Finite Group Databasebeta

← #28610058

Subgroups — #28610058

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#21427099C211yesmaximal, characteristic
3#21426971C211yesmaximal, characteristic
4#21424834C211yesmaximal, characteristic, complemented
5#21424786C211yesmaximal, characteristic, complemented
6#21424731C211yesmaximal, characteristic, complemented
7#21424668C211yesmaximal, characteristic, complemented
8#21424668C211yesmaximal, characteristic, complemented
9#21424519C211yesmaximal, characteristic
10#21424367C211yesmaximal, characteristic, complemented
11#21424291C211yesmaximal, characteristic
12#21423993C211yesmaximal, characteristic, complemented
13#21423927C211yesmaximal, characteristic, complemented
14C2×#16837165C211yesmaximal, characteristic
15C59×#1752C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837738C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28C2×#14273652C2211yescharacteristic
29#16837308C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837299—21no—
34#16837289—21no—
35#16837285—21no—
36#16837285—21no—
37#16837274—21no—
38#16837262C2211yescharacteristic
39#16837260—21no—
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C22⋊C4×Dic59—21no—
44C22⋊C4×Dic59—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C23×C8×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C4×C22⋊C4×C59C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C23×Dic59—21no—
62C23×Dic59—21no—
63#14273706—22no—
64#14273706C2×C414yes—
65C4×Dic118—22no—
66C4×Dic118—41no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—22no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—41no—
72C2×C4⋊Dic59C2×C414yes—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×C22⋊Dic59—21no—
86C2×C22⋊Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91#14273661C2×C414yescomplemented
92#14273661C2×C414yescomplemented
93#14273659—22no—
94#14273659—41no—
95#14273652—22no—
96#14273652—41no—
97#14273652C2×C414yescomplemented
98#14273652C2×C414yescomplemented
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⋊#95—21no—
108C23×C4×C59C2311yescharacteristic
109C22×C8×C59—22no—
110C22×C8×C59—22no—
111C22×C8×C59—41no—
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C22⋊C8×C59C2×C414yes—
121#97×C59—22no—
122#97×C59—41no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59C22×C412yes—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59#4112yes—
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—24no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—42no—
148C22×Dic59—42no—
149C22×Dic59—81no—
150C22×Dic59—81no—
151C22×Dic59—81no—
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×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—44no—
168C2×C8×C59C22×C412yes—
169C2×C8×C59C22×C412yes—
170C2×C8×C59#4112yes—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—22no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C22⋊Dic59—24no—
178C22⋊Dic59—42no—
179C4⋊Dic59—24no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—42no—
191C4⋊Dic59C22×C412yes—
192C4⋊Dic59C22×C412yes—
193C4⋊Dic59C22×C412yes—
194C4⋊Dic59#4112yes—
195Dic59⋊C4—24no—
196Dic59⋊C4—42no—
197Dic59⋊C4—81no—
198Dic59⋊C4—81no—
199Dic59⋊C4—81no—
200C4×Dic59—42no—
201C4×Dic59—81no—
202C4×Dic59—81no—
203C4×Dic59—81no—
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—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59C4×D412yes—
231C2×C4×C59C23×C411yescharacteristic
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—22no—
241C2×Dic59—22no—
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×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—84no—
262C2×Dic59—84no—
263Dic118—22no—
264Dic118—22no—
265Dic118—41no—
266Dic118—41no—
267C8×C59—22no—
268C8×C59—22no—
269C8×C59—22no—
270C8×C59—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—41no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—41no—
291C4×C59C4×#4111yescharacteristic
292C4×C59#52011yescharacteristic
293C4×C59#55111yescharacteristic
294C4×C59#55211yescharacteristic
295Dic59—42no—
296Dic59—42no—
297Dic59—42no—
298Dic59—42no—
299Dic59—42no—
300Dic59—81no—
301Dic59—81no—
302Dic59—81no—
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#1853—591no—
314#1776—591no—
315#1763—591no—
316#1752D5911yescharacteristic
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×C8C2×D5911yescharacteristic
327C2×C22⋊Q8—591no—
328C2×C4×Q8—591no—
329C22×C4⋊C4—591no—
330C22×C4⋊C4—591no—
331#484—598no—
332C2×#107—591no—
333C2×#107—591no—
334C2×#106—591no—
335C2×#106—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×C22⋊C8C2×D5911yescharacteristic
341C2×#97C2×D5911yescharacteristic
342C2×#97C2×D5911yescharacteristic
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—591no—
