Finite Group Databasebeta

← #28599082

Subgroups — #28599082

Order 15104.

830 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837425C211yesmaximal, characteristic
2#21426498C211yesmaximal, characteristic
3#21426491C211yesmaximal, characteristic
4#21424867C211yesmaximal, characteristic, complemented
5#21424865C211yesmaximal, characteristic, complemented
6#21424863C211yesmaximal, characteristic, complemented
7#21424861C211yesmaximal, characteristic, complemented
8#21424858C211yesmaximal, characteristic, complemented
9#21424856C211yesmaximal, characteristic, complemented
10#21424853C211yesmaximal, characteristic, complemented
11#21424851C211yesmaximal, characteristic, complemented
12#21424129C211yesmaximal, characteristic
13#21424127C211yesmaximal, characteristic
14#21424127C211yesmaximal, characteristic
15C59×#1809C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273683—21no—
18C22×C4×Dic59C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25#16837425C418yescomplemented
26C2×Dic59⋊C8C2211yescharacteristic
27C2×Dic59⋊C8C2211yescharacteristic
28C2×Dic59⋊C8C2211yescharacteristic
29C2×Dic59⋊C8C2211yescharacteristic
30#16837271C2211yescharacteristic
31#16837271C2211yescharacteristic
32#16837269—21no—
33#16837267C2211yescharacteristic
34#16837267C2211yescharacteristic
35#16837266—21no—
36#16837263—21no—
37#16837262C2211yescharacteristic
38#16837262C2211yescharacteristic
39#16837260—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44#16837253—21no—
45C22⋊C4×Dic59—21no—
46#16837247—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C22×M16×C59C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×#106×C59C2211yescharacteristic
56C59×#380C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C23×Dic59D411yescharacteristic
61C23×Dic59D411yescharacteristic
62C2×#11745664—41no—
63C2×#11745664—41no—
64C2×#11745664C2×C414yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273683—44no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59C2×C414yes—
83C2×C4×Dic59D411yescharacteristic
84C2×C4×Dic59D411yescharacteristic
85C2×C4×Dic59Q814yes—
86C2×C4×Dic59C2311yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88#14273661—41no—
89#14273660—41no—
90#14273660—41no—
91#14273659—41no—
92#14273658—41no—
93#14273658—41no—
94Dic59⋊C8C2×C414yes—
95Dic59⋊C8C2×C414yes—
96Dic59⋊C8C2×C414yes—
97Dic59⋊C8C2×C414yes—
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—
107C59⋊#95—21no—
108C23×C4×C59C2311yescharacteristic
109C2×M16×C59—41no—
110C2×M16×C59—41no—
111C2×M16×C59C2×C414yes—
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120#107×C59—41no—
121#106×C59—41no—
122#95×C59—21no—
123#95×C59—21no—
124C24×C59C2×D411yescharacteristic
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59C22×C412yes—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×Q812yes—
136C22×C4×C59C2411yescharacteristic
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—41no—
144C22×Dic59—41no—
145C22×Dic59C4⋊C414yes—
146C22×Dic59C4⋊C414yes—
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151M16×C59—44no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59C22×C412yes—
157C2×C8×C59C22×C412yes—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C22⋊C4×C59—42no—
167C22⋊C4×C59—42no—
168C22⋊Dic59—42no—
169C22⋊Dic59—42no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178Dic59⋊C4—42no—
179Dic59⋊C4—42no—
180Dic59⋊C4—42no—
181Dic59⋊C4—42no—
182C4×Dic59—42no—
183C4×Dic59—42no—
184C4×Dic59—42no—
185C4×Dic59—42no—
186C4×Dic59C4⋊C414yes—
187C4×Dic59C4⋊C414yes—
188C4×Dic59C22×C412yes—
189C4×Dic59C22×C412yes—
