Finite Group Databasebeta

← #12272353

Subgroups — #12272353

Order 1280.

1978 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×C4×C5×C22⋊C4C211yesmaximal, characteristic
2#10615762C212yesmaximal
3#10613848C212yesmaximal
4#10613824C212yesmaximal
5#10613805C212yesmaximal, complemented
6C4×#4070C212yesmaximal, complemented
7#10605273C214yesmaximal, complemented
8#10605267C214yesmaximal, complemented
9#10605259C214yesmaximal, complemented
10#10605229C214yesmaximal, complemented
11#10605092C211yesmaximal, characteristic, complemented
12#10605092C211yesmaximal, characteristic, complemented
13#10605065C212yesmaximal, complemented
14C24×C4×C5C2211yescharacteristic
15C22×C5×C22⋊C4C2211yescharacteristic
16C22×C5×C22⋊C4C2211yescharacteristic
17C22×C42×C5C2211yescharacteristic
18C22×C42×C5C2211yescharacteristic
19C2×#4070C2212yes—
20C2×#4070C2212yes—
21C2×#4070C2212yes—
22C2×#4070C2212yes—
23C2×#4016C2212yescomplemented
24C2×C4×Dic10C2212yescomplemented
25C4×C5×C22⋊C4C2214yes—
26C4×C5×C22⋊C4C2214yes—
27C2×C5×#95C2211yescharacteristic
28C2×C5×#95C2211yescharacteristic
29#65536C2214yes—
30#65501C2212yes—
31#65501C2212yes—
32#65501C2212yes—
33#65501C2212yes—
34#65501C2212yes—
35#65501C2212yes—
36#65499C2214yes—
37#65499C2214yes—
38#65499C2214yes—
39#65499C2214yes—
40C4×C22⋊Dic5C2214yes—
41#65240C2214yes—
42#65240C2214yes—
43#65240C2214yes—
44#65240C2214yes—
45#65238C2214yes—
46#65237C2214yes—
47#65237C2214yes—
48#65236C2214yes—
49#65225C2212yescomplemented
50C4×C4⋊Dic5C2212yescomplemented
51#65222C2214yescomplemented
52#65222C2214yescomplemented
53#65222C2214yescomplemented
54C4×#683C2214yescomplemented
55#65218C2212yescomplemented
56#65217C2212yescomplemented
57#65217C2212yescomplemented
58#65217C2212yescomplemented
59#64950C2214yescomplemented
60#64950C2214yescomplemented
61#64945C2214yescomplemented
62#64945C2214yescomplemented
63#64944C2214yescomplemented
64#64943C2214yescomplemented
65#64936C2214yescomplemented
66#64935C2214yescomplemented
67#35012—51nomaximal, Sylow
68C25×C5C2311yescharacteristic
69C23×C4×C5—24no—
70C23×C4×C5—42no—
71C23×C4×C5—42no—
72C23×C4×C5C2311yescharacteristic
73C23×C4×C5C2312yes—
74C23×C4×C5C2312yes—
75C23×C4×C5C2311yescharacteristic
76C23×C4×C5C2311yescharacteristic
77C23×C4×C5C2311yescharacteristic
78C23×C4×C5C2311yescharacteristic
79C23×C4×C5C2311yescharacteristic
80C22×Dic10C2312yes—
81C22×Dic10C2312yes—
82C2×C5×C22⋊C4—24no—
83C2×C5×C22⋊C4Q818yescomplemented
84C2×C5×C22⋊C4C2314yes—
85C2×C5×C22⋊C4C2314yes—
86C2×C42×C5—42no—
87C2×C42×C5—42no—
88C2×C42×C5C2312yes—
89C2×C42×C5C2312yes—
90C2×C42×C5C2312yes—
91C2×C42×C5C2312yes—
92C2×C22⋊Dic5C2314yes—
