Finite Group Databasebeta

← #28616120

Subgroups — #28616120

Order 15104.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437501C211yesmaximal, characteristic, complemented
2C2×#16837420C211yesmaximal, characteristic
3C2×#16837418C211yesmaximal, characteristic
4C2×#16837375C211yesmaximal, characteristic
5#21425267C218yesmaximal, complemented
6C2×#16837174C211yesmaximal, characteristic, complemented
7#21423805C211yesmaximal, characteristic, complemented
8C2×C59×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
9C22×#11745685C2211yescharacteristic
10C22×#11745664C2211yescharacteristic
11#16838033—22no—
12#16838030—22no—
13#16838030—22no—
14#16838028—22no—
15#16837959—22no—
16#16837955—22no—
17#16837955C2214yescomplemented
18C2×#14273686C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837420—22no—
21#16837420C2214yes—
22#16837418—22no—
23#16837418—22no—
24C2×#14273655—22no—
25C2×#14273653C2214yes—
26#16837375—22no—
27#16837375C2214yes—
28C2×#14273641—22no—
29#16837174C2214yescomplemented
30#16837161C2214yescomplemented
31C22×M16×C59C2211yescharacteristic
32C2×#117×C59C2211yescharacteristic
33C2×#117×C59C2211yescharacteristic
34C59×H(3,ℤ/4)C2214yescomplemented
35C2×#11745685—21no—
36C2×#11745685—21no—
37C2×#11745685—21no—
38C2×#11745685—21no—
39C2×#11745685—21no—
40C2×#11745685—21no—
41C2×#11745685—21no—
42C2×#11745685—21no—
43C2×#11745685—21no—
44C2×#11745685—21no—
45C2×#11745685Q812yescomplemented
46C2×#11745685C2312yes—
47C22×Dic118Q811yescharacteristic, complemented
48C22×D236Q811yescharacteristic, complemented
49C22×C22⋊D59—21no—
50C22×C4×D59—21no—
51C2×#11745664—21no—
52C2×#11745664—21no—
53C2×#11745664—21no—
54C2×#11745664—21no—
55C2×#11745664—21no—
56C2×#11745664C2312yes—
57C22×C59⋊C8—21no—
58#14273727—24no—
59#14273726—24no—
60#14273726—24no—
61#14273719—24no—
62#14273718—24no—
63#14273718—24no—
64#14273706—28no—
65#14273704—28no—
66#14273704—28no—
67C4×Dic118—28no—
68#14273686—21no—
69#14273686—21no—
70#14273686—21no—
71#14273686—21no—
72#14273686D412yes—
73#14273686D412yes—
74#14273686D412yes—
75#14273686C2312yes—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59D412yes—
80C2×C4×Dic59—21no—
81C2×C4×Dic59C2312yes—
82#14273655—28no—
83#14273653—28no—
84#14273641—28no—
85C23×C4×C59C2311yescharacteristic
86C2×M16×C59—21no—
87C2×M16×C59—21no—
88C2×M16×C59—21no—
89C2×M16×C59—21no—
90C2×M16×C59—21no—
91C2×M16×C59C2312yes—
92C22×C8×C59—21no—
93#117×C59—21no—
94#117×C59—21no—
95#117×C59—21no—
96#117×C59—21no—
97#117×C59—21no—
98#117×C59D412yes—
99#117×C59C2312yes—
100#117×C59C2312yes—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—21no—
105C2×C42×C59—21no—
106C2×C42×C59D412yes—
107C24×C59C2×D411yescharacteristic
108C23×D59—21no—
109C22×C4×C59—22no—
110C22×C4×C59—41no—
111C22×C4×C59C2×D411yescharacteristic
112C22×C4×C59C2×D411yescharacteristic
113C22×C4×C59C2×D411yescharacteristic
114C22×C4×C59C2×D411yescharacteristic
115C22×C4×C59C2×D411yescharacteristic
