Finite Group Databasebeta

← #27361420

Subgroups — #27361420

Order 13568.

1210 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285630C211yesmaximal, characteristic, complemented
2C2×#16810971C211yesmaximal, characteristic, complemented
3#20284195C211yesmaximal, characteristic, complemented
4#20273818C214yesmaximal, complemented
5#20273804C214yesmaximal, complemented
6#20273387C211yesmaximal, characteristic, complemented
7#20273166C211yesmaximal, characteristic, complemented
8#20273097C211yesmaximal, characteristic, complemented
9C53×#3499C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C2×#13133214C2211yescharacteristic
12#16810999—21no—
13#16810971—22no—
14#16810971—21no—
15#16810971C2214yes—
16#16810823—21no—
17#16810823C2212yes—
18#16810818—21no—
19#16810818C2212yes—
20#16810818C2214yes—
21#16810756—22no—
22#16810676C2211yescharacteristic
23#16810647C2211yescharacteristic
24#16810627—21no—
25C2×#13133137—22no—
26C2×C4×#88322C2211yescharacteristic
27#16810369C2214yescomplemented
28Q8⋊#88322C2212yescomplemented
29D4⋊#88322C2214yescomplemented
30D4⋊#88322C2212yescomplemented
31C2×2−1+4×C53—21no—
32D4×Q8×C53—21no—
33D4×Q8×C53C2212yescomplemented
34C53×#547—22no—
35C53×#547—21no—
36C53×#547—22no—
37C53×#547C2212yescomplemented
38C53×#536—22no—
39C53×#536—21no—
40C53×#532—21no—
41C53×#532C2211yescharacteristic
42C2×C4⋊Q8×C53—21no—
43C2×C4⋊Q8×C53C2211yescharacteristic
44C4×#41×C53C2211yescharacteristic
45C22×D212—21no—
46C22×D212—21no—
47C22×D212—21no—
48#13133242—44no—
49C2×#11719924—22no—
50C2×#11719922—22no—
51C2×#11719922—24no—
52C2×#11719917—21no—
53C22×#88322—21no—
54#13133214—21no—
55#13133214—21no—
56#13133214—21no—
57#13133214C2312yes—
58C4⋊#88322—21no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61#13133137—24no—
62#13133137—22no—
63#13133137—22no—
64#13133137—28no—
65C22⋊#88322—21no—
66C4×#88322—21no—
67C4×#88322—21no—
68C4×#88322C2312yes—
692−1+4×C53—44no—
702−1+4×C53—44no—
71C2×#41×C53—22no—
72C2×#41×C53—24no—
73C2×#41×C53—22no—
74C2×#41×C53—21no—
75C2×#41×C53D411yescharacteristic
76C22×Q8×C53—22no—
77C22×Q8×C53—22no—
78C22×Q8×C53—21no—
79C4⋊Q8×C53—22no—
80C4⋊Q8×C53—22no—
81C4⋊Q8×C53—22no—
82C4⋊Q8×C53—21no—
83C4⋊Q8×C53—21no—
84C4⋊Q8×C53D411yescharacteristic
85C4⋊Q8×C53D412yes—
86C4⋊Q8×C53C2312yes—
87#127×C53D411yescharacteristic
88#124×C53—22no—
89#124×C53—21no—
90#123×C53—24no—
91#123×C53—24no—
92#123×C53—24no—
93C22⋊Q8×C53—24no—
94C22⋊Q8×C53—24no—
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97#121×C53—22no—
98#121×C53—22no—
99#121×C53—21no—
100#121×C53D412yes—
101C4×Q8×C53D411yescharacteristic
102C4×Q8×C53C2311yescharacteristic
103C4×D4×C53—22no—
104C4×D4×C53—21no—
105C4×D4×C53—22no—
106C4×D4×C53—21no—
107C4×D4×C53—21no—
108C4×D4×C53D411yescharacteristic
109C4×D4×C53C2311yescharacteristic
110C4×D4×C53C2312yes—
111#117×C53—21no—
112#117×C53D411yescharacteristic
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53D412yes—
117C2×C42×C53—21no—
118C2×C42×C53C2311yescharacteristic
119C23×D53—41no—
120#41×C53—22no—
121#41×C53—48no—
122#41×C53—416no—
123#41×C53—44no—
124#41×C53—44no—
125#41×C53—42no—
126#41×C53—42no—
127#41×C53D814yescomplemented
128C2×Q8×C53—24no—
129C2×Q8×C53—22no—
130C2×Q8×C53—21no—
131C2×Q8×C53—22no—
132C2×Q8×C53—48no—
133C2×Q8×C53—44no—
134C2×Q8×C53—44no—
135C2×Q8×C53—42no—
136C2×Q8×C53—42no—
137C2×Q8×C53C2×D411yescharacteristic
138C2×D4×C53—21no—
139C2×D4×C53—24no—
140C2×D4×C53—22no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53C2×D412yes—
144C2×D4×C53C2×D411yescharacteristic
145C22×C4×C53—21no—
146C22×C4×C53—24no—
147C22×C4×C53—24no—
148C22×C4×C53—22no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—22no—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C2×D212—22no—
156C2×D212—22no—
157C2×D212—22no—
158C2×D212—44no—
159C2×D212—42no—
160C2×D212—42no—
161C2×D212—41no—
162C2×D212—41no—
163C2×D212—41no—
164C2×D212—82no—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C2×D412yes—
