Finite Group Databasebeta

← #12021504

Subgroups — #12021504

Order 1280.

1818 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#95C211yesmaximal, characteristic
2C2×C4×C22⋊Dic5C211yesmaximal, characteristic
3C2×#64940C211yesmaximal, characteristic, complemented
4#10602920C211yesmaximal, characteristic
5#10602920C212yesmaximal
6#10602920C212yesmaximal
7#10600211C212yesmaximal, complemented
8#10600210C212yesmaximal, complemented
9#10600196C211yesmaximal, characteristic, complemented
10#10600195C211yesmaximal, characteristic, complemented
11#10600194C211yesmaximal, characteristic, complemented
12C24×C4×C5C2212yes—
13C24×C4×C5C2211yescharacteristic
14C22×C22⋊Dic5C2212yes—
15C22×C22⋊Dic5C2211yescharacteristic
16C22×C22⋊Dic5C2211yescharacteristic
17C22×C4×Dic5C2211yescharacteristic
18C22×C4×Dic5C2211yescharacteristic
19C2×C5×#95—24no—
20C2×C5×#95C2211yescharacteristic
21C2×C5×#95C2212yes—
22C2×C5×#95C2211yescharacteristic
23C4×C22⋊Dic5C418yescomplemented
24C2×#3963C2211yescharacteristic
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2211yescharacteristic
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2211yescharacteristic
36C2×#3963C2211yescharacteristic
37#64940—24no—
38#13901—51nomaximal, Sylow
39C25×C5C2311yescharacteristic
40C23×C4×C5—24no—
41C23×C4×C5—24no—
42C23×C4×C5—24no—
43C23×C4×C5—22no—
44C23×C4×C5—24no—
45C23×C4×C5—22no—
46C23×C4×C5—24no—
47C23×C4×C5—24no—
48C23×C4×C5—42no—
49C23×C4×C5—42no—
50C23×C4×C5—42no—
51C23×C4×C5C2×C414yes—
52C23×C4×C5C2311yescharacteristic
53C23×C4×C5C2312yes—
54C23×C4×C5C2312yes—
55C23×C4×C5C2311yescharacteristic
56C23×Dic5C2312yes—
57C23×Dic5C2312yes—
58C23×Dic5C2311yescharacteristic
59C23×Dic5C2311yescharacteristic
60C23×Dic5C2311yescharacteristic
61C23×Dic5C2311yescharacteristic
62C2×C22⋊Dic5—24no—
63C2×C22⋊Dic5—22no—
64C2×C22⋊Dic5—22no—
65C2×C22⋊Dic5—24no—
66C2×C22⋊Dic5—24no—
67C2×C22⋊Dic5—24no—
68C2×C22⋊Dic5C2×C414yes—
69C2×C22⋊Dic5C2×C414yes—
70C2×C4×Dic5—22no—
71C2×C4×Dic5—22no—
72C2×C4×Dic5—42no—
73C2×C4×Dic5C2×C414yescomplemented
74C2×C4×Dic5C2×C414yescomplemented
75C5×#95—24no—
76C5×#95—22no—
77C5×#95D414yes—
78#3963—24no—
79#3963—24no—
80#3963—22no—
81#3963—24no—
82#3963—24no—
83#3963—22no—
84#3963—42no—
85#3963—42no—
86#3963—42no—
87#3963—42no—
88#3963—42no—
89#3963C2×C414yescomplemented
90#3963C2×C414yescomplemented
91C2×#383—51no—
92C2×C4×C22⋊C4—51no—
93C22×#95D511yescharacteristic
94#1443—52no—
95#1443—52no—
96#1434—51no—
97#1434—51no—
98#1434—51no—
99C22⋊#95—52no—
100C22⋊#95—52no—
101C22⋊#95—51no—
102C24×C5—21no—
103C24×C5—21no—
104C24×C5—22no—
105C24×C5—22no—
106C24×C5—21no—
107C24×C5—21no—
108C24×C5C22×C412yes—
109C24×C5#4114yes—
110C24×C5#4114yes—
111C24×C5#4112yes—
112C24×C5#4112yes—
113C24×C5C2411yescharacteristic
114C22×C4×C5—22no—
115C22×C4×C5—22no—
116C22×C4×C5—216no—
117C22×C4×C5—28no—
118C22×C4×C5—21no—
119C22×C4×C5—22no—
120C22×C4×C5—21no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—28no—
135C22×C4×C5—44no—
136C22×C4×C5—48no—
137C22×C4×C5—48no—
138C22×C4×C5—48no—
139C22×C4×C5—48no—
140C22×C4×C5—48no—
141C22×C4×C5—48no—
142C22×C4×C5—48no—
143C22×C4×C5—48no—
144C22×C4×C5C22×C412yes—
