Finite Group Databasebeta

← #28595860

Subgroups — #28595860

Order 15104.

888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837740C211yesmaximal, characteristic
2C22⋊Q8×Dic59C211yesmaximal, characteristic
3#117×Dic59C211yesmaximal, characteristic
4#21424740C211yesmaximal, characteristic, complemented
5#21424733C211yesmaximal, characteristic, complemented
6#21424677C211yesmaximal, characteristic, complemented
7#21424670C211yesmaximal, characteristic, complemented
8#21424295C211yesmaximal, characteristic
9#21424192C211yesmaximal, characteristic
10#21423988C211yesmaximal, characteristic, complemented
11#21423984C211yesmaximal, characteristic, complemented
12#103×Dic59C211yesmaximal, characteristic, complemented
13#103×Dic59C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic59C211yesmaximal, characteristic
15C59×#1873C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C22×C4×Dic59C2211yescharacteristic
19#16837740C418yescomplemented
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×C22⋊(C59⋊C8)C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837309C2211yescharacteristic
30#16837301C2211yescharacteristic
31#16837288—21no—
32#16837277—21no—
33#16837262C2211yescharacteristic
34C4⋊C4×Dic59—21no—
35C4⋊C4×Dic59C2211yescharacteristic
36C4⋊C4×Dic59C2211yescharacteristic
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C22⋊C4×Dic59—21no—
41C22⋊C4×Dic59—21no—
42C22⋊C4×Dic59—21no—
43#16837248—21no—
44C42×Dic59—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C22⋊Q8×C59C2211yescharacteristic
54C2×#117×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#103×C59C2211yescharacteristic
57C2×C22⋊C8×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C23×Dic59D411yescharacteristic
61C23×Dic59D411yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64Q8×Dic59—22no—
65Q8×Dic59—22no—
66#14273686—22no—
67#14273686C2×C414yes—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C22⋊Dic59—21no—
75C2×C22⋊Dic59—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—22no—
78C2×C4×Dic59—22no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59D411yescharacteristic
86C2×C4×Dic59D411yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89#14273660—22no—
90#14273660C2×C414yescomplemented
91#14273658—22no—
92#14273658C2×C414yescomplemented
93#14273651C2×C414yescomplemented
94#14273651C2×C414yescomplemented
95C22⋊(C59⋊C8)C2×C414yes—
96C8×Dic59—22no—
97C8×Dic59—22no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C22×Q8×C59C2311yescharacteristic
108C23×C4×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C22⋊Q8×C59—22no—
112C22⋊Q8×C59C2×C414yes—
113#117×C59—22no—
114#117×C59—22no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C42×C59—21no—
122#103×C59—22no—
123#103×C59—22no—
124C22⋊C8×C59—22no—
125C24×C59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59C22×C412yes—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59#4112yes—
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59—22no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—41no—
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156C22×Dic59C2×D411yescharacteristic
157C22×Dic59C2×D411yescharacteristic
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59C22×C412yes—
169C4⋊C4×C59#4112yes—
170C4⋊C4×C59#4112yes—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—24no—
173C42×C59—24no—
174C22⋊Dic59—24no—
175C22⋊Dic59—42no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—42no—
183C4⋊Dic59—42no—
184C4⋊Dic59C22×C412yes—
185C4⋊Dic59C22×C412yes—
186C4×Dic59—24no—
187C4×Dic59—21no—
188C4×Dic59—21no—
189C4×Dic59—22no—
190C4×Dic59—22no—
