Finite Group Databasebeta

← #28646858

Subgroups — #28646858

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429117C211yesmaximal, characteristic, complemented
2#21427067C211yesmaximal, characteristic
3#21425483C211yesmaximal, characteristic
4#21425469C211yesmaximal, characteristic
5#21425243C211yesmaximal, characteristic
6#21425233C211yesmaximal, characteristic
7#21425051C211yesmaximal, characteristic, complemented
8#21425047C211yesmaximal, characteristic, complemented
9#21424866C211yesmaximal, characteristic, complemented
10#21424689C211yesmaximal, characteristic, complemented
11#21424682C211yesmaximal, characteristic, complemented
12#21424227C211yesmaximal, characteristic, complemented
13#21424107C211yesmaximal, characteristic
14C4×#14273649C211yesmaximal, characteristic
15C59×#1939C211yesmaximal, characteristic, complemented
16C2×#14273727C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273649C2211yescharacteristic
22C2×#14273649C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25#16837335—21no—
26#16837322—21no—
27#16837306—21no—
28#16837304C2211yescharacteristic
29#16837304C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837299—21no—
33#16837292—21no—
34#16837292—21no—
35#16837268C2211yescharacteristic
36#16837268C2211yescharacteristic
37#16837264—21no—
38C4×C4⋊Dic59C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41#16837249—21no—
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C2×C4⋊C8×C59C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×C4⋊C8×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×Dic118—21no—
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62#14273727—22no—
63#14273727—41no—
64#14273719—22no—
65#14273719—41no—
66C4×D236—22no—
67C4×D236Q814yescomplemented
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88#14273661—22no—
89#14273659—22no—
90#14273649—22no—
91#14273649—22no—
92#14273649—41no—
93#14273649—41no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×C8×C59C2311yescharacteristic
104C22×C8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112C4⋊C8×C59—22no—
113C4⋊C8×C59—41no—
114C4⋊C8×C59—41no—
115C4×C8×C59—22no—
116#95×C59D411yescharacteristic
117#95×C59D411yescharacteristic
118C23×D59—21no—
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2411yescharacteristic
126C22×Dic59—21no—
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59—21no—
132C22×Dic59—21no—
133C2×D236—21no—
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236C2×Q812yes—
139C2×C4×D59—24no—
140C2×C4×D59—81no—
141C2×C4×D59—81no—
142C2×Dic118—42no—
143C2×Dic118—42no—
144C2×Dic118—81no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—41no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59C2×Q812yes—
164C42×C59#4112yes—
165(C2×D59)⋊C4—24no—
166(C2×D59)⋊C4—42no—
167(C2×D59)⋊C4—42no—
168(C2×D59)⋊C4—81no—
169(C2×D59)⋊C4—81no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—42no—
182C4⋊Dic59C2×Q812yes—
183C4⋊Dic59#4112yes—
184C4⋊Dic59#4112yes—
185Dic59⋊C4—42no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—81no—
189C4×Dic59—81no—
190C4×Dic59—81no—
191C4×Dic59—81no—
192C23×C59C22×D411yescharacteristic
193C22×D59—22no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C22×D59—41no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
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—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59C22⋊Q812yes—
216C2×C4×C59C22⋊Q812yes—
217C2×C4×C59#13512yes—
218C2×C4×C59C22×Q811yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59H(5,𝔽2)11yescharacteristic
223C2×C4×C592−1+411yescharacteristic
224C2×Dic59—22no—
225C2×Dic59—22no—
226C2×Dic59—22no—
227C2×Dic59—41no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
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—
245D236—22no—
246D236—22no—
247D236—41no—
248D236—41no—
249C4×D59—84no—
250Dic118—84no—
251C8×C59—41no—
252C8×C59—41no—
253C8×C59—41no—
254C8×C59—41no—
255#26028—591nomaximal, Sylow
256C22×C59C2×C22⋊Q811yescharacteristic
257C22×C59#54611yescharacteristic
258C22×C59#54811yescharacteristic
259C22×C59C2×#13511yescharacteristic
260C22×C59#57511yescharacteristic
261C22×C59#57611yescharacteristic
262C22×C59#57711yescharacteristic
263C2×D59—42no—
264C2×D59—42no—
265C2×D59—42no—
266C2×D59—81no—
267C2×D59—81no—
268C2×D59—81no—
269C2×D59—81no—
270C4×C59—22no—
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—41no—
276C4×C59#53311yescharacteristic
277C4×C59#54411yescharacteristic
278C4×C59#55511yescharacteristic
279C4×C59#55611yescharacteristic
280Dic59—42no—
281Dic59—42no—
282Dic59—42no—
283Dic59—81no—
284Dic59—81no—
285Dic59—81no—
286Dic59—81no—
287#2550—591no—
288#2357—591no—
289#2067—591no—
290#2065—591no—
291#2025—591no—
292#2023—591no—
293#1989—591no—
294#1988—591no—
295#1949—591no—
296#1939D5911yescharacteristic, complemented
297#1921—591no—
298#1920—591no—
299#1835—591no—
300#1804—591no—
301C4×#102—591no—
302C2×C59#243911yescharacteristic
303C2×C59#307611yescharacteristic
304C2×C59#308111yescharacteristic
305C2×C59#329711yescharacteristic
306C2×C59#332611yescharacteristic
307C2×C59#336511yescharacteristic
308C2×C59#337011yescharacteristic
309D59—82no—
310C2×C4⋊Q8—591no—
