Finite Group Databasebeta

← #27469477

Subgroups — #27469477

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×#16810944C211yesmaximal, characteristic, complemented
2C2×#16810934C211yesmaximal, characteristic, complemented
3C2×#16810665C211yesmaximal, characteristic, complemented
4#20279208C211yesmaximal, characteristic, complemented
5#20279178C211yesmaximal, characteristic, complemented
6#20276241C211yesmaximal, characteristic, complemented
7#20276238C211yesmaximal, characteristic, complemented
8#20276238C211yesmaximal, characteristic, complemented
9#20276234C211yesmaximal, characteristic, complemented
10#20275592C211yesmaximal, characteristic, complemented
11#20275582C211yesmaximal, characteristic, complemented
12#20274880C211yesmaximal, characteristic, complemented
13#20274876C211yesmaximal, characteristic, complemented
14#20274496C211yesmaximal, characteristic, complemented
15C2×C53×#469C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18#16810944—24no—
19#16810934—24no—
20#16810665—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133127C2211yescharacteristic
35C2×C8×Dic53C2211yescharacteristic
36#16810548—22no—
37#16810537—22no—
38#16810526C2211yescharacteristic
39#16810526C2211yescharacteristic
40#16810526C2211yescharacteristic
41#16810526C2211yescharacteristic
42#16810511—22no—
43#16810503—22no—
44#16810492C2211yescharacteristic
45#16810488C2211yescharacteristic
46#16810488C2211yescharacteristic
47#16810488C2211yescharacteristic
48C4×C22⋊Dic53C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51C22×D8×C53C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C2×#121×C53C2211yescharacteristic
54C2×#121×C53C2211yescharacteristic
55C53×#469—24no—
56C2×#106×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133176—22no—
63#13133176—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—22no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—22no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—22no—
74C2×C22⋊Dic53—22no—
75C2×C22⋊Dic53—22no—
76C2×C22⋊Dic53—22no—
77C2×C22⋊Dic53D412yes—
78C2×C4×Dic53—21no—
79C2×C4×Dic53C2311yescharacteristic
80C2×C4×Dic53C2311yescharacteristic
81#13133151—22no—
82#13133151—22no—
83#13133150—22no—
84#13133150—22no—
85#13133150—22no—
86#13133149—22no—
87#13133149—22no—
88#13133148—22no—
89#13133138—22no—
90#13133138—22no—
91#13133138—22no—
92#13133138—22no—
93C22⋊#88322D414yes—
94#13133127—22no—
95C8×Dic53—22no—
96#13133124—22no—
97#13133124—22no—
98#13133124—22no—
99#13133124D412yes—
100C22×D4×C53—21no—
101C22×D4×C53—21no—
102C22×D4×C53C2311yescharacteristic
103C22×D4×C53C2311yescharacteristic
104C23×C4×C53C2311yescharacteristic
105C2×D8×C53—22no—
106C2×D8×C53—22no—
107C2×D8×C53—22no—
108C2×M16×C53—22no—
109C2×M16×C53—22no—
110C2×M16×C53D414yes—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113#121×C53—22no—
114#121×C53—22no—
115#121×C53—22no—
116#121×C53—22no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121#106×C53—22no—
122#102×C53—22no—
123#102×C53—22no—
124C24×C53—21no—
125C24×C53—21no—
126C24×C53C2×D411yescharacteristic
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—44no—
140C2×D4×C53—42no—
141C2×D4×C53—42no—
142C2×D4×C53—82no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—42no—
158C22×Dic53—82no—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D412yes—
161D8×C53—44no—
162M16×C53—28no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53C2×D412yes—
166C2×C8×C53C2×D412yes—
167C2×C8×C53#4112yes—
168C2×C8×C53#4112yes—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—42no—
175C22⋊Dic53—42no—
176C22⋊Dic53—44no—
177C22⋊Dic53—42no—
178C22⋊Dic53—82no—
179C22⋊Dic53—82no—
180C22⋊Dic53—82no—
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—42no—
190#11719919—82no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—42no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C2×D412yes—
201C2×#88322C2×D412yes—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53—42no—
216C23×C53#12712yes—
217C23×C53C22×D411yescharacteristic
218D4×C53—41no—
219D4×C53—41no—
220D4×C53—41no—
221D4×C53—41no—
222D4×C53—41no—
223D4×C53—41no—
224D4×C53—41no—
225D4×C53—41no—
226D4×C53—84no—
227D4×C53—84no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53#12712yes—
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×C4×C532−1+411yescharacteristic
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—84no—
260C8×C53—22no—
261C8×C53#12314yes—
262#88322—22no—
263#88322—22no—
264#27942—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—82no—
271C22×C53—82no—
272C22×C53—82no—
273C22×C53—82no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53C2×#12711yescharacteristic
281C22×C53#54711yescharacteristic
282C22×C53#54711yescharacteristic
283C22×C53#57512yes—
284C22×C53#57612yes—
285C4×C53—22no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53C2×#12311yescharacteristic
289C4×C53#53611yescharacteristic
290C4×C53#54711yescharacteristic
291C4×C53#54711yescharacteristic
292Dic53—42no—
293Dic53—42no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298C2×#482—531no—
299C2×#480—531no—
