Finite Group Databasebeta

← #28619167

Subgroups — #28619167

Order 15104.

704 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426838C211yesmaximal, characteristic, complemented
2#21426733C211yesmaximal, characteristic
3#21425660C211yesmaximal, characteristic, complemented
4#21425624C211yesmaximal, characteristic
5#21425593C211yesmaximal, characteristic, complemented
6#21425588C211yesmaximal, characteristic, complemented
7#21425480C211yesmaximal, characteristic
8#21425466C211yesmaximal, characteristic
9#21425431C211yesmaximal, characteristic, complemented
10#21423992C211yesmaximal, characteristic, complemented
11#21423988C211yesmaximal, characteristic, complemented
12#21423952C211yesmaximal, characteristic, complemented
13#21423923C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#2025C211yesmaximal, characteristic
16C2×D4×Dic59C2211yescharacteristic
17C2×#14273660C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273637C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27#16837305C2211yescharacteristic
28#16837301—21no—
29#16837301—21no—
30#16837301C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837296—21no—
35#16837290—21no—
36#16837288—21no—
37#16837288—21no—
38#16837272C2211yescharacteristic
39#16837263—21no—
40#16837260—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44#16837248C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52C2×#125×C59C2211yescharacteristic
53C2×#102×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62D4×Dic59—22no—
63D4×Dic59Q814yes—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
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×C22⋊Dic59—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59C2311yescharacteristic
81C2×C4×Dic59C2311yescharacteristic
82#14273660—22no—
83#14273660—22no—
84#14273660—22no—
85#14273658—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
106#125×C59—22no—
107#125×C59—22no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C42×C59C2311yescharacteristic
116#102×C59—22no—
117#102×C59—22no—
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—24no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—81no—
137C22×Dic59—81no—
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C4⋊C4×C59—24no—
149C4⋊C4×C59—24no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59#4112yes—
155C4⋊C4×C59#4112yes—
156C22⋊C4×C59—81no—
157C22⋊C4×C59—81no—
158C42×C59—21no—
159C42×C59—21no—
160C22⋊Dic59—24no—
161C22⋊Dic59—81no—
162C22⋊Dic59—81no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—81no—
171C4⋊Dic59—81no—
172C4⋊Dic59—81no—
173C4⋊Dic59—81no—
174C4⋊Dic59C2×Q812yes—
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#4112yes—
189C2×C59⋊C8#4112yes—
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—22no—
205C2×C4×C59—22no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
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—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—41no—
235C2×Dic59—41no—
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—
248C8×C59—41no—
249C8×C59—41no—
250C59⋊C8—22no—
251C59⋊C8—22no—
252#23791—591nomaximal, Sylow
253C22×C59—42no—
254C22×C59—42no—
255C22×C59—42no—
256C22×C59—81no—
257C22×C59—81no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59C2×C22⋊Q811yescharacteristic
261C22×C59#52511yescharacteristic
262C22×C59#53211yescharacteristic
263C22×C59#57611yescharacteristic
264C22×C59#57611yescharacteristic
265C22×C59#57711yescharacteristic
266C22×C59#57811yescharacteristic
267C4×C59—41no—
268C4×C59—42no—
269C4×C59—41no—
270C4×C59—42no—
271C4×C59—81no—
272C4×C59#52911yescharacteristic
273C4×C59#52911yescharacteristic
274C4×C59#53311yescharacteristic
275C4×C59#53311yescharacteristic
276Dic59—22no—
277Dic59—22no—
278Dic59—42no—
279Dic59—41no—
280Dic59—41no—
281Dic59—81no—
282Dic59—81no—
283Dic59—81no—
284#2330—591no—
285#2319—591no—
286#2100—591no—
287#2093—591no—
288#2087—591no—
289#2086—591no—
290#2067—591no—
291#2065—591no—
292#2059—591no—
293#2025D5911yescharacteristic
294#1776—591no—
295C4×#107—591no—
296#1768—591no—
297#1762—591no—
298C4×#102—591no—
299C2×C59—82no—
300C2×C59#239611yescharacteristic
301C2×C59#308111yescharacteristic
302C2×C59#308311yescharacteristic
303C2×C59#311911yescharacteristic
304C2×C59#312411yescharacteristic
305C2×C59#317611yescharacteristic
306C2×C59#317811yescharacteristic
307C2×#125C2×D5911yescharacteristic
308C2×C4×D4—591no—
309C2×#107—591no—
310C2×#107—591no—
311C2×#107—591no—
312C2×#106—591no—
313C2×C4⋊C8—591no—
314C2×C4⋊C8—591no—
315C2×#102—591no—
316C2×#102—591no—
317C2×#102C2×D5911yescharacteristic
318C2×#102C2×D5911yescharacteristic
319C2×#97—591no—
320C2×C4×C8—591no—
321#390—591no—
322#390C2×D5911yescharacteristic
