Finite Group Databasebeta

← #28661507

Subgroups — #28661507

Order 15104.

742 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427132C211yesmaximal, characteristic
2#21425500C211yesmaximal, characteristic, complemented
3#21425481C211yesmaximal, characteristic, complemented
4#21425278C211yesmaximal, characteristic
5#21425278C211yesmaximal, characteristic
6#21424852C211yesmaximal, characteristic, complemented
7#21424826C211yesmaximal, characteristic, complemented
8#21424704C211yesmaximal, characteristic, complemented
9#21424688C211yesmaximal, characteristic, complemented
10#21424671C211yesmaximal, characteristic, complemented
11#21424356C211yesmaximal, characteristic, complemented
12#21424293C211yesmaximal, characteristic, complemented
13#21424188C211yesmaximal, characteristic, complemented
14C4×#14273647C211yesmaximal, characteristic, complemented
15C59×#2497C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837302—21no—
33#16837300C2211yescharacteristic
34#16837291—21no—
35#16837278—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37#16837249—21no—
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C59×#400—21no—
49C59×#396—21no—
50C59×#389C2211yescharacteristic
51C59×#389C2211yescharacteristic
52C59×#388—21no—
53C59×#387—21no—
54C59×#384C2211yescharacteristic
55C59×#380C2211yescharacteristic
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C22⋊D59—21no—
61C22×C4×D59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273701—22no—
67#14273701—22no—
68C4×D236—22no—
69C4×D236—41no—
70#14273686—22no—
71#14273686—41no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×C22⋊Dic59—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83#14273660—22no—
84#14273660—22no—
85#14273660—41no—
86#14273658—22no—
87C4⋊(C59⋊C8)—22no—
88#14273647—22no—
89#14273647—22no—
90#14273647—22no—
91#14273647—41no—
92C22⋊(C59⋊C8)—22no—
93C22⋊(C59⋊C8)—41no—
94C4×C59⋊C8—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C23×C4×C59C2311yescharacteristic
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59D411yescharacteristic
109C2×C22⋊C4×C59D411yescharacteristic
110C2×C42×C59—21no—
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
120C24×C59C2×D411yescharacteristic
121C23×D59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C2×C22⋊D59—24no—
139C2×C22⋊D59—81no—
140C2×C22⋊D59—81no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236#4112yes—
150C2×C4×D59—42no—
151C2×C4×D59—42no—
152C2×C4×D59—81no—
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—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—42no—
162C4⋊C4×C59—81no—
163C4⋊C4×C59—81no—
164C4⋊C4×C59—81no—
165C22⋊C4×C59—22no—
166C22⋊C4×C59—22no—
167C22⋊C4×C59—81no—
168C22⋊C4×C59—81no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—81no—
174C22⋊Dic59—42no—
175(C2×D59)⋊C4—24no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—81no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59#4112yes—
184C4⋊Dic59#4112yes—
185Dic59⋊C4—81no—
186C4×Dic59—42no—
187C4×Dic59—81no—
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—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8#4112yes—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59C22×D411yescharacteristic
208C22×D59—22no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—41no—
214C22×D59—41no—
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—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59#13512yes—
245C2×C4×C59#13712yes—
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59H(5,𝔽2)11yescharacteristic
251C2×C4×C592−1+411yescharacteristic
252C22⋊D59—84no—
253C2×Dic59—22no—
254C2×Dic59—22no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262D236—22no—
263D236—22no—
264D236—41no—
265D236—41no—
266C4×D59—84no—
267C59⋊C8—22no—
268C59⋊C8—41no—
269C59⋊C8—41no—
270C59⋊C8—41no—
271#27041—591nomaximal, Sylow
272C22×C59—41no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59#52511yescharacteristic
280C22×C59#54611yescharacteristic
281C22×C59#54711yescharacteristic
282C22×C59C2×#13511yescharacteristic
283C22×C59C2×#13711yescharacteristic
284C22×C59#57611yescharacteristic
285C22×C59#57611yescharacteristic
286C2×D59—42no—
287C2×D59—42no—
288C2×D59—42no—
289C2×D59—81no—
290C2×D59—81no—
291C2×D59—81no—
292C2×D59—81no—
293C4×C59—22no—
294C4×C59—22no—
295C4×C59—41no—
296C4×C59—41no—
297C4×C59—41no—
298C4×C59—81no—
299C4×C59—81no—
300C4×C59—81no—
301C4×C59—81no—
302C4×C59#52911yescharacteristic
303C4×C59#53911yescharacteristic
304C4×C59#55311yescharacteristic
305C4×C59#55611yescharacteristic
306Dic59—42no—
307Dic59—42no—
308Dic59—81no—
309#2497D5911yescharacteristic, complemented
310#2363—591no—
311#2071—591no—
312#2067—591no—
313#2033—591no—
314#2033—591no—
315#1947—591no—
316#1942—591no—
317#1923—591no—
318#1921—591no—
319#1919—591no—
320#1864—591no—
321#1854—591no—
322#1825—591no—
323C4×#102—591no—
324C2×C59—81no—
325C2×C59#246611yescharacteristic
326C2×C59#310711yescharacteristic
327C2×C59#312211yescharacteristic
328C2×C59#330511yescharacteristic
329C2×C59#331411yescharacteristic
330C2×C59#333911yescharacteristic
331C2×C59#335711yescharacteristic
332D59—82no—
333C2×#121—591no—
