Finite Group Databasebeta

← #28591097

Subgroups — #28591097

Order 15104.

1218 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837444C211yesmaximal, characteristic, complemented
2C2×#16837404C211yesmaximal, characteristic
3C2×#16837404C211yesmaximal, characteristic
4#21426586C211yesmaximal, characteristic
5C2×C4×C4⋊Dic59C211yesmaximal, characteristic, complemented
6#21424738C212yesmaximal, complemented
7#21424731C212yesmaximal, complemented
8#21424724C212yesmaximal, complemented
9#21424717C212yesmaximal, complemented
10#21423816C211yesmaximal, characteristic, complemented
11C59×#1728C211yesmaximal, characteristic, complemented
12C22×#11745685C2211yescharacteristic
13C2×#14273686C2211yescharacteristic
14C22×C4⋊Dic59C2211yescharacteristic
15C22×C4⋊Dic59C2211yescharacteristic
16C22×C4×Dic59—21no—
17#16837444—24no—
18C2×#14273661C2212yes—
19C2×#14273659C2212yes—
20#16837404—24no—
21#16837404C418yescomplemented
22C2×#14273652C2212yes—
23C2×#14273652C2212yes—
24C2×#14273649C2212yes—
25C2×#14273649C2212yes—
26#16837308C2211yescharacteristic
27#16837308C2211yescharacteristic
28#16837304C2211yescharacteristic
29#16837304C2211yescharacteristic
30#16837281—21no—
31#16837281—21no—
32C4×C4⋊Dic59—21no—
33C4×C4⋊Dic59—21no—
34C4×C4⋊Dic59C2212yes—
35C4×C4⋊Dic59C2212yes—
36C4×(C2×D59)⋊C4—21no—
37C4×(C2×D59)⋊C4—21no—
38C2×C59⋊#95—21no—
39#16837165C2212yes—
40#16837165C2212yes—
41C22×M16×C59C2211yescharacteristic
42C22×M16×C59C2211yescharacteristic
43C22×C42×C59C2211yescharacteristic
44C59×#336C2212yes—
45C59×#336C2212yes—
46C2×#11745685—22no—
47C2×#11745685—22no—
48C2×#11745685—22no—
49C2×#11745685—22no—
50C2×#11745685—22no—
51C2×#11745685C2×C414yes—
52C22×Dic118C2311yescharacteristic
53C22×D236C2311yescharacteristic
54C22×C22⋊D59—21no—
55C23×Dic59—21no—
56C23×Dic59—21no—
57C22×C4×D59—21no—
58#14273686—22no—
59#14273686—22no—
60C2×C4⋊Dic59—21no—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59—22no—
63C2×C4⋊Dic59—22no—
64C2×C4⋊Dic59C2×C414yes—
65C2×C4⋊Dic59D414yes—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2312yes—
69C2×C4⋊Dic59C2312yes—
70C2×C22⋊Dic59—21no—
71C2×(C2×D59)⋊C4—21no—
72C2×(C2×D59)⋊C4—21no—
73C2×(C2×D59)⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—22no—
77C2×C4×Dic59—22no—
78C2×C4×Dic59—44no—
79#14273661—24no—
80#14273659—24no—
81#14273652—24no—
82#14273652C2×C418yescomplemented
83#14273649—24no—
84#14273649C2×C418yescomplemented
85C59⋊#95—22no—
86C59⋊#95—22no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C23×C4×C59D411yescharacteristic
92C23×C4×C59D411yescharacteristic
93C23×C4×C59C2311yescharacteristic
94C2×M16×C59—22no—
95C2×M16×C59—22no—
96C2×M16×C59—22no—
97C2×M16×C59—22no—
98C2×M16×C59—22no—
99C2×M16×C59C2×C414yes—
100C22×C8×C59C2312yes—
101C22×C8×C59C2312yes—
102C2×C42×C59—22no—
103C2×C42×C59—22no—
104C2×C42×C59D411yescharacteristic
105C2×C42×C59D411yescharacteristic
106C2×C42×C59C2311yescharacteristic
107C2×C42×C59C2311yescharacteristic
108C24×C59C2×D411yescharacteristic
109C23×D59—21no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—41no—
115C22×C4×C59—41no—
116C22×C4×C59C22×C412yes—
117C22×C4×C59C2×D411yescharacteristic
118C22×C4×C59C2×D411yescharacteristic
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D412yes—
123C22×C4×C59#4112yes—