311C2×#125—591no—
312C2×C4×D4—591no—
313C2×#107—591no—
314C2×#106—591no—
315C2×C4⋊C8C2×D5911yescharacteristic
316C2×C4⋊C8C2×D5911yescharacteristic
317C2×C4⋊C8C2×D5911yescharacteristic
318C2×#102—591no—
319C2×#102—591no—
320C2×#102—591no—
321C2×#102—591no—
322C2×C4×C8C2×D5911yescharacteristic
323#398—1181no—
324#395—1181no—
325#391—1181no—
326#390—591no—
327#390—591no—
328#389—591no—
329#389—591no—
330#389—1181no—
331#388—1181no—
332#388—1181no—
333#382—591no—
334#382—591no—
335#381—1181no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4C2×D5911yescharacteristic
340C4×C22⋊C4—1181no—
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#340—591no—
350#340—591no—
351#336C2×D5911yescharacteristic
352#336C2×D5911yescharacteristic
353C59#2602811yescharacteristic, complemented, Sylow
354C22×Q8—1181no—
355C22×D4—591no—
356C23×C4—1181no—
357C22×C8C22×D5911yescharacteristic
358C22×C8C22×D5911yescharacteristic
359C22×C8C22×D5911yescharacteristic
360C22×C8C22×D5911yescharacteristic
361C4⋊Q8—1182no—
362C4⋊Q8—2361no—
363#125—1182no—
364#125—2361no—
365C4×D4—594no—
366C4×D4—1182no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4D23611yescharacteristic
380C2×C4⋊C4D23611yescharacteristic
381C2×C22⋊C4—1181no—
382C2×C22⋊C4—1181no—
383C2×C22⋊C4—1181no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C42—1181no—
387C2×C42—1181no—
388C2×C42—1181no—
389C2×C42C22×D5911yescharacteristic
390C2×C42C22×D5911yescharacteristic
391C2×C42C22×D5911yescharacteristic
392#107—1182no—
393#106—1182no—
394C4⋊C8—22no—
395C4⋊C8—41no—
396C4⋊C8—41no—
397#102—1182no—
398#102—1182no—
399#102—2361no—
400#102—2361no—
401C4×C8—22no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95D23611yescharacteristic
412#95D23611yescharacteristic
413C24—1181no—
414C2×Q8—4721no—
415C2×Q8—2362no—
416C2×Q8—2362no—
417C2×D4—592no—
418C2×D4—1181no—
419C2×D4—1181no—
420C2×D4—1181no—
421C2×D4—1181no—
422C2×D4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—1184no—
427C22×C4—1181no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—4721no—
432C22×C4—4721no—
433C22×C4C2×D23611yescharacteristic
434C22×C4C2×D23611yescharacteristic
435C22×C4C2×D23611yescharacteristic
436C22×C4C2×D23611yescharacteristic
437C22×C4C2×D23611yescharacteristic
438C22×C4C2×D23611yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C4⋊C4—22no—
453C4⋊C4—41no—
454C4⋊C4—592no—
455C4⋊C4—592no—
456C4⋊C4—592no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—4721no—
469C4⋊C4—2362no—
470C4⋊C4—2362no—
471C4⋊C4—2362no—
472C4⋊C4—2362no—
473C22⋊C4—1184no—
474C22⋊C4—4721no—
475C22⋊C4—4721no—
476C22⋊C4—2362no—
477C22⋊C4—2362no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—4721no—
483C42—4721no—
484C42—4721no—
485C42#1174568712yes—
486C42Q8×D5912yes—
487C23—1182no—
488C23—2361no—
489C23—2361no—
490C23—2361no—
491C23—2361no—
492C23—2361no—
493C23—2361no—
494C23C22×D23611yescharacteristic
495Q8—4724no—
496D4—1182no—
497D4—1182no—
498D4—2361no—
499D4—2361no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—1182no—
516C2×C4—1182no—
517C2×C4—1182no—
518C2×C4—4724no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
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×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4#1427370712yes—
538C2×C4#1427370712yes—
539C2×C4#1427374912yes—
540C2×C4C2×Q8×D5911yescharacteristic
541C2×C4#41×D5911yescharacteristic
542C2×C4#41×D5911yescharacteristic
543C2×C4C2×#1174568711yescharacteristic
544C2×C4#1427377211yescharacteristic
545C2×C4#1427377511yescharacteristic
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C22—4721no—
551C22—4721no—
552C22—4721no—
553C22—4721no—
554C22—2362no—
555C22—2362no—
556C22—2362no—
557C22C2×#1427370711yescharacteristic
558C22#1683804711yescharacteristic
559C22#1683806411yescharacteristic
560C22C2×#1427374911yescharacteristic
561C22#1683820211yescharacteristic
562C22#1683820711yescharacteristic
563C22#1683821411yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—4721no—
571C4—4721no—
572C4—4721no—
573C4—4721no—
574C4—2362no—
575C4—2362no—
576C4—2362no—
577C4#1683795811yescharacteristic
578C4#1683802911yescharacteristic
579C4#1683811611yescharacteristic
580C4#1683812311yescharacteristic
581C2—4722no—
582C2#2142778511yescharacteristic
583C2#2143388611yescharacteristic
584C2#2143392811yescharacteristic
585C2#2143594811yescharacteristic
586C2#2143635111yescharacteristic
587C2#2143679411yescharacteristic
588C2#2143685111yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1939C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#1939D59
hypocenterC59#26028
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
1. #21429117C2semidirect
7. #21425051C2semidirect
8. #21425047C2semidirect
9. #21424866C2semidirect
10. #21424689C2semidirect
11. #21424682C2semidirect
12. #21424227C2semidirect
15. C59×#1939C2semidirect
67. C4×D236Q8semidirect
296. #1939D59semidirect
353. C59#26028semidirect

Maximal subgroups

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

subgroupindex
#214291172
#214270672
#214254832
#214254692
#214252432
#214252332
#214250512
#214250472
#214248662
#214246892
#214246822
#214242272
#214241072
C4×#142736492
C59×#19392
#2602859