Finite Group Databasebeta

← #28600563

Subgroups — #28600563

Order 15104.

712 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434011C211yesmaximal, characteristic
2#21432795C211yesmaximal, characteristic
3#21432659C211yesmaximal, characteristic, complemented
4#21424847C211yesmaximal, characteristic, complemented
5#21424842C211yesmaximal, characteristic, complemented
6#21424805C211yesmaximal, characteristic, complemented
7#21424798C211yesmaximal, characteristic, complemented
8#21424694C211yesmaximal, characteristic, complemented
9#21424694C211yesmaximal, characteristic, complemented
10#21424398C211yesmaximal, characteristic
11#21424368C211yesmaximal, characteristic, complemented
12#21424368C211yesmaximal, characteristic, complemented
13#21424193C211yesmaximal, characteristic, complemented
14#21423882C211yesmaximal, characteristic, complemented
15C59×#1855C211yesmaximal, characteristic, complemented
16C2×#14273735C2211yescharacteristic
17C22×#11745664C2211yescharacteristic
18#16837913C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837744C411yescharacteristic, complemented
21#16837744C411yescharacteristic, complemented
22#16837738C411yescharacteristic, complemented
23#16837733C411yescharacteristic, complemented
24#16837727C411yescharacteristic, complemented
25#16837727C411yescharacteristic, complemented
26#16837721C411yescharacteristic, complemented
27#16837716C411yescharacteristic, complemented
28#16837477—21no—
29#16837477—21no—
30#16837476—21no—
31#16837476—21no—
32#16837440C2211yescharacteristic
33#16837440C2211yescharacteristic
34#16837430C2211yescharacteristic
35#16837430C2211yescharacteristic
36#16837430C2211yescharacteristic
37#16837430C2211yescharacteristic
38#16837430C2211yescharacteristic
39#16837428—21no—
40#16837428—21no—
41#16837425—21no—
42#16837425—21no—
43C2×#14273653C2211yescharacteristic
44C2×#14273653C2211yescharacteristic
45#16837399C2211yescharacteristic
46#16837399C2211yescharacteristic
47#16837370C2211yescharacteristic
48#16837360C2211yescharacteristic
49C2×#14273637C2211yescharacteristic
50C2×C8×Dic59C2211yescharacteristic
51C4×C22⋊Dic59C2211yescharacteristic
52#16837163C2211yescharacteristic
53#16837163C2211yescharacteristic
54#16837163C2211yescharacteristic
55#16837163C2211yescharacteristic
56#16837163C2211yescharacteristic
57#16837163C2211yescharacteristic
58#16837156C2211yescharacteristic
59#16837156C2211yescharacteristic
60C22×M16×C59C2211yescharacteristic
61C2×#108×C59C2211yescharacteristic
62C2×#108×C59C2211yescharacteristic
63C59×#407C2211yescharacteristic
64C2×C22⋊C8×C59C2211yescharacteristic
65C59×#341C2211yescharacteristic
66C59×#341C2211yescharacteristic
67C2×#11745685C2311yescharacteristic
68#14273735—41no—
69#14273735—41no—
70C2×#11745664—41no—
71C2×#11745664—41no—
72C2×#11745664D411yescharacteristic
