Finite Group Databasebeta

← #28595806

Subgroups — #28595806

Order 15104.

976 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837734C211yesmaximal, characteristic
2#21426626C211yesmaximal, characteristic
3C4×C4⋊C4×D59C211yesmaximal, characteristic
4#21424731C211yesmaximal, characteristic, complemented
5#21424730C211yesmaximal, characteristic, complemented
6#21424678C211yesmaximal, characteristic, complemented
7#21424402C211yesmaximal, characteristic
8#21424290C211yesmaximal, characteristic
9#21424181C211yesmaximal, characteristic
10#106×Dic59C211yesmaximal, characteristic, complemented
11#21423984C211yesmaximal, characteristic, complemented
12#21423915C211yesmaximal, characteristic, complemented
13#21423915C211yesmaximal, characteristic, complemented
14#21423853C211yesmaximal, characteristic
15C59×#1920C211yesmaximal, characteristic, complemented
16C2×#14273707C2211yescharacteristic
17C2×#14273692C2211yescharacteristic
18C2×C4⋊C4×D59C2211yescharacteristic
19C2×C4⋊C4×D59C2211yescharacteristic
20C2×C42×D59—21no—
21C2×C42×D59C2211yescharacteristic
22#16837734C418yescomplemented
23C2×#14273659C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×(C2×D59)⋊C8C2211yescharacteristic
31C2×(C2×D59)⋊C8C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33C2×C8×Dic59C2211yescharacteristic
34#16837308C2211yescharacteristic
35#16837285—21no—
36#16837273—21no—
37#16837259C2211yescharacteristic
38C4⋊C4×Dic59—21no—
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4—21no—
43C4×C4⋊Dic59C2211yescharacteristic
44#16837253—21no—
45#16837249—21no—
46#95×D59—21no—
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837157C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59—21no—
63C22×C4×D59D411yescharacteristic
64C22×C4×D59D411yescharacteristic
65C22×C4×D59C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273707—22no—
69#14273707C2×C414yes—
70#14273692—22no—
71#14273692—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C4⋊C4×D59—22no—
80C4⋊C4×D59—22no—
81C4⋊C4×D59—22no—
82C4⋊C4×D59C2×C414yes—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59D411yescharacteristic
91C2×C4×Dic59D411yescharacteristic
92C2×C4×Dic59C2311yescharacteristic
93C42×D59—22no—
94C42×D59—22no—
95C42×D59—44no—
96#14273659—22no—
97#14273658—22no—
98#14273658C2×C414yescomplemented
99#14273652—22no—
100#14273652C2×C414yescomplemented
101#14273650—22no—
102#14273650C2×C414yescomplemented
103(C2×D59)⋊C8—22no—
104(C2×D59)⋊C8C2×C414yes—
105C8×Dic59—22no—
106C8×Dic59—22no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C42×C59—21no—
122C2×C42×C59C2311yescharacteristic
123#106×C59C2×C414yescomplemented
124#95×C59—21no—
125#95×C59—21no—
126C23×D59C2×D411yescharacteristic
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C2×C4×D59—21no—
142C2×C4×D59—21no—
143C2×C4×D59—21no—
144C2×C4×D59—21no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—24no—
148C2×C4×D59—24no—
149C2×C4×D59—22no—
150C2×C4×D59—22no—
151C2×C4×D59—41no—
152C2×C4×D59—42no—
153C2×C4×D59—42no—
154C2×C4×D59—42no—
155C2×C4×D59—42no—
156C2×C4×D59SD1614yes—
157C2×C4×D59C22×C412yes—
158C2×C4×D59#4112yes—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118C22×C412yes—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59C22×C412yes—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59C22×C412yes—
178C4⋊C4×C59C22×C412yes—
179C4⋊C4×C59#4112yes—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—42no—
183(C2×D59)⋊C4—24no—
184(C2×D59)⋊C4—42no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—24no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59C22×C412yes—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—24no—
193Dic59⋊C4—24no—
194Dic59⋊C4—42no—
195C4×Dic59—22no—
196C4×Dic59—22no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59—42no—
200C4×Dic59—42no—
201C4×Dic59—42no—
202C4×Dic59#4112yes—
203C4×Dic59#4112yes—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8C22×C412yes—
210C23×C59C22×D411yescharacteristic
