Finite Group Databasebeta

← #28652249

Subgroups — #28652249

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#21427787C211yesmaximal, characteristic, complemented
2#21425664C211yesmaximal, characteristic, complemented
3#21425597C211yesmaximal, characteristic, complemented
4#21425470C211yesmaximal, characteristic, complemented
5#21425232C211yesmaximal, characteristic, complemented
6#21425048C211yesmaximal, characteristic, complemented
7#21424688C211yesmaximal, characteristic, complemented
8#21424525C211yesmaximal, characteristic
9#21424419C211yesmaximal, characteristic
10#21424343C211yesmaximal, characteristic, complemented
11#21424295C211yesmaximal, characteristic, complemented
12#21424276C211yesmaximal, characteristic, complemented
13#21424106C211yesmaximal, characteristic, complemented
14#21423812C211yesmaximal, characteristic, complemented
15C59×#2363C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28#16837325—21no—
29#16837314—21no—
30#16837306—21no—
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837289—21no—
34#16837268C2211yescharacteristic
35#16837262C2211yescharacteristic
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×#121×C59C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×#117×C59C2211yescharacteristic
51C59×#390—21no—
52C59×#390C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#386—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C22⋊C4×C59—21no—
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×Dic118—21no—
60C22×C59⋊C8—22no—
61C22×C59⋊C8—41no—
62C22×C59⋊C8—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273727—22no—
68#14273727—41no—
69#14273686—22no—
70#14273686—41no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C4×Dic59—21no—
81#14273660—22no—
82#14273658—41no—
83C4⋊(C59⋊C8)—41no—
84C4⋊(C59⋊C8)—41no—
85#14273648—22no—
86#14273648—41no—
87#14273648—41no—
88#14273648—41no—
89C22⋊(C59⋊C8)—22no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C22×D4×C59—21no—
96C22×D4×C59C2311yescharacteristic
97C23×C4×C59—21no—
98C23×C4×C59C2311yescharacteristic
99#121×C59—22no—
100#121×C59Q814yescomplemented
101C4×D4×C59—22no—
102C4×D4×C59—41no—
103#117×C59—22no—
104#117×C59—41no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C22⋊C4×C59D411yescharacteristic
116C2×C42×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#95×C59—21no—
121#95×C59—21no—
122#95×C59D411yescharacteristic
123#95×C59D411yescharacteristic
124C24×C59—21no—
125C24×C59C2×D411yescharacteristic
126C2×D4×C59—24no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—81no—
133C2×D4×C59—81no—
134C2×D4×C59C2×Q812yes—
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—21no—
142C22×C4×C59—42no—
143C22×C4×C59—42no—
144C22×C4×C59—81no—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×Q812yes—
151C22×C4×C59#4112yes—
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C2×Dic118—42no—
158C2×Dic118—42no—
159C2×Dic118—81no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59C2×Q812yes—
166C4⋊C4×C59#4112yes—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—41no—
170C22⋊C4×C59—42no—
171C22⋊C4×C59—42no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—81no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—42no—
179C42×C59—81no—
180C22⋊Dic59—42no—
181C22⋊Dic59—81no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—42no—
190C4⋊Dic59#4112yes—
191Dic59⋊C4—42no—
192Dic59⋊C4—81no—
193C4×Dic59—42no—
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×C59—22no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59C22⋊Q812yes—
221C23×C59C22×D411yescharacteristic
222D4×C59—22no—
223D4×C59—22no—
224D4×C59—41no—
225D4×C59—41no—
226D4×C59—84no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—22no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
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×C59—41no—
253C2×C4×C59—84no—
254C2×C4×C59C22⋊Q812yes—
255C2×C4×C59#13512yes—
256C2×C4×C59C22×Q811yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59H(5,𝔽2)11yescharacteristic
261C2×C4×C59H(5,𝔽2)11yescharacteristic
262C2×Dic59—22no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274Dic118—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×C59—42no—
288C22×C59—42no—
289C22×C59—42no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59C2×C22⋊Q811yescharacteristic
295C22×C59#54611yescharacteristic
296C22×C59#54611yescharacteristic
297C22×C59C2×#13511yescharacteristic
298C22×C59#57511yescharacteristic
299C22×C59#57611yescharacteristic
300C22×C59#57711yescharacteristic
301C4×C59—22no—
302C4×C59—22no—
303C4×C59—42no—
304C4×C59—41no—
305C4×C59—41no—
306C4×C59—42no—
307C4×C59—41no—
308C4×C59—81no—
309C4×C59#52911yescharacteristic