145C22×C4×C5C22×C412yes—
146C22×C4×C5C2×D414yes—
147C22×C4×C5C2×D412yes—
148C22×C4×C5#4114yes—
149C22×C4×C5#4112yes—
150C22×Dic5—21no—
151C22×Dic5—21no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—21no—
161C22×Dic5—21no—
162C22×Dic5—21no—
163C22×Dic5—21no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—21no—
167C22×Dic5—21no—
168C22×Dic5—22no—
169C22×Dic5—22no—
170C22×Dic5—22no—
171C22×Dic5—22no—
172C22×Dic5—22no—
173C22×Dic5—22no—
174C22×Dic5—22no—
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—22no—
179C22×Dic5—22no—
180C22×Dic5—22no—
181C22×Dic5—22no—
182C22×Dic5C22×C412yes—
183C22×Dic5C22×C412yes—
184C22×Dic5C22×C412yes—
185C22×Dic5C22×C412yes—
186C22⋊Dic5—28no—
187C22⋊Dic5—28no—
188C22⋊Dic5—48no—
189C4×Dic5—28no—
190C4×Dic5—44no—
191C24×C4C2×D512yes—
192C24×C4C2×D511yescharacteristic
193C22×C22⋊C4—51no—
194C22×C22⋊C4—51no—
195C22×C22⋊C4—52no—
196C22×C42—51no—
197C22×C42—51no—
198#383—104no—
199C4×C22⋊C4—58no—
200C2×#95—24no—
201C2×#95—51no—
202C2×#95—51no—
203C2×#95—51no—
204C2×#95—52no—
205C2×#95—52no—
206C2×#95—51no—
207C2×#95—52no—
208C2×#95—52no—
209C2×#95—52no—
210C2×#95—52no—
211C2×#95—52no—
212C2×#95—52no—
213C2×#95—52no—
214C2×#95C2×D511yescharacteristic
215C2×#95C2×D512yes—
216C2×#95C2×D511yescharacteristic
217C23×C5—22no—
218C23×C5—22no—
219C23×C5—22no—
220C23×C5—22no—
221C23×C5—24no—
222C23×C5—24no—
223C23×C5—24no—
224C23×C5—24no—
225C23×C5—24no—
226C23×C5—22no—
227C23×C5—22no—
228C23×C5—24no—
229C23×C5—24no—
230C23×C5—24no—
231C23×C5—24no—
232C23×C5—22no—
233C23×C5—22no—
234C23×C5—24no—
235C23×C5#11714yes—
236C23×C5#11714yes—
237C23×C5#11714yes—
238C23×C5#12618yes—
239C23×C5#12618yes—
240C23×C5C23×C411yescharacteristic
241C23×C5C22×D411yescharacteristic
242C23×C5C2×#4112yes—
243C23×C5C2×#4111yescharacteristic
244C23×C5C2×#4112yes—
245C23×C5C2×#4112yes—
246C23×C5C2×#4111yescharacteristic
247C23×C5C2×#4111yescharacteristic
248C23×C5H(5,𝔽2)12yes—
249C23×C5H(5,𝔽2)11yescharacteristic
250C23×C52−1+411yescharacteristic
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—24no—
259C2×C4×C5—24no—
260C2×C4×C5—24no—
261C2×C4×C5—24no—
262C2×C4×C5—24no—
263C2×C4×C5—24no—
264C2×C4×C5—416no—
265C2×C4×C5—432no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5C4×D414yes—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—22no—
281C2×Dic5—22no—
282C2×Dic5—22no—
283C2×Dic5—22no—
284C2×Dic5—22no—
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—22no—
288C2×Dic5—22no—
289C2×Dic5—42no—
290C2×Dic5—42no—
291C2×Dic5—42no—
292C2×Dic5—42no—
293C2×Dic5—41no—
294C2×Dic5—41no—
295C2×Dic5—42no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5—42no—
303C2×Dic5—42no—
304C2×Dic5—42no—
305C2×Dic5—42no—
306C2×Dic5—42no—
307C2×Dic5—42no—
308C2×Dic5—42no—
309C2×Dic5—42no—
310C2×Dic5—42no—
311C2×Dic5—42no—
312C2×Dic5—42no—
313C25C22×D511yescharacteristic
314C23×C4—24no—
315C23×C4—24no—
316C23×C4—24no—
317C23×C4—22no—
318C23×C4—24no—
319C23×C4—22no—
320C23×C4—24no—
321C23×C4—24no—
322C23×C4—42no—
323C23×C4—42no—
324C23×C4—42no—
325C23×C4—51no—
326C23×C4—51no—
327C23×C4—51no—
328C23×C4—51no—
329C23×C4—52no—
330C23×C4—52no—
331C23×C4C4×D514yes—