73C2×#11745664D411yescharacteristic
74C2×#11745664D411yescharacteristic
75C2×#11745664D411yescharacteristic
76C2×#11745659—21no—
77C22×C59⋊C8C2311yescharacteristic
78C22×C59⋊C8C2311yescharacteristic
79C2×C8×D59—21no—
80#14273710—21no—
81#14273706C2×C411yescharacteristic
82#14273706C2×C411yescharacteristic
83#14273701C2×C411yescharacteristic
84#14273701C2×C411yescharacteristic
85#14273696—21no—
86C4×C22⋊D59—21no—
87C4×C22⋊D59—21no—
88#14273686—21no—
89#14273686—21no—
90#14273686C2×C411yescharacteristic
91#14273686C2×C411yescharacteristic
92#14273686C2311yescharacteristic
93#14273686C2311yescharacteristic
94#14273686C2311yescharacteristic
95C2×C4⋊Dic59C2×C411yescharacteristic
96C2×C4⋊Dic59C2×C411yescharacteristic
97C2×C22⋊Dic59—21no—
98C2×C22⋊Dic59—21no—
99C2×C4×Dic59—21no—
100C2×C4×Dic59C2311yescharacteristic
101C2×C4×Dic59C2311yescharacteristic
102#14273661C2×C411yescharacteristic
103#14273661C2×C411yescharacteristic
104#14273661C2×C411yescharacteristic
105#14273661C2×C411yescharacteristic
106#14273659C2×C411yescharacteristic
107#14273659C2×C411yescharacteristic
108#14273659C2×C411yescharacteristic
109#14273659C2×C411yescharacteristic
110Dic59⋊C8—21no—
111Dic59⋊C8—21no—
112Dic59⋊C8—21no—
113Dic59⋊C8—21no—
114Dic59⋊C8—21no—
115Dic59⋊C8—21no—
116Dic59⋊C8—21no—
117Dic59⋊C8—21no—
118Dic59⋊C8—21no—
119Dic59⋊C8—21no—
120#14273653—41no—
121#14273653—41no—
122#14273652C2×C411yescharacteristic
123#14273652C2×C411yescharacteristic
124#14273652C2×C411yescharacteristic
125#14273652C2×C411yescharacteristic
126#14273649C2×C411yescharacteristic
127#14273649C2×C411yescharacteristic
128#14273649C2×C411yescharacteristic
129#14273649C2×C411yescharacteristic
130(C2×D59)⋊C8—21no—
131(C2×D59)⋊C8—21no—
132#14273637—21no—
133#14273637—41no—
134C8×Dic59—21no—
135C8×Dic59—41no—
136C59⋊#95—21no—
137C23×C4×C59C2311yescharacteristic
138C2×M16×C59—21no—
139C2×M16×C59—41no—
140C2×M16×C59—41no—
141C2×M16×C59C2311yescharacteristic
142C2×M16×C59C2311yescharacteristic
143C2×M16×C59C2311yescharacteristic
144C2×M16×C59C2311yescharacteristic
145C22×C8×C59—21no—
146C22×C8×C59C2311yescharacteristic
147C22×C8×C59C2311yescharacteristic
148#108×C59—41no—
149#108×C59—41no—
150C22⋊C8×C59—21no—
151C22⋊C8×C59—21no—
152C22⋊C8×C59C2×C411yescharacteristic
153C22⋊C8×C59C2×C411yescharacteristic
154C22⋊C8×C59C2×C411yescharacteristic
155C22⋊C8×C59C2×C411yescharacteristic
156C24×C59#4111yescharacteristic
157C22×C4×C59—21no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59—21no—
161C22×C4×C59C22×C411yescharacteristic
162C22×C4×C59C22×C411yescharacteristic
163C22×C4×C59C2×D411yescharacteristic
164C22×C4×C59C2×D411yescharacteristic
165C22×C4×C59#4111yescharacteristic
166C22×C4×C59C2411yescharacteristic
167C2×C22⋊D59—21no—
168C2×C22⋊D59—21no—
169C2×C22⋊D59—21no—
