Finite Group Databasebeta

← #12021287

Subgroups — #12021287

Order 1280.

2222 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×C4⋊C4C211yesmaximal, characteristic
2C2×#64956C211yesmaximal, characteristic
3C2×#64939C212yesmaximal, complemented
4C2×#64955C211yesmaximal, characteristic
5C5×#1442C211yesmaximal, characteristic, complemented
6#10602360C211yesmaximal, characteristic, complemented
7#10600218C212yesmaximal
8#10600218C212yesmaximal
9#10600208C211yesmaximal, characteristic, complemented
10#10600205C211yesmaximal, characteristic, complemented
11#10600197C212yesmaximal, complemented
12C23×C4×D5C2212yes—
13C23×C4×D5C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×#685C2211yescharacteristic
16C22×#685C2211yescharacteristic
17C22×#685C2212yes—
18C22×C4⋊Dic5C2211yescharacteristic
19C22×#683C2211yescharacteristic
20C22×#683C2211yescharacteristic
21C22×#683C2212yes—
22C22×C4×Dic5C2212yes—
23C2×D5×C4⋊C4—28no—
24C2×D5×C4⋊C4—24no—
25C2×C5×#95C2211yescharacteristic
26C2×C5×#95C2212yes—
27C2×C5×#95C2212yes—
28C2×C5×#95C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2211yescharacteristic
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36C2×#3963C2211yescharacteristic
37#64956—24no—
38#64955C418yescomplemented
39#64939—28no—
40#13881—51nomaximal, Sylow
41C24×D5C2311yescharacteristic
42C23×C4×C5C2311yescharacteristic
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2312yes—
45C23×C4×C5C2312yes—
46C23×C4×C5C2311yescharacteristic
47C23×Dic5C2311yescharacteristic
48C23×Dic5C2311yescharacteristic
49C23×Dic5C2312yes—
50C23×Dic5C2312yes—
51C23×Dic5C2311yescharacteristic
52C22×C4×D5—24no—
53C22×C4×D5—24no—
54C22×C4×D5—24no—
55C22×C4×D5—24no—
56C22×C4×D5—24no—
57C22×C4×D5—24no—
58C22×C4×D5—24no—
59C22×C4×D5—42no—
60C22×C4×D5—42no—
61C22×C4×D5C2×C414yes—
62C22×C4×D5D414yes—
63C22×C4×D5Q814yes—
64C2×C5×C4⋊C4—24no—
65C2×C5×C4⋊C4—22no—
66C2×#685—24no—
67C2×#685—24no—
68C2×#685—22no—
69C2×#685—22no—
70C2×#685—24no—
71C2×#685—24no—
72C2×#685C2×C418yes—
73C2×C4⋊Dic5—22no—
74C2×C4⋊Dic5—24no—
75C2×#683—24no—
76C2×#683—24no—
77C2×#683—24no—
78C2×#683—22no—
79C2×#683—22no—
80C2×#683—24no—
81C2×#683D418yes—
82C2×C4×Dic5—24no—
83C2×C4×Dic5—24no—
84C2×C4×Dic5C2×C418yescomplemented
85D5×C4⋊C4—232no—
86C5×#95—24no—
87C5×#95—22no—
88C5×#95—42no—
89C5×#95C2×C414yescomplemented
90#3963—24no—
91#3963—22no—
92#3963—24no—
93#3963—24no—
94#3963—42no—
95#3963—42no—
96#3963—42no—
97#3963C2×C414yescomplemented
98C23×C4⋊C4—51no—
99C2×#382—52no—
100C2×#380—51no—
101C2×C4×C22⋊C4—51no—
102#1442—51no—
103#1442—51no—
104#1442—51no—
105#1442D511yescharacteristic, complemented
106C4⋊#95—52no—
107C22⋊#95—52no—
108C22⋊#95—52no—
109C24×C5C2411yescharacteristic
110C23×D5—22no—
111C23×D5—22no—
112C23×D5—21no—
113C23×D5—21no—
114C23×D5—21no—
115C23×D5—21no—
116C23×D5C22×C412yes—
117C23×D5C2×D412yes—
118C23×D5C2×Q812yes—
119C23×D5#4114yes—
120C23×D5#4114yes—
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—22no—
135C22×C4×C5—22no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—22no—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5C22×C414yes—
146C22×C4×C5C22×C412yes—