323#389—591no—
324#389—591no—
325#389—591no—
326#389—591no—
327#389—1181no—
328#389—1181no—
329#388—1181no—
330#388—1181no—
331#387—1181no—
332#387—1181no—
333#384—591no—
334#381—1181no—
335#380—1181no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339#376—591no—
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340C2×D5911yescharacteristic
347#340C2×D5911yescharacteristic
348#336—591no—
349#336C2×D5911yescharacteristic
350C59#2379111yescharacteristic, complemented, Sylow
351C22×D4C22×D5911yescharacteristic
352C23×C4—1181no—
353C22×C8—591no—
354C22×C8—591no—
355C22×C8C22×D5911yescharacteristic
356C22×C8C22×D5911yescharacteristic
357#125—22no—
358#125—22no—
359C4×D4—594no—
360C4×D4—1182no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
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⋊C4—1181no—
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C4⋊C4C22×D5911yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C42—591no—
382C2×C42—591no—
383C2×C42—1181no—
384C2×C42—1181no—
385C2×C42—1181no—
386C2×C42C22×D5911yescharacteristic
387#107—1182no—
388#107—1182no—
389#107—1182no—
390#106—1182no—
391C4⋊C8—1182no—
392C4⋊C8—1182no—
393#102—22no—
394#102—22no—
395#102—1182no—
396#102—1182no—
397#97—1182no—
398C4×C8—1182no—
399#95—591no—
400#95—591no—
401#95—591no—
402#95—591no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410C24—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4Q8×D5912yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—591no—
421C22×C4—591no—
422C22×C4—591no—
423C22×C4—591no—
424C22×C4—1184no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4—1181no—
428C22×C4—1181no—
429C22×C4—4721no—
430C22×C4—4721no—
431C22×C4D4×D5911yescharacteristic
432C22×C4D4×D5911yescharacteristic
433C22×C4C23×D5911yescharacteristic
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—592no—
441C2×C8—592no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—592no—
453C4⋊C4—1181no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—4721no—
461C4⋊C4—4721no—
462C4⋊C4—4721no—
463C4⋊C4—4721no—
464C4⋊C4—4721no—
465C4⋊C4—4721no—
466C4⋊C4#1174568712yes—
467C4⋊C4#1174568712yes—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C22⋊C4—1184no—
471C22⋊C4—4721no—
472C22⋊C4—4721no—
473C42—21no—
474C42—21no—
475C42—592no—
476C42—592no—
477C42—1181no—
478C42—1181no—
479C42—4721no—
480C42—4721no—
481C42—4721no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23C2×D4×D5911yescharacteristic
490D4—22no—
491D4—22no—
492D4—41no—
493D4—41no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
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—592no—
510C2×C4—592no—
511C2×C4—1181no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1182no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—4724no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4C2×Q8×D5911yescharacteristic
535C2×C4#41×D5911yescharacteristic
536C2×C4#41×D5911yescharacteristic
537C2×C4#41×D5911yescharacteristic
538C2×C4C2×#1174568711yescharacteristic
539C2×C4C2×#1174568711yescharacteristic
540C8—41no—
541C8—41no—
542C8—1182no—
543C8—1182no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22C22⋊Q8×D5911yescharacteristic
552C22#1683790811yescharacteristic
553C22#1683795111yescharacteristic
554C22#1683820611yescharacteristic
555C22#1683820611yescharacteristic
556C22#1683820911yescharacteristic
557C22#1683821611yescharacteristic
558C4—41no—
559C4—42no—
560C4—41no—
561C4—42no—
562C4—81no—
563C4—1182no—
564C4—1182no—
565C4—4721no—
566C4—4721no—
567C4—4721no—
568C4—2362no—
569C4—2361no—
570C4—2361no—
571C4#1683793111yescharacteristic
572C4#1683793111yescharacteristic
573C4#1683795411yescharacteristic
574C4#1683795411yescharacteristic
575C2—82no—
576C2#2142736311yescharacteristic
577C2#2143392611yescharacteristic
578C2#2143394811yescharacteristic
579C2#2143429611yescharacteristic
580C2#2143435111yescharacteristic
581C2#2143474511yescharacteristic
582C2#2143476311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2025C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2025D59
hypocenterC59#23791
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2025

Complements

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

NH ≅ G/Ntype
1. #21426838C2semidirect
3. #21425660C2semidirect
5. #21425593C2semidirect
6. #21425588C2semidirect
9. #21425431C2semidirect
10. #21423992C2semidirect
11. #21423988C2semidirect
12. #21423952C2semidirect
350. C59#23791semidirect

Maximal subgroups

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

subgroupindex
#214268382
#214267332
#214256602
#214256242
#214255932
#214255882
#214254802
#214254662
#214254312
#214239922
#214239882
#214239522
#214239232
#102×Dic592
C59×#20252
#2379159