Finite Group Databasebeta

← #12021317

Subgroups — #12021317

Order 1280.

1486 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×C4⋊C4C211yesmaximal, characteristic
2C2×#65265C211yesmaximal, characteristic
3C2×#65222C212yesmaximal
4C2×C4×#683C211yesmaximal, characteristic
5#10602356C211yesmaximal, characteristic
6#10602356C211yesmaximal, characteristic
7#10602356C211yesmaximal, characteristic
8#10602329C212yesmaximal
9#10602328C211yesmaximal, characteristic
10#10600207C212yesmaximal
11#10600202C212yesmaximal
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C42×C5C2212yes—
15C22×C42×C5C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C22×C4⋊Dic5C2211yescharacteristic
18C22×#683C2212yes—
19C22×#683C2212yes—
20C22×#683C2211yescharacteristic
21C22×#683C2211yescharacteristic
22C22×C4×Dic5C2211yescharacteristic
23C22×C4×Dic5C2211yescharacteristic
24C4×C5×C4⋊C4—24no—
25C2×C5×#95C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2211yescharacteristic
35C2×#3963C2211yescharacteristic
36#65265—24no—
37#65222—28no—
38C4×#683C418yescomplemented
39#13884—51nomaximal, Sylow
40C23×C4×C5C2311yescharacteristic
41C23×C4×C5C2312yes—
42C23×C4×C5C2312yes—
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2311yescharacteristic
45C23×Dic5C2312yes—
46C23×Dic5C2312yes—
47C23×Dic5C2311yescharacteristic
48C23×Dic5C2311yescharacteristic
49C23×Dic5C2311yescharacteristic
50C23×Dic5C2311yescharacteristic
51C2×C5×C4⋊C4—22no—
52C2×C5×C4⋊C4—22no—
53C2×C5×C4⋊C4—22no—
54C2×C5×C4⋊C4—22no—
55C2×C5×C4⋊C4D414yes—
56C2×C5×C4⋊C4Q814yes—
57C2×C42×C5—22no—
58C2×C42×C5—22no—
59C2×C42×C5—24no—
60C2×C42×C5—24no—
61C2×C42×C5—42no—
62C2×C42×C5C2×C414yes—
63C2×C4⋊Dic5—22no—
64C2×C4⋊Dic5—22no—
65C2×C4⋊Dic5—22no—
66C2×C4⋊Dic5—22no—
67C2×C4⋊Dic5—22no—
68C2×C4⋊Dic5—22no—
69C2×#683—22no—
70C2×#683—22no—
71C2×#683—24no—
72C2×#683—24no—
73C2×#683—24no—
74C2×#683—24no—
75C2×#683—42no—
76C2×#683—42no—
77C2×#683C2×C414yes—
78C2×#683C2×C414yes—
79C2×#683D414yes—
80C2×#683Q814yes—
81C2×C4×Dic5—22no—
82C2×C4×Dic5—22no—
83C2×C4×Dic5—22no—
84C2×C4×Dic5—22no—
85C2×C4×Dic5C2×C414yes—
86C2×C4×Dic5C2×C414yes—
87C5×#95—24no—
88#3963—24no—
89#3963—24no—
90#3963—24no—
91#3963—24no—
92#3963—24no—
93#3963—42no—
94#3963—42no—
95#3963—42no—
96#3963C2×C414yes—
97#3963C2×C414yes—
98C2×#387—52no—
99C2×#384—51no—
100C2×C4×C4⋊C4—51no—
101C2×C4×C4⋊C4D511yescharacteristic
102#1442—52no—
103#1441—52no—
104#1436—52no—
105C4⋊#95—51no—
106C4⋊#95—51no—
107C4⋊#95—51no—
108C4⋊#95—51no—
109C24×C5C2411yescharacteristic
110C22×C4×C5—21no—
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5—21no—
114C22×C4×C5—22no—
115C22×C4×C5—22no—
116C22×C4×C5—22no—
117C22×C4×C5—21no—
118C22×C4×C5—21no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—21no—
124C22×C4×C5—21no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—21no—
129C22×C4×C5—22no—
130C22×C4×C5—21no—
131C22×C4×C5—22no—
132C22×C4×C5C22×C412yes—
133C22×C4×C5C22×C412yes—
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2×D412yes—
136C22×C4×C5C2×D414yes—
137C22×C4×C5C2×D412yes—
138C22×C4×C5C2×Q814yes—
139C22×C4×C5C2×Q812yes—
140C22×C4×C5C2×Q812yes—
141C22×C4×C5#4112yes—
142C22×C4×C5#4112yes—
143C22×C4×C5#4112yes—
144C22×C4×C5#4112yes—
145C22×Dic5—21no—
146C22×Dic5—21no—
147C22×Dic5—21no—
148C22×Dic5—21no—
149C22×Dic5—21no—
150C22×Dic5—21no—
151C22×Dic5—21no—
152C22×Dic5—21no—
