Finite Group Databasebeta

← #28560716

Subgroups — #28560716

Order 15104.

564 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434679C211yesmaximal, characteristic
2#21433196C211yesmaximal, characteristic
3#21433194C211yesmaximal, characteristic
4#21423003C211yesmaximal, characteristic, complemented
5#21423002C211yesmaximal, characteristic, complemented
6#21422948C211yesmaximal, characteristic, complemented
7#21422947C211yesmaximal, characteristic, complemented
8#21422945C211yesmaximal, characteristic, complemented
9#21422913C211yesmaximal, characteristic, complemented
10#21422912C211yesmaximal, characteristic, complemented
11#21422843C211yesmaximal, characteristic
12#21422515C211yesmaximal, characteristic
13#21422515C211yesmaximal, characteristic
14#21422384C211yesmaximal, characteristic
15C59×#1584C211yesmaximal, characteristic, complemented
16#16837956C2211yescharacteristic
17#16837956C2211yescharacteristic
18#16837806C411yescharacteristic
19#16837780C411yescharacteristic
20#16837751C411yescharacteristic
21#16837751C411yescharacteristic
22#16837644C411yescharacteristic
23#16837642C411yescharacteristic
24#16837642C411yescharacteristic
25#16837640C411yescharacteristic
26#16837533C2211yescharacteristic
27#16837532C2211yescharacteristic
28#16837531—21no—
29#16837530—21no—
30#16837529C2211yescharacteristic
31Q8×C59⋊C8C2211yescharacteristic
32C8×Dic118—21no—
33C8×Dic118C2211yescharacteristic
34#16837526—21no—
35#16837526C2211yescharacteristic
36C4×Dic236C2211yescharacteristic
37C4×#11745672C2211yescharacteristic
38C4×#11745672C2211yescharacteristic
39#16837464—21no—
40#16837462—21no—
41#16837462—21no—
42#16837461—21no—
43#16837454C2211yescharacteristic
44#16837454C2211yescharacteristic
45#16837453C2211yescharacteristic
46C4×#11745659C2211yescharacteristic
47C4×C8×D59C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837152C2211yescharacteristic
50#16837152C2211yescharacteristic
51#16837134C2211yescharacteristic
52#16837134C2211yescharacteristic
53Q8⋊(C59⋊C8)C2211yescharacteristic
54Q8⋊(C59⋊C8)C2211yescharacteristic
55#16837132C2211yescharacteristic
56#16837132C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837124C2211yescharacteristic
59#16837124C2211yescharacteristic
60C59×#446C2211yescharacteristic
61C8×Q8×C59C2211yescharacteristic
62C4×Q16×C59C2211yescharacteristic
63C59×#335C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×Q8⋊C8C2211yescharacteristic
66C59×#322C2211yescharacteristic
67C2×Dic236C2×C411yescharacteristic
68C2×#11745672C2×C411yescharacteristic
69C2×#11745672C2×C411yescharacteristic
70C2×#11745659—21no—
71C2×C8×D59C2×C411yescharacteristic
72C2×C8×D59C2×C411yescharacteristic
73#14273726C2×C411yescharacteristic
74#14273726C2×C411yescharacteristic
75#14273725C2×C411yescharacteristic
76#14273725C2×C411yescharacteristic
77#14273717—21no—
78#14273717—21no—
79#14273707C2×C411yescharacteristic
80#14273707C2×C411yescharacteristic
81#14273705—21no—
82#14273705—21no—
83#14273703C2×C411yescharacteristic
84#14273703C2×C411yescharacteristic
85C4×Dic118—21no—
86C4×Dic118—21no—
87C4×Dic118C2311yescharacteristic
88C4×Dic118C2311yescharacteristic
89#14273684—21no—
90#14273684—21no—
91C42×D59C2311yescharacteristic
92#14273661C2×C411yescharacteristic
93#14273660C2×C411yescharacteristic
94#14273660C2×C411yescharacteristic
