Finite Group Databasebeta

← #28599894

Subgroups — #28599894

Order 15104.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837536C211yesmaximal, characteristic
2C4×#14273701C211yesmaximal, characteristic, complemented
3#21424971C212yesmaximal, complemented
4#21424796C212yesmaximal, complemented
5#21424720C212yesmaximal, complemented
6#21424447C212yesmaximal, complemented
7#21424413C211yesmaximal, characteristic, complemented
8#21424289C211yesmaximal, characteristic, complemented
9#21424027C211yesmaximal, characteristic, complemented
10#21423858C211yesmaximal, characteristic, complemented
11C2×C59×#390C211yesmaximal, characteristic, complemented
12C22×#11745669C2212yes—
13C2×#14273701C2211yescharacteristic
14C2×C4×D236C2211yescharacteristic, complemented
15C2×C4×C22⋊D59—21no—
16#16837536C418yescomplemented
17C2×#14273658C2212yescomplemented
18C2×#14273648C2212yes—
19C2×#14273648C2212yes—
20C2×#14273648C2212yes—
21C2×#14273647C2212yescomplemented
22C2×C22⋊(C59⋊C8)C2211yescharacteristic
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273638C2212yescomplemented
25#16837304C2211yescharacteristic, complemented
26#16837300C2211yescharacteristic, complemented
27#16837293—21no—
28#16837281—21no—
29#16837262C2211yescharacteristic
30C4×C4⋊Dic59C2211yescharacteristic, complemented
31C4×(C2×D59)⋊C4—21no—
32#16837164C2212yescomplemented
33#16837164C2212yescomplemented
34#16837157C2212yescomplemented
35C23×D4×C59C2211yescharacteristic
36C22×C4⋊C4×C59C2211yescharacteristic
37C22×C22⋊C4×C59—21no—
38C22×C22⋊C4×C59—21no—
39C22×C42×C59C2211yescharacteristic
40C59×#390—21no—
41C59×#390—21no—
42C59×#390C2212yes—
43C59×#390C2212yes—
44C22×D236C2311yescharacteristic
45C22×C22⋊D59—21no—
46C22×C4×D59—21no—
47C2×#11745669—42no—
48C2×#11745669—42no—
49C2×#11745669C2×C418yescomplemented
50C22×C59⋊C8C2312yes—
51C22×C59⋊C8C2312yes—
52#14273701—22no—
53#14273701C2×C414yes—
54C4×D236—42no—
55C4×C22⋊D59—44no—
56C2×C4⋊Dic59C2311yescharacteristic
57C2×C4⋊Dic59C2311yescharacteristic
58C2×C4⋊Dic59C2311yescharacteristic
59C2×Dic59⋊C4—21no—
60C2×C22⋊Dic59—21no—
61C2×(C2×D59)⋊C4—21no—
62C2×(C2×D59)⋊C4—21no—
63C2×(C2×D59)⋊C4—21no—
64C2×(C2×D59)⋊C4—21no—
65C2×C4×Dic59—21no—
66#14273658—42no—
67#14273648—42no—
68#14273648—42no—
69#14273648C2×C418yescomplemented
70#14273647—42no—
71C22⋊(C59⋊C8)—22no—
72C22⋊(C59⋊C8)C2×C414yescomplemented
73#14273638—42no—
74C59⋊#95—21no—
75C59⋊#95—21no—
76C59⋊#95—21no—
77C25×C59—21no—
78C22×D4×C59—21no—
79C22×D4×C59—22no—
80C22×D4×C59—42no—
81C22×D4×C59—42no—
82C22×D4×C59C2×C414yescomplemented
83C22×D4×C59C2312yes—
84C23×C4×C59—21no—
85C23×C4×C59D411yescharacteristic
86C23×C4×C59D411yescharacteristic
87C23×C4×C59C2311yescharacteristic
88C2×C4⋊C4×C59—21no—
89C2×C4⋊C4×C59—42no—
90C2×C4⋊C4×C59C2312yes—
91C2×C22⋊C4×C59—22no—
92C2×C22⋊C4×C59—22no—
93C2×C22⋊C4×C59—22no—
94C2×C22⋊C4×C59—22no—
95C2×C22⋊C4×C59—44no—
96C2×C22⋊C4×C59—44no—
97C2×C42×C59—42no—
98C2×C42×C59D411yescharacteristic
99C2×C42×C59D411yescharacteristic
100C2×C42×C59C2311yescharacteristic
101C2×C42×C59C2311yescharacteristic
102C24×C59—24no—
103C24×C59—22no—
104C24×C59—42no—
105C24×C59—42no—
