Finite Group Databasebeta

← #28588492

Subgroups — #28588492

Order 15104.

1470 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273685C211yesmaximal, characteristic
2C2×#16837365C211yesmaximal, characteristic
3#21426475C211yesmaximal, characteristic
4#21424190C214yesmaximal, complemented
5#21424189C214yesmaximal, complemented
6#21423817C211yesmaximal, characteristic, complemented
7#21423816C211yesmaximal, characteristic, complemented
8#21423815C211yesmaximal, characteristic, complemented
9C59×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D59C2211yescharacteristic
11C22×#11745664C2211yescharacteristic
12C2×#14273686C2211yescharacteristic
13C2×#14273685—24no—
14C2×#14273685—21no—
15C2×#14273685—21no—
16C2×#14273685C418yescomplemented
17C2×#14273685C2212yes—
18C2×#14273685C2212yes—
19C22×C4⋊Dic59C2211yescharacteristic
20C22×(C2×D59)⋊C4—21no—
21C22×C4×Dic59—21no—
22#16837365—24no—
23C2×(C2×D59)⋊C8C2212yes—
24C2×(C2×D59)⋊C8C2212yes—
25#16837263—21no—
26#16837259C2212yes—
27#16837259C2212yes—
28#16837257—21no—
29C4×(C2×D59)⋊C4—21no—
30C4×(C2×D59)⋊C4—21no—
31#16837165C2214yescomplemented
32#16837164C2214yescomplemented
33#16837164C2214yescomplemented
34C22×M16×C59C2211yescharacteristic
35C2×#117×C59C2211yescharacteristic
36C22×C4⋊C4×C59C2211yescharacteristic
37C59×#340C2214yescomplemented
38C24×D59D411yescharacteristic
39C23×Dic59—21no—
40C23×Dic59D411yescharacteristic
41C22×C4×D59—22no—
42C22×C4×D59—22no—
43C22×C4×D59—22no—
44C22×C4×D59—22no—
45C22×C4×D59C2×C414yes—
46C22×C4×D59D411yescharacteristic
47C22×C4×D59D411yescharacteristic
48C22×C4×D59D414yes—
49C22×C4×D59C2311yescharacteristic
50C22×C4×D59C2311yescharacteristic
51C2×#11745664—22no—
52C2×#11745664—22no—
53C2×#11745664—22no—
54C22×C59⋊C8C2312yes—
55#14273686—22no—
56#14273686—22no—
57#14273685—28no—
58#14273685—28no—
59#14273685—24no—
60#14273685—24no—
61#14273685C2×C418yes—
62#14273685C2×C418yes—
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—22no—
65C2×C4⋊Dic59C2×C414yes—
66C2×C4⋊Dic59C2312yes—
67C2×C4⋊Dic59C2312yes—
68C2×C22⋊Dic59—21no—
69C2×(C2×D59)⋊C4—28no—
70C2×(C2×D59)⋊C4—22no—
71C2×(C2×D59)⋊C4—22no—
72C2×(C2×D59)⋊C4—22no—
73C2×(C2×D59)⋊C4—22no—
74C2×C4×Dic59—28no—
75C2×C4×Dic59—22no—
76C2×C4×Dic59—22no—
77C2×C4×Dic59—21no—
78(C2×D59)⋊C8—24no—
79(C2×D59)⋊C8—24no—
80C59⋊#95—22no—
81C59⋊#95—22no—
82C23×C4×C59—21no—
83C23×C4×C59C2311yescharacteristic
84C2×M16×C59—22no—
85C2×M16×C59—22no—
86C2×M16×C59—22no—
87C22×C8×C59C2312yes—
88#117×C59—22no—
89#117×C59—22no—
90C2×C4⋊C4×C59—21no—
91C2×C4⋊C4×C59—22no—
92C2×C4⋊C4×C59C2×C414yes—
