Finite Group Databasebeta

← #28606019

Subgroups — #28606019

Order 15104.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837411C211yesmaximal, characteristic
2#21426979C211yesmaximal, characteristic
3#21424789C211yesmaximal, characteristic, complemented
4#21424775C211yesmaximal, characteristic, complemented
5#21424734C211yesmaximal, characteristic, complemented
6#21424727C211yesmaximal, characteristic, complemented
7#21424364C211yesmaximal, characteristic, complemented
8#21424350C211yesmaximal, characteristic, complemented
9#21424188C211yesmaximal, characteristic
10#21424127C211yesmaximal, characteristic
11#21424031C211yesmaximal, characteristic
12#21423926C211yesmaximal, characteristic, complemented
13C4×#14273651C211yesmaximal, characteristic, complemented
14#21423880C211yesmaximal, characteristic
15Q8×C22⋊C4×C59C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20#16837411C418yescomplemented
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837290—21no—
35#16837290—21no—
36#16837266—21no—
37#16837262C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40#16837247—21no—
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C23×Q8×C59C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C59×#391—21no—
53C59×#391—21no—
54C59×#391C2211yescharacteristic
55C59×#391C2211yescharacteristic
56C59×#380—21no—
57C59×#380C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C4×C22⋊C4×C59C2211yescharacteristic
60C2×#11745664—41no—
61C2×#11745664—41no—
62C2×#11745664C2×C414yes—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273686—22no—
68#14273686C2×C414yes—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—41no—
81#14273658—41no—
82#14273651—22no—
83#14273651—22no—
84#14273651—41no—
85#14273651—41no—
86#14273651C2×C414yes—
87#14273651C2×C414yes—
88#14273651C2×C414yes—
89#14273651C2×C414yes—
90C22⋊(C59⋊C8)—22no—
91#14273638—41no—
92C4×C59⋊C8—41no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C22×Q8×C59—21no—
99C22×Q8×C59—22no—
100C22×Q8×C59—22no—
101C22×Q8×C59—22no—
102C22×Q8×C59—41no—
103C22×Q8×C59—41no—
104C22×Q8×C59C2×C414yes—
105C22×Q8×C59C2311yescharacteristic
106C22×Q8×C59C2311yescharacteristic
107C23×C4×C59—21no—
108C23×C4×C59C2311yescharacteristic
109C4×Q8×C59—22no—
110C4×Q8×C59—22no—
111C4×Q8×C59—41no—
112C4×Q8×C59—41no—
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×C59D411yescharacteristic
119C2×C22⋊C4×C59D411yescharacteristic
120C2×C42×C59—21no—
121C2×C42×C59—21no—
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59D411yescharacteristic
130#95×C59D411yescharacteristic
131C24×C59C2×D411yescharacteristic
132C2×Q8×C59—24no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—28no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—42no—
143C2×Q8×C59—42no—
144C2×Q8×C59—44no—
145C2×Q8×C59—44no—
146C2×Q8×C59—44no—
147C2×Q8×C59C22×C412yes—
148C2×Q8×C59C22×C412yes—
149C2×Q8×C59#4112yes—
150C2×Q8×C59#4112yes—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59—21no—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59—24no—
157C22×C4×C59—21no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59—42no—
161C22×C4×C59—42no—
162C22×C4×C59C22×C412yes—
163C22×C4×C59C2×D411yescharacteristic
164C22×C4×C59C2×D411yescharacteristic
165C22×C4×C59C2×D411yescharacteristic
166C22×C4×C59C2×D411yescharacteristic
167C22×C4×C59C2×D411yescharacteristic
168C22×C4×C59#4112yes—
169C22×C4×C59C2411yescharacteristic
170C22×Dic59—21no—
171C22×Dic59—21no—
172C22×Dic59—21no—
173C22×Dic59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59—42no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—41no—
182C22⋊C4×C59—42no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—42no—
188C42×C59—42no—
189C22⋊Dic59—24no—
