Finite Group Databasebeta

← #28661537

Subgroups — #28661537

Order 15104.

706 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#21427121C211yesmaximal, characteristic
2#21425505C211yesmaximal, characteristic, complemented
3#21425488C211yesmaximal, characteristic, complemented
4#21425299C211yesmaximal, characteristic
5#21425299C211yesmaximal, characteristic
6#21424852C211yesmaximal, characteristic, complemented
7#21424826C211yesmaximal, characteristic, complemented
8#21424704C211yesmaximal, characteristic
9#21424678C211yesmaximal, characteristic, complemented
10#21424667C211yesmaximal, characteristic, complemented
11#21424363C211yesmaximal, characteristic, complemented
12#21424293C211yesmaximal, characteristic
13#21424188C211yesmaximal, characteristic
14C4×#14273650C211yesmaximal, characteristic, complemented
15C59×#2496C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837308C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837296—21no—
34#16837290—21no—
35#16837284—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37#16837253—21no—
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#126×C59—21no—
49C59×#400—21no—
50C59×#397—21no—
51C59×#389C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#387—21no—
54C59×#384C2211yescharacteristic
55C59×#380C2211yescharacteristic
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273706—22no—
65#14273706—22no—
66C4×Dic118—22no—
67C4×Dic118—41no—
68#14273686—22no—
69#14273686—41no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81#14273660—22no—
82#14273660—22no—
83#14273660—41no—
84#14273658—22no—
85C4⋊(C59⋊C8)—22no—
86#14273650—22no—
87#14273650—22no—
88#14273650—22no—
89#14273650—41no—
90C22⋊(C59⋊C8)—22no—
91C22⋊(C59⋊C8)—41no—
92C4×C59⋊C8—41no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C23×C4×C59C2311yescharacteristic
98#126×C59—44no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59D411yescharacteristic
110C2×C22⋊C4×C59D411yescharacteristic
111C2×C42×C59—21no—
112C2×C42×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59—21no—
119#95×C59D411yescharacteristic
120#95×C59D411yescharacteristic
121C24×C59C2×D411yescharacteristic
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—21no—
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—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—42no—
157C4⋊C4×C59—42no—
158C4⋊C4×C59—42no—
159C4⋊C4×C59—42no—
160C22⋊C4×C59—22no—
161C22⋊C4×C59—22no—
162C22⋊C4×C59—42no—
163C22⋊C4×C59—42no—
164C42×C59—21no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—42no—
169C22⋊Dic59—24no—
170C22⋊Dic59—42no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59#4112yes—
176C4⋊Dic59#4112yes—
177Dic59⋊C4—24no—
178Dic59⋊C4—42no—
179Dic59⋊C4—42no—
180C4×Dic59—42no—
181C4×Dic59—42no—
182C4×Dic59—42no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
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#4112yes—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59C22×D411yescharacteristic
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—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×C59AGL(1,ℤ/8)12yes—
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—22no—
241C2×Dic59—22no—
242C2×Dic59—22no—
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—
252Dic118—22no—
253Dic118—22no—
254Dic118—41no—
255Dic118—41no—
256C59⋊C8—22no—
257C59⋊C8—41no—
258C59⋊C8—41no—
259C59⋊C8—41no—
260#27043—591nomaximal, Sylow
261C22×C59—41no—
262C22×C59—41no—
263C22×C59—41no—
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×AGL(1,ℤ/8)11yescharacteristic
273C22×C59#57811yescharacteristic
274C22×C59#57811yescharacteristic
275C4×C59—22no—
276C4×C59—22no—
277C4×C59—41no—
278C4×C59—41no—
279C4×C59—41no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59#52911yescharacteristic
285C4×C59#53911yescharacteristic
286C4×C59#55311yescharacteristic
287C4×C59#55611yescharacteristic
288Dic59—42no—
289Dic59—42no—
290Dic59—42no—
291Dic59—81no—
292#2496D5911yescharacteristic
293#2362—591no—
294#2072—591no—
295#2068—591no—
296#2036—591no—
297#2036—591no—
298#1947—591no—
299#1942—591no—
300#1923—591no—
301#1920—591no—
302#1918—591no—
303#1865—591no—
304#1854—591no—
305#1825—591no—
306C4×#103—591no—
307C2×C59—81no—
308C2×C59#246611yescharacteristic
309C2×C59#310611yescharacteristic
310C2×C59#312711yescharacteristic
311C2×C59#330211yescharacteristic
312C2×C59#330811yescharacteristic
313C2×C59#333611yescharacteristic
314C2×C59#335311yescharacteristic
315C2×#126—21no—
316C2×C22⋊Q8—591no—
317C2×C4×Q8—591no—