93C2×C4⋊C4×C59C2312yes—
94C2×C4⋊C4×C59C2312yes—
95C2×C22⋊C4×C59—21no—
96C2×C42×C59—21no—
97C24×C59C2×D411yescharacteristic
98C23×D59—22no—
99C23×D59—22no—
100C23×D59—42no—
101C23×D59—42no—
102C23×D59C22⋊C414yes—
103C23×D59C2×D411yescharacteristic
104C23×D59C2×D411yescharacteristic
105C22×C4×C59—24no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—44no—
109C22×C4×C59C22×C412yes—
110C22×C4×C59C2×D411yescharacteristic
111C22×C4×C59C2×D412yes—
112C22×C4×C59#4112yes—
113C22×C4×C59#4112yes—
114C22×C4×C59#4112yes—
115C22×C4×C59#4112yes—
116C22×C4×C59C2411yescharacteristic
117C22×Dic59—22no—
118C22×Dic59—22no—
119C22×Dic59—24no—
120C22×Dic59—22no—
121C22×Dic59—22no—
122C22×Dic59—44no—
123C22×Dic59C22⋊C414yes—
124C22×Dic59C2×D411yescharacteristic
125C22×Dic59C2×D411yescharacteristic
126C2×C4×D59—28no—
127C2×C4×D59—28no—
128C2×C4×D59—28no—
129C2×C4×D59—28no—
130C2×C4×D59—22no—
131C2×C4×D59—22no—
132C2×C4×D59—22no—
133C2×C4×D59—22no—
134C2×C4×D59—21no—
135C2×C4×D59—21no—
136C2×C4×D59—21no—
137C2×C4×D59—21no—
138C2×C4×D59—21no—
139C2×C4×D59—21no—
140C2×C4×D59—21no—
141C2×C4×D59—21no—
142C2×C4×D59C22⋊C414yes—
143C2×C4×D59C22⋊C414yes—
144C2×C4×D59C22×C412yes—
145C2×C4×D59C22×C412yes—
146C2×C4×D59C2×D412yes—
147C2×C4×D59C2×D412yes—
148M16×C59—44no—
149C2×C8×C59—22no—
150C2×C8×C59—22no—
151C2×C8×C59—22no—
152C4⋊C4×C59—24no—
153C4⋊C4×C59—22no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59C22×C414yes—
157C22⋊C4×C59—42no—
158C42×C59—42no—
159C22⋊Dic59—42no—
160(C2×D59)⋊C4—28no—
161(C2×D59)⋊C4—28no—
162(C2×D59)⋊C4—44no—
163(C2×D59)⋊C4—44no—
164C4⋊Dic59—24no—
165C4⋊Dic59—22no—
166C4⋊Dic59—22no—
167C4⋊Dic59—22no—
168C4⋊Dic59C22×C414yes—
169C4×Dic59—28no—
170C4×Dic59—28no—
171C4×Dic59—42no—
172#11745664—44no—
173C2×C59⋊C8—22no—
174C2×C59⋊C8—22no—
175C2×C59⋊C8—22no—
176C23×C59—22no—
177C23×C59—22no—
178C23×C59C2×C22⋊C412yes—
179C23×C59#12312yes—
180C23×C59#12312yes—
181C23×C59C22×D411yescharacteristic
182C22×D59—21no—
183C22×D59—21no—
184C22×D59—21no—
185C22×D59—21no—
186C22×D59—22no—
187C22×D59—22no—
188C22×D59—22no—
189C22×D59—22no—
190C22×D59—28no—
191C22×D59—44no—
192C22×D59—44no—
193C22×D59—48no—
194C22×D59—48no—
195C22×D59C2×C22⋊C412yes—
196C22×D59C2×C22⋊C412yes—
197C2×C4×C59—24no—
198C2×C4×C59—48no—
199C2×C4×C59—44no—
200C2×C4×C59—42no—
201C2×C4×C59—42no—
202C2×C4×C59—42no—
203C2×C4×C59C2×C22⋊C412yes—
204C2×C4×C59#11714yes—
205C2×C4×C59#11714yes—
206C2×C4×C59#12314yes—
207C2×C4×C59#12312yes—
208C2×C4×C59#12314yes—
209C2×C4×C59#12312yes—
210C2×C4×C59C23×C411yescharacteristic
211C2×C4×C59C22×D411yescharacteristic
