Finite Group Databasebeta

← #28608106

Subgroups — #28608106

Order 15104.

942 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×#16837512C211yesmaximal, characteristic
2#21427043C211yesmaximal, characteristic, complemented
3#21424792C211yesmaximal, characteristic, complemented
4#21424778C211yesmaximal, characteristic, complemented
5#21424678C211yesmaximal, characteristic, complemented
6#21424671C211yesmaximal, characteristic, complemented
7#21424444C211yesmaximal, characteristic
8#21424444C211yesmaximal, characteristic
9#21424229C211yesmaximal, characteristic, complemented
10#21424103C211yesmaximal, characteristic, complemented
11#21424096C211yesmaximal, characteristic, complemented
12#21423971C211yesmaximal, characteristic, complemented
13#21423921C211yesmaximal, characteristic, complemented
14C4×#14273648C211yesmaximal, characteristic, complemented
15C59×#2349C211yesmaximal, characteristic, complemented
16C22×#11745669C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273685—21no—
19#16837512C418yescomplemented
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837294—21no—
34#16837294—21no—
35#16837272C2211yescharacteristic
36#16837259—21no—
37C4×C4⋊Dic59C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#385—21no—
55C59×#385—21no—
56C59×#382C2211yescharacteristic
57C59×#380—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C2×#95×C59—21no—
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C2×#11745669—22no—
63C2×#11745669—22no—
64C2×#11745669C2×C414yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C4×D236—22no—
70C4×D236C2×C414yescomplemented
71#14273685—44no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84#14273658C2×C414yescomplemented
85C4⋊(C59⋊C8)—22no—
86#14273648—22no—
87#14273648—22no—
88#14273648C2×C414yes—
89#14273647—41no—
90#14273647—41no—
91#14273647C2×C414yescomplemented
92#14273638—41no—
93#14273638C2×C414yescomplemented
94C4×C59⋊C8—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×D4×C59C2311yescharacteristic
98C23×C4×C59—21no—
99C23×C4×C59—21no—
100C23×C4×C59—21no—
101C4×D4×C59—22no—
102C4×D4×C59—22no—
103C4×D4×C59—22no—
104C4×D4×C59—41no—
105C4×D4×C59—41no—
106C4×D4×C59C2×C414yes—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59D411yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59—21no—
127C23×D59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59C22×C412yes—
132C2×D4×C59#4112yes—
133C2×D4×C59#4112yes—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—42no—
139C22×C4×C59—42no—
140C22×C4×C59—42no—
141C22×C4×C59—42no—
142C22×C4×C59—81no—
143C22×C4×C59—81no—
144C22×C4×C59—81no—
145C22×C4×C59—81no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236C22×C412yes—
162C2×C4×D59—24no—
163C2×C4×D59—42no—
164C2×C4×D59—42no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—41no—
172C4⋊C4×C59—81no—
173C4⋊C4×C59—81no—
174C4⋊C4×C59C22×C412yes—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59C22×C412yes—
183C42×C59#4112yes—
184#11745669—28no—
185(C2×D59)⋊C4—24no—
186(C2×D59)⋊C4—42no—
187(C2×D59)⋊C4—42no—
188(C2×D59)⋊C4—81no—
189(C2×D59)⋊C4—81no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—42no—
196C4⋊Dic59—81no—
197C4⋊Dic59C22×C412yes—
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—22no—
213C23×C59—22no—
214C23×C59—22no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59C22×D411yescharacteristic
220C22×D59—22no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225C22×D59—41no—
226C22×D59—41no—
227D4×C59—22no—
228D4×C59—22no—
229D4×C59—22no—
230D4×C59#11714yescomplemented
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—84no—
256C2×C4×C59—84no—
257C2×C4×C59—84no—
258C2×C4×C59C4×D412yes—
259C2×C4×C59C4×D412yes—
260C2×C4×C59AGL(1,ℤ/8)12yes—
