Finite Group Databasebeta

← #27436099

Subgroups — #27436099

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810923C211yesmaximal, characteristic
2C2×#16810751C211yesmaximal, characteristic, complemented
3#20280098C211yesmaximal, characteristic, complemented
4#20279637C211yesmaximal, characteristic
5#20276218C211yesmaximal, characteristic, complemented
6#20276212C211yesmaximal, characteristic, complemented
7#20276099C211yesmaximal, characteristic
8#20276099C211yesmaximal, characteristic
9#20275684C211yesmaximal, characteristic, complemented
10#20275473C211yesmaximal, characteristic, complemented
11#20275452C211yesmaximal, characteristic, complemented
12#20275447C211yesmaximal, characteristic, complemented
13#20275414C211yesmaximal, characteristic
14#20275400C211yesmaximal, characteristic
15C53×#1762C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133196—21no—
18C2×C4×Dic106C2211yescharacteristic
19#16810923—24no—
20#16810751—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133146C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32#16810555—21no—
33#16810546—21no—
34#16810546—21no—
35#16810544—21no—
36#16810526C2211yescharacteristic
37#16810526C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810522C2211yescharacteristic
40#16810522C2211yescharacteristic
41#16810522C2211yescharacteristic
42#16810521—21no—
43#16810521—21no—
44#16810519C2211yescharacteristic
45#16810519C2211yescharacteristic
46#16810493C2211yescharacteristic
47#16810483—21no—
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#97×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106D411yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133209—22no—
65#13133209—22no—
66#13133196—44no—
67C4×Dic106—22no—
68C4×Dic106—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53D411yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919D411yescharacteristic
85C2×#11719919D411yescharacteristic
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133150—22no—
93#13133150—22no—
94#13133150—22no—
95#13133148—22no—
96C4⋊#88322D414yes—
97#13133146—22no—
98#13133146—22no—
99#13133138—22no—
100#13133138—22no—
101#13133127—22no—
102C8×Dic53—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×D4×C53C2311yescharacteristic
110C23×C4×C53—21no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C4×D4×C53—22no—
114C4×D4×C53—22no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#102×C53—22no—
122#102×C53Q814yescomplemented
123#97×C53—22no—
124#95×C53—21no—
125C24×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53C2×Q812yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C2×Dic106—22no—
150C2×Dic106—22no—
151C2×Dic106—41no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53C2×Q812yes—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53C2×Q812yes—
163C4⋊C4×C53#4112yes—
164C22⋊C4×C53—42no—
165C42×C53—81no—
166C42×C53C2×D412yes—
167C42×C53#4112yes—
168C22⋊Dic53—42no—
169C22⋊Dic53—42no—
170C22⋊Dic53—81no—
171C22⋊Dic53—81no—
172C4⋊Dic53—22no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—42no—
182C4⋊Dic53—42no—
183C4⋊Dic53—42no—
184C4⋊Dic53—81no—
185C4⋊Dic53—81no—
186C4⋊Dic53—81no—
187#11719919—22no—
188#11719919—22no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192#11719919—42no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322C2×D412yes—
202C2×#88322C2×D412yes—
203C23×C53—22no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53C22×D411yescharacteristic
211D4×C53—22no—
212D4×C53—22no—
213D4×C53—41no—
214D4×C53—41no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—84no—
229C2×C4×C53#12112yes—
230C2×C4×C53#12112yes—
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×Q811yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×C4×C532−1+411yescharacteristic
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×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×Dic53AGL(1,ℤ/8)12yes—
261Dic106—44no—
262C8×C53—22no—
263C8×C53—41no—
264#88322—22no—
265#88322—22no—
266#25482—531nomaximal, Sylow
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53C2×#12111yescharacteristic
275C22×C53#54811yescharacteristic
276C22×C53D4×Q811yescharacteristic
277C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C53#57511yescharacteristic
279C22×C53#57611yescharacteristic
280C22×C53#57811yescharacteristic
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—81no—
286C4×C53#53311yescharacteristic
287C4×C53#54111yescharacteristic
288C4×C53#54711yescharacteristic
289C4×C53D4×Q811yescharacteristic
290Dic53—22no—
291Dic53—41no—
292Dic53—41no—
293Dic53—41no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298C2×#478—531no—
299C2×#446—531no—
300#2584—531no—
301#2538—531no—
302#2088—531no—
303#2087—531no—
304#2067—531no—
305#2067—531no—
306#1992—531no—
307#1947—531no—
308#1943—531no—
309#1942—531no—
310#1936—531no—
311#1934—531no—
312#1762D5311yescharacteristic
313C2×C53—82no—
