Finite Group Databasebeta

← #28559599

Subgroups — #28559599

Order 15104.

618 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434114C211yesmaximal, characteristic
2#21433100C211yesmaximal, characteristic, complemented
3#21422991C211yesmaximal, characteristic, complemented
4#21422986C211yesmaximal, characteristic, complemented
5#21422944C211yesmaximal, characteristic, complemented
6#21422932C211yesmaximal, characteristic, complemented
7#21422930C211yesmaximal, characteristic, complemented
8#21422925C211yesmaximal, characteristic, complemented
9C8×#11745670C211yesmaximal, characteristic, complemented
10C8×#11745669C211yesmaximal, characteristic, complemented
11#21422877C211yesmaximal, characteristic, complemented
12#21422554C211yesmaximal, characteristic, complemented
13#21422478C211yesmaximal, characteristic, complemented
14M16×C59⋊C8C211yesmaximal, characteristic, complemented
15C59×#2970C211yesmaximal, characteristic, complemented
16#16837936C2211yescharacteristic
17#16837743C411yescharacteristic
18#16837726C411yescharacteristic
19#16837709C411yescharacteristic
20#16837702C411yescharacteristic
21#16837624C411yescharacteristic
22#16837617C411yescharacteristic
23#16837564C411yescharacteristic
24#16837547C411yescharacteristic
25#16837532C2211yescharacteristic
26C8×Dic118C2211yescharacteristic
27C4×#11745672C2211yescharacteristic
28C4×#11745671C2211yescharacteristic
29C4×#11745670C2211yescharacteristic
30C4×#11745669C2211yescharacteristic
31#16837481C2211yescharacteristic
32#16837478—21no—
33#16837471—21no—
34C8×D236C2211yescharacteristic
35#16837464—21no—
36#16837455C2211yescharacteristic
37#16837455C2211yescharacteristic
38#16837359—21no—
39C2×C4×C59⋊C8C2211yescharacteristic
40#16837154C2211yescharacteristic
41#16837154C2211yescharacteristic
42#16837152C2211yescharacteristic
43#16837152C2211yescharacteristic
44Q8⋊(C59⋊C8)C2211yescharacteristic
45Q8⋊(C59⋊C8)C2211yescharacteristic
46#16837132C2211yescharacteristic
47#16837132C2211yescharacteristic
48D4⋊(C59⋊C8)C2211yescharacteristic
49D4⋊(C59⋊C8)C2211yescharacteristic
50#16837129C2211yescharacteristic
51#16837129C2211yescharacteristic
52#16837123C2211yescharacteristic
53#16837123C2211yescharacteristic
54C8×C59⋊C8C2211yescharacteristic
55C8×C59⋊C8C2211yescharacteristic
56C4×#41×C59C2211yescharacteristic
57C59×#446C2211yescharacteristic
58C59×#446C2211yescharacteristic
59C59×#435—21no—
60C59×#435—21no—
61C8×D4×C59C2211yescharacteristic
62C8×D4×C59C2211yescharacteristic
63C59×#432—21no—
64C59×#432C2211yescharacteristic
65C4×M16×C59C2211yescharacteristic
66C2×#97×C59—21no—
67C2×#11745672C2×C411yescharacteristic
68C2×#11745670C2×C411yescharacteristic
69C2×#11745671C2×C411yescharacteristic
70C2×#11745669C2×C411yescharacteristic
71C2×#11745659—21no—
72C22×C59⋊C8—21no—
73C2×C8×D59—21no—
74#14273721—21no—
75#14273719C2×C411yescharacteristic
76#14273715C2×C411yescharacteristic
77#14273710—21no—
78#14273706C2×C411yescharacteristic
79#14273701C2×C411yescharacteristic
80C4×Dic118C2311yescharacteristic
81C4×D236C2311yescharacteristic
82C4×C22⋊D59—21no—
83#14273684—21no—
84#14273660C2×C411yescharacteristic
85#14273660C2×C411yescharacteristic
86#14273658C2×C411yescharacteristic
87#14273658C2×C411yescharacteristic
88C4⋊(C59⋊C8)C2×C411yescharacteristic
89C4⋊(C59⋊C8)C2×C411yescharacteristic
90C4⋊(C59⋊C8)C2311yescharacteristic
91C4⋊(C59⋊C8)C2311yescharacteristic
92Dic59⋊C8—21no—
93Dic59⋊C8—21no—
94#14273651C2×C411yescharacteristic
95#14273651C2×C411yescharacteristic
96#14273650C2×C411yescharacteristic
97#14273650C2×C411yescharacteristic
