Finite Group Databasebeta

← #27391823

Subgroups — #27391823

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810926C211yesmaximal, characteristic
2#20277452C211yesmaximal, characteristic, complemented
3#20277075C211yesmaximal, characteristic
4#20275671C211yesmaximal, characteristic, complemented
5#20275470C211yesmaximal, characteristic, complemented
6#20275309C211yesmaximal, characteristic, complemented
7#20275298C211yesmaximal, characteristic, complemented
8#20274963C211yesmaximal, characteristic
9#20274959C211yesmaximal, characteristic
10#20274714C211yesmaximal, characteristic
11#20274613C211yesmaximal, characteristic, complemented
12#20274573C211yesmaximal, characteristic, complemented
13#20274543C211yesmaximal, characteristic
14C4×#13133139C211yesmaximal, characteristic
15C4×#107×C53C211yesmaximal, characteristic, complemented
16C2×#13133208C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20#16810926C418yescomplemented
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133128C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810521—21no—
32#16810520—21no—
33#16810519C2211yescharacteristic
34#16810507—21no—
35#16810506—21no—
36#16810492C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C4×#11719921—21no—
41C4×#11719921—21no—
42#16810470—21no—
43#16810469C2211yescharacteristic
44#95×D53—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—22no—
67#13133208C2×C414yescomplemented
68C4×D212—22no—
69C4×D212Q814yes—
70#13133179—22no—
71#13133179—22no—
72#13133179—22no—
73#13133179C2×C414yes—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133150—22no—
92#13133148C2×C414yescomplemented
93#13133146—22no—
94#13133146C2×C414yescomplemented
95#13133139—22no—
96#13133139C2×C414yes—
97#13133137—22no—
98#13133137C2×C414yes—
99#13133128—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118#107×C53—22no—
119#107×C53C2×C414yescomplemented
120C4×C8×C53—22no—
121#95×C53—21no—
122#95×C53—21no—
123C23×D53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212C22×C412yes—
142C2×D212C2×Q812yes—
143C2×D212#4112yes—
144C2×C4×D53—24no—
145C2×C4×D53—24no—
146C2×C4×D53—24no—
147C2×C4×D53—81no—
148C2×C4×D53—81no—
149C2×C4×D53—81no—
150C2×C4×D53—81no—
151C2×C4×D53—81no—
152C2×C4×D53—81no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53C22×C412yes—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53C22×C412yes—
165C4⋊C4×C53C22×C412yes—
166C4⋊C4×C53#4112yes—
167C42×C53—81no—
168C42×C53—81no—
169C42×C53C2×Q812yes—
170C42×C53#4112yes—
171#11719921—24no—
172#11719921—24no—
173#11719921—24no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—24no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—81no—
180C4⋊Dic53C22×C412yes—
181C4⋊Dic53C2×Q812yes—
182#11719919—24no—
183#11719919—81no—
184#11719919—81no—
185#11719919—81no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—81no—
189C4×Dic53—81no—
190C4×Dic53—81no—
191C4×Dic53C22×C412yes—
192C4×Dic53#4112yes—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322C22×C412yes—
199C23×C53C22×D411yescharacteristic
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53C22⋊Q812yes—
224C2×C4×C53C22⋊Q812yes—
225C2×C4×C53C23×C411yescharacteristic
226C2×C4×C53C22×Q811yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53C4×D412yes—
251C2×Dic53C4×D412yes—
252D212—22no—
253D212—22no—
254D212—22no—
255D212C4×Q814yes—
256C4×D53—84no—
257C4×D53—84no—
258C4×D53—84no—
259C8×C53—22no—
260C8×C53—41no—
261#88322—22no—
262#88322—41no—
263#21620—531nomaximal, Sylow
264C22×C53C2×C4×D411yescharacteristic
265C22×C53C2×C22⋊Q811yescharacteristic
266C22×C53#52511yescharacteristic
267C22×C53#57611yescharacteristic
268C22×C53#57611yescharacteristic
269C22×C53#57711yescharacteristic
270C22×C53#57811yescharacteristic
271C2×D53—24no—
272C2×D53—42no—
273C2×D53—42no—
274C2×D53—42no—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—42no—
281C4×C53—42no—
282C4×C53—42no—
283C4×C53—81no—
284C4×C53C2×C4×Q811yescharacteristic
285C4×C53C4×#4111yescharacteristic
286C4×C53#52911yescharacteristic
287C4×C53#53311yescharacteristic
288Dic53—22no—
289Dic53—22no—
290Dic53—42no—
291Dic53—41no—
292Dic53—41no—
293Dic53—42no—
294Dic53—81no—
295C2×#478—531no—
296#2330—531no—
297#2275—531no—
298#1989—531no—
299#1947—531no—
300#1921—531no—
301#1920—531no—
302#1862—531no—
303#1862—531no—
304#1802—531no—
305#1776—531no—
306C4×#107D5311yescharacteristic, complemented
307#1768—531no—
308#1762—531no—
309C4×#102—531no—
310C2×C53C4×C22⋊Q811yescharacteristic
311C2×C53#294611yescharacteristic
312C2×C53#295111yescharacteristic
313C2×C53#308111yescharacteristic
314C2×C53#308311yescharacteristic
315C2×C53#312011yescharacteristic
316C2×C53#312211yescharacteristic
317D53—44no—
318C2×#125—531no—
319C2×C4×D4—531no—
320C2×C4×D4—531no—
321C2×C4×D4—531no—
322#478—538no—
323C2×#107—531no—
324C2×#107C2×D5311yescharacteristic
325C2×#107C2×D5311yescharacteristic
326C2×#106—531no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×#102—531no—
