Finite Group Databasebeta

← #28637211

Subgroups — #28637211

Order 15104.

864 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837718C211yesmaximal, characteristic
2C2×#16837646C211yesmaximal, characteristic
3#21428581C211yesmaximal, characteristic, complemented
4#21425577C211yesmaximal, characteristic, complemented
5#21425509C211yesmaximal, characteristic, complemented
6#21425474C211yesmaximal, characteristic, complemented
7#21425413C211yesmaximal, characteristic, complemented
8#21424864C211yesmaximal, characteristic, complemented
9#21424730C211yesmaximal, characteristic, complemented
10#21424430C211yesmaximal, characteristic, complemented
11#21424408C211yesmaximal, characteristic, complemented
12#21424281C211yesmaximal, characteristic, complemented
13#21424106C211yesmaximal, characteristic, complemented
14#21423817C211yesmaximal, characteristic, complemented
15C59×#2363C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C22×#11745664C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837718—24no—
21#16837646—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837342—21no—
35#16837342—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837268C2211yescharacteristic
39#16837268C2211yescharacteristic
40#16837263—21no—
41#16837263—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#121×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C59×#390—21no—
54C59×#390C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#386—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59—21no—
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63C2×#11745670—22no—
64C2×#11745670—22no—
65C2×#11745670—41no—
66C2×#11745664—22no—
67C2×#11745664—22no—
68C2×#11745664—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273706—22no—
74#14273706—22no—
75C2×C4⋊Dic59—22no—
76C2×C4⋊Dic59—22no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83#14273660—22no—
84#14273658—22no—
85#14273658—22no—
86C4⋊(C59⋊C8)—22no—
87C4⋊(C59⋊C8)—41no—
88#14273650—22no—
89#14273650—22no—
90#14273650—41no—
91#14273648—22no—
92#14273648—22no—
93#14273648—22no—
94C22⋊(C59⋊C8)—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×D4×C59—21no—
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106C23×C4×C59C2311yescharacteristic
107#121×C59—22no—
108#121×C59D414yes—
109C4×D4×C59—22no—
110C4×D4×C59—41no—
111#117×C59—22no—
112#117×C59—41no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59D411yescharacteristic
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C42×C59—21no—
125C2×C42×C59—21no—
126C2×C42×C59C2311yescharacteristic
127C2×C42×C59C2311yescharacteristic
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59—21no—
133C24×C59C2×D411yescharacteristic
134C2×D4×C59—24no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59—81no—
141C2×D4×C59—81no—
142C2×D4×C59C2×D412yes—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—42no—
151C22×C4×C59—42no—
152C22×C4×C59—81no—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D412yes—
159C22×C4×C59#4112yes—
160C22×C4×C59C2411yescharacteristic
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C22×Dic59—42no—
166C22×Dic59—42no—
167C22×Dic59—42no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—81no—
179C4⋊C4×C59C2×D412yes—
180C4⋊C4×C59#4112yes—
181C22⋊C4×C59—24no—
182C22⋊C4×C59—41no—
183C22⋊C4×C59—41no—
184C22⋊C4×C59—42no—
185C22⋊C4×C59—42no—
186C22⋊C4×C59—42no—
187C22⋊C4×C59—81no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—21no—
191C42×C59—21no—
192C42×C59—42no—
193C42×C59—81no—
194C22⋊Dic59—42no—
195#11745670—44no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—42no—
203Dic59⋊C4—42no—
204#11745664—44no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8#4112yes—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—22no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59#12112yes—
231C23×C59C22×D411yescharacteristic
