Finite Group Databasebeta

← #28618676

Subgroups — #28618676

Order 15104.

880 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic59C211yesmaximal, characteristic
2#21426498C211yesmaximal, characteristic, complemented
3#21425657C211yesmaximal, characteristic, complemented
4#21425622C211yesmaximal, characteristic, complemented
5#21425570C211yesmaximal, characteristic, complemented
6#21425522C211yesmaximal, characteristic, complemented
7#21425519C211yesmaximal, characteristic, complemented
8#21425502C211yesmaximal, characteristic, complemented
9#21425275C211yesmaximal, characteristic
10#107×Dic59C211yesmaximal, characteristic, complemented
11#21423988C211yesmaximal, characteristic, complemented
12#102×Dic59C211yesmaximal, characteristic, complemented
13#102×Dic59C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic59C211yesmaximal, characteristic, complemented
15C59×#2033C211yesmaximal, characteristic
16C2×D4×Dic59—21no—
17C2×D4×Dic59C2211yescharacteristic
18C2×#14273683—21no—
19C22×C4×Dic59C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28#16837305C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837295—21no—
31#16837282—21no—
32#16837271C2211yescharacteristic
33#16837271C2211yescharacteristic
34#16837266—21no—
35#16837262C2211yescharacteristic
36#16837262C2211yescharacteristic
37#16837260—21no—
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41#16837253—21no—
42C22⋊C4×Dic59—21no—
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#121×C59C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C23×Dic59—21no—
60C23×Dic59D411yescharacteristic
61C23×Dic59D411yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64D4×Dic59—22no—
65D4×Dic59—22no—
66D4×Dic59—44no—
67#14273683—44no—
68C2×C4⋊Dic59—21no—
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×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C4×Dic59—22no—
83C2×C4×Dic59—22no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59D411yescharacteristic
88C2×C4×Dic59D411yescharacteristic
89C2×C4×Dic59C2311yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91#14273661—22no—
92#14273660—22no—
93#14273660—22no—
94#14273648—22no—
95#14273648—22no—
96C22⋊(C59⋊C8)—22no—
97C8×Dic59—22no—
98C8×Dic59—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C22×D4×C59—21no—
106C22×D4×C59C2311yescharacteristic
107C23×C4×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110#121×C59—22no—
111#121×C59—22no—
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117#107×C59—22no—
118#102×C59—22no—
119#102×C59—22no—
120C22⋊C8×C59—22no—
121#95×C59—21no—
122C24×C59—21no—
123C24×C59C2×D411yescharacteristic
124C2×D4×C59—24no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—42no—
131C2×D4×C59—42no—
132C2×D4×C59#4112yes—
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—21no—
141C22×C4×C59—21no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—41no—
153C22×Dic59—42no—
154C22×Dic59—42no—
155C22×Dic59—42no—
156C22×Dic59D814yes—
157C22×Dic59C2×D411yescharacteristic
158C22×Dic59C2×D411yescharacteristic
159C22×Dic59C2×D411yescharacteristic
160C22×Dic59C2×D411yescharacteristic
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59#4112yes—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59#4112yes—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—42no—
175C22⋊Dic59—42no—
176C22⋊Dic59—42no—
177C22⋊Dic59—42no—
178C22⋊Dic59—42no—
179C22⋊Dic59—42no—
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⋊Dic59—42no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
192Dic59⋊C4—42no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—22no—
