Finite Group Databasebeta

← #28610702

Subgroups — #28610702

Order 15104.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427196C211yesmaximal, characteristic, complemented
2D4×C4⋊Dic59C211yesmaximal, characteristic
3#21425066C211yesmaximal, characteristic, complemented
4#21424860C211yesmaximal, characteristic, complemented
5#21424835C211yesmaximal, characteristic, complemented
6#21424825C211yesmaximal, characteristic, complemented
7#21424683C211yesmaximal, characteristic, complemented
8#21424669C211yesmaximal, characteristic, complemented
9#21424359C211yesmaximal, characteristic
10#21424359C211yesmaximal, characteristic
11#21423973C211yesmaximal, characteristic, complemented
12#21423923C211yesmaximal, characteristic
13#21423921C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C2×C59×#478C211yesmaximal, characteristic
16C2×D4×Dic59C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C22×C4⋊Dic59—21no—
19C2×#14273659C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273638C2211yescharacteristic
24C2×#14273637C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26#16837301—21no—
27#16837301C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837301C2211yescharacteristic
31#16837288—21no—
32#16837288—21no—
33#16837277—21no—
34#16837277—21no—
35#16837272C2211yescharacteristic
36#16837262—21no—
37C4⋊C4×Dic59—21no—
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#125×C59C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C59×#478C418yescomplemented
55C2×#107×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×C4⋊C8×C59C2211yescharacteristic
58C2×#102×C59C2211yescharacteristic
59C2×#102×C59C2211yescharacteristic
60C23×Dic59—21no—
61C23×Dic59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64D4×Dic59—22no—
65D4×Dic59—22no—
66D4×Dic59—41no—
67D4×Dic59—41no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—44no—
75C2×C4⋊Dic59D411yescharacteristic
76C2×C4⋊Dic59D411yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59C2311yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88#14273659—41no—
89#14273659—41no—
90#14273648—22no—
91#14273648Q814yes—
92#14273638—22no—
93#14273637—41no—
94C8×Dic59—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108#125×C59—22no—
109#125×C59C2×C414yescomplemented
110C4×D4×C59—22no—
111C4×D4×C59C2×C414yes—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C42×C59C2311yescharacteristic
119#107×C59C2×C414yescomplemented
120#106×C59C2×C414yescomplemented
121C4⋊C8×C59C2×C414yescomplemented
122#102×C59C2×C414yes—
123#102×C59C2×C414yes—
124C24×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59C22×C412yes—
129C2×D4×C59C2×Q812yes—
130C2×D4×C59#4112yes—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—24no—
135C22×C4×C59—42no—
136C22×C4×C59—42no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—42no—
145C22×Dic59—42no—
146C22×Dic59—42no—
147C22×Dic59—42no—
148C22×Dic59—81no—
149C22×Dic59—81no—
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59C22×C412yes—
159C2×C8×C59C22×C412yes—
160C4⋊C4×C59—24no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59C22×C412yes—
166C4⋊C4×C59C22×C412yes—
167C4⋊C4×C59C22×C412yes—
168C22⋊C4×C59—24no—
169C42×C59C22×C412yes—
170C42×C59#4112yes—
171C22⋊Dic59—42no—
172C22⋊Dic59—42no—
173C4⋊Dic59—21no—
174C4⋊Dic59—22no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—41no—
182C4⋊Dic59—42no—
183C4⋊Dic59—42no—
184C4⋊Dic59—42no—
185C4⋊Dic59—42no—
186C4⋊Dic59—81no—
187C4⋊Dic59—81no—
188C4⋊Dic59C2×Q812yes—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—42no—
194C4×Dic59—81no—
195C4×Dic59—81no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8C2×Q812yes—
