Finite Group Databasebeta

← #27355212

Subgroups — #27355212

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284795C211yesmaximal, characteristic
2#20283713C211yesmaximal, characteristic
3Q8×#11719917C211yesmaximal, characteristic, complemented
4#20273621C211yesmaximal, characteristic, complemented
5#20273602C211yesmaximal, characteristic, complemented
6#20273583C211yesmaximal, characteristic, complemented
7#20273575C211yesmaximal, characteristic, complemented
8#20273568C211yesmaximal, characteristic, complemented
9#20273555C211yesmaximal, characteristic, complemented
10#20273530C211yesmaximal, characteristic, complemented
11#20273519C211yesmaximal, characteristic, complemented
12#20273503C211yesmaximal, characteristic, complemented
13#20273181C211yesmaximal, characteristic
14#20273010C211yesmaximal, characteristic, complemented
15C53×#1479C211yesmaximal, characteristic, complemented
16#16811157C2211yescharacteristic
17#16810953C411yescharacteristic
18#16810936C411yescharacteristic
19#16810930C411yescharacteristic
20#16810930C411yescharacteristic
21#16810845C411yescharacteristic
22#16810845C411yescharacteristic
23#16810779C411yescharacteristic
24#16810762C411yescharacteristic
25#16810754—21no—
26#16810754—21no—
27#16810754C2211yescharacteristic
28#16810754C2211yescharacteristic
29#16810750C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31Q8×#88322C2211yescharacteristic
32#16810747C2211yescharacteristic
33C4×#11719925C2211yescharacteristic
34C4×#11719925C2211yescharacteristic
35C4×#11719924C2211yescharacteristic
36C4×#11719924C2211yescharacteristic
37#16810700C2211yescharacteristic
38#16810698—21no—
39#16810697—21no—
40#16810688C2211yescharacteristic
41#16810682—21no—
42#16810676C2211yescharacteristic
43#16810647—21no—
44#16810647C2211yescharacteristic
45#16810577—21no—
46C4×#11719917—21no—
47C4×#11719917C2211yescharacteristic
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810373C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810355C2211yescharacteristic
54#16810352C2211yescharacteristic
55#16810352C2211yescharacteristic
56#16810345C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C2×C4×Q8×C53C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C53×#431C2211yescharacteristic
63C53×Q8⋊C8C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×Q8⋊C8C2211yescharacteristic
67C2×#11719925C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719924C2×C411yescharacteristic
70C2×#11719924C2×C411yescharacteristic
71C2×#11719917—21no—
72C2×#11719917D411yescharacteristic
73C2×#11719917D411yescharacteristic
74C2×#11719912—21no—
75C2×C8×D53—21no—
76#13133211—21no—
77#13133209C2×C411yescharacteristic
78#13133205C2×C411yescharacteristic
79#13133200—21no—
80#13133196C2×C411yescharacteristic
81#13133191C2×C411yescharacteristic
82C4×Dic106C2311yescharacteristic
83C4×D212C2311yescharacteristic
84C4×C22⋊D53—21no—
85#13133174—21no—
86#13133150C2×C411yescharacteristic
87#13133150C2×C411yescharacteristic
88#13133148C2×C411yescharacteristic
89#13133148C2×C411yescharacteristic
90C4⋊#88322—21no—
91C4⋊#88322—21no—
92C4⋊#88322—21no—
93C4⋊#88322—21no—
94C4⋊#88322—21no—
95C4⋊#88322C2×C411yescharacteristic
96C4⋊#88322C2×C411yescharacteristic
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99#13133146—21no—
100#13133146—21no—
101#13133141C2×C411yescharacteristic
102#13133141C2×C411yescharacteristic
103#13133141C2×C411yescharacteristic
104#13133141C2×C411yescharacteristic
105#13133140C2×C411yescharacteristic
106#13133140C2×C411yescharacteristic
107#13133137C2×C411yescharacteristic
108#13133137C2×C411yescharacteristic
109C22⋊#88322C2×C411yescharacteristic
110C22⋊#88322C2×C411yescharacteristic
111#13133129—21no—
112#13133129—21no—
113#13133128—21no—
114#13133128—21no—
115#13133128D411yescharacteristic
116#13133128D411yescharacteristic
117#13133127—21no—
118C4×#88322—21no—
119C4×#88322—21no—
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C4×#88322C2311yescharacteristic
124C8×Dic53—21no—
125C22×Q8×C53C2×C411yescharacteristic
126C4×Q8×C53—21no—
127C4×Q8×C53—21no—
128C4×Q8×C53C2×C411yescharacteristic
129C4×Q8×C53C2×C411yescharacteristic
130C4×Q8×C53C2311yescharacteristic
131C4×Q8×C53C2311yescharacteristic