232D4×C59—22no—
233D4×C59—22no—
234D4×C59—41no—
235D4×C59—41no—
236D4×C59—84no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—22no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—84no—
265C2×C4×C59#12112yes—
266C2×C4×C59C22×D411yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59C2×#4111yescharacteristic
270C2×C4×C59H(5,𝔽2)11yescharacteristic
271C2×C4×C592−1+411yescharacteristic
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—84no—
285Dic118—41no—
286Dic118—41no—
287Dic118—41no—
288Dic118—41no—
289C59⋊C8—22no—
290C59⋊C8—41no—
291C59⋊C8—41no—
292C59⋊C8—41no—
293#25225—591nomaximal, Sylow
294C22×C59—22no—
295C22×C59—22no—
296C22×C59—22no—
297C22×C59—41no—
298C22×C59—41no—
299C22×C59—41no—
300C22×C59—41no—
301C22×C59—42no—
302C22×C59—42no—
303C22×C59—42no—
304C22×C59—81no—
305C22×C59—81no—
306C22×C59—81no—
307C22×C59—81no—
308C22×C59C2×#12111yescharacteristic
309C22×C59#54611yescharacteristic
310C22×C59#54711yescharacteristic
311C22×C59#57511yescharacteristic
312C22×C59#57511yescharacteristic
313C22×C59#57711yescharacteristic
314C22×C59#57711yescharacteristic
315C4×C59—22no—
316C4×C59—41no—
317C4×C59—41no—
318C4×C59—42no—
319C4×C59—41no—
320C4×C59—41no—
321C4×C59—42no—
322C4×C59—81no—
323C4×C59#52911yescharacteristic
324C4×C59#53911yescharacteristic
325C4×C59#54611yescharacteristic
326C4×C59#54711yescharacteristic
327Dic59—81no—
328Dic59—81no—
329Dic59—81no—
330Dic59—81no—
331C2×#481—591no—
332C2×#468—591no—
333#2503—591no—
334#2363D5911yescharacteristic
335#2085—591no—
336#2073—591no—
337#2066—591no—
338#2055—591no—
339#1949—591no—
340#1927—591no—
341#1877—591no—
342#1873—591no—
343#1851—591no—
344#1804—591no—
345#1736—591no—
346C2×C59—42no—
347C2×C59—82no—
348C2×C59#242511yescharacteristic
349C2×C59#304011yescharacteristic
350C2×C59#304211yescharacteristic
351C2×C59#331711yescharacteristic
352C2×C59#332311yescharacteristic
353C2×C59#334811yescharacteristic
354C2×C59#335111yescharacteristic
355C22×SD16—591no—
356C22×M16—591no—
357C2×C22⋊Q8—591no—
358C2×#121C2×D5911yescharacteristic
359C2×C4×D4C2×D5911yescharacteristic
360C2×#117C2×D5911yescharacteristic
361C22×C4⋊C4—591no—
362#481—1184no—
363#468—1184no—
364C2×#107—591no—
365C2×#106—591no—
366C2×#106—591no—
367C2×C4⋊C8—591no—
368C2×C4⋊C8—591no—
369C2×#103—591no—
370C2×#103—591no—
371C2×#103—591no—
372C2×#102—591no—
373C2×#102—591no—
374C2×#102—591no—
375C2×C22⋊C8—591no—
376#400—1181no—
377#400—1181no—
378#391—591no—
379#391—591no—
380#390—21no—
381#390C2×D5911yescharacteristic
382#389C2×D5911yescharacteristic
383#388—21no—
384#386—21no—
385#382—591no—
386#382—591no—
387#381—1181no—
388#381—1181no—
389C4×C4⋊C4C2×D5911yescharacteristic
390C4×C22⋊C4—21no—
391C4×C22⋊C4C2×D5911yescharacteristic
392#340—591no—
393#340—591no—
394#340—591no—
395#340—591no—
396#340—591no—
397#340—591no—
398#336—591no—
399#336—591no—
400C59#2522511yescharacteristic, complemented, Sylow
401C22×Q8—591no—
402C22×D4—21no—
403C22×D4C22×D5911yescharacteristic
404C23×C4—21no—
405C23×C4—1181no—
406C23×C4C22×D5911yescharacteristic
407C2×SD16—1182no—
408C2×SD16—1182no—
409C2×SD16—2361no—
410C2×M16—1182no—
411C2×M16—1182no—
412C2×M16—2361no—
413C22×C8—591no—
414C22×C8—591no—
415C22×C8—591no—
416C22×C8—591no—
417C22⋊Q8—1182no—
418C22⋊Q8—1182no—
419#121—22no—
420#121D23614yes—
421C4×D4—22no—
422C4×D4—41no—
423#117—22no—
424#117—41no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—591no—
427C2×C4⋊C4—591no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—1182no—
430C2×C4⋊C4—1182no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4C22×D5911yescharacteristic
434C2×C4⋊C4C22×D5911yescharacteristic
435C2×C4⋊C4C22×D5911yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—1181no—
442C2×C22⋊C4C22⋊D5911yescharacteristic
443C2×C22⋊C4C22⋊D5911yescharacteristic
444C2×C42—21no—
445C2×C42—21no—
446C2×C42C22×D5911yescharacteristic
447C2×C42C22×D5911yescharacteristic
