Finite Group Databasebeta

← #28596688

Subgroups — #28596688

Order 15104.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426838C211yesmaximal, characteristic, complemented
2#21426809C211yesmaximal, characteristic
3#21425066C211yesmaximal, characteristic, complemented
4#21424913C211yesmaximal, characteristic
5#21424795C211yesmaximal, characteristic
6#21424788C211yesmaximal, characteristic, complemented
7#21424781C211yesmaximal, characteristic
8#21424774C211yesmaximal, characteristic, complemented
9#21424224C211yesmaximal, characteristic, complemented
10#21423928C211yesmaximal, characteristic, complemented
11#21423923C211yesmaximal, characteristic
12#103×Dic59C211yesmaximal, characteristic, complemented
13#102×Dic59C211yesmaximal, characteristic
14#21423782C211yesmaximal, characteristic, complemented
15C2×C59×#488C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C2×#14273651C2211yescharacteristic
19C2×#14273651C2211yescharacteristic
20C2×#14273648C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273638C2211yescharacteristic
23C2×#14273637C2211yescharacteristic
24C2×#14273637C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27#16837309C2211yescharacteristic
28#16837305C2211yescharacteristic
29#16837301—21no—
30#16837301—21no—
31#16837301C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837295—21no—
34#16837295—21no—
35#16837288—21no—
36#16837288—21no—
37#16837277—21no—
38#16837277—21no—
39#16837272C2211yescharacteristic
40#16837272C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43#16837248C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52C2×#125×C59C2211yescharacteristic
53C2×#124×C59C2211yescharacteristic
54C59×#488C418yescomplemented
55C2×#103×C59C2211yescharacteristic
56C2×#103×C59C2211yescharacteristic
57C2×#102×C59C2211yescharacteristic
58C2×#102×C59C2211yescharacteristic
59C2×C4×C8×C59C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63Q8×Dic59—22no—
64Q8×Dic59—22no—
65D4×Dic59—22no—
66D4×Dic59—22no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59C2311yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86#14273651—22no—
87#14273651—22no—
88#14273648—22no—
89#14273648—22no—
90#14273638—22no—
91#14273637—22no—
92#14273637—22no—
93C8×Dic59—22no—
94C8×Dic59—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95D411yescharacteristic
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C59⋊#95D411yescharacteristic
105C22×Q8×C59C2311yescharacteristic
106C22×D4×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109#125×C59—22no—
110#125×C59C2×C414yescomplemented
111#124×C59—22no—
112#124×C59C2×C414yes—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C42×C59C2311yescharacteristic
120#103×C59C2×C414yescomplemented
121#103×C59C2×C414yescomplemented
122#102×C59C2×C414yes—
123#102×C59C2×C414yes—
124C4×C8×C59C2×C414yescomplemented
125C24×C59—21no—
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59C22×C412yes—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—42no—
149C22×Dic59—81no—
150C22×Dic59—81no—
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59C22×C412yes—
160C2×C8×C59C22×C412yes—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59C22×C412yes—
166C4⋊C4×C59C22×C412yes—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—24no—
169C42×C59C22×C412yes—
170C42×C59#4112yes—
171C22⋊Dic59—42no—
172C22⋊Dic59—81no—
173C22⋊Dic59—81no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—42no—
183C4⋊Dic59—81no—
184C4⋊Dic59—81no—
185C4⋊Dic59—81no—
186C4⋊Dic59—81no—
187C4⋊Dic59—81no—
188C4⋊Dic59—81no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—42no—
192C4×Dic59—81no—
193C4×Dic59—81no—
194C4×Dic59#4112yes—
195C4×Dic59#4112yes—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8#4112yes—
201C2×C59⋊C8#4112yes—
202C23×C59—22no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59C22×D411yescharacteristic
210Q8×C59—22no—
211Q8×C59—22no—
212Q8×C59—41no—
213Q8×C59—41no—
214D4×C59—22no—
215D4×C59—22no—
216D4×C59—41no—
217D4×C59—41no—
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—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59C4×D412yes—
232C2×C4×C59C4×D412yes—
233C2×C4×C59C23×C411yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
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—84no—
262C2×Dic59#12412yes—
263C2×Dic59#12412yes—
264C8×C59—22no—
265C8×C59—22no—
266C59⋊C8—22no—
267C59⋊C8—22no—
268#21643—591nomaximal, Sylow
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—81no—
273C22×C59—81no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59C2×C4×D411yescharacteristic
277C22×C59C2×#12411yescharacteristic
278C22×C59#53211yescharacteristic
279C22×C59#57611yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57811yescharacteristic
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—42no—
286C4×C59—42no—
287C4×C59—42no—
288C4×C59C4×#4111yescharacteristic
289C4×C59C4×#4111yescharacteristic
290C4×C59#52911yescharacteristic
291C4×C59#52911yescharacteristic
292Dic59—22no—
293Dic59—22no—
294Dic59—41no—
295Dic59—41no—
296Dic59—81no—
297Dic59—81no—
298Dic59—81no—
299Dic59—81no—
300C2×#488D5911yescharacteristic
301#2330—591no—
302#2326—591no—
303#1992—591no—
304#1959—591no—
305#1936—591no—
306#1935—591no—
307#1934—591no—
308#1933—591no—
309#1834—591no—
310#1763—591no—
311#1762—591no—
312C4×#103—591no—
313C4×#102—591no—
314#1726—591no—
315C2×C59—82no—