190C22⋊Dic59—81no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59C22×C412yes—
198C4⋊Dic59C22×C412yes—
199Dic59⋊C4—81no—
200Dic59⋊C4—81no—
201C4×Dic59—24no—
202C4×Dic59—42no—
203C4×Dic59—42no—
204#11745664—44no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8C22×C412yes—
216C2×C59⋊C8C22×C412yes—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59C4×D412yes—
224C23×C59C22×D411yescharacteristic
225Q8×C59—22no—
226Q8×C59—22no—
227Q8×C59—22no—
228Q8×C59—22no—
229Q8×C59—22no—
230Q8×C59—48no—
231Q8×C59—41no—
232Q8×C59—41no—
233Q8×C59#11714yes—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—22no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—48no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59C4×D412yes—
265C2×C4×C59#13712yes—
266C2×C4×C59C23×C411yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59C2×#4111yescharacteristic
270C2×C4×C59H(5,𝔽2)11yescharacteristic
271C2×C4×C59H(5,𝔽2)11yescharacteristic
272C2×Dic59—22no—
273C2×Dic59—22no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C59⋊C8—22no—
285C59⋊C8—22no—
286C59⋊C8—41no—
287C59⋊C8—41no—
288#22585—591nomaximal, Sylow
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59C2×C4×D411yescharacteristic
297C22×C59#54611yescharacteristic
298C22×C59#54611yescharacteristic
299C22×C59C2×#13711yescharacteristic
300C22×C59#57511yescharacteristic
301C22×C59#57711yescharacteristic
302C22×C59#57811yescharacteristic
303C4×C59—22no—
304C4×C59—24no—
305C4×C59—22no—
306C4×C59—41no—
307C4×C59—42no—
308C4×C59—41no—
309C4×C59—41no—
310C4×C59—81no—
311C4×C59—81no—
312C4×C59C2×#11711yescharacteristic
313C4×C59#51811yescharacteristic
314C4×C59#55111yescharacteristic
315C4×C59#55111yescharacteristic
316Dic59—42no—
317Dic59—42no—
318Dic59—81no—
319Dic59—81no—
320C2×#419—591no—
321#2346—591no—
322Q8×C22⋊C4D5911yescharacteristic
323#1935—591no—
324#1933—591no—
325#1928—591no—
326#1927—591no—
327#1865—591no—
328#1863—591no—
329#1825—591no—
330#1809—591no—
331#1787—591no—
332#1763—591no—
333C4×#103—591no—
334#1753—591no—
335C2×C59—42no—
336C2×C59#234211yescharacteristic
337C2×C59C4×#13711yescharacteristic
338C2×C59#294711yescharacteristic
339C2×C59#331211yescharacteristic
340C2×C59#332111yescharacteristic
341C2×C59#332711yescharacteristic
342C2×C59#332911yescharacteristic
343C23×Q8C2×D5911yescharacteristic
344C22×M16—591no—
345C2×C4×Q8C2×D5911yescharacteristic
346C2×C4×Q8C2×D5911yescharacteristic
347C2×#117—591no—
348C2×#107—591no—
349C2×#106—591no—
350#419—598no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#103—591no—
358C2×#103—591no—
359C2×C22⋊C8—591no—
360C2×#97—591no—
361C2×C4×C8—591no—
362#391—21no—
363#391—21no—
364#391C2×D5911yescharacteristic
365#391C2×D5911yescharacteristic
366#389—591no—
367#389—591no—
368#388—1181no—
369#388—1181no—
370#382—1181no—
371#380—21no—
372#380—591no—
373#380C2×D5911yescharacteristic
374C4×C4⋊C4—591no—
375C4×C4⋊C4—591no—
376C4×C22⋊C4—21no—
377C4×C22⋊C4C2×D5911yescharacteristic
378#376—1181no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#336—591no—
386#336—591no—
387C59#2258511yescharacteristic, complemented, Sylow
388C22×Q8—22no—
389C22×Q8—21no—
390C22×Q8—22no—
391C22×Q8—22no—
392C22×Q8—41no—
393C22×Q8—41no—
394C22×Q8C4×D5914yes—
395C22×Q8C22×D5911yescharacteristic
396C22×Q8C22×D5911yescharacteristic
397C23×C4—21no—
398C23×C4C22×D5911yescharacteristic
399C2×M16—594no—
400C2×M16—2361no—
401C2×M16—2361no—
402C22×C8—591no—
403C22×C8—591no—
404C22×C8—591no—
405C22×C8—591no—
406C4×Q8—22no—
407C4×Q8—22no—
408C4×Q8—41no—
409C4×Q8—41no—
410#117—594no—
411#117—1182no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—1181no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C4⋊C4C22×D5911yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4C22⋊D5911yescharacteristic