93C2×C22⋊Dic5C2314yes—
94C2×C22⋊Dic5C2314yes—
95C2×C22⋊Dic5C2314yes—
96C2×C22⋊Dic5C2314yes—
97C2×C22⋊Dic5C2314yes—
98C2×C4⋊Dic5C2312yes—
99C2×C4⋊Dic5C2312yes—
100C2×C4⋊Dic5C2312yes—
101C2×C4⋊Dic5C2312yes—
102C2×C4⋊Dic5C2312yes—
103C2×C4⋊Dic5C2312yes—
104C2×C4⋊Dic5C2312yes—
105C2×C4⋊Dic5C2312yes—
106C2×C4⋊Dic5C2312yes—
107C2×C4⋊Dic5C2312yes—
108#4070—28no—
109#4070—28no—
110#4070—44no—
111#4070—44no—
112C2×#683C2314yes—
113C2×#683C2314yes—
114C2×#683C2314yes—
115C2×#683C2314yes—
116C2×#683C2314yes—
117C2×#683C2314yes—
118C2×C4×Dic5C2314yes—
119C2×C4×Dic5C2314yes—
120#4016—44no—
121C4×Dic10—44no—
122C5×#95C2314yes—
123C5×#95C2314yes—
124#3963C2314yes—
125#3963C2314yes—
126#3963C2314yes—
127#3963C2314yes—
128#3963C2314yes—
129#3963C2314yes—
130#3963C2314yes—
131#3963C2314yes—
132#3963C2314yes—
133#3963C2314yes—
134#2851—52no—
135#2848—52no—
136#2659—51no—
137#2659—51no—
138#2626—52no—
139#2574—54no—
140#2556—54no—
141#2522—54no—
142#2465—54no—
143#2329—52no—
144#2312—52no—
145C4×C22⋊Q8—52no—
146C2×C4×C22⋊C4D511yescharacteristic
147C24×C5—22no—
148C24×C5—22no—
149C24×C5—41no—
150C24×C5—41no—
151C24×C5—41no—
152C24×C5—41no—
153C24×C5C2×Q814yes—
154C24×C5C2412yes—
155C24×C5C2411yescharacteristic
156C22×C4×C5—22no—
157C22×C4×C5—22no—
158C22×C4×C5—22no—
159C22×C4×C5—22no—
160C22×C4×C5—24no—
161C22×C4×C5—24no—
162C22×C4×C5—24no—
163C22×C4×C5—24no—
164C22×C4×C5—41no—
165C22×C4×C5—41no—
166C22×C4×C5—41no—
167C22×C4×C5—41no—
168C22×C4×C5—41no—
169C22×C4×C5—41no—
170C22×C4×C5—42no—
171C22×C4×C5—42no—
172C22×C4×C5—41no—
173C22×C4×C5—41no—
174C22×C4×C5—41no—
175C22×C4×C5—41no—
176C22×C4×C5—416no—
177C22×C4×C5C2×Q814yes—
178C22×C4×C5C2×Q814yes—
179C22×C4×C5#4114yes—
180C22×C4×C5#4114yes—
181C22×C4×C5C2412yes—
182C22×C4×C5C2412yes—
183C22×C4×C5C2412yes—
184C22×C4×C5C2412yes—
185C22×C4×C5C2412yes—
186C22×C4×C5C2411yescharacteristic
187C22×C4×C5C2411yescharacteristic
188C22×Dic5C2414yes—
189C22×Dic5C2414yes—
190C22×Dic5C2414yes—
191C22×Dic5C2414yes—
192C2×Dic10—24no—
193C2×Dic10—24no—
194C2×Dic10—42no—
195C2×Dic10—42no—
196C2×Dic10—42no—
197C2×Dic10—42no—
198C5×C22⋊C4—28no—
199C5×C22⋊C4C2×Q8116yes—
200C42×C5—24no—
201C42×C5—24no—
202C42×C5—42no—
203C42×C5—42no—
204C22⋊Dic5—28no—
205C22⋊Dic5—28no—
206C22⋊Dic5—44no—
207C22⋊Dic5—44no—
208C22⋊Dic5—44no—
209C22⋊Dic5—44no—
210C4⋊Dic5—24no—
211C4⋊Dic5—24no—
212C4⋊Dic5—42no—
213C4⋊Dic5—42no—
214C4⋊Dic5—42no—
215C4⋊Dic5—42no—
216C4⋊Dic5—42no—
217C4⋊Dic5—42no—
218C4⋊Dic5—42no—
219C4⋊Dic5—42no—
220#683—28no—
221#683—28no—
222#683—28no—
223#683—28no—
224#683—44no—
225#683—44no—
226C4×Dic5—44no—
227C4×Dic5—44no—
228C24×C4C2×D511yescharacteristic
229C2×#125—52no—