106C24×C59—42no—
107C24×C59—42no—
108C24×C59—42no—
109C24×C59—42no—
110C24×C59C2×D411yescharacteristic
111C23×D59—21no—
112C2×D4×C59—24no—
113C2×D4×C59—22no—
114C2×D4×C59—22no—
115C2×D4×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59—22no—
118C2×D4×C59—48no—
119C2×D4×C59—42no—
120C2×D4×C59—42no—
121C2×D4×C59—48no—
122C2×D4×C59C22×C414yes—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—41no—
130C22×C4×C59—41no—
131C22×C4×C59—41no—
132C22×C4×C59—42no—
133C22×C4×C59—42no—
134C22×C4×C59—42no—
135C22×C4×C59C22×C412yes—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59#4112yes—
142C22×C4×C59C2411yescharacteristic
143C2×C22⋊D59—24no—
144C2×C22⋊D59—42no—
145C2×C22⋊D59—42no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C2×D236—22no—
150C2×D236—22no—
151C2×D236—21no—
152C2×D236C22×C412yes—
153C2×C4×D59—42no—
154C2×C4×D59—42no—
155C2×C4×D59—42no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—42no—
159C22⋊C4×C59—44no—
160C22⋊C4×C59—44no—
161C22⋊C4×C59—44no—
162C22⋊C4×C59—44no—
163C42×C59—22no—
164C42×C59—22no—
165C42×C59—41no—
166C42×C59—41no—
167C22⋊Dic59—42no—
168#11745669—48no—
169(C2×D59)⋊C4—24no—
170(C2×D59)⋊C4—42no—
171(C2×D59)⋊C4—42no—
172(C2×D59)⋊C4—42no—
173C4⋊Dic59—21no—
174C4⋊Dic59—22no—
175C4⋊Dic59—22no—
176C4⋊Dic59C22×C412yes—
177Dic59⋊C4—42no—
178C4×Dic59—42no—
179C2×C59⋊C8—22no—
180C2×C59⋊C8—22no—
181C2×C59⋊C8—22no—
182C2×C59⋊C8—22no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8C22×C414yes—
185C23×C59—21no—
186C23×C59—21no—
187C23×C59—21no—
188C23×C59—21no—
189C23×C59—24no—
190C23×C59—48no—
191C23×C59—48no—
192C23×C59—42no—
193C23×C59—42no—
194C23×C59—42no—
195C23×C59—42no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59—48no—
199C23×C59—84no—
200C23×C59—84no—
201C23×C59—84no—
202C23×C59—84no—
203C23×C59C4×D412yes—
204C23×C59AGL(1,ℤ/8)12yes—
205C23×C59AGL(1,ℤ/8)12yes—
206C23×C59C22×D411yescharacteristic
207C22×D59—22no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—41no—
214D4×C59—24no—
215D4×C59—24no—
216D4×C59—48no—
217D4×C59—42no—
218D4×C59—42no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—24no—
224C2×C4×C59—24no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—44no—
235C2×C4×C59—44no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—84no—
240C2×C4×C59C4×D412yes—
241C2×C4×C59AGL(1,ℤ/8)12yes—
242C2×C4×C59C23×C411yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59H(5,𝔽2)12yes—
245C2×C4×C59H(5,𝔽2)12yes—
246C22⋊D59—48no—
247C2×Dic59—22no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256D236—24no—
257D236—42no—
258C4×D59—84no—
259C59⋊C8—24no—
260C59⋊C8—42no—
261#22046—591nomaximal, Sylow
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—44no—
269C22×C59—44no—
270C22×C59—44no—
271C22×C59—816no—
272C22×C59—82no—
273C22×C59—82no—
274C22×C59—82no—
275C22×C59—82no—
276C22×C59#44214yes—
277C22×C59C2×C4×D411yescharacteristic
278C22×C59#53411yescharacteristic
279C22×C59#53511yescharacteristic
280C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C59#57511yescharacteristic
284C2×D59—42no—
285C2×D59—42no—
286C2×D59—42no—
287C2×D59—81no—
288C2×D59—81no—
289C2×D59—81no—
