Finite Group Databasebeta

← #28606361

Subgroups — #28606361

Order 15104.

838 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#106×D59C211yesmaximal, characteristic
2#21426969C211yesmaximal, characteristic
3#21426961C211yesmaximal, characteristic
4#21424839C211yesmaximal, characteristic, complemented
5#21424837C211yesmaximal, characteristic, complemented
6#21424681C211yesmaximal, characteristic, complemented
7#21424679C211yesmaximal, characteristic, complemented
8#21424674C211yesmaximal, characteristic, complemented
9#21424672C211yesmaximal, characteristic, complemented
10#21424185C211yesmaximal, characteristic
11#21424181C211yesmaximal, characteristic
12#21424122C211yesmaximal, characteristic
13#21423996C211yesmaximal, characteristic, complemented
14#21423873C211yesmaximal, characteristic
15C59×#1776C211yesmaximal, characteristic, complemented
16C22×C8×D59C2211yescharacteristic
17C2×#14273684—21no—
18C2×C4⋊C4×D59C2211yescharacteristic
19C2×C4⋊C4×D59C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24#106×D59C418yescomplemented
25C2×(C2×D59)⋊C8C2211yescharacteristic
26C2×#14273637C2211yescharacteristic
27#16837300C2211yescharacteristic
28#16837300C2211yescharacteristic
29#16837298C2211yescharacteristic
30#16837298C2211yescharacteristic
31#16837286—21no—
32#16837286—21no—
33#16837284—21no—
34#16837284—21no—
35#16837264—21no—
36#16837261—21no—
37#16837259C2211yescharacteristic
38#16837259C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4×(C2×D59)⋊C4—21no—
42C4×(C2×D59)⋊C4C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×C4×D59—21no—
61C22×C4×D59—21no—
62C22×C4×D59C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C2×C8×D59—22no—
66C2×C8×D59—22no—
67C2×C8×D59C2×C414yes—
68#14273684—44no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C4⋊C4×D59—22no—
80C4⋊C4×D59—22no—
81C4⋊C4×D59C2×C414yes—
82C4⋊C4×D59C2×C414yes—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4D411yescharacteristic
86C2×(C2×D59)⋊C4D411yescharacteristic
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59C2311yescharacteristic
90#14273661—22no—
91#14273659C2×C414yes—
92#14273658C2×C414yes—
93#14273658C2×C414yes—
94(C2×D59)⋊C8—22no—
95#14273637—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95D411yescharacteristic
105C59⋊#95D411yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C42×C59—21no—
113C2×C42×C59—21no—
114C2×C42×C59C2311yescharacteristic
115#107×C59—22no—
116#106×C59C2×C414yes—
117#97×C59—22no—
118#95×C59—21no—
119#95×C59—21no—
120C23×D59C2×D411yescharacteristic
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2411yescharacteristic
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59—21no—
132C22×Dic59C2×D411yescharacteristic
133C22×Dic59C2×D411yescharacteristic
134C22×Dic59C2×D411yescharacteristic
135C2×C4×D59—21no—
136C2×C4×D59—21no—
137C2×C4×D59—21no—
138C2×C4×D59—21no—
139C2×C4×D59—24no—
140C2×C4×D59—24no—
141C2×C4×D59—42no—
142C2×C4×D59—42no—
143C2×C4×D59—81no—
144C2×C4×D59—81no—
145C2×C4×D59C22×C412yes—
146C2×C4×D59#4112yes—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59C22×C412yes—
151C2×C8×C59#4112yes—
152C2×C8×C59#4112yes—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59C22×C412yes—
160C4⋊C4×C59C22×C412yes—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—42no—
164C42×C59—81no—
165(C2×D59)⋊C4—22no—
166(C2×D59)⋊C4—22no—
167(C2×D59)⋊C4—42no—
168(C2×D59)⋊C4—42no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59C22×C412yes—
176C4⋊Dic59C22×C412yes—
177Dic59⋊C4—24no—
178Dic59⋊C4—24no—
179Dic59⋊C4—42no—
180Dic59⋊C4—42no—
181Dic59⋊C4—81no—
182Dic59⋊C4—81no—
183C4×Dic59—21no—
184C4×Dic59—21no—
185C4×Dic59—42no—
