Finite Group Databasebeta

← #28622609

Subgroups — #28622609

Order 15104.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837679C211yesmaximal, characteristic, complemented
2C2×C4×#11745671C211yesmaximal, characteristic, complemented
3#21429009C211yesmaximal, characteristic
4#21425296C212yesmaximal, complemented
5#21425097C212yesmaximal, complemented
6#21425058C211yesmaximal, characteristic, complemented
7#21424891C211yesmaximal, characteristic, complemented
8#21424418C212yesmaximal, complemented
9#21424295C212yesmaximal, complemented
10#21424101C211yesmaximal, characteristic, complemented
11C2×C59×#556C211yesmaximal, characteristic, complemented
12C22×#11745671C2211yescharacteristic
13C2×#14273701C2212yes—
14C2×#14273698—21no—
15C2×C4×D236C2211yescharacteristic
16#16837679—24no—
17C4×#11745671—24no—
18C2×#14273660C2212yes—
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2212yes—
23C2×#14273647C2211yescharacteristic
24C2×#14273647C2212yes—
25C2×C22⋊(C59⋊C8)C2212yes—
26C2×C22⋊(C59⋊C8)C2212yes—
27C2×C4×C59⋊C8C2211yescharacteristic
28#16837318—22no—
29#16837272C2211yescharacteristic
30#16837262C2212yes—
31#16837259—21no—
32#16837164C2212yes—
33#16837164C2212yes—
34#16837164C2212yes—
35C59×#556—24no—
36C59×#556—24no—
37C2×C4⋊Q8×C59C2211yescharacteristic
38C2×#126×C59—22no—
39C2×#124×C59—21no—
40C2×C22⋊Q8×C59C2212yes—
41C2×C4×Q8×C59—21no—
42C2×C4×Q8×C59C2211yescharacteristic
43C2×C4×D4×C59C2211yescharacteristic
44C2×#117×C59C2212yes—
45C22×D236C2311yescharacteristic
46C22×C22⋊D59—22no—
47C22×C4×D59—21no—
48C2×#11745671—22no—
49C2×#11745671—22no—
50C2×#11745671—22no—
51C22×C59⋊C8C2311yescharacteristic
52C22×C59⋊C8C2312yes—
53C22×C59⋊C8C2311yescharacteristic
54#14273701—24no—
55#14273701—42no—
56#14273698—44no—
57C4×D236—22no—
58C4×D236—22no—
59C2×C4⋊Dic59—21no—
60C2×C4⋊Dic59—21no—
61C2×C4⋊Dic59C2312yes—
62C2×C4⋊Dic59C2311yescharacteristic
63C2×C22⋊Dic59—22no—
64C2×(C2×D59)⋊C4—22no—
65C2×(C2×D59)⋊C4—21no—
66#14273660—24no—
67#14273658—22no—
68C4⋊(C59⋊C8)—22no—
69#14273651—22no—
70#14273651—42no—
71#14273647—24no—
72#14273647—22no—
73C22⋊(C59⋊C8)—24no—
74C22⋊(C59⋊C8)—42no—
75C4×C59⋊C8—22no—
76C59⋊#95—22no—
77C22×Q8×C59—21no—
78C22×Q8×C59C2311yescharacteristic
79C22×D4×C59D411yescharacteristic
80C23×C4×C59C2312yes—
81C4⋊Q8×C59—22no—
82C4⋊Q8×C59D414yescomplemented
83#126×C59—216no—
84#124×C59—28no—
85C22⋊Q8×C59—24no—
86C22⋊Q8×C59—24no—
87C4×Q8×C59—28no—
88C4×Q8×C59—22no—
89C4×Q8×C59D414yescomplemented
90C4×D4×C59—22no—
91C4×D4×C59—22no—
92#117×C59—24no—
93#117×C59—24no—
94C2×C4⋊C4×C59—22no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59D411yescharacteristic
98C2×C4⋊C4×C59C2311yescharacteristic
99C2×C4⋊C4×C59C2312yes—
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59D412yes—
103C2×C42×C59—22no—
104C2×C42×C59—21no—
105C2×C42×C59C2311yescharacteristic
106C24×C59C2×D412yes—
107C23×D59—21no—
108C2×Q8×C59—24no—
109C2×Q8×C59—21no—
110C2×Q8×C59—21no—
111C2×Q8×C59—21no—
112C2×Q8×C59—21no—
113C2×Q8×C59—21no—
114C2×Q8×C59—42no—
115C2×Q8×C59—42no—
