Finite Group Databasebeta

← #28610067

Subgroups — #28610067

Order 15104.

858 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837726C211yesmaximal, characteristic
2#21427134C211yesmaximal, characteristic
3#21426989C211yesmaximal, characteristic, complemented
4#21424838C211yesmaximal, characteristic, complemented
5#21424794C211yesmaximal, characteristic, complemented
6#21424714C211yesmaximal, characteristic, complemented
7#21424684C211yesmaximal, characteristic, complemented
8#21424665C211yesmaximal, characteristic, complemented
9#21424524C211yesmaximal, characteristic
10#21424358C211yesmaximal, characteristic, complemented
11#21424295C211yesmaximal, characteristic, complemented
12#21423992C211yesmaximal, characteristic, complemented
13#21423920C211yesmaximal, characteristic, complemented
14#21423872C211yesmaximal, characteristic, complemented
15C59×#1735C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×#14273689C2211yescharacteristic
18#16837726C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30#16837303C2211yescharacteristic
31#16837302—21no—
32#16837301C2211yescharacteristic
33#16837301C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837295—21no—
36#16837295—21no—
37#16837293—21no—
38#16837278—21no—
39#16837270—21no—
40#16837262C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4×C4⋊Dic59—21no—
45C4×(C2×D59)⋊C4—21no—
46C4×C22⋊Dic59C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C23×C8×C59C2211yescharacteristic
54C2×#117×C59C2211yescharacteristic
55C2×#117×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C22⋊D59—21no—
62C22×C4×D59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273701—22no—
66#14273701C2×C414yes—
67#14273689—22no—
68#14273689—41no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59D411yescharacteristic
77C2×C22⋊Dic59D411yescharacteristic
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2311yescharacteristic
86C2×C4×Dic59C2311yescharacteristic
87#14273660—41no—
88#14273660C2×C414yescomplemented
89#14273658—22no—
90#14273658C2×C414yescomplemented
91#14273649—22no—
92#14273649—41no—
93#14273647C2×C414yes—
94#14273647C2×C414yes—
95C22⋊(C59⋊C8)C2×C414yescomplemented
96#14273637—22no—
97#14273637—41no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C23×C4×C59C2311yescharacteristic
108C22×C8×C59—22no—
109C22×C8×C59—22no—
110C22×C8×C59—41no—
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113#117×C59—22no—
114#117×C59—22no—
115#117×C59—41no—
116#117×C59C2×C414yescomplemented
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59—21no—
125C24×C59C2×D411yescharacteristic
126C23×D59—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—21no—
135C22×C4×C59C22×C412yes—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59#4112yes—
138C22×C4×C59C2411yescharacteristic
139C2×C22⋊D59—24no—
140C2×C22⋊D59—81no—
141C2×C22⋊D59—81no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C2×D236—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236C22×C412yes—
155C2×C4×D59—42no—
156C2×C4×D59—42no—
157C2×C4×D59—81no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—44no—
164C2×C8×C59#4112yes—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59C22×C412yes—
171C4⋊C4×C59C22×C412yes—
172C4⋊C4×C59#4112yes—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—42no—
175C42×C59—24no—
176C42×C59—42no—
177C22⋊Dic59—41no—
178C22⋊Dic59—41no—
179C22⋊Dic59—81no—
180(C2×D59)⋊C4—24no—
181(C2×D59)⋊C4—42no—
182(C2×D59)⋊C4—42no—
183(C2×D59)⋊C4—81no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—42no—
190C4⋊Dic59—42no—
191C4⋊Dic59C22×C412yes—
192Dic59⋊C4—81no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C4×Dic59—21no—
197C4×Dic59—81no—
198C4×Dic59—81no—
199C4×Dic59—81no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8C22×C412yes—
205C2×C59⋊C8C22×C412yes—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59C4×D412yes—
