Finite Group Databasebeta

← #28652251

Subgroups — #28652251

Order 15104.

802 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427132C211yesmaximal, characteristic
2#21425663C211yesmaximal, characteristic, complemented
3#21425598C211yesmaximal, characteristic, complemented
4#21425467C211yesmaximal, characteristic, complemented
5#21425229C211yesmaximal, characteristic, complemented
6#21425049C211yesmaximal, characteristic, complemented
7#21424685C211yesmaximal, characteristic, complemented
8#21424522C211yesmaximal, characteristic
9#21424418C211yesmaximal, characteristic
10#21424342C211yesmaximal, characteristic, complemented
11#21424293C211yesmaximal, characteristic, complemented
12#21424276C211yesmaximal, characteristic, complemented
13#21424103C211yesmaximal, characteristic, complemented
14#21423812C211yesmaximal, characteristic, complemented
15C59×#2439C211yesmaximal, characteristic, complemented
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273701C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29#16837304C2211yescharacteristic
30#16837302—21no—
31#16837300C2211yescharacteristic
32#16837291—21no—
33#16837278—21no—
34C4×C4⋊Dic59C2211yescharacteristic
35#16837249—21no—
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×#117×C59C2211yescharacteristic
50C59×#396—21no—
51C59×#393—21no—
52C59×#391—21no—
53C59×#389C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#387—21no—
56C59×#382C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C22⋊D59—21no—
61C22×C4×D59—21no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8—41no—
64C22×C59⋊C8—41no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273701—22no—
70#14273701Q814yescomplemented
71C4×D236—22no—
72C4×D236—41no—
73#14273686—22no—
74#14273686—41no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86#14273660—22no—
87#14273658—41no—
88C4⋊(C59⋊C8)—41no—
89C4⋊(C59⋊C8)—41no—
90#14273647—22no—
91#14273647—41no—
92#14273647—41no—
93#14273647—41no—
94C22⋊(C59⋊C8)—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×Q8×C59—21no—
98C23×C4×C59C2311yescharacteristic
99C4⋊Q8×C59—22no—
100C4⋊Q8×C59—41no—
101#117×C59—22no—
102#117×C59—41no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
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×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59D411yescharacteristic
123#95×C59D411yescharacteristic
124C24×C59C2×D411yescharacteristic
125C23×D59—21no—
126C2×Q8×C59—42no—
127C2×Q8×C59—42no—
128C2×Q8×C59—81no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×Q812yes—
143C22×C4×C59#4112yes—
144C22×C4×C59C2411yescharacteristic
145C2×C22⋊D59—24no—
146C2×C22⋊D59—81no—
147C2×C22⋊D59—81no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×D236C2×Q812yes—
157C2×C4×D59—42no—
158C2×C4×D59—42no—
159C2×C4×D59—81no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59—81no—
170C4⋊C4×C59#4112yes—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—41no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—81no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—42no—
180C22⋊Dic59—42no—
181(C2×D59)⋊C4—24no—
182(C2×D59)⋊C4—42no—
183(C2×D59)⋊C4—42no—
184(C2×D59)⋊C4—81no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59C2×Q812yes—
190C4⋊Dic59#4112yes—
191Dic59⋊C4—81no—
192C4×Dic59—42no—
193C4×Dic59—81no—
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—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—44no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59C22⋊Q812yes—
214C23×C59C22×D411yescharacteristic
215C22×D59—22no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222Q8×C59—84no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
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×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59C22⋊Q812yes—
253C2×C4×C59#13512yes—
254C2×C4×C59C22×Q811yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59H(5,𝔽2)11yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C22⋊D59—84no—
261C2×Dic59—22no—
262C2×Dic59—22no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270D236—22no—
271D236—22no—
272D236—41no—
273D236—41no—
274C4×D59—84no—
275C59⋊C8—41no—
276C59⋊C8—41no—
277C59⋊C8—41no—
278C59⋊C8—41no—
279#26409—591nomaximal, Sylow
280C22×C59—22no—
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59C2×C22⋊Q811yescharacteristic
288C22×C59#54611yescharacteristic
289C22×C59#54611yescharacteristic
290C22×C59C2×#13511yescharacteristic
291C22×C59#57511yescharacteristic
292C22×C59#57611yescharacteristic
293C22×C59#57711yescharacteristic
294C2×D59—42no—
295C2×D59—42no—
296C2×D59—42no—
297C2×D59—81no—
298C2×D59—81no—
299C2×D59—81no—
300C2×D59—81no—
301C4×C59—22no—
302C4×C59—22no—
303C4×C59—42no—
304C4×C59—41no—
305C4×C59—41no—
306C4×C59—41no—
307C4×C59—81no—
308C4×C59—81no—
309C4×C59—81no—
310C4×C59#52911yescharacteristic
311C4×C59#53811yescharacteristic
312C4×C59#55411yescharacteristic
313C4×C59#55411yescharacteristic
314Dic59—42no—
315Dic59—42no—
316Dic59—81no—
317#2439D5911yescharacteristic, complemented
318#2363—591no—
319#2101—591no—
320#2088—591no—
321#2065—591no—
322#2023—591no—
323#1988—591no—
324#1921—591no—
325#1893—591no—
326#1875—591no—
327#1862—591no—
328#1854—591no—
329#1850—591no—
330#1804—591no—
331#1735—591no—
332C2×C59—42no—
333C2×C59#245011yescharacteristic
334C2×C59#307611yescharacteristic
335C2×C59#308111yescharacteristic
