Finite Group Databasebeta

← #28479211

Subgroups — #28479211

Order 15104.

710 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837291C211yesmaximal, characteristic
2#21423860C211yesmaximal, characteristic, complemented
3C22⋊C8×Dic59C211yesmaximal, characteristic, complemented
4C2×#16837125C211yesmaximal, characteristic
5C2×#16837125C211yesmaximal, characteristic
6#21421939C211yesmaximal, characteristic, complemented
7C59×#1280C211yesmaximal, characteristic, complemented
8C22×Dic59⋊C4C411yescharacteristic
9C22×(C2×D59)⋊C4C411yescharacteristic
10C22×(C2×D59)⋊C4C411yescharacteristic
11C22×(C2×D59)⋊C4C2211yescharacteristic
12C22×C4×Dic59C2211yescharacteristic
13C2×C22⋊(C59⋊C8)C2211yescharacteristic
14C2×C22⋊(C59⋊C8)C2211yescharacteristic
15C2×#14273637—21no—
16C2×C8×Dic59—21no—
17#16837291—21no—
18#16837291—21no—
19#16837291—21no—
20#16837291—21no—
21C2×C59⋊#95C411yescharacteristic
22#16837156—21no—
23#16837156—21no—
24#16837125C414yescomplemented
25#16837125C414yescomplemented
26C2×C22⋊C8×C59C2211yescharacteristic
27C2×C22⋊C8×C59C2211yescharacteristic
28C2×#95×C59C2211yescharacteristic
29C24×D59C2×C411yescharacteristic
30C23×Dic59C2×C411yescharacteristic
31C23×Dic59C2×C411yescharacteristic
32C23×Dic59C2×C411yescharacteristic
33C22×C59⋊C8—21no—
34C22×C59⋊C8—21no—
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—42no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×(C2×D59)⋊C4—21no—
43C2×(C2×D59)⋊C4—21no—
44C2×(C2×D59)⋊C4—21no—
45C2×(C2×D59)⋊C4—21no—
46C2×(C2×D59)⋊C4—42no—
47C2×(C2×D59)⋊C4—42no—
48C2×(C2×D59)⋊C4C2×C412yes—
49C2×(C2×D59)⋊C4C2×C412yes—
50C2×C4×Dic59—42no—
51C2×C4×Dic59D411yescharacteristic
52C2×C4×Dic59D411yescharacteristic
53C2×C4×Dic59D411yescharacteristic
54C2×C4×Dic59D411yescharacteristic
55C22⋊(C59⋊C8)—21no—
56C22⋊(C59⋊C8)—21no—
57C22⋊(C59⋊C8)C2×C412yes—
58C22⋊(C59⋊C8)C2×C412yes—
59#14273637—42no—
60C8×Dic59—42no—
61C59⋊#95—21no—
62C59⋊#95—21no—
63C23×C4×C59C2×C411yescharacteristic
64C23×C4×C59C2×C411yescharacteristic
65C23×C4×C59C2311yescharacteristic
66C22×C8×C59—21no—
67C22×C8×C59—21no—
68C22⋊C8×C59—21no—
69C22⋊C8×C59—21no—
70C22⋊C8×C59C2×C412yes—
71C22⋊C8×C59C2×C412yes—
72#95×C59—21no—
73#95×C59—21no—
74C24×C59C22×C411yescharacteristic
75C23×D59—21no—
76C23×D59—21no—
77C23×D59—21no—
78C23×D59—21no—
79C23×D59—21no—
80C23×D59—21no—
81C23×D59—41no—
82C23×D59—41no—
83C23×D59—41no—
84C23×D59—41no—
85C23×D59C4212yes—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—21no—
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59—41no—
93C22×C4×C59—41no—
94C22×C4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59—41no—
98C22×C4×C59C22×C411yescharacteristic
99C22×C4×C59C22×C411yescharacteristic
100C22×C4×C59C2×D411yescharacteristic
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59#4111yescharacteristic
103C22×C4×C59#4111yescharacteristic
104C22×Dic59—21no—
105C22×Dic59—21no—
106C22×Dic59—21no—
107C22×Dic59—21no—
108C22×Dic59—21no—
109C22×Dic59—21no—
110C22×Dic59—22no—
111C22×Dic59—22no—
112C22×Dic59—41no—