147C22×C4×C5C2×D414yes—
148C22×C4×C5C2×D412yes—
149C22×C4×C5C2×Q812yes—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—21no—
164C22×Dic5—21no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—21no—
169C22×Dic5—21no—
170C22×Dic5—21no—
171C22×Dic5—21no—
172C22×Dic5C22×C414yes—
173C22×Dic5C22×C412yes—
174C22×Dic5C2×D414yes—
175C22×Dic5C2×D412yes—
176C22×Dic5C2×D414yes—
177C22×Dic5C2×Q812yes—
178C22×Dic5#4114yes—
179C2×C4×D5—216no—
180C2×C4×D5—28no—
181C2×C4×D5—216no—
182C2×C4×D5—28no—
183C2×C4×D5—216no—
184C2×C4×D5—48no—
185C2×C4×D5—48no—
186C2×C4×D5—48no—
187C2×C4×D5—48no—
188C2×C4×D5—48no—
189C2×C4×D5—48no—
190C2×C4×D5C4⋊C4116yes—
191C5×C4⋊C4—28no—
192#685—216no—
193#685—44no—
194#685—44no—
195C4⋊Dic5—28no—
196#683—216no—
197#683—44no—
198#683—44no—
199C4×Dic5—216no—
200C24×C4—51no—
201C24×C4—52no—
202C22×C4⋊C4—52no—
203C22×C4⋊C4—51no—
204C22×C4⋊C4—51no—
205C22×C4⋊C4—51no—
206C22×C4⋊C4—108no—
207C22×C4⋊C4—104no—
208C22×C4⋊C4C2×D511yescharacteristic
209C22×C22⋊C4—52no—
210C22×C22⋊C4—51no—
211C22×C22⋊C4—51no—
212C22×C42—52no—
213#382—108no—
214#380—104no—
215C4×C22⋊C4—58no—
216C2×#95—51no—
217C2×#95—52no—
218C2×#95—52no—
219C2×#95—52no—
220C2×#95—51no—
221C2×#95—52no—
222C2×#95—52no—
223C2×#95—52no—
224C2×#95C2×D512yes—
225C2×#95C2×D511yescharacteristic
226C2×#95C2×D512yes—
227C2×#95C2×D511yescharacteristic
228C23×C5C23×C411yescharacteristic
229C23×C5C22×D411yescharacteristic
230C23×C5C22×D412yes—
231C23×C5C22×Q811yescharacteristic
232C23×C5C2×#4112yes—
233C23×C5C2×#4112yes—
234C23×C5C2×#4112yes—
235C23×C5H(5,𝔽2)11yescharacteristic
236C23×C5H(5,𝔽2)11yescharacteristic
237C23×C52−1+411yescharacteristic
238C23×C52−1+411yescharacteristic
239C22×D5—24no—
240C22×D5—22no—
241C22×D5—22no—
242C22×D5—22no—
243C22×D5—24no—
244C22×D5—22no—
245C22×D5—22no—
246C22×D5—22no—
247C22×D5—22no—
248C22×D5—22no—
249C22×D5—24no—
250C22×D5—24no—
251C22×D5—24no—
252C22×D5—24no—
253C22×D5—24no—
254C22×D5—24no—
255C22×D5—24no—
256C22×D5—24no—
257C22×D5C2×C4⋊C414yes—
258C22×D5#11718yes—
259C22×D5#12414yes—
260C22×D5#12414yes—
261C22×D5#12518yes—
262C2×C4×C5—24no—
263C2×C4×C5—22no—
264C2×C4×C5—24no—
265C2×C4×C5—22no—
266C2×C4×C5—24no—
267C2×C4×C5—24no—
268C2×C4×C5—24no—
269C2×C4×C5—24no—
270C2×C4×C5—24no—
271C2×C4×C5—24no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—42no—
287C2×C4×C5—42no—
288C2×C4×C5—42no—
289C2×C4×C5—42no—
290C2×C4×C5C2×C4⋊C414yes—
291C2×Dic5—24no—
292C2×Dic5—24no—
293C2×Dic5—24no—
294C2×Dic5—24no—
295C2×Dic5—24no—
296C2×Dic5—24no—
297C2×Dic5—42no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5—42no—
303C2×Dic5—42no—
304C2×Dic5—42no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—42no—
312C2×Dic5—42no—
313C2×Dic5—42no—
314C2×Dic5—42no—
315C2×Dic5C2×C4⋊C414yes—
316C2×Dic5C4×D418yes—
317C2×Dic5#12014yes—
318C2×Dic5C22⋊Q818yes—
319C2×Dic5#12314yes—
320C4×D5—232no—
321C4×D5—432no—
322C25—51no—
323C23×C4—51no—
324C23×C4—54no—
325C23×C4—54no—
326C23×C4—54no—
327C23×C4—52no—
328C23×C4—52no—
329C23×C4—51no—
330C23×C4—51no—
331C23×C4—104no—
332C23×C4—104no—
333C23×C4—104no—
334C23×C4—104no—
335C23×C4—104no—
336C23×C4—104no—
337C23×C4—104no—
338C23×C4—202no—
