Finite Group Databasebeta

← #28595910

Subgroups — #28595910

Order 15104.

960 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837723C211yesmaximal, characteristic
2C4×#14273686C211yesmaximal, characteristic, complemented
3#21424720C211yesmaximal, characteristic, complemented
4#21424713C211yesmaximal, characteristic, complemented
5#21424671C211yesmaximal, characteristic, complemented
6#21424664C211yesmaximal, characteristic, complemented
7#21424419C211yesmaximal, characteristic
8#21424293C211yesmaximal, characteristic, complemented
9#21424188C211yesmaximal, characteristic, complemented
10#21423994C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12#21423921C211yesmaximal, characteristic, complemented
13#21423921C211yesmaximal, characteristic, complemented
14#21423858C211yesmaximal, characteristic, complemented
15C4×#121×C59C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18#16837723C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C4×C4⋊Dic59C2211yescharacteristic
32C4×C4⋊Dic59C2211yescharacteristic
33C4×C4⋊Dic59C2211yescharacteristic
34C4×C4⋊Dic59C2211yescharacteristic
35C4×C22⋊Dic59—21no—
36C4×C22⋊Dic59—21no—
37#16837248—21no—
38C42×Dic59—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×#121×C59C2211yescharacteristic
50C2×C4×D4×C59—21no—
51C2×C4×D4×C59C2211yescharacteristic
52C22×C42×C59C2211yescharacteristic
53C59×#390C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#386—21no—
57C59×#380C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273686—22no—
65#14273686—22no—
66#14273686—22no—
67#14273686C2×C414yescomplemented
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C22⋊Dic59—21no—
73C2×C22⋊Dic59—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—22no—
81#14273660C2×C414yescomplemented
82#14273658—22no—
83#14273658C2×C414yescomplemented
84#14273648—22no—
85#14273648—22no—
86#14273648C2×C414yescomplemented
87#14273648C2×C414yescomplemented
88C22⋊(C59⋊C8)—22no—
89C22⋊(C59⋊C8)C2×C414yescomplemented
90#14273638—22no—
91#14273638—22no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×D4×C59—21no—
97C22×D4×C59C2311yescharacteristic
98C23×C4×C59—21no—
99C23×C4×C59D411yescharacteristic
100C23×C4×C59D411yescharacteristic
101C23×C4×C59C2311yescharacteristic
102#121×C59—22no—
103#121×C59C2×C414yes—
104C4×D4×C59—22no—
105C4×D4×C59—22no—
106C4×D4×C59—44no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C42×C59—22no—
117C2×C42×C59—22no—
118C2×C42×C59—21no—
119C2×C42×C59D411yescharacteristic
120C2×C42×C59D411yescharacteristic
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123#95×C59—21no—
124#95×C59—21no—
125#95×C59—21no—
126C24×C59—21no—
127C24×C59C2×D411yescharacteristic
128C2×D4×C59—24no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—42no—
135C2×D4×C59—42no—
136C2×D4×C59C22×C412yes—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—41no—
148C22×C4×C59—41no—
149C22×C4×C59—41no—
150C22×C4×C59—42no—
151C22×C4×C59—42no—
152C22×C4×C59—42no—
153C22×C4×C59C22×C412yes—
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59#4112yes—
160C22×C4×C59C2411yescharacteristic
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59C22×C412yes—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—42no—
175C22⋊C4×C59—42no—
176C22⋊C4×C59—42no—
177C42×C59—22no—
178C42×C59—22no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—42no—
182C42×C59#4112yes—
183C42×C59#4112yes—
184C22⋊Dic59—24no—
185C22⋊Dic59—24no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59C22×C412yes—
191C4⋊Dic59C22×C412yes—
192C4⋊Dic59#4112yes—
193C4⋊Dic59#4112yes—
194C4×Dic59—24no—
195C4×Dic59—24no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59—42no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8C22×C412yes—