174C4⋊C4×C53C2×D414yes—
175C4⋊C4×C53C2×D411yescharacteristic
176C4⋊C4×C53C2×D411yescharacteristic
177C22⋊C4×C53—24no—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—21no—
183C22⋊C4×C53—22no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53C2×D411yescharacteristic
190C42×C53C2411yescharacteristic
191#11719924—44no—
192#11719922—44no—
193#11719922—84no—
194#11719917—42no—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—42no—
199C23×C53—21no—
200C23×C53—22no—
201C23×C53#12011yescharacteristic
202C22×D53—42no—
203C22×D53—82no—
204C22×D53—82no—
205C22×D53—82no—
206Q8×C53—21no—
207Q8×C53—48no—
208Q8×C53—42no—
209Q8×C53—42no—
210Q8×C53—42no—
211Q8×C53—44no—
212Q8×C53C2×D812yes—
213D4×C53—24no—
214D4×C53—21no—
215D4×C53—24no—
216D4×C53—48no—
217D4×C53—44no—
218D4×C53—42no—
219D4×C53—42no—
220D4×C53C2×D812yes—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—24no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—48no—
230C2×C4×C53—48no—
231C2×C4×C53—44no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53#12011yescharacteristic
237C2×C4×C53#12011yescharacteristic
238C2×C4×C53#12011yescharacteristic
239C2×C4×C53C2×D812yes—
240C2×C4×C53AGL(1,ℤ/8)12yes—
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×D411yescharacteristic
244D212—44no—
245D212—42no—
246D212—42no—
247D212—44no—
248D212—82no—
249D212—82no—
250D212—82no—
251#88322—42no—
252#88322—42no—
253#88322—42no—
254#18640—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—22no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53#44712yes—
264C22×C53C2×#12011yescharacteristic
265C2×D53—82no—
266C2×D53—82no—
267C2×D53—82no—
268C2×D53—164no—
269C4×C53—22no—
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—44no—
273C4×C53—42no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53#44712yes—
277C4×C53#45412yes—
278C4×C53C22×D811yescharacteristic
279C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
280C4×C53#57712yes—
281#3499D5311yescharacteristic, complemented
282#3205—531no—
283C2×#486—531no—
284#3038—531no—
285#1627—534no—
286#1625—534no—
287#1546—531no—
288#1492—531no—
289#1476—531no—
290C2×C53—22no—
291C2×C53—42no—
292C2×C53—82no—
293C2×C53C2×#44711yescharacteristic
294C2×C53#301011yescharacteristic
295C2×C53C2×#45411yescharacteristic
296D53—162no—
297C2×2−1+4—21no—
298C2×AGL(1,ℤ/8)—1061no—
299D4×Q8—21no—
300D4×Q8C2×D5312yescomplemented
301#547—22no—
302#547—21no—
303#547—22no—
304#547C2×D5312yescomplemented
305#536—22no—
306#536—21no—
307#532—21no—
308#532C2×D5311yescharacteristic
309C2×C4⋊Q8—21no—
310C2×C4⋊Q8C2×D5311yescharacteristic
311C2×#127—531no—
312C4×#41C2×D5311yescharacteristic
313#493—1061no—
314#486—534no—
315#486—1062no—
316#486—1061no—
317#460—532no—
318#460—1061no—
319#459—532no—
320#459—534no—
321#459—1061no—
322#447—1062no—
323#431—531no—
324#423—531no—
325#418—1061no—
326C2×#102—1062no—
327C2×C4×C8—531no—
328#332—534no—
329Q8⋊C8—532no—
330D4⋊C8—532no—
331D4⋊C8—534no—
332C53#1864011yescharacteristic, complemented, Sylow
3332−1+4—44no—
3342−1+4—44no—
335C2×#41—22no—
336C2×#41—24no—
337C2×#41—22no—
338C2×#41—21no—
339C2×#41C22⋊D5311yescharacteristic
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—21no—
343C22×D4—1061no—
344C22×D4—1061no—
345C22×D4—1061no—
346AGL(1,ℤ/8)—2124no—
347C2×SD16—1062no—
348C2×D8—1064no—
349C2×D8—1062no—
350C2×M16—1061no—
351C22×C8—1061no—
352C4⋊Q8—22no—
353C4⋊Q8—22no—
354C4⋊Q8—22no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8C22⋊D5311yescharacteristic
358C4⋊Q8C22⋊D5312yes—
359C4⋊Q8C22×D5312yes—
360#127—532no—
361#127—1061no—
362#127—1061no—
363#127—1061no—
364#127C22⋊D5311yescharacteristic
365#124—22no—
366#124—21no—
367#123—24no—
368#123—24no—
369#123—24no—
370C22⋊Q8—24no—
371C22⋊Q8—22no—
372C22⋊Q8—24no—
373C22⋊Q8—22no—
374#121—22no—
375#121—22no—
376#121—21no—
377#121C22⋊D5312yes—
378C4×Q8C22⋊D5311yescharacteristic
379C4×Q8C22×D5311yescharacteristic