290C2×D59—81no—
291C4×C59—22no—
292C4×C59—22no—
293C4×C59—41no—
294C4×C59—41no—
295C4×C59—41no—
296C4×C59—82no—
297C4×C59#51812yes—
298C4×C59#53812yes—
299Dic59—42no—
300Dic59—81no—
301Dic59—81no—
302C2×#447—591no—
303C4×#121—591no—
304C2×#390D5911yescharacteristic, complemented
305#1972—592no—
306#1936—592no—
307#1926—592no—
308#1879—592no—
309#1874—591no—
310#1853—591no—
311#1786—591no—
312#1749—591no—
313C2×C59—24no—
314C2×C59—84no—
315C2×C59#232111yescharacteristic
316C2×C59C2×#44211yescharacteristic
317C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C59#319211yescharacteristic
319C2×C59#319411yescharacteristic
320C2×C59#321311yescharacteristic
321C2×C59#321711yescharacteristic
322D59—82no—
323C23×D4C2×D5911yescharacteristic
324C22×D8—592no—
325C2×#121—591no—
326C2×C4×D4—591no—
327C2×C4×D4—1181no—
328C22×C4⋊C4C2×D5911yescharacteristic
329C22×C22⋊C4—21no—
330C22×C22⋊C4—21no—
331C22×C42C2×D5911yescharacteristic
332#447—598no—
333C2×#106—592no—
334C2×#102—592no—
335C2×#102—592no—
336C2×#102—592no—
337C2×#102—592no—
338C2×C22⋊C8—591no—
339C2×C22⋊C8—591no—
340C2×#97—592no—
341#390—21no—
342#390—21no—
343#390—591no—
344#390C2×D5912yes—
345#390C2×D5912yes—
346#389—591no—
347#388—1181no—
348#386—1181no—
349#380—591no—
350C4×C4⋊C4—591no—
351C4×C22⋊C4—1181no—
352#340—592no—
353#340—592no—
354#336—592no—
355C59#2204611yescharacteristic, complemented, Sylow
356C25—21no—
357C22×D4—22no—
358C22×D4—21no—
359C22×D4—42no—
360C22×D4—42no—
361C22×D4—591no—
362C22×D4—1181no—
363C22×D4C4×D5914yescomplemented
364C22×D4C22×D5912yes—
365C23×C4—21no—
366C23×C4—1181no—
367C23×C4C22⋊D5911yescharacteristic
368C23×C4C22⋊D5911yescharacteristic
369C23×C4C22×D5911yescharacteristic
370C2×D8—598no—
371C2×D8—2362no—
372C2×D8—2362no—
373C22×C8—592no—
374C22×C8—592no—
375#121—594no—
376#121—1182no—
377C4×D4—2362no—
378C4×D4—2364no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4C22×D5912yes—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—44no—
391C2×C22⋊C4—44no—
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C22⋊C4—1181no—
395C2×C22⋊C4—1181no—
396C2×C22⋊C4—1181no—
397C2×C42—42no—
398C2×C42—1181no—
399C2×C42C22⋊D5911yescharacteristic
400C2×C42C22⋊D5911yescharacteristic
401C2×C42C22×D5911yescharacteristic
402C2×C42C22×D5911yescharacteristic
403#106—2362no—
404#102—598no—
405#102—2362no—
406#102—2362no—
407#102—2362no—
408C22⋊C8—594no—
409C22⋊C8—1182no—
410#97—2362no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414C24—24no—
415C24—22no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—1181no—
423C24C2×C22⋊D5911yescharacteristic
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—48no—
434C2×D4—592no—
435C2×D4—1184no—
436C2×D4—1181no—
437C2×D4—1182no—
438C2×D4—1182no—
439C2×D4—2362no—
440C2×D4—2362no—
441C2×D4C2×C4×D5914yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4—2362no—
460C22×C4C2×C4×D5912yes—
461C22×C4#1174568512yes—
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C23×D5911yescharacteristic
468D8—2368no—
469C2×C8—594no—
470C2×C8—1182no—
471C2×C8—1182no—
472C2×C8—1182no—
473C2×C8—1182no—
474C2×C8—1182no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—592no—
