Finite Group Databasebeta

← #27469537

Subgroups — #27469537

Order 13568.

888 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic
2C2×#16810961C211yesmaximal, characteristic, complemented
3C2×#16810665C211yesmaximal, characteristic, complemented
4#20279201C211yesmaximal, characteristic, complemented
5#20279178C211yesmaximal, characteristic
6#20276261C211yesmaximal, characteristic, complemented
7#20276256C211yesmaximal, characteristic, complemented
8#20276238C211yesmaximal, characteristic, complemented
9#20276234C211yesmaximal, characteristic, complemented
10#20275598C211yesmaximal, characteristic, complemented
11#20275571C211yesmaximal, characteristic, complemented
12#20274882C211yesmaximal, characteristic, complemented
13#20274878C211yesmaximal, characteristic, complemented
14#20274496C211yesmaximal, characteristic, complemented
15C2×C53×#468C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18#16810961—24no—
19#16810939—24no—
20#16810665—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133127C2211yescharacteristic
35C2×C8×Dic53C2211yescharacteristic
36#16810562—22no—
37#16810537—22no—
38#16810530C2211yescharacteristic
39#16810530C2211yescharacteristic
40#16810526C2211yescharacteristic
41#16810526C2211yescharacteristic
42#16810511—22no—
43#16810492C2211yescharacteristic
44#16810488C2211yescharacteristic
45#16810488C2211yescharacteristic
46#16810488C2211yescharacteristic
47#16810484—22no—
48C4×C22⋊Dic53C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51C22×SD16×C53C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C2×C22⋊Q8×C53C2211yescharacteristic
54C2×#121×C53C2211yescharacteristic
55C53×#468—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#103×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⋊Dic53D412yes—
76C2×C4×Dic53—21no—
77C2×C4×Dic53C2311yescharacteristic
78C2×C4×Dic53C2311yescharacteristic
79#13133151—22no—
80#13133151—22no—
81#13133150—22no—
82#13133149—22no—
83#13133149—22no—
84#13133148—22no—
85#13133148—22no—
86#13133148—22no—
87#13133141—22no—
88#13133141—22no—
89#13133138—22no—
90#13133138—22no—
91C22⋊#88322D414yes—
92#13133127—22no—
93C8×Dic53—22no—
94#13133124—22no—
95#13133124—22no—
96#13133124—22no—
97#13133124—22no—
98#13133124—22no—
99#13133124D412yes—
100C22×Q8×C53C2311yescharacteristic
101C22×D4×C53—21no—
102C22×D4×C53C2311yescharacteristic
103C23×C4×C53C2311yescharacteristic
104C2×SD16×C53—22no—
105C2×SD16×C53—22no—
106C2×SD16×C53—22no—
107C2×M16×C53—22no—
108C2×M16×C53—22no—
109C2×M16×C53D414yes—
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22⋊Q8×C53—22no—
113C22⋊Q8×C53—22no—
114#121×C53—22no—
115#121×C53—22no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121#107×C53—22no—
122#103×C53—22no—
123#102×C53—22no—
124C24×C53—21no—
125C24×C53C2×D411yescharacteristic
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×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—42no—
139C2×D4×C53—82no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D412yes—
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—44no—
156C22×Dic53—42no—
157C22×Dic53C2×D412yes—
158C22×Dic53C2×D412yes—
159SD16×C53—44no—
160M16×C53—28no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C2×D412yes—
164C2×C8×C53C2×D412yes—
165C2×C8×C53#4112yes—
166C2×C8×C53#4112yes—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—42no—
172C22⋊C4×C53—42no—
173C22⋊C4×C53—42no—
174C22⋊Dic53—42no—
175C22⋊Dic53—44no—
176C22⋊Dic53—42no—
177C22⋊Dic53—82no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—42no—
187#11719919—82no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—42no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322C2×D412yes—
198C2×#88322C2×D412yes—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53#12712yes—
209C23×C53C22×D411yescharacteristic
210Q8×C53—41no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53—41no—
214D4×C53—41no—
215D4×C53—41no—
216D4×C53—41no—
217D4×C53—41no—
218D4×C53—84no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53#12712yes—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C532−1+411yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—84no—
254C8×C53—22no—
255C8×C53#12314yes—
256#88322—22no—
257#88322—22no—
258#27946—531nomaximal, Sylow
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—82no—
265C22×C53—82no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53C2×#12711yescharacteristic
270C22×C53#54711yescharacteristic
271C22×C53#54711yescharacteristic
272C22×C53#57512yes—
273C22×C53#57712yes—
274C4×C53—22no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53C2×#12311yescharacteristic
279C4×C53#53611yescharacteristic
280C4×C53#54711yescharacteristic
281C4×C53#54711yescharacteristic
282Dic53—42no—
283Dic53—42no—
284Dic53—81no—
285Dic53—81no—
286Dic53—81no—
287Dic53—81no—
288C2×#481—531no—
289C2×#485—531no—
290C2×#468D5311yescharacteristic
291C2×#428—531no—
292#2503—531no—
293#2500—531no—
294#2097—531no—
295#2096—531no—
296#2092—531no—
297#2092—531no—
298#1973—531no—
299#1968—531no—
300#1847—531no—
301#1846—531no—
302#1753—531no—
303C2×C53—42no—
304C2×C53—161no—
305C2×C53#250111yescharacteristic
