Finite Group Databasebeta

← #28595939

Subgroups — #28595939

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×#16837720C211yesmaximal, characteristic
2#21426785C211yesmaximal, characteristic
3#21426453C211yesmaximal, characteristic, complemented
4#21424721C211yesmaximal, characteristic, complemented
5#21424714C211yesmaximal, characteristic, complemented
6#21424679C211yesmaximal, characteristic, complemented
7#21424672C211yesmaximal, characteristic, complemented
8#21424416C211yesmaximal, characteristic
9#21424293C211yesmaximal, characteristic, complemented
10#21423992C211yesmaximal, characteristic, complemented
11#21423984C211yesmaximal, characteristic, complemented
12#21423920C211yesmaximal, characteristic, complemented
13#21423910C211yesmaximal, characteristic, complemented
14#21423864C211yesmaximal, characteristic, complemented
15C59×#1823C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×#14273689C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837720C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×#14273637C2211yescharacteristic
31C2×C8×Dic59C2211yescharacteristic
32#16837303C2211yescharacteristic
33#16837302—21no—
34#16837298C2211yescharacteristic
35#16837294—21no—
36#16837278—21no—
37#16837276—21no—
38C4⋊C4×Dic59—21no—
39C4⋊C4×Dic59—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4⋊C4×Dic59C2211yescharacteristic
45C4×C22⋊Dic59—21no—
46C4×C22⋊Dic59—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C59×#380C2211yescharacteristic
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—22no—
69#14273686—22no—
70#14273686—22no—
71#14273686—22no—
72#14273686—22no—
73C2×C4⋊Dic59D411yescharacteristic
74C2×C4⋊Dic59D411yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59D411yescharacteristic
81C2×C22⋊Dic59D411yescharacteristic
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59C2311yescharacteristic
93C2×C4×Dic59C2311yescharacteristic
94#14273660—22no—
95#14273658—22no—
96#14273658C2×C414yescomplemented
97#14273658C2×C414yescomplemented
98#14273649—22no—
99#14273649—22no—
100#14273647C2×C414yes—
101#14273647C2×C414yes—
102C22⋊(C59⋊C8)C2×C414yescomplemented
103#14273637—22no—
104C8×Dic59—22no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C23×C4×C59—21no—
112C23×C4×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—22no—
117C2×C4⋊C4×C59C2×C414yescomplemented
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C22⋊C4×C59—21no—
123C22⋊C8×C59—22no—
124#95×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—24no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—81no—
137C22×C4×C59—81no—
138C22×C4×C59C22×C412yes—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59#4112yes—
141C22×C4×C59C2411yescharacteristic
142C2×C22⋊D59—24no—
143C2×C22⋊D59—81no—
144C2×C22⋊D59—81no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236C22×C412yes—
158C2×C4×D59—42no—
159C2×C4×D59—81no—
160C2×C4×D59—81no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C4⋊C4×C59—24no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59C22×C412yes—
175C4⋊C4×C59C22×C412yes—
176C22⋊C4×C59—81no—
177C22⋊Dic59—24no—
178C22⋊Dic59—22no—
179C22⋊Dic59—22no—
180(C2×D59)⋊C4—24no—
181(C2×D59)⋊C4—42no—
182(C2×D59)⋊C4—81no—
183(C2×D59)⋊C4—81no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—22no—
188C4⋊Dic59—22no—
189C4⋊Dic59C22×C412yes—
190C4⋊Dic59#4112yes—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—81no—
193C4×Dic59—24no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C4×Dic59—81no—
197C4×Dic59—81no—
198C4×Dic59—81no—
199C4×Dic59—81no—
200C4×Dic59—81no—
201C4×Dic59#4112yes—
202C4×Dic59#4112yes—
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×C59C4×D412yes—
216C23×C59C22×D411yescharacteristic
