Finite Group Databasebeta

← #28610534

Subgroups — #28610534

Order 15104.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837702C211yesmaximal, characteristic
2#21427190C211yesmaximal, characteristic, complemented
3#21425060C211yesmaximal, characteristic, complemented
4#21424862C211yesmaximal, characteristic, complemented
5#21424826C211yesmaximal, characteristic, complemented
6#21424821C211yesmaximal, characteristic, complemented
7#21424685C211yesmaximal, characteristic, complemented
8#21424678C211yesmaximal, characteristic, complemented
9#21424357C211yesmaximal, characteristic
10#21424343C211yesmaximal, characteristic
11C4⋊C4×C59⋊C8C211yesmaximal, characteristic, complemented
12C4×#14273648C211yesmaximal, characteristic
13C4×#14273648C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C59×#2349C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17#16837702C418yescomplemented
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C4×C59⋊C8C2211yescharacteristic
28C2×C4×C59⋊C8C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837297—21no—
33#16837284—21no—
34#16837283—21no—
35C4×C4⋊Dic59C2211yescharacteristic
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837253—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#385—21no—
55C59×#385—21no—
56C59×#382C2211yescharacteristic
57C59×#380—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C2×#95×C59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273719—22no—
65#14273719C2×C414yescomplemented
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77#14273660—41no—
78#14273660C2×C414yescomplemented
79#14273658—41no—
80#14273658C2×C414yescomplemented
81C4⋊(C59⋊C8)C2×C414yescomplemented
82#14273648—22no—
83#14273648C2×C414yes—
84#14273648C2×C414yes—
85#14273648Q814yes—
86C4×C59⋊C8—22no—
87C4×C59⋊C8—41no—
88C4×C59⋊C8—41no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C22×D4×C59C2311yescharacteristic
95C23×C4×C59—21no—
96C23×C4×C59—21no—
97C23×C4×C59—21no—
98C4×D4×C59—22no—
99C4×D4×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—41no—
102C4×D4×C59—41no—
103C4×D4×C59C2×C414yes—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C42×C59C2311yescharacteristic
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117#95×C59—21no—
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59D411yescharacteristic
122#95×C59D411yescharacteristic
123C24×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59C22×C412yes—
128C2×D4×C59C2×Q812yes—
129C2×D4×C59#4112yes—
130C22×C4×C59—21no—
131C22×C4×C59—24no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—42no—
135C22×C4×C59—42no—
136C22×C4×C59—42no—
137C22×C4×C59—42no—
138C22×C4×C59—81no—
139C22×C4×C59—81no—
140C22×C4×C59—81no—
141C22×C4×C59—81no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C4⋊C4×C59—22no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—81no—
161C4⋊C4×C59—81no—
162C4⋊C4×C59C22×C412yes—
163C22⋊C4×C59—24no—
164C22⋊C4×C59—42no—
165C22⋊C4×C59—42no—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59C22×C412yes—
171C42×C59#4112yes—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59C22×C412yes—
178C4⋊Dic59C22×C412yes—
179C4⋊Dic59C2×Q812yes—
180Dic59⋊C4—24no—
181Dic59⋊C4—24no—
182Dic59⋊C4—42no—
183Dic59⋊C4—81no—
184C4×Dic59—42no—
185C4×Dic59—81no—
186C4×Dic59—81no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8C22×C412yes—
197C2×C59⋊C8C22×C412yes—
198C2×C59⋊C8C2×Q812yes—
199C23×C59—22no—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59C22×D411yescharacteristic
207D4×C59—22no—
208D4×C59—22no—
209D4×C59—22no—
210D4×C59C4×Q814yes—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
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—41no—
233C2×C4×C59—41no—
234C2×C4×C59—84no—
235C2×C4×C59—84no—
236C2×C4×C59—84no—
237C2×C4×C59C4×D412yes—
238C2×C4×C59C4×D412yes—
239C2×C4×C59#13512yes—
240C2×C4×C59#13512yes—
241C2×C4×C59C23×C411yescharacteristic
242C2×C4×C59C22×Q811yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59H(5,𝔽2)11yescharacteristic
246C2×C4×C592−1+411yescharacteristic
247C2×Dic59—22no—
248C2×Dic59—22no—
249C2×Dic59—22no—
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—
259C59⋊C8—22no—
260C59⋊C8—22no—
261C59⋊C8—22no—
262C59⋊C8—41no—
263#22966—591nomaximal, Sylow
264C22×C59—24no—
265C22×C59—42no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59C2×C4×D411yescharacteristic
272C22×C59#54611yescharacteristic
273C22×C59D4×Q811yescharacteristic
274C22×C59C2×#13511yescharacteristic
275C22×C59C2×#13511yescharacteristic
276C22×C59#57611yescharacteristic
277C22×C59#57711yescharacteristic
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—22no—
281C4×C59—22no—
282C4×C59—42no—
283C4×C59—41no—
284C4×C59—41no—
285C4×C59—81no—
286C4×C59—81no—
287C4×C59C2×C4×Q811yescharacteristic
288C4×C59#52011yescharacteristic
289C4×C59#55211yescharacteristic
290C4×C59#55411yescharacteristic
291Dic59—42no—
292Dic59—42no—
293Dic59—42no—
294Dic59—81no—
295C2×#478—591no—
296#2370—591no—
297#2349D5911yescharacteristic
298#1991—591no—
299#1949—591no—
300#1942—591no—
301#1941—591no—
