Finite Group Databasebeta

← #28610720

Subgroups — #28610720

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837681C211yesmaximal, characteristic
2#21427183C211yesmaximal, characteristic, complemented
3#21427005C211yesmaximal, characteristic
4#21425067C211yesmaximal, characteristic, complemented
5#21424867C211yesmaximal, characteristic, complemented
6#21424831C211yesmaximal, characteristic, complemented
7#21424830C211yesmaximal, characteristic, complemented
8#21424691C211yesmaximal, characteristic, complemented
9#21424358C211yesmaximal, characteristic
10#21424346C211yesmaximal, characteristic
11#21423969C211yesmaximal, characteristic, complemented
12#21423924C211yesmaximal, characteristic
13#21423920C211yesmaximal, characteristic
14#21423910C211yesmaximal, characteristic
15C59×#1918C211yesmaximal, characteristic, complemented
16C2×#14273726C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273689C2211yescharacteristic
19C2×#14273689C2211yescharacteristic
20#16837681C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×#14273637C2211yescharacteristic
31C2×C8×Dic59C2211yescharacteristic
32#16837308—21no—
33#16837306—21no—
34#16837301C2211yescharacteristic
35#16837301C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837299—21no—
38#16837276—21no—
39#16837273—21no—
40#16837267C2211yescharacteristic
41#16837259—21no—
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4×Dic59⋊C4—21no—
45C4×Dic59⋊C4C2211yescharacteristic
46#95×D59—21no—
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C22×C4×D59—21no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273726—22no—
68#14273726C2×C414yescomplemented
69C4×D236—22no—
70C4×D236—41no—
71#14273689—22no—
72#14273689—22no—
73#14273689—41no—
74#14273689C2×C414yes—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4D411yescharacteristic
83C2×Dic59⋊C4D411yescharacteristic
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59C2311yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91#14273661—41no—
92#14273660C2×C414yescomplemented
93#14273658—41no—
94Dic59⋊C8C2×C414yescomplemented
95#14273649—22no—
96#14273649C2×C414yes—
97#14273647C2×C414yes—
98#14273647Q814yescomplemented
99#14273638—41no—
100#14273637—22no—
101C8×Dic59—41no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95D411yescharacteristic
110C59⋊#95D411yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120#107×C59C2×C414yescomplemented
121#95×C59—21no—
122#95×C59—21no—
123C23×D59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2411yescharacteristic
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59C2×D411yescharacteristic
135C22×Dic59C2×D411yescharacteristic
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C2×D236—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236C22×C412yes—
142C2×D236C2×Q812yes—
143C2×D236#4112yes—
144C2×C4×D59—24no—
145C2×C4×D59—42no—
146C2×C4×D59—42no—
147C2×C4×D59—42no—
148C2×C4×D59—42no—
149C2×C4×D59—81no—
150C2×C4×D59—81no—
151C2×C4×D59—81no—
152C2×C4×D59—81no—
153C2×Dic118—24no—
154C2×Dic118—81no—
155C2×Dic118—81no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59C22×C412yes—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—81no—
168C4⋊C4×C59C22×C412yes—
169C4⋊C4×C59C22×C412yes—
170C4⋊C4×C59C2×Q812yes—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—81no—
174(C2×D59)⋊C4—24no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—42no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—81no—
183C4⋊Dic59—81no—
184C4⋊Dic59C22×C412yes—
185Dic59⋊C4—24no—
186Dic59⋊C4—22no—
187Dic59⋊C4—41no—
188Dic59⋊C4—81no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—42no—
192C4×Dic59—81no—
193C4×Dic59C22×C412yes—
