Finite Group Databasebeta

← #28651787

Subgroups — #28651787

Order 15104.

848 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837699C211yesmaximal, characteristic
2C2×C4×#11745671C211yesmaximal, characteristic
3#21427007C211yesmaximal, characteristic
4#21425644C211yesmaximal, characteristic, complemented
5#21425577C211yesmaximal, characteristic, complemented
6#21425477C211yesmaximal, characteristic, complemented
7#21425431C211yesmaximal, characteristic, complemented
8#21424821C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424433C211yesmaximal, characteristic
11#21424433C211yesmaximal, characteristic
12#21424356C211yesmaximal, characteristic
13#21424350C211yesmaximal, characteristic, complemented
14#21424342C211yesmaximal, characteristic
15C59×#2453C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20#16837699—24no—
21C4×#11745671—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×#14273647C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837273—21no—
38#16837273—21no—
39#16837268C2211yescharacteristic
40#16837261—21no—
41#95×D59—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#125×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C59×#400—21no—
53C59×#398—21no—
54C59×#391C2211yescharacteristic
55C59×#391C2211yescharacteristic
56C59×#389—21no—
57C59×#389C2211yescharacteristic
58C59×#389C2211yescharacteristic
59C59×#387—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C22×C4×D59—21no—
65C2×#11745671—22no—
66C2×#11745671—22no—
67C2×#11745671Q814yes—
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C4×D236—22no—
73C4×D236—22no—
74C4×D236—22no—
75C4×D236—22no—
76C4×D236—41no—
77C4×D236—41no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86#14273660—22no—
87#14273660—41no—
88#14273658—22no—
89#14273658—22no—
90#14273658—22no—
91C4⋊(C59⋊C8)—22no—
92#14273651—22no—
93#14273651—22no—
94#14273651—41no—
95#14273647—22no—
96#14273647—22no—
97#14273647D414yes—
98C4×C59⋊C8—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104#125×C59—22no—
105#125×C59—41no—
106C4×Q8×C59—22no—
107C4×Q8×C59—41no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59D411yescharacteristic
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59D411yescharacteristic
129#95×C59D411yescharacteristic
130C23×D59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59C2×Q812yes—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2411yescharacteristic
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236C2×D412yes—
155C2×D236C2×Q812yes—
156C2×D236#4112yes—
157C2×C4×D59—24no—
158C2×C4×D59—42no—
159C2×C4×D59—42no—
160C2×C4×D59—42no—
161C2×C4×D59—42no—
162C2×C4×D59—81no—
163C2×C4×D59—81no—
164C2×C4×D59—81no—
165C2×C4×D59—81no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—41no—
173C4⋊C4×C59—42no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59—42no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59C2×D412yes—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—42no—
186#11745671—28no—
187(C2×D59)⋊C4—24no—
188(C2×D59)⋊C4—42no—
189(C2×D59)⋊C4—42no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59#4112yes—
196Dic59⋊C4—81no—
197Dic59⋊C4—81no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8C2×D412yes—
209C2×C59⋊C8C2×Q812yes—
210C23×C59C22×D411yescharacteristic
211C22×D59—22no—
212C22×D59—22no—
213C22×D59—22no—
214C22×D59—41no—
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218Q8×C59—22no—
219Q8×C59—22no—
220Q8×C59—41no—