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—21no—
161C22×Dic5—21no—
162C22×Dic5—21no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
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×Dic5C22×C412yes—
178C22×Dic5C22×C412yes—
179C22×Dic5C22×C412yes—
180C22×Dic5C22×C412yes—
181C22×Dic5C2×D412yes—
182C22×Dic5C2×D412yes—
183C22×Dic5C2×Q812yes—
184C22×Dic5C2×Q812yes—
185C22×Dic5#4112yes—
186C22×Dic5#4112yes—
187C5×C4⋊C4—28no—
188C5×C4⋊C4—28no—
189C42×C5—28no—
190C42×C5—48no—
191C4⋊Dic5—44no—
192C4⋊Dic5—44no—
193#683—28no—
194#683—48no—
195#683—48no—
196#683C4⋊C4116yescomplemented
197C4×Dic5—28no—
198C4×Dic5—28no—
199C22×C4⋊C4—51no—
200C22×C4⋊C4—51no—
201C22×C4⋊C4—51no—
202C22×C4⋊C4—51no—
203C22×C4⋊C4—52no—
204C22×C4⋊C4—52no—
205C22×C4⋊C4C2×D511yescharacteristic
206C22×C4⋊C4C2×D511yescharacteristic
207C22×C42—51no—
208C22×C42—51no—
209C22×C42C2×D512yes—
210C22×C42C2×D511yescharacteristic
211#387—108no—
212#384—104no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—58no—
215C2×#95—51no—
216C2×#95—51no—
217C2×#95—52no—
218C2×#95—52no—
219C2×#95—52no—
220C2×#95—52no—
221C2×#95—52no—
222C2×#95—52no—
223C2×#95—52no—
224C2×#95—52no—
225C2×#95C2×D512yes—
226C23×C5C23×C411yescharacteristic
227C23×C5C22×D411yescharacteristic
228C23×C5C22×D411yescharacteristic
229C23×C5C22×Q811yescharacteristic
230C23×C5C22×Q811yescharacteristic
231C23×C5C2×#4111yescharacteristic
232C23×C5C2×#4111yescharacteristic
233C23×C5C2×#4111yescharacteristic
234C23×C5C2×#4111yescharacteristic
235C23×C5C2×#4111yescharacteristic
236C23×C5C2×#4111yescharacteristic
237C23×C5H(5,𝔽2)11yescharacteristic
238C23×C5H(5,𝔽2)11yescharacteristic
239C23×C52−1+411yescharacteristic
240C23×C52−1+411yescharacteristic
241C2×C4×C5—22no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—24no—
251C2×C4×C5—24no—
252C2×C4×C5—24no—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—24no—
256C2×C4×C5—42no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—41no—
262C2×C4×C5—41no—
263C2×C4×C5—42no—
264C2×C4×C5—42no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—42no—
268C2×C4×C5C2×C4⋊C414yes—
269C2×C4×C5#11714yes—
270C2×C4×C5C4×D414yes—
271C2×C4×C5C4×Q814yes—
272C2×C4×C5C22⋊Q814yes—
273C2×C4×C5#12314yes—
274C2×C4×C5#12514yes—
275C2×C4×C5#12614yes—
276C2×Dic5—22no—
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—22no—
290C2×Dic5—22no—
291C2×Dic5—22no—
292C2×Dic5—22no—
293C2×Dic5—22no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—41no—
300C2×Dic5—41no—
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—
313C2×Dic5—42no—
314C2×Dic5—42no—
315C2×Dic5C2×C4⋊C414yes—
316C2×Dic5C2×C4⋊C414yes—
317C2×Dic5#11714yes—
318C23×C4—51no—
319C23×C4—51no—
320C23×C4—51no—
321C23×C4—51no—
322C23×C4—52no—
323C23×C4—52no—
324C23×C4C22×D512yes—
325C23×C4C22×D511yescharacteristic
326C23×C4C22×D512yes—
327C23×C4C22×D511yescharacteristic
328C23×C4C22×D511yescharacteristic
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—54no—
334C2×C4⋊C4—54no—
335C2×C4⋊C4—54no—
336C2×C4⋊C4—54no—
337C2×C4⋊C4—102no—
338C2×C4⋊C4—102no—
339C2×C4⋊C4—104no—
340C2×C4⋊C4—104no—
341C2×C4⋊C4—102no—
342C2×C4⋊C4—102no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—104no—
347C2×C4⋊C4—104no—
348C2×C4⋊C4—102no—
349C2×C4⋊C4—202no—
350C2×C4⋊C4—202no—
351C2×C4⋊C4Dic1014yes—
352C2×C4⋊C4C22⋊D514yes—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—42no—
358C2×C42—54no—
359C2×C42—54no—
