Finite Group Databasebeta

← #12022531

Subgroups — #12022531

Order 1280.

1470 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
1C2×C5×#389C211yesmaximal, characteristic
2C2×#65262C211yesmaximal, characteristic
3C2×#65266C211yesmaximal, characteristic
4C2×C4⋊C4×Dic5C211yesmaximal, characteristic
5C2×#65217C211yesmaximal, characteristic
6#10602359C212yesmaximal
7#10602357C212yesmaximal
8#10602356C211yesmaximal, characteristic
9C4×#3963C211yesmaximal, characteristic
10#10600206C212yesmaximal
11#10600197C212yesmaximal
12C22×C5×C4⋊C4C2212yes—
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C42×C5C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C22×C4⋊Dic5C2211yescharacteristic
18C22×#683C2212yes—
19C22×#683C2212yes—
20C22×C4×Dic5C2212yes—
21C22×C4×Dic5C2211yescharacteristic
22C22×C4×Dic5C2211yescharacteristic
23C5×#389—24no—
24C2×C5×#95C2212yes—
25C2×#3963C2211yescharacteristic
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2211yescharacteristic
33C2×#3963C2211yescharacteristic
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36#65266—24no—
37C4⋊C4×Dic5—24no—
38#65262C418yescomplemented
39#65217—24no—
40#14006—51nomaximal, Sylow
41C23×C4×C5C2312yes—
42C23×C4×C5C2312yes—
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2311yescharacteristic
45C23×C4×C5C2311yescharacteristic
46C23×Dic5C2312yes—
47C23×Dic5C2311yescharacteristic
48C23×Dic5C2311yescharacteristic
49C23×Dic5C2311yescharacteristic
50C23×Dic5C2312yes—
51C23×Dic5C2311yescharacteristic
52C2×C5×C4⋊C4—22no—
53C2×C5×C4⋊C4—22no—
54C2×C5×C4⋊C4—22no—
55C2×C5×C4⋊C4—22no—
56C2×C5×C4⋊C4—24no—
57C2×C5×C4⋊C4—24no—
58C2×C5×C4⋊C4—42no—
59C2×C5×C4⋊C4C2×C414yes—
60C2×C5×C4⋊C4D414yes—
61C2×C42×C5—22no—
62C2×C42×C5—22no—
63C2×C42×C5—22no—
64C2×C4⋊Dic5—22no—
65C2×C4⋊Dic5—22no—
66C2×C4⋊Dic5—22no—
67C2×C4⋊Dic5C2×C414yes—
68C2×C4⋊Dic5D414yes—
69C2×C4⋊Dic5Q814yes—
70C2×#683—24no—
71C2×#683—24no—
72C2×#683—42no—
73C2×#683—42no—
74C2×#683—42no—
75C2×#683C2×C418yes—
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×Dic5C2×C414yes—
84C2×C4×Dic5Q814yes—
85C5×#95—24no—
86#3963—24no—
87#3963—24no—
88#3963—22no—
89#3963—22no—
90#3963—24no—
91#3963—24no—
92#3963—22no—
93#3963—22no—
94#3963—24no—
95#3963—42no—
96#3963C2×C418yes—
97C2×#389—51no—
98C2×#389—51no—
99C2×#389D511yescharacteristic
100C2×#382—51no—
101C2×C4×C4⋊C4—51no—
102#1442—52no—
103#1441—52no—
104#1436—52no—
105C4⋊#95—52no—
106C4⋊#95—51no—
107C4×#95—51no—
108C24×C5C2411yescharacteristic
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5—22no—
114C22×C4×C5—22no—
115C22×C4×C5—21no—
116C22×C4×C5—21no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—21no—
123C22×C4×C5—21no—
124C22×C4×C5—22no—
125C22×C4×C5—21no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
131C22×C4×C5—22no—
132C22×C4×C5—21no—
133C22×C4×C5C22×C414yes—
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2×D412yes—
136C22×C4×C5C2×D414yes—
137C22×C4×C5C2×D412yes—
138C22×C4×C5C2×Q812yes—
139C22×C4×C5C2×Q812yes—
140C22×C4×C5#4112yes—
141C22×C4×C5#4112yes—
142C22×C4×C5#4112yes—
143C22×C4×C5#4112yes—
144C22×Dic5—21no—
145C22×Dic5—21no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—21no—
149C22×Dic5—22no—
150C22×Dic5—22no—
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—21no—
164C22×Dic5—21no—
165C22×Dic5—21no—
166C22×Dic5—21no—
167C22×Dic5—21no—
168C22×Dic5—21no—
169C22×Dic5—21no—