261C2×C4×C59C23×C411yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59H(5,𝔽2)11yescharacteristic
266C2×C4×C592−1+411yescharacteristic
267C2×Dic59—22no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276D236—22no—
277D236—22no—
278D236—41no—
279D236—41no—
280C4×D59—48no—
281C59⋊C8—22no—
282C59⋊C8—22no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285#22780—591nomaximal, Sylow
286C22×C59—24no—
287C22×C59—42no—
288C22×C59—42no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59—42no—
293C22×C59C2×C4×D411yescharacteristic
294C22×C59#54611yescharacteristic
295C22×C59#54811yescharacteristic
296C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C59#57511yescharacteristic
298C22×C59#57611yescharacteristic
299C22×C59#57711yescharacteristic
300C2×D59—42no—
301C2×D59—42no—
302C2×D59—42no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C2×D59—81no—
307C4×C59—22no—
308C4×C59—22no—
309C4×C59—22no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—41no—
313C4×C59—41no—
314C4×C59—81no—
315C4×C59—81no—
316C4×C59C2×#11711yescharacteristic
317C4×C59#52011yescharacteristic
318C4×C59#55211yescharacteristic
319C4×C59#55311yescharacteristic
320Dic59—42no—
321Dic59—81no—
322Dic59—81no—
323C2×#442—591no—
324#2354—591no—
325#2349D5911yescharacteristic, complemented
326#1936—591no—
327#1934—591no—
328#1920—591no—
329#1919—591no—
330#1879—591no—
331#1879—591no—
332#1836—591no—
333#1804—591no—
334#1802—591no—
335#1771—591no—
336#1762—591no—
337C4×#102—591no—
338C2×C59—44no—
339C2×C59#235211yescharacteristic
340C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
341C2×C59#294611yescharacteristic
342C2×C59#330611yescharacteristic
343C2×C59#331911yescharacteristic
344C2×C59#336611yescharacteristic
345C2×C59#337611yescharacteristic
346D59—82no—
347C22×D8—591no—
348C2×C4×D4—591no—
349C2×C4×D4C2×D5911yescharacteristic
350C2×C4×D4C2×D5911yescharacteristic
351C2×C4×D4C2×D5911yescharacteristic
352C2×#117—1181no—
353#442—598no—
354C2×#106—591no—
355C2×C4⋊C8—591no—
356C2×#102—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×#102—591no—
362C2×#97—591no—
363C2×#97—591no—
364C2×C4×C8—591no—
365#390—591no—
366#390—591no—
367#389C2×D5911yescharacteristic
368#389C2×D5911yescharacteristic
369#388—1181no—
370#388—1181no—
371#385—21no—
372#385—21no—
373#384—591no—
374#382C2×D5911yescharacteristic
375#380—21no—
376#380—1181no—
377C4×C4⋊C4—591no—
378C4×C4⋊C4—591no—
379C4×C4⋊C4C2×D5911yescharacteristic
380C2×#95—21no—
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#2278011yescharacteristic, complemented, Sylow
392C22×D4—591no—
393C22×D4C22×D5911yescharacteristic
394C23×C4—21no—
395C23×C4—21no—
396C23×C4—21no—
397C23×C4—1181no—
398C2×D8—594no—
399C2×D8—1182no—
400C2×D8—1182no—
401C22×C8—591no—
402C22×C8—591no—
403C22×C8—591no—
404C22×C8—591no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—41no—
409C4×D4—41no—
410C4×D4—594no—
411C4×D4—1182no—
412C4×D4C4×D5914yes—
413#117—2364no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4C22⋊D5911yescharacteristic
422C2×C4⋊C4C22⋊D5911yescharacteristic
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C4⋊C4C22×D5911yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—1181no—
430C2×C22⋊C4—1181no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C22⋊C4—1181no—
434C2×C42—1181no—
435C2×C42—1181no—
436C2×C42C22×D5911yescharacteristic
437C2×C42C22×D5911yescharacteristic
438C2×C42C22×D5911yescharacteristic
439#106—594no—
440C4⋊C8—1182no—
441#102—594no—
442#102—594no—
443#102—1182no—
444#102—1182no—
445#102—2361no—
446#102—2361no—
447#97—594no—
448#97—2361no—
449C4×C8—2361no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1181no—
455#95—1181no—
456#95C22⋊D5911yescharacteristic
457#95C22⋊D5911yescharacteristic
458C24—21no—
459C24—1181no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—592no—
