Finite Group Databasebeta

← #28595940

Subgroups — #28595940

Order 15104.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837718C211yesmaximal, characteristic
2#21426452C211yesmaximal, characteristic, complemented
3#21424720C211yesmaximal, characteristic, complemented
4#21424713C211yesmaximal, characteristic, complemented
5#21424678C211yesmaximal, characteristic, complemented
6#21424671C211yesmaximal, characteristic, complemented
7#21424419C211yesmaximal, characteristic
8#21424293C211yesmaximal, characteristic, complemented
9#21424180C211yesmaximal, characteristic, complemented
10#21423994C211yesmaximal, characteristic, complemented
11C4×#14273658C211yesmaximal, characteristic, complemented
12#21423921C211yesmaximal, characteristic, complemented
13C4×#14273648C211yesmaximal, characteristic, complemented
14#21423862C211yesmaximal, characteristic, complemented
15C59×#2324C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18#16837718C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31C4⋊C4×Dic59—21no—
32C4⋊C4×Dic59—21no—
33C4×C4⋊Dic59C2211yescharacteristic
34C4×C4⋊Dic59C2211yescharacteristic
35C4×C4⋊Dic59C2211yescharacteristic
36C4×C4⋊Dic59C2211yescharacteristic
37C22⋊C4×Dic59—21no—
38C22⋊C4×Dic59—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×#121×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C59×#390—21no—
53C59×#390C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#386—21no—
57C59×#385—21no—
58C59×#380C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273686—22no—
66#14273686—22no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—22no—
69C2×C4⋊Dic59C2×C414yescomplemented
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C22⋊Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—22no—
81#14273658—22no—
82#14273658C2×C414yescomplemented
83#14273658C2×C414yescomplemented
84#14273648—22no—
85#14273648—22no—
86#14273648C2×C414yes—
87#14273648C2×C414yes—
88C22⋊(C59⋊C8)—22no—
89C22⋊(C59⋊C8)C2×C414yescomplemented
90#14273638—22no—
91C4×C59⋊C8—22no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×D4×C59—21no—
97C22×D4×C59C2311yescharacteristic
98C23×C4×C59—21no—
99C23×C4×C59C2311yescharacteristic
100#121×C59—22no—
101#121×C59C2×C414yes—
102C4×D4×C59—22no—
103C4×D4×C59—22no—
104#117×C59—22no—
105#117×C59—22no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C22⋊C4×C59D411yescharacteristic
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125C24×C59—21no—
126C24×C59C2×D411yescharacteristic
127C2×D4×C59—24no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—81no—
134C2×D4×C59—81no—
135C2×D4×C59C22×C412yes—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—42no—
144C22×C4×C59—81no—
145C22×C4×C59—81no—
146C22×C4×C59C22×C412yes—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59#4112yes—
153C22×C4×C59C2411yescharacteristic
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—24no—
159C22×Dic59—81no—
160C22×Dic59—81no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—81no—
169C4⋊C4×C59C22×C412yes—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—81no—
175C22⋊C4×C59—81no—
176C22⋊C4×C59—81no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—81no—
180C42×C59#4112yes—
181C42×C59#4112yes—
182C22⋊Dic59—24no—
183C4⋊Dic59—24no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59C22×C412yes—
189C4⋊Dic59C22×C412yes—
190C4⋊Dic59#4112yes—
191C4⋊Dic59#4112yes—
192C4×Dic59—24no—
193C4×Dic59—81no—
194C4×Dic59—81no—
195C4×Dic59—81no—
196C4×Dic59—81no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8C22×C412yes—