316C2×C59C4×#12411yescharacteristic
317C2×C59#295111yescharacteristic
318C2×C59#295211yescharacteristic
319C2×C59#313611yescharacteristic
320C2×C59#313711yescharacteristic
321C2×C59#317811yescharacteristic
322C2×C59#317911yescharacteristic
323C2×#125C2×D5911yescharacteristic
324C2×#124C2×D5911yescharacteristic
325C2×C4×Q8—591no—
326C2×C4×D4—591no—
327#488Dic5918yescomplemented
328C2×#103—591no—
329C2×#103—591no—
330C2×#103C2×D5911yescharacteristic
331C2×#103C2×D5911yescharacteristic
332C2×#102—591no—
333C2×#102—591no—
334C2×#102C2×D5911yescharacteristic
335C2×#102C2×D5911yescharacteristic
336C2×#97—591no—
337C2×#97—591no—
338C2×#97—591no—
339C2×C4×C8—591no—
340C2×C4×C8—591no—
341C2×C4×C8C2×D5911yescharacteristic
342#391—591no—
343#390—591no—
344#389—591no—
345#389—591no—
346#389—1181no—
347#389—1181no—
348#388—1181no—
349#388—1181no—
350#387—1181no—
351#387—1181no—
352#385—1181no—
353#385—1181no—
354#384—591no—
355#384—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—591no—
358#376—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367C59#2164311yescharacteristic, complemented, Sylow
368C22×Q8C22×D5911yescharacteristic
369C22×D4C22×D5911yescharacteristic
370C23×C4—1181no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375#125—22no—
376#125C2×Dic5914yescomplemented
377#124—22no—
378#124C2×Dic5914yes—
379C4×Q8—1182no—
380C4×Q8—1182no—
381C4×D4—1182no—
382C4×D4—1182no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C42—591no—
404C2×C42—591no—
405C2×C42—1181no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42C22×D5911yescharacteristic
409#103—1182no—
410#103—1182no—
411#103C2×Dic5914yescomplemented
412#103C2×Dic5914yescomplemented
413#102—1182no—
414#102—1182no—
415#102C2×Dic5914yes—
416#102C2×Dic5914yes—
417#97—1182no—
418#97—1182no—
419#97—1182no—
420C4×C8—1182no—
421C4×C8—1182no—
422C4×C8C2×Dic5914yescomplemented
423#95—591no—
424#95—591no—
425#95—591no—
426#95—591no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433C24—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C22×Dic5912yes—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C22×Dic5912yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—591no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—4721no—
458C22×C4—4721no—
459C22×C4—2362no—
460C22×C4D4×D5911yescharacteristic
461C22×C4D4×D5911yescharacteristic
462C22×C4C23×D5911yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—592no—
468C2×C8—592no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8C22×Dic5912yes—
474C2×C8C22×Dic5912yes—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—4721no—
488C4⋊C4—4721no—
489C4⋊C4—4721no—
490C4⋊C4—4721no—
491C4⋊C4—4721no—
492C4⋊C4—4721no—
493C4⋊C4—2362no—
494C4⋊C4C22×Dic5912yes—
495C4⋊C4C22×Dic5912yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—4721no—
499C22⋊C4—4721no—
500C22⋊C4—2362no—
501C42—592no—
502C42—592no—
503C42—1181no—
504C42—1181no—
505C42—4721no—
506C42—4721no—
507C42—2362no—
508C42#1174568712yes—
509C42C22×Dic5912yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D5911yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—592no—
540C2×C4—592no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—4724no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4D4×Dic5912yes—
565C2×C4D4×Dic5912yes—
566C2×C4C23×Dic5911yescharacteristic
567C2×C4#41×D5911yescharacteristic
568C2×C4#41×D5911yescharacteristic
569C2×C4#41×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4C2×#1174568711yescharacteristic
572C8—22no—
573C8—22no—
574C8—1182no—
575C8—1182no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C2×D4×Dic5911yescharacteristic
584C22#124×D5911yescharacteristic
585C22#1683795111yescharacteristic
586C22#1683820411yescharacteristic
587C22#1683820911yescharacteristic
588C22#1683820911yescharacteristic
589C22#1683821611yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—1182no—
596C4—1182no—
597C4—4721no—
598C4—4721no—
599C4—4721no—
600C4—4721no—
601C4—2361no—
602C4—2361no—
603C4#41×Dic5911yescharacteristic
604C4#41×Dic5911yescharacteristic
605C4#1683793111yescharacteristic
606C4#1683793111yescharacteristic
607C2—82no—
608C2#124×Dic5911yescharacteristic
609C2#2143297211yescharacteristic
610C2#2143298011yescharacteristic
611C2#2143444911yescharacteristic
612C2#2143446011yescharacteristic
613C2#2143476311yescharacteristic
614C2#2143477811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#488C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC2×#488D59
hypocenterC59#21643
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#488

Complements

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

NH ≅ G/Ntype
1. #21426838C2semidirect
3. #21425066C2semidirect
6. #21424788C2semidirect
8. #21424774C2semidirect
9. #21424224C2semidirect
10. #21423928C2semidirect
12. #103×Dic59C2semidirect
14. #21423782C2semidirect
54. C59×#488C4semidirect
110. #125×C59C2×C4semidirect
120. #103×C59C2×C4semidirect
121. #103×C59C2×C4semidirect
124. C4×C8×C59C2×C4semidirect
327. #488Dic59semidirect
367. C59#21643semidirect
376. #125C2×Dic59semidirect
411. #103C2×Dic59semidirect
412. #103C2×Dic59semidirect
422. C4×C8C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214268382
#214268092
#214250662
#214249132
#214247952
#214247882
#214247812
#214247742
#214242242
#214239282
#214239232
#103×Dic592
#102×Dic592
#214237822
C2×C59×#4882
#2164359