Finite Group Databasebeta

← #28656731

Subgroups — #28656731

Order 15104.

832 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427110C211yesmaximal, characteristic
2#21425411C211yesmaximal, characteristic, complemented
3#21425278C211yesmaximal, characteristic
4#21425239C211yesmaximal, characteristic, complemented
5#21425097C211yesmaximal, characteristic
6#21425054C211yesmaximal, characteristic, complemented
7#21424775C211yesmaximal, characteristic, complemented
8#21424734C211yesmaximal, characteristic, complemented
9#21424704C211yesmaximal, characteristic, complemented
10#21424667C211yesmaximal, characteristic, complemented
11#21424433C211yesmaximal, characteristic, complemented
12#21424404C211yesmaximal, characteristic, complemented
13#21424184C211yesmaximal, characteristic, complemented
14C4×#14273647C211yesmaximal, characteristic, complemented
15C59×#2489C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×#14273701C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×(C22⋊D59)⋊C4—21no—
20C22×C4⋊Dic59C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837294—21no—
35#16837279—21no—
36#16837276—21no—
37C4×C4⋊Dic59C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#125×C59C2211yescharacteristic
49C2×C22⋊Q8×C59C2211yescharacteristic
50C59×#400—21no—
51C59×#397—21no—
52C59×#391C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#388—21no—
55C59×#382—21no—
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C22⋊D59—21no—
61C23×Dic59—21no—
62C22×C4×D59—21no—
63C2×#11745671—22no—
64C2×#11745671—41no—
65C2×#11745671—41no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273701—22no—
71#14273701—22no—
72C4×D236—22no—
73C4×D236—41no—
74(C22⋊D59)⋊C4—44no—
75C2×C4⋊Dic59—22no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—41no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×C4×Dic59—21no—
87#14273660—22no—
88C4⋊(C59⋊C8)—41no—
89#14273651—22no—
90#14273651—22no—
91#14273651—41no—
92#14273647—22no—
93#14273647—22no—
94#14273647—41no—
95C22⋊(C59⋊C8)—41no—
96C22⋊(C59⋊C8)—41no—
97C4×C59⋊C8—41no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C22×Q8×C59C2311yescharacteristic
101C23×C4×C59C2311yescharacteristic
102#125×C59—22no—
103#125×C59—41no—
104C22⋊Q8×C59—22no—
105C22⋊Q8×C59—41no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C22⋊C4×C59D411yescharacteristic
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125#95×C59D411yescharacteristic
126#95×C59D411yescharacteristic
127C24×C59C2×D411yescharacteristic
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—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
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×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2411yescharacteristic
151C2×C22⋊D59—24no—
152C2×C22⋊D59—42no—
153C2×C22⋊D59—42no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—42no—
158C22×Dic59—42no—
159C22×Dic59—81no—
160C2×D236—21no—
161C2×D236—21no—
162C2×D236—21no—
163C2×D236—21no—
164C2×D236—21no—
165C2×D236#4112yes—
166C2×C4×D59—42no—
167C2×C4×D59—42no—
168C2×C4×D59—81no—
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#4112yes—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—42no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—42no—
187#11745671—44no—
188(C2×D59)⋊C4—24no—
189(C2×D59)⋊C4—42no—
190(C2×D59)⋊C4—42no—
191(C2×D59)⋊C4—81no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—42no—
197C4⋊Dic59#4112yes—
198C4⋊Dic59#4112yes—
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—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59#13512yes—
220C23×C59C22×D411yescharacteristic
221C22×D59—22no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225C22×D59—41no—
226C22×D59—41no—
227C22×D59—41no—
228Q8×C59—41no—
229Q8×C59—41no—
230Q8×C59—41no—