95C4⋊(C59⋊C8)—21no—
96C4⋊(C59⋊C8)—21no—
97C4⋊(C59⋊C8)C2311yescharacteristic
98C4⋊(C59⋊C8)C2311yescharacteristic
99Dic59⋊C8—21no—
100Dic59⋊C8—21no—
101Dic59⋊C8—21no—
102Dic59⋊C8—21no—
103Dic59⋊C8D411yescharacteristic
104Dic59⋊C8D411yescharacteristic
105#14273652C2×C411yescharacteristic
106#14273652C2×C411yescharacteristic
107#14273651C2×C411yescharacteristic
108#14273651C2×C411yescharacteristic
109#14273650C2×C411yescharacteristic
110#14273650C2×C411yescharacteristic
111(C2×D59)⋊C8—21no—
112(C2×D59)⋊C8—21no—
113(C2×D59)⋊C8D411yescharacteristic
114(C2×D59)⋊C8D411yescharacteristic
115#14273638—21no—
116#14273637—21no—
117#14273637—21no—
118#14273637—21no—
119C4×C59⋊C8—21no—
120C4×C59⋊C8C2311yescharacteristic
121C4×C59⋊C8C2311yescharacteristic
122C4×C59⋊C8C2311yescharacteristic
123C8×Dic59—21no—
124C8×Dic59—21no—
125C8×Dic59—21no—
126C8×Dic59—21no—
127C8×Dic59C2×C411yescharacteristic
128C8×Dic59C2×C411yescharacteristic
129C2×Q16×C59C2×C411yescharacteristic
130C4×Q8×C59C2311yescharacteristic
131C4×Q8×C59C2311yescharacteristic
132#107×C59C2×C411yescharacteristic
133C4⋊C8×C59—21no—
134C4⋊C8×C59—21no—
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#103×C59C2×C411yescharacteristic
138#103×C59C2×C411yescharacteristic
139#97×C59—21no—
140C4×C8×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×Q8×C59C22×C411yescharacteristic
146C2×C4×D59C22×C411yescharacteristic
147C2×C4×D59C2×D411yescharacteristic
148C2×C4×D59#4111yescharacteristic
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118C22×C411yescharacteristic
152C2×Dic118C22×C411yescharacteristic
153Q16×C59—41no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59C22×C411yescharacteristic
158C2×C8×C59C22×C411yescharacteristic
159C2×C8×C59C2×D411yescharacteristic
160C2×C8×C59C2×D411yescharacteristic
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59C22×C411yescharacteristic
164C4⋊C4×C59C22×C411yescharacteristic
165C42×C59—21no—
166C42×C59—21no—
167C42×C59C2411yescharacteristic
168#11745672—22no—
169#11745672—22no—
170(C2×D59)⋊C4—21no—
171(C2×D59)⋊C4—21no—
172(C2×D59)⋊C4—21no—
173(C2×D59)⋊C4—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59C22×C411yescharacteristic
177C4⋊Dic59C22×C411yescharacteristic
178Dic59⋊C4—21no—
179Dic59⋊C4—21no—
180Dic59⋊C4—21no—
181Dic59⋊C4—21no—
182Dic59⋊C4—21no—
183Dic59⋊C4—21no—
184Dic59⋊C4—21no—
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188C4×Dic59—21no—
189C4×Dic59—21no—
190C4×Dic59C22×C411yescharacteristic
191C4×Dic59C2×D411yescharacteristic
192C4×Dic59#4111yescharacteristic
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8C22×C411yescharacteristic
197C2×C59⋊C8C22×C411yescharacteristic
198C2×C59⋊C8C2×D411yescharacteristic
199C2×C59⋊C8C2×D411yescharacteristic
200Dic236—41no—
201#11745659—42no—
202C8×D59—22no—
203C8×D59—41no—
204C22×D59C4×D411yescharacteristic
205Q8×C59—21no—
206Q8×C59—21no—
207Q8×C59—21no—
208Q8×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59C23×C411yescharacteristic
214C2×C4×C59C22×D411yescharacteristic
215C2×C4×C59C2×#4111yescharacteristic
216C2×Dic59—21no—
217C2×Dic59—21no—
218C2×Dic59—21no—
219C2×Dic59—21no—
220C2×Dic59C4×D411yescharacteristic
221C2×Dic59C4×D411yescharacteristic
222C2×Dic59C4×D411yescharacteristic
223C4×D59—21no—
224C4×D59—21no—
225C4×D59—21no—
