Finite Group Databasebeta

← #28646916

Subgroups — #28646916

Order 15104.

736 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427043C211yesmaximal, characteristic
2#21425467C211yesmaximal, characteristic
3#21425467C211yesmaximal, characteristic
4#21425239C211yesmaximal, characteristic
5#21425229C211yesmaximal, characteristic
6#21425054C211yesmaximal, characteristic, complemented
7#21425049C211yesmaximal, characteristic, complemented
8#21424859C211yesmaximal, characteristic, complemented
9#21424820C211yesmaximal, characteristic, complemented
10#21424678C211yesmaximal, characteristic, complemented
11#21424671C211yesmaximal, characteristic, complemented
12#21424229C211yesmaximal, characteristic, complemented
13#21424103C211yesmaximal, characteristic
14#21423924C211yesmaximal, characteristic
15C59×#2550C211yesmaximal, characteristic, complemented
16C2×C4×D236C2211yescharacteristic
17C2×#14273685—21no—
18C2×#14273658C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273647C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28#16837304C2211yescharacteristic
29#16837304C2211yescharacteristic
30#16837294—21no—
31#16837294—21no—
32#16837272C2211yescharacteristic
33#16837259—21no—
34C4×C4⋊Dic59C2211yescharacteristic
35C4×C4⋊Dic59C2211yescharacteristic
36#16837164C2211yescharacteristic
37#16837164C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4⋊Q8×C59C2211yescharacteristic
49C2×#125×C59C2211yescharacteristic
50C59×#398—21no—
51C59×#395—21no—
52C59×#391—21no—
53C59×#389—21no—
54C59×#389C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C4×D236—22no—
66C4×D236Q814yescomplemented
67#14273685—44no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×(C2×D59)⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273658—22no—
81#14273658—22no—
82C4⋊(C59⋊C8)—22no—
83C4⋊(C59⋊C8)—41no—
84C4⋊(C59⋊C8)—41no—
85#14273647—22no—
86#14273647—22no—
87#14273647—41no—
88#14273647—41no—
89#14273638—22no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C22×Q8×C59—21no—
93C4⋊Q8×C59—22no—
94C4⋊Q8×C59—41no—
95#125×C59—22no—
96#125×C59—41no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59D411yescharacteristic
103C2×C4⋊C4×C59D411yescharacteristic
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C42×C59—21no—
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119C23×D59—21no—
120C2×Q8×C59—42no—
121C2×Q8×C59—42no—
122C2×Q8×C59—81no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236C2×Q812yes—
143C2×C4×D59—24no—
144C2×C4×D59—42no—
145C2×C4×D59—42no—
146C4⋊C4×C59—22no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—41no—
154C4⋊C4×C59—42no—
155C4⋊C4×C59—42no—
156C4⋊C4×C59—42no—
157C4⋊C4×C59—42no—
158C4⋊C4×C59—81no—
159C4⋊C4×C59#4112yes—
160C4⋊C4×C59#4112yes—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—21no—
165C42×C59—81no—
166C42×C59C2×Q812yes—
167C42×C59#4112yes—
168(C2×D59)⋊C4—24no—
169(C2×D59)⋊C4—42no—
170(C2×D59)⋊C4—42no—
171(C2×D59)⋊C4—81no—
172(C2×D59)⋊C4—81no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—42no—
179C4⋊Dic59—81no—
180C4⋊Dic59C2×Q812yes—
181C4×Dic59—42no—
182C4×Dic59—42no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C23×C59C22×D411yescharacteristic
196C22×D59—22no—
197C22×D59—41no—
198C22×D59—41no—
199C22×D59—41no—
200C22×D59—41no—
201C22×D59—41no—
202C22×D59—41no—
203Q8×C59—84no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59C22⋊Q812yes—
232C2×C4×C59C22⋊Q812yes—
233C2×C4×C59AGL(1,ℤ/8)12yes—
234C2×C4×C59C22×Q811yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—22no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249D236—22no—
250D236—22no—
251D236—41no—
252D236—41no—
253C4×D59—48no—
254C59⋊C8—41no—
255C59⋊C8—41no—
256C59⋊C8—41no—
257C59⋊C8—41no—
258#26032—591nomaximal, Sylow
259C22×C59C2×C22⋊Q811yescharacteristic
260C22×C59#54611yescharacteristic
261C22×C59#54811yescharacteristic
262C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C59#57511yescharacteristic
264C22×C59#57711yescharacteristic
265C22×C59#57711yescharacteristic
266C2×D59—42no—
267C2×D59—42no—
268C2×D59—42no—
269C2×D59—81no—
270C2×D59—81no—
271C2×D59—81no—
272C2×D59—81no—
273C4×C59—22no—
274C4×C59—22no—
275C4×C59—22no—
276C4×C59—42no—
277C4×C59—41no—
278C4×C59—42no—
279C4×C59—41no—
280C4×C59—41no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59#53311yescharacteristic
284C4×C59#54411yescharacteristic
285C4×C59#55511yescharacteristic
286C4×C59#55611yescharacteristic
287Dic59—42no—
288Dic59—81no—
289Dic59—81no—
290#2550D5911yescharacteristic, complemented
291#2354—591no—
292#2065—591no—
293#2065—591no—
294#2025—591no—
295#2023—591no—
296#1989—591no—
297#1988—591no—
298#1948—591no—
299#1940—591no—
300#1920—591no—
301#1919—591no—
302#1836—591no—
303#1804—591no—
304#1762—591no—
305C2×C59#243911yescharacteristic
306C2×C59#307711yescharacteristic
307C2×C59#308411yescharacteristic
308C2×C59#330611yescharacteristic
309C2×C59#332111yescharacteristic
310C2×C59#336811yescharacteristic
311C2×C59#337611yescharacteristic
312D59—82no—
313C2×C4⋊Q8C2×D5911yescharacteristic
