Finite Group Databasebeta

← #27276164

Subgroups — #27276164

Order 13568.

1116 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275522C211yesmaximal, characteristic, complemented
2#20274720C211yesmaximal, characteristic, complemented
3#20274715C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6C2×D4⋊#88322C211yesmaximal, characteristic, complemented
7C2×C53×#550C211yesmaximal, characteristic
8C2×C4⋊#88322C2211yescharacteristic
9C2×C4⋊#88322C2211yescharacteristic
10C2×#13133141—21no—
11C2×#13133138—21no—
12C2×#13133138—22no—
13C2×C4×#88322C2211yescharacteristic
14#16810493C2211yescharacteristic
15#16810385—21no—
16#16810385—21no—
17#16810367—22no—
18Q8⋊#88322—22no—
19D4⋊#88322—22no—
20C22×#41×C53—21no—
21C53×#550—22no—
22C53×#550—22no—
23C53×#550—22no—
24C53×#550C414yescomplemented
25C2×#127×C53—21no—
26C2×#127×C53C2211yescharacteristic
27C2×#121×C53—22no—
28C2×#121×C53—21no—
29C2×C4×Q8×C53C2211yescharacteristic
30C2×C4×D4×C53—21no—
31C2×C4×D4×C53C2211yescharacteristic
32C22×#88322—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42#13133141—42no—
43#13133138—42no—
44#13133138—44no—
45C4×#88322—21no—
46C4×#88322—21no—
47C2×#41×C53—24no—
48C2×#41×C53—24no—
49C2×#41×C53—42no—
50C2×#41×C53—42no—
51C2×#41×C53—42no—
52C2×#41×C53—42no—
53C22×Q8×C53D411yescharacteristic
54C22×D4×C53—22no—
55C22×D4×C53—22no—
56C22×D4×C53—21no—
57C22×D4×C53—21no—
58C22×D4×C53D411yescharacteristic
59C22×D4×C53D412yes—
60C23×C4×C53—22no—
61C23×C4×C53—21no—
62#127×C53—24no—
63#127×C53—21no—
64#127×C53—21no—
65#127×C53—24no—
66#127×C53C2×C412yes—
67#127×C53D412yes—
68#121×C53—28no—
69#121×C53—24no—
70#121×C53—24no—
71#121×C53—28no—
72C4×Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53C2×C412yes—
75C4×Q8×C53D412yes—
76C4×D4×C53—24no—
77C4×D4×C53—24no—
78C4×D4×C53—21no—
79C4×D4×C53—21no—
80C4×D4×C53C2×C412yes—
81C4×D4×C53D412yes—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53D411yescharacteristic
84C2×C4⋊C4×C53D411yescharacteristic
85C2×C22⋊C4×C53—22no—
86C2×C22⋊C4×C53—21no—
87C2×C42×C53—21no—
88C2×C42×C53C2311yescharacteristic
89C24×C53—21no—
90C24×C53—22no—
91#41×C53—48no—
92#41×C53—48no—
93#41×C53—48no—
94#41×C53—48no—
95C2×Q8×C53—21no—
96C2×Q8×C53—21no—
97C2×Q8×C53—21no—
98C2×Q8×C53C22⋊C412yes—
99C2×Q8×C53D812yes—
100C2×Q8×C53SD1612yes—
101C2×D4×C53—28no—
102C2×D4×C53—28no—
103C2×D4×C53—22no—
104C2×D4×C53—22no—
105C2×D4×C53—21no—
106C2×D4×C53—22no—
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—44no—
114C2×D4×C53—44no—
115C2×D4×C53—42no—
116C2×D4×C53—42no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53—82no—
126C2×D4×C53—82no—
127C2×D4×C53C22⋊C412yes—
128C2×D4×C53C22⋊C414yes—
129C22×C4×C53—21no—
130C22×C4×C53—28no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—44no—
136C22×C4×C53—44no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—41no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×Dic53—41no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53C22⋊C412yes—
150C4⋊C4×C53C22⋊C412yes—
151C22⋊C4×C53—28no—
152C22⋊C4×C53—22no—
153C22⋊C4×C53—22no—
154C42×C53—22no—
155C42×C53—22no—
156C42×C53C22×C411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C4⋊Dic53—42no—
161C4⋊Dic53—41no—
162C4⋊Dic53—41no—
163C4⋊Dic53—41no—
164C4⋊Dic53—41no—
165C2×#88322—22no—
166C2×#88322—41no—
167C2×#88322—41no—
168C2×#88322—41no—
169C2×#88322—41no—
170C2×#88322—41no—
171C2×#88322—41no—
172C2×#88322—41no—
173C2×#88322—41no—
174C23×C53—24no—
175C23×C53—22no—
176C23×C53—42no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—41no—
182C23×C53—41no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—41no—
186C23×C53—42no—
187C23×C53—41no—
188C23×C53#12011yescharacteristic
189Q8×C53—22no—
190Q8×C53—22no—
191Q8×C53—22no—
192Q8×C53#10214yescomplemented
193D4×C53—22no—
194D4×C53—22no—
195D4×C53—44no—
