Finite Group Databasebeta

← #28637273

Subgroups — #28637273

Order 15104.

808 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837737C211yesmaximal, characteristic
2C2×#16837658C211yesmaximal, characteristic
3#21427121C211yesmaximal, characteristic
4#21425578C211yesmaximal, characteristic, complemented
5#21425515C211yesmaximal, characteristic, complemented
6#21425491C211yesmaximal, characteristic, complemented
7#21425418C211yesmaximal, characteristic, complemented
8#21424862C211yesmaximal, characteristic, complemented
9#21424734C211yesmaximal, characteristic, complemented
10#21424440C211yesmaximal, characteristic, complemented
11#21424409C211yesmaximal, characteristic
12#21424281C211yesmaximal, characteristic
13#21424108C211yesmaximal, characteristic, complemented
14#21423815C211yesmaximal, characteristic
15C59×#2503C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745664C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C2×#14273686C2211yescharacteristic
21#16837737—24no—
22#16837658—24no—
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×#14273650C2211yescharacteristic
34C2×C22⋊(C59⋊C8)C2211yescharacteristic
35#16837308C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837296—21no—
38#16837290—21no—
39#16837284—21no—
40C4×C4⋊Dic59C2211yescharacteristic
41#16837253—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C22×C4⋊C4×C59C2211yescharacteristic
52C59×#400—21no—
53C59×#400—21no—
54C59×#391C2211yescharacteristic
55C59×#391C2211yescharacteristic
56C59×#382C2211yescharacteristic
57C59×#382C2211yescharacteristic
58C59×#381—21no—
59C59×#381—21no—
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745672—22no—
63C2×#11745672—22no—
64C2×#11745672—41no—
65C2×#11745664—22no—
66C2×#11745664—22no—
67C2×#11745664—41no—
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72#14273706—22no—
73#14273706D414yes—
74C4×Dic118—22no—
75C4×Dic118—41no—
76#14273686—22no—
77#14273686—41no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×C22⋊Dic59—21no—
85C2×C22⋊Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89#14273660—22no—
90#14273660—22no—
91#14273658—22no—
92C4⋊(C59⋊C8)—22no—
93C4⋊(C59⋊C8)—41no—
94#14273651—22no—
95#14273651—22no—
96#14273651—41no—
97#14273650—22no—
98#14273650—22no—
99#14273650—22no—
100C22⋊(C59⋊C8)—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C22×Q8×C59C2311yescharacteristic
106C23×C4×C59—21no—
107C23×C4×C59C2311yescharacteristic
108C22⋊Q8×C59—22no—
109C22⋊Q8×C59—22no—
110C2×C4⋊C4×C59—22no—
111C2×C4⋊C4×C59—22no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59D411yescharacteristic
119C2×C22⋊C4×C59D411yescharacteristic
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×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59C2×D411yescharacteristic
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—42no—
148C22×C4×C59—42no—
149C22×C4×C59—42no—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D412yes—
156C22×C4×C59#4112yes—
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118C2×D412yes—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59—42no—
176C22⋊C4×C59—41no—
177C22⋊C4×C59—41no—
178C22⋊C4×C59—42no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C22⋊Dic59—24no—
184C22⋊Dic59—42no—
185#11745672—44no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59C2×D412yes—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—24no—
193Dic59⋊C4—42no—
194Dic59⋊C4—42no—
195C4×Dic59—42no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198#11745664—44no—
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⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8#4112yes—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59#12112yes—
218C23×C59C22×D411yescharacteristic
219Q8×C59—41no—
220Q8×C59—41no—
221Q8×C59—41no—
222Q8×C59—41no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
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—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—41no—
252C2×C4×C59—41no—
253C2×C4×C59—84no—
254C2×C4×C59#12112yes—
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—22no—
262C2×Dic59—22no—
263C2×Dic59—22no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273Dic118—22no—
274Dic118—22no—
275Dic118—41no—
276Dic118—41no—
277C59⋊C8—22no—
278C59⋊C8—41no—
279C59⋊C8—41no—
280C59⋊C8—41no—
281#25229—591nomaximal, Sylow
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—22no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59C2×#12111yescharacteristic
290C22×C59#54611yescharacteristic
291C22×C59#54711yescharacteristic
292C22×C59#57511yescharacteristic
293C22×C59#57511yescharacteristic
294C22×C59#57811yescharacteristic
295C22×C59#57811yescharacteristic
296C4×C59—22no—
297C4×C59—41no—
298C4×C59—41no—
299C4×C59—41no—
300C4×C59—41no—
301C4×C59—81no—
302C4×C59—81no—
303C4×C59—81no—
304C4×C59—81no—
305C4×C59#52911yescharacteristic
306C4×C59#53911yescharacteristic
307C4×C59#54611yescharacteristic
308C4×C59#54711yescharacteristic
309Dic59—42no—
310Dic59—42no—
311Dic59—42no—
312Dic59—81no—
313C2×#484—591no—
314C2×#470—591no—
315#2503D5911yescharacteristic
316#2362—591no—
317#2085—591no—
318#2074—591no—
319#2068—591no—
320#2056—591no—
321#1949—591no—
322#1928—591no—
323#1878—591no—
