Finite Group Databasebeta

← #28316776

Subgroups — #28316776

Order 13568.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291660C211yesmaximal, characteristic
2#20291608C211yesmaximal, characteristic, complemented
3#20290342C211yesmaximal, characteristic, complemented
4#20290340C211yesmaximal, characteristic, complemented
5#20290328C211yesmaximal, characteristic, complemented
6#20290322C211yesmaximal, characteristic, complemented
7#20290308C211yesmaximal, characteristic, complemented
8#20290305C211yesmaximal, characteristic, complemented
9Q16×#88324C211yesmaximal, characteristic, complemented
10#20290299C211yesmaximal, characteristic, complemented
11#20290291C211yesmaximal, characteristic, complemented
12#20290043C211yesmaximal, characteristic, complemented
13#20290028C211yesmaximal, characteristic
14#20289936C211yesmaximal, characteristic, complemented
15#20284796C211yesmaximal, characteristic
16#16811674C2211yescharacteristic
17Q8×#88324—21no—
18Q8×#88324—21no—
19Q8×#88324C2211yescharacteristic
20Q8×#88324C2211yescharacteristic
21Q8×#88324C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811657C2211yescharacteristic
25#16811656—21no—
26#16811655—21no—
27#16811651C2211yescharacteristic
28#16811648—21no—
29#16811640C2211yescharacteristic
30#16811639—21no—
31#16811639—21no—
32#16811639C2211yescharacteristic
33#16811639C2211yescharacteristic
34#16811630—21no—
35#16811626C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40Q8⋊#88324C2211yescharacteristic
41Q8⋊#88324C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46#16811471C2211yescharacteristic
47#16811471C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811159C2211yescharacteristic
53C2×Q8×Dic53C2211yescharacteristic
54#16810951C411yescharacteristic
55#16810937C411yescharacteristic
56#16810932C411yescharacteristic
57#16810932C411yescharacteristic
58#16810847C411yescharacteristic
59#16810847C411yescharacteristic
60#16810762C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62Q16×Dic53C2211yescharacteristic
63#16810708C2211yescharacteristic
64#16810708C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#451C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719940—21no—
69C2×D53⋊C8—21no—
70#13133284—21no—
71#13133284—21no—
72#13133284—21no—
73C4⋊#88324—21no—
74C4⋊#88324—21no—
75C4⋊#88324—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324D411yescharacteristic
79C4⋊#88324D411yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88C22⋊#88324D411yescharacteristic
89C22⋊#88324D411yescharacteristic
90#13133272—21no—
91C4×#88324—21no—
92C4×#88324—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719924C2×C411yescharacteristic
102C2×#11719924C2×C411yescharacteristic
103#13133212—21no—
104#13133208C2×C411yescharacteristic
105#13133206C2×C411yescharacteristic
106#13133201—21no—
107#13133191C2×C411yescharacteristic
108Q8×Dic53—21no—
109Q8×Dic53—21no—
110Q8×Dic53C2×C411yescharacteristic
111Q8×Dic53C2×C411yescharacteristic
112Q8×Dic53C2311yescharacteristic
113Q8×Dic53C2311yescharacteristic
114Q8×Dic53C2311yescharacteristic
115#13133179C2311yescharacteristic
116#13133177—21no—
117#13133175—21no—
118C2×C4⋊Dic53—21no—
119C2×C4⋊Dic53C2×C411yescharacteristic
120C2×C4×Dic53—21no—
121C2×C4×Dic53C2311yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133151C2×C411yescharacteristic
124#13133148C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133141C2×C411yescharacteristic
132#13133139C2×C411yescharacteristic
133#13133139C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C22×Q8×C53C2×C411yescharacteristic
138C2×Q16×C53C2×C411yescharacteristic
139C2×Q16×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144C2×Q8×C53—22no—
145C2×Q8×C53—22no—
146C2×Q8×C53—21no—
147C2×Q8×C53C22×C411yescharacteristic