191C4×Dic59—42no—
192C4×Dic59—42no—
193C4×Dic59—42no—
194C4×Dic59—42no—
195C4×Dic59—42no—
196C4×Dic59—42no—
197C4×Dic59#4112yes—
198C4×Dic59#4112yes—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8C22×C412yes—
204C2×C59⋊C8C22×C412yes—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59C4×D412yes—
212C23×C59C22×D411yescharacteristic
213Q8×C59—22no—
214Q8×C59—22no—
215Q8×C59—41no—
216Q8×C59—41no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59C4×D412yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—22no—
247C2×Dic59—22no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—44no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—44no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59#12312yes—
264C2×Dic59#12312yes—
265C2×Dic59#13512yes—
266C2×Dic59#13512yes—
267C2×Dic59#13712yes—
268C8×C59—41no—
269C8×C59—41no—
270C59⋊C8—22no—
271C59⋊C8—22no—
272C4×#485—591nomaximal, Sylow
273C22×C59—22no—
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59C2×C4×D411yescharacteristic
281C22×C59C2×#12311yescharacteristic
282C22×C59#52511yescharacteristic
283C22×C59C2×#13511yescharacteristic
284C22×C59C2×#13511yescharacteristic
285C22×C59C2×#13711yescharacteristic
286C22×C59#57511yescharacteristic
287C4×C59—22no—
288C4×C59—42no—
289C4×C59—42no—
290C4×C59—42no—
291C4×C59—42no—
292C4×C59C4×#4111yescharacteristic
293C4×C59C4×#4111yescharacteristic
294C4×C59#52911yescharacteristic
295C4×C59#52911yescharacteristic
296Dic59—22no—
297Dic59—22no—
298Dic59—41no—
299Dic59—42no—
300Dic59—42no—
301Dic59—81no—
302Dic59—81no—
303Dic59#48514yescomplemented
304C2×#485—591no—
305C4×C22⋊Q8—591no—
306C4×#117—591no—
307#1928—591no—
308#1927—591no—
309#1919—591no—
310#1918—591no—
311#1873D5911yescharacteristic
312#1854—591no—
313#1825—591no—
314C4×#107—591no—
315C4×#106—591no—
316C4×#103—591no—
317C4×#103—591no—
318C4×C22⋊C8—591no—
319C2×C59—42no—
320C2×C59C4×#12311yescharacteristic
321C2×C59C4×#13511yescharacteristic
322C2×C59C4×#13711yescharacteristic
323C2×C59C2×#48511yescharacteristic
324C2×C59#309111yescharacteristic
325C2×C59#310211yescharacteristic
326C2×C59#310711yescharacteristic
327C2×C22⋊Q8C2×D5911yescharacteristic
328C2×C4×Q8—591no—
329C2×#117—591no—
330C2×#117C2×D5911yescharacteristic
331C22×C42—591no—
332#485—598no—
333C2×#107—591no—
334C2×#107—591no—
335C2×#106—591no—
336C2×#106—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#103C2×D5911yescharacteristic
340C2×#103C2×D5911yescharacteristic
341C2×C22⋊C8—591no—
342C2×C22⋊C8C2×D5911yescharacteristic
343C2×C4×C8—591no—
344C2×C4×C8—591no—
345#391—591no—
346#389—591no—
347#387—1181no—
348#385—1181no—
349#380—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—1181no—
356C4×C22⋊C4—1181no—
357C4×C22⋊C4—1181no—
358C4×C22⋊C4—1181no—
359#376—1181no—
360C43—1181no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340C2×D5911yescharacteristic
368#340C2×D5911yescharacteristic
369#336—591no—
370#336—591no—
371C59C4×#48511yescharacteristic, complemented, Sylow
372C22×Q8C22×D5911yescharacteristic
373C23×C4—591no—
374C23×C4—591no—
375C23×C4C22×D5911yescharacteristic
376C22×C8—591no—
377C22×C8—591no—
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C22⋊Q8—22no—
381C22⋊Q8C4×D5914yes—
382C4×Q8—1182no—
383C4×Q8—1182no—
384#117—22no—
385#117—22no—
386#117—594no—
387#117—1182no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4C22×D5911yescharacteristic
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×C42—21no—
403C2×C42—591no—
404C2×C42—591no—
405C2×C42—591no—
406C2×C42—591no—
407C2×C42—1182no—
408C2×C42—1182no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42—1181no—
