Finite Group Databasebeta

← #28620921

Subgroups — #28620921

Order 15104.

1240 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
1C2×#16837557C211yesmaximal, characteristic, complemented
2C22×#14273650C211yesmaximal, characteristic, complemented
3#21427461C211yesmaximal, characteristic
4#21425244C212yesmaximal, complemented
5#21425104C212yesmaximal, complemented
6#21424520C211yesmaximal, characteristic, complemented
7#21424437C212yesmaximal, complemented
8#21424408C212yesmaximal, complemented
9#21424389C211yesmaximal, characteristic
10#21424182C211yesmaximal, characteristic, complemented
11C2×C59×#552C211yesmaximal, characteristic
12C23×Dic118C2211yescharacteristic
13C22×#11745672C2212yes—
14C23×C59⋊C8C2211yescharacteristic
15C2×#14273706C2212yes—
16#16837557—24no—
17C2×#14273651C2212yes—
18C2×#14273650—24no—
19C2×#14273650C2212yes—
20C2×#14273650C2212yes—
21C2×#14273650C2212yes—
22C2×C22⋊(C59⋊C8)C2211yescharacteristic
23C2×C22⋊(C59⋊C8)C2212yes—
24#16837322—21no—
25#16837314—21no—
26#16837308C2212yes—
27#16837282—21no—
28#16837274—21no—
29#16837262C2211yescharacteristic
30#16837164C2212yes—
31#16837164C2212yes—
32#16837164C2212yes—
33#16837157C2212yes—
34C59×#552—24no—
35C59×#552—24no—
36C2×#126×C59—21no—
37C2×#125×C59—21no—
38C2×#125×C59C2211yescharacteristic
39C2×#123×C59—21no—
40C2×C22⋊Q8×C59C2211yescharacteristic
41C2×C22⋊Q8×C59C2211yescharacteristic
42C2×C4×Q8×C59C2211yescharacteristic
43C2×C4×D4×C59C2211yescharacteristic
44C2×#117×C59—21no—
45C2×#117×C59C2211yescharacteristic
46C22×C4⋊C4×C59C2211yescharacteristic
47C22×Dic118—22no—
48C22×Dic118—24no—
49C22×Dic118—22no—
50C22×Dic118—21no—
51C22×Dic118—42no—
52C22×Dic118C2312yes—
53C23×Dic59—21no—
54C2×#11745672—24no—
55C2×#11745672—24no—
56C2×#11745672—42no—
57C22×C59⋊C8—22no—
58C22×C59⋊C8—42no—
59C22×C59⋊C8C2312yes—
60C22×C59⋊C8C2312yes—
61#14273706—24no—
62#14273706—42no—
63C2×C4⋊Dic59C2312yes—
64C2×Dic59⋊C4—22no—
65C2×C22⋊Dic59—22no—
66C2×C22⋊Dic59—21no—
67#14273651—42no—
68#14273650—24no—
69#14273650—24no—
70#14273650—24no—
71C22⋊(C59⋊C8)—22no—
72C22⋊(C59⋊C8)—42no—
73C59⋊#95—21no—
74C59⋊#95—22no—
75C59⋊#95—22no—
76C22×Q8×C59C2311yescharacteristic
77C22×D4×C59D411yescharacteristic
78C23×C4×C59—21no—
79C23×C4×C59C2311yescharacteristic
80C23×C4×C59C2311yescharacteristic
81#126×C59—28no—
82#125×C59—28no—
83#125×C59—22no—
84#125×C59—22no—
85#123×C59—28no—
86C22⋊Q8×C59—22no—
87C22⋊Q8×C59—22no—
88C22⋊Q8×C59—22no—
89C22⋊Q8×C59D414yes—
90C4×Q8×C59—22no—
91C4×Q8×C59—22no—
92C4×D4×C59—22no—
93C4×D4×C59—22no—
94#117×C59—28no—
95#117×C59—22no—
96#117×C59—22no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—22no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59D411yescharacteristic
105C2×C4⋊C4×C59D414yes—
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59D411yescharacteristic
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117C24×C59C2×D411yescharacteristic
118C24×C59C2×D411yescharacteristic
119C2×Q8×C59—21no—
120C2×Q8×C59—22no—
121C2×Q8×C59—22no—
122C2×Q8×C59C2×D412yes—
123C2×D4×C59—22no—
124C2×D4×C59—22no—
125C2×D4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—24no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D412yes—
144C22×C4×C59C2×D412yes—
145C22×C4×C59C2411yescharacteristic
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—44no—
149C22×Dic59—42no—
150C2×Dic118—28no—
151C2×Dic118—22no—
152C2×Dic118—22no—
153C2×Dic118—22no—
154C2×Dic118—22no—
155C2×Dic118—22no—
156C2×Dic118—48no—
157C2×Dic118—44no—
158C2×Dic118—44no—
159C2×Dic118—42no—
160C2×Dic118#4114yes—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—24no—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—24no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59C2×D412yes—
