Finite Group Databasebeta

← #28633918

Subgroups — #28633918

Order 15104.

678 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#21429454C211yesmaximal, characteristic
2#21425663C211yesmaximal, characteristic, complemented
3#21425653C211yesmaximal, characteristic, complemented
4#21425510C211yesmaximal, characteristic
5#21425500C211yesmaximal, characteristic
6#21425033C211yesmaximal, characteristic, complemented
7#21424864C211yesmaximal, characteristic, complemented
8#21424723C211yesmaximal, characteristic
9#21424716C211yesmaximal, characteristic
10#21424688C211yesmaximal, characteristic, complemented
11#21424674C211yesmaximal, characteristic, complemented
12#21424226C211yesmaximal, characteristic, complemented
13#21424103C211yesmaximal, characteristic
14C4×#14273647C211yesmaximal, characteristic
15C59×#2355C211yesmaximal, characteristic, complemented
16C2×C4×D236C2211yescharacteristic
17C2×#14273660C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273647C2211yescharacteristic
21C2×#14273647C2211yescharacteristic
22C2×#14273647C2211yescharacteristic
23C2×#14273647C2211yescharacteristic
24C2×C4×C59⋊C8C2211yescharacteristic
25#16837337—21no—
26#16837313—21no—
27#16837304C2211yescharacteristic
28#16837304C2211yescharacteristic
29#16837300C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837273—21no—
32#16837273—21no—
33#16837268C2211yescharacteristic
34#16837164C2211yescharacteristic
35#16837164C2211yescharacteristic
36#16837164C2211yescharacteristic
37#16837164C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4×Q8×C59C2211yescharacteristic
49C59×#391—21no—
50C59×#387—21no—
51C59×#387—21no—
52C59×#384C2211yescharacteristic
53C59×#382C2211yescharacteristic
54C59×#382C2211yescharacteristic
55C4×C4⋊C4×C59—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C4×D236—22no—
66C4×D236—22no—
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×(C2×D59)⋊C4—21no—
73C2×(C2×D59)⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77#14273660—22no—
78#14273658—22no—
79C4⋊(C59⋊C8)—22no—
80#14273647—22no—
81#14273647—22no—
82#14273647—22no—
83#14273647—22no—
84C4×C59⋊C8—22no—
85C59⋊#95—21no—
86C59⋊#95—21no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C22×Q8×C59D411yescharacteristic
90C4×Q8×C59—22no—
91C4×Q8×C59—22no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59D411yescharacteristic
95C2×C4⋊C4×C59D411yescharacteristic
96C2×C4⋊C4×C59D411yescharacteristic
97C2×C4⋊C4×C59C2311yescharacteristic
98C2×C4⋊C4×C59C2311yescharacteristic
99C2×C4⋊C4×C59C2311yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C42×C59—21no—
102C2×C42×C59—21no—
103C2×C42×C59—21no—
104C2×C42×C59C2311yescharacteristic
105C2×C42×C59C2311yescharacteristic
106C2×C42×C59C2311yescharacteristic
107#95×C59—21no—
108#95×C59—21no—
109#95×C59—21no—
110#95×C59—21no—
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114C23×D59—21no—
115C2×Q8×C59—22no—
116C2×Q8×C59—22no—
117C2×Q8×C59—41no—
118C22×C4×C59—21no—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2411yescharacteristic
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59—21no—
132C2×D236—21no—
133C2×D236—21no—
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236#4112yes—
138C2×C4×D59—42no—
139C2×C4×D59—81no—
140C2×C4×D59—81no—
141C4⋊C4×C59—22no—
142C4⋊C4×C59—22no—
143C4⋊C4×C59—22no—
144C4⋊C4×C59—21no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—42no—
152C4⋊C4×C59—81no—
153C4⋊C4×C59#4112yes—
154C42×C59—21no—
155C42×C59—21no—
156C42×C59—21no—
157C42×C59—21no—
158C42×C59—42no—
159C42×C59—81no—
160C42×C59—81no—
161C42×C59#4112yes—
162C42×C59#4112yes—
163(C2×D59)⋊C4—42no—
164(C2×D59)⋊C4—42no—
165(C2×D59)⋊C4—42no—
166(C2×D59)⋊C4—81no—