170C22×Dic59—21no—
171C22×Dic59—21no—
172C22×Dic59—21no—
173C22×Dic59—21no—
174#11745685—21no—
175#11745685—21no—
176#11745685—21no—
177#11745685—21no—
178C2×D236C22×C411yescharacteristic
179C2×C4×D59—21no—
180C2×Dic118C22×C411yescharacteristic
181M16×C59—21no—
182M16×C59—21no—
183M16×C59—21no—
184M16×C59—21no—
185M16×C59—21no—
186M16×C59—21no—
187M16×C59—21no—
188M16×C59—21no—
189M16×C59—42no—
190C2×C8×C59—21no—
191C2×C8×C59—21no—
192C2×C8×C59—21no—
193C2×C8×C59—21no—
194C2×C8×C59—21no—
195C2×C8×C59—21no—
196C2×C8×C59—21no—
197C2×C8×C59—21no—
198C2×C8×C59—42no—
199C2×C8×C59C22×C411yescharacteristic
200C2×C8×C59C22×C411yescharacteristic
201C2×C8×C59C22×C411yescharacteristic
202C2×C8×C59C22×C411yescharacteristic
203C22⋊Dic59—21no—
204C22⋊Dic59—21no—
205C22⋊Dic59—21no—
206C22⋊Dic59—21no—
207C22⋊Dic59—21no—
208C22⋊Dic59—21no—
209C22⋊Dic59—21no—
210C22⋊Dic59—42no—
211(C2×D59)⋊C4—21no—
212(C2×D59)⋊C4—21no—
213C4⋊Dic59—21no—
214C4⋊Dic59—21no—
215C4⋊Dic59C22×C411yescharacteristic
216C4⋊Dic59C22×C411yescharacteristic
217C4⋊Dic59C22×C411yescharacteristic
218C4⋊Dic59C22×C411yescharacteristic
219C4⋊Dic59C22×C411yescharacteristic
220C4⋊Dic59C22×C411yescharacteristic
221Dic59⋊C4—21no—
222Dic59⋊C4—21no—
223C4×Dic59—21no—
224C4×Dic59—21no—
225C4×Dic59—21no—
226C4×Dic59—21no—
227C4×Dic59—21no—
228C4×Dic59—21no—
229C4×Dic59—21no—
230C4×Dic59—21no—
231C4×Dic59—21no—
232#11745664—41no—
233#11745664—41no—
234#11745664—41no—
235#11745664—41no—
236C2×C59⋊C8—21no—
237C2×C59⋊C8—21no—
238C2×C59⋊C8—21no—
239C2×C59⋊C8—21no—
240C2×C59⋊C8C2×D411yescharacteristic
241C2×C59⋊C8C2×D411yescharacteristic
242C2×C59⋊C8C2×D411yescharacteristic
243C2×C59⋊C8C2×D411yescharacteristic
244#11745659—42no—
245C8×D59—42no—
246C23×C59—21no—
247C23×C59—21no—
248C23×C59—41no—
249C23×C59—41no—
250C23×C59C4×D411yescharacteristic
251C23×C59C4×D411yescharacteristic
252C23×C59C2×#4111yescharacteristic
253C22×D59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—21no—
258C2×C4×C59—21no—
259C2×C4×C59—21no—
260C2×C4×C59—21no—
261C2×C4×C59—21no—
262C2×C4×C59—21no—
263C2×C4×C59—21no—
264C2×C4×C59C4×D411yescharacteristic
265C2×C4×C59C4×D411yescharacteristic
266C2×C4×C59C23×C411yescharacteristic
267C2×C4×C59C22×D411yescharacteristic
268C2×C4×C59H(5,𝔽2)11yescharacteristic
269C2×C4×C59H(5,𝔽2)11yescharacteristic
270C2×C4×C592−1+411yescharacteristic
271C2×C4×C592−1+411yescharacteristic
272C22⋊D59—41no—
273C22⋊D59—41no—
274C22⋊D59—81no—
275C22⋊D59—81no—
276C2×Dic59—22no—
277C2×Dic59—21no—
278C2×Dic59—21no—
279C2×Dic59—21no—
280C2×Dic59—21no—
281C2×Dic59—21no—
282C2×Dic59—21no—
283C2×Dic59—21no—
284C2×Dic59—41no—
285C2×Dic59—41no—
286C2×Dic59—41no—
287C2×Dic59—41no—
288C2×Dic59—41no—
289C2×Dic59—81no—
290C2×Dic59—81no—
291D236—21no—
292D236—21no—
293C4×D59—41no—
294C4×D59—41no—
295Dic118—21no—
296Dic118—21no—
