Finite Group Databasebeta

← #12022724

Subgroups — #12022724

Order 1280.

1506 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×C4⋊C4C211yesmaximal, characteristic
2C2×#65263C211yesmaximal, characteristic
3C2×#65263C211yesmaximal, characteristic
4C2×#65218C211yesmaximal, characteristic
5C2×C4×#683C211yesmaximal, characteristic
6#10602356C211yesmaximal, characteristic
7#10602356C211yesmaximal, characteristic
8#10602329C212yesmaximal
9#10602327C212yesmaximal
10#10600211C212yesmaximal
11#10600206C212yesmaximal
12C23×Dic10C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C42×C5C2212yes—
16C22×C42×C5C2211yescharacteristic
17C22×C4⋊Dic5C2211yescharacteristic
18C22×#683C2211yescharacteristic
19C22×#683C2211yescharacteristic
20C22×C4×Dic5C2211yescharacteristic
21C22×C4×Dic5C2211yescharacteristic
22C4×C5×C4⋊C4—24no—
23C2×C5×#95C2212yes—
24C2×#3963C2212yes—
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2211yescharacteristic
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36#65263—24no—
37#65263C418yescomplemented
38C4×#683—24no—
39#65218—24no—
40#14021—51nomaximal, Sylow
41C23×C4×C5C2312yes—
42C23×C4×C5C2311yescharacteristic
43C23×C4×C5C2312yes—
44C23×C4×C5C2311yescharacteristic
45C23×C4×C5C2311yescharacteristic
46C23×Dic5C2312yes—
47C23×Dic5C2311yescharacteristic
48C23×Dic5C2312yes—
49C23×Dic5C2311yescharacteristic
50C23×Dic5C2311yescharacteristic
51C23×Dic5C2311yescharacteristic
52C22×Dic10—22no—
53C22×Dic10—22no—
54C22×Dic10—22no—
55C22×Dic10—22no—
56C22×Dic10—22no—
57C22×Dic10—22no—
58C22×Dic10C2×C414yes—
59C2×C5×C4⋊C4—22no—
60C2×C5×C4⋊C4—22no—
61C2×C5×C4⋊C4—22no—
62C2×C5×C4⋊C4—22no—
63C2×C5×C4⋊C4C2×C414yes—
64C2×C5×C4⋊C4D414yes—
65C2×C42×C5—22no—
66C2×C42×C5—22no—
67C2×C42×C5—24no—
68C2×C42×C5—24no—
69C2×C42×C5—42no—
70C2×C42×C5D414yes—
71C2×C4⋊Dic5—22no—
72C2×C4⋊Dic5—22no—
73C2×C4⋊Dic5—22no—
74C2×#683—22no—
75C2×#683—22no—
76C2×#683—22no—
77C2×#683—22no—
78C2×#683—22no—
79C2×#683Q814yescomplemented
80C2×C4×Dic5—22no—
81C2×C4×Dic5—22no—
82C2×C4×Dic5—22no—
83C2×C4×Dic5—22no—
84C2×C4×Dic5—22no—
85C2×C4×Dic5C2×C414yes—
86C5×#95—24no—
87#3963—24no—
88#3963—24no—
89#3963—22no—
90#3963—22no—
91#3963—24no—
92#3963—24no—
93#3963—24no—
94#3963—24no—
95#3963—42no—
96#3963—42no—
97#3963C2×C418yes—
98#3963C2×C418yes—
99C2×#391—51no—
100C2×#391—51no—
101C2×#391—51no—
102C2×C4×C4⋊C4—51no—
103C2×C4×C4⋊C4D511yescharacteristic
104#1443—52no—
105#1442—52no—
106#1436—52no—
107C4⋊#95—51no—
108C4⋊#95—51no—
109#1434—52no—
110C24×C5C2411yescharacteristic
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—22no—
116C22×C4×C5—21no—
117C22×C4×C5—21no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—21no—
126C22×C4×C5—21no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—22no—
133C22×C4×C5C22×C414yes—
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2×D412yes—
136C22×C4×C5C2×D412yes—
