Finite Group Databasebeta

← #28647558

Subgroups — #28647558

Order 15104.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837808C211yesmaximal, characteristic, complemented
2C2×#16837445C211yesmaximal, characteristic
3#21427820C211yesmaximal, characteristic
4#21425578C211yesmaximal, characteristic, complemented
5#21425577C211yesmaximal, characteristic, complemented
6#21425434C211yesmaximal, characteristic, complemented
7#21425421C211yesmaximal, characteristic, complemented
8#21424831C211yesmaximal, characteristic, complemented
9#21424826C211yesmaximal, characteristic, complemented
10#21424295C211yesmaximal, characteristic
11#21424283C211yesmaximal, characteristic
12#21424258C211yesmaximal, characteristic, complemented
13#21424127C211yesmaximal, characteristic
14#21423880C211yesmaximal, characteristic
15C59×#2439C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19#16837808—24no—
20#16837445—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273660C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31C2×C22⋊(C59⋊C8)C2211yescharacteristic
32C2×#14273638C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837344—21no—
35#16837336—21no—
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837274—21no—
39#16837268C2211yescharacteristic
40#16837262C2211yescharacteristic
41#16837260—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4⋊Q8×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C59×#396—21no—
53C59×#393—21no—
54C59×#391—21no—
55C59×#389C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#387—21no—
58C59×#382C2211yescharacteristic
59C59×#380C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C23×Dic59—21no—
62C2×#11745664—22no—
63C2×#11745664—22no—
64C2×#11745664Q814yescomplemented
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273719—22no—
70#14273719—41no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—41no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83#14273660—22no—
84#14273660—22no—
85#14273660—22no—
86#14273660—41no—
87#14273660—41no—
88#14273660D414yes—
89#14273658—22no—
90#14273658—22no—
91#14273658—22no—
92#14273658—22no—
93C22⋊(C59⋊C8)Q814yescomplemented
94#14273638—22no—
95C4×C59⋊C8—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×Q8×C59—21no—
102C23×C4×C59C2311yescharacteristic
103C4⋊Q8×C59—22no—
104C4⋊Q8×C59—41no—
105#117×C59—22no—
106#117×C59—41no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125#95×C59—21no—
126#95×C59D411yescharacteristic
127#95×C59D411yescharacteristic
128C24×C59C2×D411yescharacteristic
129C2×Q8×C59—42no—
130C2×Q8×C59—42no—
131C2×Q8×C59—81no—
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×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×Q812yes—
146C22×C4×C59C2×Q812yes—
147C22×C4×C59C2411yescharacteristic
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—42no—
153C22×Dic59—42no—
154C22×Dic59—81no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—42no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59C2×D412yes—
166C22⋊C4×C59—41no—
167C22⋊C4×C59—41no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—81no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—42no—
175C22⋊Dic59—42no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—42no—
184C4⋊Dic59C2×D412yes—
185Dic59⋊C4—42no—
186Dic59⋊C4—42no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189#11745664—28no—
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⋊C8C2×D412yes—
197C2×C59⋊C8C2×Q812yes—
198C2×C59⋊C8C2×Q812yes—
199C2×C59⋊C8C2×Q812yes—
200C2×C59⋊C8C2×Q812yes—
201C2×C59⋊C8#4112yes—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59C4⋊Q812yes—
209C23×C59C22×D411yescharacteristic
210Q8×C59—84no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
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—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59C4⋊Q812yes—
242C2×C4×C59C22×D411yescharacteristic
243C2×C4×C59C22×Q811yescharacteristic
244C2×C4×C59C22×Q811yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59H(5,𝔽2)11yescharacteristic
247C2×C4×C592−1+411yescharacteristic
248C2×Dic59—22no—
249C2×Dic59—41no—
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—
259C2×Dic59—41no—
260C2×Dic59—84no—
261C59⋊C8—22no—
262C59⋊C8—22no—
263C59⋊C8—22no—
264C59⋊C8C22⋊Q814yes—
265#26079—591nomaximal, Sylow
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—22no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59—41no—
273C22×C59C2×C4⋊Q811yescharacteristic
274C22×C59D4211yescharacteristic
275C22×C59#54711yescharacteristic
276C22×C59#57511yescharacteristic
277C22×C59#57511yescharacteristic
278C22×C59#57611yescharacteristic
279C22×C59#57811yescharacteristic
280C4×C59—22no—
281C4×C59—41no—
282C4×C59—41no—
283C4×C59—41no—
284C4×C59—42no—
285C4×C59—41no—
286C4×C59—81no—
287C4×C59—81no—
288C4×C59—81no—
289C4×C59C2×C22⋊Q811yescharacteristic
290C4×C59#54411yescharacteristic
291C4×C59D4×Q811yescharacteristic
292C4×C59#55411yescharacteristic
293Dic59—42no—
294Dic59—81no—
295Dic59—81no—
296Dic59—81no—
297C2×#501—591no—
298C2×#428—591no—
299#2442—591no—
300#2439D5911yescharacteristic
301#2085—591no—
