Finite Group Databasebeta

← #28323620

Subgroups — #28323620

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290889C211yesmaximal, characteristic, complemented
2#20290884C211yesmaximal, characteristic, complemented
3#20290791C211yesmaximal, characteristic
4#20290790C211yesmaximal, characteristic
5#20290787C211yesmaximal, characteristic
6#20290786C211yesmaximal, characteristic
7#20290775C211yesmaximal, characteristic, complemented
8#20290770C211yesmaximal, characteristic, complemented
9#20290769C211yesmaximal, characteristic, complemented
10#20290768C211yesmaximal, characteristic, complemented
11#20290607C211yesmaximal, characteristic
12#20290514C211yesmaximal, characteristic, complemented
13#20290507C211yesmaximal, characteristic, complemented
14C4×#13133274C211yesmaximal, characteristic
15#20277253C211yesmaximal, characteristic
16C2×#13133284C2211yescharacteristic
17C2×#13133284C2211yescharacteristic
18C2×C4⋊#88324C2211yescharacteristic
19C2×C4⋊#88324C2211yescharacteristic
20C2×#13133274C2211yescharacteristic
21C2×#13133274C2211yescharacteristic
22C2×#13133274C2211yescharacteristic
23C2×#13133274C2211yescharacteristic
24C2×#13133272C2211yescharacteristic
25C2×#13133272C2211yescharacteristic
26C2×C4×#88324C2211yescharacteristic
27C2×C4×#88324C2211yescharacteristic
28C2×C4×#88324C2211yescharacteristic
29C2×C4×#88324C2211yescharacteristic
30#16811489C2211yescharacteristic
31#16811489C2211yescharacteristic
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×#13133193C411yescharacteristic
45C2×Q8×Dic53C2211yescharacteristic
46C2×C42×D53C2211yescharacteristic
47#16810521C411yescharacteristic
48#16810510C411yescharacteristic
49#16810510C411yescharacteristic
50#16810497C411yescharacteristic
51#16810497C411yescharacteristic
52#16810480C411yescharacteristic
53C4⋊C4×Dic53C2211yescharacteristic
54#16810474C2211yescharacteristic
55#16810474C2211yescharacteristic
56#16810470C2211yescharacteristic
57#16810470C2211yescharacteristic
58C53×#391C411yescharacteristic
59C22×#88324C2311yescharacteristic
60C22×#88324C2311yescharacteristic
61C22×#88324C2311yescharacteristic
62C22×#88324C2311yescharacteristic
63C22×#88324C2311yescharacteristic
64C22×#88324C2311yescharacteristic
65C22×#88324C2311yescharacteristic
66C22×#88324C2311yescharacteristic
67#13133284—22no—
68#13133284—41no—
69C4⋊#88324—22no—
70C4⋊#88324—41no—
71#13133274—22no—
72#13133274—22no—
73#13133274—41no—
74#13133274—41no—
75#13133272—22no—
76#13133272—41no—
77C4×#88324—22no—
78C4×#88324—41no—
79C4×#88324—41no—
80C4×#88324—41no—
81C22×C4×D53C2×C411yescharacteristic
82C22×C4×D53C2×C411yescharacteristic
83C22×C4×D53C2311yescharacteristic
84#13133193—24no—
85Q8×Dic53—41no—
86Q8×Dic53—41no—
87C2×C4⋊Dic53C2×C411yescharacteristic
88C2×C4⋊Dic53C2×C411yescharacteristic
89C2×C4⋊Dic53C2×C411yescharacteristic
90C2×C4⋊Dic53C2×C411yescharacteristic
91C2×#11719919C2×C411yescharacteristic
92C2×#11719919C2×C411yescharacteristic
93C2×#11719919C2×C411yescharacteristic
94C2×#11719919C2×C411yescharacteristic
95C2×#11719921C2×C411yescharacteristic
96C2×#11719921C2×C411yescharacteristic
97C2×#11719921C2×C411yescharacteristic
98C2×#11719921C2×C411yescharacteristic
99C2×C4×Dic53C2×C411yescharacteristic
100C2×C4×Dic53C2311yescharacteristic
101C2×C4×Dic53C2311yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103C2×C4×Dic53C2311yescharacteristic
104C2×C4×Dic53C2311yescharacteristic
105C2×C4×Dic53C2311yescharacteristic
106C42×D53—41no—
107C42×D53—41no—
108#13133124C2×C411yescharacteristic
109#13133124C2×C411yescharacteristic
110#13133124C2×C411yescharacteristic
111#13133124C2×C411yescharacteristic
112#13133124C2×C411yescharacteristic
113#13133124C2×C411yescharacteristic
114C22×Q8×C53C2×C411yescharacteristic
115C2×C4⋊C4×C53C2×C411yescharacteristic
116C2×C42×C53C2×C411yescharacteristic
117#95×C53C2×C411yescharacteristic
118#95×C53C2×C411yescharacteristic
119#95×C53C2×C411yescharacteristic
120#95×C53C2×C411yescharacteristic
121C23×D53C22×C411yescharacteristic
122C2×Q8×C53—22no—
123C2×Q8×C53—22no—
124C2×Q8×C53—41no—