196C4×Dic59—22no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59—42no—
200C4×Dic59#4112yes—
201C4×Dic59#4112yes—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8#4112yes—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—22no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59C2×D812yes—
222C23×C59C22×D411yescharacteristic
223D4×C59—22no—
224D4×C59—22no—
225D4×C59—48no—
226D4×C59—41no—
227D4×C59—41no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—28no—
257C2×Dic59—44no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—84no—
271C2×Dic59#12312yes—
272C2×Dic59#12312yes—
273C2×Dic59C2×D812yes—
274C2×Dic59C2×D812yes—
275C2×Dic59#13512yes—
276C2×Dic59#13712yes—
277C8×C59—22no—
278C8×C59—41no—
279C59⋊C8—22no—
280C59⋊C8—41no—
281#23754—591nomaximal, Sylow
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—22no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59C2×#12311yescharacteristic
297C22×C59#52511yescharacteristic
298C22×C59#53211yescharacteristic
299C22×C59C22×D811yescharacteristic
300C22×C59C2×#13511yescharacteristic
301C22×C59C2×#13711yescharacteristic
302C22×C59#57511yescharacteristic
303C4×C59—42no—
304C4×C59—41no—
305C4×C59—81no—
306C4×C59—81no—
307C4×C59#52911yescharacteristic
308C4×C59#52911yescharacteristic
309C4×C59#52911yescharacteristic
310C4×C59#52911yescharacteristic
311Dic59—22no—
312Dic59—22no—
313Dic59—22no—
314Dic59—81no—
315Dic59—81no—
316Dic59—81no—
317Dic59—81no—
318Dic59#48014yes—
319C4×#121—591no—
320#2285—591no—
321#2099—591no—
322#2093—591no—
323#2083—591no—
324#2075—591no—
325#2075—591no—
326#2071—591no—
327#2033—591no—
328#2033D5911yescharacteristic
329C4×#107—591no—
330C4×#107—591no—
331C4×#102—591no—
332C4×#102—591no—
333C4×C22⋊C8—591no—
334C2×C59—42no—
335C2×C59—82no—
336C2×C59#239511yescharacteristic
337C2×C59C2×#48011yescharacteristic
338C2×C59#309111yescharacteristic
339C2×C59#310011yescharacteristic
340C2×C59#310711yescharacteristic
341C2×C59#316911yescharacteristic
342C2×C59#317211yescharacteristic
343C2×#121C2×D5911yescharacteristic
344C2×C4×D4—591no—
345C2×C4×D4—1181no—
346C2×#117—1181no—
347C22×C42—591no—
348C2×#107—591no—
349C2×#107—591no—
350C2×#107—591no—
351C2×#107C2×D5911yescharacteristic
352C2×#102—591no—
353C2×#102—591no—
354C2×#102C2×D5911yescharacteristic
355C2×#102C2×D5911yescharacteristic
356C2×C22⋊C8—591no—
357C2×C22⋊C8C2×D5911yescharacteristic
358C2×C4×C8—591no—
359C2×C4×C8—591no—
360#390—591no—
361#389—591no—
362#388—1181no—
363#386—1181no—
364#384—591no—
365#384—591no—
366#382—1181no—
367#380—591no—
368#380—591no—
369#380—1181no—
370#380C2×D5911yescharacteristic
371C4×C4⋊C4—591no—
372C4×C4⋊C4—591no—
373C4×C4⋊C4—591no—
374C4×C4⋊C4—1181no—
375C4×C22⋊C4—1181no—
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340C2×D5911yescharacteristic
384#336—591no—
385#336—591no—
386C59#2375411yescharacteristic, complemented, Sylow
387C22×D4—21no—
388C22×D4C22×D5911yescharacteristic
389C23×C4—591no—
390C23×C4—591no—
391C23×C4—1181no—
392C23×C4C22×D5911yescharacteristic
393C22×C8—591no—
394C22×C8—591no—
395C22×C8C22×D5911yescharacteristic
396C22×C8C22×D5911yescharacteristic
397#121—22no—
398#121—22no—
399C4×D4—1182no—
400C4×D4—1182no—
401C4×D4—2364no—
402#117—2364no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4C22×D5911yescharacteristic
413C2×C4⋊C4C22×D5911yescharacteristic
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C42—591no—
423C2×C42—591no—
424C2×C42—591no—
425C2×C42—591no—
426C2×C42—1182no—
427C2×C42—1182no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42—1181no—
431#107—22no—
432#107—1182no—
433#107—1182no—
434#107—1182no—
