Finite Group Databasebeta

← #27435672

Subgroups — #27435672

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811027C211yesmaximal, characteristic, complemented
2C2×#16810911C211yesmaximal, characteristic
3#20279886C211yesmaximal, characteristic
4#20277801C211yesmaximal, characteristic, complemented
5#20276201C211yesmaximal, characteristic, complemented
6#20276196C211yesmaximal, characteristic, complemented
7#20276101C211yesmaximal, characteristic
8#20275661C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275480C211yesmaximal, characteristic, complemented
11#20275430C211yesmaximal, characteristic, complemented
12#20275338C211yesmaximal, characteristic
13#20275332C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C53×#2067C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19#16811027—24no—
20#16810911—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133146C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810535—21no—
32#16810535—21no—
33#16810528—21no—
34#16810527—21no—
35#16810526C2211yescharacteristic
36#16810519C2211yescharacteristic
37#16810518—21no—
38#16810489C2211yescharacteristic
39#16810488C2211yescharacteristic
40#16810481—21no—
41#16810481—21no—
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×#11719919C2211yescharacteristic
45C4×C4⋊Dic53C2211yescharacteristic
46#16810474—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C53×#390C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106—21no—
62C22×Dic106—21no—
63C23×Dic53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133216—22no—
67#13133216—22no—
68#13133216—22no—
69#13133216D414yes—
70D4×Dic53—22no—
71D4×Dic53—22no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919D411yescharacteristic
80C2×#11719919D411yescharacteristic
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133151—22no—
90#13133150—22no—
91#13133150—22no—
92#13133149—22no—
93#13133148—22no—
94#13133146—22no—
95#13133146—22no—
96#13133138—22no—
97#13133138Q814yescomplemented
98C8×Dic53—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×D4×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53C2311yescharacteristic
119#107×C53—22no—
120C4⋊C8×C53—22no—
121#102×C53—22no—
122#102×C53—22no—
123#95×C53—21no—
124C24×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53C2×Q812yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—42no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×Dic106—24no—
149C2×Dic106—42no—
150C2×Dic106—81no—
151C2×Dic106—81no—
152C2×Dic106—81no—
153C2×Dic106—81no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53#4112yes—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—81no—
166C4⋊C4×C53C2×D412yes—
167C22⋊C4×C53—81no—
168C22⋊C4×C53—81no—
169C42×C53—21no—
170C42×C53—21no—
171C22⋊Dic53—42no—
172C22⋊Dic53—81no—
173C22⋊Dic53—81no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C2×D412yes—
181C4⋊Dic53C2×Q812yes—
182#11719919—24no—
183#11719919—22no—
184#11719919—22no—
185#11719919—42no—
186#11719919—81no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—42no—
190C4×Dic53—81no—
191C4×Dic53—81no—
192C4×Dic53C2×D412yes—
193C4×Dic53#4112yes—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C2×Q812yes—
200C23×C53—22no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53C22×D411yescharacteristic
208D4×C53—22no—
209D4×C53—22no—
210D4×C53—41no—
211D4×C53—41no—
212C2×C4×C53—22no—
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—41no—
220C2×C4×C53—41no—
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×C53C22×D411yescharacteristic
228C2×C4×C53C22×Q811yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C2×C4×C532−1+411yescharacteristic
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—22no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—22no—
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×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—84no—
254C2×Dic53#12112yes—
255C2×Dic53#12112yes—
256C2×Dic53#13512yes—
257C2×Dic53#13712yes—
258Dic106—84no—
259Dic106—84no—
260C8×C53—22no—
261C8×C53—41no—
262#88322—22no—
263#88322—41no—
264#25450—531nomaximal, Sylow
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53C2×#12111yescharacteristic
273C22×C53#54711yescharacteristic
274C22×C53D4×Q811yescharacteristic
275C22×C53C2×#13511yescharacteristic
276C22×C53C2×#13711yescharacteristic
277C22×C53#57611yescharacteristic
278C22×C53#57611yescharacteristic
279C4×C53—41no—
280C4×C53—41no—
281C4×C53—42no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53#53311yescharacteristic
285C4×C53#54111yescharacteristic
286C4×C53#54811yescharacteristic
287C4×C53D4×Q811yescharacteristic
288Dic53—22no—
289Dic53—41no—
290Dic53—41no—
291Dic53—42no—
292Dic53—42no—
293Dic53—81no—
294Dic53—81no—
295Dic53#46414yes—
296C2×#500—531no—
297C2×#476—531no—
298#2563—531no—
299#2369—531no—
300#2085—531no—
301#2085—531no—
302#2067—531no—
303#2067D5311yescharacteristic
