Finite Group Databasebeta

← #28605362

Subgroups — #28605362

Order 15104.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838112C211yesmaximal, characteristic
2#21427053C211yesmaximal, characteristic, complemented
3#21425069C211yesmaximal, characteristic, complemented
4#21425065C211yesmaximal, characteristic, complemented
5#21425061C211yesmaximal, characteristic, complemented
6#21425051C211yesmaximal, characteristic, complemented
7#21425051C211yesmaximal, characteristic, complemented
8#21425047C211yesmaximal, characteristic, complemented
9#21424107C211yesmaximal, characteristic
10#21424107C211yesmaximal, characteristic
11#21424097C211yesmaximal, characteristic
12#21423922C211yesmaximal, characteristic
13#21423922C211yesmaximal, characteristic
14C4×#14273649C211yesmaximal, characteristic
15C2×C59×#446C211yesmaximal, characteristic, complemented
16#16838112—24no—
17#16838112C418yescomplemented
18C2×#14273727C2211yescharacteristic
19C2×#14273719C2211yescharacteristic
20C2×#14273719C2211yescharacteristic
21C2×#14273705—21no—
22C2×#14273705—21no—
23C2×#14273703—21no—
24C2×C4×D236C2211yescharacteristic
25C2×C4×D236C2211yescharacteristic
26C2×C4×D236C2211yescharacteristic
27C2×C4⋊C4×D59—21no—
28C2×#14273649C2211yescharacteristic
29C2×#14273649C2211yescharacteristic
30C2×#14273649C2211yescharacteristic
31C2×#14273649C2211yescharacteristic
32#16837309—21no—
33#16837288—21no—
34#16837288—21no—
35#16837272C2211yescharacteristic
36#16837268C2211yescharacteristic
37#16837268C2211yescharacteristic
38C4⋊C4×Dic59—21no—
39C4×C4⋊Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837165C2211yescharacteristic
54C2×C4×Q8×C59C2211yescharacteristic
55C59×#446—24no—
56C2×C4⋊C8×C59C2211yescharacteristic
57C2×C4⋊C8×C59C2211yescharacteristic
58C2×C4⋊C8×C59C2211yescharacteristic
59C2×#97×C59C2211yescharacteristic
60C2×#97×C59C2211yescharacteristic
61C2×C4×C8×C59C2211yescharacteristic
62C22×Dic118—21no—
63C22×D236C2311yescharacteristic
64C22×C4×D59—21no—
65C22×C4×D59—21no—
66C22×C4×D59—21no—
67#14273727—22no—
68#14273727C2×C414yes—
69#14273719—22no—
70#14273719—22no—
71#14273719C2×C414yes—
72#14273719C2×C414yes—
73#14273705—28no—
74#14273705—28no—
75#14273703—28no—
76C4×D236—22no—
77C4×D236—22no—
78C4×D236—22no—
79C4×D236C2×C414yes—
80C4×D236C2×C414yes—
81C4×D236C2×C414yes—
82C2×C4⋊Dic59—21no—
83C2×C4⋊Dic59—21no—
84C2×C4⋊Dic59—21no—
85C2×C4⋊Dic59C2311yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×C4⋊Dic59C2311yescharacteristic
88C2×C4⋊Dic59C2311yescharacteristic
89C2×C4⋊Dic59C2311yescharacteristic
90C2×C4⋊Dic59C2311yescharacteristic
91C2×C4⋊Dic59C2311yescharacteristic
92C2×Dic59⋊C4—21no—
93C2×Dic59⋊C4—21no—
94C2×Dic59⋊C4—21no—
95C2×Dic59⋊C4—21no—
96C4⋊C4×D59—28no—
97C2×(C2×D59)⋊C4—21no—
98C2×(C2×D59)⋊C4—21no—
99C2×(C2×D59)⋊C4—21no—
100C2×C4×Dic59—21no—
101C2×C4×Dic59—21no—
102C2×C4×Dic59—21no—
103#14273649—22no—
104#14273649—22no—
105#14273649—22no—
106#14273649Q814yescomplemented
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C22×Q8×C59D411yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C22×C8×C59C2311yescharacteristic
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C4×Q8×C59C2×C414yes—
120C4×Q8×C59D414yes—