116C2×Q8×C59C2×D412yes—
117C2×D4×C59—22no—
118C2×D4×C59—22no—
119C2×D4×C59—22no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59C2×D412yes—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2411yescharacteristic
133C2×C22⋊D59—44no—
134C2×C22⋊D59—44no—
135C2×C22⋊D59—82no—
136C22×Dic59—21no—
137C22×Dic59—22no—
138C2×D236—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236#4112yes—
144C2×C4×D59—42no—
145C2×C4×D59—42no—
146C2×C4×D59—42no—
147C4⋊C4×C59—28no—
148C4⋊C4×C59—24no—
149C4⋊C4×C59—24no—
150C4⋊C4×C59—22no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—22no—
153C4⋊C4×C59C2×D412yes—
154C4⋊C4×C59C2×D414yes—
155C22⋊C4×C59—28no—
156C22⋊C4×C59—28no—
157C22⋊C4×C59—24no—
158C42×C59—28no—
159C42×C59—24no—
160C42×C59C2×D412yes—
161C42×C59C2×D412yes—
162C22⋊Dic59—44no—
163#11745671—28no—
164(C2×D59)⋊C4—44no—
165(C2×D59)⋊C4—42no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—22no—
169C4⋊Dic59—22no—
170C4⋊Dic59—42no—
171C4⋊Dic59—81no—
172C2×C59⋊C8—21no—
173C2×C59⋊C8—21no—
174C2×C59⋊C8—21no—
175C2×C59⋊C8—22no—
176C2×C59⋊C8—22no—
177C2×C59⋊C8—21no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8—21no—
180C2×C59⋊C8#4112yes—
181C23×C59—22no—
182C23×C59—22no—
183C23×C59—22no—
184C23×C59—22no—
185C23×C59—22no—
186C23×C59—22no—
187C23×C59—22no—
188C23×C59C22×D411yescharacteristic
189C22×D59—22no—
190C22×D59—41no—
191C22×D59—41no—
192C22×D59—41no—
193C22×D59—41no—
194C22×D59—41no—
195C22×D59—41no—
196Q8×C59—22no—
197Q8×C59—22no—
198Q8×C59—41no—
199Q8×C59—41no—
200Q8×C59—48no—
201D4×C59—44no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—22no—
207C2×C4×C59—24no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—42no—
213C2×C4×C59—42no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59#12012yes—
221C2×C4×C59#12012yes—
222C2×C4×C59#13512yes—
223C2×C4×C59AGL(1,ℤ/8)12yes—
224C2×C4×C59C22×D411yescharacteristic
225C2×C4×C59C22×D411yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59H(5,𝔽2)11yescharacteristic
229C2×C4×C592−1+411yescharacteristic
230C22⋊D59—88no—
231C2×Dic59—41no—
232C2×Dic59—42no—
233C2×Dic59—41no—
234C2×Dic59—42no—
235C2×Dic59—41no—
236C2×Dic59—42no—
237D236—22no—
238D236—22no—
239D236—41no—
240D236—41no—
241C4×D59—84no—
242C59⋊C8—22no—
243C59⋊C8—41no—
244C59⋊C8—42no—
245#24119—591nomaximal, Sylow
246C22×C59—42no—
247C22×C59—42no—
248C22×C59—42no—
249C22×C59—42no—
250C22×C59—42no—
251C22×C59—42no—
252C22×C59—42no—
253C22×C59C2×#12011yescharacteristic
254C22×C59#54611yescharacteristic
255C22×C59#54711yescharacteristic
256C22×C59C2×#13511yescharacteristic
257C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C59#57711yescharacteristic
259C22×C59#57811yescharacteristic
260C2×D59—42no—
261C2×D59—42no—
262C2×D59—42no—
263C2×D59—81no—
264C2×D59—81no—
265C2×D59—81no—
266C2×D59—81no—
267C4×C59—22no—
268C4×C59—42no—
269C4×C59—42no—
270C4×C59—44no—
271C4×C59—42no—
272C4×C59#44914yes—
273C4×C59#52511yescharacteristic
274C4×C59#53711yescharacteristic
275C4×C59#54811yescharacteristic
276C4×C59#55011yescharacteristic
277Dic59—81no—
278Dic59—82no—
