Finite Group Databasebeta

← #28600896

Subgroups — #28600896

Order 15104.

740 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
1C2×#16837690C211yesmaximal, characteristic
2#21426840C211yesmaximal, characteristic, complemented
3#21426733C211yesmaximal, characteristic
4#21425046C211yesmaximal, characteristic, complemented
5#21424865C211yesmaximal, characteristic, complemented
6#21424838C211yesmaximal, characteristic, complemented
7#21424822C211yesmaximal, characteristic, complemented
8#21424790C211yesmaximal, characteristic
9#21424781C211yesmaximal, characteristic
10#21424713C211yesmaximal, characteristic
11#21424681C211yesmaximal, characteristic, complemented
12#21424675C211yesmaximal, characteristic, complemented
13#21424104C211yesmaximal, characteristic
14#21423952C211yesmaximal, characteristic, complemented
15C59×#1926C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18#16837690C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273637C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28#16837308—21no—
29#16837306—21no—
30#16837305C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837296—21no—
36#16837290—21no—
37#16837289—21no—
38#16837286—21no—
39#16837267C2211yescharacteristic
40#16837263—21no—
41#16837260—21no—
42C4⋊C4×Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44#16837248C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837157C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C23×Dic59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273726—22no—
65#14273726C2×C414yes—
66D4×Dic59—22no—
67D4×Dic59C2×C414yes—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59C2311yescharacteristic
83C2×C4×Dic59C2311yescharacteristic
84#14273660—41no—
85#14273659C2×C414yes—
86#14273658C2×C414yes—
87Dic59⋊C8—22no—
88Dic59⋊C8C2×C414yes—
89#14273648—41no—
90#14273648C2×C414yes—
91#14273637—41no—
92C8×Dic59—41no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95D411yescharacteristic
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C59⋊#95D411yescharacteristic
105C22×D4×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C42×C59—21no—
114C2×C42×C59C2311yescharacteristic
115#106×C59—41no—
116#102×C59—41no—
117#102×C59C2×C414yes—
118#95×C59—21no—
119C24×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59C22×C412yes—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—24no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—81no—
138C22×Dic59—81no—
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C22×Dic59C2×D411yescharacteristic
143C2×Dic118—24no—
144C2×Dic118—81no—
145C2×Dic118—81no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59C22×C412yes—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59C22×C412yes—
158C22⋊C4×C59—81no—
159C22⋊C4×C59—81no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—81no—
163C22⋊Dic59—24no—
164C22⋊Dic59—81no—
165C22⋊Dic59—81no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—81no—
173C4⋊Dic59C22×C412yes—
174C4⋊Dic59C22×C412yes—
175Dic59⋊C4—24no—
176Dic59⋊C4—81no—
177Dic59⋊C4—81no—
178Dic59⋊C4—81no—
179C4×Dic59—21no—
180C4×Dic59—21no—
181C4×Dic59—81no—
182C4×Dic59—81no—
183C4×Dic59C22×C412yes—
184C4×Dic59#4112yes—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8C22×C412yes—
191C23×C59—22no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59C22×D411yescharacteristic
199D4×C59—22no—
200D4×C59—22no—
201D4×C59—41no—
202D4×C59—41no—
203C2×C4×C59—22no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
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×C59—41no—
218C2×C4×C59C23×C411yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×C4×C59H(5,𝔽2)11yescharacteristic
221C2×C4×C592−1+411yescharacteristic
222C2×C4×C592−1+411yescharacteristic
223C2×C4×C592−1+411yescharacteristic
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—21no—
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—41no—
246C2×Dic59—84no—
247C2×Dic59C4×D412yes—
248C2×Dic59C4×D412yes—
249Dic118—84no—
250C8×C59—22no—
251C8×C59—41no—
252C59⋊C8—22no—
253C59⋊C8—41no—
254#22123—591nomaximal, Sylow
255C22×C59—42no—
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59C2×C4×D411yescharacteristic
263C22×C59#53611yescharacteristic
264C22×C59#53711yescharacteristic
265C22×C59#57611yescharacteristic
266C22×C59#57711yescharacteristic
267C22×C59#57711yescharacteristic
268C22×C59#57711yescharacteristic
269C4×C59—42no—
270C4×C59—41no—
271C4×C59—41no—
272C4×C59—81no—
273C4×C59—81no—
274C4×C59#51911yescharacteristic
275C4×C59#52011yescharacteristic
276C4×C59#54111yescharacteristic
277C4×C59#54211yescharacteristic
278Dic59—22no—
279Dic59—22no—
280Dic59—42no—
281Dic59—42no—
282Dic59—41no—
283Dic59—41no—
284Dic59—81no—
285Dic59—81no—
286C2×#476—591no—
287#2331—591no—
288#2319—591no—
289#1988—591no—
290#1949—591no—
291#1944—591no—
292#1941—591no—
293#1936—591no—
294#1934—591no—
295#1926D5911yescharacteristic
296#1925—591no—
297#1920—591no—
298#1919—591no—
299#1804—591no—
300#1768—591no—
301C2×C59—82no—
302C2×C59#232311yescharacteristic
