Finite Group Databasebeta

← #28610103

Subgroups — #28610103

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×#16837727C211yesmaximal, characteristic
2#21427132C211yesmaximal, characteristic
3#21426979C211yesmaximal, characteristic, complemented
4#21424834C211yesmaximal, characteristic, complemented
5#21424793C211yesmaximal, characteristic, complemented
6#21424724C211yesmaximal, characteristic, complemented
7#21424675C211yesmaximal, characteristic, complemented
8#21424668C211yesmaximal, characteristic, complemented
9#21424524C211yesmaximal, characteristic
10#21424346C211yesmaximal, characteristic, complemented
11#21424297C211yesmaximal, characteristic, complemented
12#21423993C211yesmaximal, characteristic, complemented
13#21423922C211yesmaximal, characteristic, complemented
14#21423813C211yesmaximal, characteristic, complemented
15C59×#1752C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837727C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28C2×#14273649C2211yescharacteristic
29#16837304C2211yescharacteristic
30#16837302—21no—
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837291—21no—
35#16837290—21no—
36#16837290—21no—
37#16837278—21no—
38#16837266—21no—
39#16837262C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43#16837249—21no—
44#16837247—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×D236C2311yescharacteristic
61C22×C22⋊D59—21no—
62C22×C4×D59—21no—
63#14273701—22no—
64#14273701C2×C414yes—
65C4×D236—22no—
66C4×D236—41no—
67#14273686—22no—
68#14273686—22no—
69#14273686—41no—
70#14273686C2×C414yescomplemented
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93#14273661—22no—
94#14273661C2×C414yescomplemented
95#14273659—41no—
96#14273659C2×C414yescomplemented
97#14273649—22no—
98#14273649—41no—
99#14273649C2×C414yes—
100#14273649C2×C414yes—
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×C414yescomplemented
121#97×C59—22no—
122#97×C59—41no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C23×D59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C22×C412yes—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
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×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
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—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—41no—
172C22⋊C4×C59—41no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C22⋊Dic59—24no—
178C22⋊Dic59—42no—
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—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59C22×C412yes—
195C4⋊Dic59C22×C412yes—
196C4⋊Dic59C22×C412yes—
197C4⋊Dic59#4112yes—
198Dic59⋊C4—81no—
199Dic59⋊C4—81no—
200Dic59⋊C4—81no—
201C4×Dic59—24no—
202C4×Dic59—42no—
203C4×Dic59—42no—
204C4×Dic59—42no—
205C4×Dic59—81no—
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—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
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—22no—
249C2×Dic59—22no—
250C2×Dic59—22no—
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—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—22no—
275C8×C59—22no—
276C8×C59—41no—
277#22935—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—41no—
301C4×C59—41no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59C4×#4111yescharacteristic
305C4×C59#52011yescharacteristic
306C4×C59#55111yescharacteristic
307C4×C59#55211yescharacteristic
308Dic59—42no—
309Dic59—42no—
310Dic59—42no—
311Dic59—42no—
312Dic59—81no—
313Dic59—81no—
314Dic59—81no—
315C2×#482—591no—
316#2363—591no—
317#2346—591no—
318#1943—591no—
319#1936—591no—
320#1926—591no—
321#1919—591no—
322#1918—591no—
323#1893—591no—
324#1862—591no—
325#1854—591no—
326#1776—591no—
327#1762—591no—
328#1752D5911yescharacteristic, complemented
329#1735—591no—
330C2×C59—42no—
331C2×C59#235911yescharacteristic
332C2×C59#294311yescharacteristic
333C2×C59#294611yescharacteristic
334C2×C59#331711yescharacteristic
335C2×C59#331911yescharacteristic
336C2×C59#334711yescharacteristic
337C2×C59#334811yescharacteristic
338D59—82no—
339C23×C8C2×D5911yescharacteristic
340C2×#121—591no—
341C2×C4×D4—591no—
342C2×#117—591no—
343C2×#117—591no—
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⋊C8C2×D5911yescharacteristic
354C2×#97C2×D5911yescharacteristic
355C2×#97C2×D5911yescharacteristic
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#382—1181no—
366#380—591no—
367C4×C4⋊C4—591no—
368C4×C4⋊C4—591no—
369C4×C4⋊C4—591no—
370C4×C22⋊C4—1181no—
371C4×C22⋊C4C2×D5911yescharacteristic
372#376—1181no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#336C2×D5911yescharacteristic
