Finite Group Databasebeta

← #28624815

Subgroups — #28624815

Order 15104.

1162 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837699C212yesmaximal, complemented
2C4×#14273701C211yesmaximal, characteristic
3#21425431C212yesmaximal, complemented
4#21425239C212yesmaximal, complemented
5#21425054C212yesmaximal, complemented
6#21424413C211yesmaximal, characteristic
7#21424413C211yesmaximal, characteristic
8#21424289C211yesmaximal, characteristic, complemented
9#21424284C211yesmaximal, characteristic, complemented
10#21424076C211yesmaximal, characteristic, complemented
11C22×#125×C59C211yesmaximal, characteristic, complemented
12C2×#14273701C2211yescharacteristic
13C2×C4×D236C2211yescharacteristic
14C2×C4×C22⋊D59—21no—
15#16837699—28no—
16C2×#14273660C2212yes—
17C2×#14273658C2212yes—
18C2×C4⋊(C59⋊C8)C2212yes—
19C2×#14273647C2212yes—
20C2×#14273647C2212yes—
21C2×C22⋊(C59⋊C8)C2211yescharacteristic
22C2×C22⋊(C59⋊C8)C2211yescharacteristic
23#16837304C2211yescharacteristic
24#16837300C2211yescharacteristic
25#16837293—21no—
26#16837281—21no—
27#16837262C2211yescharacteristic
28C4×C4⋊Dic59C2211yescharacteristic
29C4×(C2×D59)⋊C4—21no—
30#16837164C2212yes—
31#16837164C2212yes—
32#16837164C2212yes—
33#16837164C2212yes—
34#16837157C2212yes—
35C2×#125×C59—28no—
36C2×#125×C59—21no—
37C2×#125×C59—21no—
38C2×#125×C59C2212yes—
39C2×#125×C59C2212yes—
40C22×C4⋊C4×C59—21no—
41C22×C4⋊C4×C59—21no—
42C22×C4⋊C4×C59C2211yescharacteristic
43C22×C4⋊C4×C59C2211yescharacteristic
44C22×C42×C59C2211yescharacteristic
45C22×D236C2311yescharacteristic
46C22×C22⋊D59—21no—
47C22×C4×D59—21no—
48C22×C59⋊C8C2312yes—
49C22×C59⋊C8C2312yes—
50#14273701—22no—
51#14273701Q814yes—
52C4×D236—24no—
53C4×C22⋊D59—44no—
54C2×C4⋊Dic59C2311yescharacteristic
55C2×C4⋊Dic59C2311yescharacteristic
56C2×C4⋊Dic59C2311yescharacteristic
57C2×Dic59⋊C4—21no—
58C2×C22⋊Dic59—21no—
59C2×(C2×D59)⋊C4—21no—
60C2×(C2×D59)⋊C4—21no—
61C2×(C2×D59)⋊C4—21no—
62C2×(C2×D59)⋊C4—21no—
63C2×C4×Dic59—21no—
64#14273660—24no—
65#14273658—24no—
66C4⋊(C59⋊C8)—24no—
67#14273647—24no—
68#14273647—24no—
69C22⋊(C59⋊C8)—22no—
70C22⋊(C59⋊C8)—22no—
71C59⋊#95—21no—
72C59⋊#95—21no—
73C59⋊#95—21no—
74C23×C4×C59—21no—
75C23×C4×C59—21no—
76C23×C4×C59D411yescharacteristic
77C23×C4×C59D411yescharacteristic
78C23×C4×C59C2311yescharacteristic
79#125×C59—28no—
80#125×C59—28no—
81#125×C59—24no—
82#125×C59—24no—
83#125×C59—24no—
84#125×C59D418yescomplemented
85C2×C4⋊C4×C59—28no—
86C2×C4⋊C4×C59—28no—
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59—22no—
89C2×C4⋊C4×C59—22no—
90C2×C4⋊C4×C59—22no—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—24no—
94C2×C4⋊C4×C59—24no—
95C2×C4⋊C4×C59C2312yes—
96C2×C4⋊C4×C59C2312yes—
97C2×C42×C59—24no—
98C2×C42×C59D411yescharacteristic
99C2×C42×C59D411yescharacteristic
100C2×C42×C59C2311yescharacteristic
101C2×C42×C59C2311yescharacteristic
102C24×C59C2×D411yescharacteristic
103C23×D59—21no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—24no—
