Finite Group Databasebeta

← #28661401

Subgroups — #28661401

Order 15104.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428514C211yesmaximal, characteristic
2#21427105C211yesmaximal, characteristic
3#21425507C211yesmaximal, characteristic, complemented
4#21425475C211yesmaximal, characteristic, complemented
5#21425289C211yesmaximal, characteristic
6#21424856C211yesmaximal, characteristic, complemented
7#21424825C211yesmaximal, characteristic, complemented
8#21424705C211yesmaximal, characteristic
9#21424684C211yesmaximal, characteristic, complemented
10#21424673C211yesmaximal, characteristic, complemented
11#21424366C211yesmaximal, characteristic, complemented
12#21424291C211yesmaximal, characteristic
13#21424184C211yesmaximal, characteristic
14#103×Dic59C211yesmaximal, characteristic, complemented
15C59×#2035C211yesmaximal, characteristic
16C2×#14273720—21no—
17C2×Q8×Dic59C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19C2×#14273659C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27#16837343—21no—
28#16837330—21no—
29#16837309C2211yescharacteristic
30#16837301—21no—
31#16837301C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837289—21no—
35#16837288—21no—
36#16837277—21no—
37#16837271C2211yescharacteristic
38#16837262C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C22⋊C4×Dic59—21no—
42C4×C22⋊Dic59C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×C22⋊Q8×C59C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C2×#103×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273720—44no—
63Q8×Dic59—22no—
64Q8×Dic59—41no—
65C2×C4⋊Dic59—22no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—41no—
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×C22⋊Dic59—21no—
79C2×C22⋊Dic59D411yescharacteristic
80C2×C22⋊Dic59D411yescharacteristic
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59C2311yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86#14273659—22no—
87#14273659—41no—
88#14273658—22no—
89Dic59⋊C8—22no—
90#14273651—22no—
91#14273651—41no—
92C22⋊(C59⋊C8)—41no—
93C8×Dic59—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⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C22×Q8×C59C2311yescharacteristic
105C23×C4×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22⋊Q8×C59—22no—
109C22⋊Q8×C59—22no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116#106×C59—22no—
117#103×C59—22no—
118#103×C59—22no—
119C22⋊C8×C59—22no—
120#95×C59—21no—
121C24×C59C2×D411yescharacteristic
122C2×Q8×C59—21no—
123C2×Q8×C59—21no—
124C2×Q8×C59—21no—
125C2×Q8×C59—21no—
126C2×Q8×C59—21no—
127C2×Q8×C59#4112yes—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—42no—
145C22×Dic59—42no—
146C22×Dic59—81no—
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59#4112yes—
157C4⋊C4×C59—24no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59#4112yes—
164C22⋊C4×C59—24no—
165C22⋊C4×C59—42no—
166C22⋊Dic59—41no—
167C22⋊Dic59—42no—
168C22⋊Dic59SD1614yes—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—42no—
177C4⋊Dic59—42no—
178C4⋊Dic59—42no—
179C4⋊Dic59—81no—
180C4⋊Dic59#4112yes—
181Dic59⋊C4—42no—
182Dic59⋊C4—42no—
183Dic59⋊C4—42no—
184C4×Dic59—21no—
185C4×Dic59—21no—
186C4×Dic59—21no—
187C4×Dic59—21no—
188C4×Dic59—42no—
189C4×Dic59—42no—
190C4×Dic59—81no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59C2×SD1612yes—
204C23×C59C22×D411yescharacteristic
205Q8×C59—22no—
206Q8×C59—22no—
207Q8×C59—41no—
208Q8×C59—41no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59H(5,𝔽2)11yescharacteristic
233C2×C4×C592−1+411yescharacteristic
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—22no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—84no—
257C2×Dic59C2×SD1612yes—
258C2×Dic59C2×SD1612yes—
259C8×C59—22no—
260C8×C59—41no—
261C59⋊C8—41no—
262C59⋊C8—41no—
263#27034—591nomaximal, Sylow
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59#52511yescharacteristic
272C22×C59#54611yescharacteristic
273C22×C59#54711yescharacteristic
274C22×C59C22×SD1611yescharacteristic
275C22×C59#57511yescharacteristic
276C22×C59#57711yescharacteristic
277C22×C59#57811yescharacteristic
278C4×C59—42no—
279C4×C59—41no—
280C4×C59—42no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59#52911yescharacteristic
284C4×C59#53911yescharacteristic
285C4×C59#55311yescharacteristic
286C4×C59#55611yescharacteristic
287Dic59—22no—
288Dic59—22no—
289Dic59—41no—
290Dic59—41no—
291Dic59—42no—
292Dic59—81no—
293Dic59—81no—
294Dic59—81no—
295#2496—591no—
296#2360—591no—
297#2072—591no—
298#2066—591no—
299#2035—591no—
300#2035D5911yescharacteristic
301#1948—591no—
302#1941—591no—
303#1923—591no—
304#1920—591no—
305#1919—591no—
306#1865—591no—
307#1853—591no—
308#1824—591no—
309C4×#103—591no—
310C2×C59—42no—
311C2×C59#246611yescharacteristic
312C2×C59#310311yescharacteristic
313C2×C59#312011yescharacteristic
314C2×C59#329811yescharacteristic
315C2×C59#331111yescharacteristic
