Finite Group Databasebeta

← #28574197

Subgroups — #28574197

Order 15104.

594 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435251C211yesmaximal, characteristic, complemented
2#21434258C211yesmaximal, characteristic
3#21433207C211yesmaximal, characteristic
4#21423552C211yesmaximal, characteristic, complemented
5#21423545C211yesmaximal, characteristic, complemented
6#21423524C211yesmaximal, characteristic, complemented
7#21423517C211yesmaximal, characteristic, complemented
8#21423488C211yesmaximal, characteristic, complemented
9#21423483C211yesmaximal, characteristic, complemented
10#21423476C211yesmaximal, characteristic, complemented
11#21423469C211yesmaximal, characteristic, complemented
12#21422874C211yesmaximal, characteristic, complemented
13#21422796C211yesmaximal, characteristic, complemented
14#21422751C211yesmaximal, characteristic, complemented
15C59×#3107C211yesmaximal, characteristic
16#16837959C2211yescharacteristic
17#16837805C2211yescharacteristic
18#16837798C2211yescharacteristic
19#16837752C2211yescharacteristic
20#16837752C2211yescharacteristic
21#16837743—21no—
22#16837740—21no—
23#16837740—21no—
24#16837709—21no—
25#16837709—21no—
26#16837697C2211yescharacteristic
27#16837692C2211yescharacteristic
28#16837612C2211yescharacteristic
29#16837607C2211yescharacteristic
30#16837564—21no—
31Q8×C59⋊C8C2211yescharacteristic
32Q8×C59⋊C8C2211yescharacteristic
33#16837507—21no—
34#16837502—21no—
35D4×C59⋊C8—21no—
36D4×C59⋊C8—21no—
37D4×C59⋊C8C2211yescharacteristic
38D4×C59⋊C8C2211yescharacteristic
39#16837459C2211yescharacteristic
40#16837455—21no—
41#16837455C2211yescharacteristic
42C2×C4⋊(C59⋊C8)—21no—
43C2×C4⋊(C59⋊C8)C2211yescharacteristic
44#16837154C2211yescharacteristic
45#16837154C2211yescharacteristic
46#16837152C2211yescharacteristic
47#16837152C2211yescharacteristic
48#16837150C2211yescharacteristic
49#16837150C2211yescharacteristic
50#16837149C2211yescharacteristic
51#16837149C2211yescharacteristic
52#16837148C2211yescharacteristic
53#16837148C2211yescharacteristic
54#16837148C2211yescharacteristic
55#16837148C2211yescharacteristic
56C59×#533C2211yescharacteristic
57C4×#41×C59C2211yescharacteristic
58C59×#485—21no—
59C59×#475C2211yescharacteristic
60C59×#474C2211yescharacteristic
61C59×#462C2211yescharacteristic
62C59×#461C2211yescharacteristic
63C59×#452—21no—
64C4×Q16×C59—21no—
65C4×SD16×C59—21no—
66C59×#431C2211yescharacteristic
67C2×#11745672—21no—
68C2×#11745670—21no—
69C22×C59⋊C8—21no—
70C22×C59⋊C8—21no—
71C22×C59⋊C8—21no—
72#14273727C2311yescharacteristic
73#14273727C2311yescharacteristic
74#14273719—21no—
75#14273706—21no—
76#14273706—21no—
77C4×Dic118—21no—
78C4×Dic118—21no—
79#14273686—21no—
80#14273660—21no—
81#14273660—21no—
82#14273660—21no—
83#14273658—21no—
84#14273658—21no—
85#14273658—21no—
86C4⋊(C59⋊C8)—21no—
87C4⋊(C59⋊C8)—21no—
88C4⋊(C59⋊C8)—21no—
89C4⋊(C59⋊C8)D411yescharacteristic
90C4⋊(C59⋊C8)D411yescharacteristic
91C4⋊(C59⋊C8)C2311yescharacteristic
92C4⋊(C59⋊C8)C2311yescharacteristic
93C4⋊(C59⋊C8)C2311yescharacteristic
94C4⋊(C59⋊C8)C2311yescharacteristic
95#14273651—21no—
96#14273651—21no—
97#14273651—21no—
98#14273651—21no—