414C2×C42—1181no—
415C2×C42—1181no—
416#107—594no—
417#107—1182no—
418#106—594no—
419#106—1182no—
420#103—22no—
421#103—22no—
422#103—594no—
423#103—594no—
424C22⋊C8—22no—
425C22⋊C8—594no—
426C4×C8—1182no—
427C4×C8—1182no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437C24D4×D5911yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D5912yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—591no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—1182no—
457C22×C4—1182no—
458C22×C4—1182no—
459C22×C4—1182no—
460C22×C4—1182no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—2361no—
466C22×C4C2×C4×D5912yes—
467C22×C4D4×D5911yescharacteristic
468C22×C4#1174568712yes—
469C22×C4C23×D5911yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—592no—
477C2×C8—592no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—592no—
487C4⋊C4—592no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C4⋊C4C2×C4×D5912yes—
497C4⋊C4#1174568712yes—
498C4⋊C4#1174568712yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—1184no—
502C22⋊C4—2362no—
503C42—24no—
504C42—592no—
505C42—592no—
506C42—1182no—
507C42—1182no—
508C42—1184no—
509C42—1181no—
510C42—1181no—
511C42—2362no—
512C42—2362no—
513C42—2362no—
514C42—2362no—
515C42—2362no—
516C42—2362no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23(C22⋊D59)⋊C412yes—
524C23C2×D4×D5911yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—592no—
548C2×C4—592no—
549C2×C4—592no—
550C2×C4—592no—
551C2×C4—592no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—1182no—
560C2×C4—1182no—
561C2×C4—2364no—
562C2×C4—2364no—
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(C22⋊D59)⋊C412yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4#41×D5911yescharacteristic
577C2×C4C2×#1174568711yescharacteristic
578C2×C4C2×#1174568711yescharacteristic
579C2×C4C2×#1174568711yescharacteristic
580C8—41no—
581C8—41no—
582C8—1182no—
583C8—1182no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×(C22⋊D59)⋊C411yescharacteristic
592C22#123×D5911yescharacteristic
593C22#1683790811yescharacteristic
594C22#135×D5911yescharacteristic
595C22#135×D5911yescharacteristic
596C22#137×D5911yescharacteristic
597C22#1683820111yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—594no—
604C4—1182no—
605C4—1182no—
606C4—4721no—
607C4—4721no—
608C4—2362no—
609C4—2362no—
610C4—2361no—
611C4C4×#1174568711yescharacteristic
612C4C4×#1174568711yescharacteristic
613C4#1683793311yescharacteristic
614C4#1683793311yescharacteristic
615C2—42no—
616C2#2142662111yescharacteristic
617C2#2143286411yescharacteristic
618C2#2143291111yescharacteristic
619C2#485×D5911yescharacteristic
620C2#2143401411yescharacteristic
621C2#2143411511yescharacteristic
622C2#2143416011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1873C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1873D59
hypocenterC59C4×#485
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
4. #21424740C2semidirect
5. #21424733C2semidirect
6. #21424677C2semidirect
7. #21424670C2semidirect
10. #21423988C2semidirect
11. #21423984C2semidirect
12. #103×Dic59C2semidirect
13. #103×Dic59C2semidirect
19. #16837740C4semidirect
90. #14273660C2×C4semidirect
92. #14273658C2×C4semidirect
93. #14273651C2×C4semidirect
94. #14273651C2×C4semidirect
303. Dic59#485semidirect
371. C59C4×#485semidirect

Maximal subgroups

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

subgroupindex
C2×#168377402
C22⋊Q8×Dic592
#117×Dic592
#214247402
#214247332
#214246772
#214246702
#214242952
#214241922
#214239882
#214239842
#103×Dic592
C22⋊C8×Dic592
C59×#18732
C4×#48559