310C4×C59#53811yescharacteristic
311C4×C59#55411yescharacteristic
312C4×C59#55411yescharacteristic
313Dic59—42no—
314Dic59—81no—
315Dic59—81no—
316Dic59—81no—
317#2439—591no—
318#2363D5911yescharacteristic
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—82no—
334C2×C59#245011yescharacteristic
335C2×C59#307611yescharacteristic
336C2×C59#308111yescharacteristic
337C2×C59#329711yescharacteristic
338C2×C59#330411yescharacteristic
339C2×C59#331911yescharacteristic
340C2×C59#332111yescharacteristic
341C23×C8—591no—
342C2×C4⋊Q8—591no—
343C2×#121C2×D5911yescharacteristic
344C2×C4×D4C2×D5911yescharacteristic
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—1181no—
357#393—1181no—
358#391—1181no—
359#390—21no—
360#390C2×D5911yescharacteristic
361#389—591no—
362#389—591no—
363#389C2×D5911yescharacteristic
364#388—21no—
365#387—1181no—
366#386—21no—
367#382—591no—
368#380—591no—
369C4×C4⋊C4C2×D5911yescharacteristic
370C4×C22⋊C4—21no—
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—1181no—
386C22×D4—21no—
387C22×D4C22×D5911yescharacteristic
388C23×C4—21no—
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—1182no—
398C4⋊Q8—2361no—
399#121—22no—
400#121Dic11814yescomplemented
401C4×D4—22no—
402C4×D4—41no—
403#117—22no—
404#117—41no—
405#117—1182no—
406#117—2361no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4C22⋊D5911yescharacteristic
426C2×C22⋊C4C22⋊D5911yescharacteristic
427C2×C42—21no—
428C2×C42—21no—
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—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95C22⋊D5911yescharacteristic
449#95C22⋊D5911yescharacteristic
450C24—21no—
451C24C2×C22⋊D5911yescharacteristic
452C2×Q8—4721no—
453C2×Q8—2362no—
454C2×Q8—2362no—
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4C2×Dic11812yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—1181no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4C2×Dic11812yes—
479C22×C4#1174568512yes—
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—81no—
504C4⋊C4—592no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—4721no—
513C4⋊C4—2362no—
514C4⋊C4—2362no—
515C4⋊C4C2×Dic11812yes—
516C4⋊C4#1174568712yes—
517C22⋊C4—24no—
518C22⋊C4—22no—
519C22⋊C4—41no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C22⋊C4—4721no—
525C22⋊C4—2362no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—81no—
532C42—2362no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23#1427370612yes—
547C23C22×C22⋊D5911yescharacteristic
548Q8—4724no—
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—84no—
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—22no—
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—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—1182no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4#1427370612yes—
594C2×C4#1427374712yes—
595C2×C4C22×Dic11811yescharacteristic
596C2×C4#41×D5911yescharacteristic
597C2×C4C2×#1174568711yescharacteristic
598C2×C4C2×#1174568511yescharacteristic
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—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×#1427370611yescharacteristic
620C22#1683804011yescharacteristic
621C22#1683804611yescharacteristic
622C22C2×#1427374711yescharacteristic
623C22#1683820011yescharacteristic
624C22#1683820511yescharacteristic
625C22#1683821311yescharacteristic
626C4—22no—
627C4—22no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—4721no—
635C4—4721no—
636C4—4721no—
637C4—2362no—
638C4#1683793211yescharacteristic
639C4#1683798511yescharacteristic
640C4#1683810911yescharacteristic
641C4#1683811111yescharacteristic
642C2—42no—
643C2—82no—
644C2#2142791411yescharacteristic
645C2#2143388411yescharacteristic
646C2#2143392711yescharacteristic
647C2#2143594011yescharacteristic
648C2#2143603511yescharacteristic
649C2#2143625211yescharacteristic
650C2#2143627611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2363C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2363D59
hypocenterC59#26409
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. #21427787C2semidirect
2. #21425664C2semidirect
3. #21425597C2semidirect
4. #21425470C2semidirect
5. #21425232C2semidirect
6. #21425048C2semidirect
7. #21424688C2semidirect
10. #21424343C2semidirect
11. #21424295C2semidirect
12. #21424276C2semidirect
13. #21424106C2semidirect
14. #21423812C2semidirect
100. #121×C59Q8semidirect
384. C59#26409semidirect
400. #121Dic118semidirect

Maximal subgroups

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

subgroupindex
#214277872
#214256642
#214255972
#214254702
#214252322
#214250482
#214246882
#214245252
#214244192
#214243432
#214242952
#214242762
#214241062
#214238122
C59×#23632
#2640959