Finite Group Databasebeta

← #28611777

Subgroups — #28611777

Order 15104.

854 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837680C211yesmaximal, characteristic
2#21427155C211yesmaximal, characteristic, complemented
3#21427044C211yesmaximal, characteristic
4#21425055C211yesmaximal, characteristic, complemented
5#21424851C211yesmaximal, characteristic, complemented
6#21424782C211yesmaximal, characteristic
7#21424781C211yesmaximal, characteristic
8#21424719C211yesmaximal, characteristic
9#21424670C211yesmaximal, characteristic, complemented
10#21424668C211yesmaximal, characteristic, complemented
11#107×Dic59C211yesmaximal, characteristic, complemented
12C4×#14273660C211yesmaximal, characteristic, complemented
13#21423957C211yesmaximal, characteristic, complemented
14#102×Dic59C211yesmaximal, characteristic
15C59×#1925C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×C4×Dic118—21no—
18C2×D4×Dic59C2211yescharacteristic
19C2×#14273683—21no—
20#16837680C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C4×C59⋊C8C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30#16837306—21no—
31#16837305C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837296—21no—
35#16837290—21no—
36#16837289—21no—
37#16837286—21no—
38#16837271C2211yescharacteristic
39#16837260—21no—
40C4⋊C4×Dic59C2211yescharacteristic
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×#107×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C23×Dic59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273726—22no—
65#14273726C2×C414yes—
66C4×Dic118—44no—
67D4×Dic59—22no—
68D4×Dic59C2×C414yes—
69#14273683—44no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4D411yescharacteristic
80C2×Dic59⋊C4D411yescharacteristic
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59C2311yescharacteristic
89C2×C4×Dic59C2311yescharacteristic
90#14273661C2×C414yes—
91#14273660—22no—
92#14273660C2×C414yes—
93Dic59⋊C8C2×C414yes—
94#14273648—22no—
95#14273648C2×C414yes—
96C4×C59⋊C8—22no—
97C8×Dic59—22no—
98C8×Dic59—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C22×D4×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117#107×C59—22no—
118#102×C59—22no—
119#102×C59C2×C414yes—
120#95×C59—21no—
121C24×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59C22×C412yes—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—24no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—42no—
140C22×Dic59—42no—
141C22×Dic59C2×D411yescharacteristic
142C22×Dic59C2×D411yescharacteristic
143C22×Dic59C2×D411yescharacteristic
144C22×Dic59C2×D411yescharacteristic
145C2×Dic118—24no—
146C2×Dic118—42no—
147C2×Dic118—42no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59C22×C412yes—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59C22×C412yes—
159C4⋊C4×C59#4112yes—
160C22⋊C4×C59—42no—
161C22⋊C4×C59—81no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—42no—
165C22⋊Dic59—24no—
166C22⋊Dic59—42no—
167C22⋊Dic59—81no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—42no—
175C4⋊Dic59C22×C412yes—
176C4⋊Dic59C22×C412yes—
177Dic59⋊C4—24no—
178Dic59⋊C4—22no—
179Dic59⋊C4—42no—
180Dic59⋊C4—42no—
181Dic59⋊C4—81no—
182Dic59⋊C4D814yes—
183C4×Dic59—21no—
184C4×Dic59—21no—
185C4×Dic59—42no—
186C4×Dic59—42no—
187C4×Dic59—42no—
188C4×Dic59—42no—
189C4×Dic59C22×C412yes—
190C4×Dic59#4112yes—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8C22×C412yes—
196C2×C59⋊C8#4112yes—
197C23×C59—22no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59C22×D411yescharacteristic
205D4×C59—22no—
206D4×C59—22no—