316C2×C59#333511yescharacteristic
317C2×C59#335211yescharacteristic
318C2×#126—1181no—
319C2×C22⋊Q8C2×D5911yescharacteristic
320C2×C4×Q8—591no—
321C22×C4⋊C4—591no—
322C2×#106—591no—
323C2×#106—591no—
324C2×#106—591no—
325C2×#106C2×D5911yescharacteristic
326C2×C4⋊C8—591no—
327C2×#103—591no—
328C2×#103—591no—
329C2×#103C2×D5911yescharacteristic
330C2×#103C2×D5911yescharacteristic
331C2×C22⋊C8—591no—
332C2×C22⋊C8C2×D5911yescharacteristic
333C2×C4×C8—591no—
334#400—1181no—
335#397—1181no—
336#391—591no—
337#389—591no—
338#389—591no—
339#389—591no—
340#389—1181no—
341#387—1181no—
342#387—1181no—
343#385—1181no—
344#384—591no—
345#380—591no—
346#380C2×D5911yescharacteristic
347C4×C4⋊C4—591no—
348C4×C4⋊C4—591no—
349C4×C22⋊C4—591no—
350C4×C22⋊C4—1181no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340C2×D5911yescharacteristic
359#336—591no—
360#336—591no—
361C59#2703411yescharacteristic, complemented, Sylow
362C22×Q8C22×D5911yescharacteristic
363C23×C4—1181no—
364C23×C4C22×D5911yescharacteristic
365C22×C8—591no—
366C22×C8—591no—
367C22×C8C22×D5911yescharacteristic
368C22×C8C22×D5911yescharacteristic
369#126—2364no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C4×Q8—1182no—
373C4×Q8—2361no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—1182no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—2361no—
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C4⋊C4C22×D5911yescharacteristic
390C2×C4⋊C4C22×D5911yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—591no—
394C2×C22⋊C4—591no—
395C2×C22⋊C4—1181no—
396C2×C42—591no—
397C2×C42—591no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42—1181no—
401#106—22no—
402#106—1182no—
403#106—1182no—
404#106—2361no—
405C4⋊C8—1182no—
406#103—22no—
407#103—22no—
408#103—1182no—
409#103—2361no—
410C22⋊C8—22no—
411C22⋊C8—2361no—
412C4×C8—2361no—
413#95—21no—
414#95—591no—
415#95—591no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424C24D4×D5911yescharacteristic
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8#1174568712yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—591no—
442C22×C4—591no—
443C22×C4—591no—
444C22×C4—591no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—4721no—
450C22×C4—2362no—
451C22×C4—2362no—
452C22×C4D4×D5911yescharacteristic
453C22×C4C23×D5911yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8#1174568712yes—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—592no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—4721no—
481C4⋊C4—2362no—
482C4⋊C4—2362no—
483C4⋊C4—2362no—
484C4⋊C4—2362no—
485C4⋊C4—2362no—
486C4⋊C4—2362no—
487C4⋊C4#1174568712yes—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—594no—
491C22⋊C4—2361no—
492C22⋊C4—2362no—
493C42—1181no—
494C42—1181no—
495C42—1181no—
496C42—1181no—
497C42—4721no—
498C42—2362no—
499C42—2362no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23SD16×D5912yes—
507C23C2×D4×D5911yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—592no—
532C2×C4—592no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1182no—
543C2×C4—1181no—
544C2×C4—4724no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4#41×D5911yescharacteristic
557C2×C4C2×#1174568711yescharacteristic
558C2×C4C2×#1174568711yescharacteristic
559C2×C4C2×#1174568711yescharacteristic
560C2×C4#1427377311yescharacteristic
561C2×C4#1427377411yescharacteristic
562C8—22no—
563C8—41no—
564C8—2361no—
565C8—2361no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#1683790811yescharacteristic
574C22#1683804411yescharacteristic
575C22#1683805211yescharacteristic
576C22C2×SD16×D5911yescharacteristic
577C22#1683820111yescharacteristic
578C22#1683820911yescharacteristic
579C22#1683821811yescharacteristic
580C4—41no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—1182no—
586C4—1182no—
587C4—4721no—
588C4—4721no—
589C4—4721no—
590C4—2361no—
591C4—2362no—
592C4—2361no—
593C4#1683793311yescharacteristic
594C4#1683799611yescharacteristic
595C4#1683810411yescharacteristic
596C4#1683812111yescharacteristic
597C2—42no—
598C2#2142812711yescharacteristic
599C2#2143412211yescharacteristic
600C2#2143430311yescharacteristic
601C2#2143595911yescharacteristic
602C2#2143613511yescharacteristic
603C2#2143644811yescharacteristic
604C2#2143666611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2035C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2035D59
hypocenterC59#27034
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #21425507C2semidirect
4. #21425475C2semidirect
6. #21424856C2semidirect
7. #21424825C2semidirect
9. #21424684C2semidirect
10. #21424673C2semidirect
11. #21424366C2semidirect
14. #103×Dic59C2semidirect
361. C59#27034semidirect

Maximal subgroups

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

subgroupindex
#214285142
#214271052
#214255072
#214254752
#214252892
#214248562
#214248252
#214247052
#214246842
#214246732
#214243662
#214242912
#214241842
#103×Dic592
C59×#20352
#2703459