Finite Group Databasebeta

← #28595377

Subgroups — #28595377

Order 15104.

954 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×#16837658C211yesmaximal, characteristic
2C2×C4×#11745672C211yesmaximal, characteristic, complemented
3C2×#16837358C211yesmaximal, characteristic
4#21426778C211yesmaximal, characteristic
5#21424966C211yesmaximal, characteristic, complemented
6#21424836C211yesmaximal, characteristic, complemented
7#21424775C211yesmaximal, characteristic, complemented
8#21424771C211yesmaximal, characteristic, complemented
9#21424520C211yesmaximal, characteristic
10#21424517C211yesmaximal, characteristic
11#21424127C211yesmaximal, characteristic
12C4×#14273658C211yesmaximal, characteristic, complemented
13#21423929C211yesmaximal, characteristic, complemented
14#21423926C211yesmaximal, characteristic, complemented
15C59×#2323C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745664C2211yescharacteristic
18C23×C59⋊C8C2211yescharacteristic
19C2×#14273706C2211yescharacteristic
20C2×C4×Dic118C2211yescharacteristic
21#16837658C418yescomplemented
22C4×#11745672—24no—
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32#16837358—24no—
33C2×#14273638C2211yescharacteristic
34C2×#14273638C2211yescharacteristic
35C2×C4×C59⋊C8C2211yescharacteristic
36C2×C4×C59⋊C8C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837297—21no—
40#16837296—21no—
41#16837290—21no—
42#16837283—21no—
43#16837262C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51C2×C22⋊Q8×C59C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C2×#117×C59C2211yescharacteristic
54C59×#391C2211yescharacteristic
55C59×#389—21no—
56C59×#389C2211yescharacteristic
57C59×#388—21no—
58C59×#388—21no—
59C59×#387—21no—
60C59×#380C2211yescharacteristic
61C4×C4⋊C4×C59C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672C2×C414yes—
66C2×#11745664—22no—
67C2×#11745664—22no—
68C2×#11745664C2×C414yes—
69C22×C59⋊C8—22no—
70C22×C59⋊C8—22no—
71C22×C59⋊C8—22no—
72C22×C59⋊C8C2311yescharacteristic
73C22×C59⋊C8C2311yescharacteristic
74C22×C59⋊C8C2311yescharacteristic
75C22×C59⋊C8C2311yescharacteristic
76#14273706—22no—
77#14273706C2×C414yes—
78C4×Dic118—22no—
79C4×Dic118—22no—
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×C22⋊Dic59—21no—
87C2×C22⋊Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90#14273660—22no—
91#14273658—22no—
92#14273658C2×C414yes—
93#14273651—22no—
94#14273651—22no—
95#14273651C2×C414yes—
96#14273650—22no—
97#14273650—22no—
98#14273650C2×C414yes—
99#14273638—22no—
100#14273638—22no—
101C4×C59⋊C8—22no—
102C4×C59⋊C8D414yes—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C22×Q8×C59C2311yescharacteristic
109C23×C4×C59C2311yescharacteristic
110C22⋊Q8×C59—22no—
111C22⋊Q8×C59C2×C414yes—
112C4×Q8×C59—22no—
113C4×Q8×C59—22no—
114#117×C59—22no—
115#117×C59—22no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59D411yescharacteristic
120C2×C4⋊C4×C59D411yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C22⋊C4×C59—21no—
125C2×C22⋊C4×C59—21no—
126C2×C22⋊C4×C59D411yescharacteristic
127C2×C22⋊C4×C59D411yescharacteristic
128C2×C42×C59—21no—
129C2×C42×C59—21no—
130C2×C42×C59C2311yescharacteristic
131C2×C42×C59C2311yescharacteristic
132#95×C59—21no—
133#95×C59—21no—
134#95×C59—21no—
135#95×C59—21no—
136#95×C59—21no—
137C24×C59C2×D411yescharacteristic
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59C22×C412yes—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59C22×C412yes—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59#4112yes—
159C22×C4×C59C2411yescharacteristic
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118C22×C412yes—
170C4⋊C4×C59—24no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59—81no—
180C4⋊C4×C59C22×C412yes—
181C22⋊C4×C59—24no—
182C22⋊C4×C59—22no—
183C22⋊C4×C59—22no—
184C22⋊C4×C59—81no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—42no—
188C42×C59—81no—
189C42×C59C2×D412yes—