98#14273648C2×C411yescharacteristic
99#14273648C2×C411yescharacteristic
100#14273647C2×C411yescharacteristic
101#14273647C2×C411yescharacteristic
102C22⋊(C59⋊C8)—21no—
103C22⋊(C59⋊C8)C2×C411yescharacteristic
104C22⋊(C59⋊C8)C2×C411yescharacteristic
105(C2×D59)⋊C8—21no—
106(C2×D59)⋊C8—21no—
107#14273637—21no—
108C4×C59⋊C8—21no—
109C4×C59⋊C8—21no—
110C4×C59⋊C8C2311yescharacteristic
111C4×C59⋊C8C2311yescharacteristic
112C4×C59⋊C8C2311yescharacteristic
113C4×C59⋊C8C2311yescharacteristic
114C8×Dic59—21no—
115C2×#41×C59C2×C411yescharacteristic
116C2×M16×C59D411yescharacteristic
117C2×M16×C59D411yescharacteristic
118C22×C8×C59—21no—
119C22×C8×C59—21no—
120C4×Q8×C59C2×C411yescharacteristic
121C4×Q8×C59C2311yescharacteristic
122C4×D4×C59—21no—
123C4×D4×C59—21no—
124C4×D4×C59C2×C411yescharacteristic
125C4×D4×C59C2311yescharacteristic
126#117×C59—21no—
127#117×C59C2×C411yescharacteristic
128C2×C42×C59—21no—
129C2×C42×C59C2311yescharacteristic
130C4⋊C8×C59—21no—
131C4⋊C8×C59—21no—
132C4⋊C8×C59C2311yescharacteristic
133C4⋊C8×C59C2311yescharacteristic
134C22⋊C8×C59—21no—
135C22⋊C8×C59—21no—
136C22⋊C8×C59—21no—
137#97×C59—21no—
138#97×C59—21no—
139#97×C59—21no—
140#97×C59D411yescharacteristic
141#97×C59D411yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144#41×C59—22no—
145#41×C59—41no—
146C2×Q8×C59C22×C411yescharacteristic
147C2×D4×C59—21no—
148C2×D4×C59C22×C411yescharacteristic
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59C22×C411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59#4111yescharacteristic
155C2×C22⋊D59—21no—
156C2×D236C22×C411yescharacteristic
157C2×C4×D59—21no—
158C2×Dic118C22×C411yescharacteristic
159M16×C59—22no—
160M16×C59—22no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—42no—
164C2×C8×C59—42no—
165C2×C8×C59C2×D411yescharacteristic
166C2×C8×C59C2×D411yescharacteristic
167C2×C8×C59C2×D411yescharacteristic
168C2×C8×C59C2×D411yescharacteristic
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59C22×C411yescharacteristic
172C4⋊C4×C59C22×C411yescharacteristic
173C22⋊C4×C59—21no—
174C22⋊C4×C59—21no—
175C22⋊C4×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59C22×C411yescharacteristic
180C42×C59C2×D411yescharacteristic
181C42×C59#4111yescharacteristic
182C42×C59C2411yescharacteristic
183C22⋊Dic59—21no—
184#11745672—41no—
185#11745671—41no—
186#11745670—22no—
187#11745669—22no—
188(C2×D59)⋊C4—21no—
189(C2×D59)⋊C4—21no—
190(C2×D59)⋊C4—21no—
191(C2×D59)⋊C4—21no—
192C4⋊Dic59C22×C411yescharacteristic
193C4⋊Dic59C22×C411yescharacteristic
194Dic59⋊C4—21no—
195Dic59⋊C4—21no—
196Dic59⋊C4—21no—
197Dic59⋊C4—21no—
198C4×Dic59—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—42no—
204C2×C59⋊C8C22×C411yescharacteristic
205C2×C59⋊C8C22×C411yescharacteristic
206C2×C59⋊C8C22×C411yescharacteristic
207C2×C59⋊C8C22×C411yescharacteristic
208#11745659—42no—
209C8×D59—42no—
210C23×C59—21no—
211C23×C59C4×D411yescharacteristic
212C22×D59—21no—
213Q8×C59—21no—
214Q8×C59—21no—
215D4×C59—21no—
216D4×C59—42no—
217D4×C59#13112yes—
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—42no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—81no—
230C2×C4×C59C4×D411yescharacteristic
231C2×C4×C59C4×D411yescharacteristic
232C2×C4×C59C4×D411yescharacteristic
233C2×C4×C59C23×C411yescharacteristic
234C2×C4×C59C22×D411yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C22⋊D59—81no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240D236—21no—
241D236—21no—
242C4×D59—41no—
243C4×D59—41no—
244Dic118—21no—
245Dic118—21no—
246C8×C59—21no—
