Finite Group Databasebeta

← #28691815

Subgroups — #28691815

Order 15104.

724 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426573C211yesmaximal, characteristic, complemented
2#21425598C211yesmaximal, characteristic, complemented
3#21425488C211yesmaximal, characteristic, complemented
4#21425484C211yesmaximal, characteristic
5#21425413C211yesmaximal, characteristic
6#21424949C211yesmaximal, characteristic
7#21424826C211yesmaximal, characteristic, complemented
8#21424782C211yesmaximal, characteristic
9#21424771C211yesmaximal, characteristic, complemented
10#21424713C211yesmaximal, characteristic
11#21424678C211yesmaximal, characteristic, complemented
12#21424454C211yesmaximal, characteristic
13#21424349C211yesmaximal, characteristic, complemented
14C4×C4⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
15C59×#2860C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837308C2211yescharacteristic
32#16837308C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837286—21no—
36#16837286—21no—
37C4×Dic59⋊C4—21no—
38C4×C4⋊Dic59C2211yescharacteristic
39#16837248—21no—
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#126×C59—21no—
49C59×#390C2211yescharacteristic
50C59×#390C2211yescharacteristic
51C59×#390C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#388—21no—
57C59×#388—21no—
58C43×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C2×#11745670—22no—
61C2×#11745670—22no—
62C2×#11745670—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C4×Dic118—22no—
68C4×Dic118—22no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77#14273660—22no—
78#14273660—41no—
79#14273658—22no—
80C4⋊(C59⋊C8)—22no—
81C4⋊(C59⋊C8)—22no—
82#14273650—22no—
83#14273650—22no—
84#14273650—41no—
85#14273648—22no—
86#14273648—22no—
87#14273648—41no—
88C4×C59⋊C8—41no—
89C4×C59⋊C8—41no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×D4×C59C2311yescharacteristic
98#126×C59—44no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C42×C59D411yescharacteristic
112C2×C42×C59D411yescharacteristic
113C2×C42×C59D411yescharacteristic
114C2×C42×C59D411yescharacteristic
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121C24×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C2×Dic118—21no—
143C2×Dic118—21no—
144C2×Dic118—21no—
145C2×Dic118—21no—
146C2×Dic118—21no—
147C2×Dic118#4112yes—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—42no—
155C4⋊C4×C59—42no—
156C4⋊C4×C59—42no—
157C22⋊C4×C59—42no—
158C22⋊C4×C59—42no—
159C22⋊C4×C59—42no—
160C22⋊C4×C59—42no—
161C22⋊C4×C59—42no—
162C22⋊C4×C59—42no—
163C42×C59—22no—
164C42×C59—22no—
165C42×C59—22no—
166C42×C59—22no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173#11745670—44no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59#4112yes—
180Dic59⋊C4—24no—
181Dic59⋊C4—42no—
182Dic59⋊C4—42no—
183C4×Dic59—24no—
184C4×Dic59—42no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8#4112yes—
196C2×C59⋊C8#4112yes—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59C22×D411yescharacteristic
205D4×C59—41no—
206D4×C59—41no—
207D4×C59—41no—
208D4×C59—41no—
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—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59#13512yes—
233C2×C4×C59#13512yes—
234C2×C4×C59AGL(1,ℤ/8)12yes—
235C2×C4×C59#13712yes—
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59H(5,𝔽2)11yescharacteristic
241C2×C4×C592−1+411yescharacteristic
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254Dic118—22no—
255Dic118—22no—
256Dic118—41no—
257Dic118—41no—
258C59⋊C8—22no—
259C59⋊C8—22no—
260C59⋊C8—41no—
261C59⋊C8—41no—
262#29435—591nomaximal, Sylow
263C22×C59—81no—
264C22×C59—81no—
265C22×C59—81no—
266C22×C59—81no—
267C22×C59—81no—
268C22×C59—81no—
269C22×C59—81no—
270C22×C59#52511yescharacteristic
271C22×C59#53211yescharacteristic
272C22×C59#53711yescharacteristic
273C22×C59C2×#13511yescharacteristic
274C22×C59C2×#13511yescharacteristic
275C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C59C2×#13711yescharacteristic
277C4×C59—22no—
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—22no—
281C4×C59—41no—
282C4×C59—41no—
283C4×C59—81no—
284C4×C59—81no—
285C4×C59—81no—
286C4×C59#55111yescharacteristic
287C4×C59#55211yescharacteristic
288C4×C59#55311yescharacteristic
289C4×C59#55611yescharacteristic
290Dic59—42no—
291Dic59—42no—
292Dic59—81no—
293Dic59—81no—
294#2860D5911yescharacteristic
295#2295—591no—
296#2088—591no—
297#2068—591no—
298#2067—591no—
299#2055—591no—
300#1968—591no—
301#1942—591no—
302#1934—591no—
303#1933—591no—
304#1925—591no—
305#1920—591no—
306#1880—591no—
307#1863—591no—
