Finite Group Databasebeta

← #28646748

Subgroups — #28646748

Order 15104.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427810C211yesmaximal, characteristic
2#21426975C211yesmaximal, characteristic
3#21425661C211yesmaximal, characteristic, complemented
4#21425595C211yesmaximal, characteristic, complemented
5#21425588C211yesmaximal, characteristic, complemented
6#21425432C211yesmaximal, characteristic, complemented
7#21425052C211yesmaximal, characteristic, complemented
8#21424851C211yesmaximal, characteristic, complemented
9#21424683C211yesmaximal, characteristic, complemented
10#21424672C211yesmaximal, characteristic, complemented
11#21424293C211yesmaximal, characteristic
12#21424279C211yesmaximal, characteristic
13#21424180C211yesmaximal, characteristic
14#21423816C211yesmaximal, characteristic
15C59×#1809C211yesmaximal, characteristic
16C2×#14273718C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24#16837338—21no—
25#16837325—21no—
26#16837301C2211yescharacteristic
27#16837299—21no—
28#16837298C2211yescharacteristic
29#16837298C2211yescharacteristic
30#16837298C2211yescharacteristic
31#16837289—21no—
32#16837287—21no—
33#16837287—21no—
34#16837271C2211yescharacteristic
35#16837260—21no—
36C4⋊C4×Dic59C2211yescharacteristic
37C4⋊C4×Dic59C2211yescharacteristic
38C22⋊C4×Dic59—21no—
39C4×C22⋊Dic59C2211yescharacteristic
40#16837165C2211yescharacteristic
41#16837165C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C22×M16×C59C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C59×#380C2211yescharacteristic
56C59×#380C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273718—22no—
63#14273718—41no—
64#14273686—22no—
65#14273686—41no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—22no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59Q814yes—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59D411yescharacteristic
83C2×C22⋊Dic59D411yescharacteristic
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89#14273660—41no—
90#14273658—22no—
91Dic59⋊C8—22no—
92Dic59⋊C8—41no—
93C22⋊(C59⋊C8)—22no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95D411yescharacteristic
101C59⋊#95D411yescharacteristic
102C23×C4×C59C2311yescharacteristic
103C2×M16×C59—22no—
104C2×M16×C59—22no—
105C2×M16×C59—41no—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114#107×C59—41no—
115#106×C59—22no—
116#95×C59—21no—
117#95×C59—21no—
118C24×C59C2×D411yescharacteristic
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×Q812yes—
129C22×C4×C59#4112yes—
130C22×C4×C59C2411yescharacteristic
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—24no—
135C22×Dic59—21no—
136C22×Dic59—81no—
137C22×Dic59—81no—
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142M16×C59—44no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59#4112yes—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C22⋊C4×C59—42no—
158C22⋊C4×C59—81no—
159C22⋊Dic59—22no—
160C22⋊Dic59—41no—
161C22⋊Dic59—42no—
162C4⋊Dic59—24no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—42no—
169C4⋊Dic59—81no—
170C4⋊Dic59C2×Q812yes—
171C4⋊Dic59C2×Q812yes—
172C4⋊Dic59#4112yes—
173Dic59⋊C4—42no—
174Dic59⋊C4—42no—
175Dic59⋊C4—42no—
176Dic59⋊C4—81no—
177Dic59⋊C4—81no—
178Dic59⋊C4—81no—
179C4×Dic59—21no—
180C4×Dic59—21no—
181C4×Dic59—21no—
182C4×Dic59—21no—
183C4×Dic59—42no—
184C4×Dic59—81no—
185C4×Dic59—81no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59C22⋊Q812yes—
199C23×C59C22×D411yescharacteristic
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—22no—
203C2×C4×C59—22no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59C22⋊Q812yes—
219C2×C4×C59C22×Q811yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59C2×#4111yescharacteristic
223C2×C4×C59H(5,𝔽2)11yescharacteristic
224C2×C4×C592−1+411yescharacteristic
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—22no—
228C2×Dic59—22no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—22no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—84no—
249C2×Dic59#13712yes—
250C8×C59—22no—
251C8×C59—41no—
252C59⋊C8—41no—
253C59⋊C8—41no—
254#26018—591nomaximal, Sylow
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59—41no—
262C22×C59C2×C22⋊Q811yescharacteristic
263C22×C59#54611yescharacteristic
264C22×C59#54711yescharacteristic
265C22×C59C2×#13711yescharacteristic
266C22×C59#57511yescharacteristic
267C22×C59#57611yescharacteristic
268C22×C59#57711yescharacteristic
269C4×C59—22no—
270C4×C59—81no—
271C4×C59—81no—
272C4×C59—81no—
273C4×C59—81no—
274C4×C59#53311yescharacteristic
275C4×C59#54411yescharacteristic
276C4×C59#55111yescharacteristic
277C4×C59#55211yescharacteristic
278Dic59—22no—
279Dic59—41no—
280Dic59—42no—
281Dic59—42no—
282Dic59—42no—
283Dic59—41no—
284Dic59—41no—
285Dic59—81no—
286#2441—591no—
287#2345—591no—
288#2100—591no—
289#2088—591no—
290#2086—591no—
291#2059—591no—
292#1989—591no—
293#1947—591no—
294#1920—591no—
295#1919—591no—
296#1854—591no—
297#1851—591no—
298#1823—591no—
299#1809D5911yescharacteristic
