Finite Group Databasebeta

← #28640432

Subgroups — #28640432

Order 15104.

764 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
1C2×#16837808C211yesmaximal, characteristic, complemented
2#21429292C211yesmaximal, characteristic
3#21427186C211yesmaximal, characteristic, complemented
4#21425580C211yesmaximal, characteristic, complemented
5#21425570C211yesmaximal, characteristic, complemented
6#21425480C211yesmaximal, characteristic
7#21425466C211yesmaximal, characteristic
8#21425051C211yesmaximal, characteristic, complemented
9#21424861C211yesmaximal, characteristic, complemented
10#21424851C211yesmaximal, characteristic, complemented
11#21424817C211yesmaximal, characteristic, complemented
12#21424726C211yesmaximal, characteristic
13#21424719C211yesmaximal, characteristic
14#21423921C211yesmaximal, characteristic
15C2×C59×#474C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17#16837808—24no—
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273638C2211yescharacteristic
27#16837325—21no—
28#16837325—21no—
29#16837309—21no—
30#16837309—21no—
31#16837305C2211yescharacteristic
32#16837305C2211yescharacteristic
33#16837301—21no—
34#16837301C2211yescharacteristic
35#16837301C2211yescharacteristic
36#16837275—21no—
37#16837275—21no—
38#16837271C2211yescharacteristic
39#16837267C2211yescharacteristic
40C4⋊C4×Dic59—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4⋊Q8×C59C2211yescharacteristic
54C2×C4×D4×C59C2211yescharacteristic
55C59×#474—24no—
56C2×#107×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×C4⋊C8×C59C2211yescharacteristic
59C2×#102×C59C2211yescharacteristic
60C2×#102×C59C2211yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273719—22no—
64#14273719—22no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59D411yescharacteristic
68C2×C4⋊Dic59D411yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59C2311yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86#14273660—22no—
87#14273660—22no—
88#14273658—22no—
89#14273658—22no—
90Dic59⋊C8—22no—
91Dic59⋊C8—22no—
92#14273648—22no—
93#14273648Q814yescomplemented
94#14273638—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C22×Q8×C59—21no—
105C22×D4×C59C2311yescharacteristic
106C23×C4×C59—21no—
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C4⋊Q8×C59—22no—
110C4⋊Q8×C59D414yes—
111C4×D4×C59—22no—
112C4×D4×C59—22no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C42×C59C2311yescharacteristic
119#107×C59—22no—
120#107×C59—22no—
121C4⋊C8×C59—22no—
122#102×C59—22no—
123#102×C59—22no—
124C24×C59—21no—
125C2×Q8×C59—24no—
126C2×Q8×C59—81no—
127C2×Q8×C59—81no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59C2×Q812yes—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—42no—
138C22×C4×C59—81no—
139C22×C4×C59—81no—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C4⋊C4×C59—24no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59C2×D412yes—
163C4⋊C4×C59C2×D412yes—
164C22⋊C4×C59—42no—
165C42×C59C2×D412yes—
166C42×C59#4112yes—
167C22⋊Dic59—42no—
168C22⋊Dic59—42no—
169C22⋊Dic59—81no—
170C22⋊Dic59—81no—
171C4⋊Dic59—21no—
172C4⋊Dic59—22no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—41no—
180C4⋊Dic59—81no—
181C4⋊Dic59—81no—
182C4⋊Dic59C2×Q812yes—
183Dic59⋊C4—42no—
184Dic59⋊C4—42no—
185Dic59⋊C4—81no—
186Dic59⋊C4—81no—
187C4×Dic59—21no—
188C4×Dic59—21no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—42no—
192C4×Dic59—81no—
193C4×Dic59—81no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C2×Q812yes—
199C2×C59⋊C8#4112yes—
200C23×C59—22no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59C22×D411yescharacteristic
208Q8×C59—84no—
209D4×C59—22no—
210D4×C59—22no—
211D4×C59—41no—
212D4×C59—41no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—84no—
227C2×C4×C59#12112yes—
228C2×C4×C59#12112yes—
229C2×C4×C59C22×D411yescharacteristic
230C2×C4×C59C22×Q811yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C592−1+411yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—22no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59#13712yes—
259C8×C59—41no—
260C8×C59—41no—
261C59⋊C8—22no—
262C59⋊C8—22no—
263#25462—591nomaximal, Sylow
264C22×C59—42no—
265C22×C59—42no—
266C22×C59—42no—
267C22×C59—81no—
268C22×C59—81no—
269C22×C59—81no—
270C22×C59—81no—
271C22×C59C2×#12111yescharacteristic
272C22×C59#54711yescharacteristic
273C22×C59D4×Q811yescharacteristic
274C22×C59C2×#13711yescharacteristic
275C22×C59#57511yescharacteristic
276C22×C59#57611yescharacteristic
277C22×C59#57611yescharacteristic
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—42no—
281C4×C59—42no—
282C4×C59—81no—
283C4×C59#53311yescharacteristic
284C4×C59#54111yescharacteristic
285C4×C59#54811yescharacteristic
286C4×C59D4×Q811yescharacteristic
287Dic59—22no—
288Dic59—41no—
289Dic59—41no—
290Dic59—41no—
291Dic59—42no—
292Dic59—81no—
293Dic59—81no—
294Dic59—81no—
295C2×#501—591no—
296C2×#474D5911yescharacteristic
297#2566—591no—
298#2369—591no—
299#2085—591no—
300#2083—591no—
301#2067—591no—
302#2065—591no—
303#1989—591no—
304#1949—591no—
305#1947—591no—
306#1940—591no—
307#1926—591no—
308#1925—591no—
309#1762—591no—
