Finite Group Databasebeta

← #28576312

Subgroups — #28576312

Order 15104.

590 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435314C211yesmaximal, characteristic, complemented
2#21435214C211yesmaximal, characteristic, complemented
3#21433256C211yesmaximal, characteristic, complemented
4#21423634C211yesmaximal, characteristic, complemented
5#21423627C211yesmaximal, characteristic, complemented
6#21423622C211yesmaximal, characteristic, complemented
7#21423617C211yesmaximal, characteristic, complemented
8#21423510C211yesmaximal, characteristic, complemented
9#21423503C211yesmaximal, characteristic, complemented
10#21423498C211yesmaximal, characteristic, complemented
11#21423493C211yesmaximal, characteristic, complemented
12#21422879C211yesmaximal, characteristic, complemented
13#21422791C211yesmaximal, characteristic, complemented
14C2×#16837150C211yesmaximal, characteristic, complemented
15C59×#3247C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17#16837812—21no—
18#16837805C2211yescharacteristic
19#16837805C2211yescharacteristic
20#16837761—21no—
21#16837748C2211yescharacteristic
22#16837748C2211yescharacteristic
23#16837730—21no—
24#16837713—21no—
25#16837713—21no—
26#16837713—21no—
27#16837697C2211yescharacteristic
28#16837697C2211yescharacteristic
29#16837692C2211yescharacteristic
30#16837692C2211yescharacteristic
31#16837682—21no—
32#16837682—21no—
33#16837533C2211yescharacteristic
34Q8×C59⋊C8C2211yescharacteristic
35#16837480C2211yescharacteristic
36#16837474—21no—
37D4×C59⋊C8—21no—
38D4×C59⋊C8C2211yescharacteristic
39#16837463—21no—
40#16837459C2211yescharacteristic
41#16837455C2211yescharacteristic
42#16837429—21no—
43C2×C4⋊(C59⋊C8)C2211yescharacteristic
44#16837154C2211yescharacteristic
45#16837154C2211yescharacteristic
46#16837152C2211yescharacteristic
47#16837152C2211yescharacteristic
48#16837151C2211yescharacteristic
49#16837151C2211yescharacteristic
50#16837150C2211yescharacteristic
51#16837150C2211yescharacteristic
52#16837150C2211yescharacteristic
53#16837150C2211yescharacteristic
54#16837146C2211yescharacteristic
55#16837146C2211yescharacteristic
56C59×#532C2211yescharacteristic
57C59×#529C2211yescharacteristic
58C59×#478—21no—
59C59×#477C2211yescharacteristic
60C59×#476C2211yescharacteristic
61C59×#466—21no—
62C59×#465—21no—
63C59×#463—21no—
64C59×#460C2211yescharacteristic
65C59×#459C2211yescharacteristic
66C2×C4⋊C8×C59C2211yescharacteristic
67C22×Dic118—21no—
68C2×#11745664—21no—
69C22×C59⋊C8—21no—
70C22×C59⋊C8—21no—
71#14273727—21no—
72#14273727—21no—
73#14273727—21no—
74#14273727C2311yescharacteristic
75#14273727C2311yescharacteristic
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78#14273660—21no—
79#14273660—21no—
80#14273660—21no—
81#14273660—21no—
82#14273660—21no—
83#14273660—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#14273648—21no—
98#14273648—21no—
99#14273648—21no—
100#14273648—21no—
101#14273648—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—
109#14273638—21no—
110C4×C59⋊C8—21no—
111C4×C59⋊C8C2311yescharacteristic