196D4×C53—416no—
197D4×C53—42no—
198D4×C53—42no—
199D4×C53—42no—
200D4×C53—42no—
201D4×C53—42no—
202D4×C53—41no—
203D4×C53—42no—
204D4×C53—42no—
205D4×C53—41no—
206D4×C53—42no—
207D4×C53—82no—
208D4×C53—88no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—416no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×C4×C53#12011yescharacteristic
229C2×C4×C53#12011yescharacteristic
230C2×C4×C53#12011yescharacteristic
231C2×C4×C53C2×D811yescharacteristic
232C2×C4×C53C2×SD1611yescharacteristic
233C2×C4×C53#13511yescharacteristic
234C2×C4×C53#13511yescharacteristic
235C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
236C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
237C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
238C2×C4×C53#13711yescharacteristic
239C2×Dic53—81no—
240C2×Dic53—81no—
241C2×Dic53—81no—
242#88322—42no—
243#88322—81no—
244#88322—81no—
245#8444—531nomaximal, Sylow
246C22×C53—44no—
247C22×C53—42no—
248C22×C53—44no—
249C22×C53—42no—
250C22×C53—42no—
251C22×C53—44no—
252C22×C53—81no—
253C22×C53—82no—
254C22×C53—81no—
255C22×C53—82no—
256C22×C53—81no—
257C22×C53—82no—
258C22×C53—81no—
259C22×C53—82no—
260C22×C53#37911yescharacteristic
261C22×C53#44711yescharacteristic
262C22×C53#44911yescharacteristic
263C22×C53#45011yescharacteristic
264C22×C53#45211yescharacteristic
265C22×C53#45311yescharacteristic
266C22×C53#45511yescharacteristic
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—24no—
270C4×C53—42no—
271C4×C53—42no—
272C4×C53C2×#10211yescharacteristic
273C4×C53#41611yescharacteristic
274C4×C53#41711yescharacteristic
275C4×C53#41811yescharacteristic
276C4×C53#44711yescharacteristic
277C4×C53#44911yescharacteristic
278C4×C53#45011yescharacteristic
279C4×C53#45211yescharacteristic
280Dic53—161no—
281C2×#550D5311yescharacteristic
282#1957—531no—
283#1803—531no—
284#1802—531no—
285C2×#331—531no—
286C2×Q8⋊C8—531no—
287C2×D4⋊C8—531no—
288C2×C53—84no—
289C2×C53—82no—
290C2×C53#162111yescharacteristic
291C2×C53#163111yescharacteristic
292C2×C53#164111yescharacteristic
293C2×C53#164211yescharacteristic
294C2×C53#178611yescharacteristic
295C2×C53#178811yescharacteristic
296C2×C53#179211yescharacteristic
297C22×#41—21no—
298#550—22no—
299#550—22no—
300#550—22no—
301#550Dic5314yescomplemented
302C2×#127—21no—
303C2×#127C2×D5311yescharacteristic
304C2×#121—22no—
305C2×#121—21no—
306C2×C4×Q8C2×D5311yescharacteristic
307C2×C4×D4—21no—
308C2×C4×D4C2×D5311yescharacteristic
309C2×C4⋊C8—531no—
310C2×C4⋊C8—531no—
311C2×#103—1061no—
312C2×#102—1061no—
313C2×#102—1062no—
314C2×C4×C8—531no—
315#384—531no—
316#340—1061no—
317#340—1061no—
318#331—1062no—
319Q8⋊C8—1062no—
320D4⋊C8—1062no—
321C53#844411yescharacteristic, complemented, Sylow
322C2×#41—24no—
323C2×#41—24no—
324C2×#41—42no—
325C2×#41—42no—
326C2×#41—42no—
327C2×#41—42no—
328C22×Q8C22⋊D5311yescharacteristic
329C22×D4—22no—
330C22×D4—22no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4C22⋊D5311yescharacteristic
334C22×D4C22⋊D5312yes—
335C23×C4—22no—
336C23×C4—21no—
337C22×C8—1061no—
338C22×C8—1061no—
339C22×C8—1061no—
340#127—24no—
341#127—24no—
342#127—21no—
343#127—21no—
344#127C2×Dic5312yes—
345#127C22⋊D5312yes—
346#121—24no—
347#121—28no—
348#121—28no—
349#121—24no—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8C2×Dic5312yes—
353C4×Q8C22⋊D5312yes—
354C4×D4—24no—
355C4×D4—24no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4C2×Dic5312yes—
359C4×D4C22⋊D5312yes—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4C22⋊D5311yescharacteristic
365C2×C4⋊C4C22⋊D5311yescharacteristic
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C42—21no—
369C2×C42C22×D5311yescharacteristic
370C4⋊C8—1061no—
371C4⋊C8—1061no—
372C4⋊C8—1061no—
373C4⋊C8—1061no—
374#103—2122no—
375#102—2124no—
376#102—2122no—
377C4×C8—1061no—