208C2×C59⋊C8C22×C412yes—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—22no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59C4×D412yes—
223C23×C59C22×D411yescharacteristic
224D4×C59—22no—
225D4×C59—22no—
226D4×C59—41no—
227D4×C59—41no—
228D4×C59—84no—
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—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—84no—
255C2×C4×C59C4×D412yes—
256C2×C4×C59#12312yes—
257C2×C4×C59#12312yes—
258C2×C4×C59C23×C411yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×Dic59—22no—
265C2×Dic59—22no—
266C2×Dic59—22no—
267C2×Dic59—22no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—84no—
277C59⋊C8—22no—
278C59⋊C8—22no—
279C59⋊C8—41no—
280C59⋊C8—41no—
281#21584—591nomaximal, Sylow
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—22no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59C2×C4×D411yescharacteristic
297C22×C59C2×#12311yescharacteristic
298C22×C59#52511yescharacteristic
299C22×C59#57611yescharacteristic
300C22×C59#57711yescharacteristic
301C22×C59#57711yescharacteristic
302C22×C59#57711yescharacteristic
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—22no—
306C4×C59—41no—
307C4×C59—41no—
308C4×C59—42no—
309C4×C59—81no—
310C4×C59—81no—
311C4×C59C4×#4111yescharacteristic
312C4×C59C4×#4111yescharacteristic
313C4×C59#52911yescharacteristic
314C4×C59#52911yescharacteristic
315Dic59—42no—
316Dic59—42no—
317Dic59—42no—
318Dic59—42no—
319C2×#481—591no—
320#2324D5911yescharacteristic
321#2274—591no—
322#1926—591no—
323#1925—591no—
324#1920—591no—
325#1919—591no—
326#1875—591no—
327#1854—591no—
328#1823—591no—
329#1776—591no—
330C4×#106—591no—
331#1762—591no—
332C4×#102—591no—
333#1750—591no—
334C2×C59—42no—
335C2×C59—82no—
336C2×C59C4×#12311yescharacteristic
337C2×C59#294611yescharacteristic
338C2×C59#294911yescharacteristic
339C2×C59#309611yescharacteristic
340C2×C59#309911yescharacteristic
341C2×C59#311911yescharacteristic
342C2×C59#312511yescharacteristic
343C2×#121C2×D5911yescharacteristic
344C2×C4×D4C2×D5911yescharacteristic
345C2×#117—591no—
346C2×#117C2×D5911yescharacteristic
347C22×C4⋊C4—591no—
348#481—598no—
349C2×#107—591no—
350C2×#106—591no—
351C2×#106—591no—
352C2×#106—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×C22⋊C8—591no—
358C2×C22⋊C8—591no—
359C2×#97—591no—
360C2×C4×C8—591no—
361#390—21no—
362#390C2×D5911yescharacteristic
363#389C2×D5911yescharacteristic
364#388—21no—
365#386—21no—
366#385—21no—
367#380C2×D5911yescharacteristic
368C4×C4⋊C4—591no—
369C4×C4⋊C4—591no—
370C4×C4⋊C4—591no—
371C4×C4⋊C4—591no—
372C4×C4⋊C4—1181no—
373C4×C4⋊C4—1181no—
374C4×C4⋊C4C2×D5911yescharacteristic
375C4×C22⋊C4—1181no—
376C4×C22⋊C4—1181no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#336—591no—
386#336—591no—
387C59#2158411yescharacteristic, complemented, Sylow
388C22×D4—21no—
389C22×D4C22×D5911yescharacteristic
390C23×C4—21no—
391C23×C4—1181no—
392C23×C4C22×D5911yescharacteristic
393C22×C8—591no—
394C22×C8—591no—
395C22×C8—591no—
396C22×C8—591no—
397#121—22no—
398#121C4×D5914yes—
399C4×D4—22no—
400C4×D4—22no—
401#117—22no—
402#117—22no—
403#117—1182no—
404#117—1182no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—594no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—1182no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4C22⋊D5911yescharacteristic
414C2×C4⋊C4C22⋊D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4C22⋊D5911yescharacteristic
425C2×C22⋊C4C22⋊D5911yescharacteristic
426C2×C42—21no—
427C2×C42—1181no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42—1181no—
431C2×C42—1181no—