226C4×D59—21no—
227C4×D59—22no—
228Dic118—21no—
229Dic118—21no—
230Dic118—21no—
231Dic118—21no—
232Dic118—42no—
233Dic118—42no—
234C8×C59—21no—
235C8×C59—21no—
236C8×C59—21no—
237C8×C59—21no—
238C8×C59—41no—
239C8×C59—41no—
240C8×C59—41no—
241C59⋊C8—21no—
242C59⋊C8—21no—
243C59⋊C8—21no—
244C59⋊C8—41no—
245C59⋊C8—41no—
246C59⋊C8—41no—
247C59⋊C8#13112yes—
248#18090—591nomaximal, Sylow
249C22×C59C2×C4×D411yescharacteristic
250C2×D59—21no—
251C2×D59—21no—
252C2×D59—21no—
253C4×C59—21no—
254C4×C59—41no—
255C4×C59—41no—
256C4×C59—41no—
257C4×C59—41no—
258C4×C59C2×#13111yescharacteristic
259C4×C59#56811yescharacteristic
260C4×C59#57811yescharacteristic
261C4×C59#57811yescharacteristic
262Dic59—21no—
263Dic59—21no—
264Dic59—41no—
265Dic59—41no—
266Dic59—41no—
267Dic59—41no—
268Dic59#44412yes—
269#3170—591no—
270#2986—591no—
271#2986—591no—
272#1593—591no—
273#1593—591no—
274#1584—591no—
275#1584—591no—
276#1584—591no—
277#1584D5911yescharacteristic, complemented
278#1579—591no—
279#1579—591no—
280#1564—591no—
281#1485—591no—
282#1485—591no—
283#1451—591no—
284C2×C59#293511yescharacteristic
285C2×C59#295011yescharacteristic
286C2×C59C2×#44411yescharacteristic
287D59—41no—
288#533—591no—
289#533—591no—
290#500—591no—
291#494—591no—
292#487—591no—
293#487—591no—
294#467—591no—
295#467—591no—
296#467—591no—
297#467—591no—
298#446—591no—
299#446—591no—
300#446—591no—
301#446—1181no—
302#446—1181no—
303#446C2×D5911yescharacteristic
304C8×Q8—591no—
305C8×Q8—591no—
306C8×Q8—591no—
307C8×Q8—1181no—
308C8×Q8—1181no—
309C8×Q8C2×D5911yescharacteristic
310C4×Q16—591no—
311C4×Q16—591no—
312C4×Q16—591no—
313C4×Q16C2×D5911yescharacteristic
314#433—1181no—
315#433—1181no—
316#433—1181no—
317#433—1181no—
318#431—591no—
319#431—591no—
320#431—591no—
321C4×M16—591no—
322C2×C4×C8—591no—
323#335—591no—
324#335C2×D5911yescharacteristic
325#334—591no—
326#334—591no—
327Q8⋊C8—591no—
328Q8⋊C8—591no—
329Q8⋊C8—591no—
330Q8⋊C8—591no—
331Q8⋊C8—591no—
332Q8⋊C8—591no—
333Q8⋊C8C2×D5911yescharacteristic
334Q8⋊C8C2×D5911yescharacteristic
335#322—591no—
336#322—591no—
337#322—591no—
338#322C2×D5911yescharacteristic
339C59#1809011yescharacteristic, complemented, Sylow
340C2×Q16—591no—
341C2×Q16—591no—
342C2×Q16—591no—
343C2×Q16C4×D5911yescharacteristic
344C2×M16—1181no—
345C22×C8—591no—
346C22×C8—591no—
347C4⋊Q8—591no—
348C4⋊Q8—591no—
349C4⋊Q8—591no—
350C4⋊Q8—591no—
351#125—1181no—
352#125—1181no—
353C22⋊Q8—591no—
354C22⋊Q8—591no—
355C22⋊Q8—591no—
356C22⋊Q8—591no—
357C22⋊Q8—1181no—
358C22⋊Q8—1181no—
359C4×Q8—591no—
360C4×Q8—591no—
361C4×Q8—1181no—
362C4×Q8—1181no—
363C4×Q8C22×D5911yescharacteristic
364C4×Q8C22×D5911yescharacteristic
365#117—1181no—
366#117—1181no—
367C2×C42—591no—
368#107—591no—
369#107—591no—
370#107—591no—
371#107C4×D5911yescharacteristic
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—591no—
375C4⋊C8—591no—
376C4⋊C8—591no—
377C4⋊C8—591no—
378C4⋊C8—1181no—
379C4⋊C8—1181no—
380C4⋊C8—1181no—
381C4⋊C8—1181no—
382C4⋊C8—1181no—
383C4⋊C8—1181no—
384C4⋊C8C22×D5911yescharacteristic
385C4⋊C8C22×D5911yescharacteristic
386#103—591no—
387#103—591no—
388#103—591no—
389#103—591no—
390#103—591no—
391#103—591no—
392#103C4×D5911yescharacteristic
393#103C4×D5911yescharacteristic
394C22⋊C8—591no—
395C22⋊C8—591no—
396C22⋊C8—1181no—
397C22⋊C8—1181no—
398#97—21no—