330C2×#102—531no—
331C2×#102—531no—
332C2×#102—531no—
333C2×#97—531no—
334C2×C4×C8C2×D5311yescharacteristic
335#389—531no—
336#389—531no—
337#389—1061no—
338#389—1061no—
339#387—1061no—
340#387—1061no—
341#384—531no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4C2×D5311yescharacteristic
346C4×C4⋊C4C2×D5311yescharacteristic
347C4×C22⋊C4—1061no—
348C4×C22⋊C4—1061no—
349C4×C22⋊C4—1061no—
350#376—531no—
351C2×#95—1061no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340C2×D5311yescharacteristic
359#340C2×D5311yescharacteristic
360#336—531no—
361#336—531no—
362C53#2162011yescharacteristic, complemented, Sylow
363C22×D4—531no—
364C23×C4—1061no—
365C23×C4—1061no—
366C23×C4—1061no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8C22×D5311yescharacteristic
370C22×C8C22×D5311yescharacteristic
371#125—534no—
372#125—1062no—
373C4×D4—534no—
374C4×D4—534no—
375C4×D4—1062no—
376C4×D4—1062no—
377C4×D4—1062no—
378C4×D4—1062no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—1061no—
400C2×C42—1061no—
401C2×C42—1061no—
402C2×C42C22×D5311yescharacteristic
403#107—22no—
404#107—1062no—
405#107C4×D5314yescomplemented
406#106—534no—
407C4⋊C8—534no—
408C4⋊C8—1062no—
409#102—534no—
410#102—534no—
411#102—1062no—
412#102—1062no—
413#97—1062no—
414C4×C8—22no—
415#95—21no—
416#95—21no—
417#95—531no—
418#95—531no—
419#95—531no—
420#95—531no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426C24—1061no—
427C2×D4—532no—
428C2×D4—532no—
429C2×D4—532no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—1064no—
442C22×C4—1064no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1064no—
446C22×C4—1061no—
447C22×C4—4241no—
448C22×C4—4241no—
449C22×C4—4241no—
450C22×C4—4241no—
451C22×C4—4241no—
452C22×C4—4241no—
453C22×C4D4×D5311yescharacteristic
454C22×C4D4×D5311yescharacteristic
455C22×C4C23×D5311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—532no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8C2×C4×D5312yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—532no—
474C4⋊C4—532no—
475C4⋊C4—1064no—
476C4⋊C4—1064no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—4241no—
482C4⋊C4—4241no—
483C4⋊C4—4241no—
484C4⋊C4—4241no—
485C4⋊C4C2×C4×D5312yes—
486C4⋊C4C2×C4×D5312yes—
487C4⋊C4#1171994512yes—
488C22⋊C4—1064no—
489C22⋊C4—1064no—
490C22⋊C4—1064no—
491C42—81no—
492C42—81no—
493C42—532no—
494C42—532no—
495C42—1061no—
496C42—1061no—
497C42—4241no—
498C42—4241no—
499C42—4241no—
500C42#1171994512yes—
501C42Q8×D5312yes—
502C23—1062no—
503C23—1062no—
504C23—1062no—
505C23—2121no—
506C23—2121no—
507C23—2121no—
508C23—2121no—
509C23C2×D4×D5311yescharacteristic
510D4—534no—
511D4—1062no—
512D4—1062no—
513D4—1062no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—532no—
531C2×C4—532no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—4244no—
545C2×C4—4244no—
546C2×C4—4244no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4#1313319512yes—
555C2×C4#1313319512yes—
556C2×C4C22×C4×D5311yescharacteristic
557C2×C4C2×Q8×D5311yescharacteristic
558C2×C4#41×D5311yescharacteristic
559C2×C4#41×D5311yescharacteristic
560C2×C4C2×#1171994511yescharacteristic
561C2×C4C2×#1171994511yescharacteristic
562C8—22no—
563C8—41no—
564C8—1062no—
565C8—2121no—
566C22—1064no—
567C22—2122no—
568C22—2122no—
569C22—2122no—
570C22—2122no—
571C22—2122no—
572C22—2122no—
573C22C4×D4×D5311yescharacteristic
574C22C2×#1313319511yescharacteristic
575C22#1681113211yescharacteristic
576C22#1681142411yescharacteristic
577C22#1681142511yescharacteristic
578C22#1681143011yescharacteristic
579C22#1681143611yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—81no—
586C4—1062no—
587C4—1062no—
588C4—4241no—
589C4—2121no—
590C4—2121no—
591C4—2122no—
592C4—2122no—
593C4C4×Q8×D5311yescharacteristic
594C4C4×#1171994511yescharacteristic
595C4#1681115111yescharacteristic
596C4#1681117711yescharacteristic
597C2—2124no—
598C2C4×#1313319511yescharacteristic
599C2#2028354011yescharacteristic
600C2#2028359011yescharacteristic
601C2#2028454911yescharacteristic
602C2#2028457111yescharacteristic
603C2#2028493511yescharacteristic
604C2#2028495511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#107×C53C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenterC4×#107D53
hypocenterC53#21620
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
2. #20277452C2semidirect
4. #20275671C2semidirect
5. #20275470C2semidirect
6. #20275309C2semidirect
7. #20275298C2semidirect
11. #20274613C2semidirect
12. #20274573C2semidirect
15. C4×#107×C53C2semidirect
20. #16810926C4semidirect
67. #13133208C2×C4semidirect
92. #13133148C2×C4semidirect
94. #13133146C2×C4semidirect
119. #107×C53C2×C4semidirect
306. C4×#107D53semidirect
362. C53#21620semidirect
405. #107C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109262
#202774522
#202770752
#202756712
#202754702
#202753092
#202752982
#202749632
#202749592
#202747142
#202746132
#202745732
#202745432
C4×#131331392
C4×#107×C532
#2162053