Finite Group Databasebeta

← #28607042

Subgroups — #28607042

Order 15104.

1010 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
1C2×#16837505C211yesmaximal, characteristic
2C2×#16837500C211yesmaximal, characteristic
3C2×#16837487C211yesmaximal, characteristic
4#21427014C211yesmaximal, characteristic, complemented
5#21426997C211yesmaximal, characteristic
6#21424831C211yesmaximal, characteristic, complemented
7#21424828C211yesmaximal, characteristic, complemented
8#21424366C211yesmaximal, characteristic, complemented
9#21424360C211yesmaximal, characteristic
10#21424340C211yesmaximal, characteristic
11#21424263C211yesmaximal, characteristic, complemented
12#21424105C211yesmaximal, characteristic
13#21424099C211yesmaximal, characteristic, complemented
14#21423969C211yesmaximal, characteristic, complemented
15C59×#1863C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C2×Q8×Dic59C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20C2×#14273689C2211yescharacteristic
21C2×#14273689C2211yescharacteristic
22C2×C4⋊C4×D59—21no—
23#16837505C418yescomplemented
24#16837500—24no—
25#16837487—24no—
26C2×#14273660C2211yescharacteristic
27C2×#14273660C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×Dic59⋊C8C2211yescharacteristic
31C2×#14273651C2211yescharacteristic
32C2×#14273649C2211yescharacteristic
33C2×#14273649C2211yescharacteristic
34C2×#14273647C2211yescharacteristic
35C2×#14273638C2211yescharacteristic
36C2×#14273637C2211yescharacteristic
37C2×C8×Dic59C2211yescharacteristic
38#16837301C2211yescharacteristic
39#16837300—21no—
40#16837300C2211yescharacteristic
41#16837298—21no—
42#16837298C2211yescharacteristic
43#16837276—21no—
44#16837273—21no—
45#16837271C2211yescharacteristic
46#16837267—21no—
47#16837267C2211yescharacteristic
48#16837261—21no—
49C4×Dic59⋊C4—21no—
50C4×Dic59⋊C4C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837156C2211yescharacteristic
55C2×C4×Q8×C59C2211yescharacteristic
56C2×#107×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C59×#389C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#336C2211yescharacteristic
62C22×D236C2311yescharacteristic
63C22×C4×D59—21no—
64C22×C4×D59—21no—
65C22×C4×D59—21no—
66C2×#11745671C2×C414yes—
67C2×#11745671D414yes—
68C2×#11745671Q814yes—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71Q8×Dic59—22no—
72Q8×Dic59—22no—
73Q8×Dic59—41no—
74Q8×Dic59C2×C414yescomplemented
75C4×D236—22no—
76C4×D236—41no—
77#14273689—22no—
78#14273689—22no—
79#14273689—41no—
80#14273689C2×C414yes—
81C2×C4⋊Dic59—21no—
82C2×C4⋊Dic59—21no—
83C2×C4⋊Dic59—21no—
84C2×C4⋊Dic59C2311yescharacteristic
85C2×C4⋊Dic59C2311yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4D411yescharacteristic
92C2×Dic59⋊C4D411yescharacteristic
93C4⋊C4×D59—44no—
94C2×(C2×D59)⋊C4—21no—
95C2×(C2×D59)⋊C4—21no—
96C2×(C2×D59)⋊C4—21no—
97C2×C4×Dic59—21no—
98C2×C4×Dic59—21no—
99C2×C4×Dic59C2311yescharacteristic
100C2×C4×Dic59C2311yescharacteristic
101#14273660—22no—
102#14273660C2×C414yescomplemented
103#14273659—41no—
104#14273658—22no—
105Dic59⋊C8—22no—
106#14273651—22no—
107#14273649—22no—
108#14273649C2×C414yes—
109#14273647—22no—
110#14273638—22no—
111#14273637C2×C414yescomplemented
112C8×Dic59—22no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C59⋊#95—21no—
116C59⋊#95—21no—
117C59⋊#95—21no—
118C59⋊#95D411yescharacteristic
119C59⋊#95D411yescharacteristic
120C22×Q8×C59C2311yescharacteristic
