Finite Group Databasebeta

← #27469475

Subgroups — #27469475

Order 13568.

930 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×#16810948C211yesmaximal, characteristic, complemented
2C2×#16810937C211yesmaximal, characteristic, complemented
3C2×#16810876C211yesmaximal, characteristic
4C2×#16810665C211yesmaximal, characteristic, complemented
5#20279207C211yesmaximal, characteristic, complemented
6#20279177C211yesmaximal, characteristic, complemented
7#20276242C211yesmaximal, characteristic, complemented
8#20276235C211yesmaximal, characteristic, complemented
9#20276235C211yesmaximal, characteristic, complemented
10#20276235C211yesmaximal, characteristic, complemented
11#20275591C211yesmaximal, characteristic, complemented
12#20275581C211yesmaximal, characteristic, complemented
13#20274883C211yesmaximal, characteristic, complemented
14#20274879C211yesmaximal, characteristic, complemented
15C53×#1753C211yesmaximal, characteristic, complemented
16C22×D424C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810948—24no—
22#16810937—24no—
23#16810876—24no—
24#16810665—24no—
25C2×#13133151C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133151C2211yescharacteristic
28C2×#13133151C2211yescharacteristic
29C2×#13133151C2211yescharacteristic
30C2×#13133149C2211yescharacteristic
31C2×#13133149C2211yescharacteristic
32C2×#13133149C2211yescharacteristic
33C2×#13133149C2211yescharacteristic
34C2×#13133139C2211yescharacteristic
35C2×#13133139C2211yescharacteristic
36C2×#13133139C2211yescharacteristic
37C2×#13133139C2211yescharacteristic
38C2×#13133139C2211yescharacteristic
39C2×#13133139C2211yescharacteristic
40#16810550—22no—
41#16810539—22no—
42#16810525C2211yescharacteristic
43#16810525C2211yescharacteristic
44#16810525C2211yescharacteristic
45#16810525C2211yescharacteristic
46#16810515—22no—
47#16810500—22no—
48#16810493C2211yescharacteristic
49#16810489C2211yescharacteristic
50#16810489C2211yescharacteristic
51#16810489C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×#97×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C22×C22⋊D53—21no—
63C23×Dic53—21no—
64C2×D424—22no—
65C2×D424—22no—
66C2×D424—22no—
67#13133191—22no—
68#13133191—22no—
69#13133191—22no—
70#13133191—22no—
71#13133176—22no—
72#13133176—22no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C22⋊Dic53—21no—
84C2×#11719921—21no—
85C2×#11719921—22no—
86C2×#11719921—21no—
87C2×#11719921—22no—
88C2×#11719921—22no—
89C2×#11719921—22no—
90C2×C4×Dic53—21no—
91#13133151—22no—
92#13133151—22no—
93#13133151—22no—
94#13133151—22no—
95#13133151—22no—
96#13133149—22no—
97#13133149—22no—
98#13133149—22no—
99#13133149—22no—
100#13133139—22no—
101#13133139—22no—
102#13133139—22no—
103#13133139—22no—
104#13133139—22no—
105#13133139—22no—
106#13133124—22no—
107#13133124—22no—
108#13133124—22no—
109C23×C4×C53C2311yescharacteristic
110C2×M16×C53—22no—
111C2×M16×C53—22no—
112C2×M16×C53D414yes—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C22⋊C4×C53D412yes—
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C22⋊C8×C53D414yes—
121#97×C53—22no—
122C4×C8×C53—22no—
123#95×C53D412yes—
124C24×C53C2×D411yescharacteristic
125C23×D53—21no—
126C23×D53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2411yescharacteristic
137C2×C22⋊D53—44no—
138C2×C22⋊D53—42no—
139C2×C22⋊D53—42no—
140C2×C22⋊D53—82no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—42no—
148C22×Dic53—82no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161M16×C53—28no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53C2×D412yes—
169C2×C8×C53C2×D412yes—
170C2×C8×C53C2×D412yes—
