Finite Group Databasebeta

← #28610253

Subgroups — #28610253

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
1C2×#16837733C211yesmaximal, characteristic
2#21427121C211yesmaximal, characteristic
3#21426971C211yesmaximal, characteristic
4#21424839C211yesmaximal, characteristic, complemented
5#21424772C211yesmaximal, characteristic, complemented
6#21424738C211yesmaximal, characteristic, complemented
7#21424682C211yesmaximal, characteristic, complemented
8#21424682C211yesmaximal, characteristic, complemented
9#21424529C211yesmaximal, characteristic
10#21424367C211yesmaximal, characteristic, complemented
11#21424297C211yesmaximal, characteristic
12C4×#14273659C211yesmaximal, characteristic, complemented
13#21423927C211yesmaximal, characteristic, complemented
14#21423816C211yesmaximal, characteristic
15C59×#1753C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837733C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28C2×#14273652C2211yescharacteristic
29#16837308C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837296—21no—
34#16837290—21no—
35#16837285—21no—
36#16837285—21no—
37#16837284—21no—
38#16837262C2211yescharacteristic
39#16837260—21no—
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43#16837253—21no—
44C22⋊C4×Dic59—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C22×M16×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C22⋊C4×C59C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C23×Dic59—21no—
62#14273706—22no—
63#14273706C2×C414yes—
64C4×Dic118—22no—
65C4×Dic118—41no—
66#14273686—22no—
67#14273686—41no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59C2×C414yes—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
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×Dic59⋊C4—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×C22⋊Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91#14273661—41no—
92#14273659—22no—
93#14273659C2×C414yes—
94#14273659C2×C414yes—
95#14273652—22no—
96#14273652—41no—
97#14273652C2×C414yes—
98#14273652C2×C414yes—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C23×C4×C59C2311yescharacteristic
109C2×M16×C59—22no—
110C2×M16×C59—22no—
111C2×M16×C59—41no—
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C22⋊C8×C59C2×C414yes—
121#97×C59—41no—
122C4×C8×C59—22no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59C22×C412yes—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59#4112yes—
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—24no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—81no—
148C22×Dic59—81no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118C22×C412yes—
155M16×C59—44no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59C22×C412yes—
166C2×C8×C59C22×C412yes—
167C2×C8×C59#4112yes—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—41no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C22⋊Dic59—24no—
175C22⋊Dic59—42no—
176C22⋊Dic59—42no—
177C4⋊Dic59—24no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59C22×C412yes—
189C4⋊Dic59C22×C412yes—
190C4⋊Dic59C22×C412yes—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—24no—
193Dic59⋊C4—42no—
194Dic59⋊C4—42no—
195Dic59⋊C4—81no—
196Dic59⋊C4—81no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59—42no—
200C4×Dic59—81no—
201C4×Dic59—81no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59C4×D412yes—
209C23×C59C22×D411yescharacteristic
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
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×C59C4×D412yes—
228C2×C4×C59#13712yes—
229C2×C4×C59C23×C411yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59H(5,𝔽2)11yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×Dic59—22no—
236C2×Dic59—22no—
237C2×Dic59—22no—
238C2×Dic59—22no—
239C2×Dic59—22no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
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—
260Dic118—22no—
261Dic118—22no—
262Dic118—41no—
263Dic118—41no—
264C8×C59—22no—
265C8×C59—22no—
266C8×C59—22no—
267C8×C59—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—22no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—41no—
288C4×C59C4×#4111yescharacteristic
289C4×C59#52011yescharacteristic
290C4×C59#55111yescharacteristic
291C4×C59#55211yescharacteristic
292Dic59—42no—
293Dic59—42no—
294Dic59—42no—
295Dic59—42no—
296Dic59—42no—
297Dic59—81no—
298Dic59—81no—
299Dic59—81no—
300C2×#483—591no—
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#1753D5911yescharacteristic
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×M16C2×D5911yescharacteristic
324C2×C22⋊Q8—591no—
325C2×C4×Q8—591no—
326C2×#117—591no—
327C22×C4⋊C4—591no—
328#483—598no—
329C2×#107—591no—
330C2×#106—591no—
331C2×#106—591no—
332C2×#106—591no—
333C2×#103—591no—
334C2×#103—591no—
335C2×#103—591no—