212C2×C4×C59C2×#4111yescharacteristic
213C2×C4×C59C2×#4111yescharacteristic
214C2×C4×C59C2×#4111yescharacteristic
215C2×C4×C59C2×#4111yescharacteristic
216C2×Dic59—21no—
217C2×Dic59—21no—
218C2×Dic59—21no—
219C2×Dic59—21no—
220C2×Dic59—28no—
221C2×Dic59—24no—
222C2×Dic59—48no—
223C2×Dic59—44no—
224C2×Dic59—42no—
225C2×Dic59—42no—
226C2×Dic59—42no—
227C2×Dic59C2×C22⋊C412yes—
228C2×Dic59C2×C22⋊C412yes—
229C4×D59—22no—
230C4×D59—22no—
231C4×D59—22no—
232C4×D59—22no—
233C4×D59—22no—
234C4×D59—22no—
235C4×D59—22no—
236C4×D59—22no—
237C4×D59—28no—
238C4×D59—28no—
239C8×C59—42no—
240C59⋊C8—42no—
241#20701—591nomaximal, Sylow
242C22×C59—24no—
243C22×C59—24no—
244C22×C59—22no—
245C22×C59—22no—
246C22×C59#38114yes—
247C22×C59C22×C22⋊C411yescharacteristic
248C22×C59C2×#12311yescharacteristic
249C22×C59C2×#12311yescharacteristic
250C22×C59#57512yes—
251C22×C59#57512yes—
252C2×D59—22no—
253C2×D59—22no—
254C2×D59—22no—
255C2×D59—22no—
256C2×D59—22no—
257C2×D59—22no—
258C2×D59—416no—
259C2×D59—42no—
260C2×D59—42no—
261C2×D59—42no—
262C2×D59—42no—
263C4×C59—44no—
264C4×C59—82no—
265C4×C59#38114yes—
266C4×C59C2×#11711yescharacteristic
267C4×C59C2×#11711yescharacteristic
268C4×C59C2×#12311yescharacteristic
269C4×C59C2×#12311yescharacteristic
270Dic59—22no—
271Dic59—22no—
272Dic59—44no—
273Dic59—82no—
274C22×#117—591no—
275C2×#407—591no—
276#2280—591no—
277#1825—594no—
278#1825—594no—
279#1736—591no—
280#1736—591no—
281#1736—591no—
282#1736D5911yescharacteristic, complemented
283C2×C59—24no—
284C2×C59C2×#38111yescharacteristic
285C2×C59#289611yescharacteristic
286C2×C59#289611yescharacteristic
287C2×C59#309312yes—
288C2×C59#309312yes—
289D59—42no—
290D59—42no—
291C24×C4—591no—
292C22×M16—591no—
293C22×M16C2×D5911yescharacteristic
294C2×#117—591no—
295C2×#117—598no—
296C2×#117—592no—
297C2×#117—592no—
298C2×#117—1181no—
299C2×#117—1181no—
300C2×#117—1184no—
301C2×#117C2×D5911yescharacteristic
302C22×C4⋊C4—591no—
303C22×C4⋊C4C2×D5911yescharacteristic
304C22×C22⋊C4—1181no—
305C22×C42—1181no—
306#407—1184no—
307C2×C22⋊C8—592no—
308C2×C22⋊C8—592no—
309#381—1181no—
310#380—592no—
311#380—592no—
312#379—1181no—
313C4×C22⋊C4—1181no—
314C4×C22⋊C4—1181no—
315#340—594no—
316#340—594no—
317#340—594no—
318#340C2×D5914yescomplemented
319C59#2070111yescharacteristic, complemented, Sylow
320C25—591no—
321C23×C4—21no—
322C23×C4—591no—
323C23×C4—594no—
324C23×C4—591no—
325C23×C4—591no—
326C23×C4—594no—
327C23×C4—591no—
328C23×C4—591no—
329C23×C4—1182no—
330C23×C4—1182no—
331C23×C4—1182no—
332C23×C4—1182no—
333C23×C4—1181no—