112C4×C59⋊C8C2311yescharacteristic
113C2×#41×C59—21no—
114C2×SD16×C59—21no—
115C2×D8×C59—21no—
116C22×C8×C59—21no—
117C4⋊Q8×C59C2311yescharacteristic
118#127×C59C2311yescharacteristic
119#126×C59—21no—
120#125×C59D411yescharacteristic
121#124×C59—21no—
122#124×C59D411yescharacteristic
123#123×C59—21no—
124C22⋊Q8×C59D411yescharacteristic
125#121×C59—21no—
126#121×C59—21no—
127#121×C59D411yescharacteristic
128C4×Q8×C59C2311yescharacteristic
129C4×D4×C59—21no—
130C4×D4×C59—21no—
131C4×D4×C59—21no—
132C4×D4×C59C2311yescharacteristic
133#117×C59—21no—
134C2×C42×C59C2311yescharacteristic
135#107×C59—21no—
136#106×C59—21no—
137C4⋊C8×C59—21no—
138C4⋊C8×C59C2311yescharacteristic
139C4⋊C8×C59C2311yescharacteristic
140#103×C59—21no—
141#102×C59—21no—
142#102×C59—21no—
143#102×C59—21no—
144#41×C59—82no—
145C2×Q8×C59—21no—
146C2×Q8×C59C2×D411yescharacteristic
147C2×D4×C59—21no—
148C2×D4×C59—21no—
149C2×D4×C59—21no—
150C2×D4×C59—21no—
151C2×D4×C59—41no—
152C2×D4×C59C2×D411yescharacteristic
153C22×C4×C59—21no—
154C22×C4×C59—41no—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59#4111yescharacteristic
158C22×Dic59—41no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—41no—
162C2×Dic118—82no—
163SD16×C59—81no—
164D8×C59—81no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—81no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59C2×D411yescharacteristic
175C4⋊C4×C59C2×D411yescharacteristic
176C22⋊C4×C59—21no—
177C22⋊C4×C59—21no—
178C22⋊C4×C59—21no—
179C22⋊C4×C59—21no—
180C22⋊C4×C59—21no—
181C22⋊C4×C59—41no—
182C22⋊C4×C59—41no—
183C42×C59—21no—
184C42×C59C2×D411yescharacteristic
185C42×C59C2×D411yescharacteristic
186C42×C59#4111yescharacteristic
187C42×C59C2411yescharacteristic
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—41no—
193C4⋊Dic59—41no—
194#11745664—81no—
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⋊C8C2×D411yescharacteristic
202C2×C59⋊C8C2×D411yescharacteristic
203C2×C59⋊C8C2×D411yescharacteristic
204C2×C59⋊C8C2×D411yescharacteristic
205C23×C59—21no—
206C23×C59—41no—
207C23×C59#12111yescharacteristic
208Q8×C59—21no—
209Q8×C59—21no—
210Q8×C59—81no—
211D4×C59—21no—
212D4×C59—21no—
213D4×C59—42no—
214D4×C59—41no—
215D4×C59—41no—
216D4×C59—41no—
217D4×C59—41no—
218D4×C59—81no—
219D4×C59—82no—
220C2×C4×C59—22no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—82no—
232C2×C4×C59#12111yescharacteristic
233C2×C4×C59#12111yescharacteristic
234C2×C4×C59#12111yescharacteristic
235C2×C4×C59C22×D411yescharacteristic
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—82no—
241Dic118—82no—
242Dic118—81no—
243Dic118—81no—
244C8×C59—41no—
245C8×C59—41no—
246C59⋊C8—21no—
247C59⋊C8—21no—
248C59⋊C8—21no—
249C59⋊C8—21no—
250C59⋊C8—41no—
251C59⋊C8—41no—
252C59⋊C8—41no—
253C59⋊C8—41no—
254#19591—591nomaximal, Sylow
255C22×C59—21no—
256C22×C59—21no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59#47212yes—
264C22×C59C2×#12111yescharacteristic
265C4×C59—21no—
266C4×C59—21no—
267C4×C59—21no—
268C4×C59—21no—
269C4×C59—41no—
270C4×C59—41no—
271C4×C59—41no—
272C4×C59—81no—
273C4×C59#57711yescharacteristic
