Finite Group Databasebeta

← #28606308

Subgroups — #28606308

Order 15104.

850 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#21426972C211yesmaximal, characteristic
3#21426961C211yesmaximal, characteristic
4#21424823C211yesmaximal, characteristic, complemented
5#21424821C211yesmaximal, characteristic, complemented
6#21424683C211yesmaximal, characteristic, complemented
7#21424675C211yesmaximal, characteristic, complemented
8#21424673C211yesmaximal, characteristic, complemented
9#21424185C211yesmaximal, characteristic
10#21424181C211yesmaximal, characteristic
11#21424120C211yesmaximal, characteristic
12C4×#14273658C211yesmaximal, characteristic, complemented
13#21423984C211yesmaximal, characteristic, complemented
14#21423866C211yesmaximal, characteristic
15C59×#1920C211yesmaximal, characteristic, complemented
16C22×C8×D59C2211yescharacteristic
17C2×#14273684—21no—
18C2×C4⋊C4×D59C2211yescharacteristic
19C2×C4⋊C4×D59C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25#106×D59C418yescomplemented
26C2×(C2×D59)⋊C8C2211yescharacteristic
27C2×C4×C59⋊C8C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837298C2211yescharacteristic
32#16837289—21no—
33#16837286—21no—
34#16837284—21no—
35#16837283—21no—
36#16837264—21no—
37#16837259—21no—
38#16837259C2211yescharacteristic
39#16837259C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×(C2×D59)⋊C4C2211yescharacteristic
44#16837249—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
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⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C4⋊C4×D59—22no—
79C4⋊C4×D59—22no—
80C4⋊C4×D59C2×C414yes—
81C4⋊C4×D59C2×C414yes—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4D411yescharacteristic
85C2×(C2×D59)⋊C4D411yescharacteristic
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59C2311yescharacteristic
90#14273659C2×C414yes—
91#14273658—22no—
92#14273658—22no—
93#14273658C2×C414yes—
94#14273658C2×C414yes—
95(C2×D59)⋊C8—22no—
96C4×C59⋊C8—22no—
97C8×Dic59—22no—
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⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117#106×C59C2×C414yes—
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×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59C22×C412yes—
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×C59—81no—
160C4⋊C4×C59C22×C412yes—
161C4⋊C4×C59C22×C412yes—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—42no—
165(C2×D59)⋊C4—22no—
166(C2×D59)⋊C4—42no—
167(C2×D59)⋊C4—42no—
168(C2×D59)⋊C4SD1614yes—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—81no—
176C4⋊Dic59C22×C412yes—
177C4⋊Dic59C22×C412yes—
178Dic59⋊C4—24no—
179Dic59⋊C4—24no—
180Dic59⋊C4—42no—
181Dic59⋊C4—42no—
182C4×Dic59—21no—
183C4×Dic59—21no—
184C4×Dic59—42no—
185C4×Dic59—81no—
186C4×Dic59—81no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8C22×C412yes—
191C2×C59⋊C8#4112yes—
192C2×C59⋊C8#4112yes—
193C8×D59—28no—
194C23×C59C22×D411yescharacteristic
195C22×D59—21no—
196C22×D59—21no—
197C22×D59—21no—
198C22×D59—21no—
199C22×D59C4×D412yes—
200C22×D59C2×SD1612yes—
201C22×D59#13512yes—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—41no—
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×C59C2×SD1612yes—
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—21no—
229C2×Dic59—21no—
230C2×Dic59—22no—
231C2×Dic59—22no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—41no—
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×Dic59C4×D412yes—
246C2×Dic59C2×SD1612yes—
247C4×D59—22no—
248C4×D59—22no—
249C4×D59—22no—
250C4×D59—22no—
251C4×D59—48no—
252C4×D59—84no—
253C8×C59—22no—
254C8×C59—41no—
255C59⋊C8—41no—
256C59⋊C8#11714yescomplemented
257#22609—591nomaximal, Sylow
258C22×C59C2×C4×D411yescharacteristic
259C22×C59#54711yescharacteristic
260C22×C59#54711yescharacteristic
261C22×C59C22×SD1611yescharacteristic
262C22×C59C2×#13511yescharacteristic
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×D59C4×SD1614yes—
272C4×C59—22no—
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—22no—
284Dic59—41no—
285Dic59—42no—
286Dic59—42no—
287Dic59—81no—
288Dic59—81no—
289C22×#106—591no—
290#2345—591no—
291#2344—591no—
292#1941—591no—
293#1941—591no—
294#1920—591no—
295#1920D5911yescharacteristic, complemented
296#1919—591no—
297#1919—591no—
298#1824—591no—
299#1823—591no—
300#1807—591no—
301C4×#106—591no—
302C4×#106—591no—
303C8×C22⋊C4—591no—
304C2×C59#234411yescharacteristic
305C2×C59C2×C4×SD1611yescharacteristic