230C2×C22⋊Q8—52no—
231C2×C22⋊Q8—52no—
232C2×C22⋊Q8—52no—
233C2×C22⋊Q8—52no—
234C2×C4×Q8—52no—
235C22×C22⋊C4C2×D511yescharacteristic
236C22×C22⋊C4C2×D511yescharacteristic
237C22×C42C2×D511yescharacteristic
238C22×C42C2×D511yescharacteristic
239#399—54no—
240#398—54no—
241#398—54no—
242#396—54no—
243#396—54no—
244#395—54no—
245#395—54no—
246#393—54no—
247#391—52no—
248#391—54no—
249#389—52no—
250#389—52no—
251#389—52no—
252#389—54no—
253#388—54no—
254#388—54no—
255#388—54no—
256#388—54no—
257#387—54no—
258#387—54no—
259#387—54no—
260#387—54no—
261#387—54no—
262#385—54no—
263#385—54no—
264#385—54no—
265#385—54no—
266#382—52no—
267#380—52no—
268#380—52no—
269#380—52no—
270#380—52no—
271#380—52no—
272#380—52no—
273#379—54no—
274C4×C4⋊C4—54no—
275C4×C4⋊C4—52no—
276C4×C22⋊C4—54no—
277C4×C22⋊C4C2×D514yes—
278C4×C22⋊C4C2×D514yes—
279C2×#95C2×D511yescharacteristic
280C2×#95C2×D511yescharacteristic
281C23×C5—21no—
282C23×C5—21no—
283C23×C5—21no—
284C23×C5—21no—
285C23×C5—28no—
286C23×C5—22no—
287C23×C5—22no—
288C23×C5—22no—
289C23×C5—22no—
290C23×C5—22no—
291C23×C5—22no—
292C23×C5—44no—
293C23×C5—44no—
294C23×C5—44no—
295C23×C5—44no—
296C23×C5—44no—
297C23×C5—44no—
298C23×C5C22×Q814yes—
299C23×C5C22×Q812yes—
300C23×C5C2×#4112yes—
301C23×C5C2×#4112yes—
302C23×C5C2511yescharacteristic
303C2×C4×C5—28no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—22no—
314C2×C4×C5—22no—
315C2×C4×C5—22no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5—44no—
320C2×C4×C5—44no—
321C2×C4×C5—44no—
322C2×C4×C5—44no—
323C2×C4×C5—48no—
324C2×C4×C5—48no—
325C2×C4×C5—44no—
326C2×C4×C5C22×Q814yes—
327C2×C4×C5C22×Q814yes—
328C2×C4×C5C2×#4114yes—
329C2×C4×C5C2×#4114yes—
330C2×C4×C5H(5,𝔽2)12yes—
331C2×C4×C5H(5,𝔽2)12yes—
332C2×Dic5—24no—
333C2×Dic5—24no—
334C2×Dic5—24no—
335C2×Dic5—24no—
336C2×Dic5—24no—
337C2×Dic5—24no—
338C2×Dic5—24no—
339C2×Dic5—24no—
340C2×Dic5—24no—
341C2×Dic5—24no—
342C2×Dic5—24no—
343C2×Dic5—24no—
344Dic10—48no—
345Dic10—84no—
346C25C22×D511yescharacteristic
347C22×Q8—52no—
348C22×Q8—52no—
349C23×C4—24no—
350C23×C4—42no—
351C23×C4—42no—
352C23×C4C22×D511yescharacteristic
353C23×C4C22×D512yes—
354C23×C4C22×D512yes—
355C23×C4C22×D511yescharacteristic
356C23×C4C22×D511yescharacteristic
357C23×C4C22×D511yescharacteristic
358C23×C4C22×D511yescharacteristic
359C23×C4C22×D511yescharacteristic
360#125—204no—
361C22⋊Q8—108no—
362C22⋊Q8—108no—
363C22⋊Q8—204no—
364C22⋊Q8—204no—
365C4×Q8—204no—
366C2×C4⋊C4—54no—
367C2×C4⋊C4—54no—
368C2×C4⋊C4—54no—
369C2×C4⋊C4—52no—
370C2×C4⋊C4—52no—
371C2×C4⋊C4—54no—
372C2×C4⋊C4—52no—
373C2×C4⋊C4—52no—
374C2×C4⋊C4—52no—
375C2×C4⋊C4—52no—
376C2×C4⋊C4—52no—
377C2×C4⋊C4—54no—
