Finite Group Databasebeta

← #28629994

Subgroups — #28629994

Order 15104.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837679C211yesmaximal, characteristic, complemented
2C2×#16837459C211yesmaximal, characteristic, complemented
3C4×#14273701C211yesmaximal, characteristic
4#21425527C212yesmaximal, complemented
5#21425285C212yesmaximal
6#21425049C211yesmaximal, characteristic, complemented
7#21424857C211yesmaximal, characteristic, complemented
8#21424836C212yesmaximal, complemented
9#21424792C212yesmaximal, complemented
10#21424093C211yesmaximal, characteristic, complemented
11C59×#2857C211yesmaximal, characteristic, complemented
12C2×#14273701C2211yescharacteristic
13C2×C4×D236C2211yescharacteristic
14C2×C4×C22⋊D59—21no—
15#16837679—24no—
16#16837459—24no—
17C2×#14273660C2212yes—
18C2×#14273658C2212yes—
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273647C2212yes—
22C2×#14273647C2212yes—
23C2×C22⋊(C59⋊C8)C2212yes—
24C2×#14273638C2212yes—
25#16837304C2211yescharacteristic
26#16837300C2211yescharacteristic
27#16837293—21no—
28#16837281—21no—
29#16837262C2211yescharacteristic
30C4×C4⋊Dic59C2211yescharacteristic
31C4×(C2×D59)⋊C4—21no—
32#16837164C2212yes—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837164C2212yes—
36C2×C4⋊Q8×C59C2211yescharacteristic
37C2×C22⋊Q8×C59—21no—
38C22×C42×C59C2211yescharacteristic
39C59×#391—21no—
40C59×#389—21no—
41C59×#389C2212yes—
42C59×#384C2211yescharacteristic
43C59×#380—21no—
44C59×#380C2212yes—
45C22×D236C2311yescharacteristic
46C22×C22⋊D59—21no—
47C22×C4×D59—21no—
48C22×C59⋊C8C2312yes—
49C22×C59⋊C8C2312yes—
50#14273701—22no—
51#14273701—22no—
52C4×D236—22no—
53C4×D236—22no—
54C4×C22⋊D59—44no—
55C2×C4⋊Dic59C2311yescharacteristic
56C2×C4⋊Dic59C2311yescharacteristic
57C2×C4⋊Dic59C2311yescharacteristic
58C2×Dic59⋊C4—21no—
59C2×C22⋊Dic59—21no—
60C2×(C2×D59)⋊C4—21no—
61C2×(C2×D59)⋊C4—21no—
62C2×(C2×D59)⋊C4—21no—
63C2×(C2×D59)⋊C4—21no—
64C2×C4×Dic59—21no—
65#14273660—24no—
66#14273658—24no—
67C4⋊(C59⋊C8)—22no—
68C4⋊(C59⋊C8)D414yescomplemented
69#14273647—24no—
70#14273647—24no—
71C22⋊(C59⋊C8)—24no—
72#14273638—24no—
73C59⋊#95—21no—
74C59⋊#95—21no—
75C59⋊#95—21no—
76C22×Q8×C59—21no—
77C23×C4×C59D411yescharacteristic
78C23×C4×C59D411yescharacteristic
79C23×C4×C59C2311yescharacteristic
80C4⋊Q8×C59—22no—
81C4⋊Q8×C59—22no—
82C22⋊Q8×C59—44no—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59—21no—
86C2×C4⋊C4×C59—22no—
87C2×C4⋊C4×C59C2312yes—
88C2×C4⋊C4×C59C2312yes—
89C2×C22⋊C4×C59—22no—
90C2×C42×C59—22no—
91C2×C42×C59—22no—
92C2×C42×C59D411yescharacteristic
93C2×C42×C59D411yescharacteristic
94C2×C42×C59C2311yescharacteristic
95C2×C42×C59C2311yescharacteristic
96#95×C59—22no—
97#95×C59—22no—
98C24×C59C2×D411yescharacteristic
99C23×D59—21no—
100C2×Q8×C59—42no—
101C2×Q8×C59—42no—
102C2×Q8×C59—42no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—21no—
109C22×C4×C59—21no—
110C22×C4×C59—21no—
111C22×C4×C59—21no—
112C22×C4×C59—21no—
113C22×C4×C59—21no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—41no—