99#14273651—21no—
100#14273650—21no—
101#14273650—21no—
102#14273648—21no—
103C22⋊(C59⋊C8)—21no—
104C22⋊(C59⋊C8)—21no—
105C22⋊(C59⋊C8)—21no—
106C22⋊(C59⋊C8)D411yescharacteristic
107C22⋊(C59⋊C8)D411yescharacteristic
108#14273638—21no—
109C4×C59⋊C8—21no—
110C4×C59⋊C8—21no—
111C4×C59⋊C8C2311yescharacteristic
112C4×C59⋊C8C2311yescharacteristic
113C2×#41×C59D411yescharacteristic
114C2×Q16×C59—21no—
115C2×SD16×C59—21no—
116C4⋊Q8×C59C2311yescharacteristic
117C4⋊Q8×C59C2311yescharacteristic
118#125×C59—21no—
119C22⋊Q8×C59—21no—
120C22⋊Q8×C59—21no—
121C4×Q8×C59—21no—
122C4×Q8×C59—21no—
123C4×Q8×C59D411yescharacteristic
124C4×Q8×C59C2311yescharacteristic
125C4×D4×C59—21no—
126C4×D4×C59—21no—
127C4×D4×C59D411yescharacteristic
128C4×D4×C59C2311yescharacteristic
129#117×C59—21no—
130#117×C59—21no—
131#117×C59D411yescharacteristic
132C2×C42×C59—21no—
133C2×C42×C59C2311yescharacteristic
134#107×C59—21no—
135#106×C59—21no—
136C4⋊C8×C59C2311yescharacteristic
137C4⋊C8×C59C2311yescharacteristic
138#103×C59—21no—
139#103×C59—21no—
140#103×C59—21no—
141#102×C59—21no—
142C22⋊C8×C59—21no—
143C4×C8×C59—21no—
144#41×C59—22no—
145#41×C59—22no—
146C2×Q8×C59—21no—
147C2×Q8×C59—21no—
148C2×Q8×C59C2×D411yescharacteristic
149C2×D4×C59—21no—
150C2×D4×C59C2×D411yescharacteristic
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59—21no—
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59#4111yescharacteristic
157C2×Dic118—21no—
158C2×Dic118—21no—
159Q16×C59—42no—
160SD16×C59—42no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—41no—
170C4⋊C4×C59—41no—
171C4⋊C4×C59C2×D411yescharacteristic
172C4⋊C4×C59C2×D411yescharacteristic
173C22⋊C4×C59—21no—
174C22⋊C4×C59—21no—
175C22⋊C4×C59—21no—
176C22⋊C4×C59—41no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—41no—
181C42×C59C2×D411yescharacteristic
182C42×C59C2×D411yescharacteristic
183C42×C59#4111yescharacteristic
184C42×C59C2411yescharacteristic
185C22⋊Dic59—41no—
186#11745672—42no—
187#11745670—42no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192Dic59⋊C4—41no—
193Dic59⋊C4—41no—
194C4×Dic59—41no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—81no—
200C2×C59⋊C8—81no—
201C2×C59⋊C8—81no—
202C2×C59⋊C8C2×D411yescharacteristic
203C2×C59⋊C8C2×D411yescharacteristic
204C2×C59⋊C8C2×D411yescharacteristic
205C2×C59⋊C8C2×D411yescharacteristic
206C23×C59—21no—
207C23×C59#12111yescharacteristic
208Q8×C59—21no—
209Q8×C59—41no—
210Q8×C59—41no—
211Q8×C59—41no—
212Q8×C59—41no—
213Q8×C59#13512yes—
214D4×C59—24no—
215D4×C59—21no—
216D4×C59#13512yes—
217C2×C4×C59—24no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—81no—
231C2×C4×C59#12111yescharacteristic
232C2×C4×C59#12111yescharacteristic
233C2×C4×C59#12111yescharacteristic
234C2×C4×C59C22×D411yescharacteristic
235C2×C4×C59C22×D411yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×Dic59—41no—
238C2×Dic59—41no—
239Dic118—41no—
240Dic118—41no—
241Dic118—81no—
242C8×C59—41no—
243C8×C59—41no—
244C59⋊C8—21no—
245C59⋊C8—21no—
246C59⋊C8—21no—
247C59⋊C8—21no—
248C59⋊C8—41no—
249C59⋊C8—41no—
250C59⋊C8—41no—
251C59⋊C8—41no—
252#19418—591nomaximal, Sylow
253C22×C59—21no—
254C22×C59—21no—
255C22×C59—22no—
