125—531no—
348C2×#121—531no—
349C2×C4×D4—531no—
350C2×#117—531no—
351C2×#117—531no—
352#478—1064no—
353#427—1064no—
354C2×#107—531no—
355C2×#107—531no—
356C2×#107—531no—
357C2×#106—531no—
358C2×#106—531no—
359C2×#106—531no—
360C2×C4⋊C8C2×D5311yescharacteristic
361C2×C4⋊C8C2×D5311yescharacteristic
362C2×#102—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#102—531no—
366C2×C22⋊C8C2×D5311yescharacteristic
367#399—1061no—
368#396—1061no—
369#390—531no—
370#390—1061no—
371#389—531no—
372#389—531no—
373#389—531no—
374#389—1061no—
375#388—1061no—
376#387—1061no—
377#386—1061no—
378#384—531no—
379#380—531no—
380C4×C4⋊C4—531no—
381C4×C22⋊C4—1061no—
382C4×C22⋊C4C2×D5311yescharacteristic
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#336C2×D5311yescharacteristic
390#336C2×D5311yescharacteristic
391C53#2634511yescharacteristic, complemented, Sylow
392C22×D4—531no—
393C22×D4—1061no—
394C23×C4—1061no—
395C23×C4C22×D5311yescharacteristic
396C22×C8—22no—
397C22×C8—22no—
398C22×C8—22no—
399C22×C8C22×D5311yescharacteristic
400C22×C8C22×D5311yescharacteristic
401C22×C8C22×D5311yescharacteristic
402C22×C8C22×D5311yescharacteristic
403#125—1062no—
404#125—2121no—
405#121—534no—
406#121—1062no—
407C4×D4—1062no—
408C4×D4—2121no—
409#117—1062no—
410#117—1062no—
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—531no—
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×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4D21211yescharacteristic
435C2×C22⋊C4D21211yescharacteristic
436C2×C42—1061no—
437C2×C42—1061no—
438C2×C42—1061no—
439C2×C42C22×D5311yescharacteristic
440C2×C42C22×D5311yescharacteristic
441#107—1062no—
442#107—1062no—
443#107—1062no—
444#106—534no—
445#106—1062no—
446#106—2121no—
447C4⋊C8—22no—
448C4⋊C8—22no—
449#102—1062no—
450#102—1062no—
451#102—1062no—
452#102—2121no—
453C22⋊C8—22no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95D21211yescharacteristic
460#95D21211yescharacteristic
461C24—1061no—
462C24C2×D21211yescharacteristic
463C2×D4—532no—
464C2×D4—1064no—
465C2×D4—1061no—
466C2×D4—1061no—
467C2×D4—1061no—
468C2×D4—1061no—
469C2×D4—1061no—
470C2×D4—4241no—
471C2×D4—4241no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—4241no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4C2×D21211yescharacteristic
487C22×C4C2×D21211yescharacteristic
488C22×C4C2×D21211yescharacteristic
489C22×C4C2×D21211yescharacteristic
490C22×C4C2×D21211yescharacteristic
491C22×C4#1171994712yes—
492C22×C4Q8×D5312yes—
493C22×C4C23×D5311yescharacteristic
494C2×C8—28no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8D4×D5312yes—
506C2×C8#1171994712yes—
507C4⋊C4—532no—
508C4⋊C4—532no—
509C4⋊C4—532no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—4241no—
522C4⋊C4—4241no—
523C4⋊C4—4241no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C22⋊C4—22no—
529C22⋊C4—41no—
530C22⋊C4—1064no—
531C22⋊C4—4241no—
532C22⋊C4—4241no—
533C22⋊C4—2122no—
534C22⋊C4—2122no—
535C22⋊C4—2122no—
536C22⋊C4—2122no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—4241no—
542C42—2122no—
543C42—2122no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—1062no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23#1313319712yes—
558C23C22×D21211yescharacteristic
559D4—1062no—
560D4—1062no—
561D4—4244no—
562D4—2121no—
563D4—2121no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—4244no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
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#1313319712yes—
604C2×C4#1313323912yes—
605C2×C4C2×D4×D5311yescharacteristic
606C2×C4C2×Q8×D5311yescharacteristic
607C2×C4C2×#1171994711yescharacteristic
608C2×C4C2×#1171994711yescharacteristic
609C2×C4#1313326311yescharacteristic
610C2×C4#1313326411yescharacteristic
611C8—22no—
612C8—22no—
613C8—41no—
614C8—41no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—4241no—
623C22—4241no—
624C22—4241no—
625C22—4241no—
626C22—2122no—
627C22—2122no—
628C22—2122no—
629C22C2×#1313319711yescharacteristic
630C22D4×D21211yescharacteristic
631C22#1681127711yescharacteristic
632C22C2×#1313323911yescharacteristic
633C22#1681142311yescharacteristic
634C22#1681142811yescharacteristic
635C22#1681143511yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—4241no—
642C4—4241no—
643C4—4241no—
644C4—4241no—
645C4—2122no—
646C4—2122no—
647C4—2122no—
648C4#1681112911yescharacteristic
649C4#1681121711yescharacteristic
650C4#1681128611yescharacteristic
651C4#1681133111yescharacteristic
652C2—42no—
653C2—4242no—
654C2#2027852211yescharacteristic
655C2#2028450511yescharacteristic
656C2#2028454711yescharacteristic
657C2#2028626011yescharacteristic
658C2#2028644711yescharacteristic
659C2#2028711811yescharacteristic
660C2#2028723211yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1850C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1850D53
hypocenterC53#26345
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. C2×#16810924C2semidirect
2. C2×#16810661C2semidirect
3. #20278428C2semidirect
5. #20276247C2semidirect
6. #20276213C2semidirect
7. #20276102C2semidirect
8. #20276054C2semidirect
9. #20275474C2semidirect
10. #20275294C2semidirect
13. #20274965C2semidirect
14. #20274916C2semidirect
15. C53×#1850C2semidirect
103. #13133149D4semidirect
336. #1850D53semidirect
391. C53#26345semidirect

Maximal subgroups

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

subgroupindex
C2×#168109242
C2×#168106612
#202784282
#202777512
#202762472
#202762132
#202761022
#202760542
#202754742
#202752942
#202751432
#202750352
#202749652
#202749162
C53×#18502
#2634553