172C4⋊C4×C59C2×D412yes—
173C4⋊C4×C59C2×D412yes—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—24no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C42×C59—24no—
180C42×C59—24no—
181C42×C59—21no—
182C42×C59—21no—
183C22⋊Dic59—44no—
184C22⋊Dic59—42no—
185#11745672—48no—
186C4⋊Dic59—22no—
187C4⋊Dic59—22no—
188Dic59⋊C4—44no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8—22no—
194C2×C59⋊C8—22no—
195C2×C59⋊C8—44no—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—21no—
199C23×C59—22no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—22no—
203C23×C59#12012yes—
204C23×C59#13512yes—
205C23×C59#13712yes—
206C23×C59C22×D411yescharacteristic
207Q8×C59—24no—
208Q8×C59—41no—
209Q8×C59—41no—
210D4×C59—44no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—24no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—42no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—42no—
232C2×C4×C59—48no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—42no—
236C2×C4×C59#12012yes—
237C2×C4×C59C22×D411yescharacteristic
238C2×C4×C59C22×D411yescharacteristic
239C2×C4×C59C2×#4112yes—
240C2×C4×C59H(5,𝔽2)12yes—
241C2×Dic59—24no—
242C2×Dic59—48no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248Dic118—24no—
249Dic118—24no—
250Dic118—48no—
251Dic118—42no—
252Dic118—42no—
253C59⋊C8—42no—
254C59⋊C8—42no—
255#23967—591nomaximal, Sylow
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—24no—
259C22×C59—24no—
260C22×C59—41no—
261C22×C59—41no—
262C22×C59—41no—
263C22×C59—42no—
264C22×C59—42no—
265C22×C59#45214yes—
266C22×C59C2×#12011yescharacteristic
267C22×C59#54612yes—
268C22×C59C2×#13511yescharacteristic
269C22×C59C2×#13711yescharacteristic
270C22×C59#57812yes—
271C4×C59—22no—
272C4×C59—41no—
273C4×C59—42no—
274C4×C59—41no—
275C4×C59—42no—
276C4×C59—42no—
277C4×C59—41no—
278C4×C59—42no—
279C4×C59#52511yescharacteristic
280C4×C59#53511yescharacteristic
281C4×C59#54612yes—
282Dic59—44no—
283Dic59—82no—
284C2×#552D5911yescharacteristic
285C2×#451—591no—
286C22×#103—591no—
287#2409—591no—
288#2026—592no—
289#2001—592no—
290#1892—591no—
291#1878—592no—
292#1873—592no—
293#1869—591no—
294#1823—591no—
295C2×C59—24no—
296C2×C59—81no—
297C2×C59#240411yescharacteristic
298C2×C59C2×#45211yescharacteristic
299C2×C59#301111yescharacteristic
300C2×C59#330012yes—
301C2×C59#331212yes—
302C23×Q8—591no—
303C22×Q16—592no—
304C23×C8—591no—
305#552—24no—
306#552—24no—
307C2×#126—21no—
308C2×#125—21no—
309C2×#125C2×D5911yescharacteristic
310C2×#123—21no—
311C2×C22⋊Q8—592no—
312C2×C22⋊Q8C2×D5911yescharacteristic
313C2×C22⋊Q8C2×D5911yescharacteristic
314C2×C4×Q8C2×D5911yescharacteristic
315C2×C4×D4C2×D5911yescharacteristic
316C2×#117—21no—
317C2×#117C2×D5911yescharacteristic
318C22×C4⋊C4C2×D5911yescharacteristic
319#451—1184no—
320C2×#103—592no—
321C2×#103—592no—
322C2×#103—592no—
323C2×#103—592no—
324C2×#103—1184no—
325C2×C22⋊C8—591no—
326C2×C22⋊C8—592no—
327#395—1181no—
328#393—1181no—
329#391—592no—
330#386—1181no—
331#385—1181no—
332#380—591no—
333#340—592no—
334#340—592no—
335#340—592no—
336#336—592no—
337C59#2396711yescharacteristic, complemented, Sylow
338C22×Q8—592no—
339C22×Q8—1184no—
340C22×Q8—1181no—
341C22×Q8—1182no—
342C22×Q8—1182no—
343C22×Q8—2362no—
344C22×Q8C22×D5911yescharacteristic
345C22×D4C22⋊D5911yescharacteristic
346C23×C4—21no—
347C23×C4—1181no—
348C23×C4C22×D5911yescharacteristic
349C23×C4C22×D5911yescharacteristic
350C2×Q16—1184no—
351C2×Q16—1184no—
352C2×Q16—2362no—
353C22×C8—592no—
354C22×C8—592no—
355C22×C8—1182no—
356C22×C8—2362no—
357#126—28no—
358#125—28no—
359#125—22no—
360#125—22no—
361#123—28no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—1184no—
