Finite Group Databasebeta

← #28611303

Subgroups — #28611303

Order 15104.

982 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837642C211yesmaximal, characteristic
2C2×#16837494C211yesmaximal, characteristic, complemented
3C2×#107×D59C211yesmaximal, characteristic
4#21427106C211yesmaximal, characteristic
5#21427103C211yesmaximal, characteristic
6#21424956C211yesmaximal, characteristic, complemented
7#21424828C211yesmaximal, characteristic, complemented
8#21424738C211yesmaximal, characteristic, complemented
9#21424402C211yesmaximal, characteristic
10#21424400C211yesmaximal, characteristic
11#21424352C211yesmaximal, characteristic, complemented
12#21424347C211yesmaximal, characteristic, complemented
13C4×#14273660C211yesmaximal, characteristic, complemented
14#21423866C211yesmaximal, characteristic
15C59×#1928C211yesmaximal, characteristic, complemented
16C22×#11745672C2211yescharacteristic
17C22×C8×D59C2211yescharacteristic
18C2×#14273707C2211yescharacteristic
19C2×#14273703C2211yescharacteristic
20C2×Q8×Dic59C2211yescharacteristic
21C2×#14273692C2211yescharacteristic
22C2×C4⋊C4×D59C2211yescharacteristic
23C2×C4⋊C4×D59C2211yescharacteristic
24#16837642C418yescomplemented
25#16837494—24no—
26C2×#14273661C2211yescharacteristic
27C2×#14273660C2211yescharacteristic
28C2×#14273660C2211yescharacteristic
29C2×#14273660C2211yescharacteristic
30#107×D59—24no—
31C2×#14273652C2211yescharacteristic
32C2×#14273652C2211yescharacteristic
33C2×#14273651C2211yescharacteristic
34C2×#14273650C2211yescharacteristic
35C2×(C2×D59)⋊C8C2211yescharacteristic
36C2×C4×C59⋊C8C2211yescharacteristic
37C2×C8×Dic59C2211yescharacteristic
38#16837308C2211yescharacteristic
39#16837301C2211yescharacteristic
40#16837300—21no—
41#16837299—21no—
42#16837298C2211yescharacteristic
43#16837287—21no—
44#16837284—21no—
45#16837276—21no—
46#16837276—21no—
47C4×C4⋊Dic59C2211yescharacteristic
48C4×(C2×D59)⋊C4—21no—
49C4×(C2×D59)⋊C4C2211yescharacteristic
50#16837249—21no—
51#16837165C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837156C2211yescharacteristic
55C2×#107×C59C2211yescharacteristic
56C2×#103×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C59×#391C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#336C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C22×C4×D59—21no—
64C22×C4×D59—21no—
65C22×C4×D59C2311yescharacteristic
66C2×#11745672—22no—
67C2×#11745672—22no—
68C2×#11745672C2×C414yescomplemented
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C2×C8×D59—22no—
72C2×C8×D59—22no—
73C2×C8×D59C2×C414yes—
74#14273707—22no—
75#14273707C2×C414yes—
76#14273703—22no—
77#14273703C2×C414yes—
78Q8×Dic59—22no—
79Q8×Dic59—41no—
80#14273692—22no—
81#14273692—41no—
82C2×C4⋊Dic59—21no—
83C2×C4⋊Dic59—21no—
84C2×C4⋊Dic59—21no—
85C2×C4⋊Dic59C2311yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×C4⋊Dic59C2311yescharacteristic
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4—21no—
92C2×Dic59⋊C4—21no—
93C4⋊C4×D59—22no—
94C4⋊C4×D59—22no—
95C4⋊C4×D59—41no—
96C4⋊C4×D59—41no—
97C2×(C2×D59)⋊C4—21no—
98C2×(C2×D59)⋊C4—21no—
99C2×(C2×D59)⋊C4D411yescharacteristic
100C2×(C2×D59)⋊C4D411yescharacteristic
101C2×C4×Dic59—21no—
102C2×C4×Dic59—21no—
103C2×C4×Dic59—21no—
104C2×C4×Dic59C2311yescharacteristic
105#14273661—22no—
106#14273660—22no—
107#14273660C2×C414yescomplemented
108#14273660D414yes—
109#14273652—22no—
110#14273652C2×C414yescomplemented
111#14273651—41no—
112#14273650—41no—
113(C2×D59)⋊C8—22no—
114C4×C59⋊C8—22no—
115C8×Dic59—22no—
116C59⋊#95—21no—
117C59⋊#95—21no—
118C59⋊#95—21no—
119C59⋊#95—21no—
