Finite Group Databasebeta

← #28661355

Subgroups — #28661355

Order 15104.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428507C211yesmaximal, characteristic, complemented
2#21425509C211yesmaximal, characteristic, complemented
3#21425470C211yesmaximal, characteristic, complemented
4#21425282C211yesmaximal, characteristic
5#21425275C211yesmaximal, characteristic
6#21424864C211yesmaximal, characteristic, complemented
7#21424821C211yesmaximal, characteristic, complemented
8#21424704C211yesmaximal, characteristic, complemented
9#21424674C211yesmaximal, characteristic, complemented
10#21424664C211yesmaximal, characteristic, complemented
11#21424357C211yesmaximal, characteristic, complemented
12#21424289C211yesmaximal, characteristic, complemented
13#21424186C211yesmaximal, characteristic, complemented
14C4×#14273648C211yesmaximal, characteristic, complemented
15C59×#2361C211yesmaximal, characteristic
16C2×#14273660C2211yescharacteristic
17C2×#14273658C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26C2×C22⋊(C59⋊C8)C2211yescharacteristic
27C2×C4×C59⋊C8C2211yescharacteristic
28#16837343—21no—
29#16837330—21no—
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837290—21no—
33#16837269—21no—
34#16837268C2211yescharacteristic
35#16837262C2211yescharacteristic
36#16837262C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#121×C59C2211yescharacteristic
49C2×C4×D4×C59—21no—
50C2×C4×D4×C59C2211yescharacteristic
51C22×C4⋊C4×C59C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#386—21no—
56C59×#385—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×C59⋊C8C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C2×C4⋊Dic59C2311yescharacteristic
64C2×C4⋊Dic59C2311yescharacteristic
65C2×C4⋊Dic59C2311yescharacteristic
66C2×C4⋊Dic59C2311yescharacteristic
67C2×Dic59⋊C4—21no—
68C2×Dic59⋊C4—21no—
69C2×C22⋊Dic59—21no—
70C2×C22⋊Dic59—21no—
71C2×C4×Dic59—21no—
72#14273660—22no—
73#14273658—22no—
74#14273658—22no—
75#14273658—41no—
76C4⋊(C59⋊C8)—22no—
77#14273648—22no—
78#14273648—22no—
79#14273648—22no—
80#14273648—41no—
81C22⋊(C59⋊C8)—22no—
82C22⋊(C59⋊C8)—41no—
83C4×C59⋊C8—41no—
84C59⋊#95—21no—
85C59⋊#95—21no—
86C59⋊#95—21no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C22×D4×C59—21no—
92C22×D4×C59C2311yescharacteristic
93C23×C4×C59—21no—
94C23×C4×C59—21no—
95C23×C4×C59C2311yescharacteristic
96#121×C59—22no—
97#121×C59—22no—
98C4×D4×C59—22no—
99C4×D4×C59—41no—
100C4×D4×C59—44no—
101C2×C4⋊C4×C59—22no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—41no—
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59D411yescharacteristic
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117#95×C59—21no—
118#95×C59—21no—
119#95×C59D411yescharacteristic
120#95×C59D411yescharacteristic
121C24×C59—21no—
122C24×C59C2×D411yescharacteristic
123C2×D4×C59—24no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—42no—
130C2×D4×C59—42no—
131C2×D4×C59#4112yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×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—42no—
142C22×C4×C59—42no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—81no—
146C22×C4×C59—81no—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—42no—
162C4⋊C4×C59—42no—
163C4⋊C4×C59#4112yes—
164C4⋊C4×C59#4112yes—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—22no—
167C22⋊C4×C59—22no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C22⋊C4×C59—81no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—42no—
176C22⋊Dic59—42no—
177C22⋊Dic59—81no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186Dic59⋊C4—81no—
187Dic59⋊C4—81no—
188C4×Dic59—42no—
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—21no—
200C2×C59⋊C8#4112yes—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—22no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59#13512yes—
215C23×C59C22×D411yescharacteristic
216D4×C59—22no—
217D4×C59—22no—
218D4×C59—48no—
219D4×C59—41no—
220D4×C59—41no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—22no—
225C2×C4×C59—21no—
226C2×C4×C59—22no—
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—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
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—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—84no—
249C2×C4×C59—84no—
250C2×C4×C59AGL(1,ℤ/8)12yes—
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59H(5,𝔽2)11yescharacteristic
256C2×C4×C592−1+411yescharacteristic
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C59⋊C8—22no—
270C59⋊C8—41no—
271C59⋊C8—41no—
272C59⋊C8—41no—
273#27031—591nomaximal, Sylow
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—22no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59—42no—
282C22×C59—42no—
283C22×C59—42no—
284C22×C59—81no—
285C22×C59—81no—
286C22×C59—81no—
287C22×C59—81no—
288C22×C59#52511yescharacteristic
289C22×C59#54611yescharacteristic
290C22×C59#54711yescharacteristic
291C22×C59C2×#13511yescharacteristic
292C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C59#57711yescharacteristic
294C22×C59#57711yescharacteristic
295C4×C59—22no—
296C4×C59—41no—
297C4×C59—41no—
298C4×C59—42no—
299C4×C59—41no—
300C4×C59—41no—