116C22×C4×C59C2×Q811yescharacteristic
117C22×C4×C59C2×Q811yescharacteristic
118C22×C4×C59#4111yescharacteristic
119C22×C4×C59#4111yescharacteristic
120C22×C4×C59#4111yescharacteristic
121C22×C4×C59#4111yescharacteristic
122C22×C4×C59#4111yescharacteristic
123C22×C4×C59#4111yescharacteristic
124C22×C4×C59C2411yescharacteristic
125C2×C22⋊D59—22no—
126C2×C22⋊D59—22no—
127C2×C22⋊D59—41no—
128C2×C22⋊D59—41no—
129C2×C22⋊D59—41no—
130C2×C22⋊D59—41no—
131C22×Dic59—21no—
132C22×Dic59—22no—
133C22×Dic59—22no—
134#11745685—22no—
135#11745685—22no—
136#11745685—22no—
137#11745685—22no—
138#11745685—22no—
139#11745685—22no—
140#11745685—22no—
141#11745685—22no—
142#11745685—41no—
143#11745685—41no—
144#11745685—41no—
145#11745685—41no—
146#11745685—41no—
147#11745685—41no—
148#11745685—41no—
149#11745685—41no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236C2×Q812yes—
156C2×C4×D59—22no—
157C2×C4×D59—22no—
158C2×C4×D59—41no—
159C2×C4×D59—41no—
160C2×C4×D59—41no—
161C2×C4×D59—41no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—28no—
168C2×Dic118C2×Q812yes—
169M16×C59—22no—
170M16×C59—22no—
171M16×C59—41no—
172M16×C59—41no—
173C2×C8×C59—22no—
174C2×C8×C59—41no—
175C2×C8×C59—41no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—41no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—22no—
182C22⋊C4×C59—22no—
183C22⋊C4×C59—41no—
184C42×C59—22no—
185C42×C59—22no—
186C42×C59—41no—
187C42×C59C2×D414yes—
188C22⋊Dic59—22no—
189C22⋊Dic59—22no—
190C22⋊Dic59—22no—
191C22⋊Dic59C2×D414yes—
192C4⋊Dic59—28no—
193C4⋊Dic59—22no—
194C4⋊Dic59—22no—
195C4⋊Dic59—22no—
196C4⋊Dic59—22no—
197Dic59⋊C4—28no—
198Dic59⋊C4—28no—
199C4×Dic59—22no—
200C4×Dic59—22no—
201C4×Dic59—22no—
202C4×Dic59C2×D414yes—
203#11745664—22no—
204#11745664—22no—
205#11745664—41no—
206#11745664—41no—
207C2×C59⋊C8—22no—
208C2×C59⋊C8—41no—
209C2×C59⋊C8—41no—
210C23×C59—81no—
211C23×C59#12011yescharacteristic
212C23×C59#12011yescharacteristic
213C23×C59C22⋊Q811yescharacteristic
214C23×C59C22⋊Q811yescharacteristic
215C23×C59#12411yescharacteristic
216C23×C59#12411yescharacteristic
217C23×C59C22×D411yescharacteristic
218C22×D59—22no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—82no—
222C2×C4×C59—24no—
223C2×C4×C59—42no—
224C2×C4×C59—82no—
225C2×C4×C59—82no—
226C2×C4×C59#12011yescharacteristic
227C2×C4×C59#12011yescharacteristic
228C2×C4×C59C22⋊Q811yescharacteristic
229C2×C4×C59C22⋊Q811yescharacteristic
230C2×C4×C59C22⋊Q811yescharacteristic
231C2×C4×C59C22⋊Q811yescharacteristic
232C2×C4×C59C22⋊Q811yescharacteristic
233C2×C4×C59C22⋊Q811yescharacteristic
234C2×C4×C59C22⋊Q811yescharacteristic
235C2×C4×C59C22⋊Q811yescharacteristic
236C2×C4×C59C22⋊Q811yescharacteristic
237C2×C4×C59C22⋊Q811yescharacteristic
238C2×C4×C59#12411yescharacteristic
239C2×C4×C59#12411yescharacteristic
240C2×C4×C59#12411yescharacteristic
241C2×C4×C59#12411yescharacteristic
242C2×C4×C59#12411yescharacteristic
243C2×C4×C59#12411yescharacteristic
244C2×C4×C59#12411yescharacteristic
