Finite Group Databasebeta

← #27432331

Subgroups — #27432331

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810958C211yesmaximal, characteristic
2C2×#16810862C211yesmaximal, characteristic
3#20277718C211yesmaximal, characteristic
4#20276187C211yesmaximal, characteristic, complemented
5#20276124C211yesmaximal, characteristic, complemented
6#20276110C211yesmaximal, characteristic, complemented
7#20276037C211yesmaximal, characteristic, complemented
8#20275471C211yesmaximal, characteristic, complemented
9#20275353C211yesmaximal, characteristic, complemented
10#20275059C211yesmaximal, characteristic, complemented
11#20275023C211yesmaximal, characteristic
12#20274895C211yesmaximal, characteristic
13#20274717C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic
15C53×#2502C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810958—24no—
22#16810862—24no—
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×#13133140C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35#16810529C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810520—21no—
38#16810510—21no—
39#16810495—21no—
40C4×C4⋊Dic53C2211yescharacteristic
41C22⋊C4×Dic53—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#398—21no—
53C53×#398—21no—
54C53×#391C2211yescharacteristic
55C53×#391C2211yescharacteristic
56C53×#385—21no—
57C53×#385—21no—
58C53×#384C2211yescharacteristic
59C53×#384C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C2×#11719925—22no—
64C2×#11719925—22no—
65C2×#11719925—41no—
66C22×#88322—22no—
67C22×#88322—22no—
68C22×#88322—41no—
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73#13133196—22no—
74#13133196D414yes—
75C4×Dic106—22no—
76C4×Dic106—41no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—41no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89#13133150—22no—
90#13133150—22no—
91#13133150—22no—
92C4⋊#88322—22no—
93C4⋊#88322—41no—
94#13133141—22no—
95#13133141—22no—
96#13133141—41no—
97#13133140—22no—
98#13133140—22no—
99#13133140—22no—
100C22⋊#88322—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×Q8×C53C2311yescharacteristic
106C23×C4×C53—21no—
107C23×C4×C53C2311yescharacteristic
108C22⋊Q8×C53—22no—
109C22⋊Q8×C53—22no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—22no—
117C2×C4⋊C4×C53—22no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—42no—
148C22×C4×C53—81no—
149C22×C4×C53—81no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53#4112yes—
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—42no—
163C22×Dic53—42no—
164C22×Dic53—81no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106C2×D412yes—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53—81no—
180C4⋊C4×C53—81no—
181C4⋊C4×C53—81no—
182C4⋊C4×C53—81no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—42no—
185C22⋊C4×C53Q1614yescomplemented
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—21no—
190C22⋊Dic53—24no—
191#11719925—44no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—42no—
197C4⋊Dic53C2×D412yes—
198C4⋊Dic53#4112yes—
199#11719919—24no—
200#11719919—42no—
201#11719919—81no—
202C4×Dic53—42no—
203C4×Dic53—81no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—44no—
216C2×#88322#4112yes—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53#12112yes—
222C23×C53C2×Q1612yes—
223C23×C53#13712yes—
224C23×C53C22×D411yescharacteristic
225Q8×C53—41no—
226Q8×C53—41no—
227Q8×C53—41no—
228Q8×C53—41no—
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—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
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×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—84no—
258C2×C4×C53#12112yes—
259C2×C4×C53C2×Q1612yes—
260C2×C4×C53C2×Q1612yes—
261C2×C4×C53C22×D411yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53H(5,𝔽2)11yescharacteristic
266C2×C4×C532−1+411yescharacteristic
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—84no—
280Dic106—22no—
281Dic106—22no—
282Dic106—41no—
283Dic106—41no—
284#88322—22no—
285#88322—41no—
286#88322—41no—
287#88322—41no—
288#25206—531nomaximal, Sylow
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53#46214yes—
296C22×C53C2×#12111yescharacteristic
297C22×C53#54611yescharacteristic
298C22×C53#54711yescharacteristic
299C22×C53C22×Q1611yescharacteristic
300C22×C53C2×#13711yescharacteristic
301C22×C53#57611yescharacteristic
302C22×C53#57711yescharacteristic
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—22no—
306C4×C53—41no—
307C4×C53—41no—
308C4×C53—81no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53—81no—
312C4×C53#52911yescharacteristic
313C4×C53#53911yescharacteristic
314C4×C53#54611yescharacteristic
315C4×C53#54711yescharacteristic
316Dic53—42no—
317Dic53—42no—
318Dic53—42no—
319Dic53—81no—
320C2×#484—531no—
321C2×#467—531no—
322#2502D5311yescharacteristic
323#2360—531no—
324#2083—531no—
325#2072—531no—
326#2068—531no—
327#2056—531no—
328#1947—531no—
329#1928—531no—
330#1878—531no—
331#1872—531no—
332#1850—531no—
333#1803—531no—
334C2×#340—531no—
335C2×C53—24no—
336C2×C53#242511yescharacteristic
337C2×C53C2×#46211yescharacteristic
338C2×C53#304511yescharacteristic
339C2×C53#329911yescharacteristic
340C2×C53#331411yescharacteristic
341C2×C53#333811yescharacteristic
342C2×C53#334411yescharacteristic