300#1736—591no—
301C2×C59—42no—
302C2×C59#243811yescharacteristic
303C2×C59#307811yescharacteristic
304C2×C59#308111yescharacteristic
305C2×C59#330911yescharacteristic
306C2×C59#333111yescharacteristic
307C2×C59#334611yescharacteristic
308C2×C59#335111yescharacteristic
309C22×M16C2×D5911yescharacteristic
310C2×#125—591no—
311C2×#117—591no—
312C22×C4⋊C4—591no—
313C2×#107—591no—
314C2×#107C2×D5911yescharacteristic
315C2×#106—591no—
316C2×#106C2×D5911yescharacteristic
317C2×C4⋊C8—591no—
318C2×C4⋊C8—591no—
319C2×C22⋊C8—591no—
320#399—1181no—
321#396—1181no—
322#389—591no—
323#389—591no—
324#389—591no—
325#389—591no—
326#389—1181no—
327#387—1181no—
328#387—1181no—
329#387—1181no—
330#384—591no—
331#380—1181no—
332#380C2×D5911yescharacteristic
333#380C2×D5911yescharacteristic
334C4×C4⋊C4—591no—
335C4×C4⋊C4—591no—
336C4×C22⋊C4—591no—
337C4×C22⋊C4—1181no—
338#340—591no—
339#340—591no—
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340C2×D5911yescharacteristic
349#340C2×D5911yescharacteristic
350#336—591no—
351#336—591no—
352C59#2601811yescharacteristic, complemented, Sylow
353C23×C4—1181no—
354C23×C4C22×D5911yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—41no—
358C22×C8—591no—
359C22×C8—591no—
360C22×C8C22×D5911yescharacteristic
361C22×C8C22×D5911yescharacteristic
362#125—1182no—
363#125—2361no—
364#117—1182no—
365#117—2361no—
366C2×C4⋊C4—594no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—1182no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—591no—
388C2×C22⋊C4—591no—
389C2×C22⋊C4—1181no—
390C2×C42—591no—
391C2×C42—591no—
392C2×C42—1181no—
393C2×C42—1181no—
394C2×C42—1181no—
395#107—41no—
396#107—2361no—
397#106—22no—
398#106—1182no—
399C4⋊C8—1182no—
400C4⋊C8—2361no—
401C22⋊C8—1182no—
402#95—21no—
403#95—21no—
404#95—591no—
405#95—591no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412C24D4×D5911yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—591no—
422C22×C4—591no—
423C22×C4—591no—
424C22×C4—591no—
425C22×C4—1184no—
426C22×C4—1181no—
427C22×C4—1181no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—4721no—
431C22×C4—4721no—
432C22×C4#1174568512yes—
433C22×C4D4×D5911yescharacteristic
434C22×C4Q8×D5912yes—
435C22×C4C23×D5911yescharacteristic
436M16—44no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8#1174568712yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—592no—
458C4⋊C4—592no—
459C4⋊C4—592no—
460C4⋊C4—1184no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—4721no—
467C4⋊C4—4721no—
468C4⋊C4—4721no—
469C4⋊C4—4721no—
470C4⋊C4—2362no—
471C4⋊C4—2362no—
472C4⋊C4—2362no—
473C4⋊C4—2362no—
474C22⋊C4—42no—
475C22⋊C4—81no—
476C22⋊C4—1182no—
477C22⋊C4—2361no—
478C22⋊C4—2362no—
479C42—1181no—
480C42—1181no—
481C42—1181no—
482C42—1181no—
483C42—4721no—
484C42—4721no—
485C42—2362no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1427370512yes—
493C23C2×D4×D5911yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—592no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1182no—
516C2×C4—1182no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1182no—
526C2×C4—1181no—
527C2×C4—4724no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4#1427370512yes—
538C2×C4C2×Q8×D5911yescharacteristic
539C2×C4#41×D5911yescharacteristic
540C2×C4C2×#1174568711yescharacteristic
541C2×C4C2×#1174568511yescharacteristic
542C2×C4#1427377211yescharacteristic
543C2×C4#1427377411yescharacteristic
544C8—22no—
545C8—41no—
546C8—2361no—
547C8—2361no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1427370511yescharacteristic
556C22#1683804111yescharacteristic
557C22#1683805211yescharacteristic
558C22#137×D5911yescharacteristic
559C22#1683819811yescharacteristic
560C22#1683820611yescharacteristic
561C22#1683821111yescharacteristic
562C4—22no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—1182no—
568C4—4721no—
569C4—2362no—
570C4—2362no—
571C4—2361no—
572C4—2361no—
573C4—2361no—
574C4—2362no—
575C4#1683795411yescharacteristic
576C4#1683803111yescharacteristic
577C4#1683807811yescharacteristic
578C4#1683808511yescharacteristic
579C2—42no—
580C2#2142777811yescharacteristic
581C2#2143390011yescharacteristic
582C2#2143392511yescharacteristic
583C2#2143609711yescharacteristic
584C2#2143640711yescharacteristic
585C2#2143657611yescharacteristic
586C2#2143665111yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1809C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1809D59
hypocenterC59#26018
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
3. #21425661C2semidirect
4. #21425595C2semidirect
5. #21425588C2semidirect
6. #21425432C2semidirect
7. #21425052C2semidirect
8. #21424851C2semidirect
9. #21424683C2semidirect
10. #21424672C2semidirect
352. C59#26018semidirect

Maximal subgroups

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

subgroupindex
#214278102
#214269752
#214256612
#214255952
#214255882
#214254322
#214250522
#214248512
#214246832
#214246722
#214242932
#214242792
#214241802
#214238162
C59×#18092
#2601859