301C4×C59—42no—
302C4×C59—81no—
303C4×C59#52911yescharacteristic
304C4×C59#53911yescharacteristic
305C4×C59#55311yescharacteristic
306C4×C59#55611yescharacteristic
307Dic59—81no—
308Dic59—81no—
309Dic59—81no—
310Dic59—81no—
311#2495—591no—
312#2361D5911yescharacteristic
313#2073—591no—
314#2065—591no—
315#2034—591no—
316#2033—591no—
317#1949—591no—
318#1941—591no—
319#1923—591no—
320#1919—591no—
321#1918—591no—
322#1864—591no—
323#1853—591no—
324#1824—591no—
325C4×#102—591no—
326C2×C59—42no—
327C2×C59—82no—
328C2×C59#246611yescharacteristic
329C2×C59#310511yescharacteristic
330C2×C59#312511yescharacteristic
331C2×C59#330111yescharacteristic
332C2×C59#331011yescharacteristic
333C2×C59#333711yescharacteristic
334C2×C59#335411yescharacteristic
335C2×#121C2×D5911yescharacteristic
336C2×C4×D4—21no—
337C2×C4×D4C2×D5911yescharacteristic
338C22×C4⋊C4C2×D5911yescharacteristic
339C2×#107—591no—
340C2×#106—591no—
341C2×#106—591no—
342C2×#106—591no—
343C2×C4⋊C8—591no—
344C2×#102—591no—
345C2×#102—591no—
346C2×#102—591no—
347C2×#102—591no—
348C2×C22⋊C8—591no—
349C2×C22⋊C8—591no—
350C2×C4×C8—591no—
351#400—1181no—
352#397—1181no—
353#390C2×D5911yescharacteristic
354#389—591no—
355#389—591no—
356#389C2×D5911yescharacteristic
357#388—21no—
358#388—1181no—
359#386—21no—
360#385—21no—
361#383—1181no—
362#382—591no—
363#380—591no—
364#380—591no—
365C4×C4⋊C4—591no—
366C4×C4⋊C4C2×D5911yescharacteristic
367C4×C22⋊C4C2×D5911yescharacteristic
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#336—591no—
377#336—591no—
378C59#2703111yescharacteristic, complemented, Sylow
379C22×D4—21no—
380C22×D4C22×D5911yescharacteristic
381C23×C4—21no—
382C23×C4—21no—
383C23×C4C22×D5911yescharacteristic
384C22×C8—591no—
385C22×C8—591no—
386C22×C8—591no—
387C22×C8—591no—
388#121—22no—
389#121—22no—
390C4×D4—22no—
391C4×D4—41no—
392C4×D4—44no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1181no—
411C2×C22⋊C4—1181no—
412C2×C22⋊C4C22⋊D5911yescharacteristic
413C2×C22⋊C4C22⋊D5911yescharacteristic
414C2×C42—21no—
415C2×C42—1181no—
416C2×C42C22×D5911yescharacteristic
417C2×C42C22×D5911yescharacteristic
418#107—1182no—
419#106—1182no—
420#106—1182no—
421#106—2361no—
422C4⋊C8—1182no—
423#102—1182no—
424#102—1182no—
425#102—1182no—
426#102—2361no—
427C22⋊C8—1182no—
428C22⋊C8—2361no—
429C4×C8—2361no—
430#95—21no—
431#95—21no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95C22⋊D5911yescharacteristic
440#95C22⋊D5911yescharacteristic
441C24—21no—
442C24C2×C22⋊D5911yescharacteristic
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4#1174568512yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C23×D5911yescharacteristic
477C2×C8—592no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—4721no—
504C4⋊C4—4721no—
505C4⋊C4#1174568512yes—
506C4⋊C4#1174568712yes—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—4721no—
514C22⋊C4—2362no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—2362no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23#1427374712yes—
535C23C22×C22⋊D5911yescharacteristic
536D4—22no—
537D4—22no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
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—41no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4#1427375312yes—
583C2×C4#41×D5911yescharacteristic
584C2×C4C2×#1174568711yescharacteristic
585C2×C4C2×#1174568511yescharacteristic
586C2×C4C2×#1174568511yescharacteristic
587C2×C4#1427377211yescharacteristic
588C2×C4#1427377411yescharacteristic
589C8—1182no—
590C8—2361no—
591C8—2361no—
592C8—2361no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#1683791311yescharacteristic
608C22#1683804611yescharacteristic
609C22#1683805111yescharacteristic
610C22C2×#1427374711yescharacteristic
611C22C2×#1427375311yescharacteristic
612C22#1683821011yescharacteristic
613C22#1683821311yescharacteristic
614C4—22no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—42no—
621C4—81no—
622C4—4721no—
623C4—4721no—
624C4—4721no—
625C4—4721no—
626C4#1683793211yescharacteristic
627C4#1683799411yescharacteristic
628C4#1683810211yescharacteristic
629C4#1683812211yescharacteristic
630C2—42no—
631C2—82no—
632C2#2142812611yescharacteristic
633C2#2143414311yescharacteristic
634C2#2143437311yescharacteristic
635C2#2143599811yescharacteristic
636C2#2143611711yescharacteristic
637C2#2143646911yescharacteristic
638C2#2143669111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2361C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2361D59
hypocenterC59#27031
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #21428507C2semidirect
2. #21425509C2semidirect
3. #21425470C2semidirect
6. #21424864C2semidirect
7. #21424821C2semidirect
8. #21424704C2semidirect
9. #21424674C2semidirect
10. #21424664C2semidirect
11. #21424357C2semidirect
12. #21424289C2semidirect
13. #21424186C2semidirect
14. C4×#14273648C2semidirect
378. C59#27031semidirect

Maximal subgroups

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

subgroupindex
#214285072
#214255092
#214254702
#214252822
#214252752
#214248642
#214248212
#214247042
#214246742
#214246642
#214243572
#214242892
#214241862
C4×#142736482
C59×#23612
#2703159