Finite Group Databasebeta

← #28661435

Subgroups — #28661435

Order 15104.

708 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#21428525C211yesmaximal, characteristic
2#21427101C211yesmaximal, characteristic
3#21425508C211yesmaximal, characteristic, complemented
4#21425492C211yesmaximal, characteristic, complemented
5#21425300C211yesmaximal, characteristic
6#21425299C211yesmaximal, characteristic
7#21424853C211yesmaximal, characteristic, complemented
8#21424828C211yesmaximal, characteristic, complemented
9#21424699C211yesmaximal, characteristic
10#21424687C211yesmaximal, characteristic, complemented
11#21424686C211yesmaximal, characteristic, complemented
12#21424351C211yesmaximal, characteristic, complemented
13#21424284C211yesmaximal, characteristic
14#21423915C211yesmaximal, characteristic, complemented
15C59×#1824C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×#14273692C2211yescharacteristic
18C2×#14273661C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273652C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837342—21no—
30#16837325—21no—
31#16837307C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837297—21no—
36#16837290—21no—
37#16837284—21no—
38#16837283—21no—
39#16837275—21no—
40#16837271C2211yescharacteristic
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×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59D411yescharacteristic
79C2×C22⋊Dic59D411yescharacteristic
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59C2311yescharacteristic
84C2×C4×Dic59C2311yescharacteristic
85#14273661—22no—
86#14273660—22no—
87#14273660—22no—
88#14273660—41no—
89Dic59⋊C8—22no—
90#14273652—22no—
91#14273652—41no—
92#14273650—22no—
93#14273650—22no—
94C22⋊(C59⋊C8)—22no—
95C8×Dic59—41no—
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—81no—
166C22⋊Dic59—24no—
167C22⋊Dic59—41no—
168C22⋊Dic59—81no—
169C22⋊Dic59Q1614yes—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—81no—
176C4⋊Dic59#4112yes—
177Dic59⋊C4—24no—
178Dic59⋊C4—42no—
179Dic59⋊C4—42no—
180Dic59⋊C4—81no—
181Dic59⋊C4—81no—
182Dic59⋊C4—81no—
183C4×Dic59—21no—
184C4×Dic59—21no—
185C4×Dic59—21no—
186C4×Dic59—21no—
187C4×Dic59—42no—
188C4×Dic59—81no—
189C4×Dic59—81no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8#4112yes—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59C2×Q1612yes—
203C23×C59C22×D411yescharacteristic
204C2×C4×C59—21no—
205C2×C4×C59—22no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
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—41no—
222C2×C4×C59—41no—
223C2×C4×C59—84no—
224C2×C4×C59C2×#4111yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59H(5,𝔽2)11yescharacteristic
229C2×C4×C592−1+411yescharacteristic
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—22no—
235C2×Dic59—22no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
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—41no—
252C2×Dic59C2×Q1612yes—
253C2×Dic59C2×Q1612yes—
254Dic118—22no—
255Dic118—22no—
256Dic118—41no—
257Dic118—41no—
258C8×C59—41no—
259C8×C59—41no—
260C59⋊C8—22no—
261C59⋊C8—41no—
262#27036—591nomaximal, Sylow
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59#52511yescharacteristic
271C22×C59#54611yescharacteristic
272C22×C59#54711yescharacteristic
273C22×C59C22×Q1611yescharacteristic
274C22×C59#57511yescharacteristic
275C22×C59#57611yescharacteristic
276C22×C59#57711yescharacteristic
277C4×C59—41no—
278C4×C59—42no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59#52911yescharacteristic
283C4×C59#53911yescharacteristic
284C4×C59#55311yescharacteristic
285C4×C59#55611yescharacteristic
286Dic59—22no—
287Dic59—22no—
288Dic59—41no—
289Dic59—42no—
290Dic59—42no—
291Dic59—41no—
292Dic59—81no—
293Dic59—81no—
294#2497—591no—
295#2360—591no—
296#2072—591no—
297#2068—591no—
298#2036—591no—
299#2036—591no—
300#1947—591no—
301#1942—591no—
302#1923—591no—
303#1921—591no—
304#1921—591no—
305#1863—591no—
306#1852—591no—
307#1824D5911yescharacteristic
308C4×#103—591no—
309C2×C59—42no—
310C2×C59#246611yescharacteristic
311C2×C59#310411yescharacteristic
312C2×C59#311911yescharacteristic
313C2×C59#329911yescharacteristic
314C2×C59#331911yescharacteristic
315C2×C59#333811yescharacteristic
316C2×C59#335111yescharacteristic
317C2×C22⋊Q8—591no—
318C2×C4×Q8—591no—
319C22×C4⋊C4C2×D5911yescharacteristic
320C2×#107—591no—
321C2×#107—591no—
322C2×#107—591no—
323C2×#107—591no—
324C2×C4⋊C8—591no—
325C2×#103—591no—
326C2×#103—591no—
327C2×#103—591no—
328C2×#103—591no—
329C2×C22⋊C8—591no—
330C2×C22⋊C8C2×D5911yescharacteristic
331C2×C4×C8—591no—