245C2×C4×C59#12411yescharacteristic
246C2×C4×C59#12411yescharacteristic
247C2×C4×C59#12411yescharacteristic
248C2×C4×C59C22×D411yescharacteristic
249C2×C4×C59C22×D411yescharacteristic
250C2×C4×C59C22×Q811yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C22⋊D59—42no—
255C22⋊D59—42no—
256C22⋊D59—42no—
257C22⋊D59—42no—
258C2×Dic59—24no—
259C2×Dic59—24no—
260C2×Dic59—22no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—82no—
264C2×Dic59—82no—
265D236—22no—
266D236—22no—
267D236—41no—
268D236—41no—
269C4×D59—42no—
270C4×D59—42no—
271C4×D59—42no—
272C4×D59—42no—
273Dic118—22no—
274Dic118—22no—
275Dic118—41no—
276Dic118—41no—
277Dic118—88no—
278C8×C59—42no—
279C59⋊C8—42no—
280#23472—591nomaximal, Sylow
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59#39311yescharacteristic
289C22×C59#39311yescharacteristic
290C22×C59#39311yescharacteristic
291C22×C59#39311yescharacteristic
292C22×C59C2×#12011yescharacteristic
293C22×C59C2×C22⋊Q811yescharacteristic
294C22×C59C2×#12411yescharacteristic
295C2×D59—42no—
296C2×D59—82no—
297C2×D59—82no—
298C2×D59—82no—
299C4×C59—44no—
300C4×C59—82no—
301C4×C59#39311yescharacteristic
302C4×C59#39311yescharacteristic
303C4×C59#39311yescharacteristic
304C4×C59#39311yescharacteristic
305C4×C59C2×C22⋊Q811yescharacteristic
306C4×C59C2×C22⋊Q811yescharacteristic
307C4×C59C2×#12411yescharacteristic
308C4×C59C2×#12411yescharacteristic
309Dic59—44no—
310Dic59—44no—
311Dic59—42no—
312#3458—591no—
313C2×#421—591no—
314C2×#421—591no—
315C2×#410—591no—
316#2031—598no—
317C2×#344—591no—
318#1733—591no—
319C2×H(3,ℤ/4)D5911yescharacteristic, complemented
320C2×C59—21no—
321C2×C59—21no—
322C2×C59—21no—
323C2×C59—21no—
324C2×C59—21no—
325C2×C59—21no—
326C2×C59—21no—
327C2×C59C2×#39311yescharacteristic
328D59—82no—
329C22×#41—591no—
330C22×M16—591no—
331C22×M16C2×D5911yescharacteristic
332#544—1182no—
333#544—1182no—
334#544—1182no—
335#543—1182no—
336#533—594no—
337#533—1182no—
338#533—1182no—
339C2×#117—591no—
340C2×#117—591no—
341C2×#117C2×D5911yescharacteristic
342C2×#117C2×D5911yescharacteristic
343#421—594no—
344#421—1182no—
345#421—1182no—
346#421—1182no—
347C2×#104—594no—
348C2×#104—1182no—
349#410—594no—
350#410—1182no—
351C2×#99—1182no—
352#344—594no—
353#339—594no—
354H(3,ℤ/4)C2×D5914yescomplemented
355C59#2347211yescharacteristic, complemented, Sylow
356C2×#41—592no—
357C2×#41—592no—
358C2×#41—1181no—
359C2×#41—1181no—
360C2×#41—1181no—
361C2×#41—1181no—
362C2×#41—1181no—
363C2×#41—1181no—
364C2×#41—1181no—
365C2×#41—1181no—
366C2×#41—1181no—
367C2×#41—1181no—
368C22×Q8—591no—
369C22×D4—591no—
370C22×D4—1181no—
371C23×C4—1181no—
372C23×C4C22×D5911yescharacteristic
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—592no—
379C2×M16—1181no—
380C2×M16—1181no—
381C2×M16—1181no—
382C2×M16—1181no—
383C2×M16—1181no—
384C2×M16C22×D5912yes—
385C22×C8—21no—
386C22×C8—1181no—
387C4⋊Q8—1184no—
388C4⋊Q8—1184no—
389C4⋊Q8—1184no—
390#125—1184no—
391#125—1184no—
