Finite Group Databasebeta

← #28618740

Subgroups — #28618740

Order 15104.

752 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273706C211yesmaximal, characteristic
2#21426494C211yesmaximal, characteristic
3#21425665C211yesmaximal, characteristic, complemented
4#21425643C211yesmaximal, characteristic, complemented
5#21425579C211yesmaximal, characteristic, complemented
6#21425529C211yesmaximal, characteristic
7#21425519C211yesmaximal, characteristic
8#21425516C211yesmaximal, characteristic, complemented
9#21425300C211yesmaximal, characteristic
10#21425293C211yesmaximal, characteristic
11C4×#14273661C211yesmaximal, characteristic, complemented
12C4×#14273659C211yesmaximal, characteristic, complemented
13C4×#14273652C211yesmaximal, characteristic, complemented
14C4×#14273652C211yesmaximal, characteristic, complemented
15C4×C22⋊C8×C59C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273661C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273652C2211yescharacteristic
23C2×#14273652C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26#16837308C2211yescharacteristic
27#16837300C2211yescharacteristic
28#16837286—21no—
29#16837275—21no—
30#16837270—21no—
31#16837268C2211yescharacteristic
32#16837268C2211yescharacteristic
33#16837265—21no—
34#16837262C2211yescharacteristic
35#16837262C2211yescharacteristic
36#16837262C2211yescharacteristic
37C4×Dic59⋊C4—21no—
38C4×C4⋊Dic59C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C22⋊Dic59—21no—
42C4×C22⋊Dic59—21no—
43#16837248—21no—
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C22×C42×C59C2211yescharacteristic
53C2×C22⋊C8×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60#14273706—22no—
61#14273706—22no—
62C4×Dic118—22no—
63C4×Dic118—22no—
64C2×C4⋊Dic59C2311yescharacteristic
65C2×C4⋊Dic59C2311yescharacteristic
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C22⋊Dic59—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82#14273661—22no—
83#14273661—22no—
84#14273659—22no—
85#14273659—22no—
86#14273652—22no—
87#14273652—22no—
88#14273652—22no—
89#14273652—22no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C23×C4×C59D411yescharacteristic
104C23×C4×C59D411yescharacteristic
105C23×C4×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C42×C59—22no—
111C2×C42×C59—22no—
112C2×C42×C59D411yescharacteristic
113C2×C42×C59D411yescharacteristic
114C2×C42×C59C2311yescharacteristic
115C2×C42×C59C2311yescharacteristic
116C22⋊C8×C59—22no—
117C22⋊C8×C59—22no—
118C4×C8×C59—22no—
119C4×C8×C59—22no—
120C24×C59C2×D411yescharacteristic
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—41no—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118#4112yes—
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—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59#4112yes—
164C2×C8×C59#4112yes—
165C42×C59—22no—
166C42×C59—22no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59#4112yes—
170C42×C59#4112yes—
171C22⋊Dic59—24no—
172C22⋊Dic59—42no—
173C22⋊Dic59—42no—
174C22⋊Dic59—42no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59#4112yes—
189Dic59⋊C4—24no—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
192C4×Dic59—42no—
193C4×Dic59—42no—
194C4×Dic59—42no—
195C4×Dic59—42no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59#13512yes—
203C23×C59C22×D411yescharacteristic
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—44no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—44no—
221C2×C4×C59#12312yes—
222C2×C4×C59#12312yes—
223C2×C4×C59#13512yes—
224C2×C4×C59#13712yes—
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×Dic59—22no—
232C2×Dic59—22no—
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—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255Dic118—22no—
256Dic118—22no—
257Dic118—41no—
258Dic118—41no—
259C8×C59—22no—
260C8×C59—22no—
261C8×C59—41no—
262C8×C59—41no—
263#23759—591nomaximal, Sylow
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59C2×#12311yescharacteristic
272C22×C59#52511yescharacteristic
273C22×C59#53211yescharacteristic
274C22×C59C2×#13511yescharacteristic
275C22×C59C2×#13511yescharacteristic
276C22×C59C2×#13711yescharacteristic
277C22×C59#57511yescharacteristic
278C4×C59—22no—
279C4×C59—41no—
280C4×C59—41no—
281C4×C59—41no—
282C4×C59#48514yes—
283C4×C59#52911yescharacteristic
284C4×C59#52911yescharacteristic
285C4×C59#52911yescharacteristic
286C4×C59#52911yescharacteristic
287Dic59—42no—
288Dic59—42no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292Dic59—81no—
293Dic59—81no—
294Dic59—81no—
295C4×C22⋊Q8—591no—
296#2283—591no—
297#2101—591no—
298#2097—591no—
299#2085—591no—
300#2077—591no—
301#2075—591no—
302#2074—591no—
303#2036—591no—
304#2035—591no—
305C4×#107—591no—
306C4×#106—591no—
307C4×#103—591no—
308C4×#103—591no—
309C4×C22⋊C8D5911yescharacteristic
310C2×C59—42no—
311C2×C59#239511yescharacteristic
312C2×C59C2×#48511yescharacteristic
313C2×C59#309111yescharacteristic
314C2×C59#310211yescharacteristic
315C2×C59#310811yescharacteristic
316C2×C59#316811yescharacteristic
317C2×C59#317211yescharacteristic
