Finite Group Databasebeta

← #28662015

Subgroups — #28662015

Order 15104.

802 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428682C211yesmaximal, characteristic, complemented
2#21425634C211yesmaximal, characteristic, complemented
3#21425411C211yesmaximal, characteristic, complemented
4#21425332C211yesmaximal, characteristic, complemented
5#21425173C211yesmaximal, characteristic
6#21424961C211yesmaximal, characteristic
7#21424716C211yesmaximal, characteristic, complemented
8#21424713C211yesmaximal, characteristic, complemented
9#21424688C211yesmaximal, characteristic, complemented
10#21424655C211yesmaximal, characteristic, complemented
11#21424451C211yesmaximal, characteristic, complemented
12#21424444C211yesmaximal, characteristic
13#21424096C211yesmaximal, characteristic, complemented
14C4×#14273651C211yesmaximal, characteristic, complemented
15C59×#2367C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C22×#11745669C2211yescharacteristic
18C2×#14273722—21no—
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837329—21no—
32#16837324—21no—
33#16837304C2211yescharacteristic
34#16837304C2211yescharacteristic
35#16837304C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837294—21no—
38#16837272C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#124×C59C2211yescharacteristic
49C2×C4×Q8×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C59×#391C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#385—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C22×D236C2311yescharacteristic
60C2×#11745671—22no—
61C2×#11745671—41no—
62C2×#11745671—41no—
63C2×#11745669—22no—
64C2×#11745669—22no—
65C2×#11745669—41no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273722—44no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
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×(C2×D59)⋊C4—21no—
83C2×C4×Dic59—21no—
84#14273660—22no—
85#14273660—41no—
86C4⋊(C59⋊C8)—41no—
87#14273651—22no—
88#14273651—22no—
89#14273651—41no—
90#14273648—22no—
91#14273648—41no—
92#14273648—41no—
93#14273647—22no—
94#14273647—41no—
95C4×C59⋊C8—41no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C22×Q8×C59C2311yescharacteristic
99C22×D4×C59C2311yescharacteristic
100C23×C4×C59—21no—
101#124×C59—22no—
102#124×C59—22no—
103C4×Q8×C59—22no—
104C4×Q8×C59—41no—
105C4×D4×C59—22no—
106C4×D4×C59—41no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125#95×C59D411yescharacteristic
126#95×C59D411yescharacteristic
127C24×C59—21no—
128C23×D59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59#4112yes—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59#4112yes—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59—81no—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2411yescharacteristic
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236—21no—
162C2×D236—21no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59#4112yes—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—81no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—42no—
182C42×C59—81no—
183#11745671—44no—
184#11745669—44no—
185(C2×D59)⋊C4—42no—
186(C2×D59)⋊C4—42no—
187(C2×D59)⋊C4—42no—
188(C2×D59)⋊C4—42no—
189(C2×D59)⋊C4—81no—
190(C2×D59)⋊C4—81no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—42no—
198C4⋊Dic59—81no—
199Dic59⋊C4—42no—
200C4×Dic59—42no—
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—21no—
212C2×C59⋊C8#4112yes—
213C23×C59—22no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59C22×D411yescharacteristic
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225C22×D59—41no—
226C22×D59—41no—
227C22×D59—41no—
228Q8×C59—22no—
229Q8×C59—22no—
230Q8×C59—41no—
231Q8×C59—41no—
232D4×C59—22no—
233D4×C59—22no—
234D4×C59—41no—
235D4×C59—41no—
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—22no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—22no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—84no—
262C2×C4×C59#13512yes—
263C2×C4×C59AGL(1,ℤ/8)12yes—
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59H(5,𝔽2)11yescharacteristic
269C2×C4×C59H(5,𝔽2)11yescharacteristic
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—
279D236—41no—
280D236—41no—
281D236—41no—
282D236—41no—
283C59⋊C8—22no—
284C59⋊C8—41no—
285C59⋊C8—41no—
286C59⋊C8—41no—
287#27075—591nomaximal, Sylow
288C22×C59—42no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—81no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59#52511yescharacteristic
296C22×C59#54611yescharacteristic
297C22×C59#55011yescharacteristic
298C22×C59C2×#13511yescharacteristic
299C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C59#57611yescharacteristic
301C22×C59#57811yescharacteristic
302C2×D59—81no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C2×D59—81no—
307C2×D59—81no—
308C2×D59—81no—
309C4×C59—22no—
310C4×C59—22no—
311C4×C59—41no—
312C4×C59—41no—
313C4×C59—42no—
314C4×C59—41no—
315C4×C59—42no—
316C4×C59—41no—
317C4×C59—81no—
318C4×C59#52911yescharacteristic
319C4×C59#53811yescharacteristic
320C4×C59#55111yescharacteristic
321C4×C59#55311yescharacteristic
322Dic59—81no—
323Dic59—81no—
324Dic59—81no—
325#2512—591no—
326#2367D5911yescharacteristic, complemented
327#2095—591no—
328#2055—591no—
329#2042—591no—
330#2012—591no—
331#1970—591no—
332#1925—591no—
333#1925—591no—
334#1921—591no—
335#1916—591no—
336#1880—591no—
337#1879—591no—
338#1802—591no—
339C4×#103—591no—
340C2×C59—82no—
341C2×C59#246711yescharacteristic
342C2×C59#310611yescharacteristic
343C2×C59#312411yescharacteristic
