Finite Group Databasebeta

← #28610557

Subgroups — #28610557

Order 15104.

914 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837680C211yesmaximal, characteristic
2#21427149C211yesmaximal, characteristic, complemented
3D4×Dic59⋊C4C211yesmaximal, characteristic
4#21425055C211yesmaximal, characteristic, complemented
5#21424856C211yesmaximal, characteristic, complemented
6#21424832C211yesmaximal, characteristic, complemented
7#21424831C211yesmaximal, characteristic, complemented
8#21424689C211yesmaximal, characteristic, complemented
9#21424686C211yesmaximal, characteristic, complemented
10#21424355C211yesmaximal, characteristic
11#21423962C211yesmaximal, characteristic, complemented
12#21423923C211yesmaximal, characteristic
13#21423921C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#1864C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×#14273692—21no—
18C2×D4×Dic59C2211yescharacteristic
19C2×D4×Dic59C2211yescharacteristic
20C22×Dic59⋊C4—21no—
21#16837680C418yescomplemented
22C2×#14273661C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×Dic59⋊C8C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31#16837306—21no—
32#16837300C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837284—21no—
36#16837283—21no—
37#16837275—21no—
38#16837274—21no—
39#16837271C2211yescharacteristic
40#16837260—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4×Dic59⋊C4C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C23×Dic59—21no—
62C23×Dic59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273726—22no—
66#14273726C2×C414yescomplemented
67#14273692—44no—
68D4×Dic59—22no—
69D4×Dic59—22no—
70D4×Dic59—41no—
71D4×Dic59C2×C414yes—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—44no—
83C2×Dic59⋊C4D411yescharacteristic
84C2×Dic59⋊C4D411yescharacteristic
85C2×C22⋊Dic59—21no—
86C2×C22⋊Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59C2311yescharacteristic
91C2×C4×Dic59C2311yescharacteristic
92#14273661C2×C414yescomplemented
93#14273660C2×C414yescomplemented
94#14273658—41no—
95Dic59⋊C8C2×C414yescomplemented
96#14273648C2×C414yes—
97#14273648Q814yescomplemented
98#14273638—41no—
99#14273637—41no—
100C8×Dic59—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C22×D4×C59C2311yescharacteristic
110C23×C4×C59—21no—
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C4×D4×C59—22no—
114C4×D4×C59—41no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#106×C59—41no—
122#102×C59—22no—
123#102×C59C2×C414yes—
124#95×C59—21no—
125C24×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59C22×C412yes—
130C2×D4×C59C2×Q812yes—
131C2×D4×C59#4112yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—42no—
136C22×C4×C59—42no—
137C22×C4×C59—81no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—24no—
145C22×Dic59—21no—
146C22×Dic59—42no—
147C22×Dic59—42no—
148C22×Dic59—42no—
149C22×Dic59—42no—
150C22×Dic59—81no—
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C2×Dic118—24no—
156C2×Dic118—42no—
157C2×Dic118—42no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C22×C412yes—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—42no—
170C4⋊C4×C59C22×C412yes—
171C22⋊C4×C59—42no—
172C42×C59—21no—
173C42×C59—21no—
174C22⋊Dic59—24no—
175C22⋊Dic59—42no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C22×C412yes—
183C4⋊Dic59C2×Q812yes—
184Dic59⋊C4—24no—
185Dic59⋊C4—22no—
186Dic59⋊C4—22no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191Dic59⋊C4—81no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—42no—
195C4×Dic59—42no—
196C4×Dic59—42no—
197C4×Dic59C22×C412yes—
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⋊C8C2×Q812yes—
205C23×C59—22no—
