Finite Group Databasebeta

← #28652298

Subgroups — #28652298

Order 15104.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427798C211yesmaximal, characteristic
2#21427099C211yesmaximal, characteristic
3#21425655C211yesmaximal, characteristic, complemented
4#21425594C211yesmaximal, characteristic, complemented
5#21425476C211yesmaximal, characteristic, complemented
6#21425238C211yesmaximal, characteristic, complemented
7#21425047C211yesmaximal, characteristic, complemented
8#21424675C211yesmaximal, characteristic, complemented
9#21424519C211yesmaximal, characteristic
10#21424401C211yesmaximal, characteristic
11#21424367C211yesmaximal, characteristic, complemented
12#21424291C211yesmaximal, characteristic
13#21424102C211yesmaximal, characteristic, complemented
14C2×#16837165C211yesmaximal, characteristic
15C59×#1850C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×#14273704—21no—
19C2×C4×Dic118C2211yescharacteristic
20C22×C4⋊Dic59C2211yescharacteristic
21C22×C4⋊Dic59C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28#16837334—21no—
29#16837323—21no—
30#16837308C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837299—21no—
35#16837289—21no—
36#16837277—21no—
37#16837274—21no—
38#16837272C2211yescharacteristic
39#16837262C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41C22⋊C4×Dic59—21no—
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C23×C8×C59C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×C4⋊C8×C59C2211yescharacteristic
55C2×C22⋊C8×C59C2211yescharacteristic
56C4×C22⋊C4×C59C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×Dic118—21no—
60C22×Dic118C2311yescharacteristic
61C23×Dic59—21no—
62C23×Dic59—21no—
63#14273727—22no—
64#14273727—41no—
65#14273706—22no—
66#14273706Q814yescomplemented
67#14273704—44no—
68C4×Dic118—22no—
69C4×Dic118—41no—
70C2×C4⋊Dic59—22no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—41no—
78C2×C4⋊Dic59—41no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×C4⋊Dic59C2311yescharacteristic
85C2×C4⋊Dic59C2311yescharacteristic
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×C22⋊Dic59—21no—
92C2×C22⋊Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59—21no—
95#14273659—22no—
96#14273659—41no—
97#14273652—22no—
98#14273652—41no—
99#14273652—41no—
100#14273652—41no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C23×C4×C59C2311yescharacteristic
109C22×C8×C59—22no—
110C22×C8×C59—41no—
111C22×C8×C59—41no—
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C4⋊C8×C59—41no—
121C4⋊C8×C59—41no—
122C22⋊C8×C59—22no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×Q812yes—
136C22×C4×C59#4112yes—
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—42no—
147C22×Dic59—42no—
148C22×Dic59—42no—
149C22×Dic59—42no—
150C22×Dic59—81no—
151C22×Dic59—81no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—42no—
158C2×Dic118—42no—
159C2×Dic118—81no—
160C2×Dic118C2×Q812yes—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—44no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59SD1614yes—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C22⋊Dic59—24no—
181C22⋊Dic59—42no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—42no—
194C4⋊Dic59—42no—
195C4⋊Dic59—42no—
196C4⋊Dic59—42no—
197C4⋊Dic59—81no—
198C4⋊Dic59C2×Q812yes—
199C4⋊Dic59#4112yes—
200C4⋊Dic59#4112yes—
201Dic59⋊C4—24no—
202Dic59⋊C4—42no—
203Dic59⋊C4—42no—
204Dic59⋊C4—42no—
205Dic59⋊C4—81no—
206C4×Dic59—42no—
207C4×Dic59—81no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59C22⋊Q812yes—
213C23×C59C2×SD1612yes—
214C23×C59#13712yes—