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C22×C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C22×C411yescharacteristic
132C22×Dic53C22×C411yescharacteristic
133C22×Dic53C22×C411yescharacteristic
134C22×Dic53C22×C411yescharacteristic
135C22×Dic53C22×C411yescharacteristic
136C22×Dic53C22×C411yescharacteristic
137C22×Dic53C22×C411yescharacteristic
138C22×Dic53C2411yescharacteristic
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—21no—
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—41no—
150C2×C4×D53—41no—
151C2×#88324—21no—
152C2×#88324—21no—
153C2×#88324—21no—
154C2×#88324—21no—
155C2×#88324—21no—
156C2×#88324—21no—
157C2×#88324—21no—
158C2×#88324—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#4112yes—
174C2×#88324#4112yes—
175C4⋊C4×C53—22no—
176C42×C53—41no—
177#11719921—22no—
178#11719921—22no—
179#11719921—22no—
180#11719921—22no—
181C4⋊Dic53—22no—
182C4⋊Dic53—41no—
183C4⋊Dic53—41no—
184C4⋊Dic53—41no—
185#11719919—22no—
186#11719919—22no—
187#11719919—22no—
188#11719919M1614yes—
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—21no—
201C4×Dic53—41no—
202C23×C53C23×C411yescharacteristic
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—21no—
209C22×D53#13112yes—
210Q8×C53—44no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
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—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53C2×M1612yes—
230C2×C4×C53#13112yes—
231C2×C4×C53#13112yes—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53C2×M1612yes—
249C2×Dic53C2×M1612yes—
250C2×Dic53C2×#4111yescharacteristic
251C2×Dic53C2×#4111yescharacteristic
252C2×Dic53H(5,𝔽2)11yescharacteristic
253C2×Dic53H(5,𝔽2)11yescharacteristic
254C2×Dic532−1+411yescharacteristic
255C2×Dic532−1+411yescharacteristic
256C4×D53—44no—
257C4×D53—44no—
258C4×D53—41no—
259C4×D53—41no—
260C4×D53—41no—
261C4×D53—41no—
262#88324—22no—
263#88324—22no—
264#88324—41no—
265#88324—41no—
266#88324—41no—
267#88324—41no—
268#88324—41no—
269#88324—41no—
270#22321—531nomaximal, Sylow
271C22×C53C4×#4111yescharacteristic
272C22×C53#51811yescharacteristic
273C22×C53#51911yescharacteristic
274C22×C53C22×M1611yescharacteristic
275C22×C53C2×#13111yescharacteristic
276C22×C53C2×#13111yescharacteristic
277C22×C53C2×#13111yescharacteristic
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—41no—
282C2×D53—41no—
283C2×D53—41no—
284C2×D53—41no—
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—41no—
291C4×C53—41no—
292Dic53—22no—
293Dic53—22no—
294Dic53—41no—
295Dic53—41no—
296Dic53—41no—
297Dic53—41no—
298Dic53#53911yescharacteristic
299Dic53#53911yescharacteristic
300Dic53#54211yescharacteristic
301Dic53#54211yescharacteristic
302C4×C22⋊Q8—531no—
303#1985—531no—
304#1984—531no—
305#1924—531no—
306#1924—531no—
307#1923—531no—
308#1923—531no—
309#1917—531no—
310#1916—531no—
311#1916—531no—
312#1916—531no—
313#1817—531no—
314#1771—531no—
315C8×C4⋊C4—531no—
316C4×C22⋊C8—531no—
317C2×C53#233211yescharacteristic
318C2×C53#296511yescharacteristic
319C2×C53#296611yescharacteristic
320C2×C53#297411yescharacteristic
321C2×C53#297711yescharacteristic
322C2×C53#298411yescharacteristic
323C2×C53#298611yescharacteristic
324D53—42no—
325C2×C22⋊Q8—531no—
326C2×C4×Q8—531no—
327C22×C42—531no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8—531no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—531no—
332C2×C22⋊C8—531no—
333C2×C22⋊C8—531no—
334C2×C22⋊C8—531no—
335C2×C22⋊C8—531no—
336C2×#97—531no—
337C2×#97—531no—
338C2×C4×C8—531no—
339C2×C4×C8—531no—
340C2×C4×C8—531no—
341C2×C4×C8—531no—
342#391#615711yescharacteristic
343#389—531no—
344#387—531no—
345#387—531no—
346#385—531no—
347#385—531no—
348#380—531no—
349C4×C4⋊C4—531no—
350C4×C4⋊C4—531no—
351C4×C4⋊C4—531no—
352C4×C22⋊C4—531no—
353C4×C22⋊C4—531no—
354#336—531no—
355#336—531no—
356#336—531no—
357#336—531no—
358#336—531no—
359#336—531no—
360#336—531no—
361#336—531no—
362#336—531no—
363#336—531no—
364#336—531no—
365#336—531no—
366#336—531no—
367#336—531no—
368C53#2232111yescharacteristic, complemented, Sylow
