Finite Group Databasebeta

← #28636956

Subgroups — #28636956

Order 15104.

842 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837730C211yesmaximal, characteristic
2C2×#16837641C211yesmaximal, characteristic
3#21428581C211yesmaximal, characteristic
4#21425577C211yesmaximal, characteristic, complemented
5#21425514C211yesmaximal, characteristic, complemented
6#21425474C211yesmaximal, characteristic, complemented
7#21425418C211yesmaximal, characteristic, complemented
8#21424864C211yesmaximal, characteristic, complemented
9#21424730C211yesmaximal, characteristic, complemented
10#21424437C211yesmaximal, characteristic, complemented
11#21424403C211yesmaximal, characteristic
12#21424276C211yesmaximal, characteristic
13#21424111C211yesmaximal, characteristic, complemented
14C2×#16837164C211yesmaximal, characteristic
15C59×#2360C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C23×C59⋊C8C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837730—24no—
21#16837641—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837342—21no—
35#16837342—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837268C2211yescharacteristic
39#16837268C2211yescharacteristic
40#16837263—21no—
41#16837263—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C22×C4⋊C4×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#389—21no—
55C59×#389C2211yescharacteristic
56C59×#387—21no—
57C59×#385—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59—21no—
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672—41no—
66C22×C59⋊C8—22no—
67C22×C59⋊C8—22no—
68C22×C59⋊C8—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273706—22no—
74#14273706—22no—
75C2×C4⋊Dic59—22no—
76C2×C4⋊Dic59—22no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83#14273660—22no—
84#14273658—22no—
85#14273658—22no—
86C4⋊(C59⋊C8)—22no—
87C4⋊(C59⋊C8)—41no—
88#14273651—22no—
89#14273651—22no—
90#14273651—22no—
91#14273650—22no—
92#14273650—22no—
93#14273650—41no—
94C22⋊(C59⋊C8)—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104C23×C4×C59—21no—
105C23×C4×C59C2311yescharacteristic
106C22⋊Q8×C59—22no—
107C22⋊Q8×C59D414yes—
108C4×Q8×C59—22no—
109C4×Q8×C59—41no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—22no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—41no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C22⋊C4×C59D411yescharacteristic
122C2×C42×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59C2×D411yescharacteristic
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59C2×D412yes—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—42no—
148C22×C4×C59—42no—
149C22×C4×C59—81no—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D412yes—
156C22×C4×C59#4112yes—
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—42no—
163C22×Dic59—42no—
164C22×Dic59—42no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C4⋊C4×C59—24no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—81no—
179C4⋊C4×C59C2×D412yes—
180C4⋊C4×C59#4112yes—
181C22⋊C4×C59—24no—
182C22⋊C4×C59—22no—
183C22⋊C4×C59—22no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—42no—
189C42×C59—81no—
190C22⋊Dic59—42no—
191#11745672—44no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—42no—
199Dic59⋊C4—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⋊C8—21no—
211C2×C59⋊C8—44no—
212C2×C59⋊C8#4112yes—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59#12112yes—
218C23×C59#13512yes—
219C23×C59#13712yes—
220C23×C59C22×D411yescharacteristic
221Q8×C59—22no—
222Q8×C59—22no—
223Q8×C59—41no—
224Q8×C59—41no—
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—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—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
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—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—84no—
256C2×C4×C59#12112yes—
257C2×C4×C59C22×D411yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59H(5,𝔽2)11yescharacteristic
262C2×C4×C592−1+411yescharacteristic
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
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×Dic59—41no—
275C2×Dic59—84no—
276Dic118—41no—
277Dic118—41no—
278Dic118—41no—
279Dic118—41no—
280C59⋊C8—22no—
281C59⋊C8—41no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284#25208—591nomaximal, Sylow
285C22×C59—22no—
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59#46414yes—
292C22×C59C2×#12111yescharacteristic
293C22×C59#54611yescharacteristic
294C22×C59#54711yescharacteristic
295C22×C59C2×#13511yescharacteristic
296C22×C59C2×#13711yescharacteristic
297C22×C59#57711yescharacteristic
298C22×C59#57811yescharacteristic
299C4×C59—22no—
300C4×C59—41no—
301C4×C59—41no—
302C4×C59—42no—
303C4×C59—41no—
304C4×C59—42no—
305C4×C59—42no—
306C4×C59—41no—
307C4×C59—81no—
308C4×C59#52911yescharacteristic
309C4×C59#53911yescharacteristic
310C4×C59#54611yescharacteristic
311C4×C59#54711yescharacteristic
312Dic59—81no—
313Dic59—81no—
314Dic59—81no—
315Dic59—81no—
316C2×#483—591no—
317C2×#467—591no—
318#2503—591no—
319#2360D5911yescharacteristic
320#2085—591no—
321#2074—591no—
322#2066—591no—
323#2056—591no—
324#1949—591no—
325#1927—591no—
326#1878—591no—
327#1872—591no—
328#1850—591no—
329#1805—591no—
330C2×#340—591no—
331C2×C59—24no—
332C2×C59#242511yescharacteristic
333C2×C59C2×#46411yescharacteristic
334C2×C59#304611yescharacteristic
335C2×C59#329611yescharacteristic