121C22×C8×C59C2311yescharacteristic
122C22×C8×C59C2311yescharacteristic
123C4×Q8×C59—22no—
124C4×Q8×C59—41no—
125C2×C4⋊C4×C59—21no—
126C2×C4⋊C4×C59—21no—
127C2×C4⋊C4×C59C2311yescharacteristic
128C2×C4⋊C4×C59C2311yescharacteristic
129C2×C4⋊C4×C59C2311yescharacteristic
130C2×C42×C59—21no—
131C2×C42×C59C2311yescharacteristic
132#107×C59—41no—
133#103×C59—22no—
134#103×C59C2×C414yescomplemented
135#95×C59—21no—
136C23×D59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59C22×C412yes—
141C2×Q8×C59C2×D412yes—
142C2×Q8×C59C2×Q812yes—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2411yescharacteristic
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156C22×Dic59C2×D411yescharacteristic
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236C22×C412yes—
161C2×D236C2×D412yes—
162C2×D236C2×Q812yes—
163C2×C4×D59—24no—
164C2×C4×D59—42no—
165C2×C4×D59—42no—
166C2×C4×D59—42no—
167C2×C4×D59—42no—
168C2×C4×D59—42no—
169C2×C4×D59—42no—
170C2×C4×D59—81no—
171C2×C4×D59—81no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59C22×C412yes—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—42no—
184C4⋊C4×C59—42no—
185C4⋊C4×C59C22×C412yes—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—42no—
189#11745671C4⋊C4116yescomplemented
190(C2×D59)⋊C4—24no—
191(C2×D59)⋊C4—42no—
192(C2×D59)⋊C4—42no—
193C4⋊Dic59—21no—
194C4⋊Dic59—24no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—42no—
200C4⋊Dic59—42no—
201C4⋊Dic59C22×C412yes—
202Dic59⋊C4—22no—
203Dic59⋊C4—41no—
204Dic59⋊C4—42no—
205Dic59⋊C4—42no—
206Dic59⋊C4—42no—
207Dic59⋊C4—81no—
208C4×Dic59—21no—
209C4×Dic59—21no—
210C4×Dic59—24no—
211C4×Dic59—42no—
212C4×Dic59C22×C412yes—
213C4×Dic59#4112yes—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8C22×C412yes—
218C2×C59⋊C8C2×D412yes—
219C2×C59⋊C8C2×Q812yes—
220C23×C59C22×D411yescharacteristic
221C22×D59—22no—
222C22×D59—22no—
223C22×D59—22no—
224C22×D59—41no—
225C22×D59—41no—
226C22×D59—41no—
227C22×D59—41no—
228Q8×C59—22no—
229Q8×C59—22no—
230Q8×C59—22no—
231Q8×C59C2×C4⋊C414yescomplemented
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
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×C59C23×C411yescharacteristic
251C2×C4×C59C22×D411yescharacteristic
252C2×C4×C59C22×Q811yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59H(5,𝔽2)11yescharacteristic
255C2×C4×C592−1+411yescharacteristic
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—22no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59C4×D412yes—
275C2×Dic59C4×D412yes—
276C2×Dic59AGL(1,ℤ/8)12yes—
277D236—22no—
278D236—22no—
279D236—22no—
280D236C2×C4⋊C414yes—
281C4×D59—48no—
282C4×D59—84no—
283C4×D59—84no—
284C8×C59—22no—
285C8×C59—41no—
286C59⋊C8—41no—
287C59⋊C8C2×C4⋊C414yescomplemented
288#22677—591nomaximal, Sylow
289C22×C59C2×C4×D411yescharacteristic
290C22×C59D4211yescharacteristic
291C22×C59D4×Q811yescharacteristic
292C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C59#57511yescharacteristic
294C22×C59#57711yescharacteristic
295C22×C59#57811yescharacteristic
296C2×D59—24no—
297C2×D59—42no—
298C2×D59—42no—
299C2×D59—42no—
300C2×D59—42no—
301C2×D59—42no—
302C2×D59—42no—
303C4×C59—24no—
304C4×C59—42no—
305C4×C59—41no—
306C4×C59—41no—
307C4×C59—81no—
308C4×C59—81no—
309C4×C59C22×C4⋊C411yescharacteristic
310C4×C59#52011yescharacteristic
311C4×C59#54711yescharacteristic
312C4×C59#55411yescharacteristic
313Dic59—22no—
314Dic59—22no—
315Dic59—22no—
316Dic59—42no—
317Dic59—41no—