124C22×C4×C59#4112yes—
125C22×C4×C59#4112yes—
126C22×C4×C59#4112yes—
127C22×C4×C59C2411yescharacteristic
128C2×C22⋊D59—24no—
129C2×C22⋊D59—42no—
130C2×C22⋊D59—42no—
131C22×Dic59—24no—
132C22×Dic59—24no—
133C22×Dic59—21no—
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—42no—
139C22×Dic59—42no—
140C22×Dic59—42no—
141C22×Dic59—42no—
142#11745685—28no—
143#11745685—28no—
144#11745685—44no—
145#11745685—44no—
146C2×D236—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236C22×C412yes—
152C2×C4×D59—24no—
153C2×C4×D59—42no—
154C2×C4×D59—42no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118C22×C412yes—
161M16×C59—28no—
162M16×C59—44no—
163C2×C8×C59—22no—
164C2×C8×C59—22no—
165C2×C8×C59—22no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59C22×C414yes—
169C42×C59—22no—
170C42×C59—22no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C22⋊Dic59—42no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—42no—
179(C2×D59)⋊C4—42no—
180C4⋊Dic59—24no—
181C4⋊Dic59—24no—
182C4⋊Dic59—22no—
183C4⋊Dic59—22no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59C22×C414yes—
189C4⋊Dic59C2×D414yes—
190C4×Dic59—44no—
191C4×Dic59—42no—
192C4×Dic59—44no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59C4×D412yes—
198C23×C59#12112yes—
199C23×C59#12312yes—
200C23×C59C22×D411yescharacteristic
201C22×D59—22no—
202C22×D59—41no—
203C22×D59—41no—
204C22×D59—41no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208C2×C4×C59—28no—
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—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—44no—
220C2×C4×C59#11714yes—
221C2×C4×C59C4×D414yes—
222C2×C4×C59C4×D412yes—
223C2×C4×C59#12112yes—
224C2×C4×C59#12312yes—
225C2×C4×C59#12414yes—
226C2×C4×C59#12614yes—
227C2×C4×C59AGL(1,ℤ/8)12yes—
228C2×C4×C59#13712yes—
229C2×C4×C59C23×C411yescharacteristic
230C2×C4×C59C22×D411yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C22⋊D59—48no—
236C2×Dic59—24no—
237C2×Dic59—22no—
238C2×Dic59—24no—
239C2×Dic59—48no—
240C2×Dic59—48no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253D236—22no—
254D236—22no—
255D236—41no—
256D236—41no—
257C4×D59—48no—
258Dic118—22no—
259Dic118—22no—
260Dic118—41no—
261Dic118—41no—
262C8×C59—24no—
263C8×C59—42no—
264#21059—591nomaximal, Sylow
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—22no—
269C22×C59—22no—
270C22×C59—22no—
271C22×C59#38814yes—
272C22×C59C2×C4×D411yescharacteristic
273C22×C59C2×#12111yescharacteristic
274C22×C59C2×#12311yescharacteristic
275C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C59C2×#13711yescharacteristic
277C22×C59#57511yescharacteristic
278C22×C59#57511yescharacteristic
279C2×D59—42no—
280C2×D59—42no—
281C2×D59—42no—
282C2×D59—81no—
283C2×D59—81no—
284C2×D59—81no—
285C2×D59—81no—
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59#38814yes—
291C4×C59C2×#11711yescharacteristic
292C4×C59C2×C4×D411yescharacteristic
293C4×C59C2×#12411yescharacteristic
294C4×C59C2×#12611yescharacteristic
295Dic59—44no—
296Dic59—44no—
297Dic59—42no—
298Dic59—81no—
299Dic59—81no—
300C2×#428—591no—
301C2×#417—591no—
302C2×#417—591no—
303#2297—591no—
304C2×C4×C4⋊C4—591no—
305#1928—592no—
306#1927—592no—
307#1926—592no—
308#1925—592no—
309#1736—591no—
310#1728D5911yescharacteristic, complemented
311C2×C59—24no—