318C2×#117—591no—
319C2×#107—591no—
320C2×#107—591no—
321C2×#107—591no—
322C2×#106—591no—
323C2×C4⋊C8—591no—
324C2×#103—591no—
325C2×#103—591no—
326C2×#103—591no—
327C2×#103—591no—
328C2×C22⋊C8—591no—
329C2×C22⋊C8—591no—
330C2×C4×C8—591no—
331#400—21no—
332#397—21no—
333#391—591no—
334#389—591no—
335#389C2×D5911yescharacteristic
336#389C2×D5911yescharacteristic
337#388—1181no—
338#388—1181no—
339#387—21no—
340#387—1181no—
341#384C2×D5911yescharacteristic
342#380C2×D5911yescharacteristic
343#380C2×D5911yescharacteristic
344C4×C4⋊C4—591no—
345C4×C4⋊C4—1181no—
346C4×C4⋊C4C2×D5911yescharacteristic
347C4×C22⋊C4C2×D5911yescharacteristic
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#336—591no—
357#336—591no—
358C59#2704311yescharacteristic, complemented, Sylow
359C22×Q8—591no—
360C23×C4C22×D5911yescharacteristic
361C22×C8—591no—
362C22×C8—591no—
363C22×C8—591no—
364C22×C8—591no—
365#126—44no—
366C22⋊Q8—1182no—
367C22⋊Q8—1182no—
368C4×Q8—1182no—
369C4×Q8—2361no—
370#117—1182no—
371#117—2361no—
372C2×C4⋊C4—21no—
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×C4⋊C4C22×D5911yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C22⋊C4C22⋊D5911yescharacteristic
391C2×C22⋊C4C22⋊D5911yescharacteristic
392C2×C42—21no—
393C2×C42—1181no—
394C2×C42—1181no—
395C2×C42—1181no—
396C2×C42C22×D5911yescharacteristic
397C2×C42C22×D5911yescharacteristic
398#107—1182no—
399#107—1182no—
400#107—2361no—
401#106—1182no—
402C4⋊C8—1182no—
403#103—1182no—
404#103—1182no—
405#103—1182no—
406#103—2361no—
407C22⋊C8—1182no—
408C22⋊C8—2361no—
409C4×C8—2361no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95C22⋊D5911yescharacteristic
420#95C22⋊D5911yescharacteristic
421C24C2×C22⋊D5911yescharacteristic
422C2×Q8—592no—
423C2×Q8—1181no—
424C2×Q8—1181no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C2×Q8—1181no—
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—21no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C23×D5911yescharacteristic
448C2×C8—592no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—592no—
473C4⋊C4—592no—
474C4⋊C4—1184no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—2362no—
480C4⋊C4—2362no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—1184no—
486C22⋊C4—2362no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—2362no—
493C42—2362no—
494C42—2362no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C22×C22⋊D5911yescharacteristic
503Q8—1182no—
504Q8—1182no—
505Q8—2361no—
506Q8—2361no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
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—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—1182no—
537C2×C4—1182no—
538C2×C4—1182no—
539C2×C4—2361no—
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#1427374712yes—
549C2×C4#1427375312yes—
550C2×C4#41×D5911yescharacteristic
551C2×C4#41×D5911yescharacteristic
552C2×C4#41×D5911yescharacteristic
553C2×C4#41×D5911yescharacteristic
554C2×C4#1427377311yescharacteristic
555C2×C4#1427377411yescharacteristic
556C8—1182no—
557C8—2361no—
558C8—2361no—
559C8—2361no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1683790911yescharacteristic
568C22#1683804011yescharacteristic
569C22#1683805111yescharacteristic
570C22C2×#1427374711yescharacteristic
571C22C2×#1427375311yescharacteristic
572C22#1683821711yescharacteristic
573C22#1683821711yescharacteristic
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—4721no—
584C4—2362no—
585C4—2362no—
586C4—2362no—
587C4#1683793511yescharacteristic
588C4#1683799311yescharacteristic
589C4#1683810311yescharacteristic
590C4#1683811911yescharacteristic
591C2—81no—
592C2#2142812811yescharacteristic
593C2#2143415011yescharacteristic
594C2#2143439111yescharacteristic
595C2#2143600711yescharacteristic
596C2#2143608111yescharacteristic
597C2#2143645811yescharacteristic
598C2#2143668011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2496C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2496D59
hypocenterC59#27043
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
2. #21425505C2semidirect
3. #21425488C2semidirect
6. #21424852C2semidirect
7. #21424826C2semidirect
9. #21424678C2semidirect
10. #21424667C2semidirect
11. #21424363C2semidirect
14. C4×#14273650C2semidirect
358. C59#27043semidirect

Maximal subgroups

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

subgroupindex
#214271212
#214255052
#214254882
#214252992
#214248522
#214248262
#214247042
#214246782
#214246672
#214243632
#214242932
#214241882
C4×#142736502
C59×#24962
#2704359