Finite Group Databasebeta

← #28661450

Subgroups — #28661450

Order 15104.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428492C211yesmaximal, characteristic
2#21427101C211yesmaximal, characteristic
3#21425518C211yesmaximal, characteristic, complemented
4#21425478C211yesmaximal, characteristic, complemented
5#21425300C211yesmaximal, characteristic
6#21425292C211yesmaximal, characteristic
7#21424867C211yesmaximal, characteristic, complemented
8#21424823C211yesmaximal, characteristic, complemented
9#21424699C211yesmaximal, characteristic
10#21424687C211yesmaximal, characteristic, complemented
11#21424679C211yesmaximal, characteristic, complemented
12#21424365C211yesmaximal, characteristic, complemented
13#21424289C211yesmaximal, characteristic
14#21423915C211yesmaximal, characteristic, complemented
15C59×#1823C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×#14273692C2211yescharacteristic
18C2×#14273661C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273652C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29C59⋊#401—21no—
30#16837330—21no—
31#16837307C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837297—21no—
36#16837295—21no—
37#16837287—21no—
38#16837283—21no—
39#16837275—21no—
40#16837267C2211yescharacteristic
41#16837262C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4×C22⋊Dic59—21no—
45C4×C22⋊Dic59C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C22×C4⋊C4×C59C2211yescharacteristic
53C2×C22⋊C8×C59C2211yescharacteristic
54C59×#380C2211yescharacteristic
55C59×#340C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273706—22no—
63#14273706—22no—
64#14273692—22no—
65#14273692—41no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C22⋊Dic59—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59D411yescharacteristic
77C2×C22⋊Dic59D411yescharacteristic
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59C2311yescharacteristic
82C2×C4×Dic59C2311yescharacteristic
83#14273661—22no—
84#14273658—22no—
85#14273658—22no—
86#14273658—41no—
87Dic59⋊C8—22no—
88#14273652—22no—
89#14273652—41no—
90#14273650—22no—
91#14273650—22no—
92C22⋊(C59⋊C8)—22no—
93C8×Dic59—41no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95D411yescharacteristic
105C59⋊#95D411yescharacteristic
106C23×C4×C59—21no—
107C23×C4×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C4⋊C4×C59—22no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—41no—
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C22⋊C8×C59—41no—
119#95×C59—21no—
120C24×C59C2×D411yescharacteristic
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—42no—
132C22×C4×C59—42no—
133C22×C4×C59—81no—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2411yescharacteristic
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C22×Dic59C2×D411yescharacteristic
143C22×Dic59C2×D411yescharacteristic
144C2×Dic118—21no—
145C2×Dic118—21no—
146C2×Dic118—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118#4112yes—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59#4112yes—
165C22⋊C4×C59—42no—
166C22⋊Dic59—24no—
167C22⋊Dic59—22no—
168C22⋊Dic59—22no—
169C22⋊Dic59—81no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—42no—
176C4⋊Dic59#4112yes—
177Dic59⋊C4—24no—
178Dic59⋊C4—42no—
179Dic59⋊C4—81no—
180Dic59⋊C4—81no—
181C4×Dic59—21no—
182C4×Dic59—21no—
183C4×Dic59—21no—
184C4×Dic59—21no—
185C4×Dic59—42no—
186C4×Dic59—81no—
187C4×Dic59—81no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8#4112yes—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59#13512yes—
201C23×C59C22×D411yescharacteristic
202C2×C4×C59—21no—
203C2×C4×C59—22no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
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—84no—
222C2×C4×C59C2×#4111yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C59C2×#4111yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59H(5,𝔽2)11yescharacteristic
227C2×C4×C592−1+411yescharacteristic
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—22no—
231C2×Dic59—22no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59#13712yes—
252Dic118—22no—
253Dic118—22no—
254Dic118—41no—
255Dic118—41no—
256C8×C59—41no—
257C8×C59—41no—
258C59⋊C8—22no—
259C59⋊C8—41no—
260#27037—591nomaximal, Sylow
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59#52511yescharacteristic
269C22×C59#54611yescharacteristic
270C22×C59#54711yescharacteristic
271C22×C59C2×#13511yescharacteristic
272C22×C59C2×#13711yescharacteristic
273C22×C59#57711yescharacteristic
274C22×C59#57711yescharacteristic
275C4×C59—41no—
276C4×C59—42no—
277C4×C59—81no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59#52911yescharacteristic
281C4×C59#53911yescharacteristic
282C4×C59#55311yescharacteristic
283C4×C59#55611yescharacteristic
284Dic59—22no—
285Dic59—41no—
286Dic59—41no—
287Dic59—42no—
288Dic59—42no—
289Dic59—41no—
290Dic59—81no—
291Dic59—81no—
292#2494—591no—
293#2360—591no—
294#2074—591no—
295#2066—591no—
296#2036—591no—
297#2035—591no—
298#1949—591no—
299#1941—591no—
300#1923—591no—
301#1921—591no—
302#1920—591no—
303#1865—591no—