432C2×C42C22×D5911yescharacteristic
433C2×C42C22×D5911yescharacteristic
434#107—1182no—
435#106—594no—
436#106—594no—
437#106—1182no—
438#102—594no—
439#102—594no—
440#102—1182no—
441#102—1182no—
442C22⋊C8—594no—
443C22⋊C8—1182no—
444#97—1182no—
445C4×C8—1182no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453C24—21no—
454C24C2×C22⋊D5911yescharacteristic
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4C2×C4×D5912yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—1184no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—4721no—
480C22×C4—4721no—
481C22×C4C2×C4×D5912yes—
482C22×C4#1174568512yes—
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C23×D5911yescharacteristic
489C2×C8—592no—
490C2×C8—592no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—81no—
509C4⋊C4—592no—
510C4⋊C4—592no—
511C4⋊C4—592no—
512C4⋊C4—592no—
513C4⋊C4—1184no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4C2×C4×D5912yes—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C22⋊C4—1184no—
527C42—21no—
528C42—21no—
529C42—81no—
530C42—1184no—
531C42—4721no—
532C42—4721no—
533C42—4721no—
534C42—4721no—
535C42#1174568712yes—
536C42#1174568712yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23C4×C22⋊D5912yes—
551C23C22×C22⋊D5911yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—1182no—
584C2×C4—1182no—
585C2×C4—1182no—
586C2×C4—1182no—
587C2×C4—4724no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4C4×C22⋊D5912yes—
597C2×C4#1427370812yes—
598C2×C4#1427370812yes—
599C2×C4C22×C4×D5911yescharacteristic
600C2×C4#41×D5911yescharacteristic
601C2×C4#41×D5911yescharacteristic
602C2×C4C2×#1174568711yescharacteristic
603C2×C4C2×#1174568711yescharacteristic
604C2×C4C2×#1174568511yescharacteristic
605C8—1182no—
606C8—1182no—
607C8—2361no—
608C8—2361no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×C4×C22⋊D5911yescharacteristic
624C22C2×#1427370811yescharacteristic
625C22#1683790911yescharacteristic
626C22#1683820511yescharacteristic
627C22#1683821011yescharacteristic
628C22#1683821311yescharacteristic
629C22#1683821311yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—81no—
637C4—81no—
638C4—2362no—
639C4—2362no—
640C4—2362no—
641C4—2362no—
642C4C4×#1174568711yescharacteristic
643C4C4×#1174568711yescharacteristic
644C4#1683793211yescharacteristic
645C4#1683793211yescharacteristic
646C2—42no—
647C2—82no—
648C2C4×#1427370811yescharacteristic
649C2#2143292411yescharacteristic
650C2#2143295311yescharacteristic
651C2#2143404811yescharacteristic
652C2#2143408111yescharacteristic
653C2#2143428911yescharacteristic
654C2#2143436711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2324C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
cosocleC22×C4×C59C24
hypercenter#2324D59
hypocenterC59#21584
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
2. #21426452C2semidirect
3. #21424720C2semidirect
4. #21424713C2semidirect
5. #21424678C2semidirect
6. #21424671C2semidirect
8. #21424293C2semidirect
9. #21424180C2semidirect
10. #21423994C2semidirect
11. C4×#14273658C2semidirect
12. #21423921C2semidirect
13. C4×#14273648C2semidirect
14. #21423862C2semidirect
18. #16837718C4semidirect
69. C2×C4⋊Dic59C2×C4semidirect
82. #14273658C2×C4semidirect
83. #14273658C2×C4semidirect
89. C22⋊(C59⋊C8)C2×C4semidirect
387. C59#21584semidirect

Maximal subgroups

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

subgroupindex
C2×#168377182
#214264522
#214247202
#214247132
#214246782
#214246712
#214244192
#214242932
#214241802
#214239942
C4×#142736582
#214239212
C4×#142736482
#214238622
C59×#23242
#2158459