Finite Group Databasebeta

← #27465874

Subgroups — #27465874

Order 13568.

772 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
1C2×#16810965C211yesmaximal, characteristic
2C2×#16810954C211yesmaximal, characteristic
3#20277397C211yesmaximal, characteristic
4#20277338C211yesmaximal, characteristic
5#20276198C211yesmaximal, characteristic, complemented
6#20276193C211yesmaximal, characteristic, complemented
7#20276135C211yesmaximal, characteristic, complemented
8#20276125C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275479C211yesmaximal, characteristic, complemented
11#20274811C211yesmaximal, characteristic
12#20274806C211yesmaximal, characteristic
13#20274731C211yesmaximal, characteristic, complemented
14#20274721C211yesmaximal, characteristic, complemented
15C53×#1798C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810965—24no—
21#16810954—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30#16810529C2211yescharacteristic
31#16810522—21no—
32#16810522—21no—
33#16810521C2211yescharacteristic
34#16810518—21no—
35#16810517—21no—
36#16810511—21no—
37#16810504—21no—
38#16810498—21no—
39#16810498—21no—
40#16810493C2211yescharacteristic
41#16810493C2211yescharacteristic
42#16810489C2211yescharacteristic
43#16810489C2211yescharacteristic
44#16810483C2211yescharacteristic
45C4×C4⋊Dic53C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63#13133196—22no—
64#13133196D414yes—
65C4×Dic106—22no—
66C4×Dic106—22no—
67#13133176—22no—
68#13133176—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—22no—
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×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92#13133151—22no—
93#13133149—22no—
94#13133149—22no—
95#13133149—22no—
96#13133142—22no—
97#13133142—22no—
98#13133142—22no—
99#13133142—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114#117×C53—22no—
115#117×C53—22no—
116C2×C4⋊C4×C53D411yescharacteristic
117C2×C4⋊C4×C53D411yescharacteristic
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122C4⋊C8×C53—22no—
123C4⋊C8×C53—22no—
124C22⋊C8×C53—22no—
125C22⋊C8×C53—22no—
126C24×C53C2×D411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—42no—
148C22×Dic53—81no—
149C22×Dic53—81no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106C2×D412yes—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53#4112yes—
175C42×C53#4112yes—
176C22⋊Dic53—24no—
177C22⋊Dic53—42no—
178C22⋊Dic53—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—81no—
194C4⋊Dic53—81no—
195C4⋊Dic53—81no—
196C4⋊Dic53—81no—
197C4⋊Dic53C2×D412yes—
198#11719919—24no—
199#11719919—42no—
200#11719919—81no—
201C4×Dic53—42no—
202C4×Dic53—42no—
203C4×Dic53—81no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53#12112yes—
211C23×C53C22×D411yescharacteristic
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—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
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#12112yes—
229C2×C4×C53#12312yes—
230C2×C4×C53#12312yes—
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—84no—
262Dic106—22no—
263Dic106—22no—
264Dic106—41no—
265Dic106—41no—
266C8×C53—41no—
267C8×C53—41no—
268C8×C53—41no—
269C8×C53—41no—
270#27657—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53C2×#12111yescharacteristic
279C22×C53C2×#12311yescharacteristic
280C22×C53#53611yescharacteristic
281C22×C53#57711yescharacteristic
282C22×C53#57711yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57811yescharacteristic
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53#54711yescharacteristic
291C4×C53#54711yescharacteristic
292C4×C53#55211yescharacteristic
293C4×C53#55211yescharacteristic
294Dic53—42no—
295Dic53—42no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302C2×#485—531no—
303C2×#483—531no—
304#2323—531no—
305#2317—531no—
306#2085—531no—
307#2084—531no—
308#2074—531no—
309#2072—531no—
310#1949—531no—
311#1948—531no—
312#1825—531no—
313#1824—531no—
314#1805—531no—
315#1803—531no—
316#1798D5311yescharacteristic
317C2×C53—42no—
318C2×C53#248611yescharacteristic
319C2×C53#304311yescharacteristic
320C2×C53#304611yescharacteristic
321C2×C53#309711yescharacteristic
322C2×C53#309911yescharacteristic
323C2×C53#324211yescharacteristic
324C2×C53#324411yescharacteristic
325C2×C22⋊Q8—531no—
326C2×C4×Q8—531no—
327C2×#117—531no—
328C2×#117C2×D5311yescharacteristic
329C22×C4⋊C4—531no—
330#485—1064no—
331#483—1064no—
332C2×#107—531no—
333C2×#106—531no—
334C2×#106—531no—
335C2×#106—531no—
336C2×C4⋊C8C2×D5311yescharacteristic