336C2×#103—591no—
337C2×C22⋊C8C2×D5911yescharacteristic
338C2×#97C2×D5911yescharacteristic
339C2×C4×C8C2×D5911yescharacteristic
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—1181no—
356C4×C22⋊C4C2×D5911yescharacteristic
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#336C2×D5911yescharacteristic
366#336C2×D5911yescharacteristic
367C59#2294511yescharacteristic, complemented, Sylow
368C22×Q8—591no—
369C23×C4—1181no—
370C23×C4C22×D5911yescharacteristic
371C2×M16—22no—
372C2×M16—22no—
373C2×M16—41no—
374C22×C8C22×D5911yescharacteristic
375C22×C8C22×D5911yescharacteristic
376C22×C8C22×D5911yescharacteristic
377C22×C8C22×D5911yescharacteristic
378C22⋊Q8—594no—
379C22⋊Q8—1182no—
380C4×Q8—1182no—
381C4×Q8—2361no—
382#117—1182no—
383#117—2361no—
384C2×C4⋊C4—594no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1182no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4D23611yescharacteristic
403C2×C22⋊C4D23611yescharacteristic
404C2×C42—1181no—
405C2×C42—1181no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42C22×D5911yescharacteristic
410C2×C42C22×D5911yescharacteristic
411#107—2361no—
412#106—594no—
413#106—594no—
414#106—1182no—
415#103—594no—
416#103—594no—
417#103—1182no—
418#103—2361no—
419C22⋊C8C4×D5914yes—
420#97—41no—
421C4×C8—22no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95D23611yescharacteristic
432#95D23611yescharacteristic
433C24C2×D23611yescharacteristic
434C2×Q8—592no—
435C2×Q8—1181no—
436C2×Q8—1181no—
437C2×Q8—1181no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—1184no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—4721no—
454C22×C4—4721no—
455C22×C4C2×C4×D5912yes—
456C22×C4C2×D23611yescharacteristic
457C22×C4C2×D23611yescharacteristic
458C22×C4C2×D23611yescharacteristic
459C22×C4C2×D23611yescharacteristic
460C22×C4C2×D23611yescharacteristic
461C22×C4#1174568512yes—
462C22×C4C23×D5911yescharacteristic
463M16—44no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C2×C4×D5912yes—
474C2×C8C2×C4×D5912yes—
475C2×C8#1174568712yes—
476C4⋊C4—592no—
477C4⋊C4—592no—
478C4⋊C4—592no—
479C4⋊C4—592no—
480C4⋊C4—1184no—
481C4⋊C4—1184no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—4721no—
493C4⋊C4—4721no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C22⋊C4—22no—
497C22⋊C4—41no—
498C22⋊C4—1184no—
499C22⋊C4—2362no—
500C22⋊C4—2362no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—4721no—
506C42—4721no—
507C42—2362no—
508C42—2362no—
509C42—2362no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C4×D23612yes—
517C23C22×D23611yescharacteristic
518Q8—1182no—
519Q8—1182no—
520Q8—2361no—
521Q8—2361no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—1182no—
540C2×C4—1182no—
541C2×C4—1182no—
542C2×C4—1182no—
543C2×C4—1182no—
544C2×C4—4724no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
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—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4C4×D23612yes—
565C2×C4#1427376012yes—
566C2×C4C22×C4×D5911yescharacteristic
567C2×C4#41×D5911yescharacteristic
568C2×C4C2×#1174568711yescharacteristic
569C2×C4C2×#1174568511yescharacteristic
570C2×C4#1427377211yescharacteristic
571C2×C4#1427377411yescharacteristic
572C8—22no—
573C8—22no—
574C8—22no—
575C8—41no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×C4×D23611yescharacteristic
584C22#1683804711yescharacteristic
585C22#1683805611yescharacteristic
586C22C2×#1427376011yescharacteristic
587C22#1683820211yescharacteristic
588C22#1683821411yescharacteristic
589C22#1683821411yescharacteristic
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—4721no—
596C4—4721no—
597C4—4721no—
598C4—2362no—
599C4—2362no—
600C4—2362no—
601C4—2362no—
602C4—2362no—
603C4C4×#1174568711yescharacteristic
604C4#1683788111yescharacteristic
605C4#1683807811yescharacteristic
606C4#1683808511yescharacteristic
607C2—42no—
608C2#2142709011yescharacteristic
609C2C4×#1427376011yescharacteristic
610C2#2143295711yescharacteristic
611C2#2143635111yescharacteristic
612C2#2143641311yescharacteristic
613C2#2143655311yescharacteristic
614C2#2143664311yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1753C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1753D59
hypocenterC59#22945
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #21424839C2semidirect
5. #21424772C2semidirect
6. #21424738C2semidirect
7. #21424682C2semidirect
8. #21424682C2semidirect
10. #21424367C2semidirect
12. C4×#14273659C2semidirect
13. #21423927C2semidirect
20. #16837733C4semidirect
367. C59#22945semidirect

Maximal subgroups

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

subgroupindex
C2×#168377332
#214271212
#214269712
#214248392
#214247722
#214247382
#214246822
#214245292
#214243672
#214242972
C4×#142736592
#214239272
#214238162
C59×#17532
#2294559