217C22×D59—22no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
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×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—84no—
243C2×C4×C59C4×D412yes—
244C2×C4×C59C23×C411yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C22⋊D59—84no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—22no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—22no—
259C2×Dic59—22no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59#12312yes—
271C2×Dic59#12312yes—
272D236—22no—
273D236—22no—
274D236—41no—
275D236—41no—
276C4×D59—84no—
277C8×C59—41no—
278C8×C59—41no—
279C59⋊C8—22no—
280C59⋊C8—22no—
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×C59C2×C4×D411yescharacteristic
290C22×C59C2×#12311yescharacteristic
291C22×C59#52511yescharacteristic
292C22×C59#57611yescharacteristic
293C22×C59#57711yescharacteristic
294C22×C59#57711yescharacteristic
295C22×C59#57711yescharacteristic
296C2×D59—42no—
297C2×D59—42no—
298C2×D59—42no—
299C2×D59—81no—
300C2×D59—81no—
301C2×D59—81no—
302C2×D59—81no—
303C4×C59—22no—
304C4×C59—42no—
305C4×C59—42no—
306C4×C59—81no—
307C4×C59—81no—
308C4×C59C4×#4111yescharacteristic
309C4×C59C4×#4111yescharacteristic
310C4×C59#52911yescharacteristic
311C4×C59#52911yescharacteristic
312Dic59—22no—
313Dic59—22no—
314Dic59—41no—
315Dic59—41no—
316Dic59—42no—
317Dic59—42no—
318Dic59—42no—
319C2×#481—591no—
320#2324—591no—
321#2274—591no—
322#1926—591no—
323#1925—591no—
324#1920—591no—
325#1919—591no—
326#1875—591no—
327#1854—591no—
328#1823D5911yescharacteristic, complemented
329#1776—591no—
330C4×#106—591no—
331#1762—591no—
332C4×#102—591no—
333#1750—591no—
334C2×C59—42no—
335C2×C59C4×#12311yescharacteristic
336C2×C59#294611yescharacteristic
337C2×C59#294911yescharacteristic
338C2×C59#309611yescharacteristic
339C2×C59#309911yescharacteristic
340C2×C59#311911yescharacteristic
341C2×C59#312511yescharacteristic
342D59—82no—
343C2×#121—591no—
344C2×C4×D4—591no—
345C2×#117—591no—
346C2×#117—591no—
347C22×C4⋊C4C2×D5911yescharacteristic
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⋊C8C2×D5911yescharacteristic
359C2×#97—591no—
360C2×C4×C8—591no—
361#390—591no—
362#390—1181no—
363#389—591no—
364#388—1181no—
365#386—1181no—
366#385—1181no—
367#380C2×D5911yescharacteristic
368C4×C4⋊C4—591no—
369C4×C4⋊C4—591no—
370C4×C4⋊C4—591no—
371C4×C4⋊C4—591no—
372C4×C4⋊C4—591no—
373C4×C4⋊C4—1181no—
374C4×C4⋊C4—1181no—
375C4×C22⋊C4—1181no—
376C4×C22⋊C4—1181no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340C2×D5911yescharacteristic
382#340C2×D5911yescharacteristic
383#340C2×D5911yescharacteristic
384#340C2×D5911yescharacteristic
385#336—591no—
386#336—591no—
387C59#2158411yescharacteristic, complemented, Sylow
388C22×D4—591no—
389C22×D4—1181no—
390C23×C4—21no—
391C23×C4—1181no—
392C23×C4C22×D5911yescharacteristic
393C22×C8—591no—
394C22×C8—591no—
395C22×C8C22×D5911yescharacteristic
396C22×C8C22×D5911yescharacteristic
397#121—594no—
398#121—1182no—
399C4×D4—1182no—
400C4×D4—1182no—
401#117—1182no—
402#117—1182no—
403#117—1182no—
404#117—1182no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4C4×D5914yescomplemented
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—591no—
420C2×C22⋊C4—591no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C42—591no—
427C2×C42—591no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42—1181no—
431C2×C42—1181no—
432C2×C42—1181no—
433C2×C42—1181no—
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—22no—
443C22⋊C8—594no—
444#97—1182no—
445C4×C8—1182no—
446#95—21no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453C24—1181no—
454C24D4×D5911yescharacteristic
455C2×D4—592no—
456C2×D4—1184no—
457C2×D4—1181no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C2×D4—1181no—
