Finite Group Databasebeta

← #28691948

Subgroups — #28691948

Order 15104.

788 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432463C211yesmaximal, characteristic, complemented
2#21425597C211yesmaximal, characteristic, complemented
3#21425481C211yesmaximal, characteristic, complemented
4#21425470C211yesmaximal, characteristic, complemented
5#21425406C211yesmaximal, characteristic
6#21424971C211yesmaximal, characteristic
7#21424831C211yesmaximal, characteristic, complemented
8#21424796C211yesmaximal, characteristic, complemented
9#21424792C211yesmaximal, characteristic, complemented
10#21424723C211yesmaximal, characteristic, complemented
11#21424674C211yesmaximal, characteristic, complemented
12#21424444C211yesmaximal, characteristic
13#21424343C211yesmaximal, characteristic, complemented
14#21423949C211yesmaximal, characteristic, complemented
15C59×#2293C211yesmaximal, characteristic, complemented
16C22×#11745669C2211yescharacteristic
17C2×#14273721—21no—
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837304C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837293—21no—
38#16837293—21no—
39#16837293—21no—
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4×D4×C59C2211yescharacteristic
49C2×#117×C59—21no—
50C59×#390C2211yescharacteristic
51C59×#390C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#388—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C22⋊C4×C59—21no—
58C43×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C2×#11745669—22no—
61C2×#11745669—22no—
62C2×#11745669—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273721—44no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×(C2×D59)⋊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—
80#14273660—22no—
81#14273658—22no—
82#14273658—41no—
83C4⋊(C59⋊C8)—22no—
84C4⋊(C59⋊C8)—22no—
85#14273648—22no—
86#14273648—22no—
87#14273648—41no—
88#14273647—22no—
89#14273647—22no—
90#14273647—41no—
91#14273638—41no—
92#14273638—41no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×D4×C59C2311yescharacteristic
97C23×C4×C59—21no—
98C4×D4×C59—22no—
99C4×D4×C59—22no—
100#117×C59—44no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
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×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59D411yescharacteristic
113C2×C42×C59D411yescharacteristic
114C2×C42×C59D411yescharacteristic
115C2×C42×C59D411yescharacteristic
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122C24×C59—21no—
123C23×D59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59#4112yes—
130C22×C4×C59—24no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—42no—
135C22×C4×C59—42no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C2×D236—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—42no—
158C4⋊C4×C59—42no—
159C4⋊C4×C59#4112yes—
160C22⋊C4×C59—24no—
161C22⋊C4×C59—42no—
162C22⋊C4×C59—42no—
163C22⋊C4×C59—42no—
164C22⋊C4×C59—42no—
165C42×C59—22no—
166C42×C59—22no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—41no—
174C42×C59—41no—
175C42×C59—42no—
176#11745669—44no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—42no—
179(C2×D59)⋊C4—42no—
180(C2×D59)⋊C4—42no—
181(C2×D59)⋊C4—42no—
182(C2×D59)⋊C4—42no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
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—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8#4112yes—
203C2×C59⋊C8#4112yes—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59C22×D411yescharacteristic
212C22×D59—41no—
213C22×D59—41no—
214C22×D59—41no—
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
219D4×C59—22no—
220D4×C59—22no—
221D4×C59—41no—
222D4×C59—41no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—22no—
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—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—48no—
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—41no—
249C2×C4×C59AGL(1,ℤ/8)12yes—
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—
266D236—41no—
267D236—41no—
268D236—41no—
269D236—41no—
270C59⋊C8—22no—
271C59⋊C8—22no—
272C59⋊C8—41no—
273C59⋊C8—41no—
274#29444—591nomaximal, Sylow
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59—81no—
279C22×C59—81no—
280C22×C59—81no—
281C22×C59—81no—
282C22×C59#52511yescharacteristic
283C22×C59#53211yescharacteristic
284C22×C59#53711yescharacteristic
285C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C59#57511yescharacteristic
288C22×C59#57511yescharacteristic
289C2×D59—81no—
290C2×D59—81no—
291C2×D59—81no—
292C2×D59—81no—
293C2×D59—81no—
294C2×D59—81no—
295C2×D59—81no—
296C4×C59—22no—
297C4×C59—22no—
298C4×C59—42no—
299C4×C59—41no—
300C4×C59—41no—
301C4×C59—41no—
302C4×C59—41no—
303C4×C59—81no—
304C4×C59—81no—
305C4×C59#55111yescharacteristic
306C4×C59#55211yescharacteristic
307C4×C59#55311yescharacteristic
308C4×C59#55611yescharacteristic
309Dic59—81no—
310Dic59—81no—
311Dic59—81no—
312#2860—591no—
313#2293D5911yescharacteristic, complemented
314#2088—591no—
315#2067—591no—
316#2065—591no—
317#2054—591no—
318#1972—591no—
319#1943—591no—
320#1936—591no—
321#1936—591no—
322#1926—591no—
323#1919—591no—
324#1879—591no—
325#1862—591no—
326#1767—591no—
327C2×C59—82no—
328C2×C59#260111yescharacteristic
329C2×C59#310911yescharacteristic
330C2×C59#311111yescharacteristic