360C2×C42—102no—
361C2×C42—102no—
362C2×C42—102no—
363C2×C42—102no—
364C2×C42C4×D514yes—
365#95—24no—
366#95—54no—
367#95—54no—
368#95—104no—
369#95—104no—
370#95—104no—
371#95—104no—
372#95—104no—
373#95—202no—
374#95—202no—
375#95—202no—
376C22×C5C22×C4⋊C411yescharacteristic
377C22×C5C2×#11711yescharacteristic
378C22×C5C2×#11711yescharacteristic
379C22×C5C2×C4×D411yescharacteristic
380C22×C5C2×C4×Q811yescharacteristic
381C22×C5#52011yescharacteristic
382C22×C5#52011yescharacteristic
383C22×C5C2×C22⋊Q811yescharacteristic
384C22×C5C2×#12311yescharacteristic
385C22×C5C2×#12511yescharacteristic
386C22×C5C2×#12611yescharacteristic
387C22×C5#53211yescharacteristic
388C22×C5#53211yescharacteristic
389C22×C5#53311yescharacteristic
390C22×C5#53311yescharacteristic
391C22×C5#54011yescharacteristic
392C22×C5#54111yescharacteristic
393C22×C5#54211yescharacteristic
394C22×C5#54211yescharacteristic
395C22×C5D4211yescharacteristic
396C22×C5#54611yescharacteristic
397C22×C5#54711yescharacteristic
398C22×C5#54811yescharacteristic
399C22×C5D4×Q811yescharacteristic
400C22×C5D4×Q811yescharacteristic
401C22×C5#55011yescharacteristic
402C22×C5#55211yescharacteristic
403C22×C5#55311yescharacteristic
404C22×C5#55411yescharacteristic
405C22×C5#55411yescharacteristic
406C22×C5#55611yescharacteristic
407C22×C5#55711yescharacteristic
408C22×C5#55711yescharacteristic
409C22×C5Q8211yescharacteristic
410C22×C5#55911yescharacteristic
411C4×C5—24no—
412C4×C5—42no—
413C4×C5—44no—
414C4×C5—44no—
415C4×C5#38718yescomplemented
416Dic5—24no—
417Dic5—24no—
418Dic5—24no—
419Dic5—42no—
420Dic5—82no—
421Dic5—82no—
422C24C23×D511yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
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—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—52no—
446C22×C4—52no—
447C22×C4—52no—
448C22×C4—52no—
449C22×C4—52no—
450C22×C4—52no—
451C22×C4—52no—
452C22×C4—52no—
453C22×C4—52no—
454C22×C4—52no—
455C22×C4—101no—
456C22×C4—101no—
457C22×C4—101no—
458C22×C4—101no—
459C22×C4—101no—
460C22×C4—101no—
461C22×C4—101no—
462C22×C4—101no—
463C22×C4—101no—
464C22×C4—101no—
465C22×C4—101no—
466C22×C4—101no—
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×Dic1014yes—
488C22×C4C2×Dic1012yes—
489C22×C4C2×C4×D512yes—
490C22×C4C2×C4×D512yes—
491C22×C4C2×C4×D512yes—
492C22×C4#70912yes—
493C22×C4#70912yes—
494C22×C4#70912yes—
495C22×C4#70912yes—
496C22×C4D4×D512yes—
497C22×C4Q8×D512yes—
498C22×C4C2×C22⋊D514yes—
499C22×C4C2×C22⋊D512yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—516no—
503C4⋊C4—108no—
504C4⋊C4—208no—
505C4⋊C4—208no—
506C4⋊C4—204no—
507C4⋊C4—204no—
508C42—28no—
509C42—48no—
510C42—108no—
511C42—108no—
512C2×C5#228511yescharacteristic
513C2×C5C2×#38711yescharacteristic
514C2×C5#231411yescharacteristic
515C2×C5D4×C4⋊C411yescharacteristic
516C2×C5Q8×C4⋊C411yescharacteristic
517C2×C5#235211yescharacteristic
518C2×C5#235511yescharacteristic
519C2×C5#244511yescharacteristic
520C2×C5#244611yescharacteristic
521C2×C5#245111yescharacteristic
522C2×C5#245211yescharacteristic
523C2×C5#250611yescharacteristic
524C2×C5#250711yescharacteristic
525C2×C5#251111yescharacteristic
526C2×C5#251211yescharacteristic
527C23C22×Dic1011yescharacteristic
528C23C22×C4×D511yescharacteristic
529C23C2×#70911yescharacteristic
530C23C2×#70911yescharacteristic
531C23C2×#70911yescharacteristic
