Finite Group Databasebeta

← #28651727

Subgroups — #28651727

Order 15104.

884 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837699C211yesmaximal, characteristic
2C2×C4×#11745671C211yesmaximal, characteristic
3C4⋊C4×D236C211yesmaximal, characteristic
4#21425634C211yesmaximal, characteristic, complemented
5#21425572C211yesmaximal, characteristic, complemented
6#21425491C211yesmaximal, characteristic, complemented
7#21425431C211yesmaximal, characteristic, complemented
8#21424826C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424433C211yesmaximal, characteristic
11#21424433C211yesmaximal, characteristic
12#21424364C211yesmaximal, characteristic, complemented
13#21424342C211yesmaximal, characteristic
14#21424342C211yesmaximal, characteristic
15C59×#2453C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20C2×C4⋊C4×D59—21no—
21#16837699—24no—
22C4×#11745671—24no—
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273651C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×#14273647C2211yescharacteristic
34C2×#14273647C2211yescharacteristic
35C2×C4×C59⋊C8C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837276—21no—
39#16837276—21no—
40#16837272C2211yescharacteristic
41#16837259—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#125×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C59×#400—21no—
53C59×#398—21no—
54C59×#391C2211yescharacteristic
55C59×#391C2211yescharacteristic
56C59×#389—21no—
57C59×#389C2211yescharacteristic
58C59×#389C2211yescharacteristic
59C59×#387—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C22×C4×D59—21no—
65C2×#11745671—22no—
66C2×#11745671—22no—
67C2×#11745671Q814yes—
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C4×D236—22no—
73C4×D236—22no—
74C4×D236—22no—
75C4×D236—22no—
76C4×D236—41no—
77C4×D236—41no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C4⋊C4×D59—44no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88C2×(C2×D59)⋊C4—21no—
89#14273660—22no—
90#14273660—22no—
91#14273660—41no—
92#14273658—22no—
93#14273658—22no—
94C4⋊(C59⋊C8)—22no—
95#14273651—22no—
96#14273651—22no—
97#14273651—41no—
98#14273647—22no—
99#14273647—22no—
100#14273647D414yes—
101C4×C59⋊C8—22no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C22×Q8×C59C2311yescharacteristic
105#125×C59—22no—
106#125×C59—41no—
107C4×Q8×C59—22no—
108C4×Q8×C59—41no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59D411yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123#95×C59—21no—
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59D411yescharacteristic
130#95×C59D411yescharacteristic
131C23×D59—21no—
132C2×Q8×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59C2×Q812yes—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236C2×D412yes—
156C2×D236C2×Q812yes—
157C2×D236#4112yes—
158C2×C4×D59—24no—
159C2×C4×D59—42no—
160C2×C4×D59—42no—
161C2×C4×D59—42no—
162C2×C4×D59—42no—
163C2×C4×D59—42no—
164C2×C4×D59—42no—
165C2×C4×D59—81no—
166C2×C4×D59—81no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59—42no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59C2×D412yes—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—42no—
187#11745671—28no—
188(C2×D59)⋊C4—24no—
189(C2×D59)⋊C4—42no—
190(C2×D59)⋊C4—42no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—42no—
197C4⋊Dic59—81no—
198C4⋊Dic59#4112yes—
199Dic59⋊C4—42no—
200Dic59⋊C4—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⋊C8C2×D412yes—
212C2×C59⋊C8C2×Q812yes—
213C23×C59C22×D411yescharacteristic
214C22×D59—22no—
215C22×D59—22no—
216C22×D59—22no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221Q8×C59—22no—