399#97—1181no—
400#97—1181no—
401#97—1181no—
402#97—1181no—
403C4×C8—21no—
404C4×C8—591no—
405C4×C8—591no—
406C4×C8—591no—
407C4×C8—591no—
408C4×C8—591no—
409C4×C8—1181no—
410C4×C8—1181no—
411C4×C8—1181no—
412C4×C8—1181no—
413C4×C8—1181no—
414C4×C8C22×D5911yescharacteristic
415C4×C8C22×D5911yescharacteristic
416C4×C8C22×D5911yescharacteristic
417C2×Q8—591no—
418C2×Q8—591no—
419C2×Q8—1181no—
420C2×Q8—1181no—
421C2×Q8C2×C4×D5911yescharacteristic
422C2×Q8C2×C4×D5911yescharacteristic
423C22×C4—591no—
424C22×C4—591no—
425C22×C4—591no—
426Q16—41no—
427Q16—1182no—
428Q16—1182no—
429Q16—2361no—
430M16—2362no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—591no—
435C2×C8—591no—
436C2×C8—591no—
437C2×C8—591no—
438C2×C8—1182no—
439C2×C8—1181no—
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8—2361no—
443C2×C8C2×C4×D5911yescharacteristic
444C2×C8C2×C4×D5911yescharacteristic
445C2×C8D4×D5911yescharacteristic
446C2×C8D4×D5911yescharacteristic
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—591no—
450C4⋊C4—591no—
451C4⋊C4—1181no—
452C4⋊C4—1181no—
453C4⋊C4—1181no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4C2×C4×D5911yescharacteristic
464C4⋊C4C2×C4×D5911yescharacteristic
465C22⋊C4—1181no—
466C22⋊C4—1181no—
467C22⋊C4—1181no—
468C22⋊C4—1181no—
469C42—21no—
470C42—21no—
471C42—591no—
472C42—591no—
473C42—591no—
474C42—1181no—
475C42—1181no—
476C42C23×D5911yescharacteristic
477C23—591no—
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—1181no—
483Q8—1181no—
484Q8—1181no—
485Q8—1181no—
486Q8—2362no—
487Q8—2362no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—591no—
493C2×C4—591no—
494C2×C4—591no—
495C2×C4—1182no—
496C2×C4—1181no—
497C2×C4—1181no—
498C2×C4—1181no—
499C2×C4—1181no—
500C2×C4—1181no—
501C2×C4—1181no—
502C2×C4—1181no—
503C2×C4—1181no—
504C2×C4C22×C4×D5911yescharacteristic
505C2×C4C2×D4×D5911yescharacteristic
506C2×C4#41×D5911yescharacteristic
507C8—21no—
508C8—21no—
509C8—21no—
510C8—21no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—592no—
515C8—1181no—
516C8—1181no—
517C8—1181no—
518C8—2361no—
519C8—2361no—
520C8—2361no—
521C22—1181no—
522C22—1181no—
523C22—1181no—
524C22C4×D4×D5911yescharacteristic
525C4—21no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—592no—
531C4—1181no—
532C4—1181no—
533C4—2361no—
534C4—2361no—
535C4—2361no—
536C4—2361no—
537C4#131×D5911yescharacteristic
538C4#1683816811yescharacteristic
539C4#1683821511yescharacteristic
540C4#1683821611yescharacteristic
541C2—2361no—
542C2#2143283711yescharacteristic
543C2#2143295911yescharacteristic
544C2#444×D5911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1584C2
socleC22×C59C2×C4×D4
prime radicalC22×D59C4×D4
cosocleC42×C59C24
hypercenter#1584D59
hypocenterC59#18090
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #21423003C2semidirect
5. #21423002C2semidirect
6. #21422948C2semidirect
7. #21422947C2semidirect
8. #21422945C2semidirect
9. #21422913C2semidirect
10. #21422912C2semidirect
15. C59×#1584C2semidirect
277. #1584D59semidirect
339. C59#18090semidirect

Maximal subgroups

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

subgroupindex
#214346792
#214331962
#214331942
#214230032
#214230022
#214229482
#214229472
#214229452
#214229132
#214229122
#214228432
#214225152
#214223842
C59×#15842
#1809059