426C2×C22⋊C4C22⋊D5911yescharacteristic
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—1181no—
430C2×C42—1181no—
431C2×C42—1181no—
432C2×C42C22×D5911yescharacteristic
433C2×C42C22×D5911yescharacteristic
434#107—2361no—
435#106—2361no—
436#103—594no—
437#103—594no—
438#103—594no—
439#103—594no—
440#103—1182no—
441#103—1182no—
442#103—2361no—
443#103—2361no—
444C22⋊C8—1182no—
445#97—2361no—
446C4×C8—2361no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457#95C22⋊D5911yescharacteristic
458#95C22⋊D5911yescharacteristic
459C24C2×C22⋊D5911yescharacteristic
460C2×Q8—24no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—28no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8C2×C4×D5912yes—
476C2×Q8C2×C4×D5912yes—
477C2×Q8#1174568712yes—
478C2×Q8#1174568712yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—1181no—
491C22×C4—1181no—
492C22×C4—1181no—
493C22×C4—1181no—
494C22×C4C2×C4×D5912yes—
495C22×C4#1174568512yes—
496C22×C4C2×C22⋊D5911yescharacteristic
497C22×C4C2×C22⋊D5911yescharacteristic
498C22×C4C2×C22⋊D5911yescharacteristic
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C23×D5911yescharacteristic
502M16—2364no—
503C2×C8—592no—
504C2×C8—592no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—592no—
522C4⋊C4—592no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—4721no—
530C4⋊C4—4721no—
531C22⋊C4—22no—
532C22⋊C4—41no—
533C22⋊C4—42no—
534C22⋊C4—1184no—
535C22⋊C4—4721no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—42no—
542C42—1184no—
543C42—2362no—
544C42—2362no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C4×C22⋊D5912yes—
552C23C22×C22⋊D5911yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#1427368314yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—1182no—
593C2×C4—1182no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4C4×C22⋊D5912yes—
605C2×C4#1427375812yes—
606C2×C4C22×C4×D5911yescharacteristic
607C2×C4C2×#1174568711yescharacteristic
608C2×C4C2×#1174568711yescharacteristic
609C2×C4C2×#1174568511yescharacteristic
610C2×C4#1427377211yescharacteristic
611C2×C4#1427377211yescharacteristic
612C8—1182no—
613C8—1182no—
614C8—2361no—
615C8—2361no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D5911yescharacteristic
624C22#1683804611yescharacteristic
625C22#1683804611yescharacteristic
626C22C2×#1427375811yescharacteristic
627C22#1683820011yescharacteristic
628C22#1683821011yescharacteristic
629C22#1683821711yescharacteristic
630C4—22no—
631C4—22no—
632C4—24no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—4721no—
640C4—4721no—
641C4—2362no—
642C4—2362no—
643C4C2×#1427368311yescharacteristic
644C4#1683787511yescharacteristic
645C4#1683807811yescharacteristic
646C4#1683807811yescharacteristic
647C2—42no—
648C2#2142694111yescharacteristic
649C2C4×#1427375811yescharacteristic
650C2#2143293911yescharacteristic
651C2#2143614711yescharacteristic
652C2#2143628211yescharacteristic
653C2#2143636411yescharacteristic
654C2#2143639011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingQ8×C22⋊C4×C59C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterQ8×C22⋊C4D59
hypocenterC59#22585
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #21424789C2semidirect
4. #21424775C2semidirect
5. #21424734C2semidirect
6. #21424727C2semidirect
7. #21424364C2semidirect
8. #21424350C2semidirect
12. #21423926C2semidirect
13. C4×#14273651C2semidirect
20. #16837411C4semidirect
387. C59#22585semidirect

Maximal subgroups

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

subgroupindex
C2×#168374112
#214269792
#214247892
#214247752
#214247342
#214247272
#214243642
#214243502
#214241882
#214241272
#214240312
#214239262
C4×#142736512
#214238802
Q8×C22⋊C4×C592
#2258559