Finite Group Databasebeta

← #28661539

Subgroups — #28661539

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428518C211yesmaximal, characteristic
2#21427121C211yesmaximal, characteristic
3#21425506C211yesmaximal, characteristic, complemented
4#21425490C211yesmaximal, characteristic, complemented
5#21425300C211yesmaximal, characteristic
6#21425300C211yesmaximal, characteristic
7#21424855C211yesmaximal, characteristic, complemented
8#21424829C211yesmaximal, characteristic, complemented
9#21424682C211yesmaximal, characteristic, complemented
10#21424668C211yesmaximal, characteristic, complemented
11#21424367C211yesmaximal, characteristic, complemented
12#21424297C211yesmaximal, characteristic
13#21424192C211yesmaximal, characteristic
14C4×#14273652C211yesmaximal, characteristic, complemented
15C59×#1923C211yesmaximal, characteristic
16C2×#14273720—21no—
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273661C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28#16837342—21no—
29#16837327—21no—
30#16837308C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837296—21no—
35#16837290—21no—
36#16837287—21no—
37#16837284—21no—
38#16837272C2211yescharacteristic
39#16837262C2211yescharacteristic
40#16837262C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43#16837253—21no—
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C2×C4⋊C8×C59C2211yescharacteristic
53C2×C22⋊C8×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C4×C22⋊C4×C59C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60#14273720—44no—
61#14273706—22no—
62#14273706—22no—
63C4×Dic118—22no—
64C4×Dic118—41no—
65#14273686—22no—
66#14273686—41no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89#14273661—22no—
90#14273661—22no—
91#14273661—41no—
92#14273659—22no—
93#14273652—22no—
94#14273652—22no—
95#14273652—22no—
96#14273652—41no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C23×C4×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C2×C22⋊C4×C59D411yescharacteristic
112C2×C22⋊C4×C59D411yescharacteristic
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115C4⋊C8×C59—22no—
116C22⋊C8×C59—22no—
117C22⋊C8×C59—41no—
118C4×C8×C59—41no—
119#95×C59D411yescharacteristic
120#95×C59D411yescharacteristic
121C24×C59C2×D411yescharacteristic
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C2×Dic118—21no—
143C2×Dic118—21no—
144C2×Dic118—21no—
145C2×Dic118—21no—
146C2×Dic118—21no—
147C2×Dic118#4112yes—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59#4112yes—
160C22⋊C4×C59—22no—
161C22⋊C4×C59—22no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—21no—
165C42×C59—21no—
166C22⋊Dic59—24no—
167C22⋊Dic59—42no—
168C22⋊Dic59—42no—
169C22⋊Dic59—42no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—42no—
183C4⋊Dic59—42no—
184C4⋊Dic59#4112yes—
185C4⋊Dic59#4112yes—
186Dic59⋊C4—24no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191C4×Dic59—42no—
192C4×Dic59—42no—
193C4×Dic59—42no—
194C4×Dic59—42no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59C22×D411yescharacteristic
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59#13512yes—
221C2×C4×C59AGL(1,ℤ/8)12yes—
222C2×C4×C59C2×#4111yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C59C2×#4111yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59H(5,𝔽2)11yescharacteristic
227C2×C4×C592−1+411yescharacteristic
228C2×Dic59—22no—
229C2×Dic59—22no—
230C2×Dic59—22no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252Dic118—22no—
253Dic118—22no—
254Dic118—41no—
255Dic118—41no—
256C8×C59—22no—
257C8×C59—41no—
258C8×C59—41no—
259C8×C59—41no—
260#27043—591nomaximal, Sylow
261C22×C59—41no—
262C22×C59—41no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59#52511yescharacteristic
269C22×C59#54611yescharacteristic
270C22×C59#54711yescharacteristic
271C22×C59C2×#13511yescharacteristic
272C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C59#57811yescharacteristic
274C22×C59#57811yescharacteristic
275C4×C59—22no—
276C4×C59—22no—
277C4×C59—41no—
278C4×C59—41no—
279C4×C59—41no—
280C4×C59#52911yescharacteristic
281C4×C59#53911yescharacteristic
282C4×C59#55311yescharacteristic
283C4×C59#55611yescharacteristic
284Dic59—42no—
285Dic59—42no—
286Dic59—42no—
287Dic59—81no—
288Dic59—81no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292#2496—591no—
293#2362—591no—
294#2072—591no—
295#2068—591no—
296#2036—591no—
297#2036—591no—
298#1947—591no—
299#1942—591no—
300#1923D5911yescharacteristic
301#1920—591no—
302#1918—591no—
303#1865—591no—
304#1854—591no—
305#1825—591no—
306C4×#103—591no—
307C2×C59—81no—
308C2×C59#246611yescharacteristic
309C2×C59#310611yescharacteristic
310C2×C59#312711yescharacteristic
311C2×C59#330211yescharacteristic
312C2×C59#330811yescharacteristic
313C2×C59#333611yescharacteristic
314C2×C59#335311yescharacteristic
315C2×#126—1181no—