339C23×C4—202no—
340C23×C4C22×D511yescharacteristic
341C23×C4C22×D511yescharacteristic
342C23×C4C22×D512yes—
343C23×C4C22×D512yes—
344C23×C4C22×D511yescharacteristic
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—58no—
348C2×C4⋊C4—1032no—
349C2×C4⋊C4—104no—
350C2×C4⋊C4—104no—
351C2×C4⋊C4—104no—
352C2×C4⋊C4—102no—
353C2×C4⋊C4—102no—
354C2×C4⋊C4—102no—
355C2×C4⋊C4—104no—
356C2×C4⋊C4—104no—
357C2×C22⋊C4—58no—
358C2×C22⋊C4—104no—
359C2×C22⋊C4—104no—
360C2×C22⋊C4—102no—
361C2×C22⋊C4—102no—
362C2×C22⋊C4—104no—
363C2×C22⋊C4—104no—
364C2×C42—58no—
365C2×C42—104no—
366C2×C42—104no—
367#95—22no—
368#95—24no—
369#95—42no—
370#95—54no—
371#95—104no—
372#95—104no—
373#95—102no—
374#95—104no—
375#95—202no—
376#95—202no—
377#95—202no—
378#95C4×D514yescomplemented
379C22×C5C22×C4⋊C411yescharacteristic
380C22×C5C2×#11712yes—
381C22×C5C2×C4×D412yes—
382C22×C5#52011yescharacteristic
383C22×C5#52011yescharacteristic
384C22×C5C2×#12011yescharacteristic
385C22×C5C2×C22⋊Q812yes—
386C22×C5C2×#12311yescharacteristic
387C22×C5C2×#12411yescharacteristic
388C22×C5C2×#12411yescharacteristic
389C22×C5C2×#12512yes—
390C22×C5#53511yescharacteristic
391C22×C5#53611yescharacteristic
392C22×C5#54411yescharacteristic
393C22×C5#54411yescharacteristic
394C22×C5#54612yes—
395C22×C5#54612yes—
396C22×C5#54812yes—
397C22×C5#54812yes—
398C22×C5#55212yes—
399C22×C5#55212yes—
400C22×C5#55312yes—
401C22×C5#55312yes—
402C2×D5—24no—
403C2×D5—28no—
404C2×D5—28no—
405C2×D5—24no—
406C2×D5—24no—
407C2×D5—28no—
408C2×D5—28no—
409C2×D5—24no—
410C2×D5—24no—
411C2×D5—24no—
412C2×D5#38218yescomplemented
413C4×C5—24no—
414C4×C5—44no—
415C4×C5—44no—
416C4×C5—81no—
417C4×C5—81no—
418Dic5—44no—
419Dic5—44no—
420Dic5—81no—
421Dic5—81no—
422Dic5#38518yes—
423C24—52no—
424C24—52no—
425C24—52no—
426C24—54no—
427C24—54no—
428C24—102no—
429C24—101no—
430C24—101no—
431C24—102no—
432C24—101no—
433C24—101no—
434C24C23×D511yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—52no—
460C22×C4—54no—
461C22×C4—52no—
462C22×C4—52no—
463C22×C4—54no—
464C22×C4—516no—
465C22×C4—54no—
466C22×C4—54no—
467C22×C4—108no—
468C22×C4—1016no—
469C22×C4—1016no—
470C22×C4—108no—
471C22×C4—1016no—
472C22×C4—102no—
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—101no—
486C22×C4—101no—
487C22×C4—101no—
488C22×C4—101no—
489C22×C4—102no—
490C22×C4—102no—
491C22×C4—102no—
492C22×C4—101no—
493C22×C4—101no—
494C22×C4—208no—
495C22×C4—208no—
496C22×C4—208no—
497C22×C4—208no—
498C22×C4—208no—
499C22×C4—208no—
500C22×C4C2×C4×D514yes—
501C22×C4C2×C4×D512yes—
502C22×C4D4×D514yes—
503C22×C4D4×D512yes—
504C22×C4Q8×D512yes—
505C4⋊C4—28no—
506C4⋊C4—1016no—
507C4⋊C4—108no—
508C4⋊C4—204no—
509C4⋊C4—204no—
510C22⋊C4—1016no—
511C22⋊C4—204no—
512C22⋊C4—204no—
513C42—1016no—
514C2×C5C2×#38211yescharacteristic
515C2×C5C2×#38511yescharacteristic
516C2×C5#231711yescharacteristic
517C2×C5#235212yes—
518C2×C5#235212yes—
519C2×C5#243511yescharacteristic
520C2×C5#243611yescharacteristic
521C2×C5#244212yes—
522C2×C5#244212yes—
523C2×C5#250311yescharacteristic
524C2×C5#250411yescharacteristic
525D5—28no—
526C23—54no—
527C23—58no—
528C23—54no—
529C23—54no—
530C23—58no—