190C4×Dic59C2×Q812yes—
191C4×Dic59C2×Q812yes—
192#11745664—44no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8C22×C412yes—
198C2×C59⋊C8C22×C412yes—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59C2×C4⋊C412yes—
206C23×C59C22×D411yescharacteristic
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
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×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59C2×C4⋊C412yes—
226C2×C4×C59C23×C411yescharacteristic
227C2×C4×C59C22×Q811yescharacteristic
228C2×C4×C59H(5,𝔽2)11yescharacteristic
229C2×C4×C59H(5,𝔽2)11yescharacteristic
230C2×C4×C592−1+411yescharacteristic
231C2×C4×C592−1+411yescharacteristic
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—28no—
239C2×Dic59—44no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59C2×C4⋊C412yes—
253C2×Dic59C2×C4⋊C412yes—
254C2×Dic59C2×C4⋊C412yes—
255C2×Dic59C2×C4⋊C412yes—
256C2×Dic59AGL(1,ℤ/8)12yes—
257C2×Dic59#13712yes—
258C8×C59—22no—
259C8×C59—22no—
260C59⋊C8—22no—
261C59⋊C8—22no—
262#21963—591nomaximal, Sylow
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59C22×C4⋊C411yescharacteristic
271C22×C59#53711yescharacteristic
272C22×C59#53711yescharacteristic
273C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C59C2×#13711yescharacteristic
275C22×C59#57511yescharacteristic
276C22×C59#57511yescharacteristic
277C4×C59—22no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59#51811yescharacteristic
283C4×C59#51911yescharacteristic
284C4×C59#54411yescharacteristic
285C4×C59#54411yescharacteristic
286Dic59—22no—
287Dic59—22no—
288Dic59—22no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292Dic59—81no—
293Dic59#42814yes—
294C2×#423—591no—
295#2285—591no—
296#2283—591no—
297#1949—591no—
298#1949—591no—
299#1949—591no—
300#1949—591no—
301#1948—591no—
302#1948—591no—
303#1947—591no—
304#1947—591no—
305#1809—591no—
306#1809—591no—
307#1809—591no—
308#1809D5911yescharacteristic
309C2×C59—42no—
310C2×C59#231611yescharacteristic
311C2×C59C2×#42811yescharacteristic
312C2×C59#291111yescharacteristic
313C2×C59#325111yescharacteristic
314C2×C59#325111yescharacteristic
315C2×C59#325211yescharacteristic
316C2×C59#325211yescharacteristic
317C22×M16—591no—
318C22×M16C2×D5911yescharacteristic
319C2×#117—1181no—
320C22×C42—591no—
321C2×#107—591no—
322C2×#107—591no—
323C2×#107—591no—
324C2×#107C2×D5911yescharacteristic
325C2×#106—591no—
326C2×#106—591no—
327C2×#106—591no—
328C2×#106C2×D5911yescharacteristic
329#423—598no—
330C2×C4⋊C8—591no—
331C2×C4⋊C8—591no—
332C2×C4⋊C8—591no—
333C2×C4⋊C8—591no—
334#384—591no—
335#384—591no—
336#383—1181no—
337#382—591no—
338#382—591no—
339#382—1181no—
340#381—1181no—
341#380—591no—
342#380—591no—
343#380—1181no—
344#380C2×D5911yescharacteristic
345#380C2×D5911yescharacteristic
346C4×C4⋊C4—591no—
347C4×C4⋊C4—591no—
348C4×C4⋊C4—591no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—1181no—
351C4×C22⋊C4—1181no—
352#376—1181no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340C2×D5911yescharacteristic
360#340C2×D5911yescharacteristic
361C59#2196311yescharacteristic, complemented, Sylow
362C23×C4—591no—
363C23×C4—591no—
364C23×C4C22×D5911yescharacteristic
365C2×M16—41no—
366C2×M16—41no—
367C2×M16—594no—
368C2×M16—2361no—
369C2×M16—2361no—
370C2×M16C4×D5914yes—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375#117—2364no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
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×C4⋊C4C22×D5911yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C42—594no—
393C2×C42—591no—
394C2×C42—591no—