336C2×C59#331211yescharacteristic
337C2×C59#333611yescharacteristic
338C2×C59#334411yescharacteristic
339C22×Q16—591no—
340C23×C8—591no—
341C2×C22⋊Q8—591no—
342C2×C22⋊Q8C2×D5911yescharacteristic
343C2×C4×Q8C2×D5911yescharacteristic
344C22×C4⋊C4—591no—
345C22×C4⋊C4C2×D5911yescharacteristic
346#483—1184no—
347#467—1184no—
348C2×#107—591no—
349C2×#106—591no—
350C2×#106—591no—
351C2×C4⋊C8—591no—
352C2×C4⋊C8—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#103—591no—
358C2×#103—591no—
359C2×C22⋊C8—591no—
360#400—1181no—
361#400—1181no—
362#391—591no—
363#391—591no—
364#391C2×D5911yescharacteristic
365#389—21no—
366#389C2×D5911yescharacteristic
367#387—21no—
368#385—21no—
369#382—591no—
370#382—591no—
371#381—1181no—
372#381—1181no—
373C4×C4⋊C4C2×D5911yescharacteristic
374C4×C22⋊C4—21no—
375C4×C22⋊C4C2×D5911yescharacteristic
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#336—591no—
383#336—591no—
384C59#2520811yescharacteristic, complemented, Sylow
385C22×Q8—591no—
386C22×Q8C22×D5911yescharacteristic
387C23×C4—21no—
388C23×C4—1181no—
389C23×C4C22×D5911yescharacteristic
390C2×Q16—1182no—
391C2×Q16—1182no—
392C2×Q16—2361no—
393C22×C8—591no—
394C22×C8—591no—
395C22×C8—591no—
396C22×C8—591no—
397C22×C8—1182no—
398C22×C8—1182no—
399C22×C8—2361no—
400C22⋊Q8—22no—
401C22⋊Q8—1182no—
402C22⋊Q8—1182no—
403C22⋊Q8D23614yes—
404C4×Q8—22no—
405C4×Q8—41no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—41no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—1182no—
416C2×C4⋊C4—1182no—
417C2×C4⋊C4—1181no—
418C2×C4⋊C4—1181no—
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C4⋊C4C22×D5911yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4C22⋊D5911yescharacteristic
425C2×C22⋊C4C22⋊D5911yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
428C2×C42C22×D5911yescharacteristic
429C2×C42C22×D5911yescharacteristic
430#107—1182no—
431#106—1182no—
432#106—1182no—
433C4⋊C8—1182no—
434C4⋊C8—2361no—
435#103—1182no—
436#103—1182no—
437#103—1182no—
438#103—1182no—
439#103—1182no—
440#103—2361no—
441C22⋊C8—1182no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95C22⋊D5911yescharacteristic
455#95C22⋊D5911yescharacteristic
456C24C2×C22⋊D5911yescharacteristic
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—1181no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8C2×D23612yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—81no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—2362no—
485C22×C4—2362no—
486C22×C4—2362no—
487C22×C4C2×D23612yes—
488C22×C4#1174568512yes—
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C23×D5911yescharacteristic
495Q16—2364no—
496C2×C8—592no—
497C2×C8—1181no—
498C2×C8—1181no—
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—2364no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—2362no—
524C4⋊C4—2362no—
525C4⋊C4C2×D23612yes—
526C4⋊C4#1174568712yes—
527C22⋊C4—24no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—2362no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—81no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23#1427370112yes—
542C23#1427374712yes—
543C23#1427375812yes—
544C23C22×C22⋊D5911yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—41no—
548Q8—41no—
549Q8—2361no—
550Q8—2361no—
551Q8—2361no—
552Q8—2361no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
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—22no—
572C2×C4—21no—
573C2×C4—21no—
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—84no—
584C2×C4—4724no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4#1427370112yes—
598C2×C4C22×D23611yescharacteristic
599C2×C4#41×D5911yescharacteristic
600C2×C4C2×#1174568711yescharacteristic
601C2×C4C2×#1174568511yescharacteristic
602C2×C4#1427377211yescharacteristic
603C2×C4#1427377411yescharacteristic
604C8—1182no—
605C8—2361no—
606C8—2361no—
607C8—2361no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22#1683762414yes—
615C22C2×#1427370111yescharacteristic
616C22#1683804611yescharacteristic
617C22#1683805111yescharacteristic
618C22C2×#1427374711yescharacteristic
619C22C2×#1427375811yescharacteristic
620C22#1683821311yescharacteristic
621C22#1683821711yescharacteristic
622C4—22no—
623C4—41no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—81no—
631C4—4721no—
632C4—4721no—
633C4—4721no—
634C4—4721no—
635C4#1683793211yescharacteristic
636C4#1683799411yescharacteristic
637C4#1683804711yescharacteristic
638C4#1683805611yescharacteristic
639C2—24no—
640C2#2142764111yescharacteristic
641C2C2×#1683762411yescharacteristic
642C2#2143364611yescharacteristic
643C2#2143593111yescharacteristic
644C2#2143614711yescharacteristic
645C2#2143645811yescharacteristic
646C2#2143654511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2360C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2360D59
hypocenterC59#25208
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
4. #21425577C2semidirect
5. #21425514C2semidirect
6. #21425474C2semidirect
7. #21425418C2semidirect
8. #21424864C2semidirect
9. #21424730C2semidirect
10. #21424437C2semidirect
13. #21424111C2semidirect
384. C59#25208semidirect

Maximal subgroups

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

subgroupindex
C2×#168377302
C2×#168376412
#214285812
#214255772
#214255142
#214254742
#214254182
#214248642
#214247302
#214244372
#214244032
#214242762
#214241112
C2×#168371642
C59×#23602
#2520859