279C2×#556D5911yescharacteristic, complemented
280C2×#474—591no—
281C2×C4×SD16—591no—
282#2537—591no—
283#2036—592no—
284#2000—592no—
285#1990—591no—
286#1954—591no—
287#1875—592no—
288#1854—592no—
289#1803—591no—
290C2×C59—82no—
291C2×C59#240711yescharacteristic
292C2×C59C2×#44911yescharacteristic
293C2×C59#301311yescharacteristic
294C2×C59#330211yescharacteristic
295C2×C59#332411yescharacteristic
296C2×C59#334011yescharacteristic
297C2×C59#335311yescharacteristic
298D59—82no—
299C22×SD16—591no—
300#556—24no—
301#556—24no—
302C2×C4⋊Q8C2×D5911yescharacteristic
303C2×#126—22no—
304C2×#124—21no—
305C2×C22⋊Q8C2×D5912yes—
306C2×#121—592no—
307C2×#121—1181no—
308C2×C4×Q8—21no—
309C2×C4×Q8C2×D5911yescharacteristic
310C2×C4×D4—591no—
311C2×C4×D4C2×D5911yescharacteristic
312C2×#117C2×D5912yes—
313#474—1184no—
314C4×SD16—1184no—
315C2×#107—592no—
316C2×#106—591no—
317C2×C4⋊C8—591no—
318C2×#103—591no—
319C2×#103—592no—
320C2×#102—591no—
321C2×#102—592no—
322C2×C22⋊C8—592no—
323C2×C22⋊C8—592no—
324C2×C4×C8—591no—
325#394—1182no—
326#384—591no—
327#380—592no—
328#380—1181no—
329#340—592no—
330#340—592no—
331#340—592no—
332C59#2411911yescharacteristic, complemented, Sylow
333C22×Q8—21no—
334C22×Q8C22×D5911yescharacteristic
335C22×D4—591no—
336C22×D4—1182no—
337C22×D4C22⋊D5911yescharacteristic
338C23×C4—1181no—
339C23×C4C22×D5912yes—
340C2×SD16—1182no—
341C2×SD16—1182no—
342C2×SD16—1182no—
343C22×C8—591no—
344C22×C8—591no—
345C22×C8—592no—
346C4⋊Q8—22no—
347C4⋊Q8C22⋊D5914yescomplemented
348#126—216no—
349#124—28no—
350C22⋊Q8—24no—
351C22⋊Q8—24no—
352#121—1184no—
353#121—2362no—
354#121—2364no—
355C4×Q8—28no—
356C4×Q8—22no—
357C4×Q8C22⋊D5914yescomplemented
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—1182no—
361C4×D4—1182no—
362#117—24no—
363#117—24no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—592no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4C22⋊D5911yescharacteristic
372C2×C4⋊C4C22×D5911yescharacteristic
373C2×C4⋊C4C22×D5912yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—1182no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1182no—
379C2×C22⋊C4C22⋊D5912yes—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42C22×D5911yescharacteristic
383#107—1184no—
384#106—1182no—
385C4⋊C8—1182no—
386#103—1182no—
387#103—2362no—
388#102—1182no—
389#102—1184no—
390C22⋊C8—1184no—
391C22⋊C8—2362no—
392C4×C8—1182no—
393#95—1182no—
394C24—1181no—
395C24C2×C22⋊D5912yes—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8C2×C22⋊D5912yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—592no—
409C2×D4—1181no—
410C2×D4—1181no—
411C2×D4—1181no—
412C2×D4—1181no—
413C2×D4—1181no—
414C2×D4—4722no—
415C2×D4—2364no—
416C2×D4—2364no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—1181no—
427C22×C4—1182no—
428C22×C4—2362no—
429C22×C4—2362no—
430C22×C4—2362no—
431C22×C4C2×C22⋊D5912yes—
432C22×C4C2×C22⋊D5911yescharacteristic
433C22×C4C2×C22⋊D5911yescharacteristic
434C22×C4C23×D5911yescharacteristic
435SD16—1188no—
436C2×C8—592no—
437C2×C8—1181no—
438C2×C8—1181no—
