Finite Group Databasebeta

← #28633920

Subgroups — #28633920

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#21427053C211yesmaximal, characteristic, complemented
2#21425664C211yesmaximal, characteristic, complemented
3#21425654C211yesmaximal, characteristic, complemented
4#21425509C211yesmaximal, characteristic
5#21425499C211yesmaximal, characteristic
6#21425033C211yesmaximal, characteristic, complemented
7#21424862C211yesmaximal, characteristic, complemented
8#21424720C211yesmaximal, characteristic
9#21424713C211yesmaximal, characteristic
10#21424685C211yesmaximal, characteristic, complemented
11#21424671C211yesmaximal, characteristic, complemented
12#21424226C211yesmaximal, characteristic, complemented
13#21424106C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C59×#2580C211yesmaximal, characteristic
16C2×#14273660C2211yescharacteristic
17C2×#14273658C2211yescharacteristic
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×#14273648C2211yescharacteristic
20C2×#14273648C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×C4×C59⋊C8C2211yescharacteristic
24#16837309—21no—
25#16837288—21no—
26#16837288—21no—
27#16837272C2211yescharacteristic
28#16837268C2211yescharacteristic
29#16837268C2211yescharacteristic
30C4⋊C4×Dic59—21no—
31C4×C4⋊Dic59C2211yescharacteristic
32C4×C4⋊Dic59C2211yescharacteristic
33C4×C4⋊Dic59C2211yescharacteristic
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
49C2×C4×D4×C59C2211yescharacteristic
50C59×#399—21no—
51C59×#393—21no—
52C59×#390C2211yescharacteristic
53C59×#390C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#385—21no—
57C59×#385—21no—
58C59×#382C2211yescharacteristic
59C22×C59⋊C8C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59C2311yescharacteristic
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4×Dic59—21no—
70C2×C4×Dic59—21no—
71C2×C4×Dic59—21no—
72#14273660—22no—
73#14273658—22no—
74C4⋊(C59⋊C8)—22no—
75#14273648—22no—
76#14273648—22no—
77#14273648—22no—
78#14273648—22no—
79C4×C59⋊C8—22no—
80C59⋊#95—21no—
81C59⋊#95—21no—
82C59⋊#95—21no—
83C59⋊#95—21no—
84C59⋊#95—21no—
85C59⋊#95—21no—
86C59⋊#95—21no—
87C22×Q8×C59D411yescharacteristic
88C22×D4×C59C2311yescharacteristic
89C23×C4×C59—21no—
90C4×Q8×C59—22no—
91C4×Q8×C59—22no—
92C4×D4×C59—22no—
93C4×D4×C59—22no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59D411yescharacteristic
97C2×C4⋊C4×C59D411yescharacteristic
98C2×C4⋊C4×C59D411yescharacteristic
99C2×C4⋊C4×C59C2311yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C22⋊C4×C59—21no—
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—21no—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C42×C59C2311yescharacteristic
108C2×C42×C59C2311yescharacteristic
109C2×C42×C59C2311yescharacteristic
110#95×C59—21no—
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114C24×C59—21no—
115C2×Q8×C59—22no—
116C2×Q8×C59—22no—
117C2×Q8×C59—41no—
118C2×D4×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59#4112yes—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—42no—
128C22×C4×C59—81no—
129C22×C4×C59—81no—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2411yescharacteristic
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
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—81no—
151C4⋊C4×C59—81no—
152C22⋊C4×C59—42no—
153C22⋊C4×C59—42no—
154C22⋊C4×C59—42no—
155C22⋊C4×C59—81no—
156C22⋊C4×C59—81no—
157C42×C59—21no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59#4112yes—
162C42×C59#4112yes—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—42no—
171C4⋊Dic59—81no—
172C4⋊Dic59#4112yes—
173C4×Dic59—42no—
174C4×Dic59—81no—
175C4×Dic59—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×C59—22no—
189C23×C59—41no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59C22×D411yescharacteristic
196Q8×C59—44no—
197D4×C59—22no—
198D4×C59—22no—
199D4×C59—41no—
200D4×C59—41no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
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—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
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—41no—
225C2×C4×C59—84no—
226C2×C4×C59#12312yes—
227C2×C4×C59#12312yes—
228C2×C4×C59#13512yes—
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C592−1+411yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×Dic59—22no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C59⋊C8—41no—
248C59⋊C8—41no—
249C59⋊C8—41no—
250C59⋊C8—41no—
251#24998—591nomaximal, Sylow
252C22×C59—42no—
253C22×C59—42no—
254C22×C59—42no—
255C22×C59—81no—
256C22×C59—81no—
257C22×C59—81no—
258C22×C59—81no—
259C22×C59C2×#12311yescharacteristic
260C22×C59#54811yescharacteristic
261C22×C59#54811yescharacteristic
262C22×C59C2×#13511yescharacteristic