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53C2×C411yescharacteristic
134C2×C42×C53—21no—
135C2×C42×C53C2311yescharacteristic
136C4⋊C8×C53—21no—
137C4⋊C8×C53C2311yescharacteristic
138C4⋊C8×C53C2311yescharacteristic
139C22⋊C8×C53—21no—
140C22⋊C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53—21no—
145C2×Q8×C53—22no—
146C2×Q8×C53—41no—
147C2×Q8×C53C22×C411yescharacteristic
148C2×Q8×C53C22×C411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53#4111yescharacteristic
154C2×C22⋊D53—21no—
155C2×D212C22×C411yescharacteristic
156C2×C4×D53—21no—
157C2×Dic106C22×C411yescharacteristic
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53C22×C411yescharacteristic
168C4⋊C4×C53C22×C411yescharacteristic
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53C22×C411yescharacteristic
174C42×C53C2×D411yescharacteristic
175C42×C53#4111yescharacteristic
176C42×C53C2411yescharacteristic
177C22⋊Dic53—21no—
178#11719925—22no—
179#11719925—41no—
180#11719924—22no—
181#11719924—41no—
182#11719921—21no—
183#11719921—21no—
184#11719921—21no—
185#11719921—21no—
186C4⋊Dic53C22×C411yescharacteristic
187C4⋊Dic53C22×C411yescharacteristic
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192C4×Dic53—21no—
193#11719917—22no—
194#11719917—41no—
195#11719917—42no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322C22×C411yescharacteristic
199C2×#88322C22×C411yescharacteristic
200C2×#88322C22×C411yescharacteristic
201C2×#88322C22×C411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206#11719912—42no—
207C8×D53—42no—
208C23×C53C4×D411yescharacteristic
209C22×D53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—42no—
214Q8×C53#13112yes—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53—42no—
225C2×C4×C53—81no—
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C23×C411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C22⋊D53—81no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236D212—21no—
237D212—21no—
238C4×D53—41no—
239C4×D53—41no—
240Dic106—21no—
241Dic106—21no—
242C8×C53—41no—
243C8×C53—41no—
244C8×C53—41no—
245C8×C53—41no—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—21no—
252#88322—21no—
253#88322—41no—
254#88322—41no—
255#88322#13712yes—
256#18012—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53C2×C4×D411yescharacteristic
261C2×D53—41no—
262C2×D53—41no—
263C2×D53—41no—
264C4×C53—21no—
265C4×C53—22no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53C2×#13111yescharacteristic
273C4×C53#56811yescharacteristic
274C4×C53C2×#13711yescharacteristic
275C4×C53#57511yescharacteristic
276Dic53—41no—
277Dic53—41no—
278Dic53—41no—
279#3108—531no—
280#2969—531no—
281Q8×M16—531no—
282#1593—531no—
283#1590—531no—
284#1587—531no—
285#1585—531no—
286#1584—531no—
287#1583—531no—
288SD16⋊C8—531no—
289C8×Q16—531no—
290#1574—531no—
291#1495—531no—
292#1479D5311yescharacteristic, complemented
293#1453—531no—
294C2×C53—41no—
295C2×C53#293311yescharacteristic
296C2×C53C4×#13711yescharacteristic
297C2×C53#295711yescharacteristic
298D53—81no—
299#529—531no—
300C2×C4×Q8C2×D5311yescharacteristic
301#483—531no—
302#480—531no—
303#479—531no—
304#479—531no—
305#464—531no—
306#464—531no—
307#451—531no—
308#448—531no—
309#446—531no—
310#446—531no—
311#446—531no—
312#446—1061no—
313#446—1061no—
314C8×Q8—531no—
315C8×Q8—531no—
316C8×Q8—531no—
317C4×Q16—531no—
318C4×Q16—531no—
319C4×SD16—531no—
320C4×SD16—531no—
321#436—531no—
322#436—1061no—
323#435—1061no—
324C8×D4—531no—
325#433—1061no—
326#431—531no—
327#431C2×D5311yescharacteristic
328#431C2×D5311yescharacteristic
329#423—531no—
330#423—1061no—
331#405—1061no—
332C4×M16—531no—
333C4×M16—1061no—
334#335—531no—
335#335—531no—
336#334—531no—
337#334—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8C2×D5311yescharacteristic
341Q8⋊C8C2×D5311yescharacteristic
342Q8⋊C8C2×D5311yescharacteristic
343Q8⋊C8C2×D5311yescharacteristic
344D4⋊C8—531no—
345D4⋊C8—531no—
346#322—531no—
347#322—531no—
348#322—531no—
349C82—531no—
350C53#1801211yescharacteristic, complemented, Sylow
351C22×Q8C4×D5311yescharacteristic
352C2×Q16—531no—
353C2×Q16—531no—
354C2×SD16—531no—
355C2×SD16—531no—
356C2×M16—531no—
357C2×M16—531no—
358C2×M16—1061no—
359C2×M16—1061no—