344C2×C59#329711yescharacteristic
345C2×C59#330811yescharacteristic
346C2×C59#339311yescharacteristic
347C2×C59#339911yescharacteristic
348D59—161no—
349C22×SD16—591no—
350C22×D8—591no—
351C2×#126—1181no—
352C2×#124C2×D5911yescharacteristic
353C2×C4×Q8C2×D5911yescharacteristic
354C2×C4×D4C2×D5911yescharacteristic
355C2×#107—591no—
356C2×#107—591no—
357C2×C4⋊C8—591no—
358C2×#103—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#102—591no—
362C2×#102—591no—
363C2×#102—591no—
364C2×#102—591no—
365C2×#102—591no—
366C2×C4×C8—591no—
367#397—1181no—
368#396—1181no—
369#391C2×D5911yescharacteristic
370#390—591no—
371#390—591no—
372#390—591no—
373#390C2×D5911yescharacteristic
374#389—591no—
375#388—21no—
376#388—21no—
377#388—1181no—
378#385—21no—
379#384—591no—
380C4×C4⋊C4—591no—
381C4×C4⋊C4C2×D5911yescharacteristic
382C4×C4⋊C4C2×D5911yescharacteristic
383C4×C22⋊C4—21no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#336—591no—
389#336—591no—
390#336—591no—
391#336—591no—
392C59#2707511yescharacteristic, complemented, Sylow
393C22×Q8C22×D5911yescharacteristic
394C22×D4—591no—
395C22×D4C22×D5911yescharacteristic
396C23×C4—21no—
397C2×SD16—1182no—
398C2×SD16—2361no—
399C2×SD16—2361no—
400C2×D8—1182no—
401C2×D8—1182no—
402C2×D8—2361no—
403C22×C8—591no—
404C22×C8—591no—
405C22×C8—591no—
406C22×C8—591no—
407#126—2364no—
408#124—22no—
409#124—22no—
410C4×Q8—22no—
411C4×Q8—41no—
412C4×D4—22no—
413C4×D4—41no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—1181no—
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×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1181no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4—1181no—
435C2×C22⋊C4—1181no—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—1181no—
439C2×C42C22×D5911yescharacteristic
440C2×C42C22×D5911yescharacteristic
441C2×C42C22×D5911yescharacteristic
442#107—1182no—
443#107—2361no—
444C4⋊C8—2361no—
445#103—1182no—
446#103—1182no—
447#103—2361no—
448#102—1182no—
449#102—1182no—
450#102—2361no—
451#102—2361no—
452#102—2361no—
453C4×C8—2361no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—1181no—
458#95—1181no—
459#95C22⋊D5911yescharacteristic
460#95C22⋊D5911yescharacteristic
461C24—21no—
462C24—1181no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8#1174568512yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—1181no—
475C2×D4—1181no—
476C2×D4—1181no—
477C2×D4—1181no—
478C2×D4—1181no—
479C2×D4—1181no—
480C2×D4#1174568512yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—1181no—
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
497SD16—2364no—
498D8—2364no—
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—
510C2×C8—1181no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—4721no—
524C4⋊C4—2362no—
525C4⋊C4—2362no—
526C4⋊C4#1174568712yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—81no—
532C22⋊C4—4721no—
533C22⋊C4—4721no—
534C22⋊C4—2362no—
535C22⋊C4—2362no—
536C22⋊C4—2362no—
537C22⋊C4—2362no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—42no—
545C42—81no—
546C42—2362no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—2361no—
555C23—2361no—
556C23—2361no—
557C23—2361no—
558C23—2361no—
559C23—2361no—
560C23—2361no—
561C23C22×C22⋊D5911yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—41no—
565Q8—41no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—2361no—
571D4—2361no—
572D4—2361no—
573D4—2361no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4#1427374712yes—
610C2×C4#1427375212yes—
611C2×C4#41×D5911yescharacteristic
612C2×C4C2×#1174568711yescharacteristic
613C2×C4C2×#1174568511yescharacteristic
614C2×C4C2×#1174568511yescharacteristic
615C2×C4#1427377211yescharacteristic
616C2×C4#1427377311yescharacteristic
617C8—1182no—
618C8—2361no—
619C8—2361no—
620C8—2361no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—4721no—
629C22—4721no—
630C22—4721no—
631C22—4721no—
632C22—4721no—
633C22—4721no—
634C22—4721no—
635C22#1683791311yescharacteristic
636C22#1683804611yescharacteristic
637C22#1683807111yescharacteristic
638C22C2×#1427374711yescharacteristic
639C22C2×#1427375211yescharacteristic
640C22#1683820511yescharacteristic
641C22#1683821711yescharacteristic
642C4—22no—
643C4—22no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—81no—
651C4—4721no—
652C4—4721no—
653C4—4721no—
654C4#1683793211yescharacteristic
655C4#1683798511yescharacteristic
656C4#1683808111yescharacteristic
657C4#1683809811yescharacteristic
658C2—82no—
659C2—9441no—
660C2#2142814111yescharacteristic
661C2#2143415611yescharacteristic
662C2#2143435811yescharacteristic
663C2#2143594611yescharacteristic
664C2#2143608711yescharacteristic
665C2#2143708011yescharacteristic
666C2#2143713411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2367C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
hypercenter#2367D59
hypocenterC59#27075
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #21428682C2semidirect
2. #21425634C2semidirect
3. #21425411C2semidirect
4. #21425332C2semidirect
7. #21424716C2semidirect
8. #21424713C2semidirect
9. #21424688C2semidirect
10. #21424655C2semidirect
11. #21424451C2semidirect
13. #21424096C2semidirect
14. C4×#14273651C2semidirect
15. C59×#2367C2semidirect
326. #2367D59semidirect
392. C59#27075semidirect

Maximal subgroups

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

subgroupindex
#214286822
#214256342
#214254112
#214253322
#214251732
#214249612
#214247162
#214247132
#214246882
#214246552
#214244512
#214244442
#214240962
C4×#142736512
C59×#23672
#2707559