206C23×C59—22no—
207C23×C59—22no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59C22×D411yescharacteristic
213D4×C59—22no—
214D4×C59—22no—
215D4×C59—22no—
216D4×C59C4×Q814yescomplemented
217C2×C4×C59—22no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—41no—
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×C59—41no—
232C2×C4×C59—84no—
233C2×C4×C59C23×C411yescharacteristic
234C2×C4×C59C22×Q811yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59H(5,𝔽2)11yescharacteristic
238C2×C4×C592−1+411yescharacteristic
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—22no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—48no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—84no—
261C2×Dic59C4×D412yes—
262C2×Dic59C4×D412yes—
263C2×Dic59AGL(1,ℤ/8)12yes—
264C2×Dic59AGL(1,ℤ/8)12yes—
265Dic118—48no—
266C8×C59—22no—
267C8×C59—41no—
268C59⋊C8—22no—
269C59⋊C8—22no—
270#22968—591nomaximal, Sylow
271C22×C59—24no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59C2×C4×D411yescharacteristic
279C22×C59#54611yescharacteristic
280C22×C59D4×Q811yescharacteristic
281C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C59#57711yescharacteristic
284C22×C59#57811yescharacteristic
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—41no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59C2×C4×Q811yescharacteristic
291C4×C59#52011yescharacteristic
292C4×C59#55211yescharacteristic
293C4×C59#55411yescharacteristic
294Dic59—22no—
295Dic59—41no—
296Dic59—42no—
297Dic59—42no—
298Dic59—42no—
299Dic59—41no—
300Dic59—81no—
301Dic59#44214yes—
302C2×#474—591no—
303#2365—591no—
304D4×C4⋊C4—591no—
305#1990—591no—
306#1948—591no—
307#1943—591no—
308#1943—591no—
309#1921—591no—
310#1921—591no—
311#1864—591no—
312#1864D5911yescharacteristic
313#1770—591no—
314#1762—591no—
315#1762—591no—
316C4×#102—591no—
317C2×C59—44no—
318C2×C59#236011yescharacteristic
319C2×C59C2×#44211yescharacteristic
320C2×C59#294711yescharacteristic
321C2×C59#332411yescharacteristic
322C2×C59#333011yescharacteristic
323C2×C59#338411yescharacteristic
324C2×C59#338611yescharacteristic
325C2×C4⋊Q8—591no—
326C2×C4×Q8—1181no—
327C2×C4×D4—591no—
328C2×C4×D4—591no—
329C2×C4×D4C2×D5911yescharacteristic
330C22×C4⋊C4—1181no—
331#474—598no—
332C2×#107—591no—
333C2×#107—591no—
334C2×#106—591no—
335C2×#106C2×D5911yescharacteristic
336C2×C4⋊C8—591no—
337C2×#102—591no—
338C2×#102—591no—
339C2×#102C2×D5911yescharacteristic
340C2×#102C2×D5911yescharacteristic
341C2×#97—591no—
342C2×#97—591no—
343C2×C4×C8—591no—
344#391—1181no—
345#389—591no—
346#389—591no—
347#389—591no—
348#389C2×D5911yescharacteristic
349#387—1181no—
350#387—1181no—
351#385—1181no—
352#385—1181no—
353#384—591no—
354#380—1181no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340C2×D5911yescharacteristic
367#336—591no—
368#336C2×D5911yescharacteristic
369C59#2296811yescharacteristic, complemented, Sylow
370C22×Q8—1181no—
371C22×D4C22×D5911yescharacteristic
372C23×C4—21no—
373C23×C4—1181no—
374C23×C4—1181no—
375C22×C8—591no—
376C22×C8—591no—
377C22×C8C22×D5911yescharacteristic
378C22×C8C22×D5911yescharacteristic
379C4⋊Q8—594no—
380C4⋊Q8—1182no—
381C4×Q8—2364no—
382C4×D4—22no—
383C4×D4—41no—
384C4×D4—594no—
385C4×D4—1182no—
386C4×D4—1182no—
387C4×D4—2361no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—2364no—
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C42—591no—
409C2×C42—591no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42C22×D5911yescharacteristic
414#107—594no—
415#107—594no—
416#106—41no—
417#106—2361no—
418C4⋊C8—594no—
419#102—22no—
420#102—594no—
421#102—594no—
422#102C4×D5914yes—
423#97—2361no—
424#97—2361no—
