Finite Group Databasebeta

← #28638253

Subgroups — #28638253

Order 15104.

800 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837624C211yesmaximal, characteristic
2C2×Q8×C59⋊C8C211yesmaximal, characteristic, complemented
3#21429584C211yesmaximal, characteristic
4#21425481C211yesmaximal, characteristic
5#21425474C211yesmaximal, characteristic, complemented
6#21425418C211yesmaximal, characteristic, complemented
7#21425411C211yesmaximal, characteristic
8#21424956C211yesmaximal, characteristic, complemented
9#21424953C211yesmaximal, characteristic
10#21424905C211yesmaximal, characteristic
11#21424862C211yesmaximal, characteristic, complemented
12#21424778C211yesmaximal, characteristic
13#21424771C211yesmaximal, characteristic, complemented
14C4×#14273651C211yesmaximal, characteristic, complemented
15C59×#2581C211yesmaximal, characteristic, complemented
16C22×#11745672C2211yescharacteristic
17C22×#11745671C2211yescharacteristic
18C2×#14273715C2211yescharacteristic
19#16837624—24no—
20Q8×C59⋊C8—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×#14273647C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35C2×C4×C59⋊C8C2211yescharacteristic
36C2×C4×C59⋊C8C2211yescharacteristic
37#16837328—21no—
38#16837313—21no—
39#16837308C2211yescharacteristic
40#16837308C2211yescharacteristic
41#16837304C2211yescharacteristic
42#16837304C2211yescharacteristic
43#16837292—21no—
44#16837292—21no—
45C4×C4⋊Dic59C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C59×#400—21no—
53C59×#395—21no—
54C59×#391C2211yescharacteristic
55C59×#391C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#387—21no—
59C59×#387—21no—
60C59×#382C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C22×D236C2311yescharacteristic
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672—41no—
66C2×#11745671—22no—
67C2×#11745671—22no—
68C2×#11745671—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273715—22no—
74#14273715—22no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×C4×Dic59—21no—
84#14273660—41no—
85#14273658—41no—
86C4⋊(C59⋊C8)—22no—
87C4⋊(C59⋊C8)—22no—
88C4⋊(C59⋊C8)D414yes—
89#14273651—22no—
90#14273651—22no—
91#14273650—22no—
92#14273650—22no—
93#14273650—41no—
94#14273647—22no—
95#14273647—22no—
96#14273647—41no—
97C4×C59⋊C8—22no—
98C4×C59⋊C8—22no—
99C4×C59⋊C8—22no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C22×Q8×C59D411yescharacteristic
106C22×Q8×C59C2311yescharacteristic
107C4×Q8×C59—22no—
108C4×Q8×C59—22no—
109C4×Q8×C59—22no—
110C4×Q8×C59—22no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59D411yescharacteristic
116C2×C4⋊C4×C59D411yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132C23×D59—21no—
133C2×Q8×C59—22no—
134C2×Q8×C59—22no—
135C2×Q8×C59—22no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59#4112yes—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59—42no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—42no—
185C42×C59C2×D412yes—
186C42×C59#4112yes—
187#11745672—44no—
188#11745671—44no—
189(C2×D59)⋊C4—42no—
190(C2×D59)⋊C4—42no—
191(C2×D59)⋊C4—42no—
192(C2×D59)⋊C4—42no—
193(C2×D59)⋊C4—81no—
194(C2×D59)⋊C4—81no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4×Dic59—81no—
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⋊C8—21no—
209C2×C59⋊C8C2×D412yes—
210C2×C59⋊C8C2×D412yes—
211C2×C59⋊C8#4112yes—
212C23×C59C22×D411yescharacteristic
213C22×D59—41no—
214C22×D59—41no—
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220Q8×C59—28no—