479C4⋊C4—1181no—
480C4⋊C4—1182no—
481C4⋊C4—1182no—
482C4⋊C4—2362no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—1184no—
488C22⋊C4—2362no—
489C22⋊C4—2362no—
490C22⋊C4—2362no—
491C22⋊C4—2362no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—2362no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—48no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23—1182no—
516C23—2361no—
517C23—2361no—
518C23—2361no—
519C23—2361no—
520C23—2361no—
521C23—2361no—
522C23C4×C22⋊D5912yes—
523C23#1427375212yes—
524C23#1427375312yes—
525C23C22×C22⋊D5911yescharacteristic
526D4—24no—
527D4—24no—
528D4—48no—
529D4—42no—
530D4—42no—
531D4—1184no—
532D4—2368no—
533D4—2362no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—84no—
555C2×C4—1182no—
556C2×C4—4724no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4C4×C22⋊D5912yes—
566C2×C4#1427375312yes—
567C2×C4C22×C4×D5911yescharacteristic
568C2×C4C2×#1174568511yescharacteristic
569C2×C4#1427377212yes—
570C2×C4#1427377212yes—
571C8—1184no—
572C8—2362no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—816no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—4721no—
588C22—4721no—
589C22—4721no—
590C22—4721no—
591C22—2362no—
592C22—2362no—
593C22—2362no—
594C22#1683751214yes—
595C22C2×C4×C22⋊D5911yescharacteristic
596C22#1683796111yescharacteristic
597C22#1683796911yescharacteristic
598C22C2×#1427375211yescharacteristic
599C22C2×#1427375311yescharacteristic
600C22C2×#1427375311yescharacteristic
601C22#1683820011yescharacteristic
602C4—22no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—82no—
608C4—4721no—
609C4—4721no—
610C4—2362no—
611C4#1683787512yes—
612C4#1683799012yes—
613C2—24no—
614C2—84no—
615C2—4722no—
616C2#2142675411yescharacteristic
617C2C2×#1683751211yescharacteristic
618C2C4×#1427375311yescharacteristic
619C2#2143489311yescharacteristic
620C2#2143491011yescharacteristic
621C2#2143508311yescharacteristic
622C2#2143511411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#390C2
socleC23×C59C22×D4
prime radicalC2×#16837536C2
cosocleC22×C4×C59C24
hypercenterC2×#390D59
hypocenterC59#22046
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C4×#14273701C2semidirect
3. #21424971C2semidirect
4. #21424796C2semidirect
5. #21424720C2semidirect
6. #21424447C2semidirect
7. #21424413C2semidirect
8. #21424289C2semidirect
9. #21424027C2semidirect
10. #21423858C2semidirect
11. C2×C59×#390C2semidirect
14. C2×C4×D236C22semidirect
16. #16837536C4semidirect
17. C2×#14273658C22semidirect
21. C2×#14273647C22semidirect
24. C2×#14273638C22semidirect
25. #16837304C22semidirect
26. #16837300C22semidirect
30. C4×C4⋊Dic59C22semidirect
32. #16837164C22semidirect
33. #16837164C22semidirect
34. #16837157C22semidirect
49. C2×#11745669C2×C4semidirect
69. #14273648C2×C4semidirect
72. C22⋊(C59⋊C8)C2×C4semidirect
82. C22×D4×C59C2×C4semidirect
304. C2×#390D59semidirect
355. C59#22046semidirect
363. C22×D4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168375362
C4×#142737012
#214249712
#214247962
#214247202
#214244472
#214244132
#214242892
#214240272
#214238582
C2×C59×#3902
#2204659