Finite Group Databasebeta

← #28610255

Subgroups — #28610255

Order 15104.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427127C211yesmaximal, characteristic
2#21426973C211yesmaximal, characteristic
3#21424840C211yesmaximal, characteristic, complemented
4#21424774C211yesmaximal, characteristic, complemented
5#21424739C211yesmaximal, characteristic, complemented
6#21424683C211yesmaximal, characteristic, complemented
7#21424683C211yesmaximal, characteristic, complemented
8#21424530C211yesmaximal, characteristic
9#21424366C211yesmaximal, characteristic, complemented
10#21424297C211yesmaximal, characteristic
11#106×Dic59C211yesmaximal, characteristic, complemented
12#21423928C211yesmaximal, characteristic, complemented
13#21423882C211yesmaximal, characteristic
14#21423815C211yesmaximal, characteristic
15C2×C59×#483C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273637C2211yescharacteristic
24C2×C8×Dic59C2211yescharacteristic
25#16837309C2211yescharacteristic
26#16837301C2211yescharacteristic
27#16837301C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837295—21no—
30#16837295—21no—
31#16837288—21no—
32#16837288—21no—
33#16837288—21no—
34#16837262—21no—
35#16837262C2211yescharacteristic
36C4⋊C4×Dic59—21no—
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4×C22⋊Dic59—21no—
41C4×C22⋊Dic59C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×C22⋊Q8×C59C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C59×#483C418yescomplemented
55C2×#106×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C2×C22⋊C8×C59C2211yescharacteristic
60C2×#11745664—22no—
61C2×#11745664—22no—
62C2×#11745664—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65Q8×Dic59—22no—
66Q8×Dic59—41no—
67#14273686—22no—
68#14273686—41no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59D411yescharacteristic
80C2×C22⋊Dic59D411yescharacteristic
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59C2311yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88#14273661—41no—
89#14273659—22no—
90#14273651—22no—
91#14273651—41no—
92#14273637—41no—
93C8×Dic59—22no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95D411yescharacteristic
104C59⋊#95D411yescharacteristic
105C22×Q8×C59C2311yescharacteristic
106C23×C4×C59—21no—
107C23×C4×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C22⋊Q8×C59—22no—
111C22⋊Q8×C59C2×C414yes—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—22no—
115C2×C4⋊C4×C59C2×C414yes—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120#106×C59C2×C414yes—
121#106×C59C2×C414yes—
122#103×C59C2×C414yes—
123#103×C59C2×C414yes—
124C22⋊C8×C59C2×C414yes—
125C24×C59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
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×C59—24no—
141C22×C4×C59—81no—
142C22×C4×C59—81no—
143C22×C4×C59C22×C412yes—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59#4112yes—
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59C22×C412yes—
160C2×C8×C59C22×C412yes—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59C22×C412yes—
167C4⋊C4×C59C22×C412yes—
168C4⋊C4×C59C22×C412yes—
169C22⋊C4×C59—24no—
170C22⋊Dic59—22no—
171C22⋊Dic59—41no—
172C22⋊Dic59—42no—
173C22⋊Dic59—42no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—42no—
182C4⋊Dic59—42no—
183C4⋊Dic59—81no—
184C4⋊Dic59—81no—
185C4⋊Dic59#4112yes—
186C4×Dic59—21no—
187C4×Dic59—21no—
188C4×Dic59—21no—
189C4×Dic59—21no—
190C4×Dic59—42no—
191C4×Dic59—42no—
192C4×Dic59—42no—
193C4×Dic59—81no—
194C4×Dic59—81no—
195#11745664—44no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8#4112yes—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59C4×D412yes—
209C23×C59C22×D411yescharacteristic
210Q8×C59—22no—
211Q8×C59—22no—
212Q8×C59—41no—
213Q8×C59—41no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—84no—
233C2×C4×C59C4×D412yes—
234C2×C4×C59C23×C411yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—22no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
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—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59#13712yes—
264C8×C59—22no—
265C8×C59—22no—
266C59⋊C8—22no—
267C59⋊C8—41no—
268#22945—591nomaximal, Sylow
269C22×C59—22no—
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—41no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59C2×C4×D411yescharacteristic
277C22×C59#54611yescharacteristic
278C22×C59#54711yescharacteristic
279C22×C59C2×#13711yescharacteristic
280C22×C59#57511yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57711yescharacteristic
283C4×C59—22no—
284C4×C59—42no—
285C4×C59—42no—
286C4×C59—42no—
287C4×C59—42no—
288C4×C59C4×#4111yescharacteristic
289C4×C59#52011yescharacteristic
290C4×C59#55111yescharacteristic
291C4×C59#55211yescharacteristic
292Dic59—22no—
293Dic59—41no—
294Dic59—42no—
295Dic59—41no—
296Dic59—41no—
297Dic59—81no—
298Dic59—81no—
299Dic59—81no—
300C2×#483D5911yescharacteristic
301#2362—591no—
302#2345—591no—
303#1944—591no—
304#1933—591no—
305#1928—591no—
306#1920—591no—
307#1920—591no—
308#1894—591no—
309#1865—591no—
310#1854—591no—
311C4×#106—591no—
312#1763—591no—