207D4×C59—41no—
208D4×C59—41no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—22no—
212C2×C4×C59—21no—
213C2×C4×C59—22no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59C2×D812yes—
224C2×C4×C59C23×C411yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C592−1+411yescharacteristic
229C2×C4×C592−1+411yescharacteristic
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—22no—
235C2×Dic59—22no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—48no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59C4×D412yes—
251C2×Dic59C4×D412yes—
252C2×Dic59C2×D812yes—
253C2×Dic59C2×D812yes—
254C2×Dic59#13512yes—
255Dic118—48no—
256C8×C59—22no—
257C8×C59—41no—
258C59⋊C8—22no—
259C59⋊C8—22no—
260#23060—591nomaximal, Sylow
261C22×C59—42no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—81no—
265C22×C59—81no—
266C22×C59—81no—
267C22×C59—81no—
268C22×C59C2×C4×D411yescharacteristic
269C22×C59#54711yescharacteristic
270C22×C59#54811yescharacteristic
271C22×C59C22×D811yescharacteristic
272C22×C59C2×#13511yescharacteristic
273C22×C59#57711yescharacteristic
274C22×C59#57811yescharacteristic
275C4×C59—22no—
276C4×C59—41no—
277C4×C59—42no—
278C4×C59—42no—
279C4×C59—81no—
280C4×C59C4×#4111yescharacteristic
281C4×C59#51911yescharacteristic
282C4×C59#55211yescharacteristic
283C4×C59#55611yescharacteristic
284Dic59—22no—
285Dic59—22no—
286Dic59—41no—
287Dic59—42no—
288Dic59—42no—
289Dic59—81no—
290Dic59—81no—
291Dic59C4×D814yes—
292C2×#474—591no—
293#2365—591no—
294#2354—591no—
295#1990—591no—
296#1947—591no—
297#1934—591no—
298#1934—591no—
299#1925—591no—
300#1925D5911yescharacteristic
301#1918—591no—
302#1918—591no—
303C4×#107—591no—
304C4×#107—591no—
305C8×C4⋊C4—591no—
306C4×#102—591no—
307C2×C59—82no—
308C2×C59#236211yescharacteristic
309C2×C59C2×C4×D811yescharacteristic
310C2×C59#294711yescharacteristic
311C2×C59#333411yescharacteristic
312C2×C59#334011yescharacteristic
313C2×C59#335811yescharacteristic
314C2×C59#336411yescharacteristic
315C2×C4⋊Q8—591no—
316C2×C4×Q8—1181no—
317C2×C4×D4—591no—
318C2×#117—1181no—
319#474—598no—
320C2×#107—591no—
321C2×#107—591no—
322C2×#107—591no—
323C2×#107C2×D5911yescharacteristic
324C2×C4⋊C8—591no—
325C2×#102—591no—
326C2×#102—591no—
327C2×#102C2×D5911yescharacteristic
328C2×#102C2×D5911yescharacteristic
329C2×C4×C8—591no—
330C2×C4×C8—591no—
331C2×C4×C8—591no—
332#391—1181no—
333#390—591no—
334#390C2×D5911yescharacteristic
335#389—591no—
336#389—591no—
337#388—1181no—
338#388—1181no—
339#387—1181no—
340#387—1181no—
341#384—591no—
342#380—1181no—
343C4×C4⋊C4—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C4⋊C4—591no—
347C4×C4⋊C4—591no—
348C4×C4⋊C4C2×D5911yescharacteristic
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340C2×D5911yescharacteristic
357#336—591no—
358#336C2×D5911yescharacteristic
359C59#2306011yescharacteristic, complemented, Sylow
360C22×Q8—1181no—
361C22×D4C22×D5911yescharacteristic
362C23×C4—1181no—
363C22×C8—591no—
364C22×C8—591no—
365C22×C8C22×D5911yescharacteristic
366C22×C8C22×D5911yescharacteristic
367C4⋊Q8—594no—
368C4⋊Q8—1182no—
369C4×Q8—2364no—
370C4×D4—594no—
371C4×D4—1182no—
372#117—2364no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1181no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C42—21no—
393C2×C42—591no—
394C2×C42—591no—
395C2×C42—1181no—
396C2×C42—1181no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42C22×D5911yescharacteristic
400#107—22no—
401#107—594no—
402#107—594no—
403#107—1182no—
404C4⋊C8—594no—
405#102—22no—
406#102—594no—