334C23×C4C22×D5911yescharacteristic
335C2×M16—22no—
336C2×M16—22no—
337C2×M16—22no—
338C2×M16—1182no—
339C2×M16—1182no—
340C2×M16—1182no—
341C22×C8—592no—
342C22×C8C22×D5912yes—
343#117—22no—
344#117—22no—
345#117—598no—
346#117—598no—
347#117—1188no—
348#117—1188no—
349#117—1184no—
350#117—1184no—
351#117—1182no—
352#117—1182no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—594no—
356C2×C4⋊C4—592no—
357C2×C4⋊C4—592no—
358C2×C4⋊C4—1181no—
359C2×C4⋊C4—1182no—
360C2×C4⋊C4C4×D5914yes—
361C2×C4⋊C4C22×D5912yes—
362C2×C4⋊C4C22×D5912yes—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—1188no—
365C2×C22⋊C4—1182no—
366C2×C22⋊C4—1182no—
367C2×C22⋊C4—1182no—
368C2×C22⋊C4—1182no—
369C2×C22⋊C4—1181no—
370C2×C42—21no—
371C2×C42—1188no—
372C2×C42—1182no—
373C2×C42—1182no—
374C2×C42—1181no—
375C22⋊C8—1184no—
376C22⋊C8—1184no—
377#95—1182no—
378#95—1182no—
379C24—591no—
380C24—591no—
381C24—594no—
382C24—1182no—
383C24—1182no—
384C24—2362no—
385C24—2362no—
386C24D4×D5911yescharacteristic
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—594no—
392C22×C4—591no—
393C22×C4—591no—
394C22×C4—594no—
395C22×C4—594no—
396C22×C4—592no—
397C22×C4—592no—
398C22×C4—592no—
399C22×C4—592no—
400C22×C4—1182no—
401C22×C4—1182no—
402C22×C4—1188no—
403C22×C4—1188no—
404C22×C4—1182no—
405C22×C4—1182no—
406C22×C4—1188no—
407C22×C4—1182no—
408C22×C4—1182no—
409C22×C4—1188no—
410C22×C4—1181no—
411C22×C4—1181no—
412C22×C4—1181no—
413C22×C4—1181no—
414C22×C4—1181no—
415C22×C4—1181no—
416C22×C4—1181no—
417C22×C4—1181no—
418C22×C4—1184no—
419C22×C4—1182no—
420C22×C4—1182no—
421C22×C4—2364no—
422C22×C4C2×C4×D5912yes—
423C22×C4#1174568512yes—
424C22×C4#1174568512yes—
425C22×C4D4×D5911yescharacteristic
426C22×C4D4×D5912yes—
427C22×C4#1174568712yes—
428C22×C4#1174568912yes—
429C22×C4C23×D5911yescharacteristic
430M16—44no—
431M16—2364no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—1182no—
436C2×C8—1182no—
437C2×C8—1182no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—594no—
443C4⋊C4—1184no—
444C4⋊C4—1182no—
445C4⋊C4—1182no—
446C4⋊C4—1182no—
447C4⋊C4C2×C4×D5914yes—
448C22⋊C4—42no—
449C22⋊C4—1188no—
450C22⋊C4—1188no—
451C22⋊C4—2364no—
452C22⋊C4—2364no—
453C22⋊C4—2362no—
454C42—42no—
455C42—1188no—
456C42—1188no—
457C42—2362no—
458C23—22no—
459C23—22no—
460C23—592no—
461C23—592no—
462C23—1181no—
463C23—1181no—
464C23—1181no—
465C23—1181no—
466C23—1182no—
467C23—1182no—
468C23—1182no—
469C23—1182no—
470C23—1188no—
471C23—2364no—
472C23—2364no—
473C23—2368no—
474C23—2368no—
475C23C22⋊C4×D5912yes—
476C23#1427370912yes—
