Finite Group Databasebeta

← #28610191

Subgroups — #28610191

Order 15104.

874 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837715C211yesmaximal, characteristic
2#21427132C211yesmaximal, characteristic
3#21424831C211yesmaximal, characteristic, complemented
4#21424778C211yesmaximal, characteristic, complemented
5#21424723C211yesmaximal, characteristic, complemented
6#21424685C211yesmaximal, characteristic, complemented
7#21424667C211yesmaximal, characteristic, complemented
8#21424532C211yesmaximal, characteristic
9#21424342C211yesmaximal, characteristic, complemented
10#21424293C211yesmaximal, characteristic, complemented
11C4×#14273660C211yesmaximal, characteristic, complemented
12#21423924C211yesmaximal, characteristic, complemented
13#21423880C211yesmaximal, characteristic, complemented
14#21423815C211yesmaximal, characteristic, complemented
15C59×#2345C211yesmaximal, characteristic, complemented
16C22×#11745664C2211yescharacteristic
17C2×#14273701C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837715C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837302—21no—
34#16837300C2211yescharacteristic
35#16837291—21no—
36#16837278—21no—
37C4×C4⋊Dic59C2211yescharacteristic
38#16837249—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C22×C4⋊C4×C59C2211yescharacteristic
50C59×#389C2211yescharacteristic
51C59×#389C2211yescharacteristic
52C59×#387—21no—
53C59×#387—21no—
54C59×#380—21no—
55C59×#380C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C4×C22⋊C4×C59C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C22⋊D59—21no—
62C22×C4×D59—21no—
63C2×#11745664—22no—
64C2×#11745664—22no—
65C2×#11745664—41no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273701—22no—
71#14273701C2×C414yes—
72C4×D236—22no—
73C4×D236—41no—
74#14273686—22no—
75#14273686—41no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×C22⋊Dic59—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87#14273660—22no—
88#14273660C2×C414yescomplemented
89#14273660C2×C414yescomplemented
90#14273658—41no—
91#14273647—22no—
92#14273647—41no—
93#14273647C2×C414yes—
94#14273647C2×C414yes—
95C22⋊(C59⋊C8)C2×C414yescomplemented
96#14273638—41no—
97C4×C59⋊C8—22no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C23×C4×C59—21no—
101C23×C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—22no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59C2×C414yescomplemented
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59D411yescharacteristic
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C23×D59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—24no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—81no—
137C22×C4×C59—81no—
138C22×C4×C59C22×C412yes—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59#4112yes—
145C22×C4×C59C2411yescharacteristic
146C2×C22⋊D59—24no—
147C2×C22⋊D59—81no—
148C2×C22⋊D59—81no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236C22×C412yes—
158C2×C4×D59—42no—
159C2×C4×D59—42no—
160C2×C4×D59—81no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—81no—
169C4⋊C4×C59—81no—
170C4⋊C4×C59C22×C412yes—
171C4⋊C4×C59C22×C412yes—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—41no—
174C22⋊C4×C59—42no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—81no—
180C42×C59—81no—
181C22⋊Dic59—42no—
182(C2×D59)⋊C4—24no—
183(C2×D59)⋊C4—42no—
184(C2×D59)⋊C4—42no—
185(C2×D59)⋊C4—81no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59C22×C412yes—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—81no—
193C4×Dic59—42no—
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—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8C22×C412yes—