137C22×C4×C5C2×D412yes—
138C22×C4×C5C2×D414yes—
139C22×C4×C5C2×D412yes—
140C22×C4×C5C2×Q812yes—
141C22×C4×C5#4112yes—
142C22×C4×C5#4112yes—
143C22×C4×C5#4112yes—
144C22×C4×C5#4112yes—
145C22×C4×C5#4112yes—
146C22×Dic5—21no—
147C22×Dic5—21no—
148C22×Dic5—21no—
149C22×Dic5—21no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—21no—
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—21no—
162C22×Dic5—21no—
163C22×Dic5—21no—
164C22×Dic5—21no—
165C22×Dic5—21no—
166C22×Dic5—21no—
167C22×Dic5—21no—
168C22×Dic5—21no—
169C22×Dic5—21no—
170C22×Dic5—21no—
171C22×Dic5—21no—
172C22×Dic5—21no—
173C22×Dic5—21no—
174C22×Dic5—22no—
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—22no—
179C22×Dic5—22no—
180C22×Dic5—22no—
181C22×Dic5C22×C414yes—
182C22×Dic5C22×C412yes—
183C22×Dic5C22×C412yes—
184C22×Dic5C2×Q812yes—
185C22×Dic5C2×Q812yes—
186C22×Dic5#4112yes—
187C22×Dic5#4112yes—
188C2×Dic10—28no—
189C2×Dic10—28no—
190C2×Dic10—28no—
191C2×Dic10—44no—
192C2×Dic10—44no—
193C2×Dic10—44no—
194C2×Dic10—44no—
195C5×C4⋊C4—28no—
196C5×C4⋊C4—28no—
197C42×C5—28no—
198C42×C5—48no—
199C4⋊Dic5—44no—
200#683—28no—
201#683—28no—
202C4×Dic5—28no—
203C4×Dic5—44no—
204C23×Q8—51no—
205C22×C4⋊C4—51no—
206C22×C4⋊C4—51no—
207C22×C4⋊C4—51no—
208C22×C4⋊C4C2×D511yescharacteristic
209C22×C4⋊C4C2×D511yescharacteristic
210C22×C42—51no—
211C22×C42—51no—
212C22×C42C2×D512yes—
213C22×C42C2×D511yescharacteristic
214#391—58no—
215#391—104no—
216#391—104no—
217C4×C4⋊C4—24no—
218C4×C4⋊C4—104no—
219C2×#95—52no—
220C2×#95—52no—
221C2×#95—51no—
222C2×#95—52no—
223C2×#95—52no—
224C2×#95—52no—
225C2×#95—52no—
226C2×#95—52no—
227C2×#95—52no—
228C2×#95—51no—
229C2×#95—52no—
230C2×#95—52no—
231C2×#95C2×D512yes—
232C23×C5C23×C411yescharacteristic
233C23×C5C22×D411yescharacteristic
234C23×C5C22×D411yescharacteristic
235C23×C5C22×Q811yescharacteristic
236C23×C5C2×#4111yescharacteristic
237C23×C5C2×#4111yescharacteristic
238C23×C5C2×#4111yescharacteristic
239C23×C5C2×#4111yescharacteristic
240C23×C5C2×#4111yescharacteristic
241C23×C5C2×#4111yescharacteristic
242C23×C5C2×#4111yescharacteristic
243C23×C5H(5,𝔽2)11yescharacteristic
244C23×C52−1+411yescharacteristic
245C23×C52−1+411yescharacteristic
246C23×C52−1+411yescharacteristic
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—22no—
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—24no—
256C2×C4×C5—22no—
257C2×C4×C5—24no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—24no—
261C2×C4×C5—24no—
262C2×C4×C5—42no—
263C2×C4×C5—42no—
264C2×C4×C5—42no—
265C2×C4×C5—41no—
266C2×C4×C5—42no—
267C2×C4×C5—41no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5#11714yes—
275C2×C4×C5C4×D414yes—
276C2×C4×C5C4×D414yes—
277C2×C4×C5#12114yes—
278C2×C4×C5C22⋊Q814yes—
279C2×C4×C5#12314yes—
280C2×C4×C5#12414yes—
281C2×C4×C5#12614yes—
282C2×Dic5—22no—
283C2×Dic5—22no—
284C2×Dic5—22no—