531C23—102no—
532C23—104no—
533C23—102no—
534C23—102no—
535C23—102no—
536C23—102no—
537C23—104no—
538C23—104no—
539C23—102no—
540C23—104no—
541C23—102no—
542C23—104no—
543C23—104no—
544C23—104no—
545C23—102no—
546C23—104no—
547C23—104no—
548C23—104no—
549C23C22×C4×D511yescharacteristic
550C23C2×D4×D511yescharacteristic
551C23C2×D4×D512yes—
552C23#414411yescharacteristic
553C23C2×Q8×D511yescharacteristic
554C23#414711yescharacteristic
555C23D5×#4112yes—
556C23D5×#4112yes—
557C23D5×#4112yes—
558C23#414911yescharacteristic
559C23#415011yescharacteristic
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—54no—
589C2×C4—58no—
590C2×C4—54no—
591C2×C4—58no—
592C2×C4—54no—
593C2×C4—1032no—
594C2×C4—104no—
595C2×C4—104no—
596C2×C4—104no—
597C2×C4—104no—
598C2×C4—104no—
599C2×C4—104no—
600C2×C4—2032no—
601C2×C4—202no—
602C2×C4—202no—
603C2×C4—202no—
604C2×C4—202no—
605C2×C4—202no—
606C2×C4—202no—
607C2×C4—202no—
608C2×C4—202no—
609C2×C4—201no—
610C2×C4—201no—
611C2×C4—201no—
612C2×C4—202no—
613C2×C4—201no—
614C2×C4—202no—
615C2×C4—201no—
616C2×C4—201no—
617C2×C4—202no—
618C2×C4—202no—
619C2×C4D5×C4⋊C414yes—
620C5#1388111yescharacteristic, complemented, Sylow
621C22—58no—
622C22—108no—
623C22—104no—
624C22—104no—
625C22—104no—
626C22—108no—
627C22—108no—
628C22—108no—
629C22—104no—
630C22—104no—
631C22—104no—
632C22C2×D5×C4⋊C411yescharacteristic
633C22D5×#11712yes—
634C22C4×D4×D512yes—
635C22#6588011yescharacteristic
636C22#6590811yescharacteristic
637C22D5×#12011yescharacteristic
638C22#6594111yescharacteristic
639C22#6594412yes—
640C22D5×C22⋊Q812yes—
641C22#6596211yescharacteristic
642C22#6596712yes—
643C22D5×#12311yescharacteristic
644C22#6598812yes—
645C22#6599512yes—
646C22D5×#12411yescharacteristic
647C22D5×#12411yescharacteristic
648C22#6601112yes—
649C22D5×#12512yes—
650C22#6602311yescharacteristic
651C22#6602712yes—
652C22#6604212yes—
653C22#6604312yes—
654C22#6606311yescharacteristic
655C4—24no—
656C4—44no—
657C4—44no—
658C4—58no—
659C4—81no—
660C4—81no—
661C4—204no—
662C4—204no—
663C4—401no—
664C4—401no—
665C2—108no—
666C2D5×#38211yescharacteristic
667C2D5×#38511yescharacteristic
668C2#1060703612yes—
669C2#1060756512yes—
670C2#1060769911yescharacteristic
671C2#1060851211yescharacteristic
672C2#1060863611yescharacteristic
673C2#1060922311yescharacteristic
674C2#1060947712yes—
675C2#1060986612yes—
676C2#1060988311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×#1442C2
socle, cosocleC24×C5C24
prime radicalC24×D5C23
hypercenter#1442D5
hypocenterC5#13881
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
3. C2×#64939C2semidirect
5. C5×#1442C2semidirect
6. #10602360C2semidirect
9. #10600208C2semidirect
10. #10600205C2semidirect
11. #10600197C2semidirect
38. #64955C4semidirect
84. C2×C4×Dic5C2×C4semidirect
89. C5×#95C2×C4semidirect
97. #3963C2×C4semidirect
105. #1442D5semidirect
378. #95C4×D5semidirect
412. C2×D5#382semidirect
620. C5#13881semidirect

Maximal subgroups

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

subgroupindex
C22×D5×C4⋊C42
C2×#649562
C2×#649392
C2×#649552
C5×#14422
#106023602
#106002182
#106002082
#106002052
#106001972
#138815