221Q8×C59—22no—
222Q8×C59—22no—
223Q8×C59—41no—
224Q8×C59—41no—
225C2×C4×C59—21no—
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×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—22no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—41no—
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#12112yes—
252C2×C4×C59#12112yes—
253C2×C4×C59#13512yes—
254C2×C4×C59#13512yes—
255C2×C4×C59C22×D411yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C2×C4×C592−1+411yescharacteristic
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270D236—41no—
271D236—41no—
272D236—41no—
273D236—41no—
274Dic118—41no—
275Dic118—41no—
276Dic118—41no—
277Dic118—41no—
278C59⋊C8—22no—
279C59⋊C8—22no—
280C59⋊C8—22no—
281C59⋊C8—41no—
282#25297—591nomaximal, Sylow
283C22×C59C2×#12111yescharacteristic
284C22×C59#54811yescharacteristic
285C22×C59#55011yescharacteristic
286C22×C59C2×#13511yescharacteristic
287C22×C59C2×#13511yescharacteristic
288C22×C59#57711yescharacteristic
289C22×C59#57811yescharacteristic
290C2×D59—81no—
291C2×D59—81no—
292C2×D59—81no—
293C2×D59—81no—
294C2×D59—81no—
295C2×D59—81no—
296C2×D59—81no—
297C4×C59—22no—
298C4×C59—22no—
299C4×C59—22no—
300C4×C59—22no—
301C4×C59—42no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59—81no—
305C4×C59—81no—
306C4×C59—81no—
307C4×C59#52911yescharacteristic
308C4×C59#54211yescharacteristic
309C4×C59#54611yescharacteristic
310C4×C59#54811yescharacteristic
311Dic59—81no—
312Dic59—81no—
313Dic59—81no—
314C2×#464—591no—
315C2×C8×Q8—591no—
316#2592—591no—
317#2581D5911yescharacteristic, complemented
318#2067—591no—
319#2066—591no—
320#2056—591no—
321#2055—591no—
322#1969—591no—
323#1968—591no—
324#1958—591no—
325#1949—591no—
326#1934—591no—
327#1933—591no—
328C4×#103—591no—
329C2×C59#242611yescharacteristic
330C2×C59#305011yescharacteristic
331C2×C59#306311yescharacteristic
332C2×C59#336411yescharacteristic
333C2×C59#336511yescharacteristic
334C2×C59#339311yescharacteristic
335C2×C59#339411yescharacteristic
336D59—161no—
337C22×Q16—591no—
338C22×SD16—591no—
339C2×#124—591no—
340C2×C4×Q8C2×D5911yescharacteristic
341C2×C4×Q8C2×D5911yescharacteristic
342#464—1184no—
343C8×Q8—1184no—
344C2×#107—591no—
345C2×#106—591no—
346C2×C4⋊C8—591no—
347C2×C4⋊C8—591no—
348C2×C4⋊C8—591no—
349C2×#103—591no—
350C2×#103—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×C4×C8—591no—
358C2×C4×C8—591no—
359C2×C4×C8—591no—
360#400—21no—
361#397—1181no—
362#395—21no—
363#393—1181no—
364#391—591no—
365#391—591no—
366#391C2×D5911yescharacteristic
367#391C2×D5911yescharacteristic
368#390—591no—
369#390—591no—
370#389C2×D5911yescharacteristic
371#389C2×D5911yescharacteristic
372#388—1181no—
373#388—1181no—
374#387—21no—
375#387—21no—
376#382C2×D5911yescharacteristic
377C4×C4⋊C4—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382C59#2529711yescharacteristic, complemented, Sylow
383C22×Q8—591no—
384C22×Q8C22⋊D5911yescharacteristic
385C22×Q8C22×D5911yescharacteristic
386C22×D4—591no—
387C2×Q16—1182no—
388C2×Q16—1182no—
389C2×Q16—2361no—
390C2×SD16—1182no—
391C2×SD16—1182no—
392C2×SD16—2361no—
393C22×C8—591no—
394C22×C8—591no—
395C22×C8—591no—
396C22×C8—591no—
397#124—1182no—
398#124—1182no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4C22⋊D5911yescharacteristic
409C2×C4⋊C4C22⋊D5911yescharacteristic
410C2×C4⋊C4C22⋊D5911yescharacteristic
411C2×C4⋊C4C22×D5911yescharacteristic