120C59⋊#95—21no—
121C59⋊#95D411yescharacteristic
122C59⋊#95D411yescharacteristic
123C22×Q8×C59C2311yescharacteristic
124C22×C8×C59C2311yescharacteristic
125C22×C8×C59C2311yescharacteristic
126C2×C4⋊C4×C59C2311yescharacteristic
127C2×C4⋊C4×C59C2311yescharacteristic
128C2×C4⋊C4×C59C2311yescharacteristic
129C2×C42×C59—21no—
130C2×C42×C59C2311yescharacteristic
131#107×C59—22no—
132#103×C59—22no—
133#103×C59C2×C414yescomplemented
134#95×C59—21no—
135#95×C59—21no—
136#95×C59—21no—
137C23×D59C2×D411yescharacteristic
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59C22×C412yes—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2411yescharacteristic
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59C2×D411yescharacteristic
156C22×Dic59C2×D411yescharacteristic
157C22×Dic59C2×D411yescharacteristic
158C2×C4×D59—21no—
159C2×C4×D59—21no—
160C2×C4×D59—21no—
161C2×C4×D59—21no—
162C2×C4×D59—42no—
163C2×C4×D59—42no—
164C2×C4×D59—42no—
165C2×C4×D59—42no—
166C2×C4×D59—81no—
167C2×C4×D59—81no—
168C2×C4×D59C22×C412yes—
169C2×C4×D59#4112yes—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118—21no—
174C2×Dic118—21no—
175C2×Dic118C22×C412yes—
176C2×C8×C59—21no—
177C2×C8×C59—21no—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59—21no—
181C2×C8×C59C22×C412yes—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59C22×C412yes—
187C4⋊C4×C59C2×D412yes—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—81no—
191#11745672—28no—
192(C2×D59)⋊C4—24no—
193(C2×D59)⋊C4—24no—
194(C2×D59)⋊C4—22no—
195(C2×D59)⋊C4Q1614yescomplemented
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—24no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—42no—
202C4⋊Dic59—81no—
203C4⋊Dic59C22×C412yes—
204C4⋊Dic59C2×D412yes—
205Dic59⋊C4—24no—
206Dic59⋊C4—42no—
207Dic59⋊C4—42no—
208Dic59⋊C4—42no—
209Dic59⋊C4—81no—
210C4×Dic59—21no—
211C4×Dic59—21no—
212C4×Dic59—42no—
213C4×Dic59—42no—
214C4×Dic59—81no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8C22×C412yes—
219C2×C59⋊C8C2×D412yes—
220C2×C59⋊C8#4112yes—
221C8×D59—28no—
222C23×C59C22×D411yescharacteristic
223C22×D59—21no—
224C22×D59—21no—
225C22×D59—21no—
226C22×D59—21no—
227C22×D59C4×D412yes—
228C22×D59C2×Q1612yes—
229C22×D59#13512yes—
230Q8×C59—22no—
231Q8×C59—22no—
232Q8×C59—41no—
233Q8×C59—41no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
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×C59C2×Q1612yes—
252C2×C4×C59C23×C411yescharacteristic
253C2×C4×C59C22×D411yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59H(5,𝔽2)11yescharacteristic
257C2×C4×C592−1+411yescharacteristic
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—22no—
263C2×Dic59—22no—
264C2×Dic59—21no—
265C2×Dic59—22no—
266C2×Dic59—21no—
267C2×Dic59—21no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59C4×D412yes—
278C2×Dic59C2×Q1612yes—
279C4×D59—22no—
280C4×D59—22no—
281C4×D59—22no—
282C4×D59—22no—
283C4×D59—84no—
284C4×D59—84no—
285Dic118—22no—
286Dic118—22no—
287Dic118—41no—
288Dic118—41no—
289C8×C59—22no—
290C8×C59—41no—
291C59⋊C8—41no—
292C59⋊C8C4×D414yes—
293#23024—591nomaximal, Sylow
294C22×C59C2×C4×D411yescharacteristic
295C22×C59D4211yescharacteristic
296C22×C59#54711yescharacteristic
297C22×C59C22×Q1611yescharacteristic
298C22×C59C2×#13511yescharacteristic
299C22×C59#57611yescharacteristic
300C22×C59#57811yescharacteristic
301C2×D59—22no—
302C2×D59—22no—
303C2×D59—22no—
304C2×D59—22no—
305C2×D59—22no—
306C2×D59—22no—
307C2×D59C4×Q1614yescomplemented
308C4×C59—22no—
309C4×C59—41no—
310C4×C59—42no—
311C4×C59—42no—
312C4×C59—42no—
313C4×C59—81no—