439C2×C8—1181no—
440C2×C8—1182no—
441C2×C8—1182no—
442C2×C8—1181no—
443C2×C8—1182no—
444C2×C8—1181no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—1181no—
452C4⋊C4—1181no—
453C4⋊C4—1182no—
454C4⋊C4—1182no—
455C4⋊C4—4721no—
456C4⋊C4—2362no—
457C4⋊C4C2×C22⋊D5912yes—
458C4⋊C4C2×C22⋊D5914yes—
459C22⋊C4—28no—
460C22⋊C4—28no—
461C22⋊C4—24no—
462C22⋊C4—2364no—
463C22⋊C4—2364no—
464C22⋊C4—2362no—
465C42—28no—
466C42—24no—
467C42C2×C22⋊D5912yes—
468C42C2×C22⋊D5912yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—1182no—
477C23—2361no—
478C23—2361no—
479C23—2361no—
480C23—2361no—
481C23—2361no—
482C23—2361no—
483C23C22×C22⋊D5911yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—48no—
487Q8—41no—
488Q8—41no—
489D4—44no—
490D4—1182no—
491D4—1182no—
492D4—4728no—
493D4—2361no—
494D4—2361no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—4724no—
514C2×C4—2361no—
515C2×C4—2361no—
516C2×C4—2361no—
517C2×C4—2362no—
518C2×C4—2362no—
519C2×C4—2362no—
520C2×C4#1427369612yes—
521C2×C4#1427369612yes—
522C2×C4#1427374712yes—
523C2×C4#1427375212yes—
524C2×C4C22×C22⋊D5911yescharacteristic
525C2×C4C22×C22⋊D5911yescharacteristic
526C2×C4#41×D5911yescharacteristic
527C2×C4#41×D5911yescharacteristic
528C2×C4#1427377311yescharacteristic
529C2×C4#1427377411yescharacteristic
530C8—1182no—
531C8—2361no—
532C8—2362no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—4721no—
541C22—4721no—
542C22—4721no—
543C22—4721no—
544C22—2362no—
545C22—2362no—
546C22—2362no—
547C22C2×#1427369611yescharacteristic
548C22#1683804011yescharacteristic
549C22#1683805111yescharacteristic
550C22C2×#1427374711yescharacteristic
551C22C2×#1427375211yescharacteristic
552C22#1683821011yescharacteristic
553C22#1683821711yescharacteristic
554C4—22no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—4722no—
560C4—4721no—
561C4#1683754714yes—
562C4#1683790911yescharacteristic
563C4#1683798011yescharacteristic
564C4#1683805911yescharacteristic
565C4#1683807111yescharacteristic
566C2—82no—
567C2—4722no—
568C2#2142743511yescharacteristic
569C2C2×#1683754711yescharacteristic
570C2#2143341111yescharacteristic
571C2#2143600711yescharacteristic
572C2#2143632111yescharacteristic
573C2#2143650311yescharacteristic
574C2#2143667411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#556C2
socleC23×C59C22×D4
prime radical#21429009C2
hypercenterC2×#556D59
hypocenterC59#24119
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#16837679C2semidirect
2. C2×C4×#11745671C2semidirect
4. #21425296C2semidirect
5. #21425097C2semidirect
6. #21425058C2semidirect
7. #21424891C2semidirect
8. #21424418C2semidirect
9. #21424295C2semidirect
10. #21424101C2semidirect
11. C2×C59×#556C2semidirect
82. C4⋊Q8×C59D4semidirect
89. C4×Q8×C59D4semidirect
279. C2×#556D59semidirect
332. C59#24119semidirect
347. C4⋊Q8C22⋊D59semidirect
357. C4×Q8C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168376792
C2×C4×#117456712
#214290092
#214252962
#214250972
#214250582
#214248912
#214244182
#214242952
#214241012
C2×C59×#5562
#2411959