121C2×C4⋊C4×C59D411yescharacteristic
122C2×C4⋊C4×C59D411yescharacteristic
123C2×C4⋊C4×C59D411yescharacteristic
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126C2×C42×C59C2311yescharacteristic
127C4⋊C8×C59—22no—
128C4⋊C8×C59—22no—
129C4⋊C8×C59—22no—
130#97×C59—22no—
131#97×C59—22no—
132C4×C8×C59—22no—
133C23×D59—21no—
134C2×Q8×C59—41no—
135C2×Q8×C59—41no—
136C2×Q8×C59C22⋊C414yes—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236C22×C412yes—
155C2×D236C2×Q812yes—
156C2×D236#4112yes—
157C2×C4×D59—24no—
158C2×C4×D59—24no—
159C2×C4×D59—24no—
160C2×C4×D59—42no—
161C2×C4×D59—42no—
162C2×C4×D59—81no—
163C2×C4×D59—81no—
164C2×C4×D59—81no—
165C2×C4×D59—81no—
166C2×Dic118—24no—
167C2×Dic118—81no—
168C2×Dic118—81no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59—21no—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59C2×Q812yes—
181C4⋊C4×C59C22⋊C414yes—
182C4⋊C4×C59C22⋊C414yes—
183C4⋊C4×C59C22⋊C414yes—
184C42×C59C22×C412yes—
185C42×C59C22×C412yes—
186C42×C59C22×C412yes—
187C42×C59C2×D412yes—
188C42×C59C2×D412yes—
189C42×C59C2×D412yes—
190(C2×D59)⋊C4—24no—
191(C2×D59)⋊C4—24no—
192(C2×D59)⋊C4—24no—
193C4⋊Dic59—24no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—42no—
205C4⋊Dic59—81no—
206C4⋊Dic59C22×C412yes—
207C4⋊Dic59C22×C412yes—
208C4⋊Dic59C22×C412yes—
209C4⋊Dic59C2×Q812yes—
210Dic59⋊C4—24no—
211Dic59⋊C4—24no—
212Dic59⋊C4—24no—
213Dic59⋊C4—24no—
214C4×Dic59—42no—
215C4×Dic59—81no—
216C4×Dic59—81no—
217C23×C59C22×D411yescharacteristic
218C22×D59—22no—
219C22×D59—22no—
220C22×D59—22no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225Q8×C59—44no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
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—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59C2×C22⋊C412yes—
239C2×C4×C59C2×C22⋊C412yes—
240C2×C4×C59C2×C22⋊C412yes—
241C2×C4×C59C2×C22⋊C412yes—
242C2×C4×C59C2×C22⋊C412yes—
243C2×C4×C59C2×C22⋊C412yes—
244C2×C4×C59C23×C411yescharacteristic
245C2×C4×C59C22×D411yescharacteristic
246C2×C4×C59C22×Q811yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C592−1+411yescharacteristic
249C2×C4×C592−1+411yescharacteristic
250C2×Dic59—22no—
251C2×Dic59—22no—
252C2×Dic59—22no—
253C2×Dic59—22no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271D236—22no—
272D236—22no—
273D236—22no—
274D236C4×Q814yes—
275C4×D59—48no—
276C4×D59—84no—
277C4×D59—84no—
278Dic118—84no—
279C8×C59—22no—
280C8×C59—41no—
281C8×C59—41no—
282C8×C59—41no—
283#22526—591nomaximal, Sylow
284C22×C59C22×C22⋊C411yescharacteristic
285C22×C59#54811yescharacteristic
286C22×C59D4×Q811yescharacteristic
287C22×C59#57511yescharacteristic
288C22×C59#57511yescharacteristic
289C22×C59#57611yescharacteristic
290C22×C59#57711yescharacteristic
291C2×D59—24no—
292C2×D59—42no—
293C2×D59—42no—
294C2×D59—42no—
295C2×D59—42no—
296C2×D59—42no—
297C2×D59—42no—
298C4×C59—22no—
299C4×C59—22no—
300C4×C59—22no—
301C4×C59—22no—
302C4×C59—22no—
303C4×C59—22no—
304C4×C59C2×C4×Q811yescharacteristic
305C4×C59#51911yescharacteristic
306C4×C59#54811yescharacteristic
307C4×C59D4×Q811yescharacteristic
308Dic59—42no—
309Dic59—42no—
310Dic59—42no—
311Dic59—42no—
312Dic59—81no—
313Dic59—81no—
314Dic59—81no—