312C2×C59C2×#38811yescharacteristic
313C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C59C4×#13711yescharacteristic
315C2×C59#303611yescharacteristic
316C2×C59#303711yescharacteristic
317C2×C59#309211yescharacteristic
318C2×C59#309311yescharacteristic
319D59—82no—
320C22×#41—591no—
321C22×M16C2×D5911yescharacteristic
322C22×M16C2×D5911yescharacteristic
323C2×#117—591no—
324C22×C4⋊C4—591no—
325C22×C4⋊C4—591no—
326C22×C42—1181no—
327C22×C42C2×D5911yescharacteristic
328#428—1184no—
329C2×#107—592no—
330C2×#106—592no—
331#417—598no—
332#417—1184no—
333C2×#103—592no—
334C2×#103—592no—
335C2×#102—592no—
336C2×#102—592no—
337#391—591no—
338#391—591no—
339#390—591no—
340#390—591no—
341#386—1181no—
342#386—1181no—
343C4×C4⋊C4—592no—
344C4×C4⋊C4—592no—
345C4×C4⋊C4—1181no—
346C4×C4⋊C4—1181no—
347C4×C22⋊C4—1181no—
348C4×C22⋊C4—1181no—
349C2×#95—1181no—
350#340—592no—
351#340—592no—
352#336C2×D5912yes—
353#336C2×D5912yes—
354C59#2105911yescharacteristic, complemented, Sylow
355C2×#41—594no—
356C2×#41—1182no—
357C2×#41—1182no—
358C2×#41—1182no—
359C2×#41—1182no—
360C2×#41—1182no—
361C22×Q8—591no—
362C22×D4—591no—
363C22×D4—1181no—
364C23×C4—1181no—
365C23×C4—1181no—
366C23×C4—1181no—
367C23×C4D23611yescharacteristic
368C23×C4D23611yescharacteristic
369C23×C4C22×D5911yescharacteristic
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—22no—
373C2×M16—22no—
374C2×M16—22no—
375C2×M16C4×D5914yes—
376C22×C8C22×D5912yes—
377C22×C8C22×D5912yes—
378#117—1182no—
379#117—1182no—
380C2×C4⋊C4—594no—
381C2×C4⋊C4—594no—
382C2×C4⋊C4—592no—
383C2×C4⋊C4—592no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1182no—
389C2×C4⋊C4—1182no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C22⋊C4—1181no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—1181no—
398C2×C42—1182no—
399C2×C42—1182no—
400C2×C42—2364no—
401C2×C42D23611yescharacteristic
402C2×C42D23611yescharacteristic
403C2×C42C22×D5911yescharacteristic
404C2×C42C22×D5911yescharacteristic
405#107—1184no—
406#106—1184no—
407#103—598no—
408#103—1184no—
409#102—598no—
410#102—1184no—
411#95—1182no—
412#95—1182no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417C24—1181no—
418C24C2×D23611yescharacteristic
419#41—1188no—
420#41—1188no—
421#41—2364no—
422#41—2364no—
423C2×Q8—592no—
424C2×Q8—1181no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C2×Q8—1181no—
428C2×Q8—1181no—
429C2×D4—592no—
430C2×D4—1184no—
431C2×D4—1181no—
432C2×D4—1181no—
433C2×D4—1181no—
434C2×D4—1181no—
435C2×D4—1181no—
436C2×D4—2362no—
437C2×D4—2362no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—1184no—
445C22×C4—1184no—
446C22×C4—1181no—
447C22×C4—1182no—
448C22×C4—1182no—
449C22×C4—1184no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—2362no—
453C22×C4—2362no—
454C22×C4—2362no—
455C22×C4—2362no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4C2×C4×D5912yes—
459C22×C4C2×D23611yescharacteristic
460C22×C4C2×D23611yescharacteristic
461C22×C4C2×D23611yescharacteristic
462C22×C4C2×D23611yescharacteristic
463C22×C4C2×D23611yescharacteristic
464C22×C4#1174568512yes—
465C22×C4D4×D5912yes—
466C22×C4#1174568712yes—
467C22×C4#1174568712yes—
468C22×C4#1174568912yes—
469C22×C4C23×D5911yescharacteristic
470M16—28no—
471M16—44no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8C2×C4×D5914yes—