360C22×C8—1061no—
361#126—1061no—
362#125—531no—
363#124—531no—
364#123—1061no—
365C22⋊Q8—531no—
366#121—531no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—531no—
370C4×Q8C4×D5311yescharacteristic
371C4×Q8C4×D5311yescharacteristic
372C4×Q8C22×D5311yescharacteristic
373C4×Q8C22×D5311yescharacteristic
374C4×D4—531no—
375C4×D4—1061no—
376#117—1061no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C4×D5311yescharacteristic
379C2×C42—21no—
380C2×C42C22×D5311yescharacteristic
381#107—531no—
382#107—531no—
383#106—531no—
384#106—531no—
385C4⋊C8—21no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8C22×D5311yescharacteristic
398C4⋊C8C22×D5311yescharacteristic
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#103—531no—
404#103—531no—
405#102—531no—
406#102—531no—
407C22⋊C8—21no—
408C22⋊C8—21no—
409C22⋊C8—531no—
410C22⋊C8—531no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—531no—
414#97—531no—
415#97—1061no—
416#97—1061no—
417#97—1061no—
418C4×C8—21no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8C22×D5311yescharacteristic
427C4×C8C22×D5311yescharacteristic
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8—531no—
432C2×Q8C2×C4×D5311yescharacteristic
433C2×Q8C2×C4×D5311yescharacteristic
434C2×D4—531no—
435C2×D4—1061no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—1061no—
439C22×C4C2×C4×D5311yescharacteristic
440C22×C4D4×D5311yescharacteristic
441C22×C4#1171994711yescharacteristic
442Q16—1062no—
443Q16—2121no—
444SD16—1062no—
445SD16—2121no—
446M16—1062no—
447M16—2122no—
448M16—2121no—
449M16—2122no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—531no—
459C2×C8—531no—
460C2×C8—531no—
461C2×C8—531no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—2122no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—531no—
471C4⋊C4—531no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4C2×C4×D5311yescharacteristic
477C4⋊C4C2×C4×D5311yescharacteristic
478C22⋊C4—1061no—
479C22⋊C4—1061no—
480C22⋊C4—1061no—
481C22⋊C4—1061no—
482C22⋊C4—1061no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—1061no—
488C42C2×C4×D5311yescharacteristic
489C42D4×D5311yescharacteristic
490C42#1171994711yescharacteristic
491C42C23×D5311yescharacteristic
492C23—1061no—
493C23#1313317711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8—1061no—
499Q8—1061no—
500Q8#1313322312yes—
501D4—1061no—
502D4—1061no—
503D4—4241no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4#1313317711yescharacteristic
521C2×C4#1313317711yescharacteristic
522C2×C4#1313317711yescharacteristic
523C2×C4C22×C4×D5311yescharacteristic
524C2×C4C2×D4×D5311yescharacteristic
525C2×C4C2×#1171994711yescharacteristic
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—532no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—1061no—
535C8—1061no—
536C8—1061no—
537C8—1061no—
538C8—2121no—
539C8—2121no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—2121no—
544C22—2121no—
545C22—2121no—
546C22C2×#1313317711yescharacteristic
547C4—21no—
548C4—22no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—2121no—
556C4—2121no—
557C4—2121no—
558C4C2×#1313322311yescharacteristic
559C4#1681138911yescharacteristic
560C4#137×D5311yescharacteristic
561C4#1681142211yescharacteristic
562C2—41no—
563C2—4241no—
564C2#2028343611yescharacteristic
565C2#2028353011yescharacteristic
566C2#2028362811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1479C2
socleC22×C53C2×C4×D4
prime radical#13133191C2×C4
cosocleC42×C53C24
hypercenter#1479D53
hypocenterC53#18012
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
3. Q8×#11719917C2semidirect
4. #20273621C2semidirect
5. #20273602C2semidirect
6. #20273583C2semidirect
7. #20273575C2semidirect
8. #20273568C2semidirect
9. #20273555C2semidirect
10. #20273530C2semidirect
11. #20273519C2semidirect
12. #20273503C2semidirect
14. #20273010C2semidirect
15. C53×#1479C2semidirect
292. #1479D53semidirect
350. C53#18012semidirect

Maximal subgroups

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

subgroupindex
#202847952
#202837132
Q8×#117199172
#202736212
#202736022
#202735832
#202735752
#202735682
#202735552
#202735302
#202735192
#202735032
#202731812
#202730102
C53×#14792
#1801253