435#102—22no—
436#102—22no—
437#102—1182no—
438#102—1182no—
439C22⋊C8—22no—
440C22⋊C8—1182no—
441C4×C8—1182no—
442C4×C8—1182no—
443#95—21no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450C24—21no—
451C24D4×D5911yescharacteristic
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4#1174568712yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—591no—
471C22×C4—594no—
472C22×C4—591no—
473C22×C4—591no—
474C22×C4—591no—
475C22×C4—1182no—
476C22×C4—1182no—
477C22×C4—1182no—
478C22×C4—1182no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—2362no—
484C22×C4—2362no—
485C22×C4—2362no—
486C22×C4—2361no—
487C22×C4D4×D5911yescharacteristic
488C22×C4C23×D5911yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—592no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8#1174568712yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—2362no—
515C4⋊C4—2362no—
516C4⋊C4—2362no—
517C4⋊C4—2362no—
518C4⋊C4—2362no—
519C4⋊C4#1174568712yes—
520C22⋊C4—24no—
521C22⋊C4—42no—
522C22⋊C4—2362no—
523C22⋊C4—2362no—
524C22⋊C4—2362no—
525C22⋊C4—2362no—
526C22⋊C4—2362no—
527C42—592no—
528C42—592no—
529C42—1182no—
530C42—1182no—
531C42—1181no—
532C42—1181no—
533C42—2362no—
534C42—2362no—
535C42—2362no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23D8×D5912yes—
550C23C2×D4×D5911yescharacteristic
551D4—22no—
552D4—22no—
553D4—48no—
554D4—41no—
555D4—41no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—592no—
573C2×C4—592no—
574C2×C4—592no—
575C2×C4—592no—
576C2×C4—592no—
577C2×C4—592no—
578C2×C4—1188no—
579C2×C4—1181no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—1181no—
585C2×C4—4724no—
586C2×C4—2364no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4#41×D5911yescharacteristic
600C2×C4#41×D5911yescharacteristic
601C2×C4#41×D5911yescharacteristic
602C2×C4C2×#1174568711yescharacteristic
603C2×C4C2×#1174568711yescharacteristic
604C2×C4C2×#1174568711yescharacteristic
605C8—22no—
606C8—41no—
607C8—1182no—
608C8—2361no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22#123×D5911yescharacteristic
624C22#1683790811yescharacteristic
625C22#1683794711yescharacteristic
626C22C2×D8×D5911yescharacteristic
627C22#135×D5911yescharacteristic
628C22#137×D5911yescharacteristic
629C22#1683819811yescharacteristic
630C4—41no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—594no—
635C4—1182no—
636C4—1182no—
637C4—1182no—
638C4—4721no—
639C4—4721no—
640C4—4721no—
641C4—4721no—
642C4#1683792911yescharacteristic
643C4#1683793111yescharacteristic
644C4#1683793311yescharacteristic
645C4#1683793311yescharacteristic
646C2—42no—
647C2—82no—
648C2#2142735211yescharacteristic
649C2#480×D5911yescharacteristic
650C2#2143400811yescharacteristic
651C2#2143409111yescharacteristic
652C2#2143416011yescharacteristic
653C2#2143467011yescharacteristic
654C2#2143469511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2033C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2033D59
hypocenterC59#23754
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
2. #21426498C2semidirect
3. #21425657C2semidirect
4. #21425622C2semidirect
5. #21425570C2semidirect
6. #21425522C2semidirect
7. #21425519C2semidirect
8. #21425502C2semidirect
10. #107×Dic59C2semidirect
11. #21423988C2semidirect
12. #102×Dic59C2semidirect
13. #102×Dic59C2semidirect
14. C22⋊C8×Dic59C2semidirect
386. C59#23754semidirect

Maximal subgroups

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

subgroupindex
#121×Dic592
#214264982
#214256572
#214256222
#214255702
#214255222
#214255192
#214255022
#214252752
#107×Dic592
#214239882
#102×Dic592
C22⋊C8×Dic592
C59×#20332
#2375459