117C22×C4×C59C2×D411yescharacteristic
118C22×C4×C59C2×D411yescharacteristic
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2411yescharacteristic
123C2×C22⋊D59—24no—
124C2×C22⋊D59—42no—
125C2×C22⋊D59—42no—
126C22×Dic59—21no—
127C22×Dic59—21no—
128C22×Dic59—21no—
129C2×D236—21no—
130C2×D236—21no—
131C2×D236—21no—
132C2×D236—21no—
133C2×D236—21no—
134C2×D236#4112yes—
135C2×C4×D59—42no—
136C2×C4×D59—42no—
137C2×C4×D59—42no—
138C4⋊C4×C59—22no—
139C4⋊C4×C59—22no—
140C4⋊C4×C59—22no—
141C4⋊C4×C59—22no—
142C4⋊C4×C59—42no—
143C4⋊C4×C59—44no—
144C4⋊C4×C59—42no—
145C4⋊C4×C59—42no—
146C22⋊C4×C59—44no—
147C42×C59—22no—
148C42×C59—22no—
149C42×C59—21no—
150C42×C59—21no—
151C42×C59C2×D412yes—
152C42×C59#4112yes—
153C22⋊Dic59—42no—
154(C2×D59)⋊C4—24no—
155(C2×D59)⋊C4—42no—
156(C2×D59)⋊C4—42no—
157(C2×D59)⋊C4—42no—
158C4⋊Dic59—21no—
159C4⋊Dic59—21no—
160C4⋊Dic59—21no—
161C4⋊Dic59—21no—
162C4⋊Dic59—21no—
163C4⋊Dic59#4112yes—
164Dic59⋊C4—42no—
165C4×Dic59—42no—
166C2×C59⋊C8—22no—
167C2×C59⋊C8—22no—
168C2×C59⋊C8—22no—
169C2×C59⋊C8—22no—
170C2×C59⋊C8—22no—
171C2×C59⋊C8C2×D414yes—
172C23×C59—21no—
173C23×C59—21no—
174C23×C59—21no—
175C23×C59—21no—
176C23×C59—21no—
177C23×C59—21no—
178C23×C59#13712yes—
179C23×C59C22×D411yescharacteristic
180C22×D59—22no—
181C22×D59—41no—
182C22×D59—41no—
183C22×D59—41no—
184C22×D59—41no—
185C22×D59—41no—
186C22×D59—41no—
187Q8×C59—84no—
188C2×C4×C59—21no—
189C2×C4×C59—21no—
190C2×C4×C59—21no—
191C2×C4×C59—21no—
192C2×C4×C59—21no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—44no—
200C2×C4×C59—41no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—44no—
205C2×C4×C59—42no—
206C2×C4×C59—42no—
207C2×C4×C59—42no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59#12112yes—
212C2×C4×C59#12112yes—
213C2×C4×C59AGL(1,ℤ/8)12yes—
214C2×C4×C59AGL(1,ℤ/8)12yes—
215C2×C4×C59C22×D411yescharacteristic
216C2×C4×C59C2×#4111yescharacteristic
217C2×C4×C59C2×#4111yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C592−1+411yescharacteristic
220C2×C4×C592−1+411yescharacteristic
221C22⋊D59—48no—
222C2×Dic59—22no—
223C2×Dic59—41no—
224C2×Dic59—41no—
225C2×Dic59—41no—
226C2×Dic59—41no—
227C2×Dic59—41no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231D236—22no—
232D236—22no—
233D236—41no—
234D236—41no—
235C4×D59—84no—
236C59⋊C8—24no—
237C59⋊C8—42no—
238#24709—591nomaximal, Sylow
239C22×C59—22no—
240C22×C59—22no—
241C22×C59—22no—
242C22×C59—41no—
243C22×C59—41no—
244C22×C59—41no—
245C22×C59—41no—
246C22×C59C2×#12111yescharacteristic
247C22×C59#52511yescharacteristic
248C22×C59#53611yescharacteristic
249C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C59C2×#13711yescharacteristic
252C22×C59#57511yescharacteristic
253C2×D59—42no—
254C2×D59—42no—
255C2×D59—42no—
256C2×D59—81no—
257C2×D59—81no—
258C2×D59—81no—
259C2×D59—81no—
260C4×C59—22no—
261C4×C59—41no—
262C4×C59—41no—
263C4×C59—41no—
264C4×C59—82no—
265C4×C59—82no—
266C4×C59#46914yes—
267C4×C59#54711yescharacteristic
268C4×C59#54811yescharacteristic