324#1873—591no—
325#1851—591no—
326#1805—591no—
327#1736—591no—
328C2×C59—42no—
329C2×C59#242511yescharacteristic
330C2×C59#304011yescharacteristic
331C2×C59#304411yescharacteristic
332C2×C59#331511yescharacteristic
333C2×C59#332711yescharacteristic
334C2×C59#334911yescharacteristic
335C2×C59#335211yescharacteristic
336C22×Q16—591no—
337C22×M16—591no—
338C2×C22⋊Q8—591no—
339C2×C22⋊Q8C2×D5911yescharacteristic
340C2×C4×Q8—591no—
341C2×#117—591no—
342C22×C4⋊C4C2×D5911yescharacteristic
343#484—1184no—
344#470—1184no—
345C2×#107—591no—
346C2×#107—591no—
347C2×#106—591no—
348C2×C4⋊C8—591no—
349C2×C4⋊C8—591no—
350C2×#103—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×C22⋊C8—591no—
357#400—21no—
358#400—21no—
359#391—591no—
360#391C2×D5911yescharacteristic
361#391C2×D5911yescharacteristic
362#389—591no—
363#388—1181no—
364#388—1181no—
365#387—1181no—
366#382C2×D5911yescharacteristic
367#382C2×D5911yescharacteristic
368#381—21no—
369#381—21no—
370C4×C4⋊C4—591no—
371C4×C4⋊C4—1181no—
372C4×C22⋊C4C2×D5911yescharacteristic
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#340—591no—
379#336—591no—
380#336—591no—
381C59#2522911yescharacteristic, complemented, Sylow
382C22×Q8—591no—
383C22×Q8C22×D5911yescharacteristic
384C23×C4—21no—
385C23×C4C22×D5911yescharacteristic
386C2×Q16—1182no—
387C2×Q16—1182no—
388C2×Q16—2361no—
389C2×M16—1182no—
390C2×M16—1182no—
391C2×M16—2361no—
392C22×C8—591no—
393C22×C8—591no—
394C22×C8—591no—
395C22×C8—591no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—594no—
399C22⋊Q8—1182no—
400C4×Q8—1182no—
401C4×Q8—2361no—
402#117—1182no—
403#117—2361no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4C22⋊D5911yescharacteristic
421C2×C22⋊C4C22⋊D5911yescharacteristic
422C2×C42—1181no—
423C2×C42—1181no—
424C2×C42—1181no—
425C2×C42C22×D5911yescharacteristic
426C2×C42C22×D5911yescharacteristic
427#107—1182no—
428#107—1182no—
429#106—1182no—
430C4⋊C8—1182no—
431C4⋊C8—2361no—
432#103—1182no—
433#103—1182no—
434#103—1182no—
435#103—1182no—
436#103—1182no—
437#103—2361no—
438C22⋊C8—1182no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95C22⋊D5911yescharacteristic
452#95C22⋊D5911yescharacteristic
453C24C2×C22⋊D5911yescharacteristic
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—592no—
461C2×Q8—1181no—
462C2×Q8—1181no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4D4×D5912yes—
482C22×C4#1174568712yes—
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C23×D5911yescharacteristic
489Q16—2364no—
490M16—2364no—
491C2×C8—592no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—592no—
512C4⋊C4—592no—
513C4⋊C4—1184no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—2362no—
519C4⋊C4—2362no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—42no—
523C22⋊C4—1184no—
524C22⋊C4—2362no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—2362no—
530C42—2362no—
531C42—2362no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23#1427369812yes—
539C23C22×C22⋊D5911yescharacteristic
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—1182no—
545Q8—1182no—
546Q8—2361no—
547Q8—2361no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
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—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
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—84no—
579C2×C4—1182no—
580C2×C4—1182no—
581C2×C4—1182no—
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—2361no—
591C2×C4#1427369812yes—
592C2×C4C2×D4×D5911yescharacteristic
593C2×C4#41×D5911yescharacteristic
594C2×C4#41×D5911yescharacteristic
595C2×C4C2×#1174568711yescharacteristic
596C2×C4#1427377311yescharacteristic
597C2×C4#1427377411yescharacteristic
598C8—1182no—
599C8—2361no—
600C8—2361no—
601C8—2361no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22C2×#1427369811yescharacteristic
610C22#1683804011yescharacteristic
611C22#1683805111yescharacteristic
612C22#1683820011yescharacteristic
613C22#1683820011yescharacteristic
614C22#1683821711yescharacteristic
615C22#1683821711yescharacteristic
616C4—22no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—4721no—
626C4—2362no—
627C4—2362no—
628C4—2362no—
629C4#1683792911yescharacteristic
630C4#1683799611yescharacteristic
631C4#1683804411yescharacteristic
632C4#1683805011yescharacteristic
633C2—42no—
634C2#2142764311yescharacteristic
635C2#2143358611yescharacteristic
636C2#2143362211yescharacteristic
637C2#2143618611yescharacteristic
638C2#2143635811yescharacteristic
639C2#2143662611yescharacteristic
640C2#2143666511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2503C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2503D59
hypocenterC59#25229
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
4. #21425578C2semidirect
5. #21425515C2semidirect
6. #21425491C2semidirect
7. #21425418C2semidirect
8. #21424862C2semidirect
9. #21424734C2semidirect
10. #21424440C2semidirect
13. #21424108C2semidirect
381. C59#25229semidirect

Maximal subgroups

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

subgroupindex
C2×#168377372
C2×#168376582
#214271212
#214255782
#214255152
#214254912
#214254182
#214248622
#214247342
#214244402
#214244092
#214242812
#214241082
#214238152
C59×#25032
#2522959