Finite Group Databasebeta

← #28610054

Subgroups — #28610054

Order 15104.

846 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837738C211yesmaximal, characteristic
2#21427101C211yesmaximal, characteristic
3#21426975C211yesmaximal, characteristic
4#21424832C211yesmaximal, characteristic, complemented
5#21424784C211yesmaximal, characteristic, complemented
6#21424729C211yesmaximal, characteristic, complemented
7#21424666C211yesmaximal, characteristic, complemented
8#21424666C211yesmaximal, characteristic, complemented
9#21424517C211yesmaximal, characteristic
10#21424361C211yesmaximal, characteristic, complemented
11#21423996C211yesmaximal, characteristic, complemented
12#21423925C211yesmaximal, characteristic, complemented
13#21423875C211yesmaximal, characteristic
14C2×#16837164C211yesmaximal, characteristic
15C59×#1853C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×#14273692C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837738C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273652C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273637C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29#16837307C2211yescharacteristic
30#16837298C2211yescharacteristic
31#16837298C2211yescharacteristic
32#16837298C2211yescharacteristic
33#16837297—21no—
34#16837287—21no—
35#16837287—21no—
36#16837283—21no—
37#16837275—21no—
38#16837260—21no—
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C22⋊C4×Dic59—21no—
43C4×C22⋊Dic59—21no—
44C4×C22⋊Dic59C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×#106×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C23×Dic59—21no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8—22no—
64C22×C59⋊C8—41no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273706—22no—
68#14273706C2×C414yes—
69#14273692—22no—
70#14273692—41no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—22no—
73C2×C4⋊Dic59C2×C414yes—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59D411yescharacteristic
84C2×C22⋊Dic59D411yescharacteristic
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59C2311yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91#14273661C2×C414yescomplemented
92#14273661C2×C414yescomplemented
93#14273652C2×C414yescomplemented
94#14273652C2×C414yescomplemented
95#14273650—22no—
96#14273650—41no—
97#14273637—22no—
98#14273637—41no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C23×C4×C59—21no—
110C23×C4×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C2×C4⋊C4×C59—22no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—41no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C22⋊C4×C59—21no—
121#106×C59—22no—
122#106×C59—41no—
123C22⋊C8×C59C2×C414yes—
124#95×C59—21no—
125C24×C59C2×D411yescharacteristic
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×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—42no—
135C22×C4×C59—42no—
136C22×C4×C59—81no—
137C22×C4×C59C22×C412yes—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59#4112yes—
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—24no—
146C22×Dic59—81no—
147C22×Dic59—81no—
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118C22×C412yes—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C412yes—
163C2×C8×C59C22×C412yes—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59#4112yes—
173C22⋊C4×C59—42no—
174C22⋊Dic59—24no—
175C22⋊Dic59—22no—
176C22⋊Dic59—22no—
177C4⋊Dic59—24no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C22×C412yes—
183C4⋊Dic59C22×C412yes—
184Dic59⋊C4—24no—
185Dic59⋊C4—42no—
186Dic59⋊C4—81no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—42no—
194C4×Dic59—81no—
195C4×Dic59—81no—
196C4×Dic59—81no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—44no—
203C2×C59⋊C8#4112yes—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59C4×D412yes—
209C23×C59#13712yes—
210C23×C59#13712yes—
211C23×C59C22×D411yescharacteristic
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
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—84no—
231C2×C4×C59C4×D412yes—
232C2×C4×C59C23×C411yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59H(5,𝔽2)11yescharacteristic
237C2×C4×C592−1+411yescharacteristic
238C2×Dic59—22no—
239C2×Dic59—22no—
240C2×Dic59—22no—
241C2×Dic59—22no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—84no—
263Dic118—22no—
264Dic118—22no—
265Dic118—41no—
266Dic118—41no—
267C8×C59—22no—
268C8×C59—22no—
269C59⋊C8—22no—
270C59⋊C8—41no—
271#22932—591nomaximal, Sylow
272C22×C59—22no—
273C22×C59—22no—
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—22no—
277C22×C59—22no—
278C22×C59#44114yes—
279C22×C59C2×C4×D411yescharacteristic
280C22×C59#54611yescharacteristic
281C22×C59#54711yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59C2×#13711yescharacteristic
284C22×C59#57611yescharacteristic
285C22×C59#57711yescharacteristic
286C4×C59—22no—
287C4×C59—42no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59—81no—
291C4×C59C4×#4111yescharacteristic
292C4×C59#52011yescharacteristic
293C4×C59#55111yescharacteristic
294C4×C59#55211yescharacteristic
295Dic59—41no—
296Dic59—41no—
297Dic59—42no—
298Dic59—42no—
299Dic59—42no—
300Dic59—42no—
301Dic59—41no—
302Dic59—41no—
303C2×#484—591no—
304#2360—591no—
305#2345—591no—
306#1943—591no—
307#1935—591no—
308#1927—591no—
309#1918—591no—
310#1918—591no—
311#1892—591no—
312#1865—591no—
313#1853D5911yescharacteristic
314#1776—591no—
315#1763—591no—
316#1752—591no—
317C2×#340—591no—
318C2×C59—24no—
