Finite Group Databasebeta

← #28598244

Subgroups — #28598244

Order 15104.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838033C211yesmaximal, characteristic
2C2×#16837460C211yesmaximal, characteristic
3#21426703C211yesmaximal, characteristic
4#21425067C211yesmaximal, characteristic, complemented
5#21425063C211yesmaximal, characteristic, complemented
6#21425059C211yesmaximal, characteristic, complemented
7#21425055C211yesmaximal, characteristic, complemented
8#21425052C211yesmaximal, characteristic, complemented
9#21425050C211yesmaximal, characteristic, complemented
10#21425046C211yesmaximal, characteristic, complemented
11#21425042C211yesmaximal, characteristic, complemented
12#21424192C211yesmaximal, characteristic
13#21424188C211yesmaximal, characteristic
14#21424182C211yesmaximal, characteristic
15C59×#1824C211yesmaximal, characteristic
16#16838033—24no—
17#16838033C418yescomplemented
18C2×#14273726—21no—
19C2×#14273726C2211yescharacteristic
20C2×#14273726C2211yescharacteristic
21C2×#14273718—21no—
22C2×#14273718C2211yescharacteristic
23C2×#14273718C2211yescharacteristic
24C2×#14273706—21no—
25C2×#14273706—21no—
26C2×#14273686C2211yescharacteristic
27C2×#14273686C2211yescharacteristic
28#16837460—24no—
29C2×Dic59⋊C8C2211yescharacteristic
30C2×Dic59⋊C8C2211yescharacteristic
31C2×C22⋊(C59⋊C8)C2211yescharacteristic
32C2×#14273637C2211yescharacteristic
33C2×C8×Dic59C2211yescharacteristic
34#16837296—21no—
35#16837290—21no—
36#16837287—21no—
37#16837285—21no—
38#16837271C2211yescharacteristic
39#16837267C2211yescharacteristic
40#16837262C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837164C2211yescharacteristic
55C22×C4⋊C4×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#340C2211yescharacteristic
62C22×Dic118—21no—
63C22×Dic118—21no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273726—28no—
67#14273726—22no—
68#14273726—22no—
69#14273726C2×C414yes—
70#14273726C2×C414yes—
71#14273718—28no—
72#14273718—22no—
73#14273718—22no—
74#14273718C2×C414yes—
75#14273718C2×C414yes—
76#14273706—28no—
77#14273706—28no—
78#14273686—22no—
79#14273686C2×C414yes—
80#14273686C2×C414yes—
81#14273686D414yes—
82C2×C4⋊Dic59—21no—
83C2×C4⋊Dic59—21no—
84C2×C4⋊Dic59D411yescharacteristic
85C2×C4⋊Dic59D411yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×C4⋊Dic59C2311yescharacteristic
88C2×C4⋊Dic59C2311yescharacteristic
89C2×C4⋊Dic59C2311yescharacteristic
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4—21no—
92C2×Dic59⋊C4—21no—
93C2×Dic59⋊C4—21no—
94C2×Dic59⋊C4—21no—
95C2×Dic59⋊C4—21no—
96C2×C22⋊Dic59—21no—
97C2×C22⋊Dic59—21no—
98C2×C22⋊Dic59D411yescharacteristic
99C2×C22⋊Dic59D411yescharacteristic
100C2×C4×Dic59—21no—
101C2×C4×Dic59—21no—
102C2×C4×Dic59—21no—
103C2×C4×Dic59C2311yescharacteristic
104C2×C4×Dic59C2311yescharacteristic
105Dic59⋊C8—22no—
106Dic59⋊C8—22no—
107C22⋊(C59⋊C8)—22no—
108#14273637—22no—
109C8×Dic59—22no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C23×C4×C59—21no—
116C23×C4×C59C2311yescharacteristic
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C2×C4⋊C4×C59—21no—
120C2×C4⋊C4×C59—22no—
121C2×C4⋊C4×C59C2×C414yes—
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C4⋊C4×C59C2311yescharacteristic
126C2×C22⋊C4×C59—21no—
127C22⋊C8×C59—22no—
128#95×C59—21no—
129C24×C59C2×D411yescharacteristic
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—24no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—81no—
140C22×C4×C59—81no—
141C22×C4×C59C22×C412yes—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D412yes—
144C22×C4×C59C2411yescharacteristic
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C2×Dic118—24no—
154C2×Dic118—24no—
155C2×Dic118—81no—
156C2×Dic118—81no—
157C2×Dic118—81no—
158C2×Dic118—81no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59C22×C412yes—
173C4⋊C4×C59C22×C412yes—