302#1921—591no—
303#1920—591no—
304#1864—591no—
305#1862—591no—
306C8×C4⋊C4—591no—
307C4×#102—591no—
308C4×#102—591no—
309C4×#102—591no—
310C2×C59—44no—
311C2×C59#236011yescharacteristic
312C2×C59C4×#13511yescharacteristic
313C2×C59#294611yescharacteristic
314C2×C59#329611yescharacteristic
315C2×C59#329711yescharacteristic
316C2×C59#338011yescharacteristic
317C2×C59#338111yescharacteristic
318C2×#125—591no—
319C2×C4×D4C2×D5911yescharacteristic
320C2×C4×D4C2×D5911yescharacteristic
321C2×C4×D4C2×D5911yescharacteristic
322#478—598no—
323C2×#107—591no—
324C2×#107—591no—
325C2×#106—591no—
326C2×#106—591no—
327C2×C4⋊C8—591no—
328C2×#102—591no—
329C2×#102—591no—
330C2×#102—591no—
331C2×#102—591no—
332C2×C4×C8—591no—
333C2×C4×C8—591no—
334C2×C4×C8—591no—
335#389—591no—
336#389—591no—
337#389—1181no—
338#389C2×D5911yescharacteristic
339#389C2×D5911yescharacteristic
340#387—1181no—
341#387—1181no—
342#385—21no—
343#385—21no—
344#382C2×D5911yescharacteristic
345#380—21no—
346C4×C4⋊C4—591no—
347C4×C4⋊C4—591no—
348C4×C4⋊C4—591no—
349C4×C4⋊C4—1181no—
350C4×C4⋊C4C2×D5911yescharacteristic
351C2×#95—21no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#336—591no—
361#336—591no—
362C59#2296611yescharacteristic, complemented, Sylow
363C22×D4C22×D5911yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—21no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8—591no—
370C22×C8—591no—
371#125—594no—
372#125—1182no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—41no—
377C4×D4—41no—
378C4×D4C4×D5914yes—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4C22⋊D5911yescharacteristic
390C2×C4⋊C4C22⋊D5911yescharacteristic
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42C22×D5911yescharacteristic
401C2×C42C22×D5911yescharacteristic
402C2×C42C22×D5911yescharacteristic
403#107—594no—
404#107—2361no—
405#106—594no—
406#106—2361no—
407C4⋊C8—594no—
408#102—594no—
409#102—594no—
410#102—594no—
411#102—1182no—
412C4×C8—1182no—
413C4×C8—2361no—
414C4×C8—2361no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95C22⋊D5911yescharacteristic
425#95C22⋊D5911yescharacteristic
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C4×D5912yes—
431C2×D4#1174568712yes—
432C2×D4Q8×D5912yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4C2×C22⋊D5911yescharacteristic
450C22×C4C2×C22⋊D5911yescharacteristic
451C22×C4C2×C22⋊D5911yescharacteristic
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C23×D5911yescharacteristic
456C2×C8—592no—
457C2×C8—592no—
458C2×C8—592no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—592no—
478C4⋊C4—592no—
479C4⋊C4—592no—
480C4⋊C4—1184no—
481C4⋊C4—1184no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—4721no—
488C4⋊C4—2362no—
489C4⋊C4C2×C4×D5912yes—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—4721no—
498C42—4721no—
499C42—2362no—
500C42C2×C4×D5912yes—
501C42#1174568512yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C22×C22⋊D5911yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4#1427369214yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—1182no—
541C2×C4—1182no—
542C2×C4—1182no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4C4×C22⋊D5912yes—
553C2×C4C4×C22⋊D5912yes—
554C2×C4#1427374712yes—
555C2×C4#1427374712yes—
556C2×C4C22×C4×D5911yescharacteristic
557C2×C4C2×Q8×D5911yescharacteristic
558C2×C4C2×#1174568711yescharacteristic
559C2×C4C2×#1174568511yescharacteristic
560C2×C4#1427377211yescharacteristic
561C2×C4#1427377511yescharacteristic
562C8—1182no—
563C8—1182no—
564C8—1182no—
565C8—2361no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C2×C4×C22⋊D5911yescharacteristic
574C22#1683804611yescharacteristic
575C22Q8×C22⋊D5911yescharacteristic
576C22C2×#1427374711yescharacteristic
577C22C2×#1427374711yescharacteristic
578C22#1683820511yescharacteristic
579C22#1683821311yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—4721no—
590C4—2362no—
591C4—2362no—
592C4—2362no—
593C4C2×#1427369211yescharacteristic
594C4#1683788311yescharacteristic
595C4#1683808711yescharacteristic
596C4#1683810711yescharacteristic
597C2—44no—
598C2#2142710111yescharacteristic
599C2C4×#1427374711yescharacteristic
600C2#2143292811yescharacteristic
601C2#2143593111yescharacteristic
602C2#2143594611yescharacteristic
603C2#2143695911yescharacteristic
604C2#2143697111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2349C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#2349D59
hypocenterC59#22966
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #21427190C2semidirect
3. #21425060C2semidirect
4. #21424862C2semidirect
5. #21424826C2semidirect
6. #21424821C2semidirect
7. #21424685C2semidirect
8. #21424678C2semidirect
11. C4⋊C4×C59⋊C8C2semidirect
17. #16837702C4semidirect
65. #14273719C2×C4semidirect
78. #14273660C2×C4semidirect
80. #14273658C2×C4semidirect
81. C4⋊(C59⋊C8)C2×C4semidirect
362. C59#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#168377022
#214271902
#214250602
#214248622
#214248262
#214248212
#214246852
#214246782
#214243572
#214243432
C4⋊C4×C59⋊C82
C4×#142736482
C59×#23492
#2296659