331C2×C59#317111yescharacteristic
332C2×C59#318011yescharacteristic
333C2×C59#325111yescharacteristic
334C2×C59#325111yescharacteristic
335D59—161no—
336C22×D8—591no—
337C2×#126—1181no—
338C2×C4×D4C2×D5911yescharacteristic
339C2×#117—21no—
340C2×#107—591no—
341C2×#106—591no—
342C2×#106—591no—
343C2×C4⋊C8—591no—
344C2×C4⋊C8—591no—
345C2×#102—591no—
346C2×#102—591no—
347C2×#102—591no—
348C2×#102—591no—
349C2×#102—591no—
350C2×#102—591no—
351C2×#97—591no—
352C2×#97—591no—
353#390—591no—
354#390—591no—
355#390—591no—
356#390C2×D5911yescharacteristic
357#390C2×D5911yescharacteristic
358#389—591no—
359#389—591no—
360#389—591no—
361#389C2×D5911yescharacteristic
362#389C2×D5911yescharacteristic
363#388—21no—
364#388—21no—
365#388—1181no—
366#388—1181no—
367#388—1181no—
368C4×C4⋊C4C2×D5911yescharacteristic
369C4×C22⋊C4—21no—
370C43C2×D5911yescharacteristic
371#340—591no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#336—591no—
378#336—591no—
379C59#2944411yescharacteristic, complemented, Sylow
380C22×D4—591no—
381C22×D4C22×D5911yescharacteristic
382C23×C4—21no—
383C2×D8—1182no—
384C2×D8—1182no—
385C2×D8—2361no—
386C22×C8—591no—
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—591no—
390#126—2364no—
391C4×D4—22no—
392C4×D4—22no—
393#117—44no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C22⋊C4—21no—
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⋊C4—1181no—
413C2×C22⋊C4—1181no—
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C42—21no—
417C2×C42C22⋊D5911yescharacteristic
418C2×C42C22⋊D5911yescharacteristic
419C2×C42C22⋊D5911yescharacteristic
420C2×C42C22⋊D5911yescharacteristic
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423C2×C42C22×D5911yescharacteristic
424#107—1182no—
425#106—1182no—
426#106—2361no—
427C4⋊C8—1182no—
428C4⋊C8—1182no—
429#102—1182no—
430#102—1182no—
431#102—1182no—
432#102—1182no—
433#102—2361no—
434#102—2361no—
435#97—2361no—
436#97—2361no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443C24—21no—
444C24—1181no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—1181no—
451C2×D4—1181no—
452C2×D4—1181no—
453C2×D4—1181no—
454C2×D4—1181no—
455C2×D4—1181no—
456C2×D4#1174568512yes—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C2×C22⋊D5911yescharacteristic
469C22×C4C2×C22⋊D5911yescharacteristic
470C22×C4C2×C22⋊D5911yescharacteristic
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C23×D5911yescharacteristic
473D8—2364no—
474C2×C8—592no—
475C2×C8—592no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4#1174568512yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—2362no—
509C22⋊C4—2362no—
510C22⋊C4—2362no—
511C22⋊C4—2362no—
512C22⋊C4—2362no—
513C22⋊C4—2362no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—41no—
523C42—41no—
524C42—42no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—2361no—
533C23—2361no—
534C23—2361no—
535C23—2361no—
536C23—2361no—
537C23—2361no—
538C23—2361no—
539C23C22×C22⋊D5911yescharacteristic
540D4—22no—
541D4—22no—
542D4—41no—
543D4—41no—
544D4—2361no—
545D4—2361no—
546D4—2361no—
547D4—2361no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—48no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
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—2361no—
583C2×C4#1427375212yes—
584C2×C4#1427375312yes—
585C2×C4#41×D5911yescharacteristic
586C2×C4#41×D5911yescharacteristic
587C2×C4C2×#1174568511yescharacteristic
588C2×C4C2×#1174568511yescharacteristic
589C2×C4#1427377311yescharacteristic
590C2×C4#1427377411yescharacteristic
591C8—1182no—
592C8—1182no—
593C8—2361no—
594C8—2361no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—4721no—
603C22—4721no—
604C22—4721no—
605C22—4721no—
606C22—4721no—
607C22—4721no—
608C22—4721no—
609C22#1683791311yescharacteristic
610C22#1683794811yescharacteristic
611C22#1683798011yescharacteristic
612C22C2×#1427375211yescharacteristic
613C22C2×#1427375311yescharacteristic
614C22#1683820011yescharacteristic
615C22#1683820011yescharacteristic
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—4721no—
626C4—4721no—
627C4—4721no—
628C4#1683807611yescharacteristic
629C4#1683808511yescharacteristic
630C4#1683809611yescharacteristic
631C4#1683812011yescharacteristic
632C2—82no—
633C2—9441no—
634C2#2142969311yescharacteristic
635C2#2143418711yescharacteristic
636C2#2143420511yescharacteristic
637C2#2143469211yescharacteristic
638C2#2143478811yescharacteristic
639C2#2143545211yescharacteristic
640C2#2143545811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2293C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
hypercenter#2293D59
hypocenterC59#29444
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. #21432463C2semidirect
2. #21425597C2semidirect
3. #21425481C2semidirect
4. #21425470C2semidirect
7. #21424831C2semidirect
8. #21424796C2semidirect
9. #21424792C2semidirect
10. #21424723C2semidirect
11. #21424674C2semidirect
13. #21424343C2semidirect
14. #21423949C2semidirect
15. C59×#2293C2semidirect
313. #2293D59semidirect
379. C59#29444semidirect

Maximal subgroups

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

subgroupindex
#214324632
#214255972
#214254812
#214254702
#214254062
#214249712
#214248312
#214247962
#214247922
#214247232
#214246742
#214244442
#214243432
#214239492
C59×#22932
#2944459