Finite Group Databasebeta

← #28618065

Subgroups — #28618065

Order 15104.

844 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837507C211yesmaximal, characteristic, complemented
2C2×C4×#11745670C211yesmaximal, characteristic
3#21426748C211yesmaximal, characteristic, complemented
4#21425573C211yesmaximal, characteristic, complemented
5#21425505C211yesmaximal, characteristic, complemented
6#21425477C211yesmaximal, characteristic, complemented
7#21425470C211yesmaximal, characteristic
8#21425418C211yesmaximal, characteristic, complemented
9#21425413C211yesmaximal, characteristic
10#21425332C211yesmaximal, characteristic
11#21425190C211yesmaximal, characteristic
12#21423953C211yesmaximal, characteristic, complemented
13#21423921C211yesmaximal, characteristic
14C4×#14273651C211yesmaximal, characteristic, complemented
15C59×#2326C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745670C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19#16837507—24no—
20C4×#11745670—24no—
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×#14273638C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35#16837308C2211yescharacteristic
36#16837308C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837284—21no—
40#16837284—21no—
41#16837269—21no—
42#16837266—21no—
43C4×C4⋊Dic59C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#124×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×C4×D4×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#388—21no—
56C59×#388—21no—
57C59×#385—21no—
58C59×#385—21no—
59C59×#384C2211yescharacteristic
60C59×#376C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745672—22no—
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745670—22no—
66C2×#11745670—22no—
67C2×#11745670—22no—
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C4×Dic118—22no—
73C4×Dic118D414yes—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82#14273658—22no—
83#14273658—22no—
84C4⋊(C59⋊C8)—22no—
85C4⋊(C59⋊C8)—22no—
86#14273651—22no—
87#14273651—22no—
88#14273651—22no—
89#14273650—22no—
90#14273650—22no—
91#14273648—22no—
92#14273648—22no—
93#14273648—22no—
94#14273638—22no—
95C4×C59⋊C8—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106#124×C59—22no—
107#124×C59—22no—
108C4×Q8×C59—22no—
109C4×Q8×C59—22no—
110C4×D4×C59—22no—
111C4×D4×C59—22no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C42×C59—21no—
123C2×C42×C59C2311yescharacteristic
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#95×C59—21no—
127#95×C59—21no—
128#95×C59D411yescharacteristic
129#95×C59D411yescharacteristic
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59#4112yes—
139C2×D4×C59—21no—
140C2×D4×C59—21no—
141C2×D4×C59—21no—
142C2×D4×C59—21no—
143C2×D4×C59—21no—
144C2×D4×C59#4112yes—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—42no—
149C22×C4×C59—81no—
150C22×C4×C59—81no—
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118C2×D412yes—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59—81no—
174C4⋊C4×C59—81no—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—24no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—81no—
179C22⋊C4×C59—81no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—42no—
185C42×C59C2×D412yes—
186C42×C59#4112yes—
187#11745672—28no—
188#11745670—28no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59C2×D412yes—
195Dic59⋊C4—24no—
196Dic59⋊C4—81no—
197Dic59⋊C4—81no—
198C4×Dic59—24no—
199C4×Dic59—81no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8#4112yes—
211C2×C59⋊C8#4112yes—
212C23×C59—22no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59C22×D411yescharacteristic
220Q8×C59—22no—
221Q8×C59—22no—
222Q8×C59—41no—
223Q8×C59—41no—
224D4×C59—22no—
225D4×C59—22no—
226D4×C59—41no—
227D4×C59—41no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—84no—
254C2×C4×C59#12112yes—
255C2×C4×C59#12112yes—
256C2×C4×C59C22×D411yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×Dic59—22no—
263C2×Dic59—22no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274Dic118—22no—
275Dic118—22no—
276Dic118—41no—
277Dic118—41no—
278C59⋊C8—22no—
279C59⋊C8—22no—
280C59⋊C8—41no—
281C59⋊C8—41no—
282#23694—591nomaximal, Sylow
283C22×C59—42no—
284C22×C59—42no—
285C22×C59—42no—
286C22×C59—81no—
287C22×C59—81no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59C2×#12111yescharacteristic
291C22×C59#52511yescharacteristic
292C22×C59#53211yescharacteristic
293C22×C59#57711yescharacteristic
294C22×C59#57711yescharacteristic
295C22×C59#57711yescharacteristic
296C22×C59#57811yescharacteristic
297C4×C59—22no—
298C4×C59—22no—
299C4×C59—41no—
300C4×C59—41no—
301C4×C59—42no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59—81no—
305C4×C59—81no—
306C4×C59#52511yescharacteristic
307C4×C59#52511yescharacteristic
308C4×C59#52911yescharacteristic
309C4×C59#52911yescharacteristic
310Dic59—42no—
311Dic59—42no—
312Dic59—81no—
313Dic59—81no—
314C2×#441—591no—
315C2×C4×SD16—591no—
316#2326D5911yescharacteristic
317#2320—591no—
318#2084—591no—
319#2072—591no—
320#2066—591no—
321#2065—591no—
322#2056—591no—