407#102—1182no—
408#102C4×D5914yes—
409C4×C8—1182no—
410C4×C8—1182no—
411C4×C8—1182no—
412#95—21no—
413#95—591no—
414#95—591no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421C24—21no—
422C2×Q8—1184no—
423C2×Q8—2362no—
424C2×Q8—2362no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C4×D5912yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—591no—
435C22×C4—591no—
436C22×C4—591no—
437C22×C4—591no—
438C22×C4—1184no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4—2362no—
444C22×C4—2362no—
445C22×C4D4×D5911yescharacteristic
446C22×C4D4×D5911yescharacteristic
447C22×C4C23×D5911yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—592no—
454C2×C8—592no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8C2×C4×D5912yes—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—594no—
465C4⋊C4—592no—
466C4⋊C4—592no—
467C4⋊C4—1182no—
468C4⋊C4—1184no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—4721no—
476C4⋊C4—2362no—
477C4⋊C4—2362no—
478C4⋊C4—2362no—
479C4⋊C4C2×C4×D5912yes—
480C4⋊C4#1174568712yes—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—1184no—
484C22⋊C4—4721no—
485C22⋊C4—2362no—
486C42—21no—
487C42—21no—
488C42—42no—
489C42—592no—
490C42—592no—
491C42—1181no—
492C42—1181no—
493C42—2362no—
494C42—2362no—
495C42—2362no—
496C42—2362no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23C2×D4×D5911yescharacteristic
505Q8—2368no—
506D4—22no—
507D4—22no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—592no—
525C2×C4—592no—
526C2×C4—592no—
527C2×C4—592no—
528C2×C4—592no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1182no—
537C2×C4—1182no—
538C2×C4—2368no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4D8×D5912yes—
550C2×C4C22×C4×D5911yescharacteristic
551C2×C4#41×D5911yescharacteristic
552C2×C4#41×D5911yescharacteristic
553C2×C4C2×#1174568711yescharacteristic
554C2×C4#1427377411yescharacteristic
555C2×C4#1427377411yescharacteristic
556C8—22no—
557C8—41no—
558C8—1182no—
559C8—1182no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C4×D4×D5911yescharacteristic
568C22#1683805011yescharacteristic
569C22#1683805811yescharacteristic
570C22C2×D8×D5911yescharacteristic
571C22#135×D5911yescharacteristic
572C22#1683821211yescharacteristic
573C22#1683821811yescharacteristic
574C4—22no—
575C4—42no—
576C4—42no—
577C4—41no—
578C4—81no—
579C4—594no—
580C4—1182no—
581C4—1182no—
582C4—4721no—
583C4—4721no—
584C4—2362no—
585C4—2362no—
586C4—2361no—
587C4C4×#1174568711yescharacteristic
588C4#1683787911yescharacteristic
589C4#1683808811yescharacteristic
590C4#1683812111yescharacteristic
591C2—82no—
592C2#2142712611yescharacteristic
593C2C4×D8×D5911yescharacteristic
594C2#2143293611yescharacteristic
595C2#2143643811yescharacteristic
596C2#2143650711yescharacteristic
597C2#2143672611yescharacteristic
598C2#2143677611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1925C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1925D59
hypocenterC59#23060
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
2. #21427155C2semidirect
4. #21425055C2semidirect
5. #21424851C2semidirect
9. #21424670C2semidirect
10. #21424668C2semidirect
11. #107×Dic59C2semidirect
12. C4×#14273660C2semidirect
13. #21423957C2semidirect
20. #16837680C4semidirect
359. C59#23060semidirect

Maximal subgroups

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

subgroupindex
C2×#168376802
#214271552
#214270442
#214250552
#214248512
#214247822
#214247812
#214247192
#214246702
#214246682
#107×Dic592
C4×#142736602
#214239572
#102×Dic592
C59×#19252
#2306059