332C23×C4C22×D511yescharacteristic
333C23×C4C22×D512yes—
334C23×C4C22×D512yes—
335C23×C4C22×D511yescharacteristic
336C2×C22⋊C4—54no—
337C2×C22⋊C4—54no—
338C2×C22⋊C4—104no—
339C2×C22⋊C4—102no—
340C2×C22⋊C4—102no—
341C2×C22⋊C4—104no—
342C2×C22⋊C4—104no—
343C2×C22⋊C4—104no—
344C2×C42—54no—
345C2×C42—54no—
346C2×C42—102no—
347C2×C42—102no—
348C2×C42—202no—
349#95—24no—
350#95—22no—
351#95—54no—
352#95—54no—
353#95—104no—
354#95—104no—
355#95—102no—
356#95—104no—
357#95—104no—
358#95—102no—
359#95—202no—
360#95—202no—
361#95—202no—
362#95—202no—
363#95—202no—
364#95C22⋊D514yes—
365C22×C5—24no—
366C22×C5—24no—
367C22×C5—24no—
368C22×C5—24no—
369C22×C5—24no—
370C22×C5—28no—
371C22×C5—28no—
372C22×C5—28no—
373C22×C5—28no—
374C22×C5—24no—
375C22×C5#38318yescomplemented
376C22×C5C2×#11711yescharacteristic
377C22×C5C2×#11711yescharacteristic
378C22×C5C2×#11711yescharacteristic
379C22×C5C2×C4×D411yescharacteristic
380C22×C5C4×#4111yescharacteristic
381C22×C5#51811yescharacteristic
382C22×C5#52011yescharacteristic
383C22×C5#52512yes—
384C22×C5C2×#12612yes—
385C22×C5C2×#12612yes—
386C22×C5#52912yes—
387C22×C5#53812yes—
388C22×C5#53912yes—
389C22×C5#54612yes—
390C22×C5#54611yescharacteristic
391C22×C5#54811yescharacteristic
392C22×C5#55112yes—
393C22×C5#55112yes—
394C22×C5#55112yes—
395C22×C5#55212yes—
396C22×C5#55211yescharacteristic
397C22×C5#55311yescharacteristic
398C22×C5#55312yes—
399C4×C5—24no—
400C4×C5—42no—
401C4×C5—44no—
402C4×C5—44no—
403Dic5—42no—
404Dic5—42no—
405Dic5—42no—
406Dic5—42no—
407Dic5—82no—
408Dic5—82no—
409C24—22no—
410C24—21no—
411C24—22no—
412C24—21no—
413C24—21no—
414C24—21no—
415C24C2×C4×D512yes—
416C24#70914yes—
417C24#71114yes—
418C24#71112yes—
419C24#71312yes—
420C24C23×D511yescharacteristic
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—216no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—52no—
452C22×C4—52no—
453C22×C4—52no—
454C22×C4—52no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—101no—
458C22×C4—101no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—101no—
462C22×C4—101no—
463C22×C4—101no—
464C22×C4—101no—
465C22×C4—102no—
466C22×C4—102no—
467C22×C4—101no—
468C22×C4—101no—
469C22×C4—101no—
470C22×C4—101no—
471C22×C4—101no—
472C22×C4—101no—
473C22×C4—102no—
474C22×C4—102no—
475C22×C4—102no—
476C22×C4—102no—
477C22×C4—102no—
478C22×C4—102no—
479C22×C4—102no—
480C22×C4—102no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—102no—
484C22×C4—102no—
485C22×C4—102no—
486C22×C4—102no—
487C22×C4C2×C4×D512yes—
488C22×C4C2×C4×D512yes—
489C22×C4#70914yes—
490C22×C4#70912yes—
491C22×C4C2×C22⋊D514yes—
492C22×C4C2×C22⋊D512yes—
493C22⋊C4—108no—
494C22⋊C4—108no—
495C22⋊C4—208no—
496C42—108no—
497C42—204no—
498C2×C5—28no—
499C2×C5C2×#38311yescharacteristic
500C2×C5#229011yescharacteristic
501C2×C5#231811yescharacteristic
502C2×C5#234211yescharacteristic
503C2×C5#234311yescharacteristic
504C2×C5#235211yescharacteristic
505C2×C5#235311yescharacteristic
506C2×C5#248012yes—
507C2×C5#248112yes—
508C2×C5#249312yes—
509C2×C5#251012yes—
510C23—22no—
511C23—24no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23#401814yes—