306C2×C59#294911yescharacteristic
307C2×C59#333511yescharacteristic
308C2×C59#333511yescharacteristic
309C2×C59#333711yescharacteristic
310C2×C59#333711yescharacteristic
311D59—24no—
312C23×C8—591no—
313C2×#117—1181no—
314C22×C4⋊C4—591no—
315C22×C4⋊C4—591no—
316C2×#106—591no—
317C2×#106—591no—
318C2×#106—591no—
319C2×#106—598no—
320C2×#106—591no—
321C2×#106—591no—
322C2×#106C2×D5911yescharacteristic
323C2×C22⋊C8—591no—
324C2×C4×C8—591no—
325C2×C4×C8—591no—
326#389—591no—
327#389—591no—
328#389—591no—
329#389C2×D5911yescharacteristic
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⋊C4—591no—
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#340C2×D5911yescharacteristic
352#340C2×D5911yescharacteristic
353#340C2×D5911yescharacteristic
354#336—591no—
355#336C2×D5911yescharacteristic
356C59#2260911yescharacteristic, 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—21no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—594no—
372C2×C4⋊C4—594no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—1182no—
376C2×C4⋊C4—1182no—
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—591no—
392C2×C42—1181no—
393C2×C42—1181no—
394C2×C42—1181no—
395C2×C42C22×D5911yescharacteristic
396#106—594no—
397#106—594no—
398#106—594no—
399#106—1182no—
400#106—1182no—
401#106C4×D5914yes—
402C22⋊C8—1182no—
403C4×C8—1182no—
404C4×C8—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—591no—
424C22×C4—592no—
425C22×C4—592no—
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—21no—
448C2×C8—21no—
449C2×C8—592no—
450C2×C8—592no—
451C2×C8—592no—
452C2×C8—1188no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8C2×C4×D5912yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—81no—
464C4⋊C4—592no—
465C4⋊C4—592no—
466C4⋊C4—1184no—
467C4⋊C4—1184no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—4721no—
475C4⋊C4—2362no—
476C4⋊C4—2362no—
477C4⋊C4C2×C4×D5912yes—
478C4⋊C4C2×C4×D5912yes—
479C22⋊C4—594no—
480C22⋊C4—1182no—
481C22⋊C4—2362no—
482C22⋊C4—2362no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—1181no—
487C42—1181no—
488C42—4721no—
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—22no—
505C2×C4—22no—
506C2×C4—41no—
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—592no—
517C2×C4—592no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1182no—
523C2×C4—1182no—
524C2×C4—1182no—
525C2×C4—1182no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1182no—
530C2×C4—1182no—
531C2×C4—4724no—
532C2×C4—2368no—
533C2×C4—2361no—
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×C4SD16×D5912yes—
544C2×C4C22×C4×D5911yescharacteristic
545C2×C4#41×D5911yescharacteristic
546C2×C4#41×D5911yescharacteristic
547C2×C4#41×D5911yescharacteristic
548C2×C4#1427377411yescharacteristic
549C2×C4#1427377511yescharacteristic
550C8—22no—
551C8—41no—
552C8—594no—
553C8—2361no—
554C22—594no—
555C22—1182no—
556C22—1182no—
557C22—1182no—
558C22—1182no—
559C22—1182no—
560C22—1182no—
561C22C4×D4×D5911yescharacteristic
562C22#1683805011yescharacteristic
563C22#1683805411yescharacteristic
564C22C2×SD16×D5911yescharacteristic
565C22#135×D5911yescharacteristic
566C22#1683821111yescharacteristic
567C22#1683821211yescharacteristic
568C4—22no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—1182no—
575C4—1182no—
576C4—4721no—
577C4—4721no—
578C4—2362no—
579C4—2361no—
580C4—2362no—
581C4#117×D5911yescharacteristic
582C4#1683787811yescharacteristic
583C4#1683808411yescharacteristic
584C4#1683808811yescharacteristic
585C2—1184no—
586C2#2142696211yescharacteristic
587C2C4×SD16×D5911yescharacteristic
588C2#2143295211yescharacteristic
589C2#2143644611yescharacteristic
590C2#2143645011yescharacteristic
591C2#2143646811yescharacteristic
592C2#2143647211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1920C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1920D59
hypocenterC59#22609
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
4. #21424823C2semidirect
5. #21424821C2semidirect
6. #21424683C2semidirect
7. #21424675C2semidirect
8. #21424673C2semidirect
12. C4×#14273658C2semidirect
13. #21423984C2semidirect
15. C59×#1920C2semidirect
25. #106×D59C4semidirect
256. C59⋊C8#117semidirect
295. #1920D59semidirect
356. C59#22609semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D592
#214269722
#214269612
#214248232
#214248212
#214246832
#214246752
#214246732
#214241852
#214241812
#214241202
C4×#142736582
#214239842
#214238662
C59×#19202
#2260959