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4C22⋊D5311yescharacteristic
386C4×D4C22×D5311yescharacteristic
387C4×D4C22×D5312yes—
388#117—21no—
389#117C22⋊D5311yescharacteristic
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4C22⋊D5312yes—
394C2×C42—21no—
395C2×C42C22×D5311yescharacteristic
396C4⋊C8—532no—
397C4⋊C8—532no—
398C4⋊C8—1061no—
399#102—1068no—
400#102—1062no—
401#102—1062no—
402#102—1064no—
403C22⋊C8—1061no—
404C4×C8—532no—
405C4×C8—1061no—
406C4×C8—1061no—
407C24—2121no—
408#41—22no—
409#41—48no—
410#41—416no—
411#41—44no—
412#41—44no—
413#41—42no—
414#41—42no—
415#41#1171992214yescomplemented
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—48no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8C2×C22⋊D5311yescharacteristic
426C2×D4—21no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—1062no—
432C2×D4—1062no—
433C2×D4—1062no—
434C2×D4—4242no—
435C2×D4—2124no—
436C2×D4—2121no—
437C2×D4—2121no—
438C2×D4—2121no—
439C2×D4—2122no—
440C2×D4—2122no—
441C2×D4C2×C22⋊D5312yes—
442C2×D4C2×C22⋊D5311yescharacteristic
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453SD16—2124no—
454D8—4244no—
455D8—2124no—
456M16—2122no—
457C2×C8—1062no—
458C2×C8—1062no—
459C2×C8—1062no—
460C2×C8—2122no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4C2×C22⋊D5312yes—
470C4⋊C4C2×C22⋊D5314yes—
471C4⋊C4C2×C22⋊D5311yescharacteristic
472C4⋊C4C2×C22⋊D5311yescharacteristic
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42C2×C22⋊D5311yescharacteristic
484C42C2×C22⋊D5311yescharacteristic
485C42C2×C22⋊D5311yescharacteristic
486C42C23×D5311yescharacteristic
487C23—21no—
488C23—22no—
489C23—4242no—
490C23—4242no—
491C23—4242no—
492C23—2122no—
493C23#1313318611yescharacteristic
494Q8—21no—
495Q8—48no—
496Q8—42no—
497Q8—44no—
498Q8—42no—
499Q8—42no—
500Q8C2×#1171992212yes—
501D4—24no—
502D4—21no—
503D4—24no—
504D4—48no—
505D4—44no—
506D4—42no—
507D4—42no—
508D4—4242no—
509D4—4242no—
510D4—4242no—
511D4—2124no—
512D4—2122no—
513D4—2122no—
514D4—2124no—
515D4C2×#1171992212yes—
516C2×C4—21no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4#1313318611yescharacteristic
532C2×C4#1313318611yescharacteristic
533C2×C4#1313318611yescharacteristic
534C2×C4C2×#1171992212yes—
535C2×C4#1313324312yes—
536C2×C4C22×C22⋊D5311yescharacteristic
537C2×C4C22×C22⋊D5311yescharacteristic
538C2×C4C22×C22⋊D5311yescharacteristic
539C8—2122no—
540C8—2122no—
541C8—2122no—
542C22—21no—
543C22—21no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—8484no—
551C22—4242no—
552C22—4242no—
553C22—4242no—
554C22#1681075712yes—
555C22C2×#1313318611yescharacteristic
556C4—21no—
557C4—21no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—44no—
562C4—42no—
563C4#1681075712yes—
564C4#1681079612yes—
565C4C22×#1171992211yescharacteristic
566C4C2×#1313324311yescharacteristic
567C4#1681143112yes—
568C2—22no—
569C2—42no—
570C2—82no—
571C2—8482no—
572C2C2×#1681075711yescharacteristic
573C2#2028400111yescharacteristic
574C2C2×#1681079611yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3499C2
socleC22×C53C2×#120
hypercenter#3499D53
hypocenterC53#18640
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #20285630C2semidirect
2. C2×#16810971C2semidirect
3. #20284195C2semidirect
4. #20273818C2semidirect
5. #20273804C2semidirect
6. #20273387C2semidirect
7. #20273166C2semidirect
8. #20273097C2semidirect
9. C53×#3499C2semidirect
27. #16810369C22semidirect
28. Q8⋊#88322C22semidirect
29. D4⋊#88322C22semidirect
30. D4⋊#88322C22semidirect
33. D4×Q8×C53C22semidirect
37. C53×#547C22semidirect
127. #41×C53D8semidirect
281. #3499D53semidirect
300. D4×Q8C2×D53semidirect
304. #547C2×D53semidirect
332. C53#18640semidirect
415. #41#11719922semidirect

Maximal subgroups

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

subgroupindex
#202856302
C2×#168109712
#202841952
#202738182
#202738042
#202733872
#202731662
#202730972
C53×#34992
#1864053