148C2×Q8×C53C22×C411yescharacteristic
149C2×Q8×C53C22×C411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C2×C22⋊D53—21no—
153C22×Dic53—21no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53#4111yescharacteristic
156C2×D212C22×C411yescharacteristic
157C2×C4×D53—21no—
158#11719940—81no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324C2×D411yescharacteristic
166C2×#88324C2×D411yescharacteristic
167C2×#88324C2×D411yescharacteristic
168C2×#88324C2×D411yescharacteristic
169#11719936—81no—
170D53⋊C8—81no—
171Q16×C53—22no—
172Q16×C53—22no—
173C2×C8×C53C22×C411yescharacteristic
174C2×C8×C53C22×C411yescharacteristic
175C4⋊C4×C53—21no—
176C4⋊C4×C53C22×C411yescharacteristic
177C22⋊C4×C53—21no—
178#11719924—22no—
179#11719924—22no—
180#11719921—21no—
181#11719921—21no—
182#11719921—21no—
183#11719921—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53C22×C411yescharacteristic
192C4⋊Dic53C22×C411yescharacteristic
193C4⋊Dic53C22×C411yescharacteristic
194#11719919—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53C22×C411yescharacteristic
201C4×Dic53C2×D411yescharacteristic
202C4×Dic53#4111yescharacteristic
203C4×Dic53C2411yescharacteristic
204C2×#88322C22×C411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206C23×C53C4×D411yescharacteristic
207C22×D53—21no—
208Q8×C53—24no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53#13112yes—
214Q8×C53#13112yes—
215C2×C4×C53—24no—
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×C53C4×D411yescharacteristic
223C2×C4×C53C23×C411yescharacteristic
224C22⋊D53—81no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—41no—
231C2×Dic53—81no—
232C2×Dic53C4×D411yescharacteristic
233C2×Dic53C4×D411yescharacteristic
234C2×Dic53C22×D411yescharacteristic
235C2×Dic53C2×#4111yescharacteristic
236D212—21no—
237D212—21no—
238C4×D53—41no—
239C4×D53—41no—
240#88324—21no—
241#88324—21no—
242#88324—21no—
243#88324—21no—
244#88324—41no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248#88324—41no—
249#88324—41no—
250C8×C53—21no—
251C8×C53—21no—
252#88322—21no—
253#88322—21no—
254#17919—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53C2×C4×D411yescharacteristic
259C2×D53—41no—
260C2×D53—41no—
261C2×D53—41no—
262C4×C53—22no—
263C4×C53—22no—
264C4×C53—21no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53C2×#13111yescharacteristic
268C4×C53C2×#13111yescharacteristic
269Dic53—21no—
270Dic53—21no—
271Dic53—21no—
272Dic53—41no—
273Dic53—41no—
274Dic53—41no—
275Dic53#57711yescharacteristic
276Dic53#57811yescharacteristic
277#3108—531no—
278#2993—531no—
279#2982—531no—
280#1588—531no—
281#1587—531no—
282#1584—531no—
283#1583—531no—
284#1579—531no—
285SD16⋊C8—531no—
286C8×Q16—531no—
287C8×SD16—531no—
288#1574—531no—
289#1487—531no—
290#1482—531no—
291#1454—531no—
292C2×C53—41no—
293C2×C53#292911yescharacteristic
294C2×C53#294711yescharacteristic
295C2×C53#295711yescharacteristic
296D53—81no—
297#529—531no—
298C2×C4×Q8—531no—
299#483—531no—
300#480—531no—
301#479—531no—
302#479—531no—
303#464—531no—
304#464—531no—
305#451#615711yescharacteristic
306#448—531no—
307#446—531no—
308C8×Q8—531no—
309C8×Q8—531no—
310C8×Q8—531no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—1061no—
314C8×Q8—1061no—
315C4×Q16—531no—
316C4×Q16—531no—
317C4×SD16—531no—
318C4×SD16—531no—
319#436—531no—
320#436—1061no—
321#435—1061no—
322C8×D4—531no—
323#433—1061no—
324#431—531no—
325#431—531no—
326#431—531no—
327#431—531no—
328#431—1061no—
329#431—1061no—
330#424—1061no—
331#423—531no—
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342D4⋊C8—531no—
343D4⋊C8—531no—
344#322—531no—
345#322—531no—
346#322—531no—
347C82—531no—
348C53#1791911yescharacteristic, complemented, Sylow
349C22×Q8C2×#615711yescharacteristic
350C2×Q16C2×#615711yescharacteristic
351C2×Q16C2×#615711yescharacteristic