269C4×C59#55211yescharacteristic
270C4×C59#55611yescharacteristic
271Dic59—42no—
272Dic59—81no—
273Dic59—81no—
274C2×#474—591no—
275C2×#432—591no—
276#2857D5911yescharacteristic, complemented
277C4×#121—591no—
278#2076—592no—
279#2034—592no—
280#1988—591no—
281#1948—591no—
282#1944—592no—
283#1936—592no—
284#1802—591no—
285C2×C59—42no—
286C2×C59#241511yescharacteristic
287C2×C59C2×#46911yescharacteristic
288C2×C59#305611yescharacteristic
289C2×C59#311011yescharacteristic
290C2×C59#311511yescharacteristic
291C2×C59#323511yescharacteristic
292C2×C59#323911yescharacteristic
293D59—82no—
294C2×C4⋊Q8C2×D5911yescharacteristic
295C2×C22⋊Q8—21no—
296C2×#121—591no—
297C2×C4×D4—591no—
298C2×C4×D4—1181no—
299C22×C42C2×D5911yescharacteristic
300#474—1184no—
301#432—1184no—
302C2×#107—592no—
303C2×#106—592no—
304C2×C4⋊C8—591no—
305C2×C4⋊C8—591no—
306C2×#102—592no—
307C2×#102—592no—
308C2×C22⋊C8—592no—
309C2×#97—592no—
310#391—21no—
311#390—591no—
312#389—21no—
313#389—591no—
314#389C2×D5912yes—
315#388—1181no—
316#386—1181no—
317#384C2×D5911yescharacteristic
318#380—21no—
319#380—591no—
320#380C2×D5912yes—
321C4×C4⋊C4—591no—
322C4×C22⋊C4—1181no—
323#340—592no—
324#340—592no—
325#340—592no—
326#340—592no—
327C59#2470911yescharacteristic, complemented, Sylow
328C22×Q8—21no—
329C22×D4—591no—
330C22×D4—1181no—
331C23×C4—1181no—
332C23×C4C22⋊D5911yescharacteristic
333C23×C4C22⋊D5911yescharacteristic
334C23×C4C22×D5911yescharacteristic
335C22×C8—592no—
336C22×C8—592no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C22⋊Q8—44no—
340#121—1182no—
341#121—1182no—
342C4×D4—1182no—
343C4×D4—1182no—
344C4×D4—2364no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—591no—
350C2×C4⋊C4—591no—
351C2×C4⋊C4—591no—
352C2×C4⋊C4—1181no—
353C2×C4⋊C4C22×D5912yes—
354C2×C4⋊C4C22×D5912yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—1181no—
357C2×C22⋊C4—1181no—
358C2×C22⋊C4—1181no—
359C2×C22⋊C4—1181no—
360C2×C22⋊C4—1181no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—1181no—
364C2×C42C22⋊D5911yescharacteristic
365C2×C42C22⋊D5911yescharacteristic
366C2×C42C22×D5911yescharacteristic
367C2×C42C22×D5911yescharacteristic
368#107—1184no—
369#106—1184no—
370C4⋊C8—594no—
371C4⋊C8—1182no—
372#102—1184no—
373#102—1184no—
374C22⋊C8—1184no—
375#97—1184no—
376#95—22no—
377#95—22no—
378#95—1181no—
379#95—1181no—
380#95—1181no—
381C24—1181no—
382C24C2×C22⋊D5911yescharacteristic
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×D4—592no—
387C2×D4—1184no—
388C2×D4—1181no—
389C2×D4—1181no—
390C2×D4—1181no—
391C2×D4—1181no—
392C2×D4—1181no—
393C2×D4—2362no—
394C2×D4—2362no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—1181no—
410C22×C4—1181no—
411C22×C4—1181no—
412C22×C4—2362no—
413C22×C4—2362no—
414C22×C4—2362no—
415C22×C4C2×C22⋊D5911yescharacteristic
416C22×C4C2×C22⋊D5911yescharacteristic
417C22×C4C2×C22⋊D5911yescharacteristic
418C22×C4C2×C22⋊D5911yescharacteristic
419C22×C4C2×C22⋊D5911yescharacteristic
420C22×C4C23×D5911yescharacteristic