316C2×C22⋊Q8—591no—
317C2×C4×Q8—591no—
318C2×#117—591no—
319C2×#107—591no—
320C2×#107—591no—
321C2×#107—591no—
322C2×#106—591no—
323C2×C4⋊C8C2×D5911yescharacteristic
324C2×#103—591no—
325C2×#103—591no—
326C2×#103—591no—
327C2×#103—591no—
328C2×C22⋊C8C2×D5911yescharacteristic
329C2×C22⋊C8C2×D5911yescharacteristic
330C2×C4×C8C2×D5911yescharacteristic
331#400—1181no—
332#397—1181no—
333#391—591no—
334#389—591no—
335#389—591no—
336#389—591no—
337#388—1181no—
338#388—1181no—
339#387—1181no—
340#387—1181no—
341#384—591no—
342#380—591no—
343#380—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C4⋊C4—1181no—
347C4×C22⋊C4C2×D5911yescharacteristic
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#336C2×D5911yescharacteristic
357#336C2×D5911yescharacteristic
358C59#2704311yescharacteristic, complemented, Sylow
359C22×Q8—591no—
360C23×C4C22×D5911yescharacteristic
361C22×C8C22×D5911yescharacteristic
362C22×C8C22×D5911yescharacteristic
363C22×C8C22×D5911yescharacteristic
364C22×C8C22×D5911yescharacteristic
365#126—2364no—
366C22⋊Q8—1182no—
367C22⋊Q8—1182no—
368C4×Q8—1182no—
369C4×Q8—2361no—
370#117—1182no—
371#117—2361no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4—1181no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1181no—
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C22⋊C4D23611yescharacteristic
391C2×C22⋊C4D23611yescharacteristic
392C2×C42—1181no—
393C2×C42—1181no—
394C2×C42—1181no—
395C2×C42—1181no—
396C2×C42C22×D5911yescharacteristic
397C2×C42C22×D5911yescharacteristic
398#107—1182no—
399#107—1182no—
400#107—2361no—
401#106—1182no—
402C4⋊C8—22no—
403#103—1182no—
404#103—1182no—
405#103—1182no—
406#103—2361no—
407C22⋊C8—22no—
408C22⋊C8—41no—
409C4×C8—41no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95D23611yescharacteristic
420#95D23611yescharacteristic
421C24C2×D23611yescharacteristic
422C2×Q8—592no—
423C2×Q8—1181no—
424C2×Q8—1181no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C2×Q8—1181no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4C2×D23611yescharacteristic
443C22×C4C2×D23611yescharacteristic
444C22×C4C2×D23611yescharacteristic
445C22×C4C2×D23611yescharacteristic
446C22×C4C2×D23611yescharacteristic
447C22×C4C23×D5911yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8#1174568712yes—
460C4⋊C4—592no—
461C4⋊C4—592no—
462C4⋊C4—1184no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—2362no—
476C4⋊C4—2362no—
477C4⋊C4—2362no—
478C4⋊C4—2362no—
479C4⋊C4—2362no—
480C4⋊C4—2362no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—1184no—
484C22⋊C4—2362no—
485C22⋊C4—2362no—
486C22⋊C4—2362no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—2362no—
492C42—2362no—
493C42—2362no—
494C42—2362no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C22×D23611yescharacteristic
503Q8—1182no—
504Q8—1182no—
505Q8—2361no—
506Q8—2361no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—1182no—
525C2×C4—1182no—
526C2×C4—1182no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4#1427374912yes—
549C2×C4#1427375512yes—
550C2×C4#41×D5911yescharacteristic
551C2×C4#41×D5911yescharacteristic
552C2×C4#41×D5911yescharacteristic
553C2×C4C2×#1174568711yescharacteristic
554C2×C4#1427377211yescharacteristic
555C2×C4#1427377411yescharacteristic
556C8—22no—
557C8—41no—
558C8—41no—
559C8—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1683791411yescharacteristic
568C22#1683804711yescharacteristic
569C22#1683805611yescharacteristic
570C22C2×#1427374911yescharacteristic
571C22C2×#1427375511yescharacteristic
572C22#1683821911yescharacteristic
573C22#1683821911yescharacteristic
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—4721no—
580C4—4721no—
581C4—4721no—
582C4—4721no—
583C4—4721no—
584C4—2362no—
585C4—2362no—
586C4—2362no—
587C4#1683793111yescharacteristic
588C4#1683799511yescharacteristic
589C4#1683809511yescharacteristic
590C4#1683811911yescharacteristic
591C2—81no—
592C2#2142813011yescharacteristic
593C2#2143415711yescharacteristic
594C2#2143439711yescharacteristic
595C2#2143601411yescharacteristic
596C2#2143608911yescharacteristic
597C2#2143646511yescharacteristic
598C2#2143668111yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1923C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1923D59
hypocenterC59#27043
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
3. #21425506C2semidirect
4. #21425490C2semidirect
7. #21424855C2semidirect
8. #21424829C2semidirect
9. #21424682C2semidirect
10. #21424668C2semidirect
11. #21424367C2semidirect
14. C4×#14273652C2semidirect
358. C59#27043semidirect

Maximal subgroups

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

subgroupindex
#214285182
#214271212
#214255062
#214254902
#214253002
#214248552
#214248292
#214246822
#214246682
#214243672
#214242972
#214241922
C4×#142736522
C59×#19232
#2704359