190C42×C59#4112yes—
191C22⋊Dic59—24no—
192C22⋊Dic59—81no—
193#11745672—28no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59C22×C412yes—
200Dic59⋊C4—24no—
201Dic59⋊C4—42no—
202Dic59⋊C4—81no—
203C4×Dic59—42no—
204C4×Dic59—81no—
205#11745664—28no—
206C2×C59⋊C8—28no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8C22×C412yes—
214C2×C59⋊C8C22×C412yes—
215C2×C59⋊C8C2×D412yes—
216C2×C59⋊C8C2×D412yes—
217C2×C59⋊C8#4112yes—
218C2×C59⋊C8#4112yes—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59C4×D412yes—
226C23×C59C22×D411yescharacteristic
227Q8×C59—22no—
228Q8×C59—22no—
229Q8×C59—41no—
230Q8×C59—41no—
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—22no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
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—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59C4×D412yes—
260C2×C4×C59#12112yes—
261C2×C4×C59#12112yes—
262C2×C4×C59C23×C411yescharacteristic
263C2×C4×C59C22×D411yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×Dic59—22no—
269C2×Dic59—22no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280Dic118—22no—
281Dic118—22no—
282Dic118—41no—
283Dic118—41no—
284C59⋊C8—22no—
285C59⋊C8—22no—
286C59⋊C8—22no—
287C59⋊C8C4×D414yes—
288#21537—591nomaximal, Sylow
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59C2×C4×D411yescharacteristic
297C22×C59C2×#12111yescharacteristic
298C22×C59#53211yescharacteristic
299C22×C59#57711yescharacteristic
300C22×C59#57711yescharacteristic
301C22×C59#57811yescharacteristic
302C22×C59#57811yescharacteristic
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—22no—
306C4×C59—42no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—42no—
310C4×C59—81no—
311C4×C59—81no—
312C4×C59C2×C4×D411yescharacteristic
313C4×C59C4×#4111yescharacteristic
314C4×C59#52511yescharacteristic
315C4×C59#52911yescharacteristic
316Dic59—42no—
317Dic59—42no—
318Dic59—81no—
319Dic59—81no—
320C2×#470—591no—
321C2×C4×Q16—591no—
322C2×#405—591no—
323#2323—591no—
324#2323D5911yescharacteristic
325#1971—591no—
326#1944—591no—
327#1933—591no—
328#1933—591no—
329#1892—591no—
330#1892—591no—
331#1809—591no—
332C4×#106—591no—
333#1763—591no—
334#1763—591no—
335C2×C59—42no—
336C2×C59C4×#12111yescharacteristic
337C2×C59#294711yescharacteristic
338C2×C59#294711yescharacteristic
339C2×C59#305911yescharacteristic
340C2×C59#306011yescharacteristic
341C2×C59#317811yescharacteristic
342C2×C59#317811yescharacteristic
343C22×Q16—591no—
344C22×M16—591no—
345C23×C8—591no—
346C2×C22⋊Q8—591no—
347C2×C22⋊Q8C2×D5911yescharacteristic
348C2×C4×Q8—591no—
349C2×C4×Q8C2×D5911yescharacteristic
350C2×#117C2×D5911yescharacteristic
351#470—598no—
352C4×Q16—1184no—
353C2×#107—591no—
354C2×#106—591no—
355C2×#106—591no—
356C2×#103—591no—
357C2×#103—591no—
358C2×#103—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#103—591no—
362#405—1184no—
363C2×#97—591no—
364C2×#97—591no—
365C2×C4×C8—591no—
366C2×C4×C8—591no—
367#391—591no—
368#391C2×D5911yescharacteristic
369#389—21no—
370#389—591no—
371#389—1181no—
372#389C2×D5911yescharacteristic
373#388—21no—
374#388—21no—
375#388—1181no—
376#388—1181no—
377#387—21no—
378#387—1181no—
379#380—591no—
380#380C2×D5911yescharacteristic
381C4×C4⋊C4—591no—
382C4×C4⋊C4C2×D5911yescharacteristic
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389C59#2153711yescharacteristic, complemented, Sylow
390C22×Q8—591no—
391C22×Q8C22×D5911yescharacteristic
392C23×C4C22×D5911yescharacteristic
393C2×Q16—594no—
394C2×Q16—1182no—
395C2×Q16—1182no—
396C2×M16—594no—
397C2×M16—1182no—
398C2×M16—1182no—
399C22×C8—591no—
400C22×C8—591no—
401C22×C8—591no—
402C22×C8—591no—
403C22×C8—1182no—
404C22×C8—1182no—
405C22×C8—1182no—
406C22⋊Q8—22no—
407C22⋊Q8—594no—
408C22⋊Q8—1182no—
409C22⋊Q8C4×D5914yes—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—1182no—
413C4×Q8—1182no—
414#117—22no—
415#117—22no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4C22⋊D5911yescharacteristic