314C2×C53#242911yescharacteristic
315C2×C53#305511yescharacteristic
316C2×C53#306111yescharacteristic
317C2×C53#336611yescharacteristic
318C2×C53#337711yescharacteristic
319C2×C53#338611yescharacteristic
320C2×C53#338911yescharacteristic
321C2×#125—531no—
322C2×C22⋊Q8—1061no—
323C2×C4×Q8—531no—
324C2×C4×D4C2×D5311yescharacteristic
325#478—1064no—
326#446—1064no—
327C2×#107—531no—
328C2×#107—531no—
329C2×#107—531no—
330C2×#106—531no—
331C2×C4⋊C8—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×#102—531no—
335C2×#102—531no—
336C2×#102C2×D5311yescharacteristic
337C2×#102C2×D5311yescharacteristic
338C2×#97—531no—
339C2×#97C2×D5311yescharacteristic
340C2×C4×C8—531no—
341#398—1061no—
342#396—1061no—
343#396—1061no—
344#395—1061no—
345#390—531no—
346#390—531no—
347#389—531no—
348#389—531no—
349#389—531no—
350#389—531no—
351#389—531no—
352#389—531no—
353#389—1061no—
354#389—1061no—
355#384—531no—
356#380—1061no—
357C4×C4⋊C4C2×D5311yescharacteristic
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340C2×D5311yescharacteristic
365#340C2×D5311yescharacteristic
366C53#2548211yescharacteristic, complemented, Sylow
367C22×Q8—531no—
368C22×D4C22×D5311yescharacteristic
369C23×C4—21no—
370C22×C8—531no—
371C22×C8—531no—
372C22×C8C22×D5311yescharacteristic
373C22×C8C22×D5311yescharacteristic
374#125—1062no—
375#125—1062no—
376C22⋊Q8—2124no—
377C4×Q8—1062no—
378C4×Q8—1062no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C42—21no—
407C2×C42—531no—
408C2×C42—531no—
409C2×C42C22×D5311yescharacteristic
410#107—1062no—
411#107—1062no—
412#107—1062no—
413#106—1062no—
414C4⋊C8—534no—
415C4⋊C8—1062no—
416C4⋊C8—1062no—
417#102—22no—
418#102—1062no—
419#102—1062no—
420#102Dic10614yescomplemented
421#97—22no—
422#97—1062no—
423C4×C8—1062no—
424#95—21no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431C24—21no—
432C2×Q8—1062no—
433C2×Q8—1062no—
434C2×Q8—2121no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C2×Dic10612yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4D4×D5311yescharacteristic
456C22×C4D4×D5311yescharacteristic
457C22×C4C23×D5311yescharacteristic
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—532no—
464C2×C8—532no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8C2×Dic10612yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—1062no—
475C4⋊C4—1062no—
476C4⋊C4—1062no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—4241no—
486C4⋊C4—4241no—
487C4⋊C4—4241no—
488C4⋊C4—2122no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C4⋊C4—2122no—
492C4⋊C4—2122no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4C2×Dic10612yes—
496C4⋊C4#1171994512yes—
497C22⋊C4—42no—
498C22⋊C4—4241no—
499C22⋊C4—4241no—
500C22⋊C4—2122no—
501C22⋊C4—2122no—
502C42—81no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C42—1061no—
507C42#1171994512yes—
508C42D4×D5312yes—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C2×D4×D5311yescharacteristic
517Q8—2124no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—532no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
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#1313319012yes—
561C2×C4#1313319012yes—
562C2×C4C2×D4×D5311yescharacteristic
563C2×C4C22×Dic10611yescharacteristic
564C2×C4C2×#1171994511yescharacteristic
565C2×C4C2×#1171994511yescharacteristic
566C2×C4#1313326411yescharacteristic
567C2×C4#1313326411yescharacteristic
568C8—22no—
569C8—41no—
570C8—1062no—
571C8—1062no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22C2×#1313319011yescharacteristic
580C22#1681128211yescharacteristic
581C22D4×Dic10611yescharacteristic
582C22AGL(1,ℤ/8)×D5311yescharacteristic
583C22#1681141911yescharacteristic
584C22#1681142711yescharacteristic
585C22#1681143911yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—1062no—
592C4—4241no—
593C4—4241no—
594C4—4241no—
595C4—4241no—
596C4—2121no—
597C4—2121no—
598C4—2121no—
599C4#1681118011yescharacteristic
600C4#1681123511yescharacteristic
601C4#1681127811yescharacteristic
602C4D4×Dic10611yescharacteristic
603C2—82no—
604C2#2027831411yescharacteristic
605C2#2028433211yescharacteristic
606C2#2028438411yescharacteristic
607C2#2028742311yescharacteristic
608C2#2028755811yescharacteristic
609C2#2028763711yescharacteristic
610C2#2028767411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1762C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1762D53
hypocenterC53#25482
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16810751C2semidirect
3. #20280098C2semidirect
5. #20276218C2semidirect
6. #20276212C2semidirect
9. #20275684C2semidirect
10. #20275473C2semidirect
11. #20275452C2semidirect
12. #20275447C2semidirect
122. #102×C53Q8semidirect
366. C53#25482semidirect
420. #102Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168109232
C2×#168107512
#202800982
#202796372
#202762182
#202762122
#202760992
#202756842
#202754732
#202754522
#202754472
#202754142
#202754002
C53×#17622
#2548253