532C23C2×#70911yescharacteristic
533C23C2×D4×D511yescharacteristic
534C23#414411yescharacteristic
535C23C2×Q8×D511yescharacteristic
536C23#414711yescharacteristic
537C23D5×#4111yescharacteristic
538C23D5×#4111yescharacteristic
539C23#414911yescharacteristic
540C23#415011yescharacteristic
541C23C22×C22⋊D511yescharacteristic
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—54no—
570C2×C4—54no—
571C2×C4—54no—
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—102no—
582C2×C4—102no—
583C2×C4—102no—
584C2×C4—102no—
585C2×C4—102no—
586C2×C4—102no—
587C2×C4—102no—
588C2×C4—102no—
589C2×C4—102no—
590C2×C4—102no—
591C2×C4—102no—
592C2×C4—102no—
593C2×C4—201no—
594C2×C4—201no—
595C2×C4—201no—
596C2×C4—201no—
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×C4—202no—
606C2×C4—202no—
607C2×C4—202no—
608C2×C4—202no—
609C2×C4—202no—
610C2×C4—202no—
611C2×C4C4×Dic1014yes—
612C2×C4#401614yes—
613C2×C4#401814yes—
614C2×C4#402214yes—
615C2×C4D5×C4⋊C414yes—
616C2×C4#407014yes—
617C2×C4C4×C22⋊D514yes—
618C2×C4#407514yes—
619C5#1388411yescharacteristic, complemented, Sylow
620C22C2×C4×Dic1011yescharacteristic
621C22C2×#401611yescharacteristic
622C22C2×#401811yescharacteristic
623C22C2×#402211yescharacteristic
624C22C2×D5×C4⋊C411yescharacteristic
625C22#6583811yescharacteristic
626C22#6584111yescharacteristic
627C22#6584511yescharacteristic
628C22D5×#11711yescharacteristic
629C22#6585711yescharacteristic
630C22#6586311yescharacteristic
631C22#6586411yescharacteristic
632C22D4×Dic1011yescharacteristic
633C22#6587311yescharacteristic
634C22#6588211yescharacteristic
635C22#6588211yescharacteristic
636C22#6588711yescharacteristic
637C22#6589311yescharacteristic
638C22#6589511yescharacteristic
639C22#6589611yescharacteristic
640C22Q8×Dic1011yescharacteristic
641C22#6590111yescharacteristic
642C22#6590211yescharacteristic
643C22#6590311yescharacteristic
644C22#6591211yescharacteristic
645C22#6591211yescharacteristic
646C22#6591611yescharacteristic
647C22#6591711yescharacteristic
648C22C2×#407011yescharacteristic
649C22C2×C4×C22⋊D511yescharacteristic
650C22C2×#407511yescharacteristic
651C22D4×C22⋊D511yescharacteristic
652C22Q8×C22⋊D511yescharacteristic
653C22#6616311yescharacteristic
654C22#6616511yescharacteristic
655C4—24no—
656C4—44no—
657C4—44no—
658C4—42no—
659C4—104no—
660C4—104no—
661C4—104no—
662C4—202no—
663C4—402no—
664C4—402no—
665C4#6522218yescomplemented
666C2C4⋊C4×Dic1011yescharacteristic
667C2#1060537311yescharacteristic
668C2#1060541411yescharacteristic
669C2#1060541611yescharacteristic
670C2#1060542211yescharacteristic
671C2#1060542411yescharacteristic
672C2#1060548411yescharacteristic
673C2#1060549111yescharacteristic
674C2C2×#6522211yescharacteristic
675C2C4⋊C4×C22⋊D511yescharacteristic
676C2#1061400311yescharacteristic
677C2#1061408911yescharacteristic
678C2#1061409311yescharacteristic
679C2#1061419911yescharacteristic
680C2#1061420111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C4×C5×C4⋊C4C2
socle, prime radical, cosocleC24×C5C24
hypercenterC2×C4×C4⋊C4D5
hypocenterC5#13884
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
38. C4×#683C4semidirect
196. #683C4⋊C4semidirect
415. C4×C5#387semidirect
619. C5#13884semidirect
665. C4#65222semidirect

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×#652652
C2×#652222
C2×C4×#6832
#106023562
#106023292
#106023282
#106002072
#106002022
#138845