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⋊C8C22×C412yes—
193C8×D59—28no—
194C23×C59C22×D411yescharacteristic
195C22×D59—21no—
196C22×D59—21no—
197C22×D59—21no—
198C22×D59—21no—
199C22×D59C4×D412yes—
200C22×D59AGL(1,ℤ/8)12yes—
201C22×D59#13712yes—
202C2×C4×C59—22no—
203C2×C4×C59—22no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
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×C59C23×C411yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59C2×#4111yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C592−1+411yescharacteristic
225C2×C4×C592−1+411yescharacteristic
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—22no—
229C2×Dic59—22no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59C4×D412yes—
247C4×D59—22no—
248C4×D59—22no—
249C4×D59—22no—
250C4×D59—22no—
251C4×D59—48no—
252C4×D59—84no—
253C8×C59—41no—
254C8×C59#11714yescomplemented
255C59⋊C8—22no—
256C59⋊C8—41no—
257#22615—591nomaximal, Sylow
258C22×C59C2×C4×D411yescharacteristic
259C22×C59#54711yescharacteristic
260C22×C59#54711yescharacteristic
261C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C59C2×#13711yescharacteristic
263C22×C59#57711yescharacteristic
264C22×C59#57711yescharacteristic
265C2×D59—22no—
266C2×D59—22no—
267C2×D59—22no—
268C2×D59—22no—
269C2×D59—22no—
270C2×D59—22no—
271C2×D59#44014yes—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59—42no—
275C4×C59—42no—
276C4×C59—81no—
277C4×C59—81no—
278C4×C59C2×#11711yescharacteristic
279C4×C59#51911yescharacteristic
280C4×C59#55211yescharacteristic
281C4×C59#55211yescharacteristic
282Dic59—22no—
283Dic59—41no—
284Dic59—41no—
285Dic59—42no—
286Dic59—42no—
287Dic59—81no—
288Dic59—81no—
289C22×#106—591no—
290#2345—591no—
291#2344—591no—
292#1944—591no—
293#1944—591no—
294#1920—591no—
295#1920—591no—
296#1919—591no—
297#1919—591no—
298#1824—591no—
299#1823—591no—
300#1808—591no—
301#1776—591no—
302#1776D5911yescharacteristic, complemented
303#1752—591no—
304C2×C59#234411yescharacteristic
305C2×C59C2×#44011yescharacteristic
306C2×C59#294911yescharacteristic
307C2×C59#334411yescharacteristic
308C2×C59#334411yescharacteristic
309C2×C59#334511yescharacteristic
310C2×C59#334511yescharacteristic
311D59—24no—
312C23×C8—591no—
313C2×#117—1181no—
314C22×C4⋊C4—591no—
315C22×C4⋊C4—591no—
316C2×#107—591no—
317C2×#107C2×D5911yescharacteristic
318C2×#106—591no—
319C2×#106—591no—
320C2×#106—591no—
321C2×#106—598no—
322C2×#106C2×D5911yescharacteristic
323C2×C22⋊C8—591no—
324C2×#97—591no—
325C2×#97C2×D5911yescharacteristic
326#389—591no—
327#389—591no—
328#389—591no—
329#389—591no—
330#387—1181no—
331#387—1181no—
332#387—1181no—
333#387—1181no—
334#381—1181no—
335#380—591no—
336#380—591no—
337#380—1181no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4C2×D5911yescharacteristic
341C4×C4⋊C4C2×D5911yescharacteristic
342C4×C22⋊C4—591no—
343C4×C22⋊C4—1181no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340C2×D5911yescharacteristic
353#340C2×D5911yescharacteristic
354#336—591no—
355#336—591no—
356C59#2261511yescharacteristic, complemented, Sylow
357C23×C4—591no—
358C23×C4—1181no—
359C23×C4—1181no—
360C22×C8—591no—
361C22×C8—594no—
362C22×C8—591no—
363C22×C8—1182no—
364C22×C8—1182no—
365C22×C8C22×D5911yescharacteristic
366C22×C8C22×D5911yescharacteristic
367#117—2364no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—594no—
371C2×C4⋊C4—594no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1182no—
375C2×C4⋊C4—1182no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C22⋊C4—591no—
387C2×C22⋊C4—591no—
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—591no—
393C2×C42—1181no—
394C2×C42—1181no—
395C2×C42C22×D5911yescharacteristic
396#107—22no—
397#107—1182no—
398#106—594no—