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×C414yes—
180C22×Dic5C2×D414yes—
181C22×Dic5C2×Q812yes—
182C22×Dic5C2×Q814yes—
183C22×Dic5C2×Q812yes—
184C22×Dic5#4112yes—
185C22×Dic5#4112yes—
186C5×C4⋊C4—28no—
187C5×C4⋊C4—28no—
188C5×C4⋊C4—48no—
189C42×C5—44no—
190C4⋊Dic5—44no—
191C4⋊Dic5C4⋊C4116yescomplemented
192#683—48no—
193#683—48no—
194C4×Dic5—28no—
195C4×Dic5—28no—
196C4×Dic5—48no—
197C22×C4⋊C4—51no—
198C22×C4⋊C4—52no—
199C22×C4⋊C4—51no—
200C22×C4⋊C4—52no—
201C22×C4⋊C4C2×D512yes—
202C22×C4⋊C4C2×D511yescharacteristic
203C22×C4⋊C4C2×D511yescharacteristic
204C22×C42—51no—
205C22×C42—51no—
206C22×C42—52no—
207C22×C42C2×D511yescharacteristic
208#389—24no—
209#389—58no—
210#389—104no—
211#382—104no—
212C4×C4⋊C4—104no—
213C2×#95—52no—
214C2×#95—51no—
215C2×#95—51no—
216C2×#95—51no—
217C2×#95—51no—
218C2×#95—52no—
219C2×#95—52no—
220C2×#95—52no—
221C2×#95—52no—
222C2×#95—52no—
223C2×#95—52no—
224C2×#95C2×D512yes—
225C23×C5C23×C411yescharacteristic
226C23×C5C22×D411yescharacteristic
227C23×C5C22×D411yescharacteristic
228C23×C5C22×Q811yescharacteristic
229C23×C5C22×Q811yescharacteristic
230C23×C5C2×#4111yescharacteristic
231C23×C5C2×#4111yescharacteristic
232C23×C5C2×#4111yescharacteristic
233C23×C5C2×#4111yescharacteristic
234C23×C5C2×#4111yescharacteristic
235C23×C5C2×#4111yescharacteristic
236C23×C5H(5,𝔽2)11yescharacteristic
237C23×C5H(5,𝔽2)11yescharacteristic
238C23×C52−1+411yescharacteristic
239C23×C52−1+411yescharacteristic
240C2×C4×C5—24no—
241C2×C4×C5—24no—
242C2×C4×C5—24no—
243C2×C4×C5—24no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—24no—
249C2×C4×C5—24no—
250C2×C4×C5—22no—
251C2×C4×C5—22no—
252C2×C4×C5—41no—
253C2×C4×C5—41no—
254C2×C4×C5—42no—
255C2×C4×C5—41no—
256C2×C4×C5—42no—
257C2×C4×C5—41no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—41no—
261C2×C4×C5—41no—
262C2×C4×C5—42no—
263C2×C4×C5—41no—
264C2×C4×C5—42no—
265C2×C4×C5—41no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5C2×C4⋊C414yes—
269C2×C4×C5C4×D414yes—
270C2×C4×C5C4×Q814yes—
271C2×C4×C5#12114yes—
272C2×C4×C5C22⋊Q814yes—
273C2×C4×C5#12514yes—
274C2×C4×C5C4⋊Q814yes—
275C2×Dic5—24no—
276C2×Dic5—24no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—24no—
281C2×Dic5—24no—
282C2×Dic5—22no—
283C2×Dic5—22no—
284C2×Dic5—22no—
285C2×Dic5—24no—
286C2×Dic5—24no—
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—41no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—41no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—42no—
309C2×Dic5—42no—
310C2×Dic5—42no—
311C2×Dic5—42no—
312C2×Dic5—42no—
313C2×Dic5—42no—
314C2×Dic5—42no—
315C2×Dic5C2×C4⋊C418yes—
316C2×Dic5#12514yes—
317C23×C4—52no—
318C23×C4—51no—
319C23×C4—51no—
320C23×C4—51no—
321C23×C4—51no—
322C23×C4—52no—
323C23×C4C22×D512yes—
324C23×C4C22×D511yescharacteristic
325C23×C4C22×D511yescharacteristic
326C23×C4C22×D512yes—
327C23×C4C22×D511yescharacteristic
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—54no—
336C2×C4⋊C4—54no—
337C2×C4⋊C4—54no—
338C2×C4⋊C4—58no—
339C2×C4⋊C4—104no—
340C2×C4⋊C4—102no—
341C2×C4⋊C4—102no—
342C2×C4⋊C4—104no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—202no—
345C2×C4⋊C4—202no—
346C2×C4⋊C4—202no—
347C2×C4⋊C4C4×D514yes—
348C2×C4⋊C4C22⋊D514yes—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—54no—
353C2×C42—54no—
354C2×C42—102no—
355C2×C42—102no—
356C2×C42—102no—