314C4×C59C2×C4×D411yescharacteristic
315C4×C59#52011yescharacteristic
316C4×C59#54811yescharacteristic
317C4×C59#55311yescharacteristic
318Dic59—22no—
319Dic59—22no—
320Dic59—42no—
321Dic59—42no—
322Dic59—42no—
323Dic59—41no—
324Dic59—81no—
325C2×#467—591no—
326C2×C4×Q16—591no—
327C22×#107—591no—
328#2360—591no—
329#2360—591no—
330#1969—591no—
331#1942—591no—
332#1928—591no—
333#1928D5911yescharacteristic, complemented
334#1872—591no—
335#1872—591no—
336#1863—591no—
337#1863—591no—
338C4×#107—591no—
339C8×C22⋊C4—591no—
340C2×C59#236111yescharacteristic
341C2×C59C2×C4×Q1611yescharacteristic
342C2×C59#295111yescharacteristic
343C2×C59#328011yescharacteristic
344C2×C59D4×Q1611yescharacteristic
345C2×C59#333611yescharacteristic
346C2×C59#333811yescharacteristic
347D59—24no—
348C22×Q16—591no—
349C23×C8—591no—
350C2×C22⋊Q8—591no—
351C2×C22⋊Q8—591no—
352C2×C4×Q8—591no—
353C2×C4×Q8—591no—
354C22×C4⋊C4—591no—
355C22×C4⋊C4—591no—
356#467—598no—
357C4×Q16—1184no—
358C2×#107—591no—
359C2×#107—591no—
360C2×#107—591no—
361C2×#107—591no—
362C2×#107—1184no—
363C2×#107C2×D5911yescharacteristic
364C2×#103—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×#103—591no—
368C2×#103C2×D5911yescharacteristic
369C2×#103C2×D5911yescharacteristic
370C2×C22⋊C8—591no—
371C2×C4×C8—591no—
372C2×C4×C8—591no—
373#391—591no—
374#391C2×D5911yescharacteristic
375#389—591no—
376#389—591no—
377#389—1181no—
378#389—1181no—
379#387—1181no—
380#387—1181no—
381#385—1181no—
382#385—1181no—
383C4×C4⋊C4—591no—
384C4×C4⋊C4C2×D5911yescharacteristic
385C4×C22⋊C4—591no—
386C4×C22⋊C4—1181no—
387C4×C22⋊C4—1181no—
388#340—591no—
389#340—591no—
390#340—591no—
391#340C2×D5911yescharacteristic
392#336—591no—
393#336C2×D5911yescharacteristic
394C59#2302411yescharacteristic, complemented, Sylow
395C22×Q8—591no—
396C22×Q8C22×D5911yescharacteristic
397C23×C4—591no—
398C23×C4—1181no—
399C23×C4—1181no—
400C2×Q16—594no—
401C2×Q16—1182no—
402C2×Q16—1182no—
403C22×C8—591no—
404C22×C8—594no—
405C22×C8—591no—
406C22×C8—1182no—
407C22×C8—1182no—
408C22×C8C22×D5911yescharacteristic
409C22×C8C22×D5911yescharacteristic
410C22⋊Q8—594no—
411C22⋊Q8—594no—
412C22⋊Q8—1182no—
413C22⋊Q8—1182no—
414C4×Q8—1182no—
415C4×Q8—1182no—
416C4×Q8—2361no—
417C4×Q8—2361no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—1182no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1182no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—2361no—
432C2×C4⋊C4—2361no—
433C2×C4⋊C4C22×D5911yescharacteristic
434C2×C4⋊C4C22×D5911yescharacteristic
435C2×C4⋊C4C22×D5911yescharacteristic
436C2×C22⋊C4—591no—
437C2×C22⋊C4—591no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4—1181no—
440C2×C42—21no—
441C2×C42—591no—
442C2×C42—1181no—
443C2×C42—1181no—
444C2×C42—1181no—
445C2×C42C22×D5911yescharacteristic
446#107—22no—
447#107—594no—
448#107—594no—
449#107—1182no—
450#107—1182no—
451#103—22no—
452#103—594no—
453#103—1182no—
454#103—2361no—
455#103—2361no—
456#103C4×D5914yescomplemented
457C22⋊C8—1182no—
458C4×C8—1182no—
459C4×C8—1182no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—591no—
464#95—591no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470C24—591no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—592no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×Q8—1181no—
481C2×Q8—1181no—
482C2×Q8C2×C4×D5912yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—591no—
488C22×C4—591no—
489C22×C4—592no—
490C22×C4—592no—
491C22×C4—591no—
492C22×C4—1181no—
493C22×C4—1181no—
494C22×C4—1181no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—4721no—
501C22×C4—4721no—