171C2×C8×C53C2×D412yes—
172C2×C8×C53#4112yes—
173C2×C8×C53#4112yes—
174C22⋊C4×C53—42no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C22⋊Dic53—42no—
180#11719921—44no—
181#11719921—42no—
182#11719921—42no—
183#11719921—82no—
184#11719921—82no—
185#11719921—82no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—42no—
199C4⋊Dic53—82no—
200C4×Dic53—42no—
201D424—44no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53#12712yes—
207C23×C53C22×D411yescharacteristic
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53#12712yes—
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C22⋊D53—84no—
238C22⋊D53—84no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—84no—
252D212—41no—
253D212—41no—
254D212—41no—
255D212—41no—
256D212—41no—
257D212—41no—
258D212—41no—
259D212—41no—
260C8×C53—22no—
261C8×C53—22no—
262C8×C53—22no—
263C8×C53#12314yes—
264#27942—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53C2×#12711yescharacteristic
271C22×C53#54711yescharacteristic
272C22×C53#54711yescharacteristic
273C22×C53#57512yes—
274C22×C53#57612yes—
275C2×D53—82no—
276C2×D53—82no—
277C2×D53—82no—
278C2×D53—82no—
279C2×D53—81no—
280C2×D53—81no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—42no—
288C4×C53C2×#12311yescharacteristic
289C4×C53#53611yescharacteristic
290C4×C53#54711yescharacteristic
291C4×C53#54711yescharacteristic
292Dic53—81no—
293Dic53—81no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298C2×#482—531no—
299C2×#480—531no—
300C2×#469—531no—
301C2×#428—531no—
302#2504—531no—
303#2500—531no—
304#2093—531no—
305#2092—531no—
306#2092—531no—
307#2092—531no—
308#1972—531no—
309#1970—531no—
310#1847—531no—
311#1846—531no—
312#1753D5311yescharacteristic, complemented
313C2×C53—42no—
314C2×C53#250111yescharacteristic
315C2×C53#315311yescharacteristic
316C2×C53#315511yescharacteristic
317C2×C53#334712yes—
318C2×C53#335512yes—
319D53—161no—
320D53—161no—
321C22×D8—531no—
322C22×M16C2×D5311yescharacteristic
323C2×#121—531no—
324C2×#121—531no—
325C2×#117—531no—
326C22×C4⋊C4—531no—
327#482—1064no—
328#480—1064no—
329#469—1064no—
330#428—1064no—
331C2×#107—531no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#106—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×#102—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×C22⋊C8C2×D5311yescharacteristic
347C2×#97C2×D5311yescharacteristic
348C2×C4×C8C2×D5311yescharacteristic
349#397—1062no—
350#394—1062no—
351#390—531no—
352#390—531no—
353#390—531no—
354#390—531no—
355#388—1062no—
356#386—1062no—
357#384—531no—
358#382—531no—
359#382—531no—
360#382—531no—
361C4×C22⋊C4C2×D5311yescharacteristic
362#336C2×D5311yescharacteristic
363#336C2×D5311yescharacteristic
364C53#2794211yescharacteristic, complemented, Sylow
365C22×D4—531no—
366C22×D4—531no—
367C22×D4—1061no—
368C22×D4—1061no—
369C23×C4—1061no—
370C23×C4C22×D5311yescharacteristic
371C2×D8—1062no—
372C2×D8—1062no—
373C2×D8—1062no—
374C2×M16—22no—
375C2×M16—22no—
376C2×M16D21214yes—
377C22×C8C22×D5311yescharacteristic
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381#121—1062no—
382#121—1062no—
383#121—1062no—
384#121—1062no—
385#117—1062no—
386#117—1062no—
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—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1061no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4D21212yes—
405C2×C42—1061no—
406C2×C42C22×D5311yescharacteristic
407C2×C42C22×D5311yescharacteristic
408#107—1062no—
409#107—1062no—
410#107—1062no—
411#107—1062no—
412#107—1062no—
413#106—1062no—
414#106—1062no—
415#106—1062no—
416#106—1062no—