222Q8×C59—22no—
223Q8×C59—41no—
224Q8×C59—41no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
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—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
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#13512yes—
254C2×C4×C59AGL(1,ℤ/8)12yes—
255C2×C4×C59C22×D411yescharacteristic
256C2×C4×C59C22×Q811yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C2×C4×C592−1+411yescharacteristic
261C2×Dic59—22no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270D236—22no—
271D236—22no—
272D236—22no—
273D236C22⋊Q814yes—
274C4×D59—48no—
275C4×D59—84no—
276C4×D59—84no—
277C59⋊C8—22no—
278C59⋊C8—22no—
279C59⋊C8—41no—
280C59⋊C8—41no—
281#26374—591nomaximal, Sylow
282C22×C59#52511yescharacteristic
283C22×C59D4211yescharacteristic
284C22×C59D4×Q811yescharacteristic
285C22×C59C2×#13511yescharacteristic
286C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C59#57711yescharacteristic
288C22×C59#57811yescharacteristic
289C2×D59—24no—
290C2×D59—42no—
291C2×D59—42no—
292C2×D59—42no—
293C2×D59—42no—
294C2×D59—42no—
295C2×D59—42no—
296C4×C59—22no—
297C4×C59—22no—
298C4×C59—41no—
299C4×C59—42no—
300C4×C59—41no—
301C4×C59—41no—
302C4×C59—41no—
303C4×C59—42no—
304C4×C59—81no—
305C4×C59—81no—
306C4×C59C2×C22⋊Q811yescharacteristic
307C4×C59#53911yescharacteristic
308C4×C59#54711yescharacteristic
309C4×C59#55411yescharacteristic
310Dic59—42no—
311Dic59—81no—
312Dic59—81no—
313C2×#478—591no—
314C2×C4×SD16—591no—
315#2453D5911yescharacteristic, complemented
316D4×C4⋊C4—591no—
317#2095—591no—
318#2084—591no—
319#2068—591no—
320#2059—591no—
321#1942—591no—
322#1916—591no—
323#1877—591no—
324#1877—591no—
325#1865—591no—
326#1862—591no—
327#1862—591no—
328C2×C59#244911yescharacteristic
329C2×C59#310511yescharacteristic
330C2×C59#312011yescharacteristic
331C2×C59#326511yescharacteristic
332C2×C59#328311yescharacteristic
333C2×C59#338111yescharacteristic
334C2×C59#338611yescharacteristic
335D59—44no—
336C22×SD16—591no—
337C2×#125C2×D5911yescharacteristic
338C2×C4×Q8C2×D5911yescharacteristic
339C2×C4×D4—591no—
340C2×C4×D4—591no—
341C2×C4×D4—591no—
342C22×C4⋊C4—1181no—
343#478—1184no—
344C4×SD16—1184no—
345C2×#107—591no—
346C2×#107—591no—
347C2×#107—591no—
348C2×#106—591no—
349C2×#106—591no—
350C2×C4⋊C8—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×C4×C8—591no—
358#400—21no—
359#398—21no—
360#391C2×D5911yescharacteristic
361#391C2×D5911yescharacteristic
362#389—21no—
363#389—591no—
364#389—591no—
365#389C2×D5911yescharacteristic
366#389C2×D5911yescharacteristic
367#387—21no—
368#385—1181no—
369#385—1181no—
370#384—591no—
371#380—1181no—
372C4×C4⋊C4C2×D5911yescharacteristic
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#336—591no—
378#336—591no—
379#336—591no—
380#336—591no—
381C59#2637411yescharacteristic, complemented, Sylow
382C22×Q8C22×D5911yescharacteristic
383C22×D4—591no—
384C23×C4—1181no—
385C23×C4—1181no—
386C23×C4—1181no—
387C2×SD16—594no—
388C2×SD16—1182no—
389C2×SD16—1182no—
390C22×C8—591no—
391C22×C8—591no—
392C22×C8—591no—
393C22×C8—591no—
394#125—22no—
395#125—41no—
396C4×Q8—22no—
397C4×Q8—41no—
398C4×D4—1182no—
399C4×D4—1182no—
400C4×D4—1182no—
401C4×D4—1182no—
402C4×D4—2361no—
403C4×D4—2361no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—2364no—
417C2×C4⋊C4C22⋊D5911yescharacteristic
418C2×C4⋊C4C22⋊D5911yescharacteristic
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C4⋊C4C22×D5911yescharacteristic
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4—1181no—
425C2×C42—21no—
426C2×C42C22×D5911yescharacteristic
