Finite Group Databasebeta

← #28661522

Subgroups — #28661522

Order 15104.

678 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427121C211yesmaximal, characteristic
2#21425515C211yesmaximal, characteristic, complemented
3#21425474C211yesmaximal, characteristic, complemented
4#21425299C211yesmaximal, characteristic
5#21425292C211yesmaximal, characteristic
6#21424862C211yesmaximal, characteristic, complemented
7#21424821C211yesmaximal, characteristic, complemented
8#21424704C211yesmaximal, characteristic
9#21424685C211yesmaximal, characteristic, complemented
10#21424667C211yesmaximal, characteristic, complemented
11#21424349C211yesmaximal, characteristic, complemented
12#21424293C211yesmaximal, characteristic
13#21424188C211yesmaximal, characteristic
14C4×#14273650C211yesmaximal, characteristic, complemented
15C59×#2495C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
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
48C59×#400—21no—
49C59×#397—21no—
50C59×#389C2211yescharacteristic
51C59×#389C2211yescharacteristic
52C59×#388—21no—
53C59×#383—21no—
54C59×#382C2211yescharacteristic
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#14273658—22no—
84#14273658—41no—
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
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C22⋊C4×C59—21no—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59D411yescharacteristic
107C2×C22⋊C4×C59D411yescharacteristic
108C2×C42×C59—21no—
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
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×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—
141C2×Dic118—21no—
142C2×Dic118—21no—
143C2×Dic118—21no—
144C2×Dic118—21no—
145C2×Dic118—21no—
146C2×Dic118#4112yes—
147C4⋊C4×C59—21no—
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—81no—
156C4⋊C4×C59—81no—
157C22⋊C4×C59—22no—
158C22⋊C4×C59—41no—
159C22⋊C4×C59—42no—
160C22⋊C4×C59—81no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—21no—
165C42×C59—42no—
166C22⋊Dic59—24no—
167C22⋊Dic59—42no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59#4112yes—
173C4⋊Dic59#4112yes—
174Dic59⋊C4—24no—
175Dic59⋊C4—42no—
176Dic59⋊C4—42no—
177C4×Dic59—42no—
178C4×Dic59—42no—
179C4×Dic59—42no—
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—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8#4112yes—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59C22×D411yescharacteristic
200C2×C4×C59—21no—
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—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—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#13512yes—
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59H(5,𝔽2)11yescharacteristic
236C2×C4×C592−1+411yescharacteristic
237C2×Dic59—22no—
238C2×Dic59—22no—
239C2×Dic59—22no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249Dic118—22no—
250Dic118—22no—
251Dic118—41no—
252Dic118—41no—
253C59⋊C8—22no—
254C59⋊C8—41no—
255C59⋊C8—41no—
256C59⋊C8—41no—
257#27042—591nomaximal, Sylow
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59—41no—
262C22×C59—41no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59#52511yescharacteristic
266C22×C59#54611yescharacteristic
267C22×C59#54711yescharacteristic
268C22×C59C2×#13511yescharacteristic
269C22×C59#57511yescharacteristic
270C22×C59#57711yescharacteristic
271C22×C59#57811yescharacteristic
272C4×C59—22no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—41no—
276C4×C59—41no—
277C4×C59—81no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59#52911yescharacteristic
282C4×C59#53911yescharacteristic
283C4×C59#55311yescharacteristic
284C4×C59#55611yescharacteristic
285Dic59—42no—
286Dic59—42no—
287Dic59—42no—
288Dic59—81no—
289#2495D5911yescharacteristic
290#2362—591no—
291#2074—591no—
292#2066—591no—
293#2036—591no—
294#2035—591no—
295#1949—591no—
296#1941—591no—
297#1923—591no—
298#1921—591no—
299#1918—591no—
300#1863—591no—
301#1854—591no—
302#1825—591no—
303C4×#103—591no—
304C2×C59—81no—
305C2×C59#246611yescharacteristic
306C2×C59#310211yescharacteristic
307C2×C59#312111yescharacteristic
308C2×C59#330011yescharacteristic
309C2×C59#330711yescharacteristic
310C2×C59#334011yescharacteristic
311C2×C59#335211yescharacteristic
312C2×C22⋊Q8—591no—
313C2×C4×Q8—591no—
314C2×#117—591no—
315C2×#107—591no—
316C2×#107—591no—
317C2×#106—591no—
318C2×#106—591no—
319C2×C4⋊C8—591no—
320C2×#103—591no—
321C2×#103—591no—
322C2×#103—591no—
323C2×#103—591no—
324C2×C22⋊C8—591no—
325C2×C22⋊C8—591no—
326C2×C4×C8—591no—