332#400—1181no—
333#396—1181no—
334#391—591no—
335#389—591no—
336#389—591no—
337#389—591no—
338#389—1181no—
339#388—1181no—
340#387—1181no—
341#387—1181no—
342#385—1181no—
343#384—591no—
344#380—591no—
345#380C2×D5911yescharacteristic
346C4×C4⋊C4—591no—
347C4×C4⋊C4—591no—
348C4×C22⋊C4—591no—
349C4×C22⋊C4—1181no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340C2×D5911yescharacteristic
355#340C2×D5911yescharacteristic
356#340C2×D5911yescharacteristic
357#340C2×D5911yescharacteristic
358#336—591no—
359#336—591no—
360C59#2703611yescharacteristic, complemented, Sylow
361C22×Q8—591no—
362C23×C4—21no—
363C23×C4C22×D5911yescharacteristic
364C22×C8—591no—
365C22×C8—591no—
366C22×C8C22×D5911yescharacteristic
367C22×C8C22×D5911yescharacteristic
368C22⋊Q8—1182no—
369C22⋊Q8—1182no—
370C4×Q8—1182no—
371C4×Q8—2361no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—41no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—591no—
391C2×C22⋊C4—591no—
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C42—591no—
395C2×C42—591no—
396C2×C42—1181no—
397C2×C42—1181no—
398C2×C42—1181no—
399#107—1182no—
400#107—1182no—
401#107—1182no—
402#107—2361no—
403C4⋊C8—1182no—
404#103—1182no—
405#103—1182no—
406#103—1182no—
407#103—2361no—
408C22⋊C8—41no—
409C22⋊C8—1182no—
410C4×C8—2361no—
411#95—21no—
412#95—591no—
413#95—591no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422C24D4×D5911yescharacteristic
423C2×Q8—592no—
424C2×Q8—1181no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C2×Q8—1181no—
428C2×Q8—1181no—
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—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—591no—
443C22×C4—591no—
444C22×C4—591no—
445C22×C4—591no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4D4×D5911yescharacteristic
451C22×C4C23×D5911yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—592no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
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—4721no—
480C4⋊C4—4721no—
481C4⋊C4—4721no—
482C4⋊C4—4721no—
483C4⋊C4—2362no—
484C4⋊C4—2362no—
485C4⋊C4#1174568512yes—
486C22⋊C4—81no—
487C22⋊C4—594no—
488C22⋊C4—1184no—
489C22⋊C4—4721no—
490C22⋊C4—2361no—
491C42—1181no—
492C42—1181no—
493C42—1181no—
494C42—1181no—
495C42—4721no—
496C42—4721no—
497C42—2362no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23Q16×D5912yes—
505C23C2×D4×D5911yescharacteristic
506Q8—1182no—
507Q8—1182no—
508Q8—2361no—
509Q8—2361no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
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—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—592no—
531C2×C4—592no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1182no—
537C2×C4—1182no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4#41×D5911yescharacteristic
555C2×C4#41×D5911yescharacteristic
556C2×C4#41×D5911yescharacteristic
557C2×C4C2×#1174568511yescharacteristic
558C2×C4#1427377211yescharacteristic
559C2×C4#1427377411yescharacteristic
560C8—41no—
561C8—41no—
562C8—1182no—
563C8—2361no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#1683791011yescharacteristic
572C22#1683804211yescharacteristic
573C22#1683805011yescharacteristic
574C22C2×Q16×D5911yescharacteristic
575C22#1683819811yescharacteristic
576C22#1683820611yescharacteristic
577C22#1683820811yescharacteristic
578C4—42no—
579C4—41no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—1182no—
584C4—1182no—
585C4—4721no—
586C4—4721no—
587C4—2361no—
588C4—2362no—
589C4—2362no—
590C4—2361no—
591C4#1683793011yescharacteristic
592C4#1683799511yescharacteristic
593C4#1683809711yescharacteristic
594C4#1683812011yescharacteristic
595C2—42no—
596C2#2142813111yescharacteristic
597C2#2143413211yescharacteristic
598C2#2143429011yescharacteristic
599C2#2143596811yescharacteristic
600C2#2143625311yescharacteristic
601C2#2143648311yescharacteristic
602C2#2143666011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1824C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1824D59
hypocenterC59#27036
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. #21425508C2semidirect
4. #21425492C2semidirect
7. #21424853C2semidirect
8. #21424828C2semidirect
10. #21424687C2semidirect
11. #21424686C2semidirect
12. #21424351C2semidirect
14. #21423915C2semidirect
360. C59#27036semidirect

Maximal subgroups

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

subgroupindex
#214285252
#214271012
#214255082
#214254922
#214253002
#214252992
#214248532
#214248282
#214246992
#214246872
#214246862
#214243512
#214242842
#214239152
C59×#18242
#2703659