107C22×C4×C59—24no—
108C22×C4×C59—22no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—41no—
115C22×C4×C59—41no—
116C22×C4×C59—42no—
117C22×C4×C59—42no—
118C22×C4×C59—42no—
119C22×C4×C59—42no—
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×Q812yes—
126C22×C4×C59#4112yes—
127C22×C4×C59C2411yescharacteristic
128C2×C22⋊D59—24no—
129C2×C22⋊D59—42no—
130C2×C22⋊D59—42no—
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C2×D236—22no—
135C2×D236—22no—
136C2×D236—21no—
137C2×D236C2×Q812yes—
138C2×C4×D59—42no—
139C2×C4×D59—42no—
140C2×C4×D59—42no—
141C4⋊C4×C59—24no—
142C4⋊C4×C59—24no—
143C4⋊C4×C59—28no—
144C4⋊C4×C59—28no—
145C4⋊C4×C59—28no—
146C4⋊C4×C59—28no—
147C4⋊C4×C59—22no—
148C4⋊C4×C59—22no—
149C4⋊C4×C59C2×D414yes—
150C4⋊C4×C59C2×D414yes—
151C42×C59—22no—
152C42×C59—22no—
153C42×C59—22no—
154C42×C59C2×D414yes—
155C22⋊Dic59—42no—
156(C2×D59)⋊C4—24no—
157(C2×D59)⋊C4—42no—
158(C2×D59)⋊C4—42no—
159(C2×D59)⋊C4—42no—
160C4⋊Dic59—21no—
161C4⋊Dic59—22no—
162C4⋊Dic59—22no—
163C4⋊Dic59C2×Q812yes—
164Dic59⋊C4—42no—
165C4×Dic59—42no—
166C2×C59⋊C8—22no—
167C2×C59⋊C8—22no—
168C2×C59⋊C8—22no—
169C2×C59⋊C8—22no—
170C2×C59⋊C8—22no—
171C2×C59⋊C8—22no—
172C23×C59—21no—
173C23×C59—21no—
174C23×C59—21no—
175C23×C59—21no—
176C23×C59C22⋊Q812yes—
177C23×C59#13512yes—
178C23×C59AGL(1,ℤ/8)12yes—
179C23×C59C22×D411yescharacteristic
180C22×D59—22no—
181C22×D59—41no—
182C22×D59—41no—
183C22×D59—41no—
184C22×D59—41no—
185C22×D59—41no—
186C22×D59—41no—
187C2×C4×C59—21no—
188C2×C4×C59—21no—
189C2×C4×C59—21no—
190C2×C4×C59—21no—
191C2×C4×C59—21no—
192C2×C4×C59—21no—
193C2×C4×C59—24no—
194C2×C4×C59—24no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59—24no—
198C2×C4×C59—24no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—44no—
204C2×C4×C59—44no—
205C2×C4×C59—48no—
206C2×C4×C59—48no—
207C2×C4×C59—42no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59#12012yes—
212C2×C4×C59#12012yes—
213C2×C4×C59C22⋊Q812yes—
214C2×C4×C59C22×D412yes—
215C2×C4×C59C22×Q811yescharacteristic
216C2×C4×C59C2×#4111yescharacteristic
217C2×C4×C592−1+412yes—
218C22⋊D59—48no—
219C2×Dic59—22no—
220C2×Dic59—41no—
221C2×Dic59—41no—
222C2×Dic59—41no—
223C2×Dic59—41no—
224C2×Dic59—41no—
225C2×Dic59—41no—
226C2×Dic59—41no—
227C2×Dic59—41no—
228D236—24no—
229D236—42no—
230C4×D59—84no—
231C59⋊C8—42no—
232C59⋊C8—42no—
233#24305—591nomaximal, Sylow
234C22×C59—22no—
235C22×C59—22no—
236C22×C59—22no—
237C22×C59—22no—
238C22×C59—22no—
239C22×C59—22no—
240C22×C59#47814yescomplemented
241C22×C59C2×#12011yescharacteristic
242C22×C59C2×C22⋊Q811yescharacteristic
243C22×C59#53611yescharacteristic
244C22×C59C2×#13511yescharacteristic
245C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C59#57511yescharacteristic
247C22×C59#57511yescharacteristic
248C2×D59—42no—
249C2×D59—42no—
250C2×D59—42no—
251C2×D59—81no—
252C2×D59—81no—
253C2×D59—81no—
254C2×D59—81no—
255C4×C59—22no—