357C2×C42—102no—
358C2×C42—104no—
359C2×C42—104no—
360C2×C42—202no—
361#95—24no—
362#95—58no—
363#95—104no—
364#95—102no—
365#95—104no—
366#95—102no—
367#95—104no—
368#95—102no—
369#95—102no—
370#95—104no—
371#95—104no—
372#95—202no—
373C22×C5C22×C4⋊C411yescharacteristic
374C22×C5C2×C4×D411yescharacteristic
375C22×C5C2×C4×Q811yescharacteristic
376C22×C5C4×#4111yescharacteristic
377C22×C5C4×#4111yescharacteristic
378C22×C5#51811yescharacteristic
379C22×C5#51911yescharacteristic
380C22×C5C2×#12111yescharacteristic
381C22×C5C2×C22⋊Q811yescharacteristic
382C22×C5C2×#12511yescharacteristic
383C22×C5C2×#12511yescharacteristic
384C22×C5#52911yescharacteristic
385C22×C5#52911yescharacteristic
386C22×C5C2×C4⋊Q811yescharacteristic
387C22×C5#53311yescharacteristic
388C22×C5#54211yescharacteristic
389C22×C5#54211yescharacteristic
390C22×C5#54411yescharacteristic
391C22×C5#54411yescharacteristic
392C22×C5#54611yescharacteristic
393C22×C5#54611yescharacteristic
394C22×C5#54611yescharacteristic
395C22×C5#54711yescharacteristic
396C22×C5#54811yescharacteristic
397C22×C5#54811yescharacteristic
398C22×C5#54811yescharacteristic
399C22×C5#55011yescharacteristic
400C22×C5#55211yescharacteristic
401C22×C5#55311yescharacteristic
402C22×C5#55411yescharacteristic
403C22×C5#55411yescharacteristic
404C22×C5#55511yescharacteristic
405C22×C5#55611yescharacteristic
406C22×C5#55711yescharacteristic
407C22×C5#55711yescharacteristic
408C4×C5—44no—
409C4×C5—42no—
410C4×C5—44no—
411C4×C5—42no—
412C4×C5#38918yes—
413Dic5—28no—
414Dic5—24no—
415Dic5—42no—
416Dic5—42no—
417Dic5—42no—
418Dic5—82no—
419C24C23×D511yescharacteristic
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—54no—
445C22×C4—54no—
446C22×C4—52no—
447C22×C4—54no—
448C22×C4—52no—
449C22×C4—52no—
450C22×C4—52no—
451C22×C4—52no—
452C22×C4—52no—
453C22×C4—101no—
454C22×C4—101no—
455C22×C4—101no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—101no—
459C22×C4—102no—
460C22×C4—102no—
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—101no—
474C22×C4—101no—
475C22×C4—101no—
476C22×C4—101no—
477C22×C4—101no—
478C22×C4—101no—
479C22×C4—102no—
480C22×C4—102no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—102no—
484C22×C4—102no—
485C22×C4—102no—
486C22×C4C2×C4×D514yes—
487C22×C4C2×C4×D512yes—
488C22×C4#70912yes—
489C22×C4D4×D512yes—
490C22×C4#71112yes—
491C22×C4#71112yes—
492C22×C4Q8×D512yes—
493C22×C4Q8×D512yes—
494C22×C4#71312yes—
495C22×C4C2×C22⋊D514yes—
496C22×C4C2×C22⋊D512yes—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—48no—
500C4⋊C4—516no—
501C4⋊C4—208no—
502C4⋊C4—208no—
503C4⋊C4—204no—
504C42—44no—
505C42—108no—
506C42—108no—
507C42—208no—
508C2×C5C2×#38911yescharacteristic
509C2×C5C4×#12511yescharacteristic
510C2×C5#233111yescharacteristic
511C2×C5#234911yescharacteristic
512C2×C5#235111yescharacteristic
513C2×C5#236211yescharacteristic
514C2×C5#236311yescharacteristic
515C2×C5#242611yescharacteristic
516C2×C5#242711yescharacteristic
517C2×C5#243911yescharacteristic
518C2×C5#244211yescharacteristic
519C2×C5#250811yescharacteristic
520C2×C5#250911yescharacteristic
521C2×C5#255011yescharacteristic
522C2×C5#255211yescharacteristic
523C23C22×C4×D511yescharacteristic
524C23C2×#70911yescharacteristic
525C23C2×D4×D511yescharacteristic
526C23C2×#71111yescharacteristic
527C23C2×#71111yescharacteristic
528C23#414411yescharacteristic