310C2×C59—82no—
311C2×C59#242911yescharacteristic
312C2×C59#303711yescharacteristic
313C2×C59#304111yescharacteristic
314C2×C59#334611yescharacteristic
315C2×C59#334711yescharacteristic
316C2×C59#338511yescharacteristic
317C2×C59#338911yescharacteristic
318C2×C4⋊Q8C2×D5911yescharacteristic
319C2×#125—591no—
320C2×C4×D4C2×D5911yescharacteristic
321#501—1184no—
322#474—24no—
323C2×#107—591no—
324C2×#107—591no—
325C2×#107C2×D5911yescharacteristic
326C2×#107C2×D5911yescharacteristic
327C2×#106—591no—
328C2×#106—591no—
329C2×C4⋊C8—591no—
330C2×C4⋊C8—591no—
331C2×C4⋊C8C2×D5911yescharacteristic
332C2×#102—591no—
333C2×#102—591no—
334C2×#102C2×D5911yescharacteristic
335C2×#102C2×D5911yescharacteristic
336C2×#97—591no—
337#396—1181no—
338#396—1181no—
339#391—1181no—
340#391—1181no—
341#390—591no—
342#390—591no—
343#389—591no—
344#389—591no—
345#389—1181no—
346#385—1181no—
347#385—1181no—
348#384—591no—
349#382—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—1181no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#336—591no—
362#336—591no—
363C59#2546211yescharacteristic, complemented, Sylow
364C22×Q8—21no—
365C22×D4C22×D5911yescharacteristic
366C23×C4—21no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8C22×D5911yescharacteristic
370C22×C8C22×D5911yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8C22⋊D5914yes—
373#125—1182no—
374#125—1182no—
375C4×D4—22no—
376C4×D4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4C22×D5911yescharacteristic
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1181no—
395C2×C22⋊C4—1181no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4—1181no—
398C2×C42—591no—
399C2×C42—591no—
400C2×C42—1181no—
401C2×C42—1181no—
402C2×C42—1181no—
403C2×C42C22×D5911yescharacteristic
404#107—22no—
405#107—22no—
406#107—1182no—
407#107—1182no—
408#106—1182no—
409#106—1182no—
410C4⋊C8—22no—
411C4⋊C8—1182no—
412C4⋊C8—1182no—
413#102—22no—
414#102—22no—
415#102—594no—
416#102—1182no—
417#97—1182no—
418#95—591no—
419#95—591no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427C24—21no—
428C2×Q8—24no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D5912yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—591no—
444C22×C4—591no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4D4×D5911yescharacteristic
452C22×C4D4×D5911yescharacteristic
453C22×C4C23×D5911yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—592no—
461C2×C8—592no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—592no—
472C4⋊C4—1182no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—4721no—
481C4⋊C4—4721no—
482C4⋊C4—4721no—
483C4⋊C4—4721no—
484C4⋊C4—2361no—
485C4⋊C4—2362no—
486C4⋊C4—2362no—
487C4⋊C4C2×C22⋊D5912yes—
488C4⋊C4C2×C22⋊D5912yes—
489C22⋊C4—42no—
490C22⋊C4—4721no—
491C22⋊C4—4721no—
492C22⋊C4—2362no—
493C22⋊C4—2362no—
494C42—1181no—
495C42—1181no—
496C42—1181no—
497C42—1181no—
498C42—4721no—
499C42—4721no—
500C42—2362no—
501C42#1174568712yes—
502C42C2×C22⋊D5912yes—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D5911yescharacteristic
511Q8—84no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—592no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1182no—
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—2361no—
553C2×C4—2361no—
554C2×C4#1427369912yes—
555C2×C4#1427369912yes—
556C2×C4C22×C22⋊D5911yescharacteristic
557C2×C4C2×Q8×D5911yescharacteristic
558C2×C4#41×D5911yescharacteristic
559C2×C4C2×#1174568711yescharacteristic
560C2×C4#1427377411yescharacteristic
561C2×C4#1427377511yescharacteristic
562C8—41no—
563C8—41no—
564C8—1182no—
565C8—1182no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#1427369911yescharacteristic
574C22#1683805411yescharacteristic
575C22#1683806511yescharacteristic
576C22#137×D5911yescharacteristic
577C22#1683820111yescharacteristic
578C22#1683820311yescharacteristic
579C22#1683820311yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—81no—
585C4—1182no—
586C4—4721no—
587C4—4721no—
588C4—4721no—
589C4—2362no—
590C4—2361no—
591C4—2361no—
592C4—2361no—
593C4#1683795411yescharacteristic
594C4#1683801211yescharacteristic
595C4#1683805911yescharacteristic
596C4Q8×C22⋊D5911yescharacteristic
597C2—82no—
598C2#2142769711yescharacteristic
599C2#2143356511yescharacteristic
600C2#2143359611yescharacteristic
601C2#2143657811yescharacteristic
602C2#2143660311yescharacteristic
603C2#2143700111yescharacteristic
604C2#2143704911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#474C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenterC2×#474D59
hypocenterC59#25462
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#16837808C2semidirect
3. #21427186C2semidirect
4. #21425580C2semidirect
5. #21425570C2semidirect
8. #21425051C2semidirect
9. #21424861C2semidirect
10. #21424851C2semidirect
11. #21424817C2semidirect
93. #14273648Q8semidirect
363. C59#25462semidirect

Maximal subgroups

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

subgroupindex
C2×#168378082
#214292922
#214271862
#214255802
#214255702
#214254802
#214254662
#214250512
#214248612
#214248512
#214248172
#214247262
#214247192
#214239212
C2×C59×#4742
#2546259