256C4×C59—22no—
257C4×C59—22no—
258C4×C59—41no—
259C4×C59—44no—
260C4×C59—41no—
261C4×C59—44no—
262C4×C59#54812yes—
263C4×C59D4×Q812yes—
264Dic59—42no—
265Dic59—81no—
266Dic59—81no—
267C22×#125D5911yescharacteristic, complemented
268C2×#478—592no—
269C4×#121—591no—
270#2059—592no—
271#2025—592no—
272#1989—592no—
273#1874—591no—
274#1874—591no—
275#1853—591no—
276#1852—591no—
277#1797—591no—
278C2×C59—24no—
279C2×C59#240911yescharacteristic
280C2×C59#300211yescharacteristic
281C2×C59#300611yescharacteristic
282C2×C59C2×#47811yescharacteristic
283C2×C59#308011yescharacteristic
284C2×C59#323211yescharacteristic
285C2×C59#323411yescharacteristic
286D59—82no—
287C2×#125—28no—
288C2×#125—21no—
289C2×#125—21no—
290C2×#125C2×D5912yes—
291C2×#125C2×D5912yes—
292C2×#121—591no—
293C2×C4×D4—591no—
294C2×C4×D4—1181no—
295C22×C4⋊C4—21no—
296C22×C4⋊C4—21no—
297C22×C4⋊C4C2×D5911yescharacteristic
298C22×C4⋊C4C2×D5911yescharacteristic
299C22×C42C2×D5911yescharacteristic
300#478—1188no—
301C2×#107—592no—
302C2×#106—592no—
303C2×C4⋊C8—592no—
304C2×#102—592no—
305C2×#102—592no—
306C2×C22⋊C8—591no—
307C2×C22⋊C8—591no—
308#390—591no—
309#389—591no—
310#388—1181no—
311#386—1181no—
312#380—591no—
313C4×C4⋊C4—591no—
314C4×C22⋊C4—1181no—
315#340—592no—
316#340—592no—
317#340—592no—
318#340—592no—
319#336—592no—
320C59#2430511yescharacteristic, complemented, Sylow
321C22×D4—591no—
322C22×D4—1181no—
323C23×C4—21no—
324C23×C4—21no—
325C23×C4—1181no—
326C23×C4C22⋊D5911yescharacteristic
327C23×C4C22⋊D5911yescharacteristic
328C23×C4C22×D5911yescharacteristic
329C22×C8—592no—
330C22×C8—592no—
331#125—28no—
332#125—28no—
333#125—24no—
334#125—24no—
335#125—24no—
336#125C22⋊D5918yescomplemented
337#121—594no—
338#121—1182no—
339C4×D4—1184no—
340C4×D4—2364no—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—28no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—591no—
352C2×C4⋊C4—591no—
353C2×C4⋊C4—591no—
354C2×C4⋊C4—1181no—
355C2×C4⋊C4C22×D5912yes—
356C2×C4⋊C4C22×D5912yes—
357C2×C22⋊C4—1181no—
358C2×C22⋊C4—1181no—
359C2×C22⋊C4—1181no—
360C2×C22⋊C4—1181no—
361C2×C22⋊C4—1181no—
362C2×C42—24no—
363C2×C42—1181no—
364C2×C42C22⋊D5911yescharacteristic
365C2×C42C22⋊D5911yescharacteristic
366C2×C42C22×D5911yescharacteristic
367C2×C42C22×D5911yescharacteristic
368#107—1184no—
369#106—1184no—
370C4⋊C8—1184no—
371#102—1184no—
372#102—1184no—
373C22⋊C8—1182no—
374C22⋊C8—1182no—
375#95—1181no—
376#95—1181no—
377#95—1181no—
378C24—1181no—
379C24C2×C22⋊D5911yescharacteristic
380C2×D4—592no—
381C2×D4—1184no—
382C2×D4—1182no—
383C2×D4—1181no—
384C2×D4—1182no—
385C2×D4—2362no—
386C2×D4—2362no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—1181no—
404C22×C4—1181no—
405C22×C4—1181no—
406C22×C4—2362no—
407C22×C4—2362no—
408C22×C4—2362no—
409C22×C4Q8×D5912yes—
410C22×C4#1174568912yes—
411C22×C4C2×C22⋊D5911yescharacteristic
412C22×C4C2×C22⋊D5911yescharacteristic
413C22×C4C2×C22⋊D5911yescharacteristic