231Q8×C59—41no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—22no—
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—21no—
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—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59AGL(1,ℤ/8)12yes—
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59H(5,𝔽2)11yescharacteristic
268C2×C4×C59H(5,𝔽2)11yescharacteristic
269C22⋊D59—48no—
270C2×Dic59—22no—
271C2×Dic59—22no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—84no—
280D236—22no—
281D236—22no—
282D236—41no—
283D236—41no—
284C4×D59—84no—
285C59⋊C8—41no—
286C59⋊C8—41no—
287C59⋊C8—41no—
288C59⋊C8—41no—
289#26721—591nomaximal, Sylow
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59#52511yescharacteristic
298C22×C59#54611yescharacteristic
299C22×C59#54611yescharacteristic
300C22×C59C2×#13511yescharacteristic
301C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C59#57711yescharacteristic
303C22×C59#57811yescharacteristic
304C2×D59—42no—
305C2×D59—42no—
306C2×D59—42no—
307C2×D59—81no—
308C2×D59—81no—
309C2×D59—81no—
310C2×D59—81no—
311C4×C59—22no—
312C4×C59—41no—
313C4×C59—42no—
314C4×C59—41no—
315C4×C59—41no—
316C4×C59—41no—
317C4×C59—81no—
318C4×C59—81no—
319C4×C59—81no—
320C4×C59#52911yescharacteristic
321C4×C59#53811yescharacteristic
322C4×C59#55511yescharacteristic
323C4×C59#55911yescharacteristic
324Dic59—42no—
325Dic59—42no—
326Dic59—81no—
327#2489D5911yescharacteristic, complemented
328#2361—591no—
329#2055—591no—
330#2033—591no—
331#2025—591no—
332#2000—591no—
333#1989—591no—
334#1933—591no—
335#1928—591no—
336#1923—591no—
337#1918—591no—
338#1877—591no—
339#1872—591no—
340#1824—591no—
341C4×#102—591no—
342C2×C59—42no—
343C2×C59#245911yescharacteristic
344C2×C59#310511yescharacteristic
345C2×C59#312011yescharacteristic
346C2×C59#330011yescharacteristic
347C2×C59#330111yescharacteristic
348C2×C59#331011yescharacteristic
349C2×C59#333011yescharacteristic
350D59—82no—
351C22×SD16—591no—
352C2×#125C2×D5911yescharacteristic
353C2×C22⋊Q8C2×D5911yescharacteristic
354C2×#121—591no—
355C2×C4×D4—591no—
356C2×C4×D4—1181no—
357C22×C4⋊C4—591no—
358C2×#107—591no—
359C2×C4⋊C8—591no—
360C2×#103—591no—
361C2×#103—591no—
362C2×#103—591no—
363C2×#102—591no—
364C2×#102—591no—
365C2×#102—591no—
366C2×C22⋊C8—591no—
367C2×C22⋊C8—591no—
368C2×C4×C8—591no—
369#400—21no—
370#397—21no—
371#391C2×D5911yescharacteristic
372#391C2×D5911yescharacteristic
373#390—591no—
374#389—591no—
375#388—21no—
376#388—1181no—
377#386—1181no—
378#385—1181no—
379#382—21no—
380#380C2×D5911yescharacteristic
381C4×C4⋊C4—591no—
382C4×C4⋊C4C2×D5911yescharacteristic
383C4×C22⋊C4C2×D5911yescharacteristic
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#340—591no—
390#340—591no—
391#340—591no—
392#336—591no—
393#336—591no—
394C59#2672111yescharacteristic, complemented, Sylow
395C22×Q8C22×D5911yescharacteristic
396C22×D4—591no—
397C22×D4—1181no—
398C23×C4—1181no—
399C23×C4—1181no—
400C23×C4C22×D5911yescharacteristic
401C2×SD16—1182no—
402C2×SD16—2361no—
403C2×SD16—2361no—
404C22×C8—591no—
405C22×C8—591no—
406C22×C8—591no—
407C22×C8—591no—
408#125—22no—
409#125—41no—
410C22⋊Q8—22no—
411C22⋊Q8—41no—
412#121—1182no—
413#121—1182no—
414C4×D4—1182no—
415C4×D4—2361no—
416C4×D4—2364no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—1182no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—2361no—
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⋊C4—1181no—
435C2×C22⋊C4—1181no—
436C2×C22⋊C4C22⋊D5911yescharacteristic
437C2×C22⋊C4C22⋊D5911yescharacteristic
438C2×C42—21no—
439C2×C42—1181no—
440C2×C42C22×D5911yescharacteristic
441C2×C42C22×D5911yescharacteristic
442#107—1182no—
443C4⋊C8—2361no—
444#103—1182no—
445#103—1182no—
446#103—2361no—
447#102—1182no—
448#102—1182no—
449#102—2361no—
450C22⋊C8—2361no—