314C2×#125C2×D5911yescharacteristic
315C2×C4×D4—591no—
316C2×#117—1181no—
317C2×#106—591no—
318C2×#106—591no—
319C2×C4⋊C8—591no—
320C2×C4⋊C8—591no—
321C2×C4⋊C8—591no—
322C2×#102—591no—
323C2×#102—591no—
324C2×#102—591no—
325C2×#102—591no—
326C2×#97—591no—
327#398—21no—
328#395—21no—
329#391—21no—
330#390—591no—
331#390—591no—
332#389—21no—
333#389C2×D5911yescharacteristic
334#389C2×D5911yescharacteristic
335#388—1181no—
336#388—1181no—
337#384—591no—
338#382C2×D5911yescharacteristic
339#380—1181no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4—591no—
342C4×C4⋊C4C2×D5911yescharacteristic
343C4×C4⋊C4C2×D5911yescharacteristic
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#336—591no—
355#336—591no—
356C59#2603211yescharacteristic, complemented, Sylow
357C22×Q8—21no—
358C22×D4—591no—
359C23×C4—1181no—
360C22×C8—591no—
361C22×C8—591no—
362C22×C8—591no—
363C22×C8—591no—
364C4⋊Q8—22no—
365C4⋊Q8—41no—
366#125—22no—
367#125—41no—
368C4×D4—594no—
369C4×D4—1182no—
370#117—2364no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4C22⋊D5911yescharacteristic
382C2×C4⋊C4C22⋊D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C22⋊C4—1181no—
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C42—21no—
393C2×C42—1181no—
394C2×C42—1181no—
395C2×C42C22×D5911yescharacteristic
396C2×C42C22×D5911yescharacteristic
397C2×C42C22×D5911yescharacteristic
398#106—1182no—
399#106—1182no—
400C4⋊C8—1182no—
401C4⋊C8—2361no—
402C4⋊C8—2361no—
403#102—1182no—
404#102—1182no—
405#102—2361no—
406#102—2361no—
407#97—1182no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—1181no—
414#95—1181no—
415#95C22⋊D5911yescharacteristic
416#95C22⋊D5911yescharacteristic
417C24—1181no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—81no—
421C2×D4—592no—
422C2×D4—1181no—
423C2×D4—1181no—
424C2×D4—1181no—
425C2×D4—1181no—
426C2×D4—1181no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—1181no—
432C22×C4—1184no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—2362no—
436C22×C4—2362no—
437C22×C4C2×C22⋊D5911yescharacteristic
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C2×C22⋊D5911yescharacteristic
440C22×C4C2×C22⋊D5911yescharacteristic
441C22×C4C2×C22⋊D5911yescharacteristic
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C23×D5911yescharacteristic
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—592no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—4721no—
476C4⋊C4—2362no—
477C4⋊C4#1174568712yes—
478C4⋊C4#1174568912yes—
479C22⋊C4—1184no—
480C22⋊C4—4721no—
481C22⋊C4—4721no—
482C22⋊C4—2362no—
483C22⋊C4—2362no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—81no—
489C42—2362no—
490C42—2362no—
491C42Q8×D5912yes—
492C42#1174568912yes—
493C23—1182no—
494C23—2361no—
495C23—2361no—
496C23—2361no—
497C23—2361no—
498C23—2361no—
499C23—2361no—
500C23C22×C22⋊D5911yescharacteristic
501Q8—84no—
502D4—1182no—
503D4—1182no—
504D4—2361no—
505D4—2361no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—1182no—
534C2×C4—2368no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4#1427370312yes—
544C2×C4#1427370312yes—
545C2×C4#1427375212yes—
546C2×C4C2×Q8×D5911yescharacteristic
547C2×C4C2×#1174568711yescharacteristic
548C2×C4C2×#1174568911yescharacteristic
549C2×C4C2×#1174568911yescharacteristic
550C2×C4#1427377211yescharacteristic
551C2×C4#1427377511yescharacteristic
552C8—2361no—
553C8—2361no—
554C8—2361no—
555C8—2361no—
556C22—4721no—
557C22—4721no—
558C22—4721no—
559C22—4721no—
560C22—2362no—
561C22—2362no—
562C22—2362no—
563C22C2×#1427370311yescharacteristic
564C22#1683804611yescharacteristic
565C22#1683806311yescharacteristic
566C22C2×#1427375211yescharacteristic
567C22#1683820011yescharacteristic
568C22#1683821011yescharacteristic
569C22#1683821311yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—4721no—
581C4—4721no—
582C4—2362no—
583C4#1683795711yescharacteristic
584C4#1683802911yescharacteristic
585C4#1683811811yescharacteristic
586C4#1683812611yescharacteristic
587C2—4722no—
588C2#2142778311yescharacteristic
589C2#2143388911yescharacteristic
590C2#2143395411yescharacteristic
591C2#2143605711yescharacteristic
592C2#2143628211yescharacteristic
593C2#2143682511yescharacteristic
594C2#2143692311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2550C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2550D59
hypocenterC59#26032
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2550

Complements

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

NH ≅ G/Ntype
6. #21425054C2semidirect
7. #21425049C2semidirect
8. #21424859C2semidirect
9. #21424820C2semidirect
10. #21424678C2semidirect
11. #21424671C2semidirect
12. #21424229C2semidirect
15. C59×#2550C2semidirect
66. C4×D236Q8semidirect
290. #2550D59semidirect
356. C59#26032semidirect

Maximal subgroups

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

subgroupindex
#214270432
#214254672
#214252392
#214252292
#214250542
#214250492
#214248592
#214248202
#214246782
#214246712
#214242292
#214241032
#214239242
C59×#25502
#2603259