353#380—1181no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357C4×C22⋊C4—1181no—
358C4×C22⋊C4—1181no—
359C4×C22⋊C4C2×D5911yescharacteristic
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#336C2×D5911yescharacteristic
369#336C2×D5911yescharacteristic
370C59#2293211yescharacteristic, complemented, Sylow
371C22×Q8—591no—
372C23×C4—1181no—
373C23×C4—1181no—
374C23×C4C22×D5911yescharacteristic
375C22×C8—22no—
376C22×C8—22no—
377C22×C8—41no—
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C22×C8C22×D5911yescharacteristic
381C22×C8C22×D5911yescharacteristic
382C22⋊Q8—594no—
383C22⋊Q8—1182no—
384C4×Q8—1182no—
385C4×Q8—2361no—
386C2×C4⋊C4—594no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—1182no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1182no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—2361no—
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1181no—
406C2×C22⋊C4D23611yescharacteristic
407C2×C22⋊C4D23611yescharacteristic
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42C22×D5911yescharacteristic
413C2×C42C22×D5911yescharacteristic
414#107—594no—
415#107—594no—
416#106—1182no—
417#106—2361no—
418#103—594no—
419#103—594no—
420#103—1182no—
421#103—2361no—
422C22⋊C8C4×D5914yes—
423#97—22no—
424#97—41no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95D23611yescharacteristic
435#95D23611yescharacteristic
436C24C2×D23611yescharacteristic
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—1184no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—4721no—
457C22×C4—4721no—
458C22×C4—4721no—
459C22×C4—2362no—
460C22×C4—2362no—
461C22×C4C2×C4×D5912yes—
462C22×C4C2×D23611yescharacteristic
463C22×C4C2×D23611yescharacteristic
464C22×C4C2×D23611yescharacteristic
465C22×C4C2×D23611yescharacteristic
466C22×C4C2×D23611yescharacteristic
467C22×C4#1174568512yes—
468C22×C4C23×D5911yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—44no—
479C2×C8C2×C4×D5912yes—
480C2×C8C2×C4×D5912yes—
481C2×C8#1174568712yes—
482C4⋊C4—592no—
483C4⋊C4—592no—
484C4⋊C4—592no—
485C4⋊C4—592no—
486C4⋊C4—1184no—
487C4⋊C4—1184no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—4721no—
499C4⋊C4—4721no—
500C4⋊C4—4721no—
501C4⋊C4—2362no—
502C4⋊C4—2362no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—1184no—
506C22⋊C4—2362no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—4721no—
512C42—4721no—
513C42—4721no—
514C42—2362no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23C4×D23612yes—
520C23#1427376012yes—
521C23#1427376012yes—
522C23C22×D23611yescharacteristic
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—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—1182no—
546C2×C4—1182no—
547C2×C4—1182no—
548C2×C4—1182no—
549C2×C4—1182no—
550C2×C4—4724no—
551C2×C4—4724no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
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×C4C4×D23612yes—
572C2×C4C22×C4×D5911yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4C2×#1174568711yescharacteristic
575C2×C4C2×#1174568511yescharacteristic
576C2×C4#1427377211yescharacteristic
577C2×C4#1427377411yescharacteristic
578C8—22no—
579C8—22no—
580C8—22no—
581C8—41no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#1683751014yes—
589C22C2×C4×D23611yescharacteristic
590C22#1683804711yescharacteristic
591C22#1683805611yescharacteristic
592C22C2×#1427376011yescharacteristic
593C22C2×#1427376011yescharacteristic
594C22#1683820711yescharacteristic
595C22#1683821411yescharacteristic
596C4—22no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—4721no—
602C4—4721no—
603C4—4721no—
604C4—2362no—
605C4—2362no—
606C4—2362no—
607C4—2362no—
608C4—2362no—
609C4C4×#1174568711yescharacteristic
610C4#1683788111yescharacteristic
611C4#1683807811yescharacteristic
612C4#1683808511yescharacteristic
613C2—24no—
614C2#2142709011yescharacteristic
615C2C2×#1683751011yescharacteristic
616C2#2143293011yescharacteristic
617C2#2143617911yescharacteristic
618C2#2143626911yescharacteristic
619C2#2143655311yescharacteristic
620C2#2143658311yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1752C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1752D59
hypocenterC59#22932
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
4. #21424834C2semidirect
5. #21424786C2semidirect
6. #21424731C2semidirect
7. #21424668C2semidirect
8. #21424668C2semidirect
10. #21424367C2semidirect
12. #21423993C2semidirect
13. #21423927C2semidirect
20. #16837738C4semidirect
91. #14273661C2×C4semidirect
92. #14273661C2×C4semidirect
97. #14273652C2×C4semidirect
98. #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
#214270992
#214269712
#214248342
#214247862
#214247312
#214246682
#214245192
#214243672
#214242912
#214239932
#214239272
C2×#168371652
C59×#17522
#2293259