297C8×C59—21no—
298C8×C59—21no—
299C8×C59—21no—
300C8×C59—21no—
301C8×C59—41no—
302C8×C59—41no—
303C8×C59—41no—
304C8×C59—41no—
305C59⋊C8—21no—
306C59⋊C8—21no—
307C59⋊C8—21no—
308C59⋊C8—21no—
309#22097—591nomaximal, Sylow
310C22×C59—21no—
311C22×C59—21no—
312C22×C59—21no—
313C22×C59—21no—
314C22×C59—21no—
315C22×C59—21no—
316C22×C59—21no—
317C22×C59—21no—
318C22×C59—81no—
319C22×C59—81no—
320C22×C59C2×C4×D411yescharacteristic
321C22×C59#53911yescharacteristic
322C22×C59#53911yescharacteristic
323C2×D59—41no—
324C2×D59—41no—
325C2×D59—41no—
326C4×C59—21no—
327C4×C59—21no—
328C4×C59#52011yescharacteristic
329C4×C59#52011yescharacteristic
330C4×C59#53411yescharacteristic
331C4×C59#53611yescharacteristic
332Dic59—42no—
333Dic59—41no—
334Dic59—41no—
335Dic59—41no—
336Dic59—41no—
337Dic59—41no—
338Dic59—41no—
339Dic59—41no—
340Dic59—81no—
341#3091—591no—
342#2930—591no—
343#2905—591no—
344#1946—591no—
345#1945—591no—
346#1938—591no—
347#1937—591no—
348#1922—591no—
349#1922—591no—
350#1871—591no—
351#1866—591no—
352#1866—591no—
353#1855D5911yescharacteristic, complemented
354#1826—591no—
355#1753—591no—
356C2×C59—21no—
357C2×C59—21no—
358C2×C59—41no—
359C2×C59—41no—
360C2×C59#232211yescharacteristic
361C2×C59#295611yescharacteristic
362C2×C59#295711yescharacteristic
363D59—81no—
364C2×#131—591no—
365C22×M16—591no—
366C22×M16C2×D5911yescharacteristic
367#525—591no—
368C2×#117—591no—
369#485—591no—
370#485—591no—
371#484—591no—
372#483—591no—
373#482—591no—
374#482—591no—
375#481—591no—
376#480—591no—
377#436—1181no—
378#436—1181no—
379#435—1181no—
380#435—1181no—
381C2×#108C2×D5911yescharacteristic
382C2×#108C2×D5911yescharacteristic
383#427—591no—
384#427—591no—
385#424—591no—
386#424—591no—
387#424—591no—
388#424—591no—
389#424—591no—
390#424—1181no—
391#424—1181no—
392#423—1181no—
393#423—1181no—
394C2×#104—591no—
395C2×#104—591no—
396#416—591no—
397#416—591no—
398#408—591no—
399#407C2×D5911yescharacteristic
400C2×C22⋊C8C2×D5911yescharacteristic
401#405—591no—
402C2×#97—591no—
403C2×C4×C8—591no—
404C4×C22⋊C4—591no—
405#341C2×D5911yescharacteristic
406#341C2×D5911yescharacteristic
407#339—591no—
408#339—591no—
409#339—591no—
410#339—591no—
411#339—591no—
412#339—591no—
413#336—591no—
414#336—591no—
415C59#2209711yescharacteristic, complemented, Sylow
416C2×#41—591no—
417C23×C4C22×D5911yescharacteristic
418#131—2361no—
419#131—2361no—
420C2×M16—21no—
421C2×M16—41no—
422C2×M16—41no—
423C2×M16—591no—
424C2×M16—591no—
425C2×M16—591no—
426C2×M16—591no—
427C2×M16—1181no—
428C2×M16—2361no—
429C2×M16—2361no—
430C2×M16C22×D5911yescharacteristic
431C2×M16C22×D5911yescharacteristic
432C2×M16C22×D5911yescharacteristic
433C2×M16C22×D5911yescharacteristic
434C22×C8—21no—
435C22×C8—591no—
436C22×C8—591no—
437C22×C8—1181no—
438C22×C8C22×D5911yescharacteristic