378C2×C4⋊C4—52no—
379C2×C4⋊C4—52no—
380C2×C4⋊C4—52no—
381C2×C4⋊C4—54no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—54no—
384C2×C22⋊C4—54no—
385C2×C22⋊C4—54no—
386C2×C22⋊C4—54no—
387C2×C22⋊C4—54no—
388C2×C22⋊C4—54no—
389C2×C22⋊C4Dic1018yescomplemented
390C2×C22⋊C4C22×D514yes—
391C2×C22⋊C4C22×D514yes—
392C2×C42—42no—
393C2×C42—42no—
394C2×C42—54no—
395C2×C42—54no—
396C2×C42C22×D512yes—
397C2×C42C22×D512yes—
398C2×C42C22×D512yes—
399C2×C42C22×D512yes—
400#95—54no—
401#95—54no—
402#95—54no—
403#95—54no—
404#95—54no—
405#95—54no—
406#95—54no—
407#95—54no—
408#95—54no—
409#95—54no—
410#95C22×D514yes—
411#95C22×D514yes—
412C22×C5—22no—
413C22×C5—22no—
414C22×C5—22no—
415C22×C5—22no—
416C22×C5—22no—
417C22×C5—22no—
418C22×C5—24no—
419C22×C5—24no—
420C22×C5—24no—
421C22×C5—416no—
422C22×C5—42no—
423C22×C5—42no—
424C22×C5—42no—
425C22×C5—42no—
426C22×C5#53314yes—
427C22×C5C23×Q811yescharacteristic
428C22×C5C22×#4111yescharacteristic
429C22×C5C22×#4111yescharacteristic
430C22×C5C2×H(5,𝔽2)11yescharacteristic
431C22×C5C2×H(5,𝔽2)11yescharacteristic
432C22×C5#58511yescharacteristic
433C22×C5#58511yescharacteristic
434C4×C5—24no—
435C4×C5—24no—
436C4×C5—22no—
437C4×C5—22no—
438C4×C5—24no—
439C4×C5—24no—
440C4×C5—42no—
441Dic5—44no—
442Dic5—44no—
443Dic5—44no—
444Dic5—44no—
445C24—22no—
446C24—22no—
447C24—41no—
448C24—41no—
449C24—41no—
450C24—41no—
451C24C2×Dic1014yes—
452C24C23×D512yes—
453C24C23×D511yescharacteristic
454C2×Q8—104no—
455C2×Q8—104no—
456C2×Q8—202no—
457C2×Q8—202no—
458C2×Q8—202no—
459C2×Q8—202no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—416no—
481C22×C4—54no—
482C22×C4—54no—
483C22×C4—54no—
484C22×C4—54no—
485C22×C4C2×Dic1014yes—
486C22×C4C2×Dic1014yes—
487C22×C4#70914yes—
488C22×C4#70914yes—
489C22×C4C23×D512yes—
490C22×C4C23×D512yes—
491C22×C4C23×D512yes—
492C22×C4C23×D512yes—
493C22×C4C23×D512yes—
494C22×C4C23×D511yescharacteristic
495C22×C4C23×D511yescharacteristic
496C4⋊C4—108no—
497C4⋊C4—108no—
498C4⋊C4—104no—
499C4⋊C4—104no—
500C4⋊C4—108no—
501C4⋊C4—108no—
502C4⋊C4—202no—
503C4⋊C4—202no—
504C4⋊C4—202no—
505C4⋊C4—204no—
506C4⋊C4—202no—
507C4⋊C4—202no—
508C4⋊C4—204no—
509C4⋊C4—202no—
510C4⋊C4—202no—
511C4⋊C4—202no—
512C22⋊C4—28no—
513C22⋊C4—108no—
514C22⋊C4—108no—
515C22⋊C4—204no—
516C22⋊C4—204no—
517C22⋊C4—204no—
518C22⋊C4—204no—
519C22⋊C4C2×Dic10116yes—
520C42—24no—
521C42—24no—
522C42—42no—
523C42—42no—
524C42—204no—
525C42—204no—
526C2×C5—24no—
527C2×C5—44no—
528C2×C5C2×#53311yescharacteristic
529C2×C5#345911yescharacteristic
530C2×C5#346011yescharacteristic
531C2×C5#349011yescharacteristic
532C2×C5#349311yescharacteristic
533C2×C5#350111yescharacteristic