464C2×D4—1181no—
465C2×D4—1181no—
466C2×D4—1181no—
467C2×D4—1181no—
468C2×D4—1181no—
469C2×D4C2×C4×D5912yes—
470C2×D4#1174568712yes—
471C2×D4#1174568912yes—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—81no—
484C22×C4—1181no—
485C22×C4—1184no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—2362no—
489C22×C4—2362no—
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C23×D5911yescharacteristic
497D8—1188no—
498C2×C8—592no—
499C2×C8—592no—
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—
509C2×C8—1181no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—41no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—592no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—4721no—
526C4⋊C4—2362no—
527C4⋊C4C2×C4×D5912yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—1184no—
532C22⋊C4—4721no—
533C22⋊C4—4721no—
534C22⋊C4—2362no—
535C22⋊C4—2362no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—2362no—
541C42—2362no—
542C42C2×C4×D5912yes—
543C42#1174568512yes—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—1182no—
552C23—2361no—
553C23—2361no—
554C23—2361no—
555C23—2361no—
556C23—2361no—
557C23—2361no—
558C23C22×C22⋊D5911yescharacteristic
559D4—22no—
560D4—22no—
561D4—22no—
562D4—1182no—
563D4—1182no—
564D4—2361no—
565D4—2361no—
566D4#1427368514yescomplemented
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—1182no—
595C2×C4—2368no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4C4×C22⋊D5912yes—
605C2×C4C4×C22⋊D5912yes—
606C2×C4#1427375212yes—
607C2×C4C22×C4×D5911yescharacteristic
608C2×C4C2×#1174568711yescharacteristic
609C2×C4C2×#1174568511yescharacteristic
610C2×C4C2×#1174568911yescharacteristic
611C2×C4#1427377211yescharacteristic
612C2×C4#1427377511yescharacteristic
613C8—1182no—
614C8—1182no—
615C8—2361no—
616C8—2361no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—4721no—
625C22—4721no—
626C22—4721no—
627C22—4721no—
628C22—2362no—
629C22—2362no—
630C22—2362no—
631C22C2×C4×C22⋊D5911yescharacteristic
632C22#1683804611yescharacteristic
633C22#1683806311yescharacteristic
634C22C2×#1427375211yescharacteristic
635C22#1683820011yescharacteristic
636C22#1683820511yescharacteristic
637C22#1683821311yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—42no—
644C4—41no—
645C4—81no—
646C4—81no—
647C4—4721no—
648C4—4721no—
649C4—2362no—
650C4C2×#1427368511yescharacteristic
651C4#1683788311yescharacteristic
652C4#1683808711yescharacteristic
653C4#1683810011yescharacteristic
654C2—44no—
655C2—4722no—
656C2#2142702111yescharacteristic
657C2C4×#1427375211yescharacteristic
658C2#2143292811yescharacteristic
659C2#2143605711yescharacteristic
660C2#2143625211yescharacteristic
661C2#2143680311yescharacteristic
662C2#2143692311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2349C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
cosocleC22×C4×C59C24
hypercenter#2349D59
hypocenterC59#22780
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #21427043C2semidirect
3. #21424792C2semidirect
4. #21424778C2semidirect
5. #21424678C2semidirect
6. #21424671C2semidirect
9. #21424229C2semidirect
10. #21424103C2semidirect
11. #21424096C2semidirect
12. #21423971C2semidirect
13. #21423921C2semidirect
14. C4×#14273648C2semidirect
15. C59×#2349C2semidirect
19. #16837512C4semidirect
70. C4×D236C2×C4semidirect
84. #14273658C2×C4semidirect
91. #14273647C2×C4semidirect
93. #14273638C2×C4semidirect
230. D4×C59#117semidirect
325. #2349D59semidirect
391. C59#22780semidirect
566. D4#14273685semidirect

Maximal subgroups

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

subgroupindex
C2×#168375122
#214270432
#214247922
#214247782
#214246782
#214246712
#214244442
#214242292
#214241032
#214240962
#214239712
#214239212
C4×#142736482
C59×#23492
#2278059