502C22×C4—2362no—
503C22×C4—2362no—
504C22×C4—2362no—
505C22×C4—2362no—
506C22×C4D4×D5911yescharacteristic
507C22×C4D4×D5911yescharacteristic
508C22×C4C23×D5911yescharacteristic
509Q16—1188no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—592no—
516C2×C8—592no—
517C2×C8—592no—
518C2×C8—1188no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8C2×C4×D5912yes—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—592no—
528C4⋊C4—592no—
529C4⋊C4—1184no—
530C4⋊C4—1184no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—4721no—
536C4⋊C4—4721no—
537C4⋊C4—2362no—
538C4⋊C4—2362no—
539C4⋊C4—2362no—
540C4⋊C4—2362no—
541C4⋊C4C2×C4×D5912yes—
542C4⋊C4D4×D5912yes—
543C22⋊C4—594no—
544C22⋊C4—1182no—
545C22⋊C4—1184no—
546C22⋊C4—1184no—
547C42—21no—
548C42—21no—
549C42—81no—
550C42—1181no—
551C42—1181no—
552C42—4721no—
553C42—2362no—
554C42—2362no—
555C23—592no—
556C23—592no—
557C23—592no—
558C23—1181no—
559C23—1181no—
560C23—1181no—
561C23—1181no—
562C23C2×D4×D5911yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—1182no—
568Q8—1182no—
569Q8—2361no—
570Q8—2361no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—592no—
589C2×C4—592no—
590C2×C4—1181no—
591C2×C4—1181no—
592C2×C4—1181no—
593C2×C4—1181no—
594C2×C4—1182no—
595C2×C4—1182no—
596C2×C4—1182no—
597C2×C4—1182no—
598C2×C4—1181no—
599C2×C4—1181no—
600C2×C4—1181no—
601C2×C4—1182no—
602C2×C4—1182no—
603C2×C4—1182no—
604C2×C4—4724no—
605C2×C4—4724no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4Q16×D5912yes—
616C2×C4C22×C4×D5911yescharacteristic
617C2×C4C2×D4×D5911yescharacteristic
618C2×C4#41×D5911yescharacteristic
619C2×C4#41×D5911yescharacteristic
620C2×C4#1427377311yescharacteristic
621C2×C4#1427377411yescharacteristic
622C8—22no—
623C8—41no—
624C8—594no—
625C8—2361no—
626C22—594no—
627C22—1182no—
628C22—1182no—
629C22—1182no—
630C22—1182no—
631C22—1182no—
632C22—1182no—
633C22C4×D4×D5911yescharacteristic
634C22#1683803611yescharacteristic
635C22#1683805011yescharacteristic
636C22C2×Q16×D5911yescharacteristic
637C22#135×D5911yescharacteristic
638C22#1683820611yescharacteristic
639C22#1683821811yescharacteristic
640C4—22no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—1182no—
647C4—1182no—
648C4—4721no—
649C4—2361no—
650C4—2362no—
651C4—2362no—
652C4—2362no—
653C4C4×D4×D5911yescharacteristic
654C4#1683788211yescharacteristic
655C4#1683806011yescharacteristic
656C4#1683810311yescharacteristic
657C2—1184no—
658C2#2142710911yescharacteristic
659C2C4×Q16×D5911yescharacteristic
660C2#2143296811yescharacteristic
661C2#2143577611yescharacteristic
662C2#2143584311yescharacteristic
663C2#2143645711yescharacteristic
664C2#2143648311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1928C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1928D59
hypocenterC59#23024
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
2. C2×#16837494C2semidirect
6. #21424956C2semidirect
7. #21424828C2semidirect
8. #21424738C2semidirect
11. #21424352C2semidirect
12. #21424347C2semidirect
13. C4×#14273660C2semidirect
15. C59×#1928C2semidirect
24. #16837642C4semidirect
68. C2×#11745672C2×C4semidirect
107. #14273660C2×C4semidirect
110. #14273652C2×C4semidirect
133. #103×C59C2×C4semidirect
195. (C2×D59)⋊C4Q16semidirect
307. C2×D59C4×Q16semidirect
333. #1928D59semidirect
394. C59#23024semidirect
456. #103C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168376422
C2×#168374942
C2×#107×D592
#214271062
#214271032
#214249562
#214248282
#214247382
#214244022
#214244002
#214243522
#214243472
C4×#142736602
#214238662
C59×#19282
#2302459