318C2×C22⋊Q8—591no—
319C2×C4×Q8—591no—
320C22×C42C2×D5911yescharacteristic
321C2×#107—591no—
322C2×#107—591no—
323C2×#106—591no—
324C2×#106—591no—
325C2×#103—591no—
326C2×#103—591no—
327C2×#103—591no—
328C2×#103—591no—
329C2×C22⋊C8C2×D5911yescharacteristic
330C2×C22⋊C8C2×D5911yescharacteristic
331C2×C4×C8C2×D5911yescharacteristic
332C2×C4×C8C2×D5911yescharacteristic
333#391—591no—
334#389—591no—
335#387—1181no—
336#385—1181no—
337#383—1181no—
338#382—591no—
339#382—591no—
340#381—1181no—
341#380—591no—
342#380—591no—
343#380—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C4⋊C4—591no—
347C4×C4⋊C4—1181no—
348C4×C22⋊C4—1181no—
349C4×C22⋊C4—1181no—
350#376—1181no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#336C2×D5911yescharacteristic
360#336C2×D5911yescharacteristic
361C59#2375911yescharacteristic, complemented, Sylow
362C22×Q8—591no—
363C23×C4D23611yescharacteristic
364C23×C4D23611yescharacteristic
365C23×C4C22×D5911yescharacteristic
366C22×C8C22×D5911yescharacteristic
367C22×C8C22×D5911yescharacteristic
368C22×C8C22×D5911yescharacteristic
369C22×C8C22×D5911yescharacteristic
370C22⋊Q8—1182no—
371C22⋊Q8—1182no—
372C4×Q8—1182no—
373C4×Q8—1182no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1181no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—1181no—
391C2×C42—1181no—
392C2×C42—1181no—
393C2×C42—1181no—
394C2×C42D23611yescharacteristic
395C2×C42D23611yescharacteristic
396C2×C42C22×D5911yescharacteristic
397C2×C42C22×D5911yescharacteristic
398#107—1182no—
399#107—1182no—
400#106—1182no—
401#106—1182no—
402#103—1182no—
403#103—1182no—
404#103—1182no—
405#103—1182no—
406C22⋊C8—22no—
407C22⋊C8—22no—
408C4×C8—22no—
409C4×C8—22no—
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#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423C24C2×D23611yescharacteristic
424C2×Q8—592no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C2×Q8—1181no—
428C2×Q8—1181no—
429C2×Q8—1181no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—41no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4C2×D23611yescharacteristic
451C22×C4C2×D23611yescharacteristic
452C22×C4C2×D23611yescharacteristic
453C22×C4C2×D23611yescharacteristic
454C22×C4C2×D23611yescharacteristic
455C22×C4C23×D5911yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8#1174568512yes—
467C2×C8#1174568512yes—
468C4⋊C4—592no—
469C4⋊C4—1184no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—2362no—
484C4⋊C4—2362no—
485C22⋊C4—1184no—
486C22⋊C4—2362no—
487C22⋊C4—2362no—
488C22⋊C4—2362no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—2362no—
494C42—2362no—
495C42—2362no—
496C42—2362no—
497C42#1174568712yes—
498C42#1174568712yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#1427374912yes—
506C23C22×D23611yescharacteristic
507Q8—1182no—
508Q8—1182no—
509Q8—2361no—
510Q8—2361no—
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—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—1182no—
529C2×C4—1182no—
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—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4#1427371112yes—
553C2×C4#1427371112yes—
554C2×C4#1427374912yes—
555C2×C4#1427376012yes—
556C2×C4#41×D5911yescharacteristic
557C2×C4#41×D5911yescharacteristic
558C2×C4C2×#1174568711yescharacteristic
559C2×C4C2×#1174568711yescharacteristic
560C2×C4C2×#1174568511yescharacteristic
561C2×C4C2×#1174568511yescharacteristic
562C8—22no—
563C8—22no—
564C8—41no—
565C8—41no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#1427371111yescharacteristic
574C22#1683791411yescharacteristic
575C22#1683795211yescharacteristic
576C22C2×#1427374911yescharacteristic
577C22C2×#1427374911yescharacteristic
578C22C2×#1427376011yescharacteristic
579C22#1683820211yescharacteristic
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—4721no—
585C4—4721no—
586C4—4721no—
587C4—4721no—
588C4—4721no—
589C4—4721no—
590C4—2362no—
591C4—2362no—
592C4#1683774414yes—
593C4#1683793211yescharacteristic
594C4#1683793211yescharacteristic
595C4#1683793411yescharacteristic
596C4#1683793411yescharacteristic
597C2—42no—
598C2#2142735611yescharacteristic
599C2C2×#1683774411yescharacteristic
600C2#2143401111yescharacteristic
601C2#2143411611yescharacteristic
602C2#2143417811yescharacteristic
603C2#2143466611yescharacteristic
604C2#2143470211yescharacteristic
subgroupG/H
centerC22×C4C2×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×C22⋊C8×C59C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenterC4×C22⋊C8D59
hypocenterC59#23759
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. #21425665C2semidirect
4. #21425643C2semidirect
5. #21425579C2semidirect
8. #21425516C2semidirect
11. C4×#14273661C2semidirect
12. C4×#14273659C2semidirect
13. C4×#14273652C2semidirect
14. C4×#14273652C2semidirect
361. C59#23759semidirect

Maximal subgroups

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

subgroupindex
C4×#142737062
#214264942
#214256652
#214256432
#214255792
#214255292
#214255192
#214255162
#214253002
#214252932
C4×#142736612
C4×#142736592
C4×#142736522
C4×C22⋊C8×C592
#2375959