167(C2×D59)⋊C4—81no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174Dic59⋊C4—81no—
175Dic59⋊C4—81no—
176C2×C59⋊C8—21no—
177C2×C59⋊C8—21no—
178C2×C59⋊C8—21no—
179C2×C59⋊C8—21no—
180C2×C59⋊C8—21no—
181C2×C59⋊C8—21no—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
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—44no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—22no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59#12312yes—
225C2×C4×C59#12312yes—
226C2×C4×C59#13512yes—
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C592−1+411yescharacteristic
232C2×C4×C592−1+411yescharacteristic
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242D236—22no—
243D236—22no—
244D236—41no—
245D236—41no—
246C4×D59—84no—
247C59⋊C8—41no—
248C59⋊C8—41no—
249C59⋊C8—41no—
250C59⋊C8—41no—
251#24998—591nomaximal, Sylow
252C22×C59C2×#12311yescharacteristic
253C22×C59#54811yescharacteristic
254C22×C59#54811yescharacteristic
255C22×C59C2×#13511yescharacteristic
256C22×C59#57511yescharacteristic
257C22×C59#57611yescharacteristic
258C22×C59#57711yescharacteristic
259C2×D59—42no—
260C2×D59—42no—
261C2×D59—42no—
262C2×D59—81no—
263C2×D59—81no—
264C2×D59—81no—
265C2×D59—81no—
266C4×C59—22no—
267C4×C59—22no—
268C4×C59—22no—
269C4×C59—41no—
270C4×C59—41no—
271C4×C59—41no—
272C4×C59—42no—
273C4×C59—81no—
274C4×C59—81no—
275C4×C59—81no—
276C4×C59#52911yescharacteristic
277C4×C59#54111yescharacteristic
278C4×C59#55211yescharacteristic
279C4×C59#55611yescharacteristic
280Dic59—81no—
281Dic59—81no—
282Dic59—81no—
283#2580—591no—
284#2355D5911yescharacteristic, complemented
285#2101—591no—
286#2099—591no—
287#2073—591no—
288#2071—591no—
289#1984—591no—
290#1949—591no—
291#1926—591no—
292#1925—591no—
293#1921—591no—
294#1919—591no—
295#1835—591no—
296#1804—591no—
297C4×#102—591no—
298C2×C59#242111yescharacteristic
299C2×C59#309111yescharacteristic
300C2×C59#309611yescharacteristic
301C2×C59#336111yescharacteristic
302C2×C59#336511yescharacteristic
303C2×C59#337011yescharacteristic
304C2×C59#337611yescharacteristic
305D59—82no—
306C2×C4×Q8C2×D5911yescharacteristic
307C2×C4×D4—591no—
308C2×#107—591no—
309C2×#106—591no—
310C2×C4⋊C8—591no—
311C2×#102—591no—
312C2×#102—591no—
313C2×#102—591no—
314C2×#102—591no—
315C2×C4×C8—591no—
316#399—1181no—
317#393—1181no—
318#391—21no—
319#390—591no—
320#390—591no—
321#389—591no—
322#389—591no—
323#387—21no—
324#387—21no—
325#385—1181no—
326#385—1181no—
327#384C2×D5911yescharacteristic
328#382—591no—
329#382C2×D5911yescharacteristic
330#382C2×D5911yescharacteristic
331C4×C4⋊C4—21no—
332C4×C4⋊C4C2×D5911yescharacteristic
333C4×C4⋊C4C2×D5911yescharacteristic
334C4×C4⋊C4C2×D5911yescharacteristic
335#340—591no—
336#340—591no—
337#340—591no—
338#340—591no—
339#340—591no—
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#336—591no—
346#336—591no—
347#336—591no—
348#336—591no—
349C59#2499811yescharacteristic, complemented, Sylow
350C22×Q8C22⋊D5911yescharacteristic
351C22×D4—591no—
352C23×C4—1181no—
353C22×C8—591no—
354C22×C8—591no—
355C22×C8—591no—
356C22×C8—591no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×D4—1182no—
360C4×D4—1182no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—591no—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—1181no—
367C2×C4⋊C4—1181no—
368C2×C4⋊C4C22⋊D5911yescharacteristic
369C2×C4⋊C4C22⋊D5911yescharacteristic
370C2×C4⋊C4C22⋊D5911yescharacteristic
371C2×C4⋊C4C22×D5911yescharacteristic
372C2×C4⋊C4C22×D5911yescharacteristic
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C22⋊C4—1181no—
376C2×C22⋊C4—1181no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42C22×D5911yescharacteristic
384C2×C42C22×D5911yescharacteristic