334C2×C4×D4—591no—
335C2×#117—591no—
336C2×#107—591no—
337C2×#107—591no—
338C2×#107—591no—
339C2×#106—591no—
340C2×C4⋊C8—591no—
341C2×#102—591no—
342C2×#102—591no—
343C2×#102—591no—
344C2×#102—591no—
345C2×C22⋊C8—591no—
346C2×C22⋊C8—591no—
347C2×C4×C8—591no—
348#400—21no—
349#396—21no—
350#390—591no—
351#390—1181no—
352#389—591no—
353#389C2×D5911yescharacteristic
354#389C2×D5911yescharacteristic
355#388—21no—
356#388—1181no—
357#387—21no—
358#386—1181no—
359#384C2×D5911yescharacteristic
360#380C2×D5911yescharacteristic
361#380C2×D5911yescharacteristic
362C4×C4⋊C4—591no—
363C4×C4⋊C4C2×D5911yescharacteristic
364C4×C22⋊C4—1181no—
365C4×C22⋊C4C2×D5911yescharacteristic
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#340—591no—
373#340—591no—
374#336—591no—
375#336—591no—
376C59#2704111yescharacteristic, complemented, Sylow
377C22×D4—591no—
378C22×D4—1181no—
379C23×C4—1181no—
380C23×C4C22×D5911yescharacteristic
381C22×C8—591no—
382C22×C8—591no—
383C22×C8—591no—
384C22×C8—591no—
385#121—1182no—
386#121—1182no—
387C4×D4—1182no—
388C4×D4—2361no—
389#117—1182no—
390#117—2361no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1181no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C22⋊C4—1181no—
409C2×C22⋊C4—1181no—
410C2×C22⋊C4C22⋊D5911yescharacteristic
411C2×C22⋊C4C22⋊D5911yescharacteristic
412C2×C42—21no—
413C2×C42—1181no—
414C2×C42—1181no—
415C2×C42C22×D5911yescharacteristic
416C2×C42C22×D5911yescharacteristic
417#107—1182no—
418#107—1182no—
419#107—2361no—
420#106—1182no—
421C4⋊C8—1182no—
422#102—1182no—
423#102—1182no—
424#102—1182no—
425#102—2361no—
426C22⋊C8—1182no—
427C22⋊C8—2361no—
428C4×C8—2361no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1181no—
435#95—1181no—
436#95C22⋊D5911yescharacteristic
437#95C22⋊D5911yescharacteristic
438C24—1181no—
439C24C2×C22⋊D5911yescharacteristic
440C2×D4—592no—
441C2×D4—1184no—
442C2×D4—1181no—
443C2×D4—1181no—
444C2×D4—1181no—
445C2×D4—1181no—
446C2×D4—1181no—
447C2×D4—4721no—
448C2×D4—4721no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—4721no—
463C22×C4—2362no—
464C22×C4—2362no—
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C2×C22⋊D5911yescharacteristic
469C22×C4C2×C22⋊D5911yescharacteristic
470C22×C4C23×D5911yescharacteristic
471C2×C8—592no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—592no—
496C4⋊C4—592no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—4721no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C22⋊C4—1184no—
507C22⋊C4—4721no—
508C22⋊C4—2362no—
509C22⋊C4—2362no—
510C22⋊C4—2362no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—81no—
516C42—4721no—
517C42—2362no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—1182no—
526C23—2361no—
527C23—2361no—
528C23—2361no—
529C23—2361no—
530C23—2361no—
531C23—2361no—
532C23C22×C22⋊D5911yescharacteristic
533D4—1182no—
534D4—1182no—
535D4—4724no—
536D4—2361no—
537D4—2361no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—1182no—
568C2×C4—1182no—
569C2×C4—4724no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4#1427374712yes—
578C2×C4#1427375812yes—
579C2×C4#41×D5911yescharacteristic
580C2×C4#41×D5911yescharacteristic
581C2×C4#41×D5911yescharacteristic
582C2×C4#41×D5911yescharacteristic
583C2×C4#1427377311yescharacteristic
584C2×C4#1427377411yescharacteristic
585C8—1182no—
586C8—2361no—
587C8—2361no—
588C8—2361no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—4721no—
597C22—4721no—
598C22—4721no—
599C22—4721no—
600C22—2362no—
601C22—2362no—
602C22—2362no—
603C22#1683790911yescharacteristic
604C22#1683804011yescharacteristic
605C22#1683805111yescharacteristic
606C22C2×#1427374711yescharacteristic
607C22C2×#1427375811yescharacteristic
608C22#1683820511yescharacteristic
609C22#1683820511yescharacteristic
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—4721no—
620C4—2362no—
621C4—2362no—
622C4#1683793511yescharacteristic
623C4#1683799311yescharacteristic
624C4#1683810311yescharacteristic
625C4#1683811911yescharacteristic
626C2—81no—
627C2—4722no—
628C2#2142812811yescharacteristic
629C2#2143416111yescharacteristic
630C2#2143433411yescharacteristic
631C2#2143604011yescharacteristic
632C2#2143617111yescharacteristic
633C2#2143649211yescharacteristic
634C2#2143672211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2497C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenter#2497D59
hypocenterC59#27041
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2497

Complements

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

NH ≅ G/Ntype
2. #21425500C2semidirect
3. #21425481C2semidirect
6. #21424852C2semidirect
7. #21424826C2semidirect
8. #21424704C2semidirect
9. #21424688C2semidirect
10. #21424671C2semidirect
11. #21424356C2semidirect
12. #21424293C2semidirect
13. #21424188C2semidirect
14. C4×#14273647C2semidirect
15. C59×#2497C2semidirect
309. #2497D59semidirect
376. C59#27041semidirect

Maximal subgroups

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

subgroupindex
#214271322
#214255002
#214254812
#214252782
#214248522
#214248262
#214247042
#214246882
#214246712
#214243562
#214242932
#214241882
C4×#142736472
C59×#24972
#2704159