Finite Group Databasebeta

← #12022820

Subgroups — #12022820

Order 1280.

1470 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#384C211yesmaximal, characteristic
2C2×#65274C211yesmaximal, characteristic
3C2×#65270C212yesmaximal
4C2×C4⋊C4×Dic5C211yesmaximal, characteristic
5#10602362C212yesmaximal
6#10602358C212yesmaximal
7#10602357C212yesmaximal
8#10602356C211yesmaximal, characteristic
9#10602356C211yesmaximal, characteristic
10#10602328C211yesmaximal, characteristic
11C4×#3963C211yesmaximal, characteristic
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C4⋊C4C2212yes—
15C22×C5×C4⋊C4C2212yes—
16C22×C42×C5C2211yescharacteristic
17C22×C4⋊Dic5C2211yescharacteristic
18C22×C4⋊Dic5C2212yes—
19C22×C4⋊Dic5C2211yescharacteristic
20C22×C4×Dic5C2211yescharacteristic
21C22×C4×Dic5C2212yes—
22C22×C4×Dic5C2211yescharacteristic
23C5×#384C418yescomplemented
24C2×#3963C2211yescharacteristic
25C2×#3963C2211yescharacteristic
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2211yescharacteristic
35C2×#3963C2211yescharacteristic
36#65274—24no—
37#65270—28no—
38C4⋊C4×Dic5—24no—
39#14029—51nomaximal, Sylow
40C23×C4×C5C2312yes—
41C23×C4×C5C2312yes—
42C23×C4×C5C2311yescharacteristic
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—24no—
53C2×C5×C4⋊C4—22no—
54C2×C5×C4⋊C4—24no—
55C2×C5×C4⋊C4—22no—
56C2×C5×C4⋊C4—22no—
57C2×C5×C4⋊C4—42no—
58C2×C5×C4⋊C4—42no—
59C2×C5×C4⋊C4C2×C418yes—
60C2×C5×C4⋊C4C2×C414yes—
61C2×C5×C4⋊C4C2×C414yes—
62C2×C5×C4⋊C4C2×C418yes—
63C2×C42×C5—22no—
64C2×C42×C5—22no—
65C2×C42×C5C2×C414yes—
66C2×C4⋊Dic5—22no—
67C2×C4⋊Dic5—22no—
68C2×C4⋊Dic5—22no—
69C2×C4⋊Dic5—22no—
70C2×C4⋊Dic5—22no—
71C2×C4⋊Dic5—24no—
72C2×C4⋊Dic5—24no—
73C2×C4⋊Dic5—42no—
74C2×C4⋊Dic5D414yes—
75C2×C4×Dic5—22no—
76C2×C4×Dic5—22no—
77C2×C4×Dic5—22no—
78C2×C4×Dic5—22no—
79C2×C4×Dic5—22no—
80C2×C4×Dic5—24no—
81C2×C4×Dic5—24no—
82C2×C4×Dic5—42no—
83C2×C4×Dic5Q814yes—
84#3963—24no—
85#3963—22no—
86#3963—22no—
87#3963—24no—
88#3963—24no—
89#3963—24no—
90#3963—24no—
91#3963—24no—
92#3963—42no—
93#3963—42no—
94#3963D414yes—
95#3963Q814yes—
96C2×#389—51no—
97C2×#387—52no—
98C2×#384D511yescharacteristic
99C2×C4×C4⋊C4—51no—
100#1445—52no—
101#1441—52no—
102#1436—52no—
103C4⋊#95—51no—
104C4⋊#95—51no—
105C4⋊#95—51no—
106C4×#95—51no—
107C24×C5C2411yescharacteristic
108C22×C4×C5—21no—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—21no—
116C22×C4×C5—22no—
117C22×C4×C5—22no—
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—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5C22×C414yes—
129C22×C4×C5C22×C414yes—
130C22×C4×C5C22×C412yes—
131C22×C4×C5C22×C412yes—
132C22×C4×C5C22×C412yes—
133C22×C4×C5C2×D412yes—
134C22×C4×C5C2×D412yes—
135C22×C4×C5C2×Q812yes—
136C22×C4×C5C2×Q812yes—
137C22×C4×C5#4112yes—
138C22×C4×C5#4112yes—
139C22×C4×C5#4112yes—
140C22×C4×C5#4112yes—
141C22×C4×C5#4112yes—
142C22×C4×C5#4112yes—
143C22×Dic5—21no—
144C22×Dic5—21no—
145C22×Dic5—21no—
146C22×Dic5—21no—
147C22×Dic5—21no—
148C22×Dic5—21no—
149C22×Dic5—22no—
150C22×Dic5—21no—
151C22×Dic5—21no—
152C22×Dic5—21no—
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—22no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—21no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—21no—
164C22×Dic5—21no—
165C22×Dic5—22no—
166C22×Dic5—21no—
167C22×Dic5—21no—
168C22×Dic5—22no—
169C22×Dic5—21no—
170C22×Dic5—21no—
171C22×Dic5—22no—
172C22×Dic5—22no—
173C22×Dic5—22no—