417#102—1062no—
418#102—1062no—
419#102—1062no—
420#102—1062no—
421#102—1062no—
422#102—1062no—
423C22⋊C8D21214yes—
424#97—22no—
425C4×C8—22no—
426#95—1062no—
427#95—1062no—
428#95—1062no—
429#95D21212yes—
430C24—1061no—
431C24—1061no—
432C24C2×D21211yescharacteristic
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—4242no—
446C2×D4—2124no—
447C2×D4—2122no—
448C2×D4—2122no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—4242no—
460C22×C4—2122no—
461C22×C4C2×D21212yes—
462C22×C4C2×D21212yes—
463C22×C4C2×D21211yescharacteristic
464C22×C4C2×D21212yes—
465C22×C4C2×D21212yes—
466C22×C4C23×D5311yescharacteristic
467D8—2124no—
468M16—28no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8C2×D21212yes—
476C2×C8C2×D21212yes—
477C2×C8C2×D21212yes—
478C2×C8C2×D21212yes—
479C2×C8#1171994712yes—
480C2×C8#1171994712yes—
481C4⋊C4—1061no—
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—4242no—
494C4⋊C4—2122no—
495C22⋊C4—42no—
496C22⋊C4—4242no—
497C22⋊C4—4242no—
498C22⋊C4—4242no—
499C22⋊C4—2124no—
500C22⋊C4—2122no—
501C22⋊C4—2122no—
502C22⋊C4—2122no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—2122no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—2121no—
513C23—2121no—
514C23—2121no—
515C23—2121no—
516C23—2121no—
517C23—2121no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23#1313321412yes—
523C23C22×D21211yescharacteristic
524D4—4244no—
525D4—4244no—
526D4—2121no—
527D4—2121no—
528D4—2121no—
529D4—2121no—
530D4—2121no—
531D4—2121no—
532D4—2121no—
533D4—2121no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—4244no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4#1313321412yes—
560C2×C4C22×D21211yescharacteristic
561C2×C4C22×D21211yescharacteristic
562C2×C4C2×#1171994711yescharacteristic
563C2×C4C2×#1171994711yescharacteristic
564C2×C4#1313326411yescharacteristic
565C2×C4#1313326411yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8#1313320114yes—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—4241no—
576C22—4241no—
577C22—4242no—
578C22—4242no—
579C22—4241no—
580C22—4241no—
581C22—4242no—
582C22—4242no—
583C22—4241no—
584C22—4241no—
585C22C2×#1313321411yescharacteristic
586C22#1681127711yescharacteristic
587C22#1681127711yescharacteristic
588C22#1681142312yes—
589C22#1681142812yes—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—4241no—
594C4—4241no—
595C4—4241no—
596C4—4241no—
597C4—4241no—
598C4—4241no—
599C4C2×#1313320111yescharacteristic
600C4#1681119911yescharacteristic
601C4#1681127711yescharacteristic
602C4#1681127711yescharacteristic
603C2—42no—
604C2—8481no—
605C2—8481no—
606C2#2027918611yescharacteristic
607C2#2028518111yescharacteristic
608C2#2028520111yescharacteristic
609C2#2028721712yes—
610C2#2028732012yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1753C2
socleC23×C53C22×D4
prime radicalC2×#16810876C2
hypercenter#1753D53
hypocenterC53#27942
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#16810948C2semidirect
2. C2×#16810937C2semidirect
4. C2×#16810665C2semidirect
5. #20279207C2semidirect
6. #20279177C2semidirect
7. #20276242C2semidirect
8. #20276235C2semidirect
9. #20276235C2semidirect
10. #20276235C2semidirect
11. #20275591C2semidirect
12. #20275581C2semidirect
13. #20274883C2semidirect
14. #20274879C2semidirect
15. C53×#1753C2semidirect
312. #1753D53semidirect
364. C53#27942semidirect

Maximal subgroups

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

subgroupindex
C2×#168109482
C2×#168109372
C2×#168108762
C2×#168106652
#202792072
#202791772
#202762422
#202762352
#202755912
#202755812
#202748832
#202748792
C53×#17532
#2794253