174C22⋊C4×C59—81no—
175C22⋊Dic59—24no—
176C22⋊Dic59—81no—
177C22⋊Dic59C22⋊C414yes—
178C22⋊Dic59C22⋊C414yes—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—24no—
182C4⋊Dic59—24no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59C22⋊C414yes—
188C4⋊Dic59C22⋊C414yes—
189C4⋊Dic59C22×C412yes—
190C4⋊Dic59C22×C412yes—
191Dic59⋊C4—24no—
192Dic59⋊C4—24no—
193Dic59⋊C4—24no—
194Dic59⋊C4—24no—
195Dic59⋊C4—81no—
196Dic59⋊C4—81no—
197C4×Dic59—24no—
198C4×Dic59—81no—
199C4×Dic59—81no—
200C4×Dic59C22×C412yes—
201C4×Dic59C22×C412yes—
202C4×Dic59C2×D412yes—
203C4×Dic59C2×D412yes—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59C2×C22⋊C412yes—
217C23×C59C22×D411yescharacteristic
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
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—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—84no—
237C2×C4×C59C2×C22⋊C412yes—
238C2×C4×C59C23×C411yescharacteristic
239C2×C4×C59C22×D411yescharacteristic
240C2×C4×C592−1+411yescharacteristic
241C2×C4×C592−1+411yescharacteristic
242C2×C4×C592−1+411yescharacteristic
243C2×C4×C592−1+411yescharacteristic
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—22no—
253C2×Dic59—22no—
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—41no—
264C2×Dic59C2×C22⋊C412yes—
265C2×Dic59C2×C22⋊C412yes—
266C2×Dic59C2×C22⋊C412yes—
267C2×Dic59C2×C22⋊C412yes—
268Dic118—84no—
269Dic118—84no—
270C8×C59—41no—
271C8×C59—41no—
272C59⋊C8—41no—
273C59⋊C8—41no—
274#21843—591nomaximal, Sylow
275C22×C59—22no—
276C22×C59—22no—
277C22×C59—22no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59C22×C22⋊C411yescharacteristic
283C22×C59#53611yescharacteristic
284C22×C59#53611yescharacteristic
285C22×C59#57611yescharacteristic
286C22×C59#57711yescharacteristic
287C22×C59#57711yescharacteristic
288C22×C59#57811yescharacteristic
289C4×C59—22no—
290C4×C59—42no—
291C4×C59—42no—
292C4×C59—81no—
293C4×C59—81no—
294C4×C59#51911yescharacteristic
295C4×C59#51911yescharacteristic
296C4×C59#53611yescharacteristic
297C4×C59#53611yescharacteristic
298Dic59—22no—
299Dic59—22no—
300Dic59—22no—
301Dic59—22no—
302Dic59—42no—
303Dic59—42no—
304Dic59—81no—
305Dic59—81no—
306C2×#544—591no—
307C2×#433—591no—
308#2314—591no—
309#1993—591no—
310#1992—591no—
311#1991—591no—
312#1990—591no—
313#1989—591no—
314#1989—591no—
315#1988—591no—
316#1987—591no—
317#1825—591no—
318#1825—591no—
319#1824D5911yescharacteristic
320#1823—591no—
321C2×C59—42no—
322C2×C59#231011yescharacteristic
323C2×C59#289711yescharacteristic
324C2×C59#289811yescharacteristic
325C2×C59#324111yescharacteristic
326C2×C59#324211yescharacteristic
327C2×C59#324311yescharacteristic
328C2×C59#324411yescharacteristic
329#544—598no—
330#544—1184no—
331C2×C4⋊Q8—591no—
332C2×C4⋊Q8—591no—
333C2×C4⋊Q8—1181no—
334C2×#125—591no—
335C2×#125—591no—
336C2×#125—1181no—
337C2×C22⋊Q8—1181no—
338C2×C22⋊Q8—1181no—
339C2×#117—591no—
340C2×#117—591no—
341C22×C4⋊C4C2×D5911yescharacteristic
342#433—1184no—
343C2×C4⋊C8—591no—
344C2×C4⋊C8—591no—
345C2×C22⋊C8—591no—
346C2×C22⋊C8C2×D5911yescharacteristic
347C2×#97—591no—
348C2×C4×C8—591no—
349#388—1181no—
350#388—1181no—
351#387—1181no—
352#387—1181no—
353#384—591no—
354#382—591no—
355#380—591no—
356#380C2×D5911yescharacteristic
357C4×C4⋊C4—591no—
358C4×C4⋊C4—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340C2×D5911yescharacteristic
372#340C2×D5911yescharacteristic
373#340C2×D5911yescharacteristic
374#340C2×D5911yescharacteristic
375C59#2184311yescharacteristic, complemented, Sylow
376C22×Q8—1181no—
377C22×Q8—1181no—
378C23×C4—21no—
379C23×C4C22×D5911yescharacteristic
380C22×C8—591no—
381C22×C8—591no—
382C22×C8C22×D5911yescharacteristic
383C22×C8C22×D5911yescharacteristic
384C4⋊Q8—594no—
385C4⋊Q8—594no—
386C4⋊Q8—1182no—
387C4⋊Q8—1182no—
388C4⋊Q8—1188no—
389#125—594no—
390#125—594no—
391#125—1182no—