399#106—594no—
400#106—594no—
401#106C4×D5914yes—
402C22⋊C8—1182no—
403#97—22no—
404#97—1182no—
405#95—21no—
406#95—21no—
407#95—591no—
408#95—591no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417C24—591no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—591no—
423C22×C4—592no—
424C22×C4—592no—
425C22×C4—591no—
426C22×C4—591no—
427C22×C4—1184no—
428C22×C4—1184no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4—4721no—
438C22×C4—4721no—
439C22×C4—2362no—
440C22×C4—2362no—
441C22×C4D4×D5911yescharacteristic
442C22×C4D4×D5911yescharacteristic
443C22×C4C23×D5911yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—592no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1188no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8C2×C4×D5912yes—
455C2×C8#1174568512yes—
456C2×C8#1174568512yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—592no—
464C4⋊C4—592no—
465C4⋊C4—1184no—
466C4⋊C4—1184no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—4721no—
474C4⋊C4—4721no—
475C4⋊C4—2362no—
476C4⋊C4—2362no—
477C4⋊C4C2×C4×D5912yes—
478C4⋊C4C2×C4×D5912yes—
479C22⋊C4—1182no—
480C22⋊C4—1182no—
481C22⋊C4—2362no—
482C22⋊C4—2362no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—81no—
487C42—1181no—
488C42—1181no—
489C42—4721no—
490C42—2362no—
491C23—592no—
492C23—592no—
493C23—592no—
494C23—1181no—
495C23—1181no—
496C23—1181no—
497C23—1181no—
498C23C2×D4×D5911yescharacteristic
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—592no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1182no—
521C2×C4—1182no—
522C2×C4—1182no—
523C2×C4—1182no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1182no—
531C2×C4—1182no—
532C2×C4—4724no—
533C2×C4—2368no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4C22×C4×D5911yescharacteristic
545C2×C4#41×D5911yescharacteristic
546C2×C4C2×#1174568511yescharacteristic
547C2×C4C2×#1174568511yescharacteristic
548C2×C4#1427377411yescharacteristic
549C2×C4#1427377511yescharacteristic
550C8—41no—
551C8—1182no—
552C8—2361no—
553C8#1427368414yescomplemented
554C22—594no—
555C22—1182no—
556C22—1182no—
557C22—1182no—
558C22—1182no—
559C22—1182no—
560C22—1182no—
561C22C4×D4×D5911yescharacteristic
562C22#1683805311yescharacteristic
563C22#1683805711yescharacteristic
564C22AGL(1,ℤ/8)×D5911yescharacteristic
565C22#137×D5911yescharacteristic
566C22#1683821111yescharacteristic
567C22#1683821211yescharacteristic
568C4—42no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—1182no—
575C4—4721no—
576C4—4721no—
577C4—2362no—
578C4—2362no—
579C4—2361no—
580C4—2361no—
581C4C2×#1427368411yescharacteristic
582C4#1683787811yescharacteristic
583C4#1683809111yescharacteristic
584C4#1683809311yescharacteristic
585C2—1184no—
586C2#2142696111yescharacteristic
587C2#440×D5911yescharacteristic
588C2#2143295211yescharacteristic
589C2#2143655211yescharacteristic
590C2#2143655611yescharacteristic
591C2#2143656711yescharacteristic
592C2#2143657111yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1776C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1776D59
hypocenterC59#22615
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #21424839C2semidirect
5. #21424837C2semidirect
6. #21424681C2semidirect
7. #21424679C2semidirect
8. #21424674C2semidirect
9. #21424672C2semidirect
13. #21423996C2semidirect
15. C59×#1776C2semidirect
24. #106×D59C4semidirect
254. C8×C59#117semidirect
302. #1776D59semidirect
356. C59#22615semidirect
553. C8#14273684semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D592
#214269692
#214269612
#214248392
#214248372
#214246812
#214246792
#214246742
#214246722
#214241852
#214241812
#214241222
#214239962
#214238732
C59×#17762
#2261559