337C2×C4⋊C8C2×D5311yescharacteristic
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×C22⋊C8C2×D5311yescharacteristic
343C2×C22⋊C8C2×D5311yescharacteristic
344#391—531no—
345#389—531no—
346#389—1061no—
347#389—1061no—
348#389—1061no—
349#388—1061no—
350#388—1061no—
351#387—1061no—
352#385—1061no—
353#385—1061no—
354#384—531no—
355#384—531no—
356#382—531no—
357#382—531no—
358#380—531no—
359C4×C4⋊C4—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#336C2×D5311yescharacteristic
369#336C2×D5311yescharacteristic
370C53#2765711yescharacteristic, complemented, Sylow
371C22×Q8—531no—
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C22×C8C22×D5311yescharacteristic
375C22×C8C22×D5311yescharacteristic
376C22×C8C22×D5311yescharacteristic
377C22×C8C22×D5311yescharacteristic
378C22⋊Q8—534no—
379C22⋊Q8—1062no—
380C4×Q8—1062no—
381C4×Q8—1062no—
382#117—22no—
383#117—22no—
384#117—1062no—
385#117—1062no—
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⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4D21211yescharacteristic
404C2×C4⋊C4D21211yescharacteristic
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4D21211yescharacteristic
409C2×C22⋊C4D21211yescharacteristic
410C2×C42—1061no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42C22×D5311yescharacteristic
414C2×C42C22×D5311yescharacteristic
415#107—1062no—
416#106—1062no—
417#106—1062no—
418#106—1062no—
419C4⋊C8—22no—
420C4⋊C8—22no—
421#103—1062no—
422#103—1062no—
423#103—1062no—
424#103—1062no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24C2×D21211yescharacteristic
437C2×Q8—532no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×Q8—1061no—
442C2×Q8—1061no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—4241no—
456C22×C4—4241no—
457C22×C4—2122no—
458C22×C4C2×D21211yescharacteristic
459C22×C4C2×D21211yescharacteristic
460C22×C4C2×D21211yescharacteristic
461C22×C4C2×D21211yescharacteristic
462C22×C4C2×D21211yescharacteristic
463C22×C4D4×D5312yes—
464C22×C4#1171994912yes—
465C22×C4C23×D5311yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—532no—
481C4⋊C4—1064no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4—4241no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—1064no—
505C22⋊C4—4241no—
506C22⋊C4—2122no—
507C42—21no—
508C42—21no—
509C42—4241no—
510C42—2122no—
511C42—2122no—
512C42#1171994712yes—
513C42#1171994712yes—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23#1313319212yes—
521C23C22×D21211yescharacteristic
522Q8—1062no—
523Q8—1062no—
524Q8—2121no—
525Q8—2121no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—4244no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4#1313319212yes—
568C2×C4#1313320112yes—
569C2×C4#1313320112yes—
570C2×C4C2×D4×D5311yescharacteristic
571C2×C4C2×#1171994711yescharacteristic
572C2×C4C2×#1171994711yescharacteristic
573C2×C4C2×#1171994911yescharacteristic
574C2×C4#1313326411yescharacteristic
575C2×C4#1313326411yescharacteristic
576C8—41no—
577C8—41no—
578C8—41no—
579C8—41no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1313319211yescharacteristic
588C22C2×#1313320111yescharacteristic
589C22#1681119911yescharacteristic
590C22#1681143511yescharacteristic
591C22#1681143511yescharacteristic
592C22#1681143511yescharacteristic
593C22#1681144011yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—4241no—
600C4—4241no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—4241no—
605C4—2122no—
606C4—2122no—
607C4#1681127311yescharacteristic
608C4#1681127311yescharacteristic
609C4#1681131511yescharacteristic
610C4#1681131511yescharacteristic
611C2—42no—
612C2#2027902411yescharacteristic
613C2#2028423711yescharacteristic
614C2#2028426711yescharacteristic
615C2#2028468511yescharacteristic
616C2#2028470611yescharacteristic
617C2#2028598511yescharacteristic
618C2#2028601511yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1798C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1798D53
hypocenterC53#27657
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
5. #20276198C2semidirect
6. #20276193C2semidirect
7. #20276135C2semidirect
8. #20276125C2semidirect
9. #20275485C2semidirect
10. #20275479C2semidirect
13. #20274731C2semidirect
14. #20274721C2semidirect
370. C53#27657semidirect

Maximal subgroups

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

subgroupindex
C2×#168109652
C2×#168109542
#202773972
#202773382
#202761982
#202761932
#202761352
#202761252
#202754852
#202754792
#202748112
#202748062
#202747312
#202747212
C53×#17982
#2765753