Finite Group Databasebeta

← #28657361

Subgroups — #28657361

Order 15104.

712 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427165C211yesmaximal, characteristic
2#21425500C211yesmaximal, characteristic
3#21425481C211yesmaximal, characteristic
4#21425337C211yesmaximal, characteristic
5#21425337C211yesmaximal, characteristic
6#21424771C211yesmaximal, characteristic, complemented
7#21424737C211yesmaximal, characteristic, complemented
8#21424671C211yesmaximal, characteristic, complemented
9#21424664C211yesmaximal, characteristic, complemented
10#21424655C211yesmaximal, characteristic, complemented
11#21424363C211yesmaximal, characteristic, complemented
12#21424356C211yesmaximal, characteristic
13#21424098C211yesmaximal, characteristic, complemented
14C4×#14273647C211yesmaximal, characteristic
15C59×#2507C211yesmaximal, characteristic, complemented
16C2×#14273715C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273650C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837308C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837292—21no—
34#16837291—21no—
35#16837273—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37#16837249—21no—
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C59×#398—21no—
49C59×#398—21no—
50C59×#389—21no—
51C59×#389C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#387—21no—
56C59×#384C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273715—22no—
67#14273715—22no—
68C4×Dic118—22no—
69C4×Dic118—41no—
70C4×D236—22no—
71C4×D236—41no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82#14273660—22no—
83#14273658—22no—
84C4⋊(C59⋊C8)—41no—
85#14273650—22no—
86#14273650—22no—
87#14273650—22no—
88#14273650—41no—
89#14273647—22no—
90#14273647—22no—
91#14273647—22no—
92#14273647—41no—
93C4×C59⋊C8—41no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
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×C59—21no—
103C2×C4⋊C4×C59D411yescharacteristic
104C2×C4⋊C4×C59D411yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C42×C59—21no—
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112C2×C42×C59C2311yescharacteristic
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
120C23×D59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236—21no—
140C2×D236#4112yes—
141C2×C4×D59—42no—
142C2×C4×D59—42no—
143C2×C4×D59—81no—
144C2×Dic118—21no—
145C2×Dic118—21no—
146C2×Dic118—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118#4112yes—
150C4⋊C4×C59—22no—
151C4⋊C4×C59—22no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—42no—
160C4⋊C4×C59—81no—
161C4⋊C4×C59—81no—
162C4⋊C4×C59—81no—
163C4⋊C4×C59—81no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59#4112yes—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—81no—
173(C2×D59)⋊C4—24no—
174(C2×D59)⋊C4—24no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—81no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59#4112yes—
182Dic59⋊C4—42no—
183C4×Dic59—42no—
184C4×Dic59—81no—
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—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C23×C59C22×D411yescharacteristic
198C22×D59—22no—
199C22×D59—41no—
200C22×D59—41no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59—41no—
204C22×D59—41no—
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—21no—
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—21no—
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×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59#13512yes—
234C2×C4×C59#13712yes—
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59H(5,𝔽2)11yescharacteristic
240C2×C4×C592−1+411yescharacteristic
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250D236—22no—
251D236—22no—
252D236—41no—
253D236—41no—
254C4×D59—84no—
255Dic118—22no—
256Dic118—22no—
257Dic118—41no—
258Dic118—41no—
259C59⋊C8—41no—
260C59⋊C8—41no—
261C59⋊C8—41no—
262C59⋊C8—41no—
263#26763—591nomaximal, Sylow
264C22×C59#52511yescharacteristic
265C22×C59#54611yescharacteristic
266C22×C59#54811yescharacteristic
267C22×C59C2×#13511yescharacteristic
268C22×C59C2×#13711yescharacteristic
269C22×C59#57611yescharacteristic
270C22×C59#57711yescharacteristic
271C2×D59—42no—
272C2×D59—42no—
273C2×D59—42no—
274C2×D59—81no—
275C2×D59—81no—
276C2×D59—81no—
277C2×D59—81no—
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—42no—
281C4×C59—41no—
282C4×C59—41no—
283C4×C59—41no—
284C4×C59—41no—
285C4×C59—81no—
286C4×C59—81no—
287C4×C59—81no—
288C4×C59#52911yescharacteristic
289C4×C59#53911yescharacteristic
290C4×C59#55211yescharacteristic
291C4×C59#55311yescharacteristic
292Dic59—42no—
293Dic59—42no—
294Dic59—81no—
295#2507D5911yescharacteristic, complemented
296#2367—591no—
297#2071—591no—
298#2067—591no—
299#2042—591no—
300#2042—591no—
301#1933—591no—
302#1928—591no—
303#1919—591no—
304#1918—591no—
305#1916—591no—
306#1865—591no—
307#1864—591no—
308#1803—591no—
309C4×#102—591no—
310C2×C59#246011yescharacteristic
311C2×C59#310711yescharacteristic
312C2×C59#311911yescharacteristic
313C2×C59#330111yescharacteristic
314C2×C59#331411yescharacteristic