327#400—21no—
328#397—21no—
329#391—591no—
330#389—591no—
331#389C2×D5911yescharacteristic
332#389C2×D5911yescharacteristic
333#388—21no—
334#388—1181no—
335#388—1181no—
336#387—1181no—
337#383—21no—
338#382C2×D5911yescharacteristic
339#380C2×D5911yescharacteristic
340#380C2×D5911yescharacteristic
341C4×C4⋊C4—591no—
342C4×C4⋊C4—1181no—
343C4×C4⋊C4C2×D5911yescharacteristic
344C4×C22⋊C4C2×D5911yescharacteristic
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#336—591no—
354#336—591no—
355C59#2704211yescharacteristic, complemented, Sylow
356C22×Q8—591no—
357C23×C4C22×D5911yescharacteristic
358C22×C8—591no—
359C22×C8—591no—
360C22×C8—591no—
361C22×C8—591no—
362C22⋊Q8—1182no—
363C22⋊Q8—1182no—
364C4×Q8—1182no—
365C4×Q8—2361no—
366#117—1182no—
367#117—2361no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4C22×D5911yescharacteristic
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1181no—
383C2×C22⋊C4—1181no—
384C2×C22⋊C4C22⋊D5911yescharacteristic
385C2×C22⋊C4C22⋊D5911yescharacteristic
386C2×C42—21no—
387C2×C42—1181no—
388C2×C42—1181no—
389C2×C42—1181no—
390C2×C42C22×D5911yescharacteristic
391C2×C42C22×D5911yescharacteristic
392#107—1182no—
393#107—1182no—
394#106—1182no—
395#106—2361no—
396C4⋊C8—1182no—
397#103—1182no—
398#103—1182no—
399#103—1182no—
400#103—2361no—
401C22⋊C8—1182no—
402C22⋊C8—2361no—
403C4×C8—2361no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95C22⋊D5911yescharacteristic
416#95C22⋊D5911yescharacteristic
417C24C2×C22⋊D5911yescharacteristic
418C2×Q8—592no—
419C2×Q8—1181no—
420C2×Q8—1181no—
421C2×Q8—1181no—
422C2×Q8—1181no—
423C2×Q8—1181no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C2×C22⋊D5911yescharacteristic
440C22×C4C2×C22⋊D5911yescharacteristic
441C22×C4C2×C22⋊D5911yescharacteristic
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C23×D5911yescharacteristic
444C2×C8—592no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—592no—
467C4⋊C4—592no—
468C4⋊C4—1184no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—2362no—
474C4⋊C4—2362no—
475C22⋊C4—22no—
476C22⋊C4—41no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—1184no—
480C22⋊C4—2362no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—2362no—
487C42—2362no—
488C42—2362no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C22×C22⋊D5911yescharacteristic
497Q8—1182no—
498Q8—1182no—
499Q8—2361no—
500Q8—2361no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
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—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—41no—
530C2×C4—41no—
531C2×C4—1182no—
532C2×C4—1182no—
533C2×C4—1182no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4#1427374712yes—
544C2×C4#41×D5911yescharacteristic
545C2×C4#41×D5911yescharacteristic
546C2×C4#41×D5911yescharacteristic
547C2×C4#41×D5911yescharacteristic
548C2×C4#1427377311yescharacteristic
549C2×C4#1427377411yescharacteristic
550C8—1182no—
551C8—2361no—
552C8—2361no—
553C8—2361no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#1683790911yescharacteristic
562C22#1683804011yescharacteristic
563C22#1683805111yescharacteristic
564C22C2×#1427374711yescharacteristic
565C22#1683820011yescharacteristic
566C22#1683821311yescharacteristic
567C22#1683821711yescharacteristic
568C4—22no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—4721no—
578C4—2362no—
579C4—2362no—
580C4—2362no—
581C4#1683793511yescharacteristic
582C4#1683799311yescharacteristic
583C4#1683810311yescharacteristic
584C4#1683811911yescharacteristic
585C2—81no—
586C2#2142812811yescharacteristic
587C2#2143410811yescharacteristic
588C2#2143431911yescharacteristic
589C2#2143597711yescharacteristic
590C2#2143606611yescharacteristic
591C2#2143650911yescharacteristic
592C2#2143666511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2495C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2495D59
hypocenterC59#27042
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2495

Complements

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

NH ≅ G/Ntype
2. #21425515C2semidirect
3. #21425474C2semidirect
6. #21424862C2semidirect
7. #21424821C2semidirect
9. #21424685C2semidirect
10. #21424667C2semidirect
11. #21424349C2semidirect
14. C4×#14273650C2semidirect
355. C59#27042semidirect

Maximal subgroups

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

subgroupindex
#214271212
#214255152
#214254742
#214252992
#214252922
#214248622
#214248212
#214247042
#214246852
#214246672
#214243492
#214242932
#214241882
C4×#142736502
C59×#24952
#2704259