Finite Group Databasebeta

← #28640573

Subgroups — #28640573

Order 15104.

774 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×#16837526C211yesmaximal, characteristic, complemented
2#21429485C211yesmaximal, characteristic, complemented
3#21429450C211yesmaximal, characteristic
4#21425595C211yesmaximal, characteristic, complemented
5#21425590C211yesmaximal, characteristic, complemented
6#21425482C211yesmaximal, characteristic
7#21425468C211yesmaximal, characteristic
8#21425062C211yesmaximal, characteristic, complemented
9#21424866C211yesmaximal, characteristic, complemented
10#21424827C211yesmaximal, characteristic, complemented
11#21424822C211yesmaximal, characteristic, complemented
12#21424776C211yesmaximal, characteristic
13#21424776C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C2×C59×#478C211yesmaximal, characteristic
16C2×C4×Dic118C2211yescharacteristic
17#16837526—24no—
18C2×#14273661C2211yescharacteristic
19C2×#14273659C2211yescharacteristic
20C2×Dic59⋊C8C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×C4×C59⋊C8C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27#16837338—21no—
28#16837332—21no—
29#16837322—21no—
30#16837314—21no—
31#16837305C2211yescharacteristic
32#16837305C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837298C2211yescharacteristic
36#16837298C2211yescharacteristic
37#16837298C2211yescharacteristic
38#16837298C2211yescharacteristic
39#16837297—21no—
40#16837297—21no—
41#16837275—21no—
42#16837275—21no—
43#16837268C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#125×C59C2211yescharacteristic
54C2×C4×D4×C59C2211yescharacteristic
55C59×#478—24no—
56C2×#107×C59C2211yescharacteristic
57C2×#106×C59C2211yescharacteristic
58C2×C4⋊C8×C59C2211yescharacteristic
59C2×#102×C59C2211yescharacteristic
60C2×#102×C59C2211yescharacteristic
61C22×Dic118D411yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C4×Dic118—22no—
65C4×Dic118—22no—
66C2×C4⋊Dic59D411yescharacteristic
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×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4D411yescharacteristic
78C2×Dic59⋊C4D411yescharacteristic
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59C2311yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86#14273661—22no—
87#14273659—22no—
88Dic59⋊C8—22no—
89Dic59⋊C8—22no—
90#14273648—22no—
91#14273648Q814yescomplemented
92C4×C59⋊C8—22no—
93C8×Dic59—22no—
94C8×Dic59—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108#125×C59—22no—
109#125×C59—22no—
110C4×D4×C59—22no—
111C4×D4×C59—22no—
112C2×C4⋊C4×C59—21no—
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#106×C59—22no—
121C4⋊C8×C59D414yes—
122#102×C59—22no—
123#102×C59—22no—
124C24×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59C2×Q812yes—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—42no—
135C22×C4×C59—81no—
136C22×C4×C59—81no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C2×Dic118—22no—
149C2×Dic118—22no—
150C2×Dic118—22no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59C2×D412yes—
156C2×C8×C59C2×D412yes—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59—42no—
165C22⋊C4×C59—42no—
166C42×C59C2×D412yes—
167C42×C59#4112yes—
168C22⋊Dic59—42no—
169C22⋊Dic59—42no—
170C22⋊Dic59—81no—
171C22⋊Dic59—81no—
172C4⋊Dic59—22no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59C2×Q812yes—
180C4⋊Dic59#4112yes—
181Dic59⋊C4—22no—
182Dic59⋊C4—22no—
183Dic59⋊C4—42no—
184Dic59⋊C4—42no—
185Dic59⋊C4—81no—
186Dic59⋊C4—81no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—81no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8C2×Q812yes—
200C23×C59—22no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59C22×D411yescharacteristic
208D4×C59—22no—
209D4×C59—22no—
210D4×C59—41no—
211D4×C59—41no—
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—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—84no—
226C2×C4×C59#12112yes—
227C2×C4×C59#12112yes—
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C22×Q811yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C592−1+411yescharacteristic
233C2×C4×C592−1+411yescharacteristic
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—22no—
241C2×Dic59—22no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
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—
255C2×Dic59—41no—
256C2×Dic59#13512yes—
257C2×Dic59#13512yes—
258Dic118—28no—
259C8×C59—22no—
260C8×C59—22no—
261C59⋊C8—22no—
262C59⋊C8—41no—
263#25473—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#54811yescharacteristic
273C22×C59D4×Q811yescharacteristic
274C22×C59C2×#13511yescharacteristic
275C22×C59C2×#13511yescharacteristic
276C22×C59#57611yescharacteristic
277C22×C59#57711yescharacteristic
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59#53311yescharacteristic
284C4×C59#54111yescharacteristic
285C4×C59#54711yescharacteristic
286C4×C59D4×Q811yescharacteristic
287Dic59—22no—
288Dic59—22no—
289Dic59—42no—
290Dic59—42no—
291Dic59—41no—
292Dic59—41no—
293Dic59—81no—
294Dic59—81no—
295C2×#478D5911yescharacteristic