315C2×C59#336111yescharacteristic
316C2×C59#336911yescharacteristic
317D59—82no—
318C2×#124—591no—
319C2×C4×Q8—591no—
320C2×C4×D4—591no—
321C2×#107—591no—
322C2×#106—591no—
323C2×C4⋊C8—591no—
324C2×#103—591no—
325C2×#103—591no—
326C2×#103—591no—
327C2×#103—591no—
328C2×#102—591no—
329C2×#102—591no—
330C2×#102—591no—
331C2×#102—591no—
332C2×C4×C8—591no—
333#398—21no—
334#398—21no—
335#391—591no—
336#390—591no—
337#389—21no—
338#389C2×D5911yescharacteristic
339#389C2×D5911yescharacteristic
340#389C2×D5911yescharacteristic
341#389C2×D5911yescharacteristic
342#388—1181no—
343#388—1181no—
344#387—21no—
345#385—1181no—
346#384C2×D5911yescharacteristic
347C4×C4⋊C4—591no—
348C4×C4⋊C4C2×D5911yescharacteristic
349C4×C4⋊C4C2×D5911yescharacteristic
350C4×C22⋊C4—1181no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#336—591no—
358#336—591no—
359#336—591no—
360#336—591no—
361C59#2676311yescharacteristic, complemented, Sylow
362C22×Q8—591no—
363C22×D4—591no—
364C23×C4—1181no—
365C22×C8—591no—
366C22×C8—591no—
367C22×C8—591no—
368C22×C8—591no—
369#124—1182no—
370#124—1182no—
371C4×Q8—1182no—
372C4×Q8—2361no—
373C4×D4—1182no—
374C4×D4—2361no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4C22⋊D5911yescharacteristic
385C2×C4⋊C4C22⋊D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C4⋊C4C22×D5911yescharacteristic
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C22⋊C4—1181no—
395C2×C42—21no—
396C2×C42—1181no—
397C2×C42—1181no—
398C2×C42C22×D5911yescharacteristic
399C2×C42C22×D5911yescharacteristic
400C2×C42C22×D5911yescharacteristic
401#107—1182no—
402#106—1182no—
403C4⋊C8—2361no—
404#103—1182no—
405#103—1182no—
406#103—1182no—
407#103—2361no—
408#102—1182no—
409#102—1182no—
410#102—1182no—
411#102—2361no—
412C4×C8—2361no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95C22⋊D5911yescharacteristic
422#95C22⋊D5911yescharacteristic
423C24—1181no—
424C2×Q8—592no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C2×Q8—1181no—
428C2×Q8—1181no—
429C2×Q8—1181no—
430C2×D4—592no—
431C2×D4—1181no—
432C2×D4—1181no—
433C2×D4—1181no—
434C2×D4—1181no—
435C2×D4—1181no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4—4721no—
444C22×C4—2362no—
445C22×C4—2362no—
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C2×C22⋊D5911yescharacteristic
449C22×C4C2×C22⋊D5911yescharacteristic
450C22×C4C2×C22⋊D5911yescharacteristic
451C22×C4C2×C22⋊D5911yescharacteristic
452C22×C4C23×D5911yescharacteristic
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—592no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—2362no—
485C4⋊C4#1174568912yes—
486C22⋊C4—1184no—
487C22⋊C4—1184no—
488C22⋊C4—4721no—
489C22⋊C4—2362no—
490C22⋊C4—2362no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—81no—
498C42—4721no—
499C42—2362no—
500C23—1182no—
501C23—2361no—
502C23—2361no—
503C23—2361no—
504C23—2361no—
505C23—2361no—
506C23—2361no—
507C23C22×C22⋊D5911yescharacteristic
508Q8—1182no—
509Q8—1182no—
510Q8—2361no—
511Q8—2361no—
512D4—1182no—
513D4—1182no—
514D4—2361no—
515D4—2361no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—1182no—
545C2×C4—1182no—
546C2×C4—4724no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4#1427374712yes—
555C2×C4#1427375812yes—
556C2×C4#41×D5911yescharacteristic
557C2×C4#41×D5911yescharacteristic
558C2×C4#41×D5911yescharacteristic
559C2×C4C2×#1174568911yescharacteristic
560C2×C4#1427377311yescharacteristic
561C2×C4#1427377511yescharacteristic
562C8—2361no—
563C8—2361no—
564C8—2361no—
565C8—2361no—
566C22—4721no—
567C22—4721no—
568C22—4721no—
569C22—4721no—
570C22—2362no—
571C22—2362no—
572C22—2362no—
573C22#1683790911yescharacteristic
574C22#1683804011yescharacteristic
575C22#1683806311yescharacteristic
576C22C2×#1427374711yescharacteristic
577C22C2×#1427375811yescharacteristic
578C22#1683820511yescharacteristic
579C22#1683821311yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—4721no—
591C4—2362no—
592C4—2362no—
593C4#1683793711yescharacteristic
594C4#1683800011yescharacteristic
595C4#1683808411yescharacteristic
596C4#1683810511yescharacteristic
597C2—4722no—
598C2#2142804411yescharacteristic
599C2#2143416111yescharacteristic
600C2#2143428911yescharacteristic
601C2#2143599211yescharacteristic
602C2#2143617111yescharacteristic
603C2#2143675611yescharacteristic
604C2#2143683611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2507C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2507D59
hypocenterC59#26763
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2507

Complements

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

NH ≅ G/Ntype
6. #21424771C2semidirect
7. #21424737C2semidirect
8. #21424671C2semidirect
9. #21424664C2semidirect
10. #21424655C2semidirect
11. #21424363C2semidirect
13. #21424098C2semidirect
15. C59×#2507C2semidirect
295. #2507D59semidirect
361. C59#26763semidirect

Maximal subgroups

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

subgroupindex
#214271652
#214255002
#214254812
#214253372
#214247712
#214247372
#214246712
#214246642
#214246552
#214243632
#214243562
#214240982
C4×#142736472
C59×#25072
#2676359