313#1753—591no—
314#1736—591no—
315C2×C59—42no—
316C2×C59#235911yescharacteristic
317C2×C59C4×#13711yescharacteristic
318C2×C59#294911yescharacteristic
319C2×C59#332611yescharacteristic
320C2×C59#333111yescharacteristic
321C2×C59#334411yescharacteristic
322C2×C59#335011yescharacteristic
323C22×M16—591no—
324C2×C22⋊Q8C2×D5911yescharacteristic
325C2×C4×Q8—591no—
326C2×#117—591no—
327C22×C4⋊C4C2×D5911yescharacteristic
328#483Dic5918yescomplemented
329C2×#107—591no—
330C2×#106—591no—
331C2×#106C2×D5911yescharacteristic
332C2×#106C2×D5911yescharacteristic
333C2×#103—591no—
334C2×#103—591no—
335C2×#103C2×D5911yescharacteristic
336C2×#103C2×D5911yescharacteristic
337C2×C22⋊C8C2×D5911yescharacteristic
338C2×#97—591no—
339C2×C4×C8—591no—
340#391—591no—
341#389—591no—
342#389—591no—
343#389—591no—
344#388—1181no—
345#388—1181no—
346#387—1181no—
347#387—1181no—
348#387—1181no—
349#380—591no—
350#380—1181no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—1181no—
355C4×C22⋊C4—591no—
356C4×C22⋊C4—1181no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#336—591no—
366#336—591no—
367C59#2294511yescharacteristic, complemented, Sylow
368C22×Q8C22×D5911yescharacteristic
369C23×C4—21no—
370C23×C4C22×D5911yescharacteristic
371C2×M16—1182no—
372C2×M16—1182no—
373C2×M16—2361no—
374C22×C8—591no—
375C22×C8—591no—
376C22×C8C22×D5911yescharacteristic
377C22×C8C22×D5911yescharacteristic
378C22⋊Q8—22no—
379C22⋊Q8C2×Dic5914yes—
380C4×Q8—1182no—
381C4×Q8—2361no—
382#117—1182no—
383#117—2361no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4C2×Dic5914yes—
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—591no—
401C2×C22⋊C4—591no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4—1181no—
404C2×C42—591no—
405C2×C42—591no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411#107—2361no—
412#106—1182no—
413#106C2×Dic5914yes—
414#106C2×Dic5914yes—
415#103—1182no—
416#103—2361no—
417#103C2×Dic5914yes—
418#103C2×Dic5914yes—
419C22⋊C8C2×Dic5914yes—
420#97—2361no—
421C4×C8—1182no—
422#95—591no—
423#95—591no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433C24D4×D5911yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C22×Dic5912yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4D4×D5911yescharacteristic
460C22×C4#1174568712yes—
461C22×C4C22×Dic5912yes—
462C22×C4C23×D5911yescharacteristic
463M16—2364no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—592no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8C22×Dic5912yes—
475C2×C8C22×Dic5912yes—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—592no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—4721no—
490C4⋊C4—4721no—
491C4⋊C4—2362no—
492C4⋊C4—2362no—
493C4⋊C4C22×Dic5912yes—
494C4⋊C4C22×Dic5912yes—
495C4⋊C4C22×Dic5912yes—
496C22⋊C4—24no—
497C22⋊C4—1182no—
498C22⋊C4—2361no—
499C22⋊C4—2362no—
500C22⋊C4—2362no—
501C42—1181no—
502C42—1181no—
503C42—1181no—
504C42—1181no—
505C42—4721no—
506C42—4721no—
507C42—2362no—
508C42—2362no—
509C42—2362no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23D4×Dic5912yes—
517C23C2×D4×D5911yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4—592no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1182no—
553C2×C4—1181no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4D4×Dic5912yes—
566C2×C4C23×Dic5911yescharacteristic
567C2×C4#41×D5911yescharacteristic
568C2×C4#41×D5911yescharacteristic
569C2×C4C2×#1174568711yescharacteristic
570C2×C4#1427377311yescharacteristic
571C2×C4#1427377411yescharacteristic
572C8—22no—
573C8—22no—
574C8—1182no—
575C8—2361no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×D4×Dic5911yescharacteristic
584C22#1683804411yescharacteristic
585C22#1683805011yescharacteristic
586C22#137×D5911yescharacteristic
587C22#1683820111yescharacteristic
588C22#1683820811yescharacteristic
589C22#1683820811yescharacteristic
590C4—22no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—1182no—
596C4—4721no—
597C4—4721no—
598C4—4721no—
599C4—2361no—
600C4—2361no—
601C4—2361no—
602C4—2362no—
603C4#41×Dic5911yescharacteristic
604C4#1683788511yescharacteristic
605C4#1683807511yescharacteristic
606C4#1683808911yescharacteristic
607C2—42no—
608C2#2142709211yescharacteristic
609C2#137×Dic5911yescharacteristic
610C2#2143295611yescharacteristic
611C2#2143635211yescharacteristic
612C2#2143641011yescharacteristic
613C2#2143655411yescharacteristic
614C2#2143664411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#483C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC2×#483D59
hypocenterC59#22945
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #21424840C2semidirect
4. #21424774C2semidirect
5. #21424739C2semidirect
6. #21424683C2semidirect
7. #21424683C2semidirect
9. #21424366C2semidirect
11. #106×Dic59C2semidirect
12. #21423928C2semidirect
54. C59×#483C4semidirect
328. #483Dic59semidirect
367. C59#22945semidirect

Maximal subgroups

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

subgroupindex
#214271272
#214269732
#214248402
#214247742
#214247392
#214246832
#214245302
#214243662
#214242972
#106×Dic592
#214239282
#214238822
#214238152
C2×C59×#4832
#2294559