378C4×C8—1061no—
379C24—21no—
380C24—22no—
381#41—48no—
382#41—48no—
383#41—48no—
384#41—48no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8#1171992212yes—
389C2×Q8#1171992312yes—
390C2×Q8C22⋊Dic5312yes—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4C22⋊Dic5312yes—
418C2×D4C22⋊Dic5314yes—
419C22×C4—21no—
420C22×C4—28no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—2121no—
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C2×C8—1062no—
436C2×C8—2121no—
437C2×C8—2121no—
438C2×C8—2121no—
439C2×C8—2121no—
440C2×C8—2121no—
441C2×C8—2121no—
442C2×C8—2121no—
443C2×C8—2121no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—2122no—
449C4⋊C4—2121no—
450C4⋊C4—2121no—
451C4⋊C4—2121no—
452C4⋊C4—2121no—
453C4⋊C4C22⋊Dic5312yes—
454C4⋊C4C22⋊Dic5312yes—
455C22⋊C4—28no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C42—22no—
459C42—22no—
460C42C22×Dic5311yescharacteristic
461C42C2×C22⋊D5311yescharacteristic
462C42C2×C22⋊D5311yescharacteristic
463C42C2×C22⋊D5311yescharacteristic
464C23—22no—
465C23—24no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23#1313318611yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—22no—
482Q8#1313313814yescomplemented
483D4—22no—
484D4—22no—
485D4—44no—
486D4—416no—
487D4—42no—
488D4—41no—
489D4—42no—
490D4—42no—
491D4—41no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—82no—
498D4—88no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—416no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—4241no—
516C2×C4—4241no—
517C2×C4—4241no—
518C2×C4C2×C22⋊Dic5311yescharacteristic
519C2×C4C2×C22⋊Dic5311yescharacteristic
520C2×C4C2×C22⋊Dic5311yescharacteristic
521C2×C4#1313318611yescharacteristic
522C2×C4#1313318611yescharacteristic
523C2×C4#1313318611yescharacteristic
524C2×C4C2×#1171992211yescharacteristic
525C2×C4C2×#1171992311yescharacteristic
526C2×C4#1313323711yescharacteristic
527C2×C4#1313323711yescharacteristic
528C2×C4#1313324211yescharacteristic
529C2×C4#1313324311yescharacteristic
530C2×C4#1313324311yescharacteristic
531C2×C4#1313324711yescharacteristic
532C8—4241no—
533C8—4241no—
534C8—2122no—
535C22—44no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—42no—
541C22—81no—
542C22—82no—
543C22—81no—
544C22—82no—
545C22—81no—
546C22—82no—
547C22—81no—
548C22—82no—
549C22#1681047911yescharacteristic
550C22#1681075711yescharacteristic
551C22#1681076811yescharacteristic
552C22#1681077411yescharacteristic
553C22#1681078511yescharacteristic
554C22#1681079111yescharacteristic
555C22#1681080111yescharacteristic
556C4—24no—
557C4—21no—
558C4—21no—
559C4—42no—
560C4—42no—
561C4—8481no—
562C4C2×#1313313811yescharacteristic
563C4#1681061911yescharacteristic
564C4#1681062411yescharacteristic
565C4#1681062811yescharacteristic
566C4#1681075711yescharacteristic
567C4#1681076811yescharacteristic
568C4#1681077411yescharacteristic
569C4#1681078511yescharacteristic
570C2—82no—
571C2—84no—
572C2#2027378011yescharacteristic
573C2#2027384211yescharacteristic
574C2#2027390811yescharacteristic
575C2#2027391511yescharacteristic
576C2#2027464611yescharacteristic
577C2#2027465511yescharacteristic
578C2#2027466911yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#550C2
socleC23×C53#120
cosocleC2×C42×C53C23
hypercenterC2×#550D53
hypocenterC53#8444
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #20275522C2semidirect
2. #20274720C2semidirect
3. #20274715C2semidirect
4. C2×#16810367C2semidirect
5. C2×Q8⋊#88322C2semidirect
6. C2×D4⋊#88322C2semidirect
24. C53×#550C4semidirect
192. Q8×C53#102semidirect
301. #550Dic53semidirect
321. C53#8444semidirect
482. Q8#13133138semidirect

Maximal subgroups

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

subgroupindex
#202755222
#202747202
#202747152
C2×#168103672
C2×Q8⋊#883222
C2×D4⋊#883222
C2×C53×#5502
#844453