319C2×C59#235911yescharacteristic
320C2×C59C2×#44111yescharacteristic
321C2×C59#294611yescharacteristic
322C2×C59#331411yescharacteristic
323C2×C59#332011yescharacteristic
324C2×C59#334411yescharacteristic
325C2×C59#334611yescharacteristic
326C23×C8—591no—
327C2×C22⋊Q8—591no—
328C2×C4×Q8—591no—
329C22×C4⋊C4—591no—
330C22×C4⋊C4C2×D5911yescharacteristic
331#484—598no—
332C2×#107—591no—
333C2×#107—591no—
334C2×#106C2×D5911yescharacteristic
335C2×#106C2×D5911yescharacteristic
336C2×#103—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×C22⋊C8C2×D5911yescharacteristic
341C2×#97—591no—
342C2×#97—591no—
343#391—591no—
344#389—591no—
345#389—591no—
346#389—591no—
347#389—1181no—
348#387—1181no—
349#387—1181no—
350#387—1181no—
351#385—1181no—
352#380—1181no—
353#380C2×D5911yescharacteristic
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357C4×C22⋊C4—591no—
358C4×C22⋊C4—1181no—
359C4×C22⋊C4—1181no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340C2×D5911yescharacteristic
367#340C2×D5911yescharacteristic
368#336—591no—
369#336—591no—
370C59#2293211yescharacteristic, complemented, Sylow
371C22×Q8—591no—
372C23×C4—21no—
373C23×C4—1181no—
374C23×C4C22×D5911yescharacteristic
375C22×C8—591no—
376C22×C8—591no—
377C22×C8—1182no—
378C22×C8—1182no—
379C22×C8—2361no—
380C22×C8C22×D5911yescharacteristic
381C22×C8C22×D5911yescharacteristic
382C22⋊Q8—594no—
383C22⋊Q8—1182no—
384C4×Q8—1182no—
385C4×Q8—2361no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—41no—
389C2×C4⋊C4—594no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1182no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—591no—
406C2×C22⋊C4—591no—
407C2×C22⋊C4—1181no—
408C2×C42—591no—
409C2×C42—591no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42—1181no—
414#107—594no—
415#107—594no—
416#106—22no—
417#106—41no—
418#103—594no—
419#103—594no—
420#103—1182no—
421#103—2361no—
422C22⋊C8C4×D5914yes—
423#97—1182no—
424#97—2361no—
425#95—21no—
426#95—591no—
427#95—591no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436C24D4×D5911yescharacteristic
437C2×Q8—592no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—1181no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—81no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—591no—
458C22×C4—1184no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—4721no—
464C22×C4—4721no—
465C22×C4C2×C4×D5912yes—
466C22×C4D4×D5911yescharacteristic
467C22×C4#1174568912yes—
468C22×C4C23×D5911yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—592no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—2364no—
480C2×C8C2×C4×D5912yes—
481C2×C8C2×C4×D5912yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—592no—
491C4⋊C4—592no—
492C4⋊C4—592no—
493C4⋊C4—1184no—
494C4⋊C4—1184no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—4721no—
499C4⋊C4—4721no—
500C4⋊C4—4721no—
501C4⋊C4—2362no—
502C4⋊C4#1174568712yes—
503C22⋊C4—42no—
504C22⋊C4—1184no—
505C22⋊C4—1182no—
506C22⋊C4—1182no—
507C42—1181no—
508C42—1181no—
509C42—1181no—
510C42—1181no—
511C42—4721no—
512C42—4721no—
513C42—4721no—
514C42—2362no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#1427368912yes—
520C23#1427375612yes—
521C23#1427375612yes—
522C23C2×D4×D5911yescharacteristic
523Q8—1182no—
524Q8—1182no—
525Q8—2361no—
526Q8—2361no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1182no—
559C2×C4—1182no—
560C2×C4—1182no—
561C2×C4—1182no—
562C2×C4—4724no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4#1427368912yes—
572C2×C4C22×C4×D5911yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4C2×#1174568711yescharacteristic
575C2×C4C2×#1174568911yescharacteristic
576C2×C4#1427377211yescharacteristic
577C2×C4#1427377411yescharacteristic
578C8—22no—
579C8—22no—
580C8—1182no—
581C8—2361no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#1683750814yes—
589C22C2×#1427368911yescharacteristic
590C22#1683803911yescharacteristic
591C22#1683805011yescharacteristic
592C22C2×#1427375611yescharacteristic
593C22C2×#1427375611yescharacteristic
594C22#1683820611yescharacteristic
595C22#1683821111yescharacteristic
596C4—22no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—2361no—
602C4—2362no—
603C4—2362no—
604C4—2362no—
605C4—2361no—
606C4—2361no—
607C4—2362no—
608C4—2361no—
609C4C4×#1174568711yescharacteristic
610C4#1683788111yescharacteristic
611C4#1683807911yescharacteristic
612C4#1683809411yescharacteristic
613C2—24no—
614C2#2142708611yescharacteristic
615C2C2×#1683750811yescharacteristic
616C2#2143292611yescharacteristic
617C2#2143618111yescharacteristic
618C2#2143627111yescharacteristic
619C2#2143655511yescharacteristic
620C2#2143657911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1853C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1853D59
hypocenterC59#22932
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #21424832C2semidirect
5. #21424784C2semidirect
6. #21424729C2semidirect
7. #21424666C2semidirect
8. #21424666C2semidirect
10. #21424361C2semidirect
11. #21423996C2semidirect
12. #21423925C2semidirect
20. #16837738C4semidirect
91. #14273661C2×C4semidirect
92. #14273661C2×C4semidirect
93. #14273652C2×C4semidirect
94. #14273652C2×C4semidirect
370. C59#22932semidirect

Maximal subgroups

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

subgroupindex
C2×#168377382
#214271012
#214269752
#214248322
#214247842
#214247292
#214246662
#214245172
#214243612
#214239962
#214239252
#214238752
C2×#168371642
C59×#18532
#2293259