366C22⋊Q8—2362no—
367C22⋊Q8C22⋊D5914yes—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×D4—22no—
371C4×D4—22no—
372#117—28no—
373#117—22no—
374#117—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—592no—
383C2×C4⋊C4—1182no—
384C2×C4⋊C4C22⋊D5911yescharacteristic
385C2×C4⋊C4C22⋊D5914yes—
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1182no—
393C2×C22⋊C4—1181no—
394C2×C22⋊C4C22⋊D5911yescharacteristic
395C2×C22⋊C4C22⋊D5911yescharacteristic
396C2×C42—21no—
397C2×C42—21no—
398C2×C42C22×D5911yescharacteristic
399#103—1184no—
400#103—1184no—
401#103—1184no—
402#103—2362no—
403C22⋊C8—1182no—
404C22⋊C8—2362no—
405#95—1181no—
406#95—1182no—
407#95—1182no—
408C24C2×C22⋊D5911yescharacteristic
409C24C2×C22⋊D5911yescharacteristic
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—594no—
414C2×Q8—1188no—
415C2×Q8—1182no—
416C2×Q8—1182no—
417C2×Q8—1182no—
418C2×Q8—1182no—
419C2×Q8—1182no—
420C2×Q8—2364no—
421C2×Q8—2368no—
422C2×Q8—2364no—
423C2×Q8—2362no—
424C2×Q8C2×C22⋊D5912yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1182no—
442C22×C4—1182no—
443C22×C4—2364no—
444C22×C4—2362no—
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C2×C22⋊D5911yescharacteristic
449C22×C4C2×C22⋊D5912yes—
450C22×C4C2×C22⋊D5912yes—
451C22×C4C23×D5911yescharacteristic
452Q16—2368no—
453C2×C8—1182no—
454C2×C8—1182no—
455C2×C8—1182no—
456C2×C8—1182no—
457C2×C8—1182no—
458C2×C8—1182no—
459C2×C8—2364no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—1182no—
471C4⋊C4—1182no—
472C4⋊C4—2364no—
473C4⋊C4C2×C22⋊D5912yes—
474C4⋊C4C2×C22⋊D5912yes—
475C4⋊C4C2×C22⋊D5912yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—2364no—
482C22⋊C4—2362no—
483C42—24no—
484C42—24no—
485C42—21no—
486C42—21no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23#1427369612yes—
495C23#1427374712yes—
496C23#1427375712yes—
497C23C22×C22⋊D5911yescharacteristic
498Q8—24no—
499Q8—41no—
500Q8—41no—
501Q8—1184no—
502Q8—1184no—
503Q8—2368no—
504Q8—2362no—
505Q8—2362no—
506D4—44no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—48no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—1184no—
533C2×C4—2362no—
534C2×C4—2362no—
535C2×C4—2362no—
536C2×C4—2368no—
537C2×C4—2362no—
538C2×C4—2362no—
539C2×C4#1427369612yes—
540C2×C4C22×C22⋊D5911yescharacteristic
541C2×C4C22×C22⋊D5911yescharacteristic
542C2×C4#41×D5912yes—
543C2×C4#1427377312yes—
544C8—2362no—
545C8—2362no—
546C22—22no—
547C22—24no—
548C22—22no—
549C22—24no—
550C22—41no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—42no—
555C22#1683756414yes—
556C22C2×#1427369611yescharacteristic
557C22#1683804012yes—
558C22C2×#1427374711yescharacteristic
559C22C2×#1427375711yescharacteristic
560C22#1683821712yes—
561C4—22no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—4722no—
570C4—2364no—
571C4#1683790911yescharacteristic
572C4#1683796611yescharacteristic
573C4#1683804012yes—
574C2—24no—
575C2—81no—
576C2#2142741111yescharacteristic
577C2C2×#1683756411yescharacteristic
578C2#2143339111yescharacteristic
579C2#2143597712yes—
580C2#2143614112yes—
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#552C2
socleC23×C59C22×D4
prime radicalC22×D4×C59D4
hypercenterC2×#552D59
hypocenterC59#23967
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#16837557C2semidirect
2. C22×#14273650C2semidirect
4. #21425244C2semidirect
5. #21425104C2semidirect
6. #21424520C2semidirect
7. #21424437C2semidirect
8. #21424408C2semidirect
10. #21424182C2semidirect
337. C59#23967semidirect

Maximal subgroups

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

subgroupindex
C2×#168375572
C22×#142736502
#214274612
#214252442
#214251042
#214245202
#214244372
#214244082
#214243892
#214241822
C2×C59×#5522
#2396759