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—22no—
288C2×Dic5—24no—
289C2×Dic5—22no—
290C2×Dic5—24no—
291C2×Dic5—24no—
292C2×Dic5—22no—
293C2×Dic5—22no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—42no—
303C2×Dic5—42no—
304C2×Dic5—42no—
305C2×Dic5—42no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5—42no—
317C2×Dic5—42no—
318C2×Dic5—42no—
319C2×Dic5—42no—
320C2×Dic5—42no—
321C2×Dic5—42no—
322C2×Dic5—42no—
323C2×Dic5C4×Q814yes—
324Dic10—416no—
325C22×Q8—54no—
326C22×Q8—102no—
327C22×Q8—102no—
328C22×Q8—102no—
329C22×Q8—102no—
330C22×Q8—102no—
331C22×Q8—102no—
332C23×C4—51no—
333C23×C4—51no—
334C23×C4—51no—
335C23×C4—52no—
336C23×C4—51no—
337C23×C4—52no—
338C23×C4C22×D512yes—
339C23×C4C22×D511yescharacteristic
340C23×C4C22×D512yes—
341C23×C4C22×D511yescharacteristic
342C23×C4C22×D511yescharacteristic
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—54no—
348C2×C4⋊C4—102no—
349C2×C4⋊C4—102no—
350C2×C4⋊C4—102no—
351C2×C4⋊C4—102no—
352C2×C4⋊C4—102no—
353C2×C4⋊C4—102no—
354C2×C4⋊C4—102no—
355C2×C4⋊C4—102no—
356C2×C4⋊C4C4×D514yes—
357C2×C4⋊C4C22⋊D514yes—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—42no—
363C2×C42—54no—
364C2×C42—102no—
365C2×C42—102no—
366C2×C42—102no—
367C2×C42—102no—
368C2×C42—102no—
369C2×C42D2014yes—
370#95—24no—
371#95—58no—
372#95—58no—
373#95—104no—
374#95—104no—
375#95—102no—
376#95—102no—
377#95—104no—
378#95—104no—
379#95—104no—
380#95—104no—
381#95—202no—
382#95—202no—
383C22×C5C2×#11711yescharacteristic
384C22×C5C2×C4×D411yescharacteristic
385C22×C5C2×C4×D411yescharacteristic
386C22×C5C2×C4×Q811yescharacteristic
387C22×C5C4×#4111yescharacteristic
388C22×C5#51911yescharacteristic
389C22×C5#52011yescharacteristic
390C22×C5C2×#12111yescharacteristic
391C22×C5C2×C22⋊Q811yescharacteristic
392C22×C5C2×#12311yescharacteristic
393C22×C5#52511yescharacteristic
394C22×C5C2×#12411yescharacteristic
395C22×C5C2×#12611yescharacteristic
396C22×C5#52911yescharacteristic
397C22×C5#53311yescharacteristic
398C22×C5#53611yescharacteristic
399C22×C5#53711yescharacteristic
400C22×C5#54111yescharacteristic
401C22×C5#54211yescharacteristic
402C22×C5#54611yescharacteristic
403C22×C5#54711yescharacteristic
404C22×C5#54811yescharacteristic
405C22×C5D4×Q811yescharacteristic
406C22×C5#55211yescharacteristic
407C22×C5#55211yescharacteristic
408C22×C5#55411yescharacteristic
409C22×C5#55611yescharacteristic
410C22×C5#55611yescharacteristic
411C22×C5#55611yescharacteristic
412C22×C5#55611yescharacteristic
413C22×C5#55611yescharacteristic
414C22×C5#55711yescharacteristic
415C22×C5#55711yescharacteristic
416C22×C5#55911yescharacteristic
417C22×C5#55911yescharacteristic
418C4×C5—24no—
419C4×C5—44no—
420C4×C5—42no—
421C4×C5—44no—
422C4×C5#38818yes—
423Dic5—24no—
424Dic5—42no—
425Dic5—42no—
426Dic5—42no—
427Dic5—44no—
428Dic5—82no—
429C24C23×D511yescharacteristic