194C4×Dic59#4112yes—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8C22×C412yes—
200C2×C59⋊C8C2×Q812yes—
201C23×C59C22×D411yescharacteristic
202C22×D59—22no—
203C22×D59—22no—
204C22×D59—22no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—22no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59C23×C411yescharacteristic
228C2×C4×C59C22×Q811yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59H(5,𝔽2)11yescharacteristic
232C2×C4×C592−1+411yescharacteristic
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—22no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59C4×D412yes—
252C2×Dic59C4×D412yes—
253C2×Dic59#13712yes—
254D236—22no—
255D236—22no—
256D236—22no—
257D236C4×Q814yes—
258C4×D59—84no—
259C4×D59—84no—
260C4×D59—84no—
261Dic118—84no—
262C8×C59—22no—
263C8×C59—41no—
264C59⋊C8—22no—
265C59⋊C8—22no—
266#22979—591nomaximal, Sylow
267C22×C59C2×C4×D411yescharacteristic
268C22×C59#54611yescharacteristic
269C22×C59D4×Q811yescharacteristic
270C22×C59C2×#13711yescharacteristic
271C22×C59#57511yescharacteristic
272C22×C59#57611yescharacteristic
273C22×C59#57611yescharacteristic
274C2×D59—24no—
275C2×D59—42no—
276C2×D59—42no—
277C2×D59—42no—
278C2×D59—42no—
279C2×D59—42no—
280C2×D59—42no—
281C4×C59—42no—
282C4×C59—42no—
283C4×C59—42no—
284C4×C59—41no—
285C4×C59—41no—
286C4×C59—81no—
287C4×C59C2×C4×Q811yescharacteristic
288C4×C59#52011yescharacteristic
289C4×C59#55211yescharacteristic
290C4×C59#55411yescharacteristic
291Dic59—22no—
292Dic59—22no—
293Dic59—22no—
294Dic59—42no—
295Dic59—41no—
296Dic59—81no—
297Dic59—81no—
298C2×#474—591no—
299#2369—591no—
300#2349—591no—
301#1993—591no—
302#1949—591no—
303#1943—591no—
304#1942—591no—
305#1921—591no—
306#1918D5911yescharacteristic, complemented
307#1864—591no—
308#1862—591no—
309#1771—591no—
310#1762—591no—
311#1762—591no—
312C4×#102—591no—
313C2×C59#236011yescharacteristic
314C2×C59C4×#13711yescharacteristic
315C2×C59#294811yescharacteristic
316C2×C59#331411yescharacteristic
317C2×C59#331911yescharacteristic
318C2×C59#338511yescharacteristic
319C2×C59#338911yescharacteristic
320D59—44no—
321C2×C4⋊Q8—591no—
322C2×C4×D4—591no—
323C2×C4×D4—591no—
324C2×C4×D4—591no—
325#474—598no—
326C2×#107—591no—
327C2×#107—591no—
328C2×#107C2×D5911yescharacteristic
329C2×#106—591no—
330C2×C4⋊C8—591no—
331C2×#102—591no—
332C2×#102—591no—
333C2×#102—591no—
334C2×#102—591no—
335C2×#97—591no—
336C2×#97—591no—
337C2×C4×C8—591no—
338#391—1181no—
339#391—1181no—
340#389—591no—
341#389—591no—
342#389—591no—
343#389—1181no—
344#389C2×D5911yescharacteristic
345#385—1181no—
346#385—1181no—
347#382—591no—
348#380—1181no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—1181no—
353C4×C4⋊C4C2×D5911yescharacteristic
354C2×#95—1181no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340C2×D5911yescharacteristic
361#340C2×D5911yescharacteristic
362#340C2×D5911yescharacteristic
363#336—591no—
364#336C2×D5911yescharacteristic
365C59#2297911yescharacteristic, complemented, Sylow
366C22×Q8—1181no—
367C22×D4—591no—
368C23×C4—1181no—
369C23×C4—1181no—
370C23×C4—1181no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375C4⋊Q8—594no—
376C4⋊Q8—1182no—
377C4×D4—594no—
378C4×D4—1182no—
379C4×D4—1182no—
380C4×D4—1182no—
381C4×D4—2361no—
382C4×D4—2361no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C42—21no—
401C2×C42—591no—
402C2×C42—591no—
403C2×C42—1181no—
404C2×C42—1181no—
405C2×C42C22×D5911yescharacteristic
406#107—594no—
407#107—2361no—
408#107C4×D5914yescomplemented
409#106—2361no—
410C4⋊C8—594no—
411#102—594no—
412#102—594no—
413#102—594no—
414#102—1182no—
415#97—1182no—
416#97—2361no—
417C4×C8—2361no—
418#95—21no—
419#95—21no—
420#95—591no—
421#95—591no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429C24—1181no—