412C2×C4⋊C4C22×D5911yescharacteristic
413C2×C4⋊C4C22×D5911yescharacteristic
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C42—21no—
421C2×C42—1181no—
422C2×C42C22×D5911yescharacteristic
423C2×C42C22×D5911yescharacteristic
424C2×C42C22×D5911yescharacteristic
425#107—2361no—
426#106—2361no—
427C4⋊C8—594no—
428C4⋊C8—1182no—
429C4⋊C8—1182no—
430#103—1182no—
431#103—1182no—
432#103—1182no—
433#103—1182no—
434#103—2361no—
435#102—1182no—
436#102—1182no—
437#102—2361no—
438C4×C8—1182no—
439C4×C8—1182no—
440C4×C8—1182no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454C24—1181no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8—1181no—
469C2×Q8#1174568912yes—
470C2×D4—1181no—
471C2×D4—1181no—
472C2×D4—1181no—
473C2×D4—1181no—
474C2×D4—1181no—
475C2×D4—1181no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490Q16—2364no—
491SD16—2364no—
492C2×C8—592no—
493C2×C8—592no—
494C2×C8—592no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C22⋊C4—4721no—
521C22⋊C4—4721no—
522C22⋊C4—2362no—
523C22⋊C4—2362no—
524C22⋊C4—2362no—
525C22⋊C4—2362no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—4721no—
532C42D4×D5912yes—
533C42#1174568712yes—
534C23—2361no—
535C23—2361no—
536C23—2361no—
537C23—2361no—
538C23—2361no—
539C23—2361no—
540C23—2361no—
541C23C22×C22⋊D5911yescharacteristic
542Q8—28no—
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—2361no—
548Q8—2361no—
549Q8—2361no—
550Q8—2361no—
551D4—2361no—
552D4—2361no—
553D4—2361no—
554D4—2361no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4#1427369912yes—
591C2×C4#1427369912yes—
592C2×C4#1427374712yes—
593C2×C4#1427374712yes—
594C2×C4C2×D4×D5911yescharacteristic
595C2×C4#41×D5911yescharacteristic
596C2×C4C2×#1174568711yescharacteristic
597C2×C4C2×#1174568911yescharacteristic
598C2×C4#1427377311yescharacteristic
599C2×C4#1427377511yescharacteristic
600C8—1182no—
601C8—1182no—
602C8—1182no—
603C8—2361no—
604C22—4721no—
605C22—4721no—
606C22—4721no—
607C22—4721no—
608C22—4721no—
609C22—4721no—
610C22—4721no—
611C22C2×#1427369911yescharacteristic
612C22#1683806311yescharacteristic
613C22#1683807111yescharacteristic
614C22C2×#1427374711yescharacteristic
615C22C2×#1427374711yescharacteristic
616C22#1683821011yescharacteristic
617C22#1683821711yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—4721no—
629C4—4721no—
630C4—4721no—
631C4#1683793811yescharacteristic
632C4#1683802511yescharacteristic
633C4#1683804311yescharacteristic
634C4#1683806211yescharacteristic
635C2—9441no—
636C2#2142766411yescharacteristic
637C2#2143367411yescharacteristic
638C2#2143378311yescharacteristic
639C2#2143678111yescharacteristic
640C2#2143679211yescharacteristic
641C2#2143708011yescharacteristic
642C2#2143709211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2581C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2581D59
hypocenterC59#25297
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C59⋊C8C2semidirect
5. #21425474C2semidirect
6. #21425418C2semidirect
8. #21424956C2semidirect
11. #21424862C2semidirect
13. #21424771C2semidirect
14. C4×#14273651C2semidirect
15. C59×#2581C2semidirect
317. #2581D59semidirect
382. C59#25297semidirect

Maximal subgroups

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

subgroupindex
C2×#168376242
C2×Q8×C59⋊C82
#214295842
#214254812
#214254742
#214254182
#214254112
#214249562
#214249532
#214249052
#214248622
#214247782
#214247712
C4×#142736512
C59×#25812
#2529759