274C4×C59#57711yescharacteristic
275C4×C59#57811yescharacteristic
276C4×C59#57811yescharacteristic
277Dic59—81no—
278Dic59—81no—
279#3247D5911yescharacteristic
280#3238—591no—
281#3226—591no—
282#2994—591no—
283#1698—591no—
284#1697—591no—
285#1696—591no—
286#1695—591no—
287#1678—591no—
288#1677—591no—
289#1676—591no—
290#1675—591no—
291#1573—591no—
292#1551—591no—
293C2×#333—591no—
294C2×C59—22no—
295C2×C59—81no—
296C2×C59—161no—
297C2×C59#304611yescharacteristic
298C2×C59#306311yescharacteristic
299C2×C59C2×#47211yescharacteristic
300#532C2×D5911yescharacteristic
301C2×C4⋊Q8—591no—
302#529C2×D5911yescharacteristic
303#501—1181no—
304#500—591no—
305#500—591no—
306#489—1181no—
307#486—591no—
308#486—591no—
309#483—1181no—
310#480—1181no—
311#480—1181no—
312#480—1181no—
313#478—21no—
314#477—591no—
315#477—591no—
316#477C2×D5911yescharacteristic
317#476—591no—
318#476—591no—
319#476C2×D5911yescharacteristic
320#474—1181no—
321#474—1181no—
322#466—21no—
323#465—21no—
324#463—21no—
325#460C2×D5911yescharacteristic
326#459C2×D5911yescharacteristic
327#446—591no—
328C8×Q8—591no—
329#436—591no—
330#435—1181no—
331C8×D4—591no—
332C8×D4—1181no—
333#433—1181no—
334#432—591no—
335#431—591no—
336#424—1181no—
337C2×C4⋊C8—591no—
338C2×C4⋊C8C2×D5911yescharacteristic
339#335—591no—
340#335—591no—
341#334—591no—
342#334—591no—
343#333—591no—
344#333—591no—
345#333—591no—
346#333—591no—
347#333—591no—
348#333—591no—
349#331—591no—
350#331—591no—
351C59#1959111yescharacteristic, complemented, Sylow
352C2×#41—21no—
353C22×Q8—1181no—
354C2×SD16—21no—
355C2×D8—21no—
356C2×M16—1181no—
357C22×C8—21no—
358C22×C8—1181no—
359C22×C8—1181no—
360C4⋊Q8—591no—
361C4⋊Q8—591no—
362C4⋊Q8—1181no—
363C4⋊Q8—1181no—
364C4⋊Q8—1181no—
365C4⋊Q8C22×D5911yescharacteristic
366#127C22×D5911yescharacteristic
367#126—21no—
368#125C22⋊D5911yescharacteristic
369#124—21no—
370#124C22⋊D5911yescharacteristic
371#123—21no—
372C22⋊Q8C22⋊D5911yescharacteristic
373#121—21no—
374#121—21no—
375#121C22⋊D5911yescharacteristic
376C4×Q8C22×D5911yescharacteristic
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C22×D5911yescharacteristic
381#117—21no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C42C22×D5911yescharacteristic
385#107—21no—
386#107—1181no—
387#107—1181no—
388#107—1181no—
389#107—1181no—
390#107—1181no—
391#107—1181no—
392#106—21no—
393#106—1181no—
394#106—1181no—
395C4⋊C8—21no—
396C4⋊C8—591no—
397C4⋊C8—591no—
398C4⋊C8—591no—
399C4⋊C8—591no—
400C4⋊C8—591no—
401C4⋊C8—591no—
402C4⋊C8—1181no—
403C4⋊C8—1181no—
404C4⋊C8—1181no—
405C4⋊C8C22×D5911yescharacteristic
406C4⋊C8C22×D5911yescharacteristic
407#103—21no—
408#103—1181no—
409#103—1181no—
410#102—21no—
411#102—21no—
412#102—21no—
413#102—1181no—
414#102—1181no—
415#102—1181no—
416#102—1181no—
417#102—1181no—
418#102—1181no—
419C22⋊C8—591no—
420C22⋊C8—591no—
421C22⋊C8—1181no—
422C22⋊C8—1181no—
423C22⋊C8—1181no—
424#97—1181no—
425#97—1181no—
426C4×C8—591no—
427C4×C8—591no—
428C4×C8—1181no—
429#41—82no—
430C2×Q8—21no—
431C2×Q8—1181no—
432C2×Q8—1181no—
433C2×Q8—4722no—