529C23#402314yes—
530C23#402518yes—
531C23#403814yes—
532C23#404418yes—
533C23C22×C4×D511yescharacteristic
534C23C2×#70912yes—
535C23C2×#70912yes—
536C23C2×#70911yescharacteristic
537C23C2×#71112yes—
538C23C2×#71111yescharacteristic
539C23#414412yes—
540C23#414411yescharacteristic
541C23C2×#71311yescharacteristic
542C23#414711yescharacteristic
543C23C22×C22⋊D511yescharacteristic
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—416no—
558C2×C4—432no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—102no—
570C2×C4—102no—
571C2×C4—102no—
572C2×C4—102no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—102no—
576C2×C4—102no—
577C2×C4—102no—
578C2×C4—102no—
579C2×C4—102no—
580C2×C4—102no—
581C2×C4—202no—
582C2×C4—202no—
583C2×C4—202no—
584C2×C4—201no—
585C2×C4—202no—
586C2×C4—201no—
587C2×C4—202no—
588C2×C4—202no—
589C2×C4—201no—
590C2×C4—201no—
591C2×C4—202no—
592C2×C4—202no—
593C2×C4—202no—
594C2×C4—202no—
595C2×C4—202no—
596C2×C4—202no—
597C2×C4—202no—
598C2×C4—202no—
599C2×C4—202no—
600C2×C4—202no—
601C2×C4—202no—
602C2×C4—202no—
603C2×C4—202no—
604C2×C4—202no—
605C2×C4C4×C22⋊D514yes—
606C5#1390111yescharacteristic, complemented, Sylow
607C22—24no—
608C22—24no—
609C22—28no—
610C22—28no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—28no—
615C22—24no—
616C22—28no—
617C22#6494018yescomplemented
618C22C2×#401811yescharacteristic
619C22C4×#70911yescharacteristic
620C22#6581312yes—
621C22C2×#402311yescharacteristic
622C22C2×#402512yes—
623C22#6582011yescharacteristic
624C22#6582812yes—
625C22#6582912yes—
626C22#6582912yes—
627C22#6582912yes—
628C22C2×#403811yescharacteristic
629C22#6583811yescharacteristic
630C22C2×#404412yes—
631C22#6584712yes—
632C22#6584711yescharacteristic
633C22#6584812yes—
634C22#6584811yescharacteristic
635C22#6584912yes—
636C22C2×C4×C22⋊D511yescharacteristic
637C22#6612512yes—
638C22#6614012yes—
639C22#6614011yescharacteristic
640C22#6615111yescharacteristic
641C4—24no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—202no—
646C4—202no—
647C4—202no—
648C4—202no—
649C4—402no—
650C4—402no—
651C2—28no—
652C2C2×#6494011yescharacteristic
653C2#1060506611yescharacteristic
654C2#1060506711yescharacteristic
655C2#1060508412yes—
656C2#1060508712yes—
657C2#1060509811yescharacteristic
658C2#1061379611yescharacteristic
659C2#1061382911yescharacteristic
660C2#1061384312yes—
661C2#1061389211yescharacteristic
662C2#1061390612yes—
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC22×C5×#95C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC22×#95D5
hypocenterC5#13901
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
3. C2×#64940C2semidirect
7. #10600211C2semidirect
8. #10600210C2semidirect
9. #10600196C2semidirect
10. #10600195C2semidirect
11. #10600194C2semidirect
23. C4×C22⋊Dic5C4semidirect
73. C2×C4×Dic5C2×C4semidirect
74. C2×C4×Dic5C2×C4semidirect
89. #3963C2×C4semidirect
90. #3963C2×C4semidirect
375. C22×C5#383semidirect
606. C5#13901semidirect
617. C22#64940semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#952
C2×C4×C22⋊Dic52
C2×#649402
#106029202
#106002112
#106002102
#106001962
#106001952
#106001942
#139015