323#2055—591no—
324#2042—591no—
325#2015—591no—
326#1768—591no—
327#1762—591no—
328C4×#103—591no—
329C2×C59—82no—
330C2×C59#239211yescharacteristic
331C2×C59#304311yescharacteristic
332C2×C59#304611yescharacteristic
333C2×C59#312011yescharacteristic
334C2×C59#312511yescharacteristic
335C2×C59#317311yescharacteristic
336C2×C59#317811yescharacteristic
337C22×Q16—591no—
338C22×SD16—591no—
339C2×#124C2×D5911yescharacteristic
340C2×C4×Q8—591no—
341C2×C4×Q8C2×D5911yescharacteristic
342C2×C4×D4C2×D5911yescharacteristic
343#441—1184no—
344C4×SD16—1184no—
345C2×#106—591no—
346C2×#106—591no—
347C2×C4⋊C8—591no—
348C2×C4⋊C8—591no—
349C2×#103—591no—
350C2×#103—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×#97—591no—
358C2×C4×C8—591no—
359#391—591no—
360#391—591no—
361#391C2×D5911yescharacteristic
362#390C2×D5911yescharacteristic
363#389—591no—
364#389—591no—
365#388—21no—
366#388—21no—
367#387—1181no—
368#387—1181no—
369#385—21no—
370#385—21no—
371#384C2×D5911yescharacteristic
372#383—1181no—
373#382—1181no—
374C4×C4⋊C4—591no—
375#376C2×D5911yescharacteristic
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#336—591no—
381#336—591no—
382C59#2369411yescharacteristic, complemented, Sylow
383C22×Q8—591no—
384C22×Q8C22×D5911yescharacteristic
385C22×D4C22×D5911yescharacteristic
386C23×C4—21no—
387C2×Q16—1182no—
388C2×Q16—1182no—
389C2×Q16—1182no—
390C2×SD16—1182no—
391C2×SD16—1182no—
392C2×SD16—1182no—
393C22×C8—591no—
394C22×C8—591no—
395C22×C8—591no—
396C22×C8—591no—
397#124—22no—
398#124—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—594no—
402C4×Q8—1182no—
403C4×D4—22no—
404C4×D4—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C42—21no—
422C2×C42—1181no—
423C2×C42—1181no—
424C2×C42C22×D5911yescharacteristic
425C2×C42C22×D5911yescharacteristic
426C2×C42C22×D5911yescharacteristic
427#106—1182no—
428#106—1182no—
429C4⋊C8—1182no—
430C4⋊C8—1182no—
431#103—1182no—
432#103—1182no—
433#103—1182no—
434#103—1182no—
435#103—1182no—
436#102—1182no—
437#102—1182no—
438#102—1182no—
439#97—1182no—
440C4×C8—1182no—
441#95—21no—
442#95—21no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95C22⋊D5911yescharacteristic
451#95C22⋊D5911yescharacteristic
452#95C22⋊D5911yescharacteristic
453#95C22⋊D5911yescharacteristic
454C24—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—592no—
461C2×Q8—1181no—
462C2×Q8—1181no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×Q8#1174568512yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4#1174568512yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490Q16—1188no—
491SD16—1188no—
492C2×C8—592no—
493C2×C8—592no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—592no—
512C4⋊C4—1184no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—4721no—
519C4⋊C4—4721no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C22⋊C4—81no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—42no—
530C42—1184no—
531C42—4721no—
532C42D4×D5912yes—
533C42#1174568712yes—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×C22⋊D5911yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—1182no—
547Q8—1182no—
548Q8—2361no—
549Q8—2361no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—1182no—
581C2×C4—1182no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4#1427369812yes—
593C2×C4#1427369812yes—
594C2×C4C2×D4×D5911yescharacteristic
595C2×C4#41×D5911yescharacteristic
596C2×C4#41×D5911yescharacteristic
597C2×C4C2×#1174568711yescharacteristic
598C2×C4C2×#1174568511yescharacteristic
599C2×C4C2×#1174568511yescharacteristic
600C8—1182no—
601C8—1182no—
602C8—2361no—
603C8—2361no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#1427369811yescharacteristic
612C22#1683791311yescharacteristic
613C22#1683794811yescharacteristic
614C22#1683821011yescharacteristic
615C22#1683821311yescharacteristic
616C22#1683821311yescharacteristic
617C22#1683821711yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—4721no—
628C4—4721no—
629C4—2362no—
630C4—2362no—
631C4#1683791111yescharacteristic
632C4#1683791111yescharacteristic
633C4#1683793211yescharacteristic
634C4#1683793211yescharacteristic
635C2—82no—
636C2#2142733411yescharacteristic
637C2#2143361311yescharacteristic
638C2#2143364211yescharacteristic
639C2#2143431011yescharacteristic
640C2#2143437311yescharacteristic
641C2#2143470711yescharacteristic
642C2#2143475811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2326C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2326D59
hypocenterC59#23694
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. C2×#16837507C2semidirect
3. #21426748C2semidirect
4. #21425573C2semidirect
5. #21425505C2semidirect
6. #21425477C2semidirect
8. #21425418C2semidirect
12. #21423953C2semidirect
14. C4×#14273651C2semidirect
382. C59#23694semidirect

Maximal subgroups

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

subgroupindex
C2×#168375072
C2×C4×#117456702
#214267482
#214255732
#214255052
#214254772
#214254702
#214254182
#214254132
#214253322
#214251902
#214239532
#214239212
C4×#142736512
C59×#23262
#2369459