215C23×C59C22×D411yescharacteristic
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59C22⋊Q812yes—
233C2×C4×C59C2×SD1612yes—
234C2×C4×C59C2×SD1612yes—
235C2×C4×C59C22×Q811yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59H(5,𝔽2)11yescharacteristic
240C2×C4×C59H(5,𝔽2)11yescharacteristic
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—22no—
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—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—84no—
266C2×Dic59—84no—
267Dic118—22no—
268Dic118—22no—
269Dic118—41no—
270Dic118—41no—
271Dic118—84no—
272C8×C59—41no—
273C8×C59—41no—
274C8×C59—41no—
275C8×C59—41no—
276#26412—591nomaximal, Sylow
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—22no—
281C22×C59—22no—
282C22×C59—22no—
283C22×C59#47514yes—
284C22×C59C2×C22⋊Q811yescharacteristic
285C22×C59#54611yescharacteristic
286C22×C59#54611yescharacteristic
287C22×C59C22×SD1611yescharacteristic
288C22×C59C2×#13711yescharacteristic
289C22×C59#57611yescharacteristic
290C22×C59#57811yescharacteristic
291C4×C59—22no—
292C4×C59—22no—
293C4×C59—22no—
294C4×C59—41no—
295C4×C59—41no—
296C4×C59#52911yescharacteristic
297C4×C59#53811yescharacteristic
298C4×C59#55411yescharacteristic
299C4×C59#55411yescharacteristic
300Dic59—42no—
301Dic59—42no—
302Dic59—42no—
303Dic59—42no—
304Dic59—81no—
305Dic59—81no—
306Dic59—81no—
307Dic59—81no—
308#2440—591no—
309#2360—591no—
310#2099—591no—
311#2088—591no—
312#2066—591no—
313#2024—591no—
314#1988—591no—
315#1919—591no—
316#1892—591no—
317#1872—591no—
318#1865—591no—
319#1853—591no—
320#1850D5911yescharacteristic
321#1803—591no—
322C2×#340—591no—
323C2×C59—24no—
324C2×C59#245011yescharacteristic
325C2×C59C2×#47511yescharacteristic
326C2×C59#308311yescharacteristic
327C2×C59#329511yescharacteristic
328C2×C59#329811yescharacteristic
329C2×C59#331211yescharacteristic
330C2×C59#331411yescharacteristic
331C23×C8C2×D5911yescharacteristic
332C2×C4⋊Q8—591no—
333C2×C22⋊Q8—591no—
334C2×C22⋊Q8—1181no—
335C2×C4×Q8—591no—
336C22×C4⋊C4—591no—
337C22×C4⋊C4—591no—
338C2×#106—591no—
339C2×#106—591no—
340C2×C4⋊C8C2×D5911yescharacteristic
341C2×C4⋊C8C2×D5911yescharacteristic
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×#103—591no—
346C2×C22⋊C8C2×D5911yescharacteristic
347#398—1181no—
348#395—1181no—
349#391—591no—
350#389—591no—
351#389—591no—
352#389—591no—
353#389—1181no—
354#387—1181no—
355#385—1181no—
356#385—1181no—
357#384—591no—
358#380—591no—
359C4×C4⋊C4—591no—
360C4×C22⋊C4—1181no—
361C4×C22⋊C4C2×D5911yescharacteristic
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#336C2×D5911yescharacteristic
373#336C2×D5911yescharacteristic
374C59#2641211yescharacteristic, complemented, Sylow
375C22×Q8—591no—
376C22×Q8—1181no—
377C23×C4—1181no—
378C23×C4—1181no—
379C23×C4C22×D5911yescharacteristic
380C22×C8—22no—
381C22×C8—41no—
382C22×C8—41no—
383C22×C8C22×D5911yescharacteristic
384C22×C8C22×D5911yescharacteristic
385C22×C8C22×D5911yescharacteristic
386C22×C8C22×D5911yescharacteristic
387C4⋊Q8—1182no—
388C4⋊Q8—2361no—
389C22⋊Q8—594no—
390C22⋊Q8—1182no—
391C22⋊Q8—2364no—
392C4×Q8—1182no—
393C4×Q8—2361no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—1182no—
402C2×C4⋊C4—1182no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—2361no—
414C2×C4⋊C4—2361no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4—1181no—
417C2×C22⋊C4D23611yescharacteristic
418C2×C22⋊C4D23611yescharacteristic
419C2×C42—1181no—
420C2×C42—1181no—
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423#106—1182no—
424#106—2361no—
425C4⋊C8—41no—
426C4⋊C8—41no—