430C2×Q8—108no—
431C2×Q8—108no—
432C2×Q8—108no—
433C2×Q8—204no—
434C2×Q8—204no—
435C2×Q8—204no—
436C2×Q8—204no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
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—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—52no—
460C22×C4—52no—
461C22×C4—52no—
462C22×C4—52no—
463C22×C4—54no—
464C22×C4—52no—
465C22×C4—52no—
466C22×C4—101no—
467C22×C4—101no—
468C22×C4—101no—
469C22×C4—101no—
470C22×C4—101no—
471C22×C4—102no—
472C22×C4—101no—
473C22×C4—101no—
474C22×C4—101no—
475C22×C4—101no—
476C22×C4—101no—
477C22×C4—102no—
478C22×C4—102no—
479C22×C4—102no—
480C22×C4—102no—
481C22×C4—101no—
482C22×C4—101no—
483C22×C4—101no—
484C22×C4—101no—
485C22×C4—101no—
486C22×C4—101no—
487C22×C4—101no—
488C22×C4—101no—
489C22×C4—101no—
490C22×C4—101no—
491C22×C4—101no—
492C22×C4—101no—
493C22×C4—102no—
494C22×C4—102no—
495C22×C4—102no—
496C22×C4—102no—
497C22×C4—102no—
498C22×C4—102no—
499C22×C4—102no—
500C22×C4—102no—
501C22×C4C2×C4×D514yes—
502C22×C4C2×C4×D512yes—
503C22×C4C2×D2012yes—
504C22×C4C2×D2012yes—
505C22×C4C2×D2012yes—
506C22×C4#70912yes—
507C22×C4#70912yes—
508C22×C4#70912yes—
509C22×C4#70912yes—
510C22×C4#71112yes—
511C22×C4Q8×D512yes—
512C22×C4C2×C22⋊D514yes—
513C22×C4C2×C22⋊D512yes—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—108no—
517C4⋊C4—108no—
518C4⋊C4—204no—
519C42—28no—
520C42—48no—
521C42—108no—
522C42—204no—
523C2×C5C2×#38811yescharacteristic
524C2×C5C4×C22⋊Q811yescharacteristic
525C2×C5#232311yescharacteristic
526C2×C5#235511yescharacteristic
527C2×C5#235611yescharacteristic
528C2×C5#236511yescharacteristic
529C2×C5#236711yescharacteristic
530C2×C5#241511yescharacteristic
531C2×C5#241711yescharacteristic
532C2×C5#244611yescharacteristic
533C2×C5#244711yescharacteristic
534C2×C5#251111yescharacteristic
535C2×C5#251411yescharacteristic
536C2×C5#256811yescharacteristic
537C2×C5#259211yescharacteristic
538C23C22×C4×D511yescharacteristic
539C23C22×D2011yescharacteristic
540C23C2×#70911yescharacteristic
541C23C2×#70911yescharacteristic
542C23C2×#70911yescharacteristic
543C23C2×#70911yescharacteristic
544C23C2×#71111yescharacteristic
545C23#414411yescharacteristic
546C23C2×Q8×D511yescharacteristic
547C23#414711yescharacteristic
548C23D5×#4111yescharacteristic
549C23D5×#4111yescharacteristic
550C23#415011yescharacteristic
551C23#415011yescharacteristic
552C23C22×C22⋊D511yescharacteristic
553Q8—2016no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—41no—
570C2×C4—41no—
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—42no—
580C2×C4—41no—
581C2×C4—54no—
582C2×C4—102no—
583C2×C4—102no—
584C2×C4—102no—
585C2×C4—102no—
586C2×C4—102no—
587C2×C4—102no—
588C2×C4—104no—
589C2×C4—102no—
590C2×C4—104no—
591C2×C4—102no—
592C2×C4—102no—
593C2×C4—102no—
594C2×C4—102no—
595C2×C4—104no—
596C2×C4—102no—
597C2×C4—102no—