343C22×Q16—531no—
344C23×C8—531no—
345C2×C22⋊Q8—531no—
346C2×C22⋊Q8C2×D5311yescharacteristic
347C2×C4×Q8—531no—
348C22×C4⋊C4—531no—
349C22×C4⋊C4C2×D5311yescharacteristic
350#484—1064no—
351#467—1064no—
352C2×#107—531no—
353C2×#107—531no—
354C2×#107—531no—
355C2×C4⋊C8—531no—
356C2×C4⋊C8—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×#103—531no—
361C2×#103—531no—
362C2×#103—531no—
363C2×C22⋊C8—531no—
364#398—21no—
365#398—21no—
366#391—531no—
367#391C2×D5311yescharacteristic
368#391C2×D5311yescharacteristic
369#389—531no—
370#389—1061no—
371#387—1061no—
372#385—21no—
373#385—21no—
374#385—1061no—
375#384C2×D5311yescharacteristic
376#384C2×D5311yescharacteristic
377C4×C4⋊C4—531no—
378C4×C22⋊C4—1061no—
379C4×C22⋊C4C2×D5311yescharacteristic
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#336—531no—
387#336—531no—
388C53#2520611yescharacteristic, complemented, Sylow
389C22×Q8—531no—
390C22×Q8C22×D5311yescharacteristic
391C23×C4—21no—
392C23×C4—1061no—
393C23×C4C22×D5311yescharacteristic
394C2×Q16—1062no—
395C2×Q16—1062no—
396C2×Q16—2121no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8—531no—
401C22×C8—1062no—
402C22×C8—1062no—
403C22×C8—2121no—
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406C22⋊Q8—534no—
407C22⋊Q8—1062no—
408C4×Q8—1062no—
409C4×Q8—2121no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—1062no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—2121no—
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C4⋊C4C22×D5311yescharacteristic
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4C22⋊D5311yescharacteristic
433C2×C22⋊C4C22⋊D5311yescharacteristic
434C2×C42—1061no—
435C2×C42—1061no—
436C2×C42C22×D5311yescharacteristic
437C2×C42C22×D5311yescharacteristic
438#107—1062no—
439#107—1062no—
440#107—1062no—
441C4⋊C8—1062no—
442C4⋊C8—2121no—
443#103—1062no—
444#103—1062no—
445#103—1062no—
446#103—1062no—
447#103—1062no—
448#103—2121no—
449C22⋊C8—1062no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95C22⋊D5311yescharacteristic
459#95C22⋊D5311yescharacteristic
460C24C2×C22⋊D5311yescharacteristic
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—532no—
468C2×Q8—1061no—
469C2×Q8—1061no—
470C2×Q8—1061no—
471C2×Q8—1061no—
472C2×Q8—1061no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—81no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—4241no—
489C22×C4—2122no—
490C22×C4—2122no—
491C22×C4D4×D5312yes—
492C22×C4#1171994712yes—
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C23×D5311yescharacteristic
499Q16—2124no—
500C2×C8—532no—
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—2124no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—81no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—532no—
526C4⋊C4—532no—
527C4⋊C4—1064no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—4241no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C22⋊C4—22no—
536C22⋊C4—42no—
537C22⋊C4—1064no—
538C22⋊C4#1171992514yescomplemented
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—4241no—
544C42—2122no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#1313318812yes—
550C23C2×#1171992512yes—
551C23#1313324812yes—
552C23C22×C22⋊D5311yescharacteristic
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—41no—
557Q8—1062no—
558Q8—1062no—
559Q8—2121no—
560Q8—2121no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—4244no—
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#1313318812yes—
604C2×C4C2×#1171992512yes—
605C2×C4C2×#1171992512yes—
606C2×C4C2×D4×D5311yescharacteristic
607C2×C4#41×D5311yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4#1313326311yescharacteristic
611C2×C4#1313326411yescharacteristic
612C8—1062no—
613C8—2121no—
614C8—2121no—
615C8—2121no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22#1681083014yes—
623C22C2×#1313318811yescharacteristic
624C22#1681126111yescharacteristic
625C22#1681127211yescharacteristic
626C22C22×#1171992511yescharacteristic
627C22C2×#1313324811yescharacteristic
628C22#1681142611yescharacteristic
629C22#1681143411yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—4241no—
640C4—2122no—
641C4—2122no—
642C4—2122no—
643C4#1681115011yescharacteristic
644C4#1681121711yescharacteristic
645C4#1681126511yescharacteristic
646C4#1681127111yescharacteristic
647C2—24no—
648C2#2027826211yescharacteristic
649C2C2×#1681083011yescharacteristic
650C2#2028425011yescharacteristic
651C2#2028658511yescharacteristic
652C2#2028679011yescharacteristic
653C2#2028710311yescharacteristic
654C2#2028716411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2502C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2502D53
hypocenterC53#25206
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
4. #20276187C2semidirect
5. #20276124C2semidirect
6. #20276110C2semidirect
7. #20276037C2semidirect
8. #20275471C2semidirect
9. #20275353C2semidirect
10. #20275059C2semidirect
13. #20274717C2semidirect
185. C22⋊C4×C53Q16semidirect
388. C53#25206semidirect
538. C22⋊C4#11719925semidirect

Maximal subgroups

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

subgroupindex
C2×#168109582
C2×#168108622
#202777182
#202761872
#202761242
#202761102
#202760372
#202754712
#202753532
#202750592
#202750232
#202748952
#202747172
C2×#168103852
C53×#25022
#2520653