392#125—1184no—
393C22⋊Q8—1188no—
394C22⋊Q8—1188no—
395C22⋊Q8—1188no—
396C4×Q8—1188no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—592no—
403#117—592no—
404#117—592no—
405#117—592no—
406#117—1181no—
407#117—1181no—
408#117—1181no—
409#117—1181no—
410#117D23612yes—
411#117C22×D5912yes—
412#117C22×D5912yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—592no—
420C2×C22⋊C4—1181no—
421C2×C42—21no—
422C2×C42—592no—
423C2×C42—1181no—
424C2×C42D23612yes—
425#104—1188no—
426#104—1188no—
427#99—1188no—
428C24—1181no—
429C24D4×D5911yescharacteristic
430#41—1182no—
431#41—1182no—
432#41—1182no—
433#41—1182no—
434#41—1182no—
435#41—1182no—
436#41—1182no—
437#41—1182no—
438#41—2361no—
439#41—2361no—
440#41—2361no—
441#41—2361no—
442#41—2361no—
443#41—2361no—
444#41—2361no—
445#41—2361no—
446C2×Q8—592no—
447C2×Q8—1188no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C2×Q8—1181no—
453C2×D4—592no—
454C2×D4—1182no—
455C2×D4—1182no—
456C2×D4—1181no—
457C2×D4—1181no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C2×D4—2361no—
462C2×D4—2361no—
463C2×D4—2361no—
464C2×D4—2361no—
465C22×C4—22no—
466C22×C4—41no—
467C22×C4—1182no—
468C22×C4—1182no—
469C22×C4—1181no—
470C22×C4—1182no—
471C22×C4—1182no—
472C22×C4—2361no—
473C22×C4—2361no—
474C22×C4—2361no—
475C22×C4—2361no—
476C22×C4C2×D23611yescharacteristic
477C22×C4C2×D23611yescharacteristic
478C22×C4#1174568511yescharacteristic
479C22×C4#1174568511yescharacteristic
480C22×C4#1174568511yescharacteristic
481C22×C4#1174568511yescharacteristic
482C22×C4D4×D5911yescharacteristic
483C22×C4D4×D5911yescharacteristic
484C22×C4D4×D5911yescharacteristic
485C22×C4#1174568711yescharacteristic
486C22×C4#1174568711yescharacteristic
487C22×C4Q8×D5911yescharacteristic
488C22×C4Q8×D5911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490M16—22no—
491M16—22no—
492M16—41no—
493M16—41no—
494M16—1182no—
495M16—1182no—
496M16—2361no—
497M16—2361no—
498C2×C8—22no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—1182no—
502C2×C8—2361no—
503C2×C8—2361no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—1188no—
509C4⋊C4—1188no—
510C4⋊C4—1188no—
511C4⋊C4—1182no—
512C4⋊C4—1182no—
513C4⋊C4—1182no—
514C4⋊C4—1182no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—594no—
520C22⋊C4—1182no—
521C22⋊C4—1182no—
522C22⋊C4—1182no—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—594no—
527C42—1182no—
528C42—1182no—
529C42—1182no—
530C42C2×D23614yes—
531C23—81no—
532C23—1182no—
533C23—4722no—
534C23—2361no—
535C23—2361no—
536C23#1427369711yescharacteristic
537C23#1427369711yescharacteristic
538C23#1427370511yescharacteristic
539C23#1427370511yescharacteristic
540C23#1427371311yescharacteristic
541C23#1427371311yescharacteristic
542C23C2×D4×D5911yescharacteristic
543Q8—1182no—
544Q8—1182no—
545Q8—4728no—
546Q8—2361no—
547Q8—2361no—
548D4—1182no—
549D4—1182no—
550D4—2362no—
551D4—2362no—
552D4—2362no—
553D4—2362no—
554D4—2361no—