300C2×#469D5311yescharacteristic
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#1753—531no—
313C2×C53—42no—
314C2×C53—161no—
315C2×C53—161no—
316C2×C53#250111yescharacteristic
317C2×C53#315311yescharacteristic
318C2×C53#315511yescharacteristic
319C2×C53#334712yes—
320C2×C53#335512yes—
321C22×D8C2×D5311yescharacteristic
322C22×M16C2×D5311yescharacteristic
323C2×#121C2×D5311yescharacteristic
324C2×#121C2×D5311yescharacteristic
325C2×#117—531no—
326C22×C4⋊C4—531no—
327#482—1064no—
328#480—1064no—
329#469—24no—
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×#106C2×D5311yescharacteristic
340C2×#102—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×#102C2×D5311yescharacteristic
345C2×#102C2×D5311yescharacteristic
346C2×C22⋊C8—531no—
347C2×#97—531no—
348C2×C4×C8—531no—
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⋊C4—531no—
362#336—531no—
363#336—531no—
364C53#2794211yescharacteristic, complemented, Sylow
365C22×D4—21no—
366C22×D4—21no—
367C22×D4C22×D5311yescharacteristic
368C22×D4C22×D5311yescharacteristic
369C23×C4—1061no—
370C23×C4C22×D5311yescharacteristic
371C2×D8—22no—
372C2×D8—22no—
373C2×D8—22no—
374C2×M16—22no—
375C2×M16—22no—
376C2×M16C22⋊D5314yes—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381#121—22no—
382#121—22no—
383#121—22no—
384#121—22no—
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—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—532no—
400C2×C22⋊C4—1062no—
401C2×C22⋊C4—1062no—
402C2×C22⋊C4—1062no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—1061no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—1061no—
408#107—1062no—
409#107—1062no—
410#107—1062no—
411#107—1062no—
412#107—1062no—
413#106—22no—
414#106—1062no—
415#106—1062no—
416#106—1062no—
417#102—22no—
418#102—22no—
419#102—1062no—
420#102—1062no—
421#102—1062no—
422#102—1062no—
423C22⋊C8—534no—
424#97—1062no—
425C4×C8—1062no—
426#95—532no—
427#95—1062no—
428#95—1062no—
429#95—1062no—
430C24—21no—
431C24—21no—
432C24D4×D5311yescharacteristic
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—82no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—4242no—
462C22×C4—2122no—
463C22×C4D4×D5311yescharacteristic
464C22×C4D4×D5312yes—
465C22×C4C2×C22⋊D5312yes—
466C22×C4C23×D5311yescharacteristic
467D8—44no—
468M16—28no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—532no—
472C2×C8—532no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8#1171994712yes—
478C2×C8#1171994712yes—
479C2×C8C2×C22⋊D5312yes—
480C2×C8C2×C22⋊D5312yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
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—42no—
497C22⋊C4—4242no—
498C22⋊C4—4242no—
499C22⋊C4—4242no—
500C22⋊C4—2122no—
501C22⋊C4—2124no—
502C22⋊C4—2122no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C42—1061no—
507C42—2122no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23#1313321312yes—
523C23C2×D4×D5311yescharacteristic
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—84no—
533D4—84no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—4244no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4#1313321312yes—
560C2×C4C2×D4×D5311yescharacteristic
561C2×C4C22×C22⋊D5311yescharacteristic
562C2×C4C2×#1171994711yescharacteristic
563C2×C4C2×#1171994711yescharacteristic
564C2×C4#1313326411yescharacteristic
565C2×C4#1313326411yescharacteristic
566C8—22no—
567C8—1062no—
568C8—1062no—
569C8#1313319814yes—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—82no—
578C22—82no—
579C22—81no—
580C22—81no—
581C22—82no—
582C22—82no—
583C22—81no—
584C22—81no—
585C22C2×#1313321311yescharacteristic
586C22#1681127311yescharacteristic
587C22#1681127311yescharacteristic
588C22#1681141912yes—
589C22#1681142712yes—
590C4—22no—
591C4—81no—
592C4—81no—
593C4—4241no—
594C4—4241no—
595C4—4241no—
596C4—4241no—
597C4—2122no—
598C4—2122no—
599C4C2×#1313319811yescharacteristic
600C4#1681119411yescharacteristic
601C4#1681127311yescharacteristic
602C4#1681127311yescharacteristic
603C2—42no—
604C2—161no—
605C2—161no—
606C2#2027918711yescharacteristic
607C2#2028517711yescharacteristic
608C2#2028520011yescharacteristic
609C2#2028721012yes—
610C2#2028731612yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#469C2
socleC23×C53C22×D4
hypercenterC2×#469D53
hypocenterC53#27942
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#469

Complements

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

NH ≅ G/Ntype
1. C2×#16810944C2semidirect
2. C2×#16810934C2semidirect
3. C2×#16810665C2semidirect
4. #20279208C2semidirect
5. #20279178C2semidirect
6. #20276241C2semidirect
7. #20276238C2semidirect
8. #20276238C2semidirect
9. #20276234C2semidirect
10. #20275592C2semidirect
11. #20275582C2semidirect
12. #20274880C2semidirect
13. #20274876C2semidirect
14. #20274496C2semidirect
364. C53#27942semidirect

Maximal subgroups

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

subgroupindex
C2×#168109442
C2×#168109342
C2×#168106652
#202792082
#202791782
#202762412
#202762382
#202762342
#202755922
#202755822
#202748802
#202748762
#202744962
C2×C53×#4692
#2794253