529C23C2×Q8×D511yescharacteristic
530C23C2×Q8×D511yescharacteristic
531C23C2×#71311yescharacteristic
532C23#414711yescharacteristic
533C23D5×#4111yescharacteristic
534C23D5×#4111yescharacteristic
535C23#414911yescharacteristic
536C23#415011yescharacteristic
537C23C22×C22⋊D511yescharacteristic
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—58no—
567C2×C4—54no—
568C2×C4—102no—
569C2×C4—104no—
570C2×C4—102no—
571C2×C4—104no—
572C2×C4—102no—
573C2×C4—104no—
574C2×C4—102no—
575C2×C4—102no—
576C2×C4—104no—
577C2×C4—104no—
578C2×C4—102no—
579C2×C4—104no—
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—201no—
590C2×C4—201no—
591C2×C4—201no—
592C2×C4—201no—
593C2×C4—201no—
594C2×C4—201no—
595C2×C4—201no—
596C2×C4—201no—
597C2×C4—201no—
598C2×C4—201no—
599C2×C4—201no—
600C2×C4—201no—
601C2×C4—202no—
602C2×C4—202no—
603C2×C4—202no—
604C2×C4—202no—
605C2×C4—202no—
606C2×C4—202no—
607C2×C4—202no—
608C2×C4#403314yes—
609C2×C4#403514yes—
610C2×C4D5×C4⋊C414yes—
611C2×C4#404214yes—
612C2×C4C4×C22⋊D514yes—
613C2×C4#408514yes—
614C2×C4#409014yes—
615C5#1400611yescharacteristic, complemented, Sylow
616C22C2×#403311yescharacteristic
617C22C2×#403511yescharacteristic
618C22C2×D5×C4⋊C411yescharacteristic
619C22C2×#404211yescharacteristic
620C22C4×#71111yescharacteristic
621C22#6587911yescharacteristic
622C22#6589211yescharacteristic
623C22#6589711yescharacteristic
624C22#6590611yescharacteristic
625C22C4×#71311yescharacteristic
626C22#6591511yescharacteristic
627C22#6591811yescharacteristic
628C22#6593211yescharacteristic
629C22#6593411yescharacteristic
630C22#6593711yescharacteristic
631C22#6594611yescharacteristic
632C22#6595811yescharacteristic
633C22#6596411yescharacteristic
634C22#6596711yescharacteristic
635C22#6597711yescharacteristic
636C22#6601911yescharacteristic
637C22D5×#12511yescharacteristic
638C22#6602311yescharacteristic
639C22#6602411yescharacteristic
640C22#6605511yescharacteristic
641C22#6606211yescharacteristic
642C22#6606311yescharacteristic
643C22#6606411yescharacteristic
644C22C2×C4×C22⋊D511yescharacteristic
645C22C2×#408511yescharacteristic
646C22#6614011yescharacteristic
647C22C2×#409011yescharacteristic
648C22#6615111yescharacteristic
649C22#6615811yescharacteristic
650C22#6616811yescharacteristic
651C4—44no—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—108no—
656C4—104no—
657C4—202no—
658C4—202no—
659C4—202no—
660C4—402no—
661C4#6526118yes—
662C2#1060769711yescharacteristic
663C2#1060770511yescharacteristic
664C2#1060771311yescharacteristic
665C2#1060773311yescharacteristic
666C2#1060777011yescharacteristic
667C2#1060777411yescharacteristic
668C2#1060793411yescharacteristic
669C2#1060795411yescharacteristic
670C2C2×#6526111yescharacteristic
671C2#1061407211yescharacteristic
672C2#1061419511yescharacteristic
673C2#1061437811yescharacteristic
674C2#1061448611yescharacteristic
675C2#1061476811yescharacteristic
676C2#1061497211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#389C2
socle, prime radical, cosocleC24×C5C24
hypercenterC2×#389D5
hypocenterC5#14006
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
38. #65262C4semidirect
191. C4⋊Dic5C4⋊C4semidirect
615. C5#14006semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3892
C2×#652622
C2×#652662
C2×C4⋊C4×Dic52
C2×#652172
#106023592
#106023572
#106023562
C4×#39632
#106002062
#106001972
#140065