Finite Group Databasebeta

← #28619094

Subgroups — #28619094

Order 15104.

712 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426842C211yesmaximal, characteristic, complemented
2#21426733C211yesmaximal, characteristic
3#21425663C211yesmaximal, characteristic, complemented
4#21425595C211yesmaximal, characteristic, complemented
5#21425585C211yesmaximal, characteristic, complemented
6#21425468C211yesmaximal, characteristic
7#21425468C211yesmaximal, characteristic
8#21425430C211yesmaximal, characteristic, complemented
9#21425232C211yesmaximal, characteristic
10#21423995C211yesmaximal, characteristic, complemented
11#106×Dic59C211yesmaximal, characteristic, complemented
12#21423952C211yesmaximal, characteristic, complemented
13#21423923C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#2092C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C2×#14273659C2211yescharacteristic
19C2×#14273659C2211yescharacteristic
20C2×Dic59⋊C8C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273637C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26#16837306—21no—
27#16837306—21no—
28#16837305C2211yescharacteristic
29#16837300C2211yescharacteristic
30#16837298C2211yescharacteristic
31#16837298C2211yescharacteristic
32#16837298C2211yescharacteristic
33#16837296—21no—
34#16837290—21no—
35#16837287—21no—
36#16837287—21no—
37#16837263—21no—
38#16837260—21no—
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42#16837248C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52C2×#107×C59C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C59×#382C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×Dic118—21no—
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273727—22no—
64#14273727—22no—
65D4×Dic59—22no—
66D4×Dic59Q814yescomplemented
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59C2311yescharacteristic
83C2×C4×Dic59C2311yescharacteristic
84#14273659—22no—
85#14273659—22no—
86Dic59⋊C8—22no—
87Dic59⋊C8—22no—
88#14273648—22no—
89#14273648—22no—
90#14273637—22no—
91C8×Dic59—22no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95D411yescharacteristic
100C59⋊#95D411yescharacteristic
101C59⋊#95D411yescharacteristic
102C59⋊#95D411yescharacteristic
103C22×D4×C59C2311yescharacteristic
104C22×C8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C42×C59C2311yescharacteristic
112#107×C59—22no—
113#106×C59—22no—
114#102×C59—22no—
115#102×C59—22no—
116#95×C59—21no—
117#95×C59—21no—
118C24×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59C2×Q812yes—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
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
142C2×Dic118—24no—
143C2×Dic118—81no—
144C2×Dic118—81no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59#4112yes—
150C2×C8×C59#4112yes—
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—81no—
158C22⋊C4×C59—81no—
159C42×C59—21no—
160C42×C59—21no—
161C22⋊Dic59—24no—
162C22⋊Dic59—81no—
163C22⋊Dic59—81no—
164C4⋊Dic59—24no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59C2×Q812yes—
171C4⋊Dic59#4112yes—
172C4⋊Dic59#4112yes—
173Dic59⋊C4—81no—
174Dic59⋊C4—81no—
175Dic59⋊C4—81no—
176Dic59⋊C4—81no—
177C4×Dic59—21no—
178C4×Dic59—21no—
179C4×Dic59—81no—
180C4×Dic59—81no—
181C4×Dic59—81no—
182C4×Dic59C2×Q812yes—
183C4×Dic59#4112yes—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C23×C59—22no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59C22×D411yescharacteristic
198D4×C59—22no—
199D4×C59—22no—
200D4×C59—41no—
201D4×C59—41no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
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×C59C22×Q811yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59C2×#4111yescharacteristic
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—22no—
228C2×Dic59—22no—
229C2×Dic59—22no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—84no—
246C2×Dic59C22⋊Q812yes—
247C2×Dic59C22⋊Q812yes—
248Dic118—84no—
249C8×C59—22no—
250C8×C59—22no—
251C59⋊C8—41no—
252C59⋊C8—41no—
253#23786—591nomaximal, Sylow
254C22×C59—42no—
255C22×C59—42no—
256C22×C59—42no—
257C22×C59—81no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59C2×C22⋊Q811yescharacteristic
262C22×C59#52511yescharacteristic
263C22×C59#53211yescharacteristic
264C22×C59#57611yescharacteristic
265C22×C59#57711yescharacteristic
266C22×C59#57711yescharacteristic
267C22×C59#57711yescharacteristic
268C4×C59—41no—
269C4×C59—41no—
270C4×C59—81no—
271C4×C59—81no—
272C4×C59—81no—
273C4×C59#52911yescharacteristic
274C4×C59#52911yescharacteristic
275C4×C59#53311yescharacteristic
276C4×C59#53311yescharacteristic
277Dic59—22no—
278Dic59—22no—
279Dic59—42no—
280Dic59—42no—
281Dic59—42no—
282Dic59—41no—
283Dic59—41no—
284Dic59—81no—
285#2331—591no—
286#2319—591no—
287#2101—591no—
288#2092D5911yescharacteristic
289#2088—591no—
290#2086—591no—
291#2065—591no—
292#2065—591no—
293#2058—591no—
294#2023—591no—
295#1776—591no—
296C4×#106—591no—
297#1768—591no—
298#1762—591no—
299C4×#102—591no—
300C2×C59—82no—
301C2×C59#239611yescharacteristic
302C2×C59#308111yescharacteristic
303C2×C59#308411yescharacteristic
304C2×C59#311811yescharacteristic
305C2×C59#312611yescharacteristic