315C2×#554—591no—
316C2×#446D5911yescharacteristic, complemented
317#2355—591no—
318#1993—591no—
319#1992—591no—
320#1991—591no—
321#1989—591no—
322#1989—591no—
323#1988—591no—
324#1804—591no—
325#1804—591no—
326#1802—591no—
327#1762—591no—
328#1762—591no—
329C4×#102—591no—
330C2×C59Q8×C22⋊C411yescharacteristic
331C2×C59#289611yescharacteristic
332C2×C59#289711yescharacteristic
333C2×C59#337011yescharacteristic
334C2×C59#337611yescharacteristic
335C2×C59#338811yescharacteristic
336C2×C59#338911yescharacteristic
337D59—44no—
338#554—598no—
339#554—1184no—
340C2×C4⋊Q8—591no—
341C2×#125—591no—
342C2×#125—591no—
343C2×C22⋊Q8—1181no—
344C2×C22⋊Q8—1181no—
345C2×C22⋊Q8—1181no—
346C2×C4×Q8C2×D5911yescharacteristic
347C2×C4×D4—591no—
348C2×C4×D4—591no—
349C2×C4×D4—591no—
350C22×C4⋊C4—1181no—
351#446—24no—
352C2×C4⋊C8C2×D5911yescharacteristic
353C2×C4⋊C8C2×D5911yescharacteristic
354C2×C4⋊C8C2×D5911yescharacteristic
355C2×#102—591no—
356C2×#102—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#97C2×D5911yescharacteristic
360C2×#97C2×D5911yescharacteristic
361C2×C4×C8C2×D5911yescharacteristic
362#391—1181no—
363#387—1181no—
364#387—1181no—
365#384—591no—
366#382—591no—
367#382—591no—
368C4×C4⋊C4—591no—
369C4×C4⋊C4—591no—
370C4×C4⋊C4—591no—
371C4×C4⋊C4—1181no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384C59#2252611yescharacteristic, complemented, Sylow
385C22×Q8—1181no—
386C22×Q8D23611yescharacteristic
387C22×D4—591no—
388C23×C4—1181no—
389C23×C4—1181no—
390C23×C4—1181no—
391C22×C8C22×D5911yescharacteristic
392C22×C8C22×D5911yescharacteristic
393C22×C8C22×D5911yescharacteristic
394C22×C8C22×D5911yescharacteristic
395C4⋊Q8—594no—
396C4⋊Q8—1182no—
397#125—594no—
398#125—594no—
399#125—1182no—
400#125—1182no—
401C22⋊Q8—1188no—
402C22⋊Q8—1188no—
403C22⋊Q8—1188no—
404C4×Q8C4×D5914yes—
405C4×Q8C22⋊D5914yes—
406C4×D4—594no—
407C4×D4—594no—
408C4×D4—594no—
409C4×D4—1182no—
410C4×D4—1182no—
411C4×D4—1182no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1188no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4D23611yescharacteristic
428C2×C4⋊C4D23611yescharacteristic
429C2×C4⋊C4D23611yescharacteristic
430C2×C22⋊C4—1181no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C42—1181no—
434C2×C42—1181no—
435C2×C42—1181no—
436C2×C42C22×D5911yescharacteristic
437C2×C42C22×D5911yescharacteristic
438C2×C42C22×D5911yescharacteristic
439C4⋊C8—22no—
440C4⋊C8—22no—
441C4⋊C8—22no—
442#102—594no—
443#102—1182no—
444#102—1182no—
445#102—1182no—
446#97—22no—
447#97—22no—
448C4×C8—22no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456C24—1181no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—1184no—
460C2×Q8—4721no—
461C2×Q8—4721no—
462C2×Q8(C2×D59)⋊C414yes—
463C2×D4—592no—
464C2×D4—592no—
465C2×D4—592no—
466C2×D4—1181no—
467C2×D4—1181no—
468C2×D4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—1184no—
474C22×C4—1184no—
475C22×C4—1184no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—4721no—
480C22×C4—4721no—
481C22×C4—4721no—
482C22×C4—4721no—
483C22×C4—2362no—
484C22×C4—2362no—
485C22×C4C2×D23611yescharacteristic
486C22×C4C2×D23611yescharacteristic
487C22×C4C2×D23611yescharacteristic
488C22×C4C2×D23611yescharacteristic