598C2×C4—102no—
599C2×C4—102no—
600C2×C4—202no—
601C2×C4—202no—
602C2×C4—201no—
603C2×C4—201no—
604C2×C4—201no—
605C2×C4—201no—
606C2×C4—201no—
607C2×C4—201no—
608C2×C4—202no—
609C2×C4—201no—
610C2×C4—201no—
611C2×C4—202no—
612C2×C4—201no—
613C2×C4—201no—
614C2×C4—201no—
615C2×C4—201no—
616C2×C4—202no—
617C2×C4—202no—
618C2×C4—202no—
619C2×C4—202no—
620C2×C4—202no—
621C2×C4—202no—
622C2×C4—202no—
623C2×C4#401814yes—
624C2×C4C4×D2014yes—
625C2×C4#402114yes—
626C2×C4#402214yes—
627C2×C4#404314yes—
628C2×C4C4×C22⋊D514yes—
629C2×C4#407514yes—
630C2×C4#407614yes—
631C5#1402111yescharacteristic, complemented, Sylow
632C22C2×#401811yescharacteristic
633C22C2×C4×D2011yescharacteristic
634C22C2×#402111yescharacteristic
635C22C2×#402211yescharacteristic
636C22C2×#404311yescharacteristic
637C22#6584411yescharacteristic
638C22#6584511yescharacteristic
639C22#6584711yescharacteristic
640C22#6585811yescharacteristic
641C22#6586011yescharacteristic
642C22#6586311yescharacteristic
643C22#6586711yescharacteristic
644C22C4×#71111yescharacteristic
645C22#6587611yescharacteristic
646C22#6587611yescharacteristic
647C22#6588011yescharacteristic
648C22#6589111yescharacteristic
649C22#6589311yescharacteristic
650C22#6589311yescharacteristic
651C22#6589511yescharacteristic
652C22#6590111yescharacteristic
653C22#6590111yescharacteristic
654C22#6590211yescharacteristic
655C22#6590211yescharacteristic
656C22C4×Q8×D511yescharacteristic
657C22#6590611yescharacteristic
658C22#6591211yescharacteristic
659C22#6591711yescharacteristic
660C22C2×C4×C22⋊D511yescharacteristic
661C22C2×#407511yescharacteristic
662C22C2×#407611yescharacteristic
663C22#6614011yescharacteristic
664C22Q8×C22⋊D511yescharacteristic
665C22#6615811yescharacteristic
666C22#6616511yescharacteristic
667C4—24no—
668C4—44no—
669C4—44no—
670C4—42no—
671C4—104no—
672C4—204no—
673C4—202no—
674C4—202no—
675C4—202no—
676C4—402no—
677C4#6522918yes—
678C2#1060537211yescharacteristic
679C2#1060537511yescharacteristic
680C2#1060542211yescharacteristic
681C2#1060542811yescharacteristic
682C2#1060545311yescharacteristic
683C2#1060546611yescharacteristic
684C2C4×#404311yescharacteristic
685C2#1060549311yescharacteristic
686C2C2×#6522911yescharacteristic
687C2#1061392511yescharacteristic
688C2#1061400811yescharacteristic
689C2#1061407511yescharacteristic
690C2#1061409011yescharacteristic
691C2#1061419611yescharacteristic
692C2#1061419911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C4×C5×C4⋊C4C2
socle, prime radical, cosocleC24×C5C24
hypercenterC2×C4×C4⋊C4D5
hypocenterC5#14021
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
37. #65263C4semidirect
79. C2×#683Q8semidirect
631. C5#14021semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×C4⋊C42
C2×#652632
C2×#652182
C2×C4×#6832
#106023562
#106023292
#106023272
#106002112
#106002062
#140215