336C2×C59#329711yescharacteristic
337C2×C59#330411yescharacteristic
338C2×C59#331911yescharacteristic
339C2×C59#332111yescharacteristic
340D59—82no—
341C23×C8—591no—
342C2×C4⋊Q8C2×D5911yescharacteristic
343C2×#121—591no—
344C2×C4×D4—591no—
345C2×#117—591no—
346C2×#117C2×D5911yescharacteristic
347C2×#107—591no—
348C2×#106—591no—
349C2×C4⋊C8—591no—
350C2×C4⋊C8—591no—
351C2×#102—591no—
352C2×#102—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×C22⋊C8—591no—
356#396—21no—
357#393—21no—
358#391—21no—
359#390—591no—
360#390—1181no—
361#389—591no—
362#389C2×D5911yescharacteristic
363#389C2×D5911yescharacteristic
364#388—1181no—
365#387—21no—
366#386—1181no—
367#382C2×D5911yescharacteristic
368#380C2×D5911yescharacteristic
369C4×C4⋊C4—591no—
370C4×C22⋊C4—1181no—
371C4×C22⋊C4C2×D5911yescharacteristic
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#336—591no—
383#336—591no—
384C59#2640911yescharacteristic, complemented, Sylow
385C22×Q8—21no—
386C22×D4—591no—
387C22×D4—1181no—
388C23×C4—1181no—
389C23×C4C22×D5911yescharacteristic
390C22×C8—591no—
391C22×C8—591no—
392C22×C8—591no—
393C22×C8—591no—
394C22×C8—1182no—
395C22×C8—2361no—
396C22×C8—2361no—
397C4⋊Q8—22no—
398C4⋊Q8—41no—
399#121—594no—
400#121—1182no—
401C4×D4—1182no—
402C4×D4—2361no—
403#117—22no—
404#117—41no—
405#117—1182no—
406#117—2361no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4C22⋊D5911yescharacteristic
426C2×C22⋊C4C22⋊D5911yescharacteristic
427C2×C42—21no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42C22×D5911yescharacteristic
431C2×C42C22×D5911yescharacteristic
432#107—1182no—
433#106—2361no—
434C4⋊C8—2361no—
435C4⋊C8—2361no—
436#102—1182no—
437#102—2361no—
438#102—2361no—
439#102—2361no—
440C22⋊C8—1182no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—1181no—
447#95—1181no—
448#95C22⋊D5911yescharacteristic
449#95C22⋊D5911yescharacteristic
450C24—1181no—
451C24C2×C22⋊D5911yescharacteristic
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×D4—592no—
456C2×D4—1184no—
457C2×D4—1181no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C2×D4—1181no—
462C2×D4—4721no—
463C2×D4—4721no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—4721no—
476C22×C4—2362no—
477C22×C4—2362no—
478C22×C4Q8×D5912yes—
479C22×C4#1174568912yes—
480C22×C4C2×C22⋊D5911yescharacteristic
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C23×D5911yescharacteristic
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—2364no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—592no—
510C4⋊C4—592no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—4721no—
516C4⋊C4#1174568712yes—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—42no—
520C22⋊C4—81no—
521C22⋊C4—1184no—
522C22⋊C4—4721no—
523C22⋊C4—2362no—
524C22⋊C4—2362no—
525C22⋊C4—2362no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—4721no—
532C42—2362no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—1182no—
540C23—2361no—
541C23—2361no—
542C23—2361no—
543C23—2361no—
544C23—2361no—
545C23—2361no—
546C23#1427370312yes—
547C23C22×C22⋊D5911yescharacteristic
548Q8—84no—
549D4—1182no—
550D4—1182no—
551D4—4724no—
552D4—2361no—
553D4—2361no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—1182no—
584C2×C4—1182no—
585C2×C4—4724no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4#1427370312yes—
594C2×C4#1427374712yes—
595C2×C4C2×Q8×D5911yescharacteristic
596C2×C4#41×D5911yescharacteristic
597C2×C4C2×#1174568711yescharacteristic
598C2×C4C2×#1174568911yescharacteristic
599C2×C4#1427377211yescharacteristic
600C2×C4#1427377311yescharacteristic
601C8—2361no—
602C8—2361no—
603C8—2361no—
604C8—2361no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—4721no—
613C22—4721no—
614C22—4721no—
615C22—4721no—
616C22—2362no—
617C22—2362no—
618C22—2362no—
619C22C2×#1427370311yescharacteristic
620C22#1683804011yescharacteristic
621C22#1683804611yescharacteristic
622C22C2×#1427374711yescharacteristic
623C22#1683820011yescharacteristic
624C22#1683820511yescharacteristic
625C22#1683821011yescharacteristic
626C4—22no—
627C4—22no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—4721no—
636C4—2362no—
637C4—2362no—
638C4#1683793811yescharacteristic
639C4#1683799211yescharacteristic
640C4#1683810611yescharacteristic
641C4#1683811011yescharacteristic
642C2—42no—
643C2—4722no—
644C2#2142791611yescharacteristic
645C2#2143387811yescharacteristic
646C2#2143392111yescharacteristic
647C2#2143594611yescharacteristic
648C2#2143602911yescharacteristic
649C2#2143624611yescharacteristic
650C2#2143628211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2439C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenter#2439D59
hypocenterC59#26409
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
2. #21425663C2semidirect
3. #21425598C2semidirect
4. #21425467C2semidirect
5. #21425229C2semidirect
6. #21425049C2semidirect
7. #21424685C2semidirect
10. #21424342C2semidirect
11. #21424293C2semidirect
12. #21424276C2semidirect
13. #21424103C2semidirect
14. #21423812C2semidirect
15. C59×#2439C2semidirect
70. #14273701Q8semidirect
317. #2439D59semidirect
384. C59#26409semidirect

Maximal subgroups

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

subgroupindex
#214271322
#214256632
#214255982
#214254672
#214252292
#214250492
#214246852
#214245222
#214244182
#214243422
#214242932
#214242762
#214241032
#214238122
C59×#24392
#2640959