318Dic59—81no—
319Dic59—81no—
320C2×#440—591no—
321C2×#440—591no—
322C2×C4×SD16—591no—
323#2351—591no—
324D4×C4⋊C4—591no—
325#1943—591no—
326#1942—591no—
327#1865—591no—
328#1864—591no—
329#1863D5911yescharacteristic, complemented
330#1862—591no—
331#1847—591no—
332#1804—591no—
333#1803—591no—
334#1771—591no—
335C2×C59D4×C4⋊C411yescharacteristic
336C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
337C2×C59#294711yescharacteristic
338C2×C59#326511yescharacteristic
339C2×C59#326911yescharacteristic
340C2×C59#338611yescharacteristic
341C2×C59#338811yescharacteristic
342D59—44no—
343C22×SD16—591no—
344C2×C4×Q8—591no—
345C2×C4×Q8—591no—
346C2×C4×Q8C2×D5911yescharacteristic
347C2×C4×D4—591no—
348C2×C4×D4—591no—
349C2×C4×D4—591no—
350C22×C4⋊C4—1181no—
351#440—598no—
352#440—1184no—
353C4×SD16—1184no—
354C2×#107—591no—
355C2×#107—591no—
356C2×#107C2×D5911yescharacteristic
357C2×#106—591no—
358C2×#106—591no—
359C2×C4⋊C8—591no—
360C2×#103—591no—
361C2×#103C2×D5911yescharacteristic
362C2×#103C2×D5911yescharacteristic
363C2×#102—591no—
364C2×#102—591no—
365C2×#102—591no—
366C2×#97—591no—
367C2×#97—591no—
368C2×C4×C8—591no—
369#389—591no—
370#389—591no—
371#389—591no—
372#389—1181no—
373#389—1181no—
374#389C2×D5911yescharacteristic
375#385—1181no—
376#385—1181no—
377#384—591no—
378#382—591no—
379#382—1181no—
380#380—1181no—
381C4×C4⋊C4—591no—
382C4×C4⋊C4—1181no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340C2×D5911yescharacteristic
387#336—591no—
388#336C2×D5911yescharacteristic
389C59#2267711yescharacteristic, complemented, Sylow
390C22×Q8C22×D5911yescharacteristic
391C22×D4—591no—
392C23×C4—1181no—
393C23×C4—1181no—
394C23×C4—1181no—
395C2×SD16—594no—
396C2×SD16—594no—
397C2×SD16—594no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8C22×D5911yescharacteristic
401C22×C8C22×D5911yescharacteristic
402C4×Q8—22no—
403C4×Q8—41no—
404C4×Q8—594no—
405C4×Q8—1182no—
406C4×Q8—1182no—
407C4×Q8—2361no—
408C4×D4—594no—
409C4×D4—1182no—
410C4×D4—1182no—
411C4×D4—1182no—
412C4×D4—2361no—
413C4×D4—2361no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—2364no—
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C4⋊C4C22×D5911yescharacteristic
431C2×C4⋊C4C22×D5911yescharacteristic
432C2×C22⋊C4—1181no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4—1181no—
435C2×C42—21no—
436C2×C42—591no—
437C2×C42—591no—
438C2×C42—1181no—
439C2×C42—1181no—
440C2×C42C22×D5911yescharacteristic
441#107—41no—
442#107—594no—
443#107—1182no—
444#106—1182no—
445#106—2361no—
446C4⋊C8—1182no—
447#103—22no—
448#103—1182no—
449#103C4×D5914yescomplemented
450#102—594no—
451#102—1182no—
452#102—1182no—
453#97—594no—
454#97—1182no—
455C4×C8—1182no—
456#95—21no—
457#95—591no—
458#95—591no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464C24—1181no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C2×C4×D5912yes—
469C2×Q8D4×D5912yes—
470C2×Q8Q8×D5912yes—
471C2×D4—592no—
472C2×D4—592no—
473C2×D4—592no—
474C2×D4—1181no—
475C2×D4—1181no—
476C2×D4—1181no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—591no—
482C22×C4—591no—
483C22×C4—591no—
484C22×C4—591no—
485C22×C4—1184no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—4721no—
490C22×C4—4721no—
491C22×C4—2362no—
492C22×C4—2362no—
493C22×C4—2362no—
494C22×C4—2362no—
495C22×C4—2362no—
496C22×C4—2362no—
497C22×C4D4×D5911yescharacteristic
498C22×C4D4×D5911yescharacteristic
499C22×C4C23×D5911yescharacteristic