113C22×Dic59—41no—
114C22×Dic59—41no—
115C22×Dic59—41no—
116C22×Dic59C4212yes—
117C22×Dic59C22⋊C411yescharacteristic
118C22×Dic59C22⋊C411yescharacteristic
119C22×Dic59C22⋊C411yescharacteristic
120C22×Dic59C22⋊C411yescharacteristic
121C2×C8×C59—22no—
122C2×C8×C59—22no—
123C2×C8×C59—41no—
124C2×C8×C59—41no—
125C2×C8×C59—41no—
126C2×C8×C59—41no—
127(C2×D59)⋊C4—22no—
128(C2×D59)⋊C4—22no—
129(C2×D59)⋊C4—22no—
130(C2×D59)⋊C4—22no—
131(C2×D59)⋊C4—42no—
132(C2×D59)⋊C4—42no—
133(C2×D59)⋊C4—42no—
134(C2×D59)⋊C4—42no—
135Dic59⋊C4—42no—
136Dic59⋊C4—42no—
137C4×Dic59—22no—
138C4×Dic59—22no—
139C4×Dic59—41no—
140C4×Dic59—41no—
141C2×C59⋊C8—22no—
142C2×C59⋊C8—22no—
143C2×C59⋊C8—41no—
144C2×C59⋊C8—41no—
145C2×C59⋊C8—41no—
146C2×C59⋊C8—41no—
147C23×C59—21no—
148C23×C59—21no—
149C23×C59—21no—
150C23×C59—21no—
151C23×C59C2×C4211yescharacteristic
152C23×C59C2×C22⋊C411yescharacteristic
153C23×C59#11711yescharacteristic
154C23×C59#13111yescharacteristic
155C23×C59#13111yescharacteristic
156C23×C59#13111yescharacteristic
157C23×C59#13111yescharacteristic
158C22×D59—22no—
159C22×D59—22no—
160C22×D59—22no—
161C22×D59—42no—
162C22×D59—42no—
163C22×D59—41no—
164C22×D59—41no—
165C22×D59—41no—
166C22×D59—41no—
167C22×D59—41no—
168C22×D59—41no—
169C22×D59—41no—
170C22×D59—41no—
171C22×D59—42no—
172C22×D59—42no—
173C22×D59—81no—
174C22×D59—81no—
175C22×D59—81no—
176C22×D59—81no—
177C22×D59—81no—
178C22×D59—81no—
179C22×D59—81no—
180C22×D59—81no—
181C2×C4×C59—21no—
182C2×C4×C59—21no—
183C2×C4×C59—21no—
184C2×C4×C59—21no—
185C2×C4×C59—41no—
186C2×C4×C59—41no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59—41no—
194C2×C4×C59—41no—
195C2×C4×C59—81no—
196C2×C4×C59—81no—
197C2×C4×C59—81no—
198C2×C4×C59—81no—
199C2×C4×C59C4×D411yescharacteristic
200C2×C4×C59C4×D411yescharacteristic
201C2×C4×C59C4×D411yescharacteristic
202C2×C4×C59C4×D411yescharacteristic
203C2×Dic59—21no—
204C2×Dic59—21no—
205C2×Dic59—21no—
206C2×Dic59—21no—
207C2×Dic59—22no—
208C2×Dic59—21no—
209C2×Dic59—21no—
210C2×Dic59—21no—
211C2×Dic59—21no—
212C2×Dic59—21no—
213C2×Dic59—21no—
214C2×Dic59—42no—
215C2×Dic59—42no—
216C2×Dic59—42no—
217C2×Dic59—42no—
218C2×Dic59—42no—
219C2×Dic59—41no—
220C2×Dic59—41no—
221C2×Dic59—42no—
222C2×Dic59#9912yes—
223C2×Dic59#10012yes—
224C8×C59—42no—
225C8×C59—42no—
226C59⋊C8—42no—
227C59⋊C8—42no—
228#8155—591nomaximal, Sylow
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—21no—
240C22×C59—21no—
241C22×C59C4×C22⋊C411yescharacteristic
242C22×C59#40511yescharacteristic
243C22×C59#40511yescharacteristic
244C22×C59#40811yescharacteristic
245C22×C59#40811yescharacteristic
246C22×C59C2×#9911yescharacteristic
247C22×C59#41011yescharacteristic
248C22×C59C2×#10011yescharacteristic
249C22×C59#41311yescharacteristic
250C22×C59#43311yescharacteristic
251C22×C59#43311yescharacteristic
252C2×D59—42no—
253C2×D59—42no—
254C2×D59—42no—
255C2×D59—82no—
256C2×D59—82no—
257C2×D59—81no—
258C2×D59—81no—
259C2×D59—81no—
260C2×D59—81no—
261C2×D59—82no—
262C2×D59—82no—
263C4×C59—21no—
