Finite Group Databasebeta

← #28593048

Subgroups — #28593048

Order 15104.

862 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837512C211yesmaximal, characteristic
2#21426740C211yesmaximal, characteristic, complemented
3#21424971C211yesmaximal, characteristic
4#21424961C211yesmaximal, characteristic
5#21424836C211yesmaximal, characteristic, complemented
6#21424821C211yesmaximal, characteristic, complemented
7#21424796C211yesmaximal, characteristic, complemented
8#21424792C211yesmaximal, characteristic, complemented
9#21424782C211yesmaximal, characteristic, complemented
10#21424778C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12C4×#14273647C211yesmaximal, characteristic, complemented
13C4×#14273648C211yesmaximal, characteristic, complemented
14#21423783C211yesmaximal, characteristic, complemented
15C59×#2319C211yesmaximal, characteristic, complemented
16C22×#11745669C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18#16837512C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32C2×C4×C59⋊C8C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837304C2211yescharacteristic
35#16837304C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837291—21no—
39#16837291—21no—
40#16837264—21no—
41#16837261—21no—
42C4×C4⋊Dic59C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C59×#390C2211yescharacteristic
51C59×#390C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#388—21no—
56C59×#381—21no—
57C59×#380—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C59×#376C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C2×#11745669—22no—
63C2×#11745669—22no—
64C2×#11745669C2×C414yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C4×D236—22no—
70C4×D236C2×C414yes—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×C4×Dic59—21no—
82#14273660—22no—
83#14273658—22no—
84#14273658C2×C414yes—
85#14273648—22no—
86#14273648—22no—
87#14273648C2×C414yes—
88#14273647—22no—
89#14273647—22no—
90#14273647C2×C414yes—
91#14273638—22no—
92#14273638—22no—
93#14273638C2×C414yes—
94C4×C59⋊C8—22no—
95C4×C59⋊C8—22no—
96C4×C59⋊C8—22no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C22×D4×C59C2311yescharacteristic
101C23×C4×C59—21no—
102C4×D4×C59—22no—
103C4×D4×C59C2×C414yes—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C42×C59—21no—
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59D411yescharacteristic
122#95×C59D411yescharacteristic
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59—21no—
126C23×D59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59C22×C412yes—
133C22×C4×C59—21no—
134C22×C4×C59—24no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—81no—
138C22×C4×C59—81no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2411yescharacteristic
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—24no—
156C2×C4×D59—81no—
157C2×C4×D59—81no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—81no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59C22×C412yes—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—81no—
168C22⋊C4×C59—81no—
169C22⋊C4×C59—81no—
170C22⋊C4×C59—81no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—81no—
176C42×C59C22×C412yes—
177C42×C59#4112yes—
178#11745669—28no—
179(C2×D59)⋊C4—24no—
180(C2×D59)⋊C4—81no—
181(C2×D59)⋊C4—81no—
182(C2×D59)⋊C4—81no—
183(C2×D59)⋊C4—81no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59C22×C412yes—
190Dic59⋊C4—81no—
191Dic59⋊C4—81no—
192C4×Dic59—81no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8C22×C412yes—
200C2×C59⋊C8C22×C412yes—
201C2×C59⋊C8#4112yes—
202C2×C59⋊C8#4112yes—
203C2×C59⋊C8#4112yes—
204C2×C59⋊C8#4112yes—
205C23×C59—22no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59C22×D411yescharacteristic
213C22×D59—22no—