439C22×C8C22×D5911yescharacteristic
440#123—1181no—
441C22⋊Q8—591no—
442C22⋊Q8—591no—
443#121—591no—
444#121—591no—
445#120—1181no—
446C4×D4—1181no—
447C4×D4—1181no—
448#117—591no—
449#117—591no—
450#117—591no—
451#117—591no—
452#117—591no—
453#117—1181no—
454#117—1181no—
455C2×C4⋊C4—591no—
456C2×C4⋊C4—591no—
457C2×C22⋊C4—1181no—
458C2×C22⋊C4—1181no—
459C2×C42—591no—
460C2×C42—591no—
461C2×C42—1181no—
462#108—41no—
463#108—41no—
464#107—591no—
465#107—591no—
466#107—591no—
467#107—591no—
468#106—591no—
469#106—591no—
470#106—591no—
471#106—591no—
472C4⋊C8—1181no—
473C4⋊C8—1181no—
474C4⋊C8—1181no—
475C4⋊C8—1181no—
476C4⋊C8—1181no—
477C4⋊C8—1181no—
478C4⋊C8—1181no—
479C4⋊C8—1181no—
480C4⋊C8—1181no—
481C4⋊C8—1181no—
482#104—2361no—
483#104—2361no—
484#103—591no—
485#103—591no—
486#103—591no—
487#103—591no—
488#102—591no—
489#102—591no—
490#102—591no—
491#102—591no—
492C22⋊C8—21no—
493C22⋊C8—21no—
494C22⋊C8—1181no—
495C22⋊C8—1181no—
496C22⋊C8C4×D5911yescharacteristic
497C22⋊C8C4×D5911yescharacteristic
498C22⋊C8C4×D5911yescharacteristic
499C22⋊C8C4×D5911yescharacteristic
500#97—1181no—
501#97—2361no—
502C4×C8—1181no—
503C4×C8—2361no—
504#95—1181no—
505C24#1174568911yescharacteristic
506#41—1181no—
507#41—1181no—
508#41—1181no—
509#41—1181no—
510C2×Q8—591no—
511C2×D4—591no—
512C2×D4—1181no—
513C2×D4—1181no—
514C2×D4—1181no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—1181no—
520C22×C4—1181no—
521C22×C4—1181no—
522C22×C4—1181no—
523C22×C4—1181no—
524C22×C4C2×C4×D5911yescharacteristic
525C22×C4C2×C4×D5911yescharacteristic
526C22×C4D4×D5911yescharacteristic
527C22×C4D4×D5911yescharacteristic
528C22×C4#1174568911yescharacteristic
529C22×C4C23×D5911yescharacteristic
530M16—21no—
531M16—21no—
532M16—21no—
533M16—21no—
534M16—21no—
535M16—21no—
536M16—21no—
537M16—21no—
538M16—42no—
539M16—2362no—
540M16—2361no—
541M16—2361no—
542M16—2361no—
543M16—2361no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—42no—
553C2×C8—591no—
554C2×C8—591no—
555C2×C8—591no—
556C2×C8—591no—
557C2×C8—1181no—
558C2×C8—1181no—
559C2×C8—1181no—
560C2×C8—1181no—
561C2×C8—2362no—
562C2×C8C2×C4×D5911yescharacteristic
563C2×C8C2×C4×D5911yescharacteristic
564C2×C8C2×C4×D5911yescharacteristic
565C2×C8C2×C4×D5911yescharacteristic
566C4⋊C4—591no—
567C4⋊C4—591no—
568C4⋊C4—591no—
569C4⋊C4—591no—
570C4⋊C4—591no—
571C4⋊C4—591no—
572C4⋊C4—1181no—
573C4⋊C4—1181no—
574C4⋊C4—1181no—
575C4⋊C4—1181no—
576C22⋊C4—1181no—
577C22⋊C4—1181no—
578C22⋊C4—1181no—
579C22⋊C4—1181no—
580C22⋊C4—1181no—
581C22⋊C4—1181no—
582C22⋊C4—1181no—
583C22⋊C4—1181no—
584C22⋊C4—1181no—
585C22⋊C4—2362no—
586C42—1181no—
587C42—1181no—
588C42—1181no—
589C42—1181no—
590C42—1181no—
591C42—1181no—
592C42—1181no—
593C42—1181no—
594C42—1181no—
595C23—21no—
596C23—21no—
597C23—41no—
598C23—41no—