247C8×C59—21no—
248C8×C59—41no—
249C8×C59—41no—
250C8×C59#13512yes—
251C8×C59#13512yes—
252C59⋊C8—22no—
253C59⋊C8—21no—
254C59⋊C8—21no—
255C59⋊C8—21no—
256C59⋊C8—21no—
257C59⋊C8—41no—
258C59⋊C8—41no—
259C59⋊C8—41no—
260#17996—591nomaximal, Sylow
261C22×C59—21no—
262C22×C59—22no—
263C22×C59—21no—
264C22×C59—21no—
265C22×C59—41no—
266C22×C59—41no—
267C22×C59C2×C4×D411yescharacteristic
268C2×D59—41no—
269C2×D59—41no—
270C2×D59—41no—
271C4×C59—21no—
272C4×C59—21no—
273C4×C59—21no—
274C4×C59—21no—
275C4×C59—41no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59C2×#13111yescharacteristic
279C4×C59#56811yescharacteristic
280C4×C59C2×#13511yescharacteristic
281C4×C59C2×#13511yescharacteristic
282Dic59—41no—
283Dic59—41no—
284Dic59—41no—
285#3102—591no—
286#2970D5911yescharacteristic, complemented
287#2969—591no—
288#1591—591no—
289#1590—591no—
290#1584—591no—
291#1582—591no—
292#1582—591no—
293#1581—591no—
294C8×SD16—591no—
295C8×D8—591no—
296#1572—591no—
297#1494—591no—
298#1476—591no—
299C8×M16—591no—
300C2×C59—41no—
301C2×C59—42no—
302C2×C59#293311yescharacteristic
303C2×C59C4×#13511yescharacteristic
304C2×C59#295611yescharacteristic
305D59—81no—
306#529—591no—
307C4×#41C2×D5911yescharacteristic
308#485—591no—
309#482—591no—
310#479—591no—
311#478—591no—
312#464—591no—
313#463—591no—
314#452—591no—
315#449—591no—
316#446—591no—
317#446C2×D5911yescharacteristic
318#446C2×D5911yescharacteristic
319C8×Q8—591no—
320C4×Q16—591no—
321C4×SD16—591no—
322C4×SD16—591no—
323C4×D8—591no—
324#436—591no—
325#436—1181no—
326#435—21no—
327#435—21no—
328#435—1181no—
329C8×D4—591no—
330C8×D4C2×D5911yescharacteristic
331C8×D4C2×D5911yescharacteristic
332#433—1181no—
333#432—21no—
334#432C2×D5911yescharacteristic
335#431—591no—
336#431—591no—
337#405—1181no—
338C4×M16C2×D5911yescharacteristic
339C2×#97—21no—
340C2×C4×C8—591no—
341#335—591no—
342#335—591no—
343#334—591no—
344#334—591no—
345Q8⋊C8—591no—
346Q8⋊C8—591no—
347Q8⋊C8—591no—
348Q8⋊C8—591no—
349D4⋊C8—591no—
350D4⋊C8—591no—
351D4⋊C8—591no—
352D4⋊C8—591no—
353#322—591no—
354#322—591no—
355C82—591no—
356C82—591no—
357C59#1799611yescharacteristic, complemented, Sylow
358C2×#41C4×D5911yescharacteristic
359C2×Q16—591no—
360C2×SD16—591no—
361C2×SD16—591no—
362C2×D8—591no—
363C2×M16—1181no—
364C2×M16C22⋊D5911yescharacteristic
365C2×M16C22⋊D5911yescharacteristic
366C22×C8—21no—
367C22×C8—21no—
368C22×C8—1181no—
369C22×C8—1181no—
370#126—1181no—
371#125—591no—
372#124—591no—
373#123—1181no—
374C22⋊Q8—591no—
375#121—591no—
376C4×Q8—591no—
377C4×Q8C4×D5911yescharacteristic
378C4×Q8C22×D5911yescharacteristic
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—591no—
382C4×D4—1181no—
383C4×D4C4×D5911yescharacteristic
384C4×D4C22×D5911yescharacteristic
385#117—21no—
386#117—1181no—
387#117C4×D5911yescharacteristic
388C2×C42—21no—
389C2×C42C22×D5911yescharacteristic
390#107—591no—
391#107—591no—
392#106—591no—
393#106—591no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—591no—
397C4⋊C8—591no—
398C4⋊C8—591no—
399C4⋊C8—591no—
400C4⋊C8—1181no—
401C4⋊C8—1181no—
402C4⋊C8C22×D5911yescharacteristic
403C4⋊C8C22×D5911yescharacteristic
404#103—591no—
405#103—591no—
406#103—591no—
407#103—591no—
408#102—591no—
409#102—591no—
410#102—591no—
411#102—591no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8—591no—
416C22⋊C8—591no—
417C22⋊C8—1181no—
418C22⋊C8—1181no—
419C22⋊C8—1181no—
420#97—21no—
421#97—21no—
422#97—21no—
423#97—1181no—
424#97C22⋊D5911yescharacteristic
425#97C22⋊D5911yescharacteristic
426C4×C8—591no—
427C4×C8—591no—
428C4×C8—591no—