421C2×C8—594no—
422C2×C8—1182no—
423C2×C8—1182no—
424C2×C8—1182no—
425C2×C8—1182no—
426C2×C8—1182no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—592no—
436C4⋊C4—1181no—
437C4⋊C4—1181no—
438C4⋊C4—1181no—
439C4⋊C4—1181no—
440C4⋊C4—1181no—
441C4⋊C4—2362no—
442C22⋊C4—44no—
443C22⋊C4—1184no—
444C22⋊C4—2362no—
445C22⋊C4—2362no—
446C22⋊C4—2362no—
447C22⋊C4—2362no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—2362no—
453C42D4×D5912yes—
454C42#1174568712yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—1182no—
462C23—2361no—
463C23—2361no—
464C23—2361no—
465C23—2361no—
466C23—2361no—
467C23—2361no—
468C23#1427375812yes—
469C23C22×C22⋊D5911yescharacteristic
470Q8—84no—
471D4—1182no—
472D4—1182no—
473D4—2368no—
474D4—2361no—
475D4—2361no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—1182no—
500C2×C4—4724no—
501C2×C4—2361no—
502C2×C4—2361no—
503C2×C4—2361no—
504C2×C4—2361no—
505C2×C4—2361no—
506C2×C4—2361no—
507C2×C4—2361no—
508C2×C4—2361no—
509C2×C4#1427369912yes—
510C2×C4#1427369912yes—
511C2×C4#1427375212yes—
512C2×C4#1427375312yes—
513C2×C4C2×D4×D5911yescharacteristic
514C2×C4#41×D5911yescharacteristic
515C2×C4#41×D5911yescharacteristic
516C2×C4C2×#1174568711yescharacteristic
517C2×C4#1427377511yescharacteristic
518C2×C4#1427377511yescharacteristic
519C8—1184no—
520C8—2362no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—4721no—
529C22—4721no—
530C22—4721no—
531C22—4721no—
532C22—2362no—
533C22—2362no—
534C22—2362no—
535C22C2×#1427369911yescharacteristic
536C22#1683790911yescharacteristic
537C22#1683797611yescharacteristic
538C22C2×#1427375211yescharacteristic
539C22C2×#1427375311yescharacteristic
540C22C2×#1427375811yescharacteristic
541C22#1683820011yescharacteristic
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—82no—
547C4—82no—
548C4—4721no—
549C4—4721no—
550C4—2362no—
551C4#1683765314yes—
552C4#1683805411yescharacteristic
553C4#1683806211yescharacteristic
554C4#1683808611yescharacteristic
555C4#1683812311yescharacteristic
556C2—42no—
557C2—4722no—
558C2#2142752611yescharacteristic
559C2C2×#1683765311yescharacteristic
560C2#2143372211yescharacteristic
561C2#2143419111yescharacteristic
562C2#2143424111yescharacteristic
563C2#2143529411yescharacteristic
564C2#2143532611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2857C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenter#2857D59
hypocenterC59#24709
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
1. C2×#16837679C2semidirect
2. C2×#16837459C2semidirect
4. #21425527C2semidirect
6. #21425049C2semidirect
7. #21424857C2semidirect
8. #21424836C2semidirect
9. #21424792C2semidirect
10. #21424093C2semidirect
11. C59×#2857C2semidirect
68. C4⋊(C59⋊C8)D4semidirect
276. #2857D59semidirect
327. C59#24709semidirect

Maximal subgroups

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

subgroupindex
C2×#168376792
C2×#168374592
C4×#142737012
#214255272
#214252852
#214250492
#214248572
#214248362
#214247922
#214240932
C59×#28572
#2470959