382#336C2×D5911yescharacteristic
383C59#2293511yescharacteristic, complemented, Sylow
384C22×D4—591no—
385C22×D4—1181no—
386C23×C4—1181no—
387C23×C4C22×D5911yescharacteristic
388C22×C8—22no—
389C22×C8—22no—
390C22×C8—41no—
391C22×C8C22×D5911yescharacteristic
392C22×C8C22×D5911yescharacteristic
393C22×C8C22×D5911yescharacteristic
394C22×C8C22×D5911yescharacteristic
395#121—594no—
396#121—1182no—
397C4×D4—1182no—
398C4×D4—2361no—
399#117—594no—
400#117—1182no—
401#117—1182no—
402#117—2361no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C22⋊C4—1181no—
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4D23611yescharacteristic
421C2×C22⋊C4D23611yescharacteristic
422C2×C42—1181no—
423C2×C42—1181no—
424C2×C42—1181no—
425C2×C42—1181no—
426C2×C42—1181no—
427C2×C42C22×D5911yescharacteristic
428C2×C42C22×D5911yescharacteristic
429#107—594no—
430#107—1182no—
431#106—594no—
432#106—2361no—
433#102—594no—
434#102—594no—
435#102—1182no—
436#102—2361no—
437C22⋊C8C4×D5914yescomplemented
438#97—22no—
439#97—41no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95D23611yescharacteristic
448#95D23611yescharacteristic
449C24—1181no—
450C24C2×D23611yescharacteristic
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—1181no—
465C22×C4—1181no—
466C22×C4—1181no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—4721no—
472C22×C4—2362no—
473C22×C4—2362no—
474C22×C4C2×C4×D5912yes—
475C22×C4C2×D23611yescharacteristic
476C22×C4C2×D23611yescharacteristic
477C22×C4C2×D23611yescharacteristic
478C22×C4C2×D23611yescharacteristic
479C22×C4C2×D23611yescharacteristic
480C22×C4#1174568512yes—
481C22×C4C23×D5911yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—44no—
492C2×C8C2×C4×D5912yes—
493C2×C8C2×C4×D5912yes—
494C2×C8#1174568712yes—
495C4⋊C4—592no—
496C4⋊C4—592no—
497C4⋊C4—592no—
498C4⋊C4—592no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—4721no—
510C4⋊C4—4721no—
511C4⋊C4—4721no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—1184no—
515C22⋊C4—1184no—
516C22⋊C4—4721no—
517C22⋊C4—4721no—
518C22⋊C4—2362no—
519C22⋊C4—2362no—
520C22⋊C4—2362no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—1184no—
526C42—4721no—
527C42—2362no—
528C42—2362no—
529C42—2362no—
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—
543C23C4×D23612yes—
544C23C22×D23611yescharacteristic
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—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—1182no—
569C2×C4—1182no—
570C2×C4—1182no—
571C2×C4—1182no—
572C2×C4—4724no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
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×C4C4×D23612yes—
591C2×C4C22×C4×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4C2×#1174568711yescharacteristic
594C2×C4C2×#1174568511yescharacteristic
595C2×C4#1427377211yescharacteristic
596C2×C4#1427377411yescharacteristic
597C8—22no—
598C8—22no—
599C8—22no—
600C8—41no—
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×C4×D23611yescharacteristic
616C22#1683804711yescharacteristic
617C22#1683805611yescharacteristic
618C22#1683820211yescharacteristic
619C22#1683820211yescharacteristic
620C22#1683820711yescharacteristic
621C22#1683821411yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—4721no—
628C4—4721no—
629C4—4721no—
630C4—2362no—
631C4—2362no—
632C4—2362no—
633C4—2362no—
634C4C4×#1174568711yescharacteristic
635C4#1683788111yescharacteristic
636C4#1683807811yescharacteristic
637C4#1683808511yescharacteristic
638C2—42no—
639C2—4722no—
640C2#2142709011yescharacteristic
641C2#2143289811yescharacteristic
642C2#2143293011yescharacteristic
643C2#2143622411yescharacteristic
644C2#2143625411yescharacteristic
645C2#2143659811yescharacteristic
646C2#2143661311yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1752C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenter#1752D59
hypocenterC59#22935
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
3. #21426979C2semidirect
4. #21424834C2semidirect
5. #21424793C2semidirect
6. #21424724C2semidirect
7. #21424675C2semidirect
8. #21424668C2semidirect
10. #21424346C2semidirect
11. #21424297C2semidirect
12. #21423993C2semidirect
13. #21423922C2semidirect
14. #21423813C2semidirect
15. C59×#1752C2semidirect
20. #16837727C4semidirect
70. #14273686C2×C4semidirect
94. #14273661C2×C4semidirect
96. #14273659C2×C4semidirect
120. C22⋊C8×C59C2×C4semidirect
328. #1752D59semidirect
383. C59#22935semidirect
437. C22⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377272
#214271322
#214269792
#214248342
#214247932
#214247242
#214246752
#214246682
#214245242
#214243462
#214242972
#214239932
#214239222
#214238132
C59×#17522
#2293559