211C2×C59⋊C8C22×C412yes—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—22no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59C4×D412yes—
226C23×C59C22×D411yescharacteristic
227D4×C59—22no—
228D4×C59—22no—
229D4×C59—48no—
230D4×C59—41no—
231D4×C59—41no—
232C2×C4×C59—21no—
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—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—44no—
249C2×C4×C59—44no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—84no—
259C2×C4×C59C4×D412yes—
260C2×C4×C59#12312yes—
261C2×C4×C59#12312yes—
262C2×C4×C59AGL(1,ℤ/8)12yes—
263C2×C4×C59C23×C411yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×Dic59—22no—
270C2×Dic59—22no—
271C2×Dic59—22no—
272C2×Dic59—22no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C59⋊C8—22no—
282C59⋊C8—22no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285#21582—591nomaximal, Sylow
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—42no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59—81no—
300C22×C59C2×C4×D411yescharacteristic
301C22×C59C2×#12311yescharacteristic
302C22×C59#52511yescharacteristic
303C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C59#57511yescharacteristic
305C22×C59#57511yescharacteristic
306C22×C59#57511yescharacteristic
307C4×C59—22no—
308C4×C59—22no—
309C4×C59—22no—
310C4×C59—22no—
311C4×C59—41no—
312C4×C59—42no—
313C4×C59—81no—
314C4×C59—81no—
315C4×C59C4×#4111yescharacteristic
316C4×C59C4×#4111yescharacteristic
317C4×C59#52911yescharacteristic
318C4×C59#52911yescharacteristic
319Dic59—42no—
320Dic59—42no—
321Dic59—42no—
322Dic59—42no—
323C2×#482—591no—
324C4×#121D5911yescharacteristic
325C4×#117—591no—
326#1926—591no—
327#1925—591no—
328#1919—591no—
329#1918—591no—
330#1875—591no—
331#1854—591no—
332#1825—591no—
333#1776—591no—
334#1776—591no—
335#1762—591no—
336#1762—591no—
337#1749—591no—
338C2×C59—42no—
339C2×C59—82no—
340C2×C59C4×#12311yescharacteristic
341C2×C59#294311yescharacteristic
342C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
343C2×C59#309311yescharacteristic
344C2×C59#309411yescharacteristic
345C2×C59#311111yescharacteristic
346C2×C59#311511yescharacteristic
347C2×#121C2×D5911yescharacteristic
348C2×C4×D4—21no—
349C2×C4×D4C2×D5911yescharacteristic
350C2×#117—591no—
351C2×#117—591no—
352C22×C42C2×D5911yescharacteristic
353#482—598no—
354C2×#107—591no—
355C2×#107—591no—
356C2×#106—591no—
357C2×#106—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×#102—591no—
362C2×C22⋊C8—591no—
363C2×C22⋊C8—591no—
364C2×#97—591no—
365C2×#97—591no—
366#390C2×D5911yescharacteristic
367#389C2×D5911yescharacteristic
368#388—21no—
369#386—21no—
370#380C2×D5911yescharacteristic
371C4×C4⋊C4—591no—
372C4×C4⋊C4—591no—
373C4×C4⋊C4—591no—
374C4×C4⋊C4—591no—
375C4×C4⋊C4C2×D5911yescharacteristic
376C4×C22⋊C4—21no—
377C4×C22⋊C4—1181no—
378C4×C22⋊C4—1181no—
379#376—1181no—
380C43—1181no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#336—591no—
390#336—591no—
391C59#2158211yescharacteristic, complemented, Sylow
392C22×D4—21no—
393C22×D4C22×D5911yescharacteristic
394C23×C4—21no—
395C23×C4C22⋊D5911yescharacteristic
396C23×C4C22⋊D5911yescharacteristic
397C23×C4C22×D5911yescharacteristic
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401C22×C8—591no—
402#121—22no—
403#121C4×D5914yes—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—44no—
407#117—594no—
408#117—1182no—
409#117—1182no—
410#117—1182no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C42—22no—
427C2×C42—22no—
428C2×C42—21no—
429C2×C42—1181no—
430C2×C42—1181no—
431C2×C42—1181no—
432C2×C42—1181no—
433C2×C42—1181no—
434C2×C42—1181no—
435C2×C42C22⋊D5911yescharacteristic