414C22×C4C2×C22⋊D5911yescharacteristic
415C22×C4C2×C22⋊D5911yescharacteristic
416C22×C4C23×D5911yescharacteristic
417C2×C8—1182no—
418C2×C8—1182no—
419C2×C8—1182no—
420C2×C8—1182no—
421C2×C8—1182no—
422C2×C8—1182no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—592no—
432C4⋊C4—1181no—
433C4⋊C4—1182no—
434C4⋊C4—1182no—
435C4⋊C4—2362no—
436C4⋊C4C2×C22⋊D5914yes—
437C4⋊C4C2×C22⋊D5914yes—
438C22⋊C4—1184no—
439C22⋊C4—2362no—
440C22⋊C4—2362no—
441C22⋊C4—2362no—
442C22⋊C4—2362no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—2362no—
447C42C2×C22⋊D5914yes—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—1182no—
453C23—2361no—
454C23—2361no—
455C23—2361no—
456C23—2361no—
457C23—2361no—
458C23—2361no—
459C23#1427370312yes—
460C23#1427374712yes—
461C23#1427375212yes—
462C23C22×C22⋊D5911yescharacteristic
463D4—1184no—
464D4—2368no—
465D4—2362no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—1182no—
491C2×C4—4724no—
492C2×C4—2361no—
493C2×C4—2361no—
494C2×C4—2361no—
495C2×C4—2361no—
496C2×C4—2361no—
497C2×C4—2361no—
498C2×C4—2361no—
499C2×C4—2361no—
500C2×C4#1427369612yes—
501C2×C4#1427369612yes—
502C2×C4#1427370312yes—
503C2×C4C22×C22⋊D5912yes—
504C2×C4C2×Q8×D5911yescharacteristic
505C2×C4C2×#1174568911yescharacteristic
506C2×C4#1427377512yes—
507C8—2362no—
508C8—2362no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—4721no—
516C22—4721no—
517C22—4721no—
518C22—4721no—
519C22—2362no—
520C22—2362no—
521C22—2362no—
522C22#1683769914yescomplemented
523C22C2×#1427369611yescharacteristic
524C22C2×#1427370311yescharacteristic
525C22#1683797611yescharacteristic
526C22C2×#1427374711yescharacteristic
527C22C2×#1427375211yescharacteristic
528C22#1683820011yescharacteristic
529C22#1683820011yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—41no—
536C4—44no—
537C4—4721no—
538C4—4721no—
539C4—2362no—
540C4#1683806312yes—
541C4Q8×C22⋊D5912yes—
542C2—24no—
543C2—4722no—
544C2#2142745511yescharacteristic
545C2#2143331411yescharacteristic
546C2#2143334811yescharacteristic
547C2C2×#1683769911yescharacteristic
548C2#2143391411yescharacteristic
549C2#2143526411yescharacteristic
550C2#2143528611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC22×#125×C59C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenterC22×#125D59
hypocenterC59#24305
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
1. C2×#16837699C2semidirect
3. #21425431C2semidirect
4. #21425239C2semidirect
5. #21425054C2semidirect
8. #21424289C2semidirect
9. #21424284C2semidirect
10. #21424076C2semidirect
11. C22×#125×C59C2semidirect
84. #125×C59D4semidirect
240. C22×C59#478semidirect
267. C22×#125D59semidirect
320. C59#24305semidirect
336. #125C22⋊D59semidirect
522. C22#16837699semidirect

Maximal subgroups

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

subgroupindex
C2×#168376992
C4×#142737012
#214254312
#214252392
#214250542
#214244132
#214242892
#214242842
#214240762
C22×#125×C592
#2430559