304#1987—531no—
305#1949—531no—
306#1949—531no—
307#1939—531no—
308#1925—531no—
309#1925—531no—
310C4×#102—531no—
311C2×C53—82no—
312C2×C53#242911yescharacteristic
313C2×C53C2×#46411yescharacteristic
314C2×C53#304111yescharacteristic
315C2×C53#334111yescharacteristic
316C2×C53#335711yescharacteristic
317C2×C53#338011yescharacteristic
318C2×C53#338511yescharacteristic
319C2×C4⋊Q8—531no—
320C2×C4⋊Q8—531no—
321C2×C4×D4—531no—
322#500—1064no—
323#476—1064no—
324C2×#107—531no—
325C2×#107—531no—
326C2×#107—531no—
327C2×#107C2×D5311yescharacteristic
328C2×#106—531no—
329C2×#106—531no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—531no—
332C2×C4⋊C8C2×D5311yescharacteristic
333C2×#102—531no—
334C2×#102—531no—
335C2×#102C2×D5311yescharacteristic
336C2×#102C2×D5311yescharacteristic
337C2×C4×C8—531no—
338#393—1061no—
339#393—1061no—
340#391—1061no—
341#391—1061no—
342#390—531no—
343#390C2×D5311yescharacteristic
344#389—531no—
345#389—1061no—
346#389C2×D5311yescharacteristic
347#382—531no—
348#382—531no—
349#380—1061no—
350#380—1061no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—1061no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340C2×D5311yescharacteristic
362#336—531no—
363#336—531no—
364C53#2545011yescharacteristic, complemented, Sylow
365C22×Q8—1061no—
366C22×Q8—1061no—
367C22×D4C22×D5311yescharacteristic
368C23×C4—1061no—
369C22×C8—531no—
370C22×C8—531no—
371C22×C8C22×D5311yescharacteristic
372C22×C8C22×D5311yescharacteristic
373C4⋊Q8—534no—
374C4⋊Q8—1062no—
375C4⋊Q8—1062no—
376C4⋊Q8—1062no—
377C4×D4—1062no—
378C4×D4—1062no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
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#107—1062no—
406#107—1062no—
407#106—1062no—
408#106—1062no—
409C4⋊C8—22no—
410C4⋊C8—1062no—
411C4⋊C8—1062no—
412#102—22no—
413#102—22no—
414#102—534no—
415#102—1062no—
416C4×C8—1062no—
417#95—21no—
418#95—531no—
419#95—531no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428C24—21no—
429C2×Q8—1064no—
430C2×Q8—4241no—
431C2×Q8—4241no—
432C2×Q8—4241no—
433C2×Q8—4241no—
434C2×Q8—2122no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4Q8×D5312yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4—2122no—
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—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8#1171994712yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—81no—
476C4⋊C4—532no—
477C4⋊C4—532no—
478C4⋊C4—1062no—
479C4⋊C4—1062no—
480C4⋊C4—1064no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—4241no—
488C4⋊C4—2122no—
489C4⋊C4D4×D5312yes—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—4241no—
493C22⋊C4—4241no—
494C22⋊C4—2122no—
495C42—21no—
496C42—21no—
497C42—532no—
498C42—532no—
499C42—1061no—
500C42—1061no—
501C42—4241no—
502C42—4241no—
503C42—2122no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D5311yescharacteristic
512Q8—4244no—
513Q8—4244no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—532no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—4244no—
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×C4C2×D4×D5311yescharacteristic
559C2×C4C2×Q8×D5311yescharacteristic
560C2×C4#41×D5311yescharacteristic
561C2×C4C2×#1171994711yescharacteristic
562C2×C4#1313326411yescharacteristic
563C2×C4#1313326411yescharacteristic
564C8—22no—
565C8—41no—
566C8—1062no—
567C8—2121no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22#121×D5311yescharacteristic
576C22#1681127311yescharacteristic
577C22#1681128611yescharacteristic
578C22#135×D5311yescharacteristic
579C22#137×D5311yescharacteristic
580C22#1681142411yescharacteristic
581C22#1681142711yescharacteristic
582C4—42no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—534no—
588C4—1062no—
589C4—4241no—
590C4—4241no—
591C4—2122no—
592C4—2122no—
593C4—2121no—
594C4—2121no—
595C4#1681117511yescharacteristic
596C4#1681123011yescharacteristic
597C4#1681127911yescharacteristic
598C4#1681128611yescharacteristic
599C2—82no—
600C2#2027830711yescharacteristic
601C2#464×D5311yescharacteristic
602C2#2028421311yescharacteristic
603C2#2028713411yescharacteristic
604C2#2028733611yescharacteristic
605C2#2028757711yescharacteristic
606C2#2028762011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2067C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2067D53
hypocenterC53#25450
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. C2×#16811027C2semidirect
4. #20277801C2semidirect
5. #20276201C2semidirect
6. #20276196C2semidirect
8. #20275661C2semidirect
9. #20275485C2semidirect
10. #20275480C2semidirect
11. #20275430C2semidirect
97. #13133138Q8semidirect
364. C53#25450semidirect

Maximal subgroups

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

subgroupindex
C2×#168110272
C2×#168109112
#202798862
#202778012
#202762012
#202761962
#202761012
#202756612
#202754852
#202754802
#202754302
#202753382
#202753322
#102×Dic532
C53×#20672
#2545053