302#2085—591no—
303#2059—591no—
304#2057—591no—
305#1943—591no—
306#1942—591no—
307#1854—591no—
308#1852—591no—
309#1846—591no—
310#1809—591no—
311#1753—591no—
312C2×C59—42no—
313C2×C59#244011yescharacteristic
314C2×C59#316411yescharacteristic
315C2×C59#316611yescharacteristic
316C2×C59#327311yescharacteristic
317C2×C59#327711yescharacteristic
318C2×C59#334711yescharacteristic
319C2×C59#334911yescharacteristic
320C22×M16—591no—
321C2×C4⋊Q8C2×D5911yescharacteristic
322C2×#125—591no—
323C2×#117C2×D5911yescharacteristic
324C22×C4⋊C4—591no—
325#501—1184no—
326#428—1184no—
327C2×#107—591no—
328C2×#107—591no—
329C2×#107—591no—
330C2×#107—591no—
331C2×#107—591no—
332C2×#107—591no—
333C2×#106—591no—
334C2×#106—591no—
335C2×#106—591no—
336C2×#106—591no—
337C2×C22⋊C8—591no—
338C2×#97—591no—
339C2×C4×C8—591no—
340#400—1181no—
341#398—1181no—
342#396—21no—
343#393—21no—
344#391—21no—
345#389—591no—
346#389—591no—
347#389C2×D5911yescharacteristic
348#389C2×D5911yescharacteristic
349#387—21no—
350#385—1181no—
351#382—591no—
352#382C2×D5911yescharacteristic
353#380—591no—
354#380—1181no—
355#380C2×D5911yescharacteristic
356C4×C22⋊C4C2×D5911yescharacteristic
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#336—591no—
364#336—591no—
365C59#2607911yescharacteristic, complemented, Sylow
366C22×Q8—21no—
367C23×C4—1181no—
368C23×C4C22×D5911yescharacteristic
369C2×M16—594no—
370C2×M16—1182no—
371C2×M16—1182no—
372C22×C8—591no—
373C22×C8—591no—
374C22×C8—591no—
375C22×C8—591no—
376C4⋊Q8—22no—
377C4⋊Q8—41no—
378#125—1182no—
379#125—2361no—
380#117—22no—
381#117—41no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—2361no—
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4C22⋊D5911yescharacteristic
404C2×C22⋊C4C22⋊D5911yescharacteristic
405C2×C42—21no—
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408#107—594no—
409#107—1182no—
410#107—1182no—
411#107—1182no—
412#107—2361no—
413#107—2361no—
414#106—1182no—
415#106—1182no—
416#106—1182no—
417#106—1182no—
418C22⋊C8—594no—
419#97—1182no—
420C4×C8—1182no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95C22⋊D5911yescharacteristic
432#95C22⋊D5911yescharacteristic
433C24C2×C22⋊D5911yescharacteristic
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—81no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—4721no—
450C22×C4—2362no—
451C22×C4—2362no—
452C22×C4Q8×D5912yes—
453C22×C4Q8×D5912yes—
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C2×C22⋊D5911yescharacteristic
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460M16—1188no—
461C2×C8—592no—
462C2×C8—592no—
463C2×C8—592no—
464C2×C8—592no—
465C2×C8—592no—
466C2×C8—592no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—592no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—4721no—
492C4⋊C4—4721no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C4⋊C4D4×D5912yes—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—2362no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—42no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1427372512yes—
514C23C22×C22⋊D5911yescharacteristic
515Q8—84no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
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—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—1182no—
547C2×C4—4724no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4#1427372512yes—
560C2×C4C2×D4×D5911yescharacteristic
561C2×C4C2×Q8×D5911yescharacteristic
562C2×C4C2×Q8×D5911yescharacteristic
563C2×C4#41×D5911yescharacteristic
564C2×C4#1427377211yescharacteristic
565C2×C4#1427377411yescharacteristic
566C8—594no—
567C8—1182no—
568C8—1182no—
569C8—1182no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1427372511yescharacteristic
578C22D4×C22⋊D5911yescharacteristic
579C22#1683805111yescharacteristic
580C22#1683820011yescharacteristic
581C22#1683820011yescharacteristic
582C22#1683820511yescharacteristic
583C22#1683821711yescharacteristic
584C4—22no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—4721no—
594C4—4721no—
595C4—4721no—
596C4—2362no—
597C4C22⋊Q8×D5911yescharacteristic
598C4#1683803111yescharacteristic
599C4#1683806511yescharacteristic
600C4#1683810611yescharacteristic
601C2—42no—
602C2#2142779511yescharacteristic
603C2#2143463011yescharacteristic
604C2#2143464911yescharacteristic
605C2#2143570311yescharacteristic
606C2#2143574911yescharacteristic
607C2#2143659611yescharacteristic
608C2#2143662011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2439C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2439D59
hypocenterC59#26079
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
1. C2×#16837808C2semidirect
4. #21425578C2semidirect
5. #21425577C2semidirect
6. #21425434C2semidirect
7. #21425421C2semidirect
8. #21424831C2semidirect
9. #21424826C2semidirect
12. #21424258C2semidirect
64. C2×#11745664Q8semidirect
93. C22⋊(C59⋊C8)Q8semidirect
365. C59#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#168378082
C2×#168374452
#214278202
#214255782
#214255772
#214254342
#214254212
#214248312
#214248262
#214242952
#214242832
#214242582
#214241272
#214238802
C59×#24392
#2607959