385C2×C42C22×D5911yescharacteristic
386#107—1182no—
387#106—1182no—
388C4⋊C8—1182no—
389#102—1182no—
390#102—1182no—
391#102—1182no—
392#102—1182no—
393C4×C8—1182no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—1181no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405C24—1181no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—41no—
409C2×D4—592no—
410C2×D4—1181no—
411C2×D4—1181no—
412C2×D4—1181no—
413C2×D4—1181no—
414C2×D4—1181no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—1181no—
420C22×C4—1181no—
421C22×C4—1181no—
422C22×C4—4721no—
423C22×C4—4721no—
424C22×C4—2362no—
425C22×C4C2×C22⋊D5911yescharacteristic
426C22×C4C2×C22⋊D5911yescharacteristic
427C22×C4C2×C22⋊D5911yescharacteristic
428C22×C4C2×C22⋊D5911yescharacteristic
429C22×C4C2×C22⋊D5911yescharacteristic
430C22×C4C2×C22⋊D5911yescharacteristic
431C22×C4C23×D5911yescharacteristic
432C2×C8—1181no—
433C2×C8—1181no—
434C2×C8—1181no—
435C2×C8—1181no—
436C2×C8—1181no—
437C2×C8—1181no—
438C2×C8—1181no—
439C2×C8—1181no—
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8—1181no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—81no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—4721no—
463C4⋊C4—4721no—
464C4⋊C4#1174568912yes—
465C22⋊C4—4721no—
466C22⋊C4—4721no—
467C22⋊C4—2362no—
468C22⋊C4—2362no—
469C22⋊C4—2362no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—42no—
475C42—81no—
476C42—81no—
477C42#1174568512yes—
478C42#1174568512yes—
479C23—1182no—
480C23—2361no—
481C23—2361no—
482C23—2361no—
483C23—2361no—
484C23—2361no—
485C23—2361no—
486C23C22×C22⋊D5911yescharacteristic
487Q8—44no—
488D4—1182no—
489D4—1182no—
490D4—2361no—
491D4—2361no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—4724no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4#1427371012yes—
530C2×C4#1427371012yes—
531C2×C4#1427374712yes—
532C2×C4#41×D5911yescharacteristic
533C2×C4C2×#1174568511yescharacteristic
534C2×C4C2×#1174568511yescharacteristic
535C2×C4C2×#1174568911yescharacteristic
536C2×C4#1427377411yescharacteristic
537C2×C4#1427377511yescharacteristic
538C8—2361no—
539C8—2361no—
540C8—2361no—
541C8—2361no—
542C22—4721no—
543C22—4721no—
544C22—4721no—
545C22—4721no—
546C22—2362no—
547C22—2362no—
548C22—2362no—
549C22C2×#1427371011yescharacteristic
550C22#1683805911yescharacteristic
551C22#1683806311yescharacteristic
552C22C2×#1427374711yescharacteristic
553C22#1683820011yescharacteristic
554C22#1683820511yescharacteristic
555C22#1683821011yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—4721no—
567C4—4721no—
568C4—4721no—
569C4#1683793911yescharacteristic
570C4#1683801011yescharacteristic
571C4#1683809311yescharacteristic
572C4#1683812511yescharacteristic
573C2—4722no—
574C2#2142759311yescharacteristic
575C2#2143400911yescharacteristic
576C2#2143405411yescharacteristic
577C2#2143675211yescharacteristic
578C2#2143679211yescharacteristic
579C2#2143684911yescharacteristic
580C2#2143691711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2355C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2355D59
hypocenterC59#24998
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. #21425663C2semidirect
3. #21425653C2semidirect
6. #21425033C2semidirect
7. #21424864C2semidirect
10. #21424688C2semidirect
11. #21424674C2semidirect
12. #21424226C2semidirect
15. C59×#2355C2semidirect
284. #2355D59semidirect
349. C59#24998semidirect

Maximal subgroups

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

subgroupindex
#214294542
#214256632
#214256532
#214255102
#214255002
#214250332
#214248642
#214247232
#214247162
#214246882
#214246742
#214242262
#214241032
C4×#142736472
C59×#23552
#2499859