448#107—1182no—
449#106—1182no—
450#106—1182no—
451C4⋊C8—1182no—
452C4⋊C8—2361no—
453#103—1182no—
454#103—1182no—
455#103—2361no—
456#102—1182no—
457#102—1182no—
458#102—1182no—
459C22⋊C8—1182no—
460#95—21no—
461#95—21no—
462#95—1181no—
463#95—1181no—
464#95—1181no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470#95C22⋊D5911yescharacteristic
471#95C22⋊D5911yescharacteristic
472C24—21no—
473C24C2×C22⋊D5911yescharacteristic
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×D4—24no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—81no—
487C2×D4—81no—
488C2×D4C2×D23612yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—81no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—2362no—
504C22×C4—2362no—
505C22×C4—2362no—
506C22×C4C2×D23612yes—
507C22×C4#1174568512yes—
508C22×C4C2×C22⋊D5911yescharacteristic
509C22×C4C2×C22⋊D5911yescharacteristic
510C22×C4C2×C22⋊D5911yescharacteristic
511C22×C4C2×C22⋊D5911yescharacteristic
512C22×C4C2×C22⋊D5911yescharacteristic
513C22×C4C23×D5911yescharacteristic
514SD16—2364no—
515M16—2364no—
516C2×C8—592no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C2×C8—1181no—
527C2×C8—1181no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—81no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—1181no—
536C4⋊C4—1181no—
537C4⋊C4—1181no—
538C4⋊C4—1181no—
539C4⋊C4—2362no—
540C4⋊C4—2362no—
541C4⋊C4C2×D23612yes—
542C4⋊C4#1174568712yes—
543C22⋊C4—24no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—2362no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—81no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23#1427370112yes—
571C23C22×C22⋊D5911yescharacteristic
572Q8—2361no—
573Q8—2361no—
574Q8—2361no—
575Q8—2361no—
576D4—22no—
577D4—22no—
578D4—41no—
579D4—41no—
580D4—84no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—84no—
609C2×C4—4724no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4—2361no—
622C2×C4#1427370112yes—
623C2×C4C22×D23611yescharacteristic
624C2×C4#41×D5911yescharacteristic
625C2×C4C2×#1174568711yescharacteristic
626C2×C4C2×#1174568511yescharacteristic
627C2×C4#1427377211yescharacteristic
628C2×C4#1427377411yescharacteristic
629C8—1182no—
630C8—2361no—
631C8—2361no—
632C8—2361no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22C2×#1427370111yescharacteristic
648C22#1683804611yescharacteristic
649C22#1683805111yescharacteristic
650C22#1683820011yescharacteristic
651C22#1683820011yescharacteristic
652C22#1683821311yescharacteristic
653C22#1683821311yescharacteristic
654C4—22no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—42no—
659C4—41no—
660C4—41no—
661C4—81no—
662C4—4721no—
663C4—4721no—
664C4—4721no—
665C4—4721no—
666C4#1683793211yescharacteristic
667C4#1683799411yescharacteristic
668C4#1683804711yescharacteristic
669C4#1683805611yescharacteristic
670C2—42no—
671C2—82no—
672C2#2142764111yescharacteristic
673C2#2143359011yescharacteristic
674C2#2143360811yescharacteristic
675C2#2143622211yescharacteristic
676C2#2143631211yescharacteristic
677C2#2143661111yescharacteristic
678C2#2143665011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2363C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2363D59
hypocenterC59#25225
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
3. #21428581C2semidirect
4. #21425577C2semidirect
5. #21425509C2semidirect
6. #21425474C2semidirect
7. #21425413C2semidirect
8. #21424864C2semidirect
9. #21424730C2semidirect
10. #21424430C2semidirect
11. #21424408C2semidirect
12. #21424281C2semidirect
13. #21424106C2semidirect
14. #21423817C2semidirect
400. C59#25225semidirect

Maximal subgroups

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

subgroupindex
C2×#168377182
C2×#168376462
#214285812
#214255772
#214255092
#214254742
#214254132
#214248642
#214247302
#214244302
#214244082
#214242812
#214241062
#214238172
C59×#23632
#2522559