462C2×D4—4721no—
463C2×D4—4721no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—591no—
476C22×C4—591no—
477C22×C4—591no—
478C22×C4—591no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—4721no—
483C22×C4—4721no—
484C22×C4—2362no—
485C22×C4C2×C4×D5912yes—
486C22×C4D4×D5911yescharacteristic
487C22×C4#1174568712yes—
488C22×C4C23×D5911yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—592no—
496C2×C8—592no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—592no—
509C4⋊C4—592no—
510C4⋊C4—592no—
511C4⋊C4—1182no—
512C4⋊C4—1182no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—4721no—
517C4⋊C4C2×C4×D5912yes—
518C4⋊C4C2×C4×D5912yes—
519C22⋊C4—81no—
520C22⋊C4—1182no—
521C22⋊C4—1182no—
522C22⋊C4—1184no—
523C22⋊C4—1184no—
524C22⋊C4—4721no—
525C22⋊C4—4721no—
526C22⋊C4—2362no—
527C42—592no—
528C42—592no—
529C42—1184no—
530C42—1181no—
531C42—1181no—
532C42—4721no—
533C42—4721no—
534C42—4721no—
535C42—4721no—
536C42—4721no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—1182no—
544C23—2361no—
545C23—2361no—
546C23—2361no—
547C23—2361no—
548C23—2361no—
549C23—2361no—
550C23(C22⋊D59)⋊C412yes—
551C23C2×D4×D5911yescharacteristic
552D4—1182no—
553D4—1182no—
554D4—4724no—
555D4—2361no—
556D4—2361no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—592no—
577C2×C4—592no—
578C2×C4—1181no—
579C2×C4—1181no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—1182no—
585C2×C4—1182no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1182no—
591C2×C4—4724no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4(C22⋊D59)⋊C412yes—
599C2×C4C22×C4×D5911yescharacteristic
600C2×C4#41×D5911yescharacteristic
601C2×C4#41×D5911yescharacteristic
602C2×C4#41×D5911yescharacteristic
603C2×C4#41×D5911yescharacteristic
604C2×C4C2×#1174568711yescharacteristic
605C8—41no—
606C8—41no—
607C8—1182no—
608C8—1182no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—4721no—
617C22—4721no—
618C22—4721no—
619C22—4721no—
620C22—2362no—
621C22—2362no—
622C22—2362no—
623C22C2×(C22⋊D59)⋊C411yescharacteristic
624C22#123×D5911yescharacteristic
625C22#1683791011yescharacteristic
626C22#1683820311yescharacteristic
627C22#1683820811yescharacteristic
628C22#1683821111yescharacteristic
629C22#1683821111yescharacteristic
630C4—22no—
631C4—42no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—1182no—
636C4—1182no—
637C4—2362no—
638C4—2362no—
639C4—2362no—
640C4—2361no—
641C4—2361no—
642C4#1683787011yescharacteristic
643C4#1683787011yescharacteristic
644C4#1683792911yescharacteristic
645C4#1683792911yescharacteristic
646C2—42no—
647C2—4722no—
648C2#2142661311yescharacteristic
649C2#2143292211yescharacteristic
650C2#2143295111yescharacteristic
651C2#2143404711yescharacteristic
652C2#2143408011yescharacteristic
653C2#2143429811yescharacteristic
654C2#2143436811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1823C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenter#1823D59
hypocenterC59#21584
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. #21426453C2semidirect
4. #21424721C2semidirect
5. #21424714C2semidirect
6. #21424679C2semidirect
7. #21424672C2semidirect
9. #21424293C2semidirect
10. #21423992C2semidirect
11. #21423984C2semidirect
12. #21423920C2semidirect
13. #21423910C2semidirect
14. #21423864C2semidirect
15. C59×#1823C2semidirect
20. #16837720C4semidirect
96. #14273658C2×C4semidirect
97. #14273658C2×C4semidirect
102. C22⋊(C59⋊C8)C2×C4semidirect
117. C2×C4⋊C4×C59C2×C4semidirect
328. #1823D59semidirect
387. C59#21584semidirect
413. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377202
#214267852
#214264532
#214247212
#214247142
#214246792
#214246722
#214244162
#214242932
#214239922
#214239842
#214239202
#214239102
#214238642
C59×#18232
#2158459