206C2×C59⋊C8C22×C412yes—
207C2×C59⋊C8#4112yes—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59C4×D412yes—
215C23×C59C22×D411yescharacteristic
216C22×D59—22no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—84no—
253C2×C4×C59C4×D412yes—
254C2×C4×C59#13712yes—
255C2×C4×C59C23×C411yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C2×C4×C592−1+411yescharacteristic
261C22⋊D59—84no—
262C2×Dic59—22no—
263C2×Dic59—22no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271D236—22no—
272D236—22no—
273D236—41no—
274D236—41no—
275C4×D59—84no—
276C59⋊C8—22no—
277C59⋊C8—22no—
278C59⋊C8—22no—
279C59⋊C8—41no—
280#22941—591nomaximal, Sylow
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59C2×C4×D411yescharacteristic
289C22×C59#54611yescharacteristic
290C22×C59#54711yescharacteristic
291C22×C59C2×#13711yescharacteristic
292C22×C59#57511yescharacteristic
293C22×C59#57611yescharacteristic
294C22×C59#57611yescharacteristic
295C2×D59—42no—
296C2×D59—42no—
297C2×D59—42no—
298C2×D59—81no—
299C2×D59—81no—
300C2×D59—81no—
301C2×D59—81no—
302C4×C59—22no—
303C4×C59—22no—
304C4×C59—41no—
305C4×C59—41no—
306C4×C59—42no—
307C4×C59—42no—
308C4×C59—41no—
309C4×C59—81no—
310C4×C59—81no—
311C4×C59C4×#4111yescharacteristic
312C4×C59#52011yescharacteristic
313C4×C59#55111yescharacteristic
314C4×C59#55211yescharacteristic
315Dic59—42no—
316Dic59—42no—
317Dic59—81no—
318C2×#480—591no—
319#2363—591no—
320#2345D5911yescharacteristic, complemented
321#1943—591no—
322#1934—591no—
323#1926—591no—
324#1921—591no—
325#1918—591no—
326#1895—591no—
327#1862—591no—
328#1854—591no—
329C4×#107—591no—
330#1762—591no—
331#1753—591no—
332#1736—591no—
333C2×C59—42no—
334C2×C59#235911yescharacteristic
335C2×C59C4×#13711yescharacteristic
336C2×C59#294811yescharacteristic
337C2×C59#332211yescharacteristic
338C2×C59#333311yescharacteristic
339C2×C59#334611yescharacteristic
340C2×C59#334711yescharacteristic
341D59—82no—
342C22×M16—591no—
343C2×#121—591no—
344C2×C4×D4—591no—
345C2×#117—591no—
346C22×C4⋊C4C2×D5911yescharacteristic
347#480—598no—
348C2×#107—591no—
349C2×#107—591no—
350C2×#107—591no—
351C2×#106—591no—
352C2×#102—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×C22⋊C8—591no—
357C2×#97—591no—
358C2×C4×C8—591no—
359#390—591no—
360#390—1181no—
361#389—591no—
362#389C2×D5911yescharacteristic
363#389C2×D5911yescharacteristic
364#388—1181no—
365#387—21no—
366#387—21no—
367#386—1181no—
368#380—21no—
369#380C2×D5911yescharacteristic
370C4×C4⋊C4—591no—
371C4×C4⋊C4C2×D5911yescharacteristic
372C4×C4⋊C4C2×D5911yescharacteristic
373C4×C22⋊C4—21no—
374C4×C22⋊C4—1181no—
375C4×C22⋊C4C2×D5911yescharacteristic
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#336—591no—
385#336—591no—
386C59#2294111yescharacteristic, complemented, Sylow
387C22×D4—591no—
388C22×D4—1181no—
389C23×C4—21no—
390C23×C4—1181no—
391C23×C4C22×D5911yescharacteristic
392C2×M16—1182no—
393C2×M16—1182no—
394C2×M16—2361no—
395C22×C8—591no—
396C22×C8—591no—
397C22×C8—591no—
398C22×C8—591no—
399#121—594no—
400#121—1182no—
401C4×D4—1182no—
402C4×D4—2361no—
403#117—1182no—
404#117—2361no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4C4×D5914yescomplemented
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4C22⋊D5911yescharacteristic
425C2×C22⋊C4C22⋊D5911yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42C22×D5911yescharacteristic
431C2×C42C22×D5911yescharacteristic
432#107—594no—
433#107—594no—
434#107—1182no—
435#106—2361no—
436#102—594no—
437#102—594no—
438#102—1182no—
439#102—2361no—
440C22⋊C8—594no—
441#97—2361no—
442C4×C8—1182no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1181no—
449#95—1181no—
450#95C22⋊D5911yescharacteristic