256C22×C59—21no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59C2×#12111yescharacteristic
260C4×C59—21no—
261C4×C59—21no—
262C4×C59—22no—
263C4×C59—21no—
264C4×C59—21no—
265C4×C59—41no—
266C4×C59—41no—
267C4×C59—81no—
268C4×C59—81no—
269C4×C59C2×#13511yescharacteristic
270C4×C59C2×#13511yescharacteristic
271C4×C59#57711yescharacteristic
272C4×C59#57811yescharacteristic
273Dic59—81no—
274Dic59—81no—
275#3231—591no—
276#3116—591no—
277#3107D5911yescharacteristic
278#2988—591no—
279#1684—591no—
280#1683—591no—
281#1680—591no—
282#1679—591no—
283#1674—591no—
284#1673—591no—
285#1672—591no—
286#1671—591no—
287#1572—591no—
288#1552—591no—
289#1542—591no—
290C2×C59—41no—
291C2×C59—42no—
292C2×C59#304611yescharacteristic
293C2×C59#305011yescharacteristic
294C2×C59#306911yescharacteristic
295#533—591no—
296#533C2×D5911yescharacteristic
297C4×#41C2×D5911yescharacteristic
298#500—591no—
299#498—591no—
300#487—591no—
301#487—591no—
302#485—21no—
303#485—1181no—
304#485—1181no—
305#485—1181no—
306#479—1181no—
307#479—1181no—
308#477—591no—
309#476—591no—
310#475C2×D5911yescharacteristic
311#474C2×D5911yescharacteristic
312#462—591no—
313#462C2×D5911yescharacteristic
314#461—591no—
315#461C2×D5911yescharacteristic
316#452—21no—
317#452—1181no—
318C8×Q8—591no—
319C8×Q8—591no—
320#441—1181no—
321#440—1181no—
322C4×Q16—21no—
323C4×SD16—21no—
324C8×D4—591no—
325C8×D4—591no—
326C8×D4—1181no—
327C8×D4—1181no—
328#432—591no—
329#431—591no—
330#431—1181no—
331#431C2×D5911yescharacteristic
332C2×C4⋊C8—591no—
333C2×C4⋊C8—1181no—
334#335—591no—
335#335—591no—
336#334—591no—
337#334—591no—
338#333—591no—
339#333—591no—
340#332—591no—
341#332—591no—
342#332—591no—
343#332—591no—
344#332—591no—
345#332—591no—
346C59#1941811yescharacteristic, complemented, Sylow
347C2×#41C22⋊D5911yescharacteristic
348C2×Q16—21no—
349C2×Q16—1181no—
350C2×SD16—21no—
351C2×SD16—1181no—
352C22×C8—1181no—
353C22×C8—1181no—
354C22×C8—1181no—
355C4⋊Q8—591no—
356C4⋊Q8—591no—
357C4⋊Q8C22×D5911yescharacteristic
358C4⋊Q8C22×D5911yescharacteristic
359#125—21no—
360#125—1181no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—1181no—
364C22⋊Q8—1181no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—1181no—
368C4×Q8—1181no—
369C4×Q8C22⋊D5911yescharacteristic
370C4×Q8C22×D5911yescharacteristic
371C4×D4—21no—
372C4×D4—21no—
373C4×D4C22⋊D5911yescharacteristic
374C4×D4C22×D5911yescharacteristic
375#117—21no—
376#117—21no—
377#117—1181no—
378#117C22⋊D5911yescharacteristic
379C2×C42—21no—
380C2×C42C22×D5911yescharacteristic
381#107—21no—
382#107—1181no—
383#107—1181no—
384#107—1181no—
385#106—21no—
386#106—1181no—
387#106—1181no—
388#106—1181no—
389C4⋊C8—591no—
390C4⋊C8—591no—
391C4⋊C8—591no—
392C4⋊C8—591no—
393C4⋊C8—591no—
394C4⋊C8—591no—
395C4⋊C8—1181no—
396C4⋊C8—1181no—
397C4⋊C8—1181no—
398C4⋊C8C22×D5911yescharacteristic
399C4⋊C8C22×D5911yescharacteristic
400#103—21no—
401#103—21no—
402#103—21no—
403#103—1181no—
404#103—1181no—
405#103—1181no—
406#103—1181no—
407#103—1181no—
408#103—1181no—
409#103—1181no—
410#102—21no—
411#102—1181no—
412C22⋊C8—21no—
413C22⋊C8—591no—
414C22⋊C8—591no—
415C22⋊C8—1181no—
416C22⋊C8—1181no—
417C22⋊C8—1181no—