263C22×C59#57511yescharacteristic
264C22×C59#57611yescharacteristic
265C22×C59#57711yescharacteristic
266C4×C59—22no—
267C4×C59—22no—
268C4×C59—22no—
269C4×C59—41no—
270C4×C59—41no—
271C4×C59—41no—
272C4×C59—81no—
273C4×C59—81no—
274C4×C59—81no—
275C4×C59#52911yescharacteristic
276C4×C59#54111yescharacteristic
277C4×C59#55211yescharacteristic
278C4×C59#55611yescharacteristic
279Dic59—42no—
280Dic59—81no—
281Dic59—81no—
282Dic59—81no—
283#2580D5911yescharacteristic
284#2355—591no—
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—82no—
299C2×C59#242111yescharacteristic
300C2×C59#309111yescharacteristic
301C2×C59#309611yescharacteristic
302C2×C59#336111yescharacteristic
303C2×C59#336511yescharacteristic
304C2×C59#337011yescharacteristic
305C2×C59#337611yescharacteristic
306C2×C4×Q8C2×D5911yescharacteristic
307C2×C4×D4C2×D5911yescharacteristic
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—21no—
317#393—21no—
318#391—1181no—
319#390C2×D5911yescharacteristic
320#390C2×D5911yescharacteristic
321#389C2×D5911yescharacteristic
322#389C2×D5911yescharacteristic
323#387—1181no—
324#387—1181no—
325#385—21no—
326#385—21no—
327#384—591no—
328#382—591no—
329#382—591no—
330#382C2×D5911yescharacteristic
331C4×C4⋊C4—591no—
332C4×C4⋊C4—591no—
333C4×C4⋊C4—591no—
334C4×C4⋊C4—1181no—
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×D4C22×D5911yescharacteristic
352C23×C4—21no—
353C22×C8—591no—
354C22×C8—591no—
355C22×C8—591no—
356C22×C8—591no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×D4—22no—
360C4×D4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—591no—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—1181no—
368C2×C4⋊C4—1181no—
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—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C42—1181no—
381C2×C42—1181no—
382C2×C42—1181no—
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—1181no—
399#95—1181no—
400#95—1181no—
401#95—1181no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405C24—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—41no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4#1174568912yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—1181no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
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—81no—
454C4⋊C4—81no—
455C4⋊C4—592no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—4721no—
464C4⋊C4—2362no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—4721no—
475C42—4721no—
476C42—2362no—
477C42#1174568712yes—
478C42#1174568712yes—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23C22×C22⋊D5911yescharacteristic
487Q8—44no—
488D4—22no—
489D4—22no—
490D4—41no—
491D4—41no—
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—41no—
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—84no—
517C2×C4—1182no—
518C2×C4—2361no—
519C2×C4—2361no—
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#1427370812yes—
530C2×C4#1427370812yes—
531C2×C4#1427374712yes—
532C2×C4#41×D5911yescharacteristic
533C2×C4C2×#1174568711yescharacteristic
534C2×C4C2×#1174568711yescharacteristic
535C2×C4C2×#1174568911yescharacteristic
536C2×C4#1427377411yescharacteristic
537C2×C4#1427377511yescharacteristic
538C8—2361no—
539C8—2361no—
540C8—2361no—
541C8—2361no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22C2×#1427370811yescharacteristic
550C22#1683805911yescharacteristic
551C22#1683806311yescharacteristic
552C22C2×#1427374711yescharacteristic
553C22#1683820011yescharacteristic
554C22#1683820511yescharacteristic
555C22#1683821311yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—4721no—
566C4—4721no—
567C4—4721no—
568C4—2362no—
569C4#1683793811yescharacteristic
570C4#1683801211yescharacteristic
571C4#1683809411yescharacteristic
572C4#1683812311yescharacteristic
573C2—82no—
574C2#2142759511yescharacteristic
575C2#2143400511yescharacteristic
576C2#2143404811yescharacteristic
577C2#2143675611yescharacteristic
578C2#2143678811yescharacteristic
579C2#2143684311yescharacteristic
580C2#2143692311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2580C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2580D59
hypocenterC59#24998
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. #21427053C2semidirect
2. #21425664C2semidirect
3. #21425654C2semidirect
6. #21425033C2semidirect
7. #21424862C2semidirect
10. #21424685C2semidirect
11. #21424671C2semidirect
12. #21424226C2semidirect
349. C59#24998semidirect

Maximal subgroups

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

subgroupindex
#214270532
#214256642
#214256542
#214255092
#214254992
#214250332
#214248622
#214247202
#214247132
#214246852
#214246712
#214242262
#214241062
C4×#142736482
C59×#25802
#2499859