425C4×C8—1182no—
426#95—21no—
427#95—591no—
428#95—591no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435C24—21no—
436C2×Q8—1184no—
437C2×Q8—2362no—
438C2×Q8—2362no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D5912yes—
443C2×D4#1174568712yes—
444C2×D4Q8×D5912yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—1184no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—4721no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4—2362no—
464C22×C4—2362no—
465C22×C4D4×D5911yescharacteristic
466C22×C4D4×D5911yescharacteristic
467C22×C4C23×D5911yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—592no—
474C2×C8—592no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8C2×C4×D5912yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—592no—
487C4⋊C4—592no—
488C4⋊C4—592no—
489C4⋊C4—1182no—
490C4⋊C4—1182no—
491C4⋊C4—1184no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—4721no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4C2×C4×D5912yes—
503C22⋊C4—42no—
504C22⋊C4—1184no—
505C22⋊C4—2362no—
506C42—21no—
507C42—21no—
508C42—592no—
509C42—592no—
510C42—1181no—
511C42—1181no—
512C42—2362no—
513C42—2362no—
514C42—2362no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×D4×D5911yescharacteristic
523Q8—2368no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4#1427369214yescomplemented
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—592no—
545C2×C4—592no—
546C2×C4—592no—
547C2×C4—592no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1182no—
557C2×C4—1182no—
558C2×C4—1182no—
559C2×C4—4724no—
560C2×C4—2368no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4C22×C4×D5911yescharacteristic
571C2×C4C2×Q8×D5911yescharacteristic
572C2×C4#41×D5911yescharacteristic
573C2×C4C2×#1174568711yescharacteristic
574C2×C4#1427377211yescharacteristic
575C2×C4#1427377411yescharacteristic
576C8—22no—
577C8—41no—
578C8—1182no—
579C8—1182no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22C4×D4×D5911yescharacteristic
588C22#1683803911yescharacteristic
589C22#1683806511yescharacteristic
590C22AGL(1,ℤ/8)×D5911yescharacteristic
591C22AGL(1,ℤ/8)×D5911yescharacteristic
592C22#1683820811yescharacteristic
593C22#1683821611yescharacteristic
594C4—42no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—594no—
600C4—1182no—
601C4—4721no—
602C4—2362no—
603C4—2362no—
604C4—2362no—
605C4—2361no—
606C4—2361no—
607C4C2×#1427369211yescharacteristic
608C4#1683788111yescharacteristic
609C4#1683808911yescharacteristic
610C4#1683810611yescharacteristic
611C2—44no—
612C2#2142709611yescharacteristic
613C2#442×D5911yescharacteristic
614C2#2143293211yescharacteristic
615C2#2143632011yescharacteristic
616C2#2143639411yescharacteristic
617C2#2143699011yescharacteristic
618C2#2143701211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1864C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1864D59
hypocenterC59#22968
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #21427149C2semidirect
4. #21425055C2semidirect
5. #21424856C2semidirect
6. #21424832C2semidirect
7. #21424831C2semidirect
8. #21424689C2semidirect
9. #21424686C2semidirect
11. #21423962C2semidirect
21. #16837680C4semidirect
66. #14273726C2×C4semidirect
92. #14273661C2×C4semidirect
93. #14273660C2×C4semidirect
95. Dic59⋊C8C2×C4semidirect
97. #14273648Q8semidirect
216. D4×C59C4×Q8semidirect
369. C59#22968semidirect
527. D4#14273692semidirect

Maximal subgroups

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

subgroupindex
C2×#168376802
#214271492
D4×Dic59⋊C42
#214250552
#214248562
#214248322
#214248312
#214246892
#214246862
#214243552
#214239622
#214239232
#214239212
#102×Dic592
C59×#18642
#2296859