534C2×C5#350111yescharacteristic
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23C22×Dic1014yes—
553C23C22×Dic1012yes—
554C23C2×#70912yes—
555C23C2×#70912yes—
556C23C24×D511yescharacteristic
557Q8—208no—
558Q8—404no—
559C2×C4—28no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—104no—
583C2×C4—104no—
584C2×C4—104no—
585C2×C4—104no—
586C2×C4—104no—
587C2×C4—104no—
588C2×C4—104no—
589C2×C4—104no—
590C2×C4—104no—
591C2×C4—104no—
592C2×C4—104no—
593C2×C4—104no—
594C2×C4C22×Dic1014yes—
595C2×C4C22×Dic1014yes—
596C2×C4C2×#70914yes—
597C2×C4C2×#70914yes—
598C2×C4#414412yes—
599C2×C4#414412yes—
600C5#3501211yescharacteristic, complemented, Sylow
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—416no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#6580414yes—
616C22C23×Dic1011yescharacteristic
617C22C22×#70911yescharacteristic
618C22C22×#70911yescharacteristic
619C22C2×#414411yescharacteristic
620C22C2×#414411yescharacteristic
621C22#6628311yescharacteristic
622C22#6628311yescharacteristic
623C4—24no—
624C4—24no—
625C4—24no—
626C4—22no—
627C4—22no—
628C4—24no—
629C4—42no—
630C4—204no—
631C4—204no—
632C4—204no—
633C4—204no—
634C2—24no—
635C2—44no—
636C2C2×#6580411yescharacteristic
637C2#1061759811yescharacteristic
638C2#1061764711yescharacteristic
639C2#1061766711yescharacteristic
640C2#1061766711yescharacteristic
641C2#1061767711yescharacteristic
642C2#1061767811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutator, socle, cosocleC23×C5C25
FittingC2×C4×C5×C22⋊C4C2
prime radicalC25×C5C23
hypercenterC2×C4×C22⋊C4D5
hypocenterC5#35012
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. #10613805C2semidirect
6. C4×#4070C2semidirect
7. #10605273C2semidirect
8. #10605267C2semidirect
9. #10605259C2semidirect
10. #10605229C2semidirect
11. #10605092C2semidirect
12. #10605092C2semidirect
13. #10605065C2semidirect
23. C2×#4016C22semidirect
24. C2×C4×Dic10C22semidirect
49. #65225C22semidirect
50. C4×C4⋊Dic5C22semidirect
51. #65222C22semidirect
52. #65222C22semidirect
53. #65222C22semidirect
54. C4×#683C22semidirect
55. #65218C22semidirect
56. #65217C22semidirect
57. #65217C22semidirect
58. #65217C22semidirect
59. #64950C22semidirect
60. #64950C22semidirect
61. #64945C22semidirect
62. #64945C22semidirect
63. #64944C22semidirect
64. #64943C22semidirect
65. #64936C22semidirect
66. #64935C22semidirect
83. C2×C5×C22⋊C4Q8semidirect
389. C2×C22⋊C4Dic10semidirect
600. C5#35012semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×C22⋊C42
#106157622
#106138482
#106138242
#106138052
C4×#40702
#106052732
#106052672
#106052592
#106052292
#106050922
#106050652
#350125