264C4×C59—21no—
265C4×C59—21no—
266C4×C59—21no—
267C4×C59—81no—
268C4×C59—81no—
269Dic59—22no—
270Dic59—22no—
271Dic59—41no—
272Dic59—41no—
273Dic59—42no—
274C2×#388—591no—
275#1749—591no—
276C4×C22⋊C8—591no—
277C2×#323—591no—
278C2×#323—591no—
279#1280—591no—
280#1280D5911yescharacteristic, complemented
281C2×C59—21no—
282C2×C59—21no—
283C2×C59—21no—
284C2×C59—21no—
285C2×C59#146811yescharacteristic
286C2×C59#146911yescharacteristic
287C2×C59#147111yescharacteristic
288C2×C59#147311yescharacteristic
289C2×C59#175011yescharacteristic
290C2×C59C4×#9911yescharacteristic
291C2×C59C4×#10011yescharacteristic
292D59—82no—
293C22×C4⋊C4—591no—
294C22×C22⋊C4—591no—
295C22×C22⋊C4—591no—
296C22×C22⋊C4—591no—
297C22×C42—591no—
298C2×C22⋊C8—591no—
299C2×C22⋊C8—591no—
300C2×C22⋊C8C2×D5911yescharacteristic
301C2×C22⋊C8C2×D5911yescharacteristic
302C2×#97—1181no—
303C2×C4×C8—1181no—
304#388—1181no—
305#388—1181no—
306#388—1181no—
307#388—1181no—
308C2×#95—591no—
309C2×#95C2×D5911yescharacteristic
310#336—1181no—
311#336—1181no—
312#323—594no—
313#323—594no—
314C59#815511yescharacteristic, complemented, Sylow
315C25—591no—
316C23×C4—591no—
317C23×C4—591no—
318C23×C4—591no—
319C23×C4C4×D5911yescharacteristic
320C23×C4C4×D5911yescharacteristic
321C23×C4C22×D5911yescharacteristic
322C22×C8—21no—
323C22×C8—21no—
324C22×C8—1181no—
325C22×C8—1181no—
326C2×C4⋊C4—1181no—
327C2×C4⋊C4—1181no—
328C2×C4⋊C4—2362no—
329C2×C22⋊C4—592no—
330C2×C22⋊C4—592no—
331C2×C22⋊C4—1181no—
332C2×C22⋊C4—1181no—
333C2×C22⋊C4—1181no—
334C2×C22⋊C4—1181no—
335C2×C22⋊C4—1181no—
336C2×C22⋊C4—1181no—
337C2×C22⋊C4—1181no—
338C2×C22⋊C4—1181no—
339C2×C22⋊C4—2362no—
340C2×C22⋊C4—2362no—
341C2×C42—591no—
342C2×C42—591no—
343C2×C42—591no—
344C2×C42—591no—
345C2×C42—2362no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348C22⋊C8—592no—
349C22⋊C8—592no—
350C22⋊C8—1181no—
351C22⋊C8—1181no—
352C22⋊C8C4×D5912yes—
353C22⋊C8C4×D5912yes—
354#97—2362no—
355C4×C8—2362no—
356#95—21no—
357#95—21no—
358#95—1181no—
359#95—1181no—
360C24—592no—
361C24—1181no—
362C24—1181no—
363C24—1181no—
364C24—1181no—
365C24—1181no—
366C24—1181no—
367C24—2361no—
368C24—2361no—
369C24—2361no—
370C24—2361no—
371C24C2×C4×D5911yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—591no—
385C22×C4—591no—
386C22×C4—591no—
387C22×C4—591no—
388C22×C4—592no—
389C22×C4—1182no—
390C22×C4—1182no—
391C22×C4—1181no—
392C22×C4—1181no—
393C22×C4—1181no—
394C22×C4—1181no—
395C22×C4—1181no—
396C22×C4—1181no—
397C22×C4—2361no—
398C22×C4—2361no—
399C22×C4—2361no—
400C22×C4—2361no—
401C22×C4C2×C4×D5911yescharacteristic
402C22×C4C2×C4×D5911yescharacteristic
403C22×C4D4×D5911yescharacteristic
404C22×C4D4×D5911yescharacteristic
405C22×C4#1174568711yescharacteristic
406C22×C4#1174568911yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—1182no—
414C2×C8—1182no—
415C2×C8—2361no—
416C2×C8—2361no—
417C2×C8—2361no—
418C2×C8—2361no—
419C4⋊C4—2362no—
420C4⋊C4—2362no—
421C22⋊C4—1182no—
422C22⋊C4—1182no—
423C22⋊C4—1182no—
424C22⋊C4—1182no—
425C22⋊C4—2362no—