303C2×C59#294611yescharacteristic
304C2×C59#294911yescharacteristic
305C2×C59#324111yescharacteristic
306C2×C59#324411yescharacteristic
307C2×C59#325711yescharacteristic
308C2×C59#326311yescharacteristic
309C2×C4⋊Q8—591no—
310C2×C4×D4—591no—
311#476—598no—
312C2×#107—591no—
313C2×#106—591no—
314C2×#106—591no—
315C2×#106C2×D5911yescharacteristic
316C2×C4⋊C8—591no—
317C2×C4⋊C8—591no—
318C2×#102—591no—
319C2×#102—591no—
320C2×#102C2×D5911yescharacteristic
321C2×#102C2×D5911yescharacteristic
322C2×#97—591no—
323C2×C4×C8—591no—
324#391—1181no—
325#391—1181no—
326#390—591no—
327#390C2×D5911yescharacteristic
328#389—591no—
329#389—591no—
330#389—591no—
331#389—591no—
332#388—1181no—
333#388—1181no—
334#387—1181no—
335#387—1181no—
336#382—591no—
337#381—1181no—
338#380—1181no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4C2×D5911yescharacteristic
342#376—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340C2×D5911yescharacteristic
351#336—591no—
352#336C2×D5911yescharacteristic
353C59#2212311yescharacteristic, complemented, Sylow
354C22×Q8—1181no—
355C22×D4C22×D5911yescharacteristic
356C23×C4—1181no—
357C22×C8—591no—
358C22×C8—591no—
359C22×C8C22×D5911yescharacteristic
360C22×C8C22×D5911yescharacteristic
361C4⋊Q8—594no—
362C4⋊Q8—1182no—
363C4×D4—594no—
364C4×D4—1182no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C4⋊C4C22×D5911yescharacteristic
376C2×C4⋊C4C22×D5911yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C22⋊C4—1181no—
382C2×C42—21no—
383C2×C42—591no—
384C2×C42—591no—
385C2×C42—1181no—
386C2×C42—1181no—
387C2×C42C22×D5911yescharacteristic
388#107—2361no—
389#106—41no—
390#106—594no—
391#106—594no—
392C4⋊C8—594no—
393C4⋊C8—1182no—
394#102—41no—
395#102—594no—
396#102—2361no—
397#102C4×D5914yes—
398#97—2361no—
399C4×C8—2361no—
400#95—21no—
401#95—591no—
402#95—591no—
403#95—591no—
404#95—591no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413C24—21no—
414C2×Q8—1184no—
415C2×Q8—4721no—
416C2×Q8—4721no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D5912yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—591no—
430C22×C4—1184no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—4721no—
436C22×C4—4721no—
437C22×C4D4×D5911yescharacteristic
438C22×C4D4×D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—592no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8C2×C4×D5912yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
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—1181no—
466C4⋊C4—4721no—
467C4⋊C4—4721no—
468C4⋊C4—4721no—
469C4⋊C4—4721no—
470C4⋊C4C2×C4×D5912yes—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—1184no—
474C22⋊C4—4721no—
475C22⋊C4—4721no—
476C42—21no—
477C42—21no—
478C42—81no—
479C42—592no—
480C42—592no—
481C42—1181no—
482C42—1181no—
483C42—4721no—
484C42—4721no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C2×D4×D5911yescharacteristic
493Q8—4724no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—592no—
514C2×C4—592no—
515C2×C4—1181no—
516C2×C4—1181no—
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—1182no—
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—2361no—
538C2×C4C22×C4×D5911yescharacteristic
539C2×C4#41×D5911yescharacteristic
540C2×C4#1427377211yescharacteristic
541C2×C4#1427377411yescharacteristic
542C2×C4#1427377411yescharacteristic
543C2×C4#1427377411yescharacteristic
544C8—22no—
545C8—41no—
546C8—1182no—
547C8—2361no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C4×D4×D5911yescharacteristic
556C22#1683797411yescharacteristic
557C22#1683798111yescharacteristic
558C22#1683820611yescharacteristic
559C22#1683820811yescharacteristic
560C22#1683820911yescharacteristic
561C22#1683821211yescharacteristic
562C4—42no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—1182no—
568C4—1182no—
569C4—4721no—
570C4—4721no—
571C4—2362no—
572C4—2361no—
573C4—2362no—
574C4—2361no—
575C4#1683787911yescharacteristic
576C4#1683788111yescharacteristic
577C4#1683800911yescharacteristic
578C4#1683801611yescharacteristic
579C2—82no—
580C2#2142677511yescharacteristic
581C2#2143292511yescharacteristic
582C2#2143295011yescharacteristic
583C2#2143534711yescharacteristic
584C2#2143539011yescharacteristic
585C2#2143550511yescharacteristic
586C2#2143556311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1926C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1926D59
hypocenterC59#22123
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1926

Complements

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

NH ≅ G/Ntype
2. #21426840C2semidirect
4. #21425046C2semidirect
5. #21424865C2semidirect
6. #21424838C2semidirect
7. #21424822C2semidirect
11. #21424681C2semidirect
12. #21424675C2semidirect
14. #21423952C2semidirect
18. #16837690C4semidirect
353. C59#22123semidirect

Maximal subgroups

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

subgroupindex
C2×#168376902
#214268402
#214267332
#214250462
#214248652
#214248382
#214248222
#214247902
#214247812
#214247132
#214246812
#214246752
#214241042
#214239522
C59×#19262
#2212359