451#95C22⋊D5911yescharacteristic
452C24—1181no—
453C24C2×C22⋊D5911yescharacteristic
454C2×D4—592no—
455C2×D4—1184no—
456C2×D4—1181no—
457C2×D4—1181no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C2×D4—4721no—
462C2×D4—4721no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—1181no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—4721no—
478C22×C4—2362no—
479C22×C4—2362no—
480C22×C4C2×C4×D5912yes—
481C22×C4#1174568512yes—
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C23×D5911yescharacteristic
488M16—2364no—
489C2×C8—592no—
490C2×C8—592no—
491C2×C8—592no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—592no—
511C4⋊C4—592no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—4721no—
517C4⋊C4C2×C4×D5912yes—
518C4⋊C4C2×C4×D5912yes—
519C22⋊C4—22no—
520C22⋊C4—41no—
521C22⋊C4—42no—
522C22⋊C4—1184no—
523C22⋊C4—4721no—
524C22⋊C4—2362no—
525C22⋊C4—2362no—
526C22⋊C4—2362no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—81no—
532C42—81no—
533C42—4721no—
534C42—2362no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—1182no—
542C23—2361no—
543C23—2361no—
544C23—2361no—
545C23—2361no—
546C23—2361no—
547C23—2361no—
548C23C4×C22⋊D5912yes—
549C23C22×C22⋊D5911yescharacteristic
550D4—1182no—
551D4—1182no—
552D4—4724no—
553D4—2361no—
554D4—2361no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—1182no—
586C2×C4—1182no—
587C2×C4—4724no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4C4×C22⋊D5912yes—
596C2×C4#1427375812yes—
597C2×C4C22×C4×D5911yescharacteristic
598C2×C4#41×D5911yescharacteristic
599C2×C4#41×D5911yescharacteristic
600C2×C4C2×#1174568511yescharacteristic
601C2×C4#1427377311yescharacteristic
602C2×C4#1427377411yescharacteristic
603C8—1182no—
604C8—1182no—
605C8—1182no—
606C8—2361no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—4721no—
615C22—4721no—
616C22—4721no—
617C22—4721no—
618C22—2362no—
619C22—2362no—
620C22—2362no—
621C22C2×C4×C22⋊D5911yescharacteristic
622C22#1683804011yescharacteristic
623C22#1683805111yescharacteristic
624C22C2×#1427375811yescharacteristic
625C22#1683820011yescharacteristic
626C22#1683820511yescharacteristic
627C22#1683820511yescharacteristic
628C4—22no—
629C4—22no—
630C4—41no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—81no—
636C4—81no—
637C4—4721no—
638C4—2362no—
639C4—2362no—
640C4#1683787011yescharacteristic
641C4#1683788211yescharacteristic
642C4#1683807611yescharacteristic
643C4#1683808511yescharacteristic
644C2—42no—
645C2—4722no—
646C2#2142708811yescharacteristic
647C2C4×#1427375811yescharacteristic
648C2#2143294611yescharacteristic
649C2#2143629111yescharacteristic
650C2#2143642811yescharacteristic
651C2#2143657511yescharacteristic
652C2#2143659011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2345C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenter#2345D59
hypocenterC59#22941
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
3. #21424831C2semidirect
4. #21424778C2semidirect
5. #21424723C2semidirect
6. #21424685C2semidirect
7. #21424667C2semidirect
9. #21424342C2semidirect
10. #21424293C2semidirect
11. C4×#14273660C2semidirect
12. #21423924C2semidirect
13. #21423880C2semidirect
14. #21423815C2semidirect
15. C59×#2345C2semidirect
20. #16837715C4semidirect
88. #14273660C2×C4semidirect
89. #14273660C2×C4semidirect
95. C22⋊(C59⋊C8)C2×C4semidirect
106. C2×C4⋊C4×C59C2×C4semidirect
320. #2345D59semidirect
386. C59#22941semidirect
413. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377152
#214271322
#214248312
#214247782
#214247232
#214246852
#214246672
#214245322
#214243422
#214242932
C4×#142736602
#214239242
#214238802
#214238152
C59×#23452
#2294159