477C23#1427371212yes—
478C23C2×D4×D5911yescharacteristic
479C2×C4—24no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—592no—
486C2×C4—592no—
487C2×C4—1181no—
488C2×C4—1181no—
489C2×C4—1181no—
490C2×C4—1181no—
491C2×C4—1188no—
492C2×C4—1182no—
493C2×C4—1182no—
494C2×C4—1182no—
495C2×C4—1182no—
496C2×C4—1188no—
497C2×C4—1188no—
498C2×C4—1182no—
499C2×C4—1182no—
500C2×C4—1182no—
501C2×C4—1182no—
502C2×C4—1184no—
503C2×C4—2362no—
504C2×C4—2362no—
505C2×C4—2362no—
506C2×C4—2368no—
507C2×C4—2364no—
508C2×C4C22⋊C4×D5912yes—
509C2×C4#1427368414yes—
510C2×C4#1427368514yes—
511C2×C4#1427370914yes—
512C2×C4#1427370912yes—
513C2×C4#1427371214yes—
514C2×C4#1427371212yes—
515C2×C4C22×C4×D5911yescharacteristic
516C2×C4C2×D4×D5911yescharacteristic
517C2×C4C2×#1174568711yescharacteristic
518C2×C4C2×#1174568511yescharacteristic
519C2×C4C2×#1174568511yescharacteristic
520C2×C4C2×#1174568911yescharacteristic
521C8—42no—
522C8—2362no—
523C22—22no—
524C22—22no—
525C22—24no—
526C22—24no—
527C22—1182no—
528C22—1182no—
529C22—1182no—
530C22—1182no—
531C22—1182no—
532C22—1182no—
533C22—23616no—
534C22—2362no—
535C22—2362no—
536C22—2362no—
537C22—2362no—
538C22#1683726414yes—
539C22C2×C22⋊C4×D5911yescharacteristic
540C22C2×#1427370911yescharacteristic
541C22C2×#1427371211yescharacteristic
542C22#1683819812yes—
543C22#1683820112yes—
544C4—44no—
545C4—82no—
546C4—1182no—
547C4—1182no—
548C4—4722no—
549C4—2364no—
550C4#1683726414yes—
551C4C2×#1427368411yescharacteristic
552C4C2×#1427368511yescharacteristic
553C4C2×#1427370911yescharacteristic
554C4C2×#1427371211yescharacteristic
555C2—24no—
556C2—2362no—
557C2—2362no—
558C2C2×#1683726411yescharacteristic
559C2#2143261011yescharacteristic
560C2#2143261411yescharacteristic
561C2#2143402612yes—
562C2#2143403012yes—
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1736C2
socleC23×C59C22×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenter#1736D59
hypocenterC59#20701
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #21424190C2semidirect
5. #21424189C2semidirect
6. #21423817C2semidirect
7. #21423816C2semidirect
8. #21423815C2semidirect
9. C59×#1736C2semidirect
16. C2×#14273685C4semidirect
31. #16837165C22semidirect
32. #16837164C22semidirect
33. #16837164C22semidirect
37. C59×#340C22semidirect
282. #1736D59semidirect
318. #340C2×D59semidirect
319. C59#20701semidirect

Maximal subgroups

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

subgroupindex
C22×#142736852
C2×#168373652
#214264752
#214241902
#214241892
#214238172
#214238162
#214238152
C59×#17362
#2070159