352C2×SD16—531no—
353C2×SD16—531no—
354C2×M16—1061no—
355C2×M16—1061no—
356C22×C8—1061no—
357#126—1061no—
358#125—531no—
359#124—531no—
360#123—1061no—
361C22⋊Q8C2×#615711yescharacteristic
362#121—531no—
363C4×Q8—531no—
364C4×Q8—531no—
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—531no—
368C4×Q8—1061no—
369C4×Q8—1061no—
370C4×D4—531no—
371C4×D4—1061no—
372#117—1061no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—1061no—
375C2×C42—531no—
376C2×C42—1061no—
377#107—531no—
378#107—531no—
379#106—531no—
380#106—531no—
381C4⋊C8—531no—
382C4⋊C8—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103C2×#615711yescharacteristic
402#103C2×#615711yescharacteristic
403#102—531no—
404#102—531no—
405C22⋊C8—531no—
406C22⋊C8—531no—
407C22⋊C8—531no—
408C22⋊C8—1061no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8C2×#615711yescharacteristic
413#97—1061no—
414C4×C8—531no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8—1061no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8C22×#615711yescharacteristic
430C2×Q8C22×#615711yescharacteristic
431C2×Q8C22×#615711yescharacteristic
432C2×D4—531no—
433C2×D4—1061no—
434C22×C4—21no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4C22×#615711yescharacteristic
440Q16—22no—
441Q16—22no—
442SD16—1062no—
443SD16—1062no—
444M16—4241no—
445M16—4241no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—4241no—
459C2×C8C22×#615711yescharacteristic
460C2×C8C22×#615711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—531no—
463C4⋊C4—531no—
464C4⋊C4—531no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4C22×#615711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1061no—
476C22⋊C4—1061no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C42—531no—
480C42—531no—
481C42—531no—
482C42—531no—
483C42—1061no—
484C42—1061no—
485C42—1061no—
486C42—1061no—
487C42—1061no—
488C23—1061no—
489C23D4×#615711yescharacteristic
490Q8—24no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8#1313330412yes—
496Q8#1313330412yes—
497D4—1061no—
498D4—1061no—
499D4—4241no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—531no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—531no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—4241no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4D4×#615711yescharacteristic
521C2×C4C23×#615711yescharacteristic
522C8—21no—
523C8—21no—
524C8—1061no—
525C8—1061no—
526C8—1061no—
527C8—1061no—
528C8—1061no—
529C8—1061no—
530C8—2121no—
531C8—2121no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—2121no—
540C22—2121no—
541C22—2121no—
542C22C2×D4×#615711yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—531no—
549C4—531no—
550C4—1061no—
551C4—1061no—
552C4—1061no—
553C4—2121no—
554C4—2121no—
555C4—2121no—
556C4C2×#1313330411yescharacteristic
557C4C2×#1313330411yescharacteristic
558C2—41no—
559C2—4241no—
560C2#2029143011yescharacteristic
561C2#2029147311yescharacteristic
562C2#2029149511yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#451C4
socleC22×C53C2×C4×D4
prime radical#13133191C2×C4
cosocleC4×Dic53C24
hypercenter#451#6157
hypocenterC53#17919
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #20291608C2semidirect
3. #20290342C2semidirect
4. #20290340C2semidirect
5. #20290328C2semidirect
6. #20290322C2semidirect
7. #20290308C2semidirect
8. #20290305C2semidirect
9. Q16×#88324C2semidirect
10. #20290299C2semidirect
11. #20290291C2semidirect
12. #20290043C2semidirect
14. #20289936C2semidirect
348. C53#17919semidirect

Maximal subgroups

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

subgroupindex
#202916602
#202916082
#202903422
#202903402
#202903282
#202903222
#202903082
#202903052
Q16×#883242
#202902992
#202902912
#202900432
#202900282
#202899362
#202847962
#1791953