434C2×Q8—2361no—
435C2×Q8C2×C22⋊D5911yescharacteristic
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4C2×C22⋊D5911yescharacteristic
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—2361no—
445C22×C4D4×D5911yescharacteristic
446C22×C4#1174568711yescharacteristic
447C22×C4C2×C22⋊D5911yescharacteristic
448SD16—81no—
449D8—81no—
450M16—4721no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—81no—
454C2×C8—591no—
455C2×C8—591no—
456C2×C8—591no—
457C2×C8—591no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—4721no—
463C2×C8—4721no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—2361no—
475C4⋊C4—2361no—
476C4⋊C4C2×C22⋊D5911yescharacteristic
477C4⋊C4C2×C22⋊D5911yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C42—21no—
486C42D4×D5911yescharacteristic
487C42#1174568711yescharacteristic
488C42C2×C22⋊D5911yescharacteristic
489C42C23×D5911yescharacteristic
490C23—21no—
491C23—41no—
492C23#1427369911yescharacteristic
493Q8—21no—
494Q8—21no—
495Q8—81no—
496Q8—4721no—
497Q8—4721no—
498Q8—4722no—
499D4—21no—
500D4—21no—
501D4—42no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—81no—
507D4—82no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—4722no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4#1427369911yescharacteristic
524C2×C4#1427369911yescharacteristic
525C2×C4#1427369911yescharacteristic
526C2×C4C2×D4×D5911yescharacteristic
527C2×C4C22×C22⋊D5911yescharacteristic
528C2×C4C2×#1174568711yescharacteristic
529C8—41no—
530C8—41no—
531C8—1181no—
532C8—1181no—
533C8—1181no—
534C8—1181no—
535C8—2361no—
536C8—2361no—
537C8—2361no—
538C8—2361no—
539C22—21no—
540C22—21no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22#1683767012yes—
548C22C2×#1427369911yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—81no—
557C4—4721no—
558C4—4721no—
559C4#1683820911yescharacteristic
560C4#1683821311yescharacteristic
561C4#1683821611yescharacteristic
562C4#1683821711yescharacteristic
563C2—22no—
564C2—81no—
565C2—161no—
566C2#2143364411yescharacteristic
567C2#2143378911yescharacteristic
568C2C2×#1683767011yescharacteristic
subgroupG/H
centerC22C2×#14273699
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
Fitting, prime radicalC59×#3247C2
socleC22×C59C2×#121
hypercenter#3247D59
hypocenterC59#19591
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3247

Complements

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

NH ≅ G/Ntype
1. #21435314C2semidirect
2. #21435214C2semidirect
3. #21433256C2semidirect
4. #21423634C2semidirect
5. #21423627C2semidirect
6. #21423622C2semidirect
7. #21423617C2semidirect
8. #21423510C2semidirect
9. #21423503C2semidirect
10. #21423498C2semidirect
11. #21423493C2semidirect
12. #21422879C2semidirect
13. #21422791C2semidirect
14. C2×#16837150C2semidirect
351. C59#19591semidirect

Maximal subgroups

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

subgroupindex
#214353142
#214352142
#214332562
#214236342
#214236272
#214236222
#214236172
#214235102
#214235032
#214234982
#214234932
#214228792
#214227912
C2×#168371502
C59×#32472
#1959159