489C22×C4C2×D23611yescharacteristic
490C22×C4C2×D23611yescharacteristic
491C22×C4C23×D5911yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8Q8×D5912yes—
504C4⋊C4—592no—
505C4⋊C4—592no—
506C4⋊C4—592no—
507C4⋊C4—592no—
508C4⋊C4—1184no—
509C4⋊C4—1184no—
510C4⋊C4—1184no—
511C4⋊C4—1184no—
512C4⋊C4—1184no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—4721no—
524C4⋊C4—2362no—
525C4⋊C4(C2×D59)⋊C414yes—
526C4⋊C4(C2×D59)⋊C414yes—
527C4⋊C4(C2×D59)⋊C414yes—
528C22⋊C4—1184no—
529C22⋊C4—1184no—
530C22⋊C4—1184no—
531C42—4721no—
532C42—4721no—
533C42—2362no—
534C42C2×C4×D5912yes—
535C42C2×C4×D5912yes—
536C42C2×C4×D5912yes—
537C42C2×C22⋊D5912yes—
538C42C2×C22⋊D5912yes—
539C42C2×C22⋊D5912yes—
540C23—1182no—
541C23—1182no—
542C23—1182no—
543C23—2361no—
544C23—2361no—
545C23—2361no—
546C23—2361no—
547C23C22×D23611yescharacteristic
548Q8—44no—
549Q8—4724no—
550D4—594no—
551D4—1182no—
552D4—1182no—
553D4—1182no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—1182no—
567C2×C4—1182no—
568C2×C4—1182no—
569C2×C4—1182no—
570C2×C4—4724no—
571C2×C4—4724no—
572C2×C4—2368no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
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×C4C2×(C2×D59)⋊C412yes—
591C2×C4C2×(C2×D59)⋊C412yes—
592C2×C4C2×(C2×D59)⋊C412yes—
593C2×C4C2×(C2×D59)⋊C412yes—
594C2×C4C2×(C2×D59)⋊C412yes—
595C2×C4C2×(C2×D59)⋊C412yes—
596C2×C4C22×C4×D5911yescharacteristic
597C2×C4C22×C22⋊D5911yescharacteristic
598C2×C4C2×Q8×D5911yescharacteristic
599C2×C4#41×D5911yescharacteristic
600C2×C4#1427377411yescharacteristic
601C2×C4#1427377511yescharacteristic
602C8—22no—
603C8—41no—
604C8—41no—
605C8—41no—
606C22—1184no—
607C22—2362no—
608C22—2362no—
609C22—2362no—
610C22—2362no—
611C22—2362no—
612C22—2362no—
613C22C22×(C2×D59)⋊C411yescharacteristic
614C22#1683806411yescharacteristic
615C22Q8×D23611yescharacteristic
616C22#1683820211yescharacteristic
617C22#1683820211yescharacteristic
618C22#1683820711yescharacteristic
619C22#1683821411yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—4721no—
627C4—4721no—
628C4—4721no—
629C4—2362no—
630C4—2362no—
631C4—2362no—
632C4—2362no—
633C4C4×Q8×D5911yescharacteristic
634C4#1683787811yescharacteristic
635C4#1683805911yescharacteristic
636C4Q8×C22⋊D5911yescharacteristic
637C2—2364no—
638C2Q8×(C2×D59)⋊C411yescharacteristic
639C2#2143261611yescharacteristic
640C2#2143262311yescharacteristic
641C2#2143685111yescharacteristic
642C2#2143692511yescharacteristic
643C2#2143704311yescharacteristic
644C2#2143705611yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#446C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenterC2×#446D59
hypocenterC59#22526
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #21427053C2semidirect
3. #21425069C2semidirect
4. #21425065C2semidirect
5. #21425061C2semidirect
6. #21425051C2semidirect
7. #21425051C2semidirect
8. #21425047C2semidirect
15. C2×C59×#446C2semidirect
17. #16838112C4semidirect
106. #14273649Q8semidirect
316. C2×#446D59semidirect
384. C59#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#168381122
#214270532
#214250692
#214250652
#214250612
#214250512
#214250472
#214241072
#214240972
#214239222
C4×#142736492
C2×C59×#4462
#2252659