500SD16—5916no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—592no—
507C2×C8—592no—
508C2×C8—592no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8C2×C4×D5912yes—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—592no—
521C4⋊C4—1182no—
522C4⋊C4—1184no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—4721no—
529C4⋊C4—2361no—
530C4⋊C4—2362no—
531C4⋊C4—2362no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4—2362no—
535C4⋊C4C2×C4×D5912yes—
536C22⋊C4—1184no—
537C22⋊C4—2362no—
538C22⋊C4—2362no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—592no—
543C42—592no—
544C42—1184no—
545C42—1181no—
546C42—1181no—
547C42—2362no—
548C23—1182no—
549C23—1182no—
550C23—1182no—
551C23—2361no—
552C23—2361no—
553C23—2361no—
554C23—2361no—
555C23C2×D4×D5911yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—22no—
559Q8C4⋊C4×D5914yescomplemented
560D4—594no—
561D4—1182no—
562D4—1182no—
563D4—1182no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
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—592no—
583C2×C4—592no—
584C2×C4—592no—
585C2×C4—1181no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1181no—
591C2×C4—1181no—
592C2×C4—1181no—
593C2×C4—1181no—
594C2×C4—1182no—
595C2×C4—4724no—
596C2×C4—4724no—
597C2×C4—2368no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4C22×C4×D5911yescharacteristic
607C2×C4C2×D4×D5911yescharacteristic
608C2×C4C2×Q8×D5911yescharacteristic
609C2×C4#41×D5911yescharacteristic
610C2×C4#1427377211yescharacteristic
611C2×C4#1427377411yescharacteristic
612C8—22no—
613C8—41no—
614C8—594no—
615C8—2361no—
616C22—1184no—
617C22—2362no—
618C22—2362no—
619C22—2362no—
620C22—2362no—
621C22—2362no—
622C22—2362no—
623C22C4×D4×D5911yescharacteristic
624C22#1683803411yescharacteristic
625C22#1683806511yescharacteristic
626C22AGL(1,ℤ/8)×D5911yescharacteristic
627C22#1683820111yescharacteristic
628C22#1683820911yescharacteristic
629C22#1683821511yescharacteristic
630C4—24no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—1182no—
637C4—1182no—
638C4—1182no—
639C4—4721no—
640C4—4721no—
641C4—2361no—
642C4—2362no—
643C4C2×C4⋊C4×D5911yescharacteristic
644C4#1683788111yescharacteristic
645C4#1683805011yescharacteristic
646C4#1683810611yescharacteristic
647C2—2364no—
648C2#2142699411yescharacteristic
649C2#2143290111yescharacteristic
650C2#2143293411yescharacteristic
651C2#2143559411yescharacteristic
652C2#2143565411yescharacteristic
653C2#2143702011yescharacteristic
654C2#2143704411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1863C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1863D59
hypocenterC59#22677
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
4. #21427014C2semidirect
6. #21424831C2semidirect
7. #21424828C2semidirect
8. #21424366C2semidirect
11. #21424263C2semidirect
13. #21424099C2semidirect
14. #21423969C2semidirect
15. C59×#1863C2semidirect
23. #16837505C4semidirect
74. Q8×Dic59C2×C4semidirect
102. #14273660C2×C4semidirect
111. #14273637C2×C4semidirect
134. #103×C59C2×C4semidirect
189. #11745671C4⋊C4semidirect
231. Q8×C59C2×C4⋊C4semidirect
287. C59⋊C8C2×C4⋊C4semidirect
329. #1863D59semidirect
389. C59#22677semidirect
449. #103C4×D59semidirect
559. Q8C4⋊C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168375052
C2×#168375002
C2×#168374872
#214270142
#214269972
#214248312
#214248282
#214243662
#214243602
#214243402
#214242632
#214241052
#214240992
#214239692
C59×#18632
#2267759