426C22⋊C4—2362no—
427C22⋊C4—2362no—
428C22⋊C4—2362no—
429C42—1182no—
430C42—1182no—
431C42—2361no—
432C42—2361no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—1182no—
438C23—1182no—
439C23—1182no—
440C23—4721no—
441C23—4721no—
442C23—4721no—
443C23—4721no—
444C23—4721no—
445C23—4721no—
446C23—4721no—
447C23—4721no—
448C23—2362no—
449C23—2362no—
450C23—2362no—
451C23—2362no—
452C23—2361no—
453C23—2361no—
454C23—2361no—
455C23—2361no—
456C23—2361no—
457C23—2361no—
458C23—2361no—
459C23—2361no—
460C23C42×D5911yescharacteristic
461C23C22⋊C4×D5911yescharacteristic
462C23#1427368511yescharacteristic
463C23#1427373311yescharacteristic
464C23#1427373311yescharacteristic
465C23#1427373511yescharacteristic
466C23#1427373511yescharacteristic
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—592no—
486C2×C4—592no—
487C2×C4—1181no—
488C2×C4—1181no—
489C2×C4—1181no—
490C2×C4—1181no—
491C2×C4—1181no—
492C2×C4—1181no—
493C2×C4—1182no—
494C2×C4—1181no—
495C2×C4—1181no—
496C2×C4—1181no—
497C2×C4—1181no—
498C2×C4—2362no—
499C2×C4—2362no—
500C2×C4—2362no—
501C2×C4—2362no—
502C2×C4—2362no—
503C2×C4—2361no—
504C2×C4—2361no—
505C2×C4—2362no—
506C2×C4(C22⋊D59)⋊C411yescharacteristic
507C2×C4(C22⋊D59)⋊C411yescharacteristic
508C2×C4#1427368911yescharacteristic
509C2×C4#1427368911yescharacteristic
510C8—42no—
511C8—42no—
512C8—2362no—
513C8—2362no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—4722no—
527C22—4722no—
528C22—4722no—
529C22—4722no—
530C22—4721no—
531C22—4721no—
532C22—4721no—
533C22—4721no—
534C22—2362no—
535C22—2362no—
536C22—2362no—
537C22#1683724911yescharacteristic
538C22#1683735611yescharacteristic
539C22#1683735911yescharacteristic
540C22#1683736711yescharacteristic
541C22#1683736711yescharacteristic
542C22#99×D5911yescharacteristic
543C22#1683737411yescharacteristic
544C22#100×D5911yescharacteristic
545C22#1683738511yescharacteristic
546C22#1683746211yescharacteristic
547C22#1683746211yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—81no—
553C4—81no—
554C4—1182no—
555C4—1182no—
556C4—2362no—
557C4—2361no—
558C4—2361no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—4722no—
564C2#2142244311yescharacteristic
565C2#2142244611yescharacteristic
566C2#2142245411yescharacteristic
567C2#2142246411yescharacteristic
568C2#2142386111yescharacteristic
569C2#2142388311yescharacteristic
570C2#2142388711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC23×C59C2×C42
FrattiniC23×C4C22×D59
FittingC59×#1280C2
socleC23×C59C2×C22⋊C4
prime radicalC24×D59C2×C4
cosocleC23×C4×C59C23
hypercenter#1280D59
hypocenterC59#8155
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
2. #21423860C2semidirect
3. C22⋊C8×Dic59C2semidirect
6. #21421939C2semidirect
7. C59×#1280C2semidirect
24. #16837125C4semidirect
25. #16837125C4semidirect
280. #1280D59semidirect
314. C59#8155semidirect

Maximal subgroups

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

subgroupindex
C2×#168372912
#214238602
C22⋊C8×Dic592
C2×#168371252
#214219392
C59×#12802
#815559