213C23×C59C22×D411yescharacteristic
214C22×D59—22no—
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59C4×D412yes—
240C2×C4×C59C23×C411yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59H(5,𝔽2)11yescharacteristic
245C2×C4×C592−1+411yescharacteristic
246C22⋊D59—84no—
247C2×Dic59—22no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268D236—22no—
269D236—22no—
270D236—41no—
271D236—41no—
272C4×D59—84no—
273C8×C59—22no—
274C8×C59—41no—
275C59⋊C8—22no—
276C59⋊C8—22no—
277#22933—591nomaximal, Sylow
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—22no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59C2×C4×D411yescharacteristic
286C22×C59#54611yescharacteristic
287C22×C59#54711yescharacteristic
288C22×C59#57511yescharacteristic
289C22×C59#57511yescharacteristic
290C22×C59#57611yescharacteristic
291C22×C59#57711yescharacteristic
292C2×D59—42no—
293C2×D59—42no—
294C2×D59—42no—
295C2×D59—81no—
296C2×D59—81no—
297C2×D59—81no—
298C2×D59—81no—
299C4×C59—22no—
300C4×C59—42no—
301C4×C59—42no—
302C4×C59—42no—
303C4×C59—81no—
304C4×C59C4×#4111yescharacteristic
305C4×C59#52011yescharacteristic
306C4×C59#55111yescharacteristic
307C4×C59#55211yescharacteristic
308Dic59—41no—
309Dic59—41no—
310Dic59—42no—
311Dic59—41no—
312Dic59—41no—
313Dic59—81no—
314Dic59—81no—
315C2×#482—591no—
316#2363—591no—
317#2347—591no—
318#1944—591no—
319#1936—591no—
320#1925—591no—
321#1920—591no—
322#1918—591no—
323#1893—591no—
324#1864—591no—
325#1854—591no—
326#1776—591no—
327#1762—591no—
328#1752—591no—
329#1735D5911yescharacteristic, complemented
330C2×C59—42no—
331C2×C59#235911yescharacteristic
332C2×C59#294311yescharacteristic
333C2×C59#294611yescharacteristic
334C2×C59#330911yescharacteristic
335C2×C59#331111yescharacteristic
336C2×C59#334711yescharacteristic
337C2×C59#335011yescharacteristic
338D59—82no—
339C23×C8C2×D5911yescharacteristic
340C2×#121—591no—
341C2×C4×D4—591no—
342C2×#117C2×D5911yescharacteristic
343C2×#117C2×D5911yescharacteristic
344#482—598no—
345C2×#107—591no—
346C2×#107—591no—
347C2×#106—591no—
348C2×#106—591no—
349C2×#102—591no—
350C2×#102—591no—
351C2×#102—591no—
352C2×#102—591no—
353C2×C22⋊C8—591no—
354C2×#97—591no—
355C2×#97—591no—
356#390—591no—
357#390—1181no—
358#389—591no—
359#389—591no—
360#389—591no—
361#388—1181no—
362#388—1181no—
363#388—1181no—
364#386—1181no—
365#383—1181no—
366#380—591no—
367C4×C4⋊C4—591no—
368C4×C4⋊C4—591no—
369C4×C4⋊C4—591no—
370C4×C4⋊C4—1181no—
371C4×C22⋊C4—591no—
372C4×C22⋊C4—1181no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340C2×D5911yescharacteristic
378#340C2×D5911yescharacteristic
379#340C2×D5911yescharacteristic
380#340C2×D5911yescharacteristic
381#336—591no—
382#336—591no—
383C59#2293311yescharacteristic, complemented, Sylow
384C22×D4—591no—
385C22×D4—1181no—
386C23×C4—1181no—
387C23×C4C22×D5911yescharacteristic
388C22×C8—22no—
389C22×C8—22no—
390C22×C8—41no—
391C22×C8—591no—
392C22×C8—591no—
393C22×C8C22×D5911yescharacteristic
394C22×C8C22×D5911yescharacteristic
395#121—594no—
396#121—1182no—
397C4×D4—1182no—
398C4×D4—2361no—
399#117—22no—
400#117—22no—
401#117—41no—
402#117C4×D5914yescomplemented
403C2×C4⋊C4—591no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C4⋊C4C22×D5911yescharacteristic
411C2×C4⋊C4C22×D5911yescharacteristic
412C2×C4⋊C4C22×D5911yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—591no—
416C2×C22⋊C4—591no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—591no—
425C2×C42—591no—
426C2×C42—1181no—
427C2×C42—1181no—
428C2×C42—1181no—
429#107—594no—
430#107—2361no—
431#106—594no—
432#106—1182no—
433#102—594no—
434#102—594no—
435#102—1182no—
436#102—2361no—
437C22⋊C8—594no—
438#97—1182no—
439#97—2361no—
440#95—591no—
441#95—591no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449C24—1181no—
450C24D4×D5911yescharacteristic
451C2×D4—592no—
452C2×D4—1184no—
453C2×D4—1181no—
454C2×D4—1181no—