174C22×Dic5—22no—
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5C2×D412yes—
179C22×Dic5C2×D414yes—
180C22×Dic5C2×D412yes—
181C22×Dic5C2×Q812yes—
182C22×Dic5C2×Q812yes—
183C22×Dic5C2×Q812yes—
184C22×Dic5C2×Q812yes—
185C5×C4⋊C4—216no—
186C5×C4⋊C4—28no—
187C5×C4⋊C4—28no—
188C5×C4⋊C4—48no—
189C42×C5—28no—
190C4⋊Dic5—28no—
191C4⋊Dic5—48no—
192C4⋊Dic5—44no—
193C4×Dic5—28no—
194C4×Dic5—48no—
195C4×Dic5—44no—
196C22×C4⋊C4—51no—
197C22×C4⋊C4—51no—
198C22×C4⋊C4—52no—
199C22×C4⋊C4C2×D511yescharacteristic
200C22×C4⋊C4C2×D511yescharacteristic
201C22×C4⋊C4C2×D512yes—
202C22×C4⋊C4C2×D512yes—
203C22×C42—51no—
204C22×C42—51no—
205C22×C42—52no—
206C22×C42C2×D511yescharacteristic
207#389—104no—
208#387—108no—
209#384Dic518yescomplemented
210C4×C4⋊C4—104no—
211C2×#95—51no—
212C2×#95—51no—
213C2×#95—51no—
214C2×#95—51no—
215C2×#95—52no—
216C2×#95—52no—
217C2×#95—52no—
218C2×#95—52no—
219C2×#95—52no—
220C2×#95—52no—
221C2×#95—52no—
222C2×#95—52no—
223C23×C5C23×C411yescharacteristic
224C23×C5C22×D411yescharacteristic
225C23×C5C22×D411yescharacteristic
226C23×C5C22×Q811yescharacteristic
227C23×C5C22×Q811yescharacteristic
228C23×C5C2×#4111yescharacteristic
229C23×C5C2×#4111yescharacteristic
230C23×C5C2×#4111yescharacteristic
231C23×C5C2×#4111yescharacteristic
232C23×C5C2×#4111yescharacteristic
233C23×C5C2×#4111yescharacteristic
234C23×C5H(5,𝔽2)11yescharacteristic
235C23×C5H(5,𝔽2)11yescharacteristic
236C23×C52−1+411yescharacteristic
237C23×C52−1+411yescharacteristic
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
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—42no—
262C2×C4×C5—42no—
263C2×C4×C5—42no—
264C2×C4×C5#11714yes—
265C2×C4×C5C4×D414yes—
266C2×C4×C5C4×D414yes—
267C2×C4×C5C4×Q814yes—
268C2×C4×C5C4×Q814yes—
269C2×C4×C5C22⋊Q814yes—
270C2×C4×C5#12314yes—
271C2×C4×C5#12514yes—
272C2×C4×C5#12614yes—
273C2×Dic5—22no—
274C2×Dic5—22no—
275C2×Dic5—22no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—24no—
281C2×Dic5—22no—
282C2×Dic5—22no—
283C2×Dic5—22no—
284C2×Dic5—22no—
285C2×Dic5—24no—
286C2×Dic5—22no—
287C2×Dic5—24no—
288C2×Dic5—22no—
289C2×Dic5—22no—
290C2×Dic5—22no—
291C2×Dic5—41no—
292C2×Dic5—41no—
293C2×Dic5—42no—
294C2×Dic5—41no—
295C2×Dic5—42no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—42no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—42no—
306C2×Dic5—41no—
307C2×Dic5—42no—
308C2×Dic5—42no—
309C2×Dic5—42no—
310C2×Dic5—42no—
311C2×Dic5—42no—
312C2×Dic5—42no—
313C2×Dic5—42no—
314C2×Dic5C4⋊Q814yes—
315C23×C4—51no—
316C23×C4—51no—
317C23×C4—52no—
318C23×C4—51no—
319C23×C4—51no—
320C23×C4—52no—
321C23×C4C22×D512yes—
322C23×C4C22×D512yes—
323C23×C4C22×D511yescharacteristic
324C23×C4C22×D511yescharacteristic
325C23×C4C22×D511yescharacteristic
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—54no—
335C2×C4⋊C4—102no—
336C2×C4⋊C4—102no—
337C2×C4⋊C4—104no—
338C2×C4⋊C4—104no—
339C2×C4⋊C4—102no—
340C2×C4⋊C4—102no—
341C2×C4⋊C4—102no—
342C2×C4⋊C4—202no—
343C2×C4⋊C4C2×Dic518yes—
344C2×C4⋊C4C2×Dic514yes—
345C2×C4⋊C4C2×Dic514yes—
346C2×C4⋊C4C2×Dic518yes—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—54no—
350C2×C42—102no—
351C2×C42—102no—
352C2×C42—104no—
353C2×C42—104no—
354C2×C42—102no—
355C2×C42—102no—
356C2×C42—102no—
357C2×C42—202no—
358C2×C42C2×Dic514yes—
359#95—54no—
360#95—54no—
361#95—104no—
362#95—104no—
363#95—104no—
364#95—104no—
365#95—104no—
366#95—102no—
367#95—102no—