599C23—1181no—
600C23#1427368911yescharacteristic
601C23#1427368911yescharacteristic
602C23C2×#1174568911yescharacteristic
603Q8—1181no—
604Q8—1181no—
605D4—1181no—
606D4—1181no—
607D4—4721no—
608D4—4721no—
609D4—2361no—
610D4—2361no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—1182no—
622C2×C4—1181no—
623C2×C4—1181no—
624C2×C4—1181no—
625C2×C4—1181no—
626C2×C4—1181no—
627C2×C4—1181no—
628C2×C4—1181no—
629C2×C4—4721no—
630C2×C4—4721no—
631C2×C4—2361no—
632C2×C4—2361no—
633C2×C4—2361no—
634C2×C4—2361no—
635C2×C4—2361no—
636C2×C4—2361no—
637C2×C4—2361no—
638C2×C4#1427368911yescharacteristic
639C2×C4#1427368911yescharacteristic
640C2×C4C22×C4×D5911yescharacteristic
641C2×C4C2×D4×D5911yescharacteristic
642C2×C4#1427377211yescharacteristic
643C2×C4#1427377211yescharacteristic
644C2×C4#1427377411yescharacteristic
645C2×C4#1427377411yescharacteristic
646C8—21no—
647C8—21no—
648C8—21no—
649C8—21no—
650C8—41no—
651C8—41no—
652C8—41no—
653C8—41no—
654C8—1181no—
655C8—1181no—
656C8—1181no—
657C8—1181no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—81no—
667C22—81no—
668C22—2361no—
669C22—2361no—
670C22—2361no—
671C22C2×#1427368911yescharacteristic
672C22#1683800111yescharacteristic
673C22#1683800111yescharacteristic
674C4—21no—
675C4—21no—
676C4—4721no—
677C4—2362no—
678C4—2361no—
679C4—2361no—
680C4—2361no—
681C4—2361no—
682C4—2361no—
683C4—2361no—
684C4—2361no—
685C4#1683788111yescharacteristic
686C4#1683788111yescharacteristic
687C4#1683796211yescharacteristic
688C4#1683797411yescharacteristic
689C2—21no—
690C2—21no—
691C2—41no—
692C2—41no—
693C2—4721no—
694C2#2142676411yescharacteristic
695C2#2143300511yescharacteristic
696C2#2143301211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1855C2
socleC22×C59C2×C4×D4
prime radical#16837913C22
cosocleC22×C4×C59C24
hypercenter#1855D59
hypocenterC59#22097
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1855

Complements

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

NH ≅ G/Ntype
3. #21432659C2semidirect
4. #21424847C2semidirect
5. #21424842C2semidirect
6. #21424805C2semidirect
7. #21424798C2semidirect
8. #21424694C2semidirect
9. #21424694C2semidirect
11. #21424368C2semidirect
12. #21424368C2semidirect
13. #21424193C2semidirect
14. #21423882C2semidirect
15. C59×#1855C2semidirect
20. #16837744C4semidirect
21. #16837744C4semidirect
22. #16837738C4semidirect
23. #16837733C4semidirect
24. #16837727C4semidirect
25. #16837727C4semidirect
26. #16837721C4semidirect
27. #16837716C4semidirect
353. #1855D59semidirect
415. C59#22097semidirect

Maximal subgroups

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

subgroupindex
#214340112
#214327952
#214326592
#214248472
#214248422
#214248052
#214247982
#214246942
#214243982
#214243682
#214241932
#214238822
C59×#18552
#2209759