418#97—1181no—
419C4×C8—21no—
420C4×C8—591no—
421C4×C8—591no—
422C4×C8—1181no—
423C4×C8—1181no—
424#41—22no—
425#41—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—1181no—
429C2×Q8—1181no—
430C2×Q8C2×C22⋊D5911yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D5911yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4D4×D5911yescharacteristic
437C22×C4#1174568711yescharacteristic
438C22×C4C2×C22⋊D5911yescharacteristic
439Q16—42no—
440Q16—2362no—
441SD16—42no—
442SD16—2362no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—591no—
446C2×C8—591no—
447C2×C8—591no—
448C2×C8—591no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—4721no—
454C2×C8—4721no—
455C2×C8—4721no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—2361no—
469C4⋊C4—2361no—
470C4⋊C4C2×C22⋊D5911yescharacteristic
471C4⋊C4C2×C22⋊D5911yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—41no—
476C22⋊C4—2361no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—41no—
481C42—2361no—
482C42D4×D5911yescharacteristic
483C42#1174568711yescharacteristic
484C42C2×C22⋊D5911yescharacteristic
485C42C23×D5911yescharacteristic
486C23—21no—
487C23#1427369911yescharacteristic
488Q8—21no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—4721no—
494Q8—2361no—
495Q8—2361no—
496Q8#1427374712yes—
497D4—24no—
498D4—21no—
499D4#1427374712yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—81no—
514C2×C4—2361no—
515C2×C4—2361no—
516C2×C4#1427369911yescharacteristic
517C2×C4#1427369911yescharacteristic
518C2×C4#1427369911yescharacteristic
519C2×C4C2×D4×D5911yescharacteristic
520C2×C4C22×C22⋊D5911yescharacteristic
521C2×C4C2×#1174568711yescharacteristic
522C8—41no—
523C8—41no—
524C8—1181no—
525C8—1181no—
526C8—1181no—
527C8—1181no—
528C8—2361no—
529C8—2361no—
530C8—2361no—
531C8—2361no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—21no—
536C22—41no—
537C22—41no—
538C22C2×#1427369911yescharacteristic
539C4—21no—
540C4—21no—
541C4—22no—
542C4—21no—
543C4—21no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—4721no—
549C4—4721no—
550C4C2×#1427374711yescharacteristic
551C4C2×#1427374711yescharacteristic
552C4#1683820911yescharacteristic
553C4#1683821611yescharacteristic
554C2—41no—
555C2—42no—
556C2#2143364411yescharacteristic
557C2#2143367411yescharacteristic
558C2#2143383011yescharacteristic
subgroupG/H
centerC22C2×#14273699
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#3107C2
socleC22×C59C2×#121
prime radicalC2×#41×C59D4
hypercenter#3107D59
hypocenterC59#19418
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
1. #21435251C2semidirect
4. #21423552C2semidirect
5. #21423545C2semidirect
6. #21423524C2semidirect
7. #21423517C2semidirect
8. #21423488C2semidirect
9. #21423483C2semidirect
10. #21423476C2semidirect
11. #21423469C2semidirect
12. #21422874C2semidirect
13. #21422796C2semidirect
14. #21422751C2semidirect
346. C59#19418semidirect

Maximal subgroups

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

subgroupindex
#214352512
#214342582
#214332072
#214235522
#214235452
#214235242
#214235172
#214234882
#214234832
#214234762
#214234692
#214228742
#214227962
#214227512
C59×#31072
#1941859