202C23×C59—22no—
203C23×C59—22no—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59C22×D411yescharacteristic
210D4×C59—22no—
211D4×C59—22no—
212D4×C59—22no—
213D4×C59C4×Q814yes—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—48no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59C4×D412yes—
229C2×C4×C59C4×D412yes—
230C2×C4×C59C23×C411yescharacteristic
231C2×C4×C59C22×Q811yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59H(5,𝔽2)11yescharacteristic
235C2×C4×C592−1+411yescharacteristic
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—22no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—41no—
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—84no—
260C2×Dic59—84no—
261C2×Dic59AGL(1,ℤ/8)12yes—
262C8×C59—22no—
263C8×C59—22no—
264C59⋊C8—22no—
265C59⋊C8—41no—
266#22978—591nomaximal, Sylow
267C22×C59—24no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59C2×C4×D411yescharacteristic
275C22×C59#54611yescharacteristic
276C22×C59D4×Q811yescharacteristic
277C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C59#57511yescharacteristic
279C22×C59#57711yescharacteristic
280C22×C59#57711yescharacteristic
281C4×C59—22no—
282C4×C59—22no—
283C4×C59—42no—
284C4×C59—42no—
285C4×C59—42no—
286C4×C59C2×C4×Q811yescharacteristic
287C4×C59#52011yescharacteristic
288C4×C59#55211yescharacteristic
289C4×C59#55411yescharacteristic
290Dic59—22no—
291Dic59—42no—
292Dic59—41no—
293Dic59—41no—
294Dic59—41no—
295Dic59—81no—
296Dic59—81no—
297Dic59—81no—
298C2×#478D5911yescharacteristic
299#2370—591no—
300D4×C4⋊C4—591no—
301#1992—591no—
302#1948—591no—
303#1943—591no—
304#1941—591no—
305#1920—591no—
306#1918—591no—
307#1864—591no—
308#1864—591no—
309#1771—591no—
310#1762—591no—
311#1762—591no—
312C4×#102—591no—
313C2×C59—44no—
314C2×C59#236011yescharacteristic
315C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C59#295211yescharacteristic
317C2×C59#330711yescharacteristic
318C2×C59#331011yescharacteristic
319C2×C59#338411yescharacteristic
320C2×C59#338811yescharacteristic
321C2×#125C2×D5911yescharacteristic
322C2×C4×D4—591no—
323C2×C4×D4—591no—
324C2×C4×D4C2×D5911yescharacteristic
325C22×C4⋊C4—1181no—
326#478Dic5918yescomplemented
327C2×#107C2×D5911yescharacteristic
328C2×#106—591no—
329C2×#106—591no—
330C2×#106C2×D5911yescharacteristic
331C2×C4⋊C8C2×D5911yescharacteristic
332C2×#102—591no—
333C2×#102—591no—
334C2×#102C2×D5911yescharacteristic
335C2×#102C2×D5911yescharacteristic
336C2×#97—591no—
337C2×#97—591no—
338C2×C4×C8—591no—
339#389—591no—
340#389—591no—
341#389—591no—
342#389—591no—
343#389—1181no—
344#387—1181no—
345#387—1181no—
346#385—1181no—
347#385—1181no—
348#384—591no—
349#380—1181no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—1181no—
355#340—591no—
356#340—591no—
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#2297811yescharacteristic, complemented, Sylow
366C22×D4C22×D5911yescharacteristic
367C23×C4—21no—
368C23×C4—1181no—
369C23×C4—1181no—
370C22×C8—591no—
371C22×C8—591no—
372C22×C8C22×D5911yescharacteristic
373C22×C8C22×D5911yescharacteristic
374#125—22no—
375#125C2×Dic5914yescomplemented
376C4×D4—22no—
377C4×D4—1182no—
378C4×D4—1182no—
379C4×D4—2361no—
380C4×D4—2361no—
381C4×D4C2×Dic5914yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—2364no—
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C42—591no—
404C2×C42—591no—
405C2×C42—1181no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42C22×D5911yescharacteristic
409#107C2×Dic5914yescomplemented
410#106—2361no—
411#106—2361no—
412#106C2×Dic5914yescomplemented
413C4⋊C8C2×Dic5914yescomplemented
414#102—594no—
415#102—1182no—