368#95—104no—
369#95—202no—
370#95—202no—
371C22×C5C2×#11711yescharacteristic
372C22×C5C2×C4×D411yescharacteristic
373C22×C5C2×C4×D411yescharacteristic
374C22×C5C2×C4×Q811yescharacteristic
375C22×C5C2×C4×Q811yescharacteristic
376C22×C5#51811yescharacteristic
377C22×C5#51911yescharacteristic
378C22×C5C2×C22⋊Q811yescharacteristic
379C22×C5C2×#12311yescharacteristic
380C22×C5C2×#12511yescharacteristic
381C22×C5C2×#12611yescharacteristic
382C22×C5C2×C4⋊Q811yescharacteristic
383C22×C5#53211yescharacteristic
384C22×C5#53311yescharacteristic
385C22×C5#53311yescharacteristic
386C22×C5#54211yescharacteristic
387C22×C5#54211yescharacteristic
388C22×C5#54411yescharacteristic
389C22×C5#54411yescharacteristic
390C22×C5D4211yescharacteristic
391C22×C5#54711yescharacteristic
392C22×C5#54711yescharacteristic
393C22×C5#54711yescharacteristic
394C22×C5D4×Q811yescharacteristic
395C22×C5#55011yescharacteristic
396C22×C5#55011yescharacteristic
397C22×C5#55211yescharacteristic
398C22×C5#55211yescharacteristic
399C22×C5#55311yescharacteristic
400C22×C5#55311yescharacteristic
401C22×C5#55411yescharacteristic
402C22×C5#55411yescharacteristic
403C22×C5#55611yescharacteristic
404C22×C5#55711yescharacteristic
405C22×C5#55911yescharacteristic
406C4×C5—24no—
407C4×C5—24no—
408C4×C5—44no—
409C4×C5—44no—
410C4×C5#38718yescomplemented
411Dic5—24no—
412Dic5—44no—
413Dic5—42no—
414Dic5—42no—
415Dic5—42no—
416Dic5—82no—
417C24C23×D511yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
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—52no—
439C22×C4—52no—
440C22×C4—52no—
441C22×C4—52no—
442C22×C4—54no—
443C22×C4—52no—
444C22×C4—52no—
445C22×C4—101no—
446C22×C4—101no—
447C22×C4—101no—
448C22×C4—101no—
449C22×C4—101no—
450C22×C4—101no—
451C22×C4—101no—
452C22×C4—102no—
453C22×C4—101no—
454C22×C4—101no—
455C22×C4—102no—
456C22×C4—101no—
457C22×C4—101no—
458C22×C4—101no—
459C22×C4—101no—
460C22×C4—101no—
461C22×C4—102no—
462C22×C4—101no—
463C22×C4—101no—
464C22×C4—102no—
465C22×C4—101no—
466C22×C4—101no—
467C22×C4—101no—
468C22×C4—102no—
469C22×C4—102no—
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#70912yes—
481C22×C4#70912yes—
482C22×C4D4×D512yes—
483C22×C4D4×D512yes—
484C22×C4#71112yes—
485C22×C4#71112yes—
486C22×C4Q8×D512yes—
487C22×C4Q8×D512yes—
488C22×C4#71312yes—
489C22×C4#71312yes—
490C22×C4C22×Dic514yes—
491C22×C4C22×Dic514yes—
492C22×C4C22×Dic512yes—
493C22×C4C22×Dic512yes—
494C22×C4C22×Dic512yes—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—48no—
499C4⋊C4—108no—
500C4⋊C4—208no—
501C4⋊C4—204no—
502C42—28no—
503C42—108no—
504C42—208no—
505C42—204no—
506C2×C5C2×#38711yescharacteristic
507C2×C5C4×C4⋊Q811yescharacteristic
508C2×C5#233111yescharacteristic
509C2×C5#234411yescharacteristic
510C2×C5#234511yescharacteristic
511C2×C5#236511yescharacteristic
512C2×C5#236611yescharacteristic
513C2×C5#242211yescharacteristic
514C2×C5#242311yescharacteristic
515C2×C5#244011yescharacteristic
516C2×C5#244111yescharacteristic
517C2×C5#251111yescharacteristic
518C2×C5#251211yescharacteristic
519C2×C5#255211yescharacteristic
520C2×C5#257011yescharacteristic
521C23C2×#70911yescharacteristic
522C23C2×#70911yescharacteristic
523C23C2×D4×D511yescharacteristic
524C23C2×D4×D511yescharacteristic
525C23C2×#71111yescharacteristic
526C23C2×#71111yescharacteristic
527C23#414411yescharacteristic
528C23#414411yescharacteristic
529C23C2×Q8×D511yescharacteristic