429C4×C8—591no—
430C4×C8—1181no—
431C4×C8—1181no—
432C4×C8—1181no—
433C4×C8C22×D5911yescharacteristic
434C4×C8C22×D5911yescharacteristic
435#41—22no—
436#41—41no—
437C2×Q8—591no—
438C2×Q8C2×C4×D5911yescharacteristic
439C2×D4—21no—
440C2×D4—591no—
441C2×D4—1181no—
442C2×D4C2×C4×D5911yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1181no—
447C22×C4C2×C4×D5911yescharacteristic
448C22×C4#1174568511yescharacteristic
449C22×C4C2×C22⋊D5911yescharacteristic
450Q16—2361no—
451SD16—1182no—
452SD16—2361no—
453D8—1182no—
454M16—22no—
455M16—22no—
456M16—2362no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—591no—
462C2×C8—591no—
463C2×C8—591no—
464C2×C8—591no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—2362no—
470C2×C8—2362no—
471C2×C8C2×C22⋊D5911yescharacteristic
472C2×C8C2×C22⋊D5911yescharacteristic
473C2×C8C2×C22⋊D5911yescharacteristic
474C2×C8C2×C22⋊D5911yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—591no—
478C4⋊C4—591no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4C2×C4×D5911yescharacteristic
484C4⋊C4C2×C4×D5911yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—1181no—
489C22⋊C4—1181no—
490C22⋊C4—1181no—
491C22⋊C4—1181no—
492C22⋊C4—1181no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—1181no—
497C42C2×C4×D5911yescharacteristic
498C42#1174568511yescharacteristic
499C42C2×C22⋊D5911yescharacteristic
500C42C23×D5911yescharacteristic
501C23—21no—
502C23—1181no—
503C23C4×C22⋊D5911yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—1181no—
507Q8—1181no—
508D4—21no—
509D4—42no—
510D4—1181no—
511D4—1181no—
512D4—4721no—
513D4#1427373512yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—81no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4C4×C22⋊D5911yescharacteristic
532C2×C4C4×C22⋊D5911yescharacteristic
533C2×C4C4×C22⋊D5911yescharacteristic
534C2×C4C22×C4×D5911yescharacteristic
535C2×C4C22×C22⋊D5911yescharacteristic
536C2×C4C2×#1174568511yescharacteristic
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—1182no—
542C8—1181no—
543C8—1181no—
544C8—1181no—
545C8—1181no—
546C8—2361no—
547C8—2361no—
548C8—2361no—
549C8#1427374712yes—
550C8#1427374712yes—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—41no—
556C22—41no—
557C22—2361no—
558C22—2361no—
559C22—2361no—
560C22C2×C4×C22⋊D5911yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—2361no—
569C4—2361no—
570C4—2361no—
571C4C2×#1427373511yescharacteristic
572C4#1683816811yescharacteristic
573C4C2×#1427374711yescharacteristic
574C4C2×#1427374711yescharacteristic
575C2—41no—
576C2—42no—
577C2—4721no—
578C2#2143282111yescharacteristic
579C2C4×#1427374711yescharacteristic
580C2#2143300311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#2970C2
socleC22×C59C2×C4×D4
prime radical#16837547C4
cosocleC42×C59C24
hypercenter#2970D59
hypocenterC59#17996
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #21433100C2semidirect
3. #21422991C2semidirect
4. #21422986C2semidirect
5. #21422944C2semidirect
6. #21422932C2semidirect
7. #21422930C2semidirect
8. #21422925C2semidirect
9. C8×#11745670C2semidirect
10. C8×#11745669C2semidirect
11. #21422877C2semidirect
12. #21422554C2semidirect
13. #21422478C2semidirect
14. M16×C59⋊C8C2semidirect
15. C59×#2970C2semidirect
286. #2970D59semidirect
357. C59#17996semidirect

Maximal subgroups

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

subgroupindex
#214341142
#214331002
#214229912
#214229862
#214229442
#214229322
#214229302
#214229252
C8×#117456702
C8×#117456692
#214228772
#214225542
#214224782
M16×C59⋊C82
C59×#29702
#1799659