395C2×C42—594no—
396C2×C42—591no—
397C2×C42—591no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42—1181no—
401C2×C42—1181no—
402#107—41no—
403#107—2361no—
404#107—2361no—
405#107—2361no—
406#106—41no—
407#106—2361no—
408#106—2361no—
409#106—2361no—
410C4⋊C8—594no—
411C4⋊C8—594no—
412C4⋊C8—594no—
413C4⋊C8—594no—
414#95—21no—
415#95—21no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426C24D4×D5911yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—591no—
436C22×C4—594no—
437C22×C4—591no—
438C22×C4—591no—
439C22×C4—591no—
440C22×C4—594no—
441C22×C4—1182no—
442C22×C4—1182no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—2361no—
448C22×C4—2361no—
449C22×C4C2×C4×D5912yes—
450C22×C4D4×D5911yescharacteristic
451C22×C4Q8×D5912yes—
452C22×C4C23×D5911yescharacteristic
453M16—44no—
454M16—2364no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—592no—
460C2×C8—592no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8C2×C4×D5912yes—
466C2×C8C2×C4×D5912yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—2362no—
484C4⋊C4—2362no—
485C4⋊C4—2362no—
486C4⋊C4—2362no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—2362no—
490C22⋊C4—2362no—
491C42—594no—
492C42—594no—
493C42—592no—
494C42—592no—
495C42—592no—
496C42—592no—
497C42—2362no—
498C42—2362no—
499C42—2362no—
500C42—2362no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C4⋊C4×D5912yes—
508C23C2×D4×D5911yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—592no—
528C2×C4—592no—
529C2×C4—592no—
530C2×C4—592no—
531C2×C4—592no—
532C2×C4—592no—
533C2×C4—1188no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—2364no—
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×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4C4⋊C4×D5912yes—
554C2×C4C22×C4×D5911yescharacteristic
555C2×C4C2×Q8×D5911yescharacteristic
556C2×C4#1427377211yescharacteristic
557C2×C4#1427377211yescharacteristic
558C2×C4#1427377411yescharacteristic
559C2×C4#1427377411yescharacteristic
560C8—22no—
561C8—22no—
562C8—1182no—
563C8—1182no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22C2×C4⋊C4×D5911yescharacteristic
572C22#1683797911yescharacteristic
573C22#1683797911yescharacteristic
574C22AGL(1,ℤ/8)×D5911yescharacteristic
575C22#137×D5911yescharacteristic
576C22#1683819811yescharacteristic
577C22#1683820111yescharacteristic
578C4—22no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—594no—
584C4—1182no—
585C4—1182no—
586C4—1182no—
587C4—4721no—
588C4—4721no—
589C4—4721no—
590C4—4721no—
591C4#1683787511yescharacteristic
592C4#1683787911yescharacteristic
593C4#1683803111yescharacteristic
594C4#1683803111yescharacteristic
595C2—42no—
596C2#2142671311yescharacteristic
597C2#428×D5911yescharacteristic
598C2#2143268311yescharacteristic
599C2#2143545111yescharacteristic
600C2#2143545311yescharacteristic
601C2#2143546211yescharacteristic
602C2#2143546411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1809C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1809D59
hypocenterC59#21963
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #21424867C2semidirect
5. #21424865C2semidirect
6. #21424863C2semidirect
7. #21424861C2semidirect
8. #21424858C2semidirect
9. #21424856C2semidirect
10. #21424853C2semidirect
11. #21424851C2semidirect
25. #16837425C4semidirect
361. C59#21963semidirect

Maximal subgroups

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

subgroupindex
C2×#168374252
#214264982
#214264912
#214248672
#214248652
#214248632
#214248612
#214248582
#214248562
#214248532
#214248512
#214241292
#214241272
C59×#18092
#2196359