427#103—1182no—
428#103—2361no—
429#103—2361no—
430#103—2361no—
431C22⋊C8—22no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95D23611yescharacteristic
440#95D23611yescharacteristic
441C24C2×D23611yescharacteristic
442C2×Q8—592no—
443C2×Q8—1181no—
444C2×Q8—1181no—
445C2×Q8—1181no—
446C2×Q8—1181no—
447C2×Q8—1181no—
448C2×Q8—4721no—
449C2×Q8—2362no—
450C2×Q8—2362no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—4721no—
464C22×C4—4721no—
465C22×C4—2362no—
466C22×C4—2362no—
467C22×C4—2362no—
468C22×C4—2362no—
469C22×C4C2×D23611yescharacteristic
470C22×C4C2×D23611yescharacteristic
471C22×C4C2×D23611yescharacteristic
472C22×C4C2×D23611yescharacteristic
473C22×C4C2×D23611yescharacteristic
474C22×C4#1174568712yes—
475C22×C4Q8×D5912yes—
476C22×C4C23×D5911yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—44no—
490C4⋊C4—592no—
491C4⋊C4—592no—
492C4⋊C4—592no—
493C4⋊C4—1184no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—4721no—
506C4⋊C4—4721no—
507C4⋊C4—2362no—
508C4⋊C4—2362no—
509C4⋊C4—2362no—
510C4⋊C4—2362no—
511C4⋊C4—2362no—
512C4⋊C4—2362no—
513C4⋊C4—2362no—
514C22⋊C4—22no—
515C22⋊C4—1184no—
516C22⋊C4—2362no—
517C22⋊C4#1174566014yes—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—4721no—
523C42—2362no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1427370712yes—
529C23C2×#1174566012yes—
530C23#1427376012yes—
531C23C22×D23611yescharacteristic
532Q8—1182no—
533Q8—1182no—
534Q8—4724no—
535Q8—2361no—
536Q8—2361no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—1182no—
554C2×C4—1182no—
555C2×C4—1182no—
556C2×C4—1182no—
557C2×C4—4724no—
558C2×C4—4724no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4#1427370712yes—
580C2×C4C2×#1174566012yes—
581C2×C4C2×#1174566012yes—
582C2×C4C2×Q8×D5911yescharacteristic
583C2×C4#41×D5911yescharacteristic
584C2×C4#41×D5911yescharacteristic
585C2×C4C2×#1174568711yescharacteristic
586C2×C4#1427377211yescharacteristic
587C2×C4#1427377211yescharacteristic
588C8—41no—
589C8—41no—
590C8—41no—
591C8—41no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1683768814yes—
599C22C2×#1427370711yescharacteristic
600C22#1683804711yescharacteristic
601C22#1683804711yescharacteristic
602C22C22×#1174566011yescharacteristic
603C22C2×#1427376011yescharacteristic
604C22#1683820711yescharacteristic
605C22#1683821911yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—4721no—
612C4—4721no—
613C4—4721no—
614C4—4721no—
615C4—2362no—
616C4—2362no—
617C4—2362no—
618C4—2362no—
619C4#1683792911yescharacteristic
620C4#1683798711yescharacteristic
621C4#1683810611yescharacteristic
622C4#1683810611yescharacteristic
623C2—24no—
624C2#2142791811yescharacteristic
625C2C2×#1683768811yescharacteristic
626C2#2143394911yescharacteristic
627C2#2143592111yescharacteristic
628C2#2143596211yescharacteristic
629C2#2143614911yescharacteristic
630C2#2143617911yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1850C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1850D59
hypocenterC59#26412
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
3. #21425655C2semidirect
4. #21425594C2semidirect
5. #21425476C2semidirect
6. #21425238C2semidirect
7. #21425047C2semidirect
8. #21424675C2semidirect
11. #21424367C2semidirect
13. #21424102C2semidirect
66. #14273706Q8semidirect
374. C59#26412semidirect

Maximal subgroups

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

subgroupindex
#214277982
#214270992
#214256552
#214255942
#214254762
#214252382
#214250472
#214246752
#214245192
#214244012
#214243672
#214242912
#214241022
C2×#168371652
C59×#18502
#2641259