306C2×C53#315311yescharacteristic
307C2×C53#315411yescharacteristic
308C2×C53#334812yes—
309C2×C53#335112yes—
310C22×SD16C2×D5311yescharacteristic
311C22×M16C2×D5311yescharacteristic
312C2×C22⋊Q8C2×D5311yescharacteristic
313C2×#121C2×D5311yescharacteristic
314C2×#117—531no—
315C22×C4⋊C4—531no—
316#485—1064no—
317#481—1064no—
318#468—24no—
319#428—1064no—
320C2×#107—531no—
321C2×#107—531no—
322C2×#107—531no—
323C2×#107C2×D5311yescharacteristic
324C2×#106—531no—
325C2×#106—531no—
326C2×#106—531no—
327C2×#106—531no—
328C2×#106—531no—
329C2×#103—531no—
330C2×#103—531no—
331C2×#103C2×D5311yescharacteristic
332C2×#102—531no—
333C2×#102—531no—
334C2×#102C2×D5311yescharacteristic
335C2×C22⋊C8—531no—
336C2×#97—531no—
337C2×C4×C8—531no—
338#400—1062no—
339#394—1062no—
340#391—531no—
341#391—531no—
342#390—531no—
343#390—531no—
344#388—1062no—
345#384—531no—
346#382—531no—
347#382—531no—
348#382—531no—
349#381—1062no—
350C4×C22⋊C4—531no—
351#336—531no—
352#336—531no—
353C53#2794611yescharacteristic, complemented, Sylow
354C22×Q8C22×D5311yescharacteristic
355C22×D4—21no—
356C22×D4C22×D5311yescharacteristic
357C23×C4—1061no—
358C23×C4C22×D5311yescharacteristic
359C2×SD16—22no—
360C2×SD16—22no—
361C2×SD16—22no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16C22⋊D5314yes—
365C22×C8—531no—
366C22×C8—531no—
367C22×C8C22×D5311yescharacteristic
368C22×C8C22×D5311yescharacteristic
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371#121—22no—
372#121—22no—
373#117—1062no—
374#117—1062no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—532no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—1062no—
391C2×C22⋊C4—1061no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—1061no—
395#107—22no—
396#107—1062no—
397#107—1062no—
398#107—1062no—
399#106—1062no—
400#106—1062no—
401#106—1062no—
402#106—1062no—
403#106—1062no—
404#103—22no—
405#103—1062no—
406#103—1062no—
407#102—22no—
408#102—1062no—
409#102—1062no—
410C22⋊C8—534no—
411#97—1062no—
412C4×C8—1062no—
413#95—532no—
414#95—1062no—
415#95—1062no—
416#95—1062no—
417#95—1062no—
418#95—1062no—
419C24—21no—
420C24D4×D5311yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—42no—
434C2×D4—82no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—532no—
443C22×C4—532no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—2124no—
449C22×C4—2122no—
450C22×C4D4×D5311yescharacteristic
451C22×C4D4×D5312yes—
452C22×C4C2×C22⋊D5312yes—
453C22×C4C23×D5311yescharacteristic
454SD16—44no—
455M16—28no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—532no—
459C2×C8—532no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8#1171994712yes—
465C2×C8#1171994712yes—
466C2×C8C2×C22⋊D5312yes—
467C2×C8C2×C22⋊D5312yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—4242no—
482C4⋊C4—2122no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—4242no—
486C22⋊C4—2122no—
487C22⋊C4—2124no—
488C22⋊C4—2122no—
489C42—1061no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42—2122no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23#1313321312yes—
504C23C2×D4×D5311yescharacteristic
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—84no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—4244no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4#1313321312yes—
543C2×C4C2×D4×D5311yescharacteristic
544C2×C4C22×C22⋊D5311yescharacteristic
545C2×C4C2×#1171994711yescharacteristic
546C2×C4C2×#1171994711yescharacteristic
547C2×C4#1313326411yescharacteristic
548C2×C4#1313326411yescharacteristic
549C8—22no—
550C8—1062no—
551C8—1062no—
552C8#1313319814yes—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—42no—
557C22—42no—
558C22—81no—
559C22—82no—
560C22—81no—
561C22—82no—
562C22—81no—
563C22C2×#1313321311yescharacteristic
564C22#1681127311yescharacteristic
565C22#1681127311yescharacteristic
566C22#1681142212yes—
567C22#1681143312yes—
568C4—22no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—4241no—
573C4—4241no—
574C4—4241no—
575C4—4241no—
576C4—2122no—
577C4—2122no—
578C4C2×#1313319811yescharacteristic
579C4#1681119411yescharacteristic
580C4#1681127311yescharacteristic
581C4#1681127311yescharacteristic
582C2—42no—
583C2—161no—
584C2#2027918711yescharacteristic
585C2#2028518211yescharacteristic
586C2#2028518911yescharacteristic
587C2#2028722512yes—
588C2#2028728112yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#468C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#468D53
hypocenterC53#27946
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#468

Complements

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

NH ≅ G/Ntype
2. C2×#16810961C2semidirect
3. C2×#16810665C2semidirect
4. #20279201C2semidirect
6. #20276261C2semidirect
7. #20276256C2semidirect
8. #20276238C2semidirect
9. #20276234C2semidirect
10. #20275598C2semidirect
11. #20275571C2semidirect
12. #20274882C2semidirect
13. #20274878C2semidirect
14. #20274496C2semidirect
353. C53#27946semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168109612
C2×#168106652
#202792012
#202791782
#202762612
#202762562
#202762382
#202762342
#202755982
#202755712
#202748822
#202748782
#202744962
C2×C53×#4682
#2794653