306C2×C59#317311yescharacteristic
307C2×C59#317911yescharacteristic
308C2×C4⋊Q8—591no—
309C2×C4×D4—591no—
310C2×#107C2×D5911yescharacteristic
311C2×#106—591no—
312C2×#106—591no—
313C2×#106C2×D5911yescharacteristic
314C2×C4⋊C8—591no—
315C2×C4⋊C8—591no—
316C2×#102—591no—
317C2×#102—591no—
318C2×#102C2×D5911yescharacteristic
319C2×#102C2×D5911yescharacteristic
320C2×#97—591no—
321C2×C4×C8—591no—
322#391—1181no—
323#391—1181no—
324#390—591no—
325#390C2×D5911yescharacteristic
326#389—591no—
327#389—591no—
328#389—591no—
329#389—591no—
330#388—1181no—
331#388—1181no—
332#387—1181no—
333#387—1181no—
334#382C2×D5911yescharacteristic
335#381—1181no—
336#380—1181no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—591no—
340#376—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#336—591no—
350#336C2×D5911yescharacteristic
351C59#2378611yescharacteristic, complemented, Sylow
352C22×Q8—1181no—
353C22×D4C22×D5911yescharacteristic
354C23×C4—1181no—
355C22×C8—591no—
356C22×C8—591no—
357C22×C8C22×D5911yescharacteristic
358C22×C8C22×D5911yescharacteristic
359C4⋊Q8—1182no—
360C4⋊Q8—1182no—
361C4×D4—594no—
362C4×D4—1182no—
363C2×C4⋊C4—591no—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—1181no—
368C2×C4⋊C4—1181no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4C22×D5911yescharacteristic
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C42—591no—
381C2×C42—591no—
382C2×C42—1181no—
383C2×C42—1181no—
384C2×C42—1181no—
385C2×C42C22×D5911yescharacteristic
386#107—22no—
387#106—22no—
388#106—1182no—
389#106—1182no—
390C4⋊C8—1182no—
391C4⋊C8—1182no—
392#102—22no—
393#102—22no—
394#102—1182no—
395#102—1182no—
396#97—1182no—
397C4×C8—1182no—
398#95—21no—
399#95—21no—
400#95—591no—
401#95—591no—
402#95—591no—
403#95—591no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411C24—21no—
412C2×Q8—1184no—
413C2×Q8—4721no—
414C2×Q8—4721no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4Q8×D5912yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—591no—
425C22×C4—591no—
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—1184no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—4721no—
434C22×C4—4721no—
435C22×C4D4×D5911yescharacteristic
436C22×C4D4×D5911yescharacteristic
437C22×C4C23×D5911yescharacteristic
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—
449C2×C8#1174568712yes—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—592no—
457C4⋊C4—592no—
458C4⋊C4—592no—
459C4⋊C4—1184no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—4721no—
466C4⋊C4—4721no—
467C4⋊C4—4721no—
468C4⋊C4—4721no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C22⋊C4—1184no—
472C22⋊C4—4721no—
473C22⋊C4—4721no—
474C42—21no—
475C42—21no—
476C42—592no—
477C42—592no—
478C42—1181no—
479C42—1181no—
480C42—4721no—
481C42—4721no—
482C42—4721no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C2×D4×D5911yescharacteristic
491Q8—4724no—
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
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—592no—
512C2×C4—592no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—1181no—
517C2×C4—1182no—
518C2×C4—1182no—
519C2×C4—1182no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—4724no—
527C2×C4—2361no—
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×C4C2×Q8×D5911yescharacteristic
537C2×C4#41×D5911yescharacteristic
538C2×C4#41×D5911yescharacteristic
539C2×C4#41×D5911yescharacteristic
540C2×C4C2×#1174568711yescharacteristic
541C2×C4C2×#1174568711yescharacteristic
542C8—22no—
543C8—22no—
544C8—2361no—
545C8—2361no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C22⋊Q8×D5911yescharacteristic
554C22#1683791011yescharacteristic
555C22#1683795011yescharacteristic
556C22#1683820611yescharacteristic
557C22#1683820911yescharacteristic
558C22#1683820911yescharacteristic
559C22#1683821211yescharacteristic
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—1182no—
566C4—1182no—
567C4—4721no—
568C4—2362no—
569C4—2361no—
570C4—2361no—
571C4—2362no—
572C4—2362no—
573C4#1683792911yescharacteristic
574C4#1683792911yescharacteristic
575C4#1683795411yescharacteristic
576C4#1683795411yescharacteristic
577C2—82no—
578C2#2142736511yescharacteristic
579C2#2143392611yescharacteristic
580C2#2143395911yescharacteristic
581C2#2143427511yescharacteristic
582C2#2143438311yescharacteristic
583C2#2143471011yescharacteristic
584C2#2143477611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2092C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2092D59
hypocenterC59#23786
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2092

Complements

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

NH ≅ G/Ntype
1. #21426842C2semidirect
3. #21425663C2semidirect
4. #21425595C2semidirect
5. #21425585C2semidirect
8. #21425430C2semidirect
10. #21423995C2semidirect
11. #106×Dic59C2semidirect
12. #21423952C2semidirect
66. D4×Dic59Q8semidirect
351. C59#23786semidirect

Maximal subgroups

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

subgroupindex
#214268422
#214267332
#214256632
#214255952
#214255852
#214254682
#214254302
#214252322
#214239952
#106×Dic592
#214239522
#214239232
#102×Dic592
C59×#20922
#2378659