416#102C2×Dic5914yes—
417#102C2×Dic5914yes—
418#97—1182no—
419#97—2361no—
420C4×C8—2361no—
421#95—591no—
422#95—591no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4Q8×D5912yes—
435C2×D4#1174568912yes—
436C2×D4C22×Dic5912yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—591no—
444C22×C4—591no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—4721no—
452C22×C4—4721no—
453C22×C4—2362no—
454C22×C4—2362no—
455C22×C4—2362no—
456C22×C4—2362no—
457C22×C4D4×D5911yescharacteristic
458C22×C4D4×D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—592no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8C22×Dic5912yes—
471C2×C8C22×Dic5912yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—592no—
478C4⋊C4—1182no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—4721no—
487C4⋊C4—4721no—
488C4⋊C4—2361no—
489C4⋊C4—2362no—
490C4⋊C4—2362no—
491C4⋊C4—2362no—
492C4⋊C4—2362no—
493C4⋊C4C22×Dic5912yes—
494C4⋊C4C22×Dic5912yes—
495C4⋊C4C22×Dic5912yes—
496C22⋊C4—24no—
497C22⋊C4—2362no—
498C22⋊C4—2362no—
499C42—1181no—
500C42—1181no—
501C42—1181no—
502C42—1181no—
503C42—4721no—
504C42—4721no—
505C42—2362no—
506C42#1174568712yes—
507C42C22×Dic5912yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D5911yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4Q8×Dic5914yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—48no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—592no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1182no—
547C2×C4—4724no—
548C2×C4—4724no—
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—2361no—
560C2×C4D4×Dic5912yes—
561C2×C4D4×Dic5912yes—
562C2×C4C23×Dic5911yescharacteristic
563C2×C4C2×Q8×D5911yescharacteristic
564C2×C4C2×#1174568711yescharacteristic
565C2×C4C2×#1174568911yescharacteristic
566C2×C4#1427377311yescharacteristic
567C2×C4#1427377411yescharacteristic
568C8—22no—
569C8—22no—
570C8—1182no—
571C8—2361no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×D4×Dic5911yescharacteristic
580C22#1683804311yescharacteristic
581C22#1683806511yescharacteristic
582C22AGL(1,ℤ/8)×D5911yescharacteristic
583C22#1683819811yescharacteristic
584C22#1683820911yescharacteristic
585C22#1683820911yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—1182no—
592C4—4721no—
593C4—4721no—
594C4—4721no—
595C4—2362no—
596C4—2361no—
597C4—2361no—
598C4—2361no—
599C4C2×Q8×Dic5911yescharacteristic
600C4#1683788511yescharacteristic
601C4#1683809411yescharacteristic
602C4#1683811011yescharacteristic
603C2—44no—
604C2#2142710511yescharacteristic
605C2AGL(1,ℤ/8)×Dic5911yescharacteristic
606C2#2143298011yescharacteristic
607C2#2143606911yescharacteristic
608C2#2143611411yescharacteristic
609C2#2143699811yescharacteristic
610C2#2143703811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#478C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC2×#478D59
hypocenterC59#22978
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #21427196C2semidirect
3. #21425066C2semidirect
4. #21424860C2semidirect
5. #21424835C2semidirect
6. #21424825C2semidirect
7. #21424683C2semidirect
8. #21424669C2semidirect
11. #21423973C2semidirect
54. C59×#478C4semidirect
109. #125×C59C2×C4semidirect
119. #107×C59C2×C4semidirect
120. #106×C59C2×C4semidirect
121. C4⋊C8×C59C2×C4semidirect
326. #478Dic59semidirect
365. C59#22978semidirect
375. #125C2×Dic59semidirect
409. #107C2×Dic59semidirect
412. #106C2×Dic59semidirect
413. C4⋊C8C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214271962
D4×C4⋊Dic592
#214250662
#214248602
#214248352
#214248252
#214246832
#214246692
#214243592
#214239732
#214239232
#214239212
#102×Dic592
C2×C59×#4782
#2297859