478C4⋊C4—594no—
479C4⋊C4—594no—
480C4⋊C4—1184no—
481C4⋊C4—1184no—
482C4⋊C4—1182no—
483C4⋊C4—1182no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C22⋊C4—2362no—
489C22⋊C4—2362no—
490C22⋊C4—2362no—
491C22⋊C4—2362no—
492C22⋊C4—2362no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—2364no—
500C42—2362no—
501C42—2364no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—1182no—
507C23—2361no—
508C23—2361no—
509C23—2361no—
510C23—2361no—
511C23—2361no—
512C23—2361no—
513C23C4×D23612yes—
514C23#1427370212yes—
515C23#1427371112yes—
516C23C22×D23611yescharacteristic
517Q8—1182no—
518Q8—1182no—
519Q8—2361no—
520Q8—2361no—
521D4—1182no—
522D4—1182no—
523D4—2368no—
524D4—2361no—
525D4—2361no—
526C2×C4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—1184no—
539C2×C4—1184no—
540C2×C4—1182no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2368no—
548C2×C4—2368no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2362no—
552C2×C4—2362no—
553C2×C4—2362no—
554C2×C4—2362no—
555C2×C4—2368no—
556C2×C4#1427368514yes—
557C2×C4(C22⋊D59)⋊C414yes—
558C2×C4C4×D23612yes—
559C2×C4#1427370212yes—
560C2×C4#1427371112yes—
561C2×C4#1427371314yes—
562C2×C4#1427372214yes—
563C2×C4#1427375512yes—
564C2×C4#1427376012yes—
565C2×C4C22×C4×D5911yescharacteristic
566C2×C4C2×D4×D5911yescharacteristic
567C2×C4C2×#1174568711yescharacteristic
568C2×C4C2×#1174568711yescharacteristic
569C2×C4C2×#1174568511yescharacteristic
570C2×C4C2×#1174568911yescharacteristic
571C8—24no—
572C8—42no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—4721no—
580C22—4721no—
581C22—4721no—
582C22—4721no—
583C22—2362no—
584C22—2362no—
585C22—2362no—
586C22#1683729414yes—
587C22C2×C4×D23611yescharacteristic
588C22C2×#1427370211yescharacteristic
589C22C2×#1427371111yescharacteristic
590C22C2×#1427375511yescharacteristic
591C22C2×#1427376011yescharacteristic
592C22#1683820211yescharacteristic
593C22#1683820211yescharacteristic
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—4721no—
599C4—4721no—
600C4—2364no—
601C4—2362no—
602C4—2364no—
603C4#1683729414yes—
604C4C2×#1427368511yescharacteristic
605C4C2×(C22⋊D59)⋊C411yescharacteristic
606C4C2×#1427371311yescharacteristic
607C4C2×#1427372211yescharacteristic
608C2—24no—
609C2—4722no—
610C2C2×#1683729411yescharacteristic
611C2C4×#1427375511yescharacteristic
612C2C4×#1427376011yescharacteristic
613C2#2143356211yescharacteristic
614C2#2143357011yescharacteristic
615C2#2143402111yescharacteristic
616C2#2143402911yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1728C2
socleC23×C59C22×D4
prime radicalC22×#11745685C22
cosocleC22×C4×C59C24
hypercenter#1728D59
hypocenterC59#21059
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
1. C2×#16837444C2semidirect
5. C2×C4×C4⋊Dic59C2semidirect
6. #21424738C2semidirect
7. #21424731C2semidirect
8. #21424724C2semidirect
9. #21424717C2semidirect
10. #21423816C2semidirect
11. C59×#1728C2semidirect
21. #16837404C4semidirect
82. #14273652C2×C4semidirect
84. #14273649C2×C4semidirect
310. #1728D59semidirect
354. C59#21059semidirect

Maximal subgroups

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

subgroupindex
C2×#168374442
C2×#168374042
#214265862
C2×C4×C4⋊Dic592
#214247382
#214247312
#214247242
#214247172
#214238162
C59×#17282
#2105959