Finite Group Databasebeta

← #28646348

Subgroups — #28646348

Order 15104.

774 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426971C211yesmaximal, characteristic
2#21425653C211yesmaximal, characteristic, complemented
3#21425597C211yesmaximal, characteristic, complemented
4#21425597C211yesmaximal, characteristic, complemented
5#21425426C211yesmaximal, characteristic, complemented
6#21425049C211yesmaximal, characteristic, complemented
7#21424857C211yesmaximal, characteristic, complemented
8#21424671C211yesmaximal, characteristic, complemented
9#21424671C211yesmaximal, characteristic, complemented
10#21424289C211yesmaximal, characteristic
11#21424276C211yesmaximal, characteristic
12#21424184C211yesmaximal, characteristic
13#21424119C211yesmaximal, characteristic
14C2×#16837164C211yesmaximal, characteristic
15C59×#2440C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25#16837300C2211yescharacteristic
26#16837300C2211yescharacteristic
27#16837285—21no—
28#16837285—21no—
29#16837262C2211yescharacteristic
30#16837260—21no—
31C4×C4⋊Dic59C2211yescharacteristic
32C4×C4⋊Dic59C2211yescharacteristic
33C22⋊C4×Dic59—21no—
34#16837164C2211yescharacteristic
35#16837164C2211yescharacteristic
36#16837164C2211yescharacteristic
37#16837164C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4⋊Q8×C59C2211yescharacteristic
49C2×C22⋊Q8×C59—21no—
50C22×C4⋊C4×C59C2211yescharacteristic
51C59×#398—21no—
52C59×#395—21no—
53C59×#389C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#385—21no—
56C59×#384C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8—22no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—22no—
69C2×C4⋊Dic59Q814yescomplemented
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×C22⋊Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79#14273658—22no—
80#14273658—22no—
81#14273658—41no—
82#14273658—41no—
83C4⋊(C59⋊C8)—22no—
84C4⋊(C59⋊C8)—41no—
85C22⋊(C59⋊C8)—22no—
86C59⋊#95—21no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C22×Q8×C59—21no—
92C23×C4×C59—21no—
93C23×C4×C59C2311yescharacteristic
94C4⋊Q8×C59—22no—
95C4⋊Q8×C59—41no—
96C22⋊Q8×C59—44no—
97C2×C4⋊C4×C59—22no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—41no—
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59D411yescharacteristic
111C2×C22⋊C4×C59D411yescharacteristic
112C2×C42×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119C24×C59C2×D411yescharacteristic
120C2×Q8×C59—42no—
121C2×Q8×C59—42no—
122C2×Q8×C59—81no—
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×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×Q812yes—
140C22×C4×C59#4112yes—
141C22×C4×C59C2411yescharacteristic
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—24no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—81no—
148C22×Dic59—81no—
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—
160C4⋊C4×C59—42no—
161C4⋊C4×C59—81no—
162C4⋊C4×C59#4112yes—
163C22⋊C4×C59—22no—
164C22⋊C4×C59—42no—
165C22⋊C4×C59SD1614yes—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—21no—
170C22⋊Dic59—42no—
171C4⋊Dic59—24no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59C2×Q812yes—
179C4⋊Dic59C2×Q812yes—
180Dic59⋊C4—81no—
181Dic59⋊C4—81no—
182C4×Dic59—81no—
183C4×Dic59—81no—
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—21no—
195C2×C59⋊C8—44no—
196C2×C59⋊C8#4112yes—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59C22⋊Q812yes—
202C23×C59C2×SD1612yes—
203C23×C59#13712yes—
204C23×C59C22×D411yescharacteristic
205Q8×C59—84no—
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—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
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—41no—
233C2×C4×C59—41no—
234C2×C4×C59—84no—
235C2×C4×C59C22⋊Q812yes—
236C2×C4×C59C2×SD1612yes—
237C2×C4×C59C2×SD1612yes—
238C2×C4×C59C22×Q811yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59H(5,𝔽2)11yescharacteristic
243C2×C4×C592−1+411yescharacteristic
244C2×Dic59—22no—
245C2×Dic59—22no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—84no—
257C59⋊C8—22no—
258C59⋊C8—41no—
259C59⋊C8—41no—
260C59⋊C8—41no—
261#25992—591nomaximal, Sylow
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59#47514yes—
269C22×C59C2×C22⋊Q811yescharacteristic
270C22×C59#54611yescharacteristic
271C22×C59#54711yescharacteristic
272C22×C59C22×SD1611yescharacteristic
273C22×C59C2×#13711yescharacteristic
274C22×C59#57611yescharacteristic
275C22×C59#57711yescharacteristic
276C4×C59—22no—
277C4×C59—22no—
278C4×C59—22no—
279C4×C59—41no—
280C4×C59—42no—
281C4×C59—41no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59—81no—
285C4×C59#53311yescharacteristic
286C4×C59#54411yescharacteristic
287C4×C59#55111yescharacteristic
288C4×C59#55211yescharacteristic
289Dic59—42no—
290Dic59—42no—
291Dic59—81no—
292Dic59—81no—
293#2440D5911yescharacteristic
294#2345—591no—
295#2099—591no—
296#2088—591no—
297#2088—591no—
298#2058—591no—
299#1988—591no—
300#1948—591no—
301#1919—591no—
302#1919—591no—
303#1853—591no—
304#1850—591no—
305#1824—591no—
306#1807—591no—
307C2×#340—591no—