221Q8×C59—41no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59#13512yes—
252C2×C4×C59C22×D411yescharacteristic
253C2×C4×C59C22×Q811yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59H(5,𝔽2)11yescharacteristic
257C2×C4×C592−1+411yescharacteristic
258C2×Dic59—22no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267D236—22no—
268D236—22no—
269D236—22no—
270D236C22⋊Q814yes—
271C4×D59—84no—
272C4×D59—84no—
273C4×D59—84no—
274C59⋊C8—22no—
275C59⋊C8—22no—
276C59⋊C8—41no—
277C59⋊C8—41no—
278#26378—591nomaximal, Sylow
279C22×C59#52511yescharacteristic
280C22×C59D4211yescharacteristic
281C22×C59D4×Q811yescharacteristic
282C22×C59C2×#13511yescharacteristic
283C22×C59#57511yescharacteristic
284C22×C59#57711yescharacteristic
285C22×C59#57711yescharacteristic
286C2×D59—24no—
287C2×D59—42no—
288C2×D59—42no—
289C2×D59—42no—
290C2×D59—42no—
291C2×D59—42no—
292C2×D59—42no—
293C4×C59—22no—
294C4×C59—41no—
295C4×C59—41no—
296C4×C59—42no—
297C4×C59—41no—
298C4×C59—41no—
299C4×C59—42no—
300C4×C59—41no—
301C4×C59—81no—
302C4×C59—81no—
303C4×C59C2×C22⋊Q811yescharacteristic
304C4×C59#53911yescharacteristic
305C4×C59#54711yescharacteristic
306C4×C59#55411yescharacteristic
307Dic59—42no—
308Dic59—81no—
309Dic59—81no—
310C2×#478—591no—
311C2×C4×SD16—591no—
312#2453D5911yescharacteristic, complemented
313#2349—591no—
314#2097—591no—
315#2085—591no—
316#2066—591no—
317#2059—591no—
318#1941—591no—
319#1916—591no—
320#1877—591no—
321#1877—591no—
322#1864—591no—
323#1863—591no—
324#1862—591no—
325C2×C59#244911yescharacteristic
326C2×C59#310211yescharacteristic
327C2×C59#312311yescharacteristic
328C2×C59#326711yescharacteristic
329C2×C59#328211yescharacteristic
330C2×C59#338111yescharacteristic
331C2×C59#338811yescharacteristic
332D59—44no—
333C22×SD16—591no—
334C2×#125C2×D5911yescharacteristic
335C2×C4×Q8C2×D5911yescharacteristic
336C2×C4×D4—591no—
337C2×C4×D4—591no—
338C2×C4×D4—591no—
339#478—1184no—
340C4×SD16—1184no—
341C2×#107—591no—
342C2×#107—591no—
343C2×#106—591no—
344C2×#106—591no—
345C2×#106—591no—
346C2×C4⋊C8—591no—
347C2×#103—591no—
348C2×#103—591no—
349C2×#103—591no—
350C2×#102—591no—
351C2×#102—591no—
352C2×#102—591no—
353C2×C4×C8—591no—
354#400—21no—
355#398—21no—
356#391C2×D5911yescharacteristic
357#391C2×D5911yescharacteristic
358#389—21no—
359#389—591no—
360#389—591no—
361#389C2×D5911yescharacteristic
362#389C2×D5911yescharacteristic
363#387—21no—
364#385—1181no—
365#385—1181no—
366#382—591no—
367#380—1181no—
368C4×C4⋊C4C2×D5911yescharacteristic
369C2×#95—1181no—
370#340—591no—
371#340—591no—
372#340—591no—
373#340—591no—
374#336—591no—
375#336—591no—
376#336—591no—
377#336—591no—
378C59#2637811yescharacteristic, complemented, Sylow
379C22×Q8C22×D5911yescharacteristic
380C22×D4—591no—
381C23×C4—1181no—
382C23×C4—1181no—
383C23×C4—1181no—
384C2×SD16—594no—
385C2×SD16—1182no—
386C2×SD16—1182no—
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—591no—
390C22×C8—591no—
391#125—22no—
392#125—41no—
393C4×Q8—22no—
394C4×Q8—41no—
395C4×D4—1182no—
396C4×D4—1182no—
397C4×D4—1182no—
398C4×D4—1182no—
399C4×D4—2361no—
400C4×D4—2361no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4C22⋊D5911yescharacteristic
412C2×C4⋊C4C22⋊D5911yescharacteristic
413C2×C4⋊C4C22×D5911yescharacteristic
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C22⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C42—21no—
420C2×C42C22×D5911yescharacteristic
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423#107—1182no—
424#107—2361no—