214C22×D59—41no—
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220D4×C59—22no—
221D4×C59—22no—
222D4×C59—41no—
223D4×C59—41no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—22no—
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×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—84no—
250C2×C4×C59C4×D412yes—
251C2×C4×C59C4×D412yes—
252C2×C4×C59C23×C411yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×Dic59—22no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267D236—22no—
268D236—22no—
269D236—41no—
270D236—41no—
271C4×D59—84no—
272C59⋊C8—22no—
273C59⋊C8—22no—
274C59⋊C8—22no—
275C59⋊C8#11714yes—
276#21290—591nomaximal, Sylow
277C22×C59—42no—
278C22×C59—42no—
279C22×C59—42no—
280C22×C59—81no—
281C22×C59—81no—
282C22×C59—81no—
283C22×C59—81no—
284C22×C59C2×C4×D411yescharacteristic
285C22×C59#53211yescharacteristic
286C22×C59#53211yescharacteristic
287C22×C59#57611yescharacteristic
288C22×C59#57611yescharacteristic
289C22×C59#57711yescharacteristic
290C22×C59#57711yescharacteristic
291C2×D59—42no—
292C2×D59—42no—
293C2×D59—42no—
294C2×D59—81no—
295C2×D59—81no—
296C2×D59—81no—
297C2×D59—81no—
298C4×C59—22no—
299C4×C59—22no—
300C4×C59—41no—
301C4×C59—41no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59—42no—
305C4×C59—81no—
306C4×C59—81no—
307C4×C59C2×#11711yescharacteristic
308C4×C59C4×#4111yescharacteristic
309C4×C59#52911yescharacteristic
310C4×C59#52911yescharacteristic
311Dic59—42no—
312Dic59—81no—
313Dic59—81no—
314C2×#442—591no—
315#2319—591no—
316#2319D5911yescharacteristic, complemented
317#1972—591no—
318#1970—591no—
319#1944—591no—
320#1941—591no—
321#1936—591no—
322#1936—591no—
323#1934—591no—
324#1934—591no—
325#1776—591no—
326C4×#102—591no—
327C4×#102—591no—
328#1726—591no—
329C2×C59—82no—
330C2×C59#229311yescharacteristic
331C2×C59#294611yescharacteristic
332C2×C59#294611yescharacteristic
333C2×C59#317311yescharacteristic
334C2×C59#317611yescharacteristic
335C2×C59#317911yescharacteristic
336C2×C59#317911yescharacteristic
337D59—82no—
338C22×D8—591no—
339C2×C4×D4—591no—
340C2×C4×D4C2×D5911yescharacteristic
341#442—598no—
342C2×#107—591no—
343C2×#106—591no—
344C2×#106—591no—
345C2×#102—591no—
346C2×#102—591no—
347C2×#102—591no—
348C2×#102—591no—
349C2×#102—591no—
350C2×#102—591no—
351C2×#97—591no—
352C2×#97—591no—
353C2×#97—591no—
354C2×C4×C8—591no—
355C2×C4×C8—591no—
356C2×C4×C8—591no—
357#390—591no—
358#390—591no—
359#390C2×D5911yescharacteristic
360#390C2×D5911yescharacteristic
361#389—591no—
362#389—591no—
363#389C2×D5911yescharacteristic
364#389C2×D5911yescharacteristic
365#388—21no—
366#388—21no—
367#388—1181no—
368#388—1181no—
369#381—21no—
370#381—1181no—
371#380—21no—
372#380—1181no—
373C4×C4⋊C4—591no—
374C4×C4⋊C4C2×D5911yescharacteristic
375#376C2×D5911yescharacteristic
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382C59#2129011yescharacteristic, complemented, Sylow
383C22×D4—591no—
384C22×D4C22×D5911yescharacteristic
385C23×C4—21no—
386C23×C4—1181no—
387C2×D8—594no—
388C2×D8—1182no—
389C2×D8—1182no—
390C22×C8—591no—
391C22×C8—591no—
392C22×C8—591no—
393C22×C8—591no—
394C4×D4—22no—
395C4×D4—594no—
396C4×D4—1182no—
397C4×D4C4×D5914yes—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1181no—
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C42—21no—
419C2×C42—1181no—
420C2×C42C22×D5911yescharacteristic
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423#107—1182no—
424#106—594no—
425#106—1182no—
426#102—594no—
427#102—594no—
428#102—1182no—
429#102—1182no—
430#102—1182no—
431#102—1182no—
432#97—594no—
433#97—1182no—
434#97—1182no—
435C4×C8—1182no—
436C4×C8—1182no—
437C4×C8—1182no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1181no—
442#95—1181no—
443#95—1181no—