211C22×D59—21no—
212C22×D59—21no—
213C22×D59—21no—
214C22×D59—21no—
215C22×D59C4×D412yes—
216C22×D59C2×SD1612yes—
217C22×D59#13512yes—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
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×C59C2×SD1612yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—22no—
247C2×Dic59—22no—
248C2×Dic59—22no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59C4×D412yes—
259C2×Dic59#12312yes—
260C2×Dic59#12312yes—
261C2×Dic59C2×SD1612yes—
262C2×Dic59#13712yes—
263C4×D59—22no—
264C4×D59—22no—
265C4×D59—22no—
266C4×D59—22no—
267C4×D59—28no—
268C4×D59—44no—
269C4×D59—48no—
270C4×D59—48no—
271Dic118—22no—
272Dic118—22no—
273Dic118—41no—
274Dic118—41no—
275C8×C59—22no—
276C8×C59—41no—
277C59⋊C8—22no—
278C59⋊C8—41no—
279C4×#483—591nomaximal, Sylow
280C22×C59C2×C4×D411yescharacteristic
281C22×C59C2×#12311yescharacteristic
282C22×C59#52511yescharacteristic
283C22×C59C22×SD1611yescharacteristic
284C22×C59C2×#13511yescharacteristic
285C22×C59C2×#13711yescharacteristic
286C22×C59#57511yescharacteristic
287C2×D59—22no—
288C2×D59—22no—
289C2×D59—22no—
290C2×D59—22no—
291C2×D59—22no—
292C2×D59—22no—
293C2×D59C4×SD1614yes—
294C4×C59—22no—
295C4×C59—41no—
296C4×C59—42no—
297C4×C59—42no—
298C4×C59—42no—
299C4×C59—81no—
300C4×C59C4×#4111yescharacteristic
301C4×C59C4×#4111yescharacteristic
302C4×C59#52911yescharacteristic
303C4×C59#52911yescharacteristic
304Dic59—22no—
305Dic59—22no—
306Dic59—42no—
307Dic59—42no—
308Dic59—42no—
309Dic59—81no—
310Dic59#48314yescomplemented
311C2×#483—591no—
312C4×C22⋊Q8—591no—
313C2×C4×C4⋊C4—591no—
314#1927—591no—
315#1927—591no—
316#1920—591no—
317#1920D5911yescharacteristic, complemented
318#1872—591no—
319#1853—591no—
320#1823—591no—
321C4×#106—591no—
322C4×#106—591no—
323C4×#103—591no—
324C4×#103—591no—
325C4×C22⋊C8—591no—
326C2×C59C4×#12311yescharacteristic
327C2×C59C2×C4×SD1611yescharacteristic
328C2×C59C4×#13711yescharacteristic
329C2×C59C2×#48311yescharacteristic
330C2×C59#309111yescharacteristic
331C2×C59#310111yescharacteristic
332C2×C59#310811yescharacteristic
333D59—24no—
334C2×C22⋊Q8—591no—
335C2×C4×Q8—591no—
336C22×C4⋊C4—591no—
337C22×C4⋊C4—591no—
338C22×C42—591no—
339C22×C42—1181no—
340#483—598no—
341C2×#106—591no—
342C2×#106—591no—
343C2×#106—591no—
344C2×#106C2×D5911yescharacteristic
345C2×#103—591no—
346C2×#103—591no—
347C2×#103—591no—
348C2×#103—591no—
349C2×C22⋊C8—591no—
350C2×C22⋊C8—591no—
351C2×C4×C8—591no—
352C2×C4×C8—591no—
353#391—591no—
354#389C2×D5911yescharacteristic
355#387—1181no—
356#385—1181no—
357#380—591no—
358C4×C4⋊C4—591no—
359C4×C4⋊C4—591no—
360C4×C4⋊C4—591no—
361C4×C4⋊C4—591no—
362C4×C4⋊C4—1181no—
363C4×C4⋊C4—1181no—
364C4×C4⋊C4—1181no—
365C4×C4⋊C4C2×D5911yescharacteristic
366C4×C22⋊C4—1181no—
367C2×#95—1181no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#340—591no—
373#340C2×D5911yescharacteristic
374#340C2×D5911yescharacteristic
375#340C2×D5911yescharacteristic
376#336—591no—
377#336C2×D5911yescharacteristic
378C59C4×#48311yescharacteristic, complemented, Sylow
379C22×Q8—591no—
380C23×C4—591no—
381C23×C4—591no—
382C23×C4—591no—
383C23×C4—1181no—
384C23×C4—1181no—
385C22×C8—591no—
386C22×C8—591no—
387C22×C8C22×D5911yescharacteristic
388C22×C8C22×D5911yescharacteristic
389C22⋊Q8—594no—
390C22⋊Q8—1182no—
391C4×Q8—1182no—
392C4×Q8—1182no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—594no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—1182no—
399C2×C4⋊C4—1182no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1182no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C22⋊C4—1181no—
410C2×C22⋊C4—1181no—
411C2×C42—21no—
412C2×C42—591no—
413C2×C42—591no—
414C2×C42—591no—
415C2×C42—1182no—
416C2×C42—1182no—
417C2×C42—1181no—
418C2×C42—1181no—
419C2×C42—1181no—
420C2×C42—1181no—
421C2×C42—1181no—
422C2×C42—2364no—
423C2×C42C22×D5911yescharacteristic
424#106—594no—
425#106—1182no—
426#106—1182no—
427#106C4×D5914yescomplemented
428#103—594no—