296C2×C8×Q8—591no—
297#2584—591no—
298#2580—591no—
299#2088—591no—
300#2087—591no—
301#2067—591no—
302#2065—591no—
303#1991—591no—
304#1949—591no—
305#1942—591no—
306#1941—591no—
307#1934—591no—
308#1934—591no—
309C4×#102—591no—
310C2×C59—82no—
311C2×C59#242911yescharacteristic
312C2×C59#305011yescharacteristic
313C2×C59#306211yescharacteristic
314C2×C59#336111yescharacteristic
315C2×C59#336511yescharacteristic
316C2×C59#338011yescharacteristic
317C2×C59#338111yescharacteristic
318C2×#125C2×D5911yescharacteristic
319C2×C4×Q8—591no—
320C2×C4×D4C2×D5911yescharacteristic
321#478—24no—
322C8×Q8—1184no—
323C2×#107—591no—
324C2×#107C2×D5911yescharacteristic
325C2×#106—591no—
326C2×#106C2×D5911yescharacteristic
327C2×C4⋊C8—591no—
328C2×C4⋊C8—591no—
329C2×C4⋊C8C2×D5911yescharacteristic
330C2×#102—591no—
331C2×#102—591no—
332C2×#102C2×D5911yescharacteristic
333C2×#102C2×D5911yescharacteristic
334C2×C4×C8—591no—
335C2×C4×C8—591no—
336C2×C4×C8—591no—
337#399—1181no—
338#398—1181no—
339#395—1181no—
340#393—1181no—
341#390—591no—
342#390—591no—
343#389—591no—
344#389—591no—
345#389—591no—
346#389—591no—
347#389—591no—
348#389—591no—
349#389—1181no—
350#389—1181no—
351#385—1181no—
352#385—1181no—
353#382—591no—
354C4×C4⋊C4—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363C59#2547311yescharacteristic, complemented, Sylow
364C22×Q8—591no—
365C22×D4C22×D5911yescharacteristic
366C23×C4—21no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8C22×D5911yescharacteristic
370C22×C8C22×D5911yescharacteristic
371#125—22no—
372#125—22no—
373C4×Q8—1182no—
374C4×Q8—1182no—
375C4×D4—22no—
376C4×D4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C42—591no—
401C2×C42—591no—
402C2×C42—1181no—
403C2×C42C22×D5911yescharacteristic
404#107—22no—
405#107—1182no—
406#106—22no—
407#106—1182no—
408C4⋊C8—1182no—
409C4⋊C8—1182no—
410C4⋊C8C22⋊D5914yes—
411#102—22no—
412#102—22no—
413#102—594no—
414#102—1182no—
415C4×C8—1182no—
416C4×C8—1182no—
417C4×C8—1182no—
418#95—1181no—
419#95—1181no—
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—1182no—
429C2×Q8—1182no—
430C2×Q8—1182no—
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—592no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8C2×C22⋊D5912yes—
465C2×C8C2×C22⋊D5912yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—592no—
475C4⋊C4—592no—
476C4⋊C4—1182no—
477C4⋊C4—1182no—
478C4⋊C4—1182no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—4721no—
486C4⋊C4—4721no—
487C4⋊C4—4721no—
488C4⋊C4—4721no—
489C4⋊C4—2362no—
490C4⋊C4—2362no—
491C22⋊C4—42no—
492C22⋊C4—4721no—
493C22⋊C4—4721no—
494C22⋊C4—2362no—
495C22⋊C4—2362no—
496C42—1181no—
497C42—1181no—
498C42—1181no—
499C42—1181no—
500C42—4721no—
501C42#1174568712yes—
502C42C2×C22⋊D5912yes—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D5911yescharacteristic
511Q8—1188no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
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—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—592no—
531C2×C4—592no—
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—1182no—
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#1427369812yes—
555C2×C4#1427369812yes—
556C2×C4C22×C22⋊D5911yescharacteristic
557C2×C4C2×Q8×D5911yescharacteristic
558C2×C4#41×D5911yescharacteristic
559C2×C4C2×#1174568711yescharacteristic
560C2×C4#1427377411yescharacteristic
561C2×C4#1427377511yescharacteristic
562C8—22no—
563C8—22no—
564C8—1182no—
565C8—2361no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#1427369811yescharacteristic
574C22#1683806211yescharacteristic
575C22#1683806511yescharacteristic
576C22#135×D5911yescharacteristic
577C22#135×D5911yescharacteristic
578C22#1683820611yescharacteristic
579C22#1683821211yescharacteristic
580C4—22no—
581C4—22no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—1182no—
586C4—1182no—
587C4—4721no—
588C4—4721no—
589C4—2362no—
590C4—2362no—
591C4—2361no—
592C4—2361no—
593C4#1683795411yescharacteristic
594C4#1683801211yescharacteristic
595C4#1683805111yescharacteristic
596C4Q8×C22⋊D5911yescharacteristic
597C2—82no—
598C2#2142769111yescharacteristic
599C2#2143367311yescharacteristic
600C2#2143377011yescharacteristic
601C2#2143675911yescharacteristic
602C2#2143679511yescharacteristic
603C2#2143695811yescharacteristic
604C2#2143696611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#478C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenterC2×#478D59
hypocenterC59#25473
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×#16837526C2semidirect
2. #21429485C2semidirect
4. #21425595C2semidirect
5. #21425590C2semidirect
8. #21425062C2semidirect
9. #21424866C2semidirect
10. #21424827C2semidirect
11. #21424822C2semidirect
91. #14273648Q8semidirect
363. C59#25473semidirect

Maximal subgroups

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

subgroupindex
C2×#168375262
#214294852
#214294502
#214255952
#214255902
#214254822
#214254682
#214250622
#214248662
#214248272
#214248222
#214247762
C4×#142736482
C2×C59×#4782
#2547359