425#106—1182no—
426#106—1182no—
427#106—1182no—
428C4⋊C8—1182no—
429#103—1182no—
430#103—1182no—
431#103—2361no—
432#102—594no—
433#102—1182no—
434#102—1182no—
435C4×C8—1182no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95C22⋊D5911yescharacteristic
447#95C22⋊D5911yescharacteristic
448C24—1181no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8Q8×D5912yes—
455C2×D4—592no—
456C2×D4—592no—
457C2×D4—592no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—1184no—
466C22×C4—1181no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—4721no—
470C22×C4—4721no—
471C22×C4—4721no—
472C22×C4—4721no—
473C22×C4—2362no—
474C22×C4—2362no—
475C22×C4—2362no—
476C22×C4—2362no—
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C2×C22⋊D5911yescharacteristic
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C23×D5911yescharacteristic
484SD16—1188no—
485C2×C8—592no—
486C2×C8—592no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—592no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—4721no—
516C4⋊C4—4721no—
517C4⋊C4D4×D5912yes—
518C22⋊C4—1184no—
519C22⋊C4—2362no—
520C22⋊C4—2362no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—42no—
528C23—1182no—
529C23—1182no—
530C23—1182no—
531C23—2361no—
532C23—2361no—
533C23—2361no—
534C23—2361no—
535C23C22×C22⋊D5911yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—594no—
541D4—1182no—
542D4—1182no—
543D4—1182no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1182no—
574C2×C4—4724no—
575C2×C4—4724no—
576C2×C4—4724no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4#1427374712yes—
586C2×C4C2×D4×D5911yescharacteristic
587C2×C4C2×Q8×D5911yescharacteristic
588C2×C4#41×D5911yescharacteristic
589C2×C4#41×D5911yescharacteristic
590C2×C4#1427377211yescharacteristic
591C2×C4#1427377511yescharacteristic
592C8—1182no—
593C8—1182no—
594C8—2361no—
595C8—2361no—
596C22—1184no—
597C22—2362no—
598C22—2362no—
599C22—2362no—
600C22—2362no—
601C22—2362no—
602C22—2362no—
603C22#1683790911yescharacteristic
604C22D4×C22⋊D5911yescharacteristic
605C22Q8×C22⋊D5911yescharacteristic
606C22C2×#1427374711yescharacteristic
607C22#1683820011yescharacteristic
608C22#1683821011yescharacteristic
609C22#1683821011yescharacteristic
610C4—22no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—4721no—
621C4—4721no—
622C4—2362no—
623C4C22⋊Q8×D5911yescharacteristic
624C4#1683799811yescharacteristic
625C4#1683805411yescharacteristic
626C4#1683810611yescharacteristic
627C2—2364no—
628C2#2142790211yescharacteristic
629C2#2143410811yescharacteristic
630C2#2143434311yescharacteristic
631C2#2143562211yescharacteristic
632C2#2143580411yescharacteristic
633C2#2143696711yescharacteristic
634C2#2143703511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2453C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2453D59
hypocenterC59#26378
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2453

Complements

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

NH ≅ G/Ntype
4. #21425644C2semidirect
5. #21425577C2semidirect
6. #21425477C2semidirect
7. #21425431C2semidirect
8. #21424821C2semidirect
9. #21424655C2semidirect
13. #21424350C2semidirect
15. C59×#2453C2semidirect
312. #2453D59semidirect
378. C59#26378semidirect

Maximal subgroups

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

subgroupindex
C2×#168376992
C2×C4×#117456712
#214270072
#214256442
#214255772
#214254772
#214254312
#214248212
#214246552
#214244332
#214243562
#214243502
#214243422
C59×#24532
#2637859