436C2×C42C22⋊D5911yescharacteristic
437C2×C42C22×D5911yescharacteristic
438C2×C42C22×D5911yescharacteristic
439#107—594no—
440#107—1182no—
441#106—594no—
442#106—1182no—
443#102—594no—
444#102—594no—
445#102—1182no—
446#102—1182no—
447C22⋊C8—594no—
448C22⋊C8—1182no—
449#97—1182no—
450#97—1182no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457#95—1181no—
458C24—21no—
459C24C2×C22⋊D5911yescharacteristic
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4C2×C4×D5912yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4C2×C4×D5912yes—
490C22×C4#1174568512yes—
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C23×D5911yescharacteristic
497C2×C8—592no—
498C2×C8—592no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—592no—
516C4⋊C4—592no—
517C4⋊C4—592no—
518C4⋊C4—592no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4C2×C4×D5912yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—1184no—
530C22⋊C4—1184no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—1184no—
537C42—1184no—
538C42—2362no—
539C42—2362no—
540C42—2362no—
541C42—2362no—
542C42#1174568712yes—
543C42#1174568712yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C4×C22⋊D5912yes—
558C23C22×C22⋊D5911yescharacteristic
559D4—22no—
560D4—22no—
561D4—48no—
562D4—41no—
563D4—41no—
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—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—1182no—
592C2×C4—1182no—
593C2×C4—1182no—
594C2×C4—1182no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4C4×C22⋊D5912yes—
604C2×C4#1427370812yes—
605C2×C4#1427370812yes—
606C2×C4#1427375312yes—
607C2×C4C22×C4×D5911yescharacteristic
608C2×C4#41×D5911yescharacteristic
609C2×C4#41×D5911yescharacteristic
610C2×C4C2×#1174568711yescharacteristic
611C2×C4C2×#1174568711yescharacteristic
612C2×C4C2×#1174568511yescharacteristic
613C8—1182no—
614C8—1182no—
615C8—2361no—
616C8—2361no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×C4×C22⋊D5911yescharacteristic
632C22C2×#1427370811yescharacteristic
633C22#1683790911yescharacteristic
634C22C2×#1427375311yescharacteristic
635C22#1683820011yescharacteristic
636C22#1683820011yescharacteristic
637C22#1683820011yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—2362no—
647C4—2362no—
648C4—2362no—
649C4—2362no—
650C4C4×#1174568711yescharacteristic
651C4C4×#1174568711yescharacteristic
652C4#1683793211yescharacteristic
653C4#1683793211yescharacteristic
654C2—42no—
655C2—82no—
656C2C4×#1427370811yescharacteristic
657C2#2143289411yescharacteristic
658C2C4×#1427375311yescharacteristic
659C2#2143402511yescharacteristic
660C2#2143403211yescharacteristic
661C2#2143419911yescharacteristic
662C2#2143424111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#121×C59C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
cosocleC22×C4×C59C24
hypercenterC4×#121D59
hypocenterC59#21582
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C4×#14273686C2semidirect
3. #21424720C2semidirect
4. #21424713C2semidirect
5. #21424671C2semidirect
6. #21424664C2semidirect
8. #21424293C2semidirect
9. #21424188C2semidirect
10. #21423994C2semidirect
11. #21423994C2semidirect
12. #21423921C2semidirect
13. #21423921C2semidirect
14. #21423858C2semidirect
18. #16837723C4semidirect
67. #14273686C2×C4semidirect
81. #14273660C2×C4semidirect
83. #14273658C2×C4semidirect
86. #14273648C2×C4semidirect
87. #14273648C2×C4semidirect
89. C22⋊(C59⋊C8)C2×C4semidirect
391. C59#21582semidirect

Maximal subgroups

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

subgroupindex
C2×#168377232
C4×#142736862
#214247202
#214247132
#214246712
#214246642
#214244192
#214242932
#214241882
#214239942
#214239212
#214238582
C4×#121×C592
#2158259