430C2×Q8—1184no—
431C2×Q8—4721no—
432C2×Q8—4721no—
433C2×D4—592no—
434C2×D4—592no—
435C2×D4—592no—
436C2×D4—1181no—
437C2×D4—1181no—
438C2×D4—1181no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—591no—
444C22×C4—591no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—1184no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—4721no—
452C22×C4—4721no—
453C22×C4—4721no—
454C22×C4—4721no—
455C22×C4—2362no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4D4×D5911yescharacteristic
460C22×C4D4×D5911yescharacteristic
461C22×C4C23×D5911yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—592no—
468C2×C8—592no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8C2×C4×D5912yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—81no—
480C4⋊C4—592no—
481C4⋊C4—1182no—
482C4⋊C4—1184no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—4721no—
489C4⋊C4—4721no—
490C4⋊C4—4721no—
491C4⋊C4—2361no—
492C4⋊C4C2×C4×D5912yes—
493C4⋊C4C2×C4×D5912yes—
494C4⋊C4Q8×D5912yes—
495C22⋊C4—1184no—
496C22⋊C4—2362no—
497C22⋊C4—2362no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42—592no—
502C42—592no—
503C42—1181no—
504C42—1181no—
505C42—4721no—
506C42—2362no—
507C23—1182no—
508C23—1182no—
509C23—1182no—
510C23—2361no—
511C23—2361no—
512C23—2361no—
513C23—2361no—
514C23C2×D4×D5911yescharacteristic
515Q8—4724no—
516D4—594no—
517D4—1182no—
518D4—1182no—
519D4—1182no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—592no—
539C2×C4—592no—
540C2×C4—592no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1182no—
551C2×C4—4724no—
552C2×C4—4724no—
553C2×C4—4724no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4C22×C4×D5911yescharacteristic
563C2×C4C2×Q8×D5911yescharacteristic
564C2×C4#41×D5911yescharacteristic
565C2×C4#41×D5911yescharacteristic
566C2×C4#1427377311yescharacteristic
567C2×C4#1427377411yescharacteristic
568C8—22no—
569C8—41no—
570C8—1182no—
571C8—1182no—
572C22—1184no—
573C22—2362no—
574C22—2362no—
575C22—2362no—
576C22—2362no—
577C22—2362no—
578C22—2362no—
579C22C4×D4×D5911yescharacteristic
580C22#1683804411yescharacteristic
581C22#1683806511yescharacteristic
582C22#137×D5911yescharacteristic
583C22#1683820111yescharacteristic
584C22#1683820311yescharacteristic
585C22#1683820411yescharacteristic
586C4—41no—
587C4—41no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—81no—
592C4—1182no—
593C4—1182no—
594C4—1182no—
595C4—4721no—
596C4—4721no—
597C4—2361no—
598C4—2362no—
599C4C4×Q8×D5911yescharacteristic
600C4#1683788211yescharacteristic
601C4#1683808811yescharacteristic
602C4#1683810811yescharacteristic
603C2—2364no—
604C2#2142710411yescharacteristic
605C2#2143291211yescharacteristic
606C2#2143294311yescharacteristic
607C2#2143618011yescharacteristic
608C2#2143625511yescharacteristic
609C2#2143700911yescharacteristic
610C2#2143704911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1918C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1918D59
hypocenterC59#22979
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #21427183C2semidirect
4. #21425067C2semidirect
5. #21424867C2semidirect
6. #21424831C2semidirect
7. #21424830C2semidirect
8. #21424691C2semidirect
11. #21423969C2semidirect
15. C59×#1918C2semidirect
20. #16837681C4semidirect
68. #14273726C2×C4semidirect
92. #14273660C2×C4semidirect
94. Dic59⋊C8C2×C4semidirect
98. #14273647Q8semidirect
120. #107×C59C2×C4semidirect
306. #1918D59semidirect
365. C59#22979semidirect
408. #107C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168376812
#214271832
#214270052
#214250672
#214248672
#214248312
#214248302
#214246912
#214243582
#214243462
#214239692
#214239242
#214239202
#214239102
C59×#19182
#2297959