308C4×C4⋊C8—591no—
309C2×C59—161no—
310C2×C59#260111yescharacteristic
311C2×C59#310611yescharacteristic
312C2×C59#310811yescharacteristic
313C2×C59#316811yescharacteristic
314C2×C59#318211yescharacteristic
315C2×C59#324711yescharacteristic
316C2×C59#324811yescharacteristic
317C22×SD16—591no—
318C2×#126—21no—
319C2×C4×Q8—591no—
320C2×#107—591no—
321C2×#107—591no—
322C2×#106—591no—
323C2×C4⋊C8—591no—
324C2×C4⋊C8—591no—
325C2×#103—591no—
326C2×#103—591no—
327C2×#103—591no—
328C2×#102—591no—
329C2×#102—591no—
330C2×#102—591no—
331C2×C4×C8—591no—
332C2×C4×C8—591no—
333#391—591no—
334#391—591no—
335#390C2×D5911yescharacteristic
336#390C2×D5911yescharacteristic
337#390C2×D5911yescharacteristic
338#389—591no—
339#389—591no—
340#389C2×D5911yescharacteristic
341#389C2×D5911yescharacteristic
342#389C2×D5911yescharacteristic
343#388—21no—
344#388—21no—
345#388—21no—
346#387—1181no—
347#387—1181no—
348C4×C4⋊C4—591no—
349C4×C4⋊C4—1181no—
350#376—1181no—
351C43C2×D5911yescharacteristic
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#336—591no—
359#336—591no—
360C59#2943511yescharacteristic, complemented, Sylow
361C22×Q8—591no—
362C22×D4C22×D5911yescharacteristic
363C2×SD16—1182no—
364C2×SD16—1182no—
365C2×SD16—2361no—
366C22×C8—591no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8—591no—
370#126—44no—
371C4×Q8—1182no—
372C4×Q8—1182no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—1181no—
392C2×C42—1181no—
393C2×C42C22⋊D5911yescharacteristic
394C2×C42C22⋊D5911yescharacteristic
395C2×C42C22⋊D5911yescharacteristic
396C2×C42C22⋊D5911yescharacteristic
397C2×C42C22×D5911yescharacteristic
398C2×C42C22×D5911yescharacteristic
399C2×C42C22×D5911yescharacteristic
400#107—1182no—
401#107—2361no—
402#106—1182no—
403C4⋊C8—1182no—
404C4⋊C8—1182no—
405#103—1182no—
406#103—1182no—
407#103—2361no—
408#102—1182no—
409#102—1182no—
410#102—2361no—
411C4×C8—2361no—
412C4×C8—2361no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423C24—21no—
424C2×Q8—592no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C2×Q8—1181no—
428C2×Q8—1181no—
429C2×Q8—1181no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C2×C22⋊D5911yescharacteristic
449C22×C4C23×D5911yescharacteristic
450SD16—2364no—
451C2×C8—592no—
452C2×C8—592no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—592no—
473C4⋊C4—1184no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—2362no—
480C4⋊C4—2362no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—1184no—
498C42—2362no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C22⋊D5911yescharacteristic
507Q8—1182no—
508Q8—1182no—
509Q8—2361no—
510Q8—2361no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—1182no—
539C2×C4—1182no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4#1427374712yes—
551C2×C4#1427374712yes—
552C2×C4#1427375312yes—
553C2×C4#1427375712yes—
554C2×C4#41×D5911yescharacteristic
555C2×C4#41×D5911yescharacteristic
556C2×C4#41×D5911yescharacteristic
557C2×C4#41×D5911yescharacteristic
558C2×C4#1427377311yescharacteristic
559C2×C4#1427377411yescharacteristic
560C8—1182no—
561C8—1182no—
562C8—2361no—
563C8—2361no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#1683790911yescharacteristic
572C22#1683794811yescharacteristic
573C22#1683798011yescharacteristic
574C22C2×#1427374711yescharacteristic
575C22C2×#1427374711yescharacteristic
576C22C2×#1427375311yescharacteristic
577C22C2×#1427375711yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—4721no—
588C4—4721no—
589C4—2362no—
590C4—2362no—
591C4#1683807711yescharacteristic
592C4#1683808811yescharacteristic
593C4#1683810311yescharacteristic
594C4#1683811911yescharacteristic
595C2—161no—
596C2#2142969511yescharacteristic
597C2#2143415011yescharacteristic
598C2#2143417211yescharacteristic
599C2#2143466011yescharacteristic
600C2#2143481011yescharacteristic
601C2#2143541611yescharacteristic
602C2#2143543311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2860C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2860D59
hypocenterC59#29435
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #21426573C2semidirect
2. #21425598C2semidirect
3. #21425488C2semidirect
7. #21424826C2semidirect
9. #21424771C2semidirect
11. #21424678C2semidirect
13. #21424349C2semidirect
14. C4×C4⋊(C59⋊C8)C2semidirect
360. C59#29435semidirect

Maximal subgroups

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

subgroupindex
#214265732
#214255982
#214254882
#214254842
#214254132
#214249492
#214248262
#214247822
#214247712
#214247132
#214246782
#214244542
#214243492
C4×C4⋊(C59⋊C8)2
C59×#28602
#2943559