369C22×Q8C2×#615711yescharacteristic
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—531no—
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—531no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8—531no—
381C22⋊Q8—1064no—
382C4×Q8—2121no—
383C4×Q8—2121no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4C2×#615711yescharacteristic
393C2×C22⋊C4—531no—
394C2×C22⋊C4—531no—
395C2×C22⋊C4—531no—
396C2×C22⋊C4—531no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—2121no—
405C2×C42—2121no—
406C2×C42C2×#615711yescharacteristic
407C4⋊C8—1062no—
408C4⋊C8—1062no—
409C4⋊C8—2121no—
410C4⋊C8—2121no—
411C22⋊C8—1062no—
412C22⋊C8—1062no—
413C22⋊C8—2121no—
414C22⋊C8—2121no—
415#97—1062no—
416#97—2121no—
417C4×C8—1062no—
418C4×C8—2121no—
419C4×C8—2121no—
420C4×C8—2121no—
421#95—531no—
422#95—531no—
423#95—531no—
424#95—531no—
425#95—531no—
426#95—531no—
427#95C2×#615711yescharacteristic
428#95C2×#615711yescharacteristic
429#95C2×#615711yescharacteristic
430#95C2×#615711yescharacteristic
431C24—531no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—41no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1062no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—2121no—
453C22×C4—2121no—
454C22×C4C22×#615711yescharacteristic
455C22×C4C22×#615711yescharacteristic
456C22×C4C22×#615711yescharacteristic
457C22×C4C22×#615711yescharacteristic
458C22×C4C22×#615711yescharacteristic
459C22×C4C22×#615711yescharacteristic
460C22×C4C22×#615711yescharacteristic
461C2×C8—532no—
462C2×C8—532no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C4⋊C4—22no—
486C4⋊C4—534no—
487C4⋊C4—1062no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—2121no—
492C4⋊C4—2121no—
493C4⋊C4—2121no—
494C22⋊C4—1062no—
495C22⋊C4—1062no—
496C22⋊C4—1062no—
497C22⋊C4—1062no—
498C42—41no—
499C42—1061no—
500C42—1061no—
501C42—1061no—
502C42—1061no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C42—1061no—
507C42—1061no—
508C42—1061no—
509C42—1061no—
510C42—1061no—
511C42—2121no—
512C23—532no—
513C23—1061no—
514C23—1061no—
515C23—1061no—
516C23—1061no—
517C23—1061no—
518C23—1061no—
519C23C23×#615711yescharacteristic
520Q8—44no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—531no—
540C2×C4—531no—
541C2×C4—531no—
542C2×C4—531no—
543C2×C4—531no—
544C2×C4—531no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—2124no—
564C2×C4—2124no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4D53⋊M1612yes—
570C2×C4#1313330412yes—
571C2×C4#1313330512yes—
572C8—1062no—
573C8—1062no—
574C8—2121no—
575C8—2121no—
576C8—2121no—
577C8—2121no—
578C8—2121no—
579C8—2121no—
580C22—1062no—
581C22—1062no—
582C22—1062no—
583C22—2121no—
584C22—2121no—
585C22—2121no—
586C22—2121no—
587C22#41×#615711yescharacteristic
588C22#1681169011yescharacteristic
589C22#1681169111yescharacteristic
590C22C2×D53⋊M1611yescharacteristic
591C22C2×#1313330411yescharacteristic
592C22D53⋊#13111yescharacteristic
593C22C2×#1313330511yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—531no—
602C4—531no—
603C4—531no—
604C4—531no—
605C4—1062no—
606C4—1062no—
607C4—2121no—
608C4—2121no—
609C4—2121no—
610C4—2121no—
611C2—2122no—
612C2#2029121111yescharacteristic
613C2#2029152311yescharacteristic
614C2#2029152911yescharacteristic
615C2#2029157411yescharacteristic
616C2#2029159011yescharacteristic
617C2#2029161611yescharacteristic
618C2#2029162611yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC53×#391C4
prime radicalC23×D53C22×C4
cosocleC22×Dic53C24
hypercenter#391#6157
hypocenterC53#22321
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #20290889C2semidirect
2. #20290884C2semidirect
7. #20290775C2semidirect
8. #20290770C2semidirect
9. #20290769C2semidirect
10. #20290768C2semidirect
12. #20290514C2semidirect
13. #20290507C2semidirect
368. C53#22321semidirect

Maximal subgroups

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

subgroupindex
#202908892
#202908842
#202907912
#202907902
#202907872
#202907862
#202907752
#202907702
#202907692
#202907682
#202906072
#202905142
#202905072
C4×#131332742
#202772532
#2232153