530C23C2×Q8×D511yescharacteristic
531C23C2×#71311yescharacteristic
532C23C2×#71311yescharacteristic
533C23#414711yescharacteristic
534C23#414711yescharacteristic
535C23C23×Dic511yescharacteristic
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—54no—
563C2×C4—102no—
564C2×C4—102no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—102no—
568C2×C4—102no—
569C2×C4—104no—
570C2×C4—102no—
571C2×C4—102no—
572C2×C4—104no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—102no—
576C2×C4—102no—
577C2×C4—104no—
578C2×C4—102no—
579C2×C4—102no—
580C2×C4—102no—
581C2×C4—201no—
582C2×C4—201no—
583C2×C4—202no—
584C2×C4—201no—
585C2×C4—201no—
586C2×C4—202no—
587C2×C4—201no—
588C2×C4—201no—
589C2×C4—201no—
590C2×C4—201no—
591C2×C4—202no—
592C2×C4—201no—
593C2×C4—201no—
594C2×C4—201no—
595C2×C4—202no—
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#403514yes—
605C2×C4#404014yes—
606C2×C4#404214yes—
607C2×C4#404414yes—
608C2×C4#407214yes—
609C2×C4D4×Dic514yes—
610C2×C4D4×Dic514yes—
611C2×C4Q8×Dic514yes—
612C2×C4Q8×Dic514yes—
613C5#1402911yescharacteristic, complemented, Sylow
614C22C2×#403511yescharacteristic
615C22C2×#404011yescharacteristic
616C22#6584111yescharacteristic
617C22#6584111yescharacteristic
618C22C2×#404211yescharacteristic
619C22#6584511yescharacteristic
620C22#6584511yescharacteristic
621C22C2×#404411yescharacteristic
622C22#6584711yescharacteristic
623C22#6584711yescharacteristic
624C22#6584811yescharacteristic
625C22#6584811yescharacteristic
626C22#6604711yescharacteristic
627C22#6605111yescharacteristic
628C22#6605211yescharacteristic
629C22#6605211yescharacteristic
630C22#6605511yescharacteristic
631C22D5×C4⋊Q811yescharacteristic
632C22#6605911yescharacteristic
633C22#6606011yescharacteristic
634C22#6606111yescharacteristic
635C22#6606211yescharacteristic
636C22#6606211yescharacteristic
637C22#6606311yescharacteristic
638C22#6606311yescharacteristic
639C22#6606611yescharacteristic
640C22#6606711yescharacteristic
641C22#6606711yescharacteristic
642C22C2×#407211yescharacteristic
643C22C2×D4×Dic511yescharacteristic
644C22C2×D4×Dic511yescharacteristic
645C22#6613211yescharacteristic
646C22C2×Q8×Dic511yescharacteristic
647C22C2×Q8×Dic511yescharacteristic
648C22#6614611yescharacteristic
649C4—24no—
650C4—24no—
651C4—44no—
652C4—44no—
653C4—104no—
654C4—204no—
655C4—202no—
656C4—202no—
657C4—202no—
658C4—402no—
659C4#6527018yescomplemented
660C2#1060605711yescharacteristic
661C2#1060605811yescharacteristic
662C2#1060606311yescharacteristic
663C2#1060606411yescharacteristic
664C2#1060606811yescharacteristic
665C2#1060607211yescharacteristic
666C2#1060609311yescharacteristic
667C2#1060609411yescharacteristic
668C2C2×#6527011yescharacteristic
669C2#1061389011yescharacteristic
670C2#1061389111yescharacteristic
671C2#1061490411yescharacteristic
672C2Dic5×C4⋊Q811yescharacteristic
673C2#1061495911yescharacteristic
674C2#1061496011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#384C2
socle, prime radical, cosocleC24×C5C24
hypercenterC2×#384D5
hypocenterC5#14029
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
23. C5×#384C4semidirect
209. #384Dic5semidirect
410. C4×C5#387semidirect
613. C5#14029semidirect
659. C4#65270semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3842
C2×#652742
C2×#652702
C2×C4⋊C4×Dic52
#106023622
#106023582
#106023572
#106023562
#106023282
C4×#39632
#140295