444#95C22⋊D5911yescharacteristic
445#95C22⋊D5911yescharacteristic
446#95C22⋊D5911yescharacteristic
447#95C22⋊D5911yescharacteristic
448C24—21no—
449C24—1181no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—592no—
456C2×D4—1181no—
457C2×D4—1181no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C2×D4C2×C4×D5912yes—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—1181no—
469C22×C4—1184no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—4721no—
473C22×C4—4721no—
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C23×D5911yescharacteristic
481D8—1188no—
482C2×C8—592no—
483C2×C8—592no—
484C2×C8—592no—
485C2×C8—592no—
486C2×C8—592no—
487C2×C8—592no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—592no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—4721no—
508C4⋊C4—4721no—
509C4⋊C4C2×C4×D5912yes—
510C22⋊C4—24no—
511C22⋊C4—81no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C22⋊C4—1184no—
516C22⋊C4—4721no—
517C22⋊C4—4721no—
518C22⋊C4—4721no—
519C22⋊C4—4721no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—81no—
525C42—4721no—
526C42C2×C4×D5912yes—
527C42#1174568512yes—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—1182no—
536C23—2361no—
537C23—2361no—
538C23—2361no—
539C23—2361no—
540C23—2361no—
541C23—2361no—
542C23C22×C22⋊D5911yescharacteristic
543D4—22no—
544D4—22no—
545D4—41no—
546D4—41no—
547D4—1182no—
548D4—1182no—
549D4—2361no—
550D4—2361no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
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—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—1182no—
578C2×C4—4724no—
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×C4C4×C22⋊D5912yes—
588C2×C4C4×C22⋊D5912yes—
589C2×C4C22×C4×D5911yescharacteristic
590C2×C4#41×D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4#41×D5911yescharacteristic
594C2×C4C2×#1174568511yescharacteristic
595C8—594no—
596C8—1182no—
597C8—1182no—
598C8—1182no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—4721no—
607C22—4721no—
608C22—4721no—
609C22—4721no—
610C22—2362no—
611C22—2362no—
612C22—2362no—
613C22C2×C4×C22⋊D5911yescharacteristic
614C22#1683794811yescharacteristic
615C22#1683794811yescharacteristic
616C22#1683820511yescharacteristic
617C22#1683820511yescharacteristic
618C22#1683821011yescharacteristic
619C22#1683821311yescharacteristic
620C4—22no—
621C4—22no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—4721no—
630C4—4721no—
631C4—2362no—
632C4#117×D5911yescharacteristic
633C4#1683787011yescharacteristic
634C4#1683793011yescharacteristic
635C4#1683793011yescharacteristic
636C2—82no—
637C2—4722no—
638C2#2142655511yescharacteristic
639C2#2143292411yescharacteristic
640C2#2143292811yescharacteristic
641C2#2143470711yescharacteristic
642C2#2143474011yescharacteristic
643C2#2143477311yescharacteristic
644C2#2143478111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2319C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
cosocleC22×C4×C59C24
hypercenter#2319D59
hypocenterC59#21290
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #21426740C2semidirect
5. #21424836C2semidirect
6. #21424821C2semidirect
7. #21424796C2semidirect
8. #21424792C2semidirect
9. #21424782C2semidirect
10. #21424778C2semidirect
11. #21423994C2semidirect
12. C4×#14273647C2semidirect
13. C4×#14273648C2semidirect
14. #21423783C2semidirect
15. C59×#2319C2semidirect
18. #16837512C4semidirect
316. #2319D59semidirect
382. C59#21290semidirect

Maximal subgroups

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

subgroupindex
C2×#168375122
#214267402
#214249712
#214249612
#214248362
#214248212
#214247962
#214247922
#214247822
#214247782
#214239942
C4×#142736472
C4×#142736482
#214237832
C59×#23192
#2129059