455C2×D4—1181no—
456C2×D4—1181no—
457C2×D4—1181no—
458C2×D4—4721no—
459C2×D4—4721no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—591no—
469C22×C4—591no—
470C22×C4—591no—
471C22×C4—591no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—4721no—
476C22×C4—2362no—
477C22×C4—2362no—
478C22×C4C2×C4×D5912yes—
479C22×C4D4×D5911yescharacteristic
480C22×C4#1174568712yes—
481C22×C4C23×D5911yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—44no—
488C2×C8—592no—
489C2×C8—592no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8#1174568512yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—592no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—4721no—
507C4⋊C4—2362no—
508C4⋊C4—2362no—
509C4⋊C4C2×C4×D5912yes—
510C4⋊C4C2×C4×D5912yes—
511C4⋊C4#1174568512yes—
512C22⋊C4—24no—
513C22⋊C4—42no—
514C22⋊C4—1184no—
515C22⋊C4—4721no—
516C22⋊C4—4721no—
517C22⋊C4—2362no—
518C22⋊C4—2362no—
519C22⋊C4—2361no—
520C22⋊C4—2361no—
521C42—24no—
522C42—42no—
523C42—1181no—
524C42—1181no—
525C42—1181no—
526C42—1181no—
527C42—4721no—
528C42—4721no—
529C42—4721no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—1182no—
537C23—2361no—
538C23—2361no—
539C23—2361no—
540C23—2361no—
541C23—2361no—
542C23—2361no—
543C23(C22⋊D59)⋊C412yes—
544C23C2×D4×D5911yescharacteristic
545D4—1182no—
546D4—1182no—
547D4—4724no—
548D4—2361no—
549D4—2361no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—1181no—
569C2×C4—1181no—
570C2×C4—1181no—
571C2×C4—1181no—
572C2×C4—1181no—
573C2×C4—1181no—
574C2×C4—1181no—
575C2×C4—1181no—
576C2×C4—1181no—
577C2×C4—1181no—
578C2×C4—1181no—
579C2×C4—1181no—
580C2×C4—1182no—
581C2×C4—4724no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4(C22⋊D59)⋊C412yes—
591C2×C4C22×C4×D5911yescharacteristic
592C2×C4C2×#1174568711yescharacteristic
593C2×C4C2×#1174568511yescharacteristic
594C2×C4C2×#1174568511yescharacteristic
595C2×C4#1427377211yescharacteristic
596C2×C4#1427377511yescharacteristic
597C8—22no—
598C8—41no—
599C8—1182no—
600C8—1182no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—4721no—
609C22—4721no—
610C22—4721no—
611C22—4721no—
612C22—2362no—
613C22—2362no—
614C22—2362no—
615C22C2×(C22⋊D59)⋊C411yescharacteristic
616C22#1683804211yescharacteristic
617C22#1683805311yescharacteristic
618C22#1683819911yescharacteristic
619C22#1683819911yescharacteristic
620C22#1683820411yescharacteristic
621C22#1683820911yescharacteristic
622C4—22no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—4721no—
628C4—4721no—
629C4—2362no—
630C4—2361no—
631C4—2361no—
632C4—2361no—
633C4—2361no—
634C4C4×#1174568511yescharacteristic
635C4#1683788311yescharacteristic
636C4#1683807811yescharacteristic
637C4#1683808711yescharacteristic
638C2—42no—
639C2—4722no—
640C2#2142708411yescharacteristic
641C2#2143289211yescharacteristic
642C2#2143293111yescharacteristic
643C2#2143609811yescharacteristic
644C2#2143612811yescharacteristic
645C2#2143659211yescharacteristic
646C2#2143663711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1735C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenter#1735D59
hypocenterC59#22933
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
3. #21426989C2semidirect
4. #21424838C2semidirect
5. #21424794C2semidirect
6. #21424714C2semidirect
7. #21424684C2semidirect
8. #21424665C2semidirect
10. #21424358C2semidirect
11. #21424295C2semidirect
12. #21423992C2semidirect
13. #21423920C2semidirect
14. #21423872C2semidirect
15. C59×#1735C2semidirect
18. #16837726C4semidirect
88. #14273660C2×C4semidirect
90. #14273658C2×C4semidirect
95. C22⋊(C59⋊C8)C2×C4semidirect
116. #117×C59C2×C4semidirect
329. #1735D59semidirect
383. C59#22933semidirect
402. #117C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377262
#214271342
#214269892
#214248382
#214247942
#214247142
#214246842
#214246652
#214245242
#214243582
#214242952
#214239922
#214239202
#214238722
C59×#17352
#2293359