451C22⋊C8—2361no—
452C4×C8—2361no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—1181no—
460#95—1181no—
461#95C22⋊D5911yescharacteristic
462#95C22⋊D5911yescharacteristic
463C24—1181no—
464C24C2×C22⋊D5911yescharacteristic
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×D4—592no—
472C2×D4—1184no—
473C2×D4—1181no—
474C2×D4—1181no—
475C2×D4—1181no—
476C2×D4—1181no—
477C2×D4—1181no—
478C2×D4—2362no—
479C2×D4—2362no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—1181no—
491C22×C4—1181no—
492C22×C4—1181no—
493C22×C4—4721no—
494C22×C4—4721no—
495C22×C4—2362no—
496C22×C4—2362no—
497C22×C4—2362no—
498C22×C4—2362no—
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C2×C22⋊D5911yescharacteristic
503C22×C4C2×C22⋊D5911yescharacteristic
504C22×C4C23×D5911yescharacteristic
505SD16—2364no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—592no—
527C4⋊C4—592no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4#1174568712yes—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—1184no—
540C22⋊C4—4721no—
541C22⋊C4—2362no—
542C22⋊C4—2362no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—42no—
548C42—2362no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—1182no—
556C23—2361no—
557C23—2361no—
558C23—2361no—
559C23—2361no—
560C23—2361no—
561C23—2361no—
562C23#1427374712yes—
563C23C22×C22⋊D5911yescharacteristic
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—41no—
568D4—1182no—
569D4—1182no—
570D4—2368no—
571D4—2361no—
572D4—2361no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—1182no—
604C2×C4—1182no—
605C2×C4—4724no—
606C2×C4—4724no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4#1427375212yes—
615C2×C4#41×D5911yescharacteristic
616C2×C4#41×D5911yescharacteristic
617C2×C4#41×D5911yescharacteristic
618C2×C4C2×#1174568711yescharacteristic
619C2×C4#1427377211yescharacteristic
620C2×C4#1427377311yescharacteristic
621C8—2361no—
622C8—2361no—
623C8—2361no—
624C8—2361no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—4721no—
633C22—4721no—
634C22—4721no—
635C22—4721no—
636C22—2362no—
637C22—2362no—
638C22—2362no—
639C22#1683790911yescharacteristic
640C22#1683804011yescharacteristic
641C22#1683804611yescharacteristic
642C22C2×#1427374711yescharacteristic
643C22C2×#1427375211yescharacteristic
644C22#1683821011yescharacteristic
645C22#1683821711yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—42no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—4721no—
656C4—2362no—
657C4—2362no—
658C4#1683793111yescharacteristic
659C4#1683798611yescharacteristic
660C4#1683811411yescharacteristic
661C4#1683813411yescharacteristic
662C2—42no—
663C2—4722no—
664C2#2142802911yescharacteristic
665C2#2143413911yescharacteristic
666C2#2143430411yescharacteristic
667C2#2143597711yescharacteristic
668C2#2143599811yescharacteristic
669C2#2143611111yescharacteristic
670C2#2143640111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2489C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenter#2489D59
hypocenterC59#26721
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2489

Complements

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

NH ≅ G/Ntype
2. #21425411C2semidirect
4. #21425239C2semidirect
6. #21425054C2semidirect
7. #21424775C2semidirect
8. #21424734C2semidirect
9. #21424704C2semidirect
10. #21424667C2semidirect
11. #21424433C2semidirect
12. #21424404C2semidirect
13. #21424184C2semidirect
14. C4×#14273647C2semidirect
15. C59×#2489C2semidirect
327. #2489D59semidirect
394. C59#26721semidirect

Maximal subgroups

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

subgroupindex
#214271102
#214254112
#214252782
#214252392
#214250972
#214250542
#214247752
#214247342
#214247042
#214246672
#214244332
#214244042
#214241842
C4×#142736472
C59×#24892
#2672159