555D4—2361no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—1184no—
561C2×C4—1184no—
562C2×C4—1182no—
563C2×C4—4722no—
564C2×C4—4722no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2362no—
568C2×C4—2362no—
569C2×C4—2362no—
570C2×C4—2362no—
571C2×C4#1427369711yescharacteristic
572C2×C4#1427369711yescharacteristic
573C2×C4#1427370511yescharacteristic
574C2×C4#1427370511yescharacteristic
575C2×C4#1427370511yescharacteristic
576C2×C4#1427370511yescharacteristic
577C2×C4#1427370511yescharacteristic
578C2×C4#1427370511yescharacteristic
579C2×C4#1427370711yescharacteristic
580C2×C4#1427370711yescharacteristic
581C2×C4#1427370711yescharacteristic
582C2×C4#1427370711yescharacteristic
583C2×C4#1427371311yescharacteristic
584C2×C4#1427371311yescharacteristic
585C2×C4#1427371311yescharacteristic
586C2×C4#1427371311yescharacteristic
587C2×C4#1427371311yescharacteristic
588C2×C4#1427371311yescharacteristic
589C2×C4#1427371511yescharacteristic
590C2×C4#1427371511yescharacteristic
591C2×C4#1427371511yescharacteristic
592C2×C4#1427371511yescharacteristic
593C2×C4C2×D4×D5911yescharacteristic
594C2×C4C22×D23611yescharacteristic
595C2×C4C2×Q8×D5911yescharacteristic
596C2×C4C2×#1174568711yescharacteristic
597C2×C4C2×#1174568511yescharacteristic
598C2×C4C2×#1174568511yescharacteristic
599C8—42no—
600C8—2362no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—4722no—
609C22—4722no—
610C22—4722no—
611C22—2362no—
612C22#1683731511yescharacteristic
613C22#1683731511yescharacteristic
614C22#1683731511yescharacteristic
615C22#1683731511yescharacteristic
616C22C2×#1427369711yescharacteristic
617C22C2×#1427370511yescharacteristic
618C22C2×#1427371311yescharacteristic
619C4—44no—
620C4—82no—
621C4—2364no—
622C4—2364no—
623C4—2362no—
624C4#1683731511yescharacteristic
625C4#1683731511yescharacteristic
626C4#1683731511yescharacteristic
627C4#1683731511yescharacteristic
628C4C2×#1427370511yescharacteristic
629C4C2×#1427370711yescharacteristic
630C4C2×#1427371311yescharacteristic
631C4C2×#1427371511yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—4722no—
640C2C2×#1683731511yescharacteristic
subgroupG/H
centerC4C2×#14273715
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×H(3,ℤ/4)C2
socleC2×C59C2×#393
prime radicalC22×#11745685C22
hypercenterC2×H(3,ℤ/4)D59
hypocenterC59#23472
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. #21437501C2semidirect
5. #21425267C2semidirect
6. C2×#16837174C2semidirect
7. #21423805C2semidirect
8. C2×C59×H(3,ℤ/4)C2semidirect
17. #16837955C22semidirect
29. #16837174C22semidirect
30. #16837161C22semidirect
34. C59×H(3,ℤ/4)C22semidirect
45. C2×#11745685Q8semidirect
47. C22×Dic118Q8semidirect
48. C22×D236Q8semidirect
319. C2×H(3,ℤ/4)D59semidirect
354. H(3,ℤ/4)C2×D59semidirect
355. C59#23472semidirect

Maximal subgroups

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

subgroupindex
#214375012
C2×#168374202
C2×#168374182
C2×#168373752
#214252672
C2×#168371742
#214238052
C2×C59×H(3,ℤ/4)2
#2347259