304#1853—591no—
305#1823D5911yescharacteristic
306C4×#103—591no—
307C2×C59—42no—
308C2×C59#246611yescharacteristic
309C2×C59#310811yescharacteristic
310C2×C59#312511yescharacteristic
311C2×C59#329611yescharacteristic
312C2×C59#332011yescharacteristic
313C2×C59#333711yescharacteristic
314C2×C59#335611yescharacteristic
315C2×C22⋊Q8—591no—
316C2×C4×Q8—591no—
317C22×C4⋊C4C2×D5911yescharacteristic
318C2×#107—591no—
319C2×#106—591no—
320C2×#106—591no—
321C2×#106—591no—
322C2×C4⋊C8—591no—
323C2×#103—591no—
324C2×#103—591no—
325C2×#103—591no—
326C2×#103—591no—
327C2×C22⋊C8—591no—
328C2×C22⋊C8C2×D5911yescharacteristic
329C2×C4×C8—591no—
330#401—1181no—
331#397—1181no—
332#391—591no—
333#389—591no—
334#389—591no—
335#389—591no—
336#389—1181no—
337#388—1181no—
338#387—1181no—
339#387—1181no—
340#385—1181no—
341#382—591no—
342#380—591no—
343#380C2×D5911yescharacteristic
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C22⋊C4—591no—
347C4×C22⋊C4—1181no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340C2×D5911yescharacteristic
353#340C2×D5911yescharacteristic
354#340C2×D5911yescharacteristic
355#340C2×D5911yescharacteristic
356#336—591no—
357#336—591no—
358C59#2703711yescharacteristic, complemented, Sylow
359C22×Q8—591no—
360C23×C4—21no—
361C23×C4C22×D5911yescharacteristic
362C22×C8—591no—
363C22×C8—591no—
364C22×C8C22×D5911yescharacteristic
365C22×C8C22×D5911yescharacteristic
366C22⋊Q8—1182no—
367C22⋊Q8—1182no—
368C4×Q8—1182no—
369C4×Q8—2361no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—591no—
387C2×C22⋊C4—591no—
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C42—591no—
391C2×C42—591no—
392C2×C42—1181no—
393C2×C42—1181no—
394C2×C42—1181no—
395#107—1182no—
396#106—1182no—
397#106—1182no—
398#106—2361no—
399C4⋊C8—1182no—
400#103—1182no—
401#103—1182no—
402#103—1182no—
403#103—2361no—
404C22⋊C8—41no—
405C22⋊C8—1182no—
406C4×C8—2361no—
407#95—21no—
408#95—591no—
409#95—591no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420C24D4×D5911yescharacteristic
421C2×Q8—592no—
422C2×Q8—1181no—
423C2×Q8—1181no—
424C2×Q8—1181no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—591no—
441C22×C4—591no—
442C22×C4—591no—
443C22×C4—591no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4D4×D5911yescharacteristic
449C22×C4C23×D5911yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—592no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—592no—
471C4⋊C4—1184no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—4721no—
478C4⋊C4—4721no—
479C4⋊C4—2362no—
480C4⋊C4—2362no—
481C4⋊C4#1174568512yes—
482C22⋊C4—42no—
483C22⋊C4—1182no—
484C22⋊C4—1184no—
485C22⋊C4—1182no—
486C22⋊C4—4721no—
487C42—1181no—
488C42—1181no—
489C42—1181no—
490C42—1181no—
491C42—4721no—
492C42—4721no—
493C42—2362no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1427374612yes—
501C23C2×D4×D5911yescharacteristic
502Q8—1182no—
503Q8—1182no—
504Q8—2361no—
505Q8—2361no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
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—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—592no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1182no—
530C2×C4—1182no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
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#41×D5911yescharacteristic
551C2×C4#41×D5911yescharacteristic
552C2×C4#41×D5911yescharacteristic
553C2×C4C2×#1174568511yescharacteristic
554C2×C4#1427377211yescharacteristic
555C2×C4#1427377411yescharacteristic
556C8—41no—
557C8—41no—
558C8—1182no—
559C8—2361no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1683791011yescharacteristic
568C22#1683804211yescharacteristic
569C22#1683805011yescharacteristic
570C22C2×#1427374611yescharacteristic
571C22#137×D5911yescharacteristic
572C22#1683820811yescharacteristic
573C22#1683821111yescharacteristic
574C4—42no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—1182no—
580C4—4721no—
581C4—4721no—
582C4—2362no—
583C4—2361no—
584C4—2361no—
585C4—2361no—
586C4—2362no—
587C4#1683793011yescharacteristic
588C4#1683799511yescharacteristic
589C4#1683809711yescharacteristic
590C4#1683812011yescharacteristic
591C2—42no—
592C2#2142813111yescharacteristic
593C2#2143417311yescharacteristic
594C2#2143436811yescharacteristic
595C2#2143592711yescharacteristic
596C2#2143626811yescharacteristic
597C2#2143647911yescharacteristic
598C2#2143670411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1823C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1823D59
hypocenterC59#27037
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. #21425518C2semidirect
4. #21425478C2semidirect
7. #21424867C2semidirect
8. #21424823C2semidirect
10. #21424687C2semidirect
11. #21424679C2semidirect
12. #21424365C2semidirect
14. #21423915C2semidirect
358. C59#27037semidirect

Maximal subgroups

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

subgroupindex
#214284922
#214271012
#214255182
#214254782
#214253002
#214252922
#214248672
#214248232
#214246992
#214246872
#214246792
#214243652
#214242892
#214239152
C59×#18232
#2703759