427C2×C42C22×D5911yescharacteristic
428C2×C42C22×D5911yescharacteristic
429#107—1182no—
430#107—1182no—
431#107—2361no—
432#106—1182no—
433#106—1182no—
434C4⋊C8—1182no—
435#103—1182no—
436#103—1182no—
437#103—2361no—
438#102—594no—
439#102—1182no—
440#102—1182no—
441C4×C8—1182no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1181no—
449#95—1181no—
450#95C22⋊D5911yescharacteristic
451#95C22⋊D5911yescharacteristic
452C24—1181no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8Q8×D5912yes—
459C2×D4—592no—
460C2×D4—592no—
461C2×D4—592no—
462C2×D4—1181no—
463C2×D4—1181no—
464C2×D4—1181no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—1184no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—4721no—
474C22×C4—4721no—
475C22×C4—2362no—
476C22×C4—2362no—
477C22×C4—2362no—
478C22×C4—2362no—
479C22×C4—2362no—
480C22×C4—2362no—
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C23×D5911yescharacteristic
488SD16—1188no—
489C2×C8—592no—
490C2×C8—592no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—592no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—4721no—
520C4⋊C4—2362no—
521C4⋊C4—2362no—
522C4⋊C4—2362no—
523C4⋊C4D4×D5912yes—
524C22⋊C4—1184no—
525C22⋊C4—2362no—
526C22⋊C4—2362no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—42no—
534C23—1182no—
535C23—1182no—
536C23—1182no—
537C23—2361no—
538C23—2361no—
539C23—2361no—
540C23—2361no—
541C23C22×C22⋊D5911yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546D4—594no—
547D4—1182no—
548D4—1182no—
549D4—1182no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—1182no—
579C2×C4—4724no—
580C2×C4—4724no—
581C2×C4—2368no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4#1427374712yes—
591C2×C4#1427375212yes—
592C2×C4C2×D4×D5911yescharacteristic
593C2×C4C2×Q8×D5911yescharacteristic
594C2×C4#41×D5911yescharacteristic
595C2×C4#41×D5911yescharacteristic
596C2×C4#1427377211yescharacteristic
597C2×C4#1427377511yescharacteristic
598C8—1182no—
599C8—1182no—
600C8—2361no—
601C8—2361no—
602C22—1184no—
603C22—2362no—
604C22—2362no—
605C22—2362no—
606C22—2362no—
607C22—2362no—
608C22—2362no—
609C22#1683790911yescharacteristic
610C22D4×C22⋊D5911yescharacteristic
611C22Q8×C22⋊D5911yescharacteristic
612C22C2×#1427374711yescharacteristic
613C22C2×#1427375211yescharacteristic
614C22#1683821011yescharacteristic
615C22#1683821711yescharacteristic
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—4721no—
627C4—4721no—
628C4—2362no—
629C4C22⋊Q8×D5911yescharacteristic
630C4#1683799811yescharacteristic
631C4#1683805411yescharacteristic
632C4#1683810611yescharacteristic
633C2—2364no—
634C2#2142790211yescharacteristic
635C2#2143413911yescharacteristic
636C2#2143430411yescharacteristic
637C2#2143559211yescharacteristic
638C2#2143581411yescharacteristic
639C2#2143696711yescharacteristic
640C2#2143701311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2453C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2453D59
hypocenterC59#26374
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2453

Complements

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

NH ≅ G/Ntype
4. #21425634C2semidirect
5. #21425572C2semidirect
6. #21425491C2semidirect
7. #21425431C2semidirect
8. #21424826C2semidirect
9. #21424655C2semidirect
12. #21424364C2semidirect
15. C59×#2453C2semidirect
315. #2453D59semidirect
381. C59#26374semidirect

Maximal subgroups

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

subgroupindex
C2×#168376992
C2×C4×#117456712
C4⋊C4×D2362
#214256342
#214255722
#214254912
#214254312
#214248262
#214246552
#214244332
#214243642
#214243422
C59×#24532
#2637459