429#103—594no—
430#103—1182no—
431#103—1182no—
432C22⋊C8—594no—
433C22⋊C8—1182no—
434C4×C8—1182no—
435C4×C8—1182no—
436#95—21no—
437#95—21no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445C24—591no—
446C2×Q8—592no—
447C2×Q8—1181no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—591no—
457C22×C4—591no—
458C22×C4—591no—
459C22×C4—594no—
460C22×C4—592no—
461C22×C4—592no—
462C22×C4—1182no—
463C22×C4—1182no—
464C22×C4—1182no—
465C22×C4—1184no—
466C22×C4—1182no—
467C22×C4—1184no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—1181no—
476C22×C4—2361no—
477C22×C4—2362no—
478C22×C4—2362no—
479C22×C4—2362no—
480C22×C4—2362no—
481C22×C4D4×D5911yescharacteristic
482C22×C4D4×D5911yescharacteristic
483C22×C4C23×D5911yescharacteristic
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—592no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8C2×C4×D5912yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—592no—
503C4⋊C4—592no—
504C4⋊C4—1184no—
505C4⋊C4—1184no—
506C4⋊C4—1184no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—2362no—
512C4⋊C4C2×C4×D5912yes—
513C4⋊C4C2×C4×D5912yes—
514C4⋊C4#1174568912yes—
515C22⋊C4—1184no—
516C22⋊C4—2362no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—592no—
521C42—592no—
522C42—1182no—
523C42—1182no—
524C42—2362no—
525C42—2362no—
526C42—2362no—
527C42—2362no—
528C42—2362no—
529C23—592no—
530C23—592no—
531C23—592no—
532C23—1181no—
533C23—1181no—
534C23—1181no—
535C23—1181no—
536C23C2×D4×D5911yescharacteristic
537Q8—1182no—
538Q8—1182no—
539Q8—2361no—
540Q8—2361no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—592no—
559C2×C4—592no—
560C2×C4—592no—
561C2×C4—592no—
562C2×C4—592no—
563C2×C4—1188no—
564C2×C4—1181no—
565C2×C4—1181no—
566C2×C4—1181no—
567C2×C4—1181no—
568C2×C4—1182no—
569C2×C4—1182no—
570C2×C4—1182no—
571C2×C4—1182no—
572C2×C4—1182no—
573C2×C4—1182no—
574C2×C4—1182no—
575C2×C4—2364no—
576C2×C4—2368no—
577C2×C4—2368no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4SD16×D5912yes—
588C2×C4C22×C4×D5911yescharacteristic
589C2×C4#41×D5911yescharacteristic
590C2×C4#41×D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4C2×#1174568911yescharacteristic
594C8—22no—
595C8—41no—
596C8—1182no—
597C8—2361no—
598C22—594no—
599C22—1182no—
600C22—1182no—
601C22—1182no—
602C22—1182no—
603C22—1182no—
604C22—1182no—
605C22C4×D4×D5911yescharacteristic
606C22#123×D5911yescharacteristic
607C22#1683791511yescharacteristic
608C22C2×SD16×D5911yescharacteristic
609C22#135×D5911yescharacteristic
610C22#137×D5911yescharacteristic
611C22#1683819811yescharacteristic
612C4—22no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—594no—
619C4—1182no—
620C4—1182no—
621C4—4721no—
622C4—2362no—
623C4—2362no—
624C4—2362no—
625C4#1683787011yescharacteristic
626C4C4×#1174568911yescharacteristic
627C4#1683793511yescharacteristic
628C4#1683793711yescharacteristic
629C2—1184no—
630C2#2142661711yescharacteristic
631C2C4×SD16×D5911yescharacteristic
632C2#2143291811yescharacteristic
633C2#483×D5911yescharacteristic
634C2#2143400811yescharacteristic
635C2#2143410611yescharacteristic
636C2#2143418111yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1920C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1920D59
hypocenterC59C4×#483
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
4. #21424731C2semidirect
5. #21424730C2semidirect
6. #21424678C2semidirect
10. #106×Dic59C2semidirect
11. #21423984C2semidirect
12. #21423915C2semidirect
13. #21423915C2semidirect
15. C59×#1920C2semidirect
22. #16837734C4semidirect
98. #14273658C2×C4semidirect
100. #14273652C2×C4semidirect
102. #14273650C2×C4semidirect
123. #106×C59C2×C4semidirect
310. Dic59#483semidirect
317. #1920D59semidirect
378. C59C4×#483semidirect
427. #106C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377342
#214266262
C4×C4⋊C4×D592
#214247312
#214247302
#214246782
#214244022
#214242902
#214241812
#106×Dic592
#214239842
#214239152
#214238532
C59×#19202
C4×#48359