308C2×C59—24no—
309C2×C59#243811yescharacteristic
310C2×C59C2×#47511yescharacteristic
311C2×C59#308111yescharacteristic
312C2×C59#329511yescharacteristic
313C2×C59#332011yescharacteristic
314C2×C59#333511yescharacteristic
315C2×C59#334611yescharacteristic
316C23×C8—591no—
317C2×C4⋊Q8C2×D5911yescharacteristic
318C2×C22⋊Q8—21no—
319C22×C4⋊C4—591no—
320C22×C4⋊C4C2×D5911yescharacteristic
321C2×#106—591no—
322C2×#106—591no—
323C2×#106—591no—
324C2×#106—591no—
325C2×C4⋊C8—591no—
326C2×C4⋊C8—591no—
327C2×C22⋊C8—591no—
328#398—21no—
329#395—21no—
330#389—591no—
331#389—591no—
332#389C2×D5911yescharacteristic
333#389C2×D5911yescharacteristic
334#387—1181no—
335#387—1181no—
336#385—21no—
337#384C2×D5911yescharacteristic
338#380—591no—
339#380—1181no—
340#380C2×D5911yescharacteristic
341C4×C4⋊C4—591no—
342C4×C4⋊C4—591no—
343C4×C22⋊C4—1181no—
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#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#336—591no—
358#336—591no—
359C59#2599211yescharacteristic, complemented, Sylow
360C22×Q8—21no—
361C23×C4—21no—
362C23×C4—1181no—
363C23×C4C22×D5911yescharacteristic
364C22×C8—591no—
365C22×C8—591no—
366C22×C8—591no—
367C22×C8—591no—
368C22×C8—1182no—
369C22×C8—1182no—
370C22×C8—2361no—
371C4⋊Q8—22no—
372C4⋊Q8—41no—
373C22⋊Q8—44no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4—594no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—1182no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4C22⋊D5911yescharacteristic
398C2×C22⋊C4C22⋊D5911yescharacteristic
399C2×C42—1181no—
400C2×C42—1181no—
401C2×C42C22×D5911yescharacteristic
402C2×C42C22×D5911yescharacteristic
403#106—1182no—
404#106—1182no—
405#106—2361no—
406#106—2361no—
407C4⋊C8—1182no—
408C4⋊C8—2361no—
409C22⋊C8—1182no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95C22⋊D5911yescharacteristic
419#95C22⋊D5911yescharacteristic
420C24C2×C22⋊D5911yescharacteristic
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—81no—
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—42no—
433C22×C4—42no—
434C22×C4—81no—
435C22×C4—1184no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—4721no—
441C22×C4—4721no—
442C22×C4Q8×D5912yes—
443C22×C4#1174568912yes—
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C2×C22⋊D5911yescharacteristic
449C22×C4C23×D5911yescharacteristic
450C2×C8—592no—
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—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—2364no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—592no—
477C4⋊C4—592no—
478C4⋊C4—1184no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—4721no—
486C4⋊C4—4721no—
487C4⋊C4#1174568712yes—
488C22⋊C4—22no—
489C22⋊C4—42no—
490C22⋊C4—2362no—
491C22⋊C4#1174567014yes—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—4721no—
497C42—4721no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23#1427370312yes—
503C23C2×#1174567012yes—
504C23#1427375812yes—
505C23C22×C22⋊D5911yescharacteristic
506Q8—84no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
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—84no—
536C2×C4—1182no—
537C2×C4—1182no—
538C2×C4—4724no—
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—2361no—
549C2×C4#1427370312yes—
550C2×C4C2×#1174567012yes—
551C2×C4C2×#1174567012yes—
552C2×C4C2×Q8×D5911yescharacteristic
553C2×C4#41×D5911yescharacteristic
554C2×C4C2×#1174568711yescharacteristic
555C2×C4C2×#1174568911yescharacteristic
556C2×C4#1427377211yescharacteristic
557C2×C4#1427377411yescharacteristic
558C8—1182no—
559C8—2361no—
560C8—2361no—
561C8—2361no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22#1683768414yes—
569C22C2×#1427370311yescharacteristic
570C22#1683804611yescharacteristic
571C22#1683805111yescharacteristic
572C22C22×#1174567011yescharacteristic
573C22C2×#1427375811yescharacteristic
574C22#1683820511yescharacteristic
575C22#1683821011yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—4721no—
586C4—4721no—
587C4—2362no—
588C4—2362no—
589C4#1683795411yescharacteristic
590C4#1683803111yescharacteristic
591C4#1683807911yescharacteristic
592C4#1683809411yescharacteristic
593C2—24no—
594C2#2142776811yescharacteristic
595C2C2×#1683768411yescharacteristic
596C2#2143392111yescharacteristic
597C2#2143592011yescharacteristic
598C2#2143626711yescharacteristic
599C2#2143645311yescharacteristic
600C2#2143657511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2440C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2440D59
hypocenterC59#25992
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. #21425653C2semidirect
3. #21425597C2semidirect
4. #21425597C2semidirect
5. #21425426C2semidirect
6. #21425049C2semidirect
7. #21424857C2semidirect
8. #21424671C2semidirect
9. #21424671C2semidirect
69. C2×C4⋊Dic59Q8semidirect
359. C59#25992semidirect

Maximal subgroups

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

subgroupindex
#214269712
#214256532
#214255972
#214254262
#214250492
#214248572
#214246712
#214242892
#214242762
#214241842
#214241192
C2×#168371642
C59×#24402
#2599259