426C2×C4⋊C4C22⋊D5911yescharacteristic
427C2×C4⋊C4C22×D5911yescharacteristic
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1181no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4C22⋊D5911yescharacteristic
435C2×C22⋊C4C22⋊D5911yescharacteristic
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—1181no—
439C2×C42—1181no—
440C2×C42C22×D5911yescharacteristic
441C2×C42C22×D5911yescharacteristic
442#107—1182no—
443#106—594no—
444#106—1182no—
445#103—594no—
446#103—594no—
447#103—1182no—
448#103—1182no—
449#103—1182no—
450#103—1182no—
451#97—1182no—
452#97—1182no—
453C4×C8—594no—
454C4×C8—1182no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464#95—1181no—
465C24C2×C22⋊D5911yescharacteristic
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—592no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8C2×C4×D5912yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—1181no—
490C22×C4C2×C4×D5912yes—
491C22×C4#1174568512yes—
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C2×C22⋊D5911yescharacteristic
497C22×C4C23×D5911yescharacteristic
498Q16—1188no—
499M16—1188no—
500C2×C8—592no—
501C2×C8—592no—
502C2×C8—592no—
503C2×C8—592no—
504C2×C8—592no—
505C2×C8—592no—
506C2×C8—1188no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—592no—
524C4⋊C4—1184no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—4721no—
531C4⋊C4—2362no—
532C4⋊C4C2×C4×D5912yes—
533C22⋊C4—24no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—81no—
537C22⋊C4—1184no—
538C22⋊C4—4721no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—81no—
543C42—4721no—
544C42—2362no—
545C42D4×D5912yes—
546C42#1174568712yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23C4×C22⋊D5912yes—
554C23C22×C22⋊D5911yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—1182no—
560Q8—1182no—
561Q8—2361no—
562Q8—2361no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
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—41no—
591C2×C4—1182no—
592C2×C4—1182no—
593C2×C4—2361no—
594C2×C4—2361no—
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#1427369912yes—
605C2×C4#1427369912yes—
606C2×C4C22×C4×D5911yescharacteristic
607C2×C4C2×D4×D5911yescharacteristic
608C2×C4#41×D5911yescharacteristic
609C2×C4#41×D5911yescharacteristic
610C2×C4C2×#1174568711yescharacteristic
611C2×C4C2×#1174568511yescharacteristic
612C8—594no—
613C8—1182no—
614C8—1182no—
615C8—1182no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D5911yescharacteristic
624C22C2×#1427369911yescharacteristic
625C22#1683794811yescharacteristic
626C22#1683821011yescharacteristic
627C22#1683821311yescharacteristic
628C22#1683821711yescharacteristic
629C22#1683821711yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—4721no—
640C4—4721no—
641C4—2362no—
642C4—2362no—
643C4C4×D4×D5911yescharacteristic
644C4C4×#1174568711yescharacteristic
645C4#1683791111yescharacteristic
646C4#1683793211yescharacteristic
647C2—42no—
648C2C4×#1427369911yescharacteristic
649C2#2143293511yescharacteristic
650C2#2143293911yescharacteristic
651C2#2143374811yescharacteristic
652C2#2143375511yescharacteristic
653C2#2143475811yescharacteristic
654C2#2143476611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2323C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#2323D59
hypocenterC59#21537
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11745672C2semidirect
5. #21424966C2semidirect
6. #21424836C2semidirect
7. #21424775C2semidirect
8. #21424771C2semidirect
12. C4×#14273658C2semidirect
13. #21423929C2semidirect
14. #21423926C2semidirect
21. #16837658C4semidirect
389. C59#21537semidirect

Maximal subgroups

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

subgroupindex
C2×#168376582
C2×C4×#117456722
C2×#168373582
#214267782
#214249662
#214248362
#214247752
#214247712
#214245202
#214245172
#214241272
C4×#142736582
#214239292
#214239262
C59×#23232
#2153759