392#125—1182no—
393#125—1188no—
394C22⋊Q8—1188no—
395C22⋊Q8—1188no—
396#117—594no—
397#117—594no—
398#117—594no—
399#117—1182no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4C4×D5914yes—
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—591no—
423C2×C22⋊C4—591no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C42—591no—
427C2×C42—591no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42—1181no—
431C4⋊C8—1182no—
432C4⋊C8—1182no—
433C22⋊C8—22no—
434C22⋊C8—1182no—
435#97—1182no—
436C4×C8—1182no—
437#95—21no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443C24D4×D5911yescharacteristic
444C2×Q8—1184no—
445C2×Q8—1184no—
446C2×Q8—4721no—
447C2×Q8—4721no—
448C2×Q8—4721no—
449C2×Q8—4721no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—591no—
462C22×C4—591no—
463C22×C4—591no—
464C22×C4—591no—
465C22×C4—1181no—
466C22×C4—1181no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4C2×C4×D5912yes—
470C22×C4D4×D5911yescharacteristic
471C22×C4D4×D5912yes—
472C22×C4C23×D5911yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—594no—
493C4⋊C4—594no—
494C4⋊C4—592no—
495C4⋊C4—592no—
496C4⋊C4—1184no—
497C4⋊C4—1184no—
498C4⋊C4—1184no—
499C4⋊C4—1184no—
500C4⋊C4—1184no—
501C4⋊C4—1184no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—4721no—
509C4⋊C4—4721no—
510C4⋊C4C2×C4×D5912yes—
511C4⋊C4C2×C4×D5912yes—
512C22⋊C4—81no—
513C22⋊C4—594no—
514C22⋊C4—594no—
515C22⋊C4—1184no—
516C22⋊C4—4721no—
517C42—592no—
518C42—592no—
519C42—592no—
520C42—592no—
521C42—1184no—
522C42—4721no—
523C42—4721no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C22⋊C4×D5912yes—
531C23C2×D4×D5911yescharacteristic
532Q8—4724no—
533Q8—4724no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—84no—
553C2×C4—592no—
554C2×C4—592no—
555C2×C4—592no—
556C2×C4—592no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—1181no—
560C2×C4—1181no—
561C2×C4—1181no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—1181no—
565C2×C4—1182no—
566C2×C4—1182no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4C22⋊C4×D5912yes—
578C2×C4C22×C4×D5911yescharacteristic
579C2×C4C2×D4×D5911yescharacteristic
580C2×C4#1427377411yescharacteristic
581C2×C4#1427377411yescharacteristic
582C2×C4#1427377411yescharacteristic
583C2×C4#1427377411yescharacteristic
584C8—41no—
585C8—41no—
586C8—2361no—
587C8—2361no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×C22⋊C4×D5911yescharacteristic
596C22#1683797411yescharacteristic
597C22#1683797411yescharacteristic
598C22#1683820611yescharacteristic
599C22#1683820811yescharacteristic
600C22#1683821211yescharacteristic
601C22#1683821511yescharacteristic
602C4—22no—
603C4—42no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—1182no—
608C4—1182no—
609C4—1182no—
610C4—1182no—
611C4—4721no—
612C4—4721no—
613C4—2362no—
614C4—2362no—
615C4#1683787911yescharacteristic
616C4#1683787911yescharacteristic
617C4#1683797411yescharacteristic
618C4#1683797411yescharacteristic
619C2—42no—
620C2#2142668111yescharacteristic
621C2#2143262111yescharacteristic
622C2#2143262411yescharacteristic
623C2#2143535311yescharacteristic
624C2#2143536811yescharacteristic
625C2#2143538311yescharacteristic
626C2#2143539011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1824C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1824D59
hypocenterC59#21843
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. #21425067C2semidirect
5. #21425063C2semidirect
6. #21425059C2semidirect
7. #21425055C2semidirect
8. #21425052C2semidirect
9. #21425050C2semidirect
10. #21425046C2semidirect
11. #21425042C2semidirect
17. #16838033C4semidirect
375. C59#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#168380332
C2×#168374602
#214267032
#214250672
#214250632
#214250592
#214250552
#214250522
#214250502
#214250462
#214250422
#214241922
#214241882
#214241822
C59×#18242
#2184359