Finite Group Databasebeta

← #12024601

Subgroups — #12024601

Order 1280.

2062 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
1C22×C4×Dic10C211yesmaximal, characteristic, complemented
2C2×C4×C5×C22⋊C4C211yesmaximal, characteristic
3C2×#65499C212yesmaximal
4C2×#65236C211yesmaximal, characteristic
5#10602339C211yesmaximal, characteristic
6#10602338C212yesmaximal
7#10602329C212yesmaximal, complemented
8#10602327C211yesmaximal, characteristic, complemented
9#10600217C211yesmaximal, characteristic, complemented
10#10600213C211yesmaximal, characteristic, complemented
11#10600199C212yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C23×Dic10C2211yescharacteristic
14C22×C5×C22⋊C4C2211yescharacteristic
15C22×C5×C22⋊C4C2211yescharacteristic
16C22×C42×C5C2211yescharacteristic
17C22×C42×C5C2211yescharacteristic
18C22×C22⋊Dic5C2212yes—
19C22×C22⋊Dic5C2212yes—
20C22×C4⋊Dic5C2211yescharacteristic
21C22×#683C2212yes—
22C22×#683C2212yes—
23C22×C4×Dic5C2212yes—
24C2×C4×Dic10—28no—
25C2×C4×Dic10—24no—
26C4×C5×C22⋊C4—24no—
27C2×C5×#95C2211yescharacteristic
28C2×C5×#95C2211yescharacteristic
29#65499—28no—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2211yescharacteristic
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36C2×#3963C2212yes—
37C2×#3963C2212yes—
38C2×#3963C2211yescharacteristic
39#65236C418yescomplemented
40#14172—51nomaximal, Sylow
41C25×C5C2311yescharacteristic
42C23×C4×C5—22no—
43C23×C4×C5—22no—
44C23×C4×C5—22no—
45C23×C4×C5—24no—
46C23×C4×C5—42no—
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2311yescharacteristic
49C23×C4×C5C2311yescharacteristic
50C23×C4×C5C2311yescharacteristic
51C23×C4×C5C2311yescharacteristic
52C23×C4×C5C2311yescharacteristic
53C23×Dic5C2312yes—
54C23×Dic5C2312yes—
55C23×Dic5C2312yes—
56C23×Dic5C2312yes—
57C22×Dic10—22no—
58C22×Dic10—22no—
59C22×Dic10—24no—
60C22×Dic10—24no—
61C22×Dic10C2×C414yescomplemented
62C2×C5×C22⋊C4—22no—
63C2×C5×C22⋊C4—22no—
64C2×C5×C22⋊C4—22no—
65C2×C5×C22⋊C4—22no—
66C2×C5×C22⋊C4C2×C414yes—
67C2×C5×C22⋊C4Q814yescomplemented
68C2×C42×C5—24no—
69C2×C42×C5—22no—
70C2×C42×C5—42no—
71C2×C42×C5D414yes—
72C2×C22⋊Dic5—24no—
73C2×C22⋊Dic5—24no—
74C2×C22⋊Dic5—24no—
75C2×C22⋊Dic5—24no—
76C2×C22⋊Dic5—42no—
77C2×C22⋊Dic5C2×C418yes—
78C2×C4⋊Dic5—22no—
79C2×C4⋊Dic5—24no—
80C2×#683—24no—
81C2×#683—24no—
82C2×#683—24no—
83C2×#683—24no—
84C2×#683—42no—
85C2×#683D418yes—
86C2×C4×Dic5—24no—
87C2×C4×Dic5—24no—
88C2×C4×Dic5—24no—
89C4×Dic10—232no—
90C5×#95—22no—
91C5×#95—22no—
92#3963—24no—
93#3963—24no—
94#3963—22no—
95#3963—24no—
96#3963—24no—
97#3963—42no—
98#3963—42no—
99#3963C2×C414yescomplemented
100#3963C2×C418yescomplemented
101C22×C4×Q8—51no—
102C2×#393—51no—
103C2×#385—52no—
104C2×C4×C22⋊C4D511yescharacteristic
105#1446—51no—
106#1444—51no—
107#1439—51no—
108#1437—52no—
109#1436—52no—
110#1436—52no—
111#1434—51no—
112C24×C5—22no—
113C24×C5—21no—
114C24×C5—22no—
115C24×C5—21no—
116C24×C5—21no—
117C24×C5—21no—
118C24×C5—22no—
119C24×C5—22no—
120C24×C5C22×C412yes—
121C24×C5C2×Q812yes—
122C24×C5#4112yes—
123C24×C5C2411yescharacteristic
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—21no—
128C22×C4×C5—21no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—28no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—22no—
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—48no—
145C22×C4×C5—48no—
146C22×C4×C5—44no—
147C22×C4×C5—44no—
148C22×C4×C5C22×C412yes—
149C22×C4×C5C22×C412yes—
150C22×C4×C5C2×D414yes—
151C22×C4×C5C2×D412yes—
152C22×C4×C5C2×D412yes—
153C22×C4×C5C2×D412yes—
154C22×C4×C5C2×Q812yes—
155C22×C4×C5C2×Q812yes—
156C22×C4×C5#4112yes—
157C22×C4×C5#4112yes—
158C22×C4×C5#4112yes—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
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×Dic5—22no—
178C22×Dic5—22no—
179C22×Dic5—22no—
180C22×Dic5—22no—
181C22×Dic5—22no—
182C22×Dic5C22×C414yes—
183C22×Dic5C22×C414yes—
184C22×Dic5C2×D414yes—
185C22×Dic5C2×D414yes—
186C22×Dic5#4114yes—
187C2×Dic10—216no—
188C2×Dic10—28no—
189C2×Dic10—28no—
190C2×Dic10—28no—
191C2×Dic10—216no—
192C5×C22⋊C4—28no—
193C5×C22⋊C4—28no—
194C42×C5—28no—
195C42×C5—44no—
196C22⋊Dic5—216no—
197C22⋊Dic5—48no—
198C4⋊Dic5—28no—
199#683—216no—
200#683—48no—
201C4×Dic5—216no—
202C23×Q8—51no—
203C24×C4C2×D511yescharacteristic
204C2×C4×Q8—108no—
205C2×C4×Q8—104no—
206C22×C4⋊C4—52no—
207C22×C4⋊C4—51no—
208C22×C4⋊C4—52no—
209C22×C22⋊C4—52no—
210C22×C22⋊C4—52no—
211C22×C22⋊C4C2×D511yescharacteristic
212C22×C22⋊C4C2×D511yescharacteristic
213C22×C42—52no—
214C22×C42C2×D511yescharacteristic
215C22×C42C2×D511yescharacteristic
216#393—58no—
217#385—108no—
218C4×C22⋊C4—24no—
219C2×#95—51no—
220C2×#95—52no—
221C2×#95—52no—
222C2×#95—52no—
223C2×#95—51no—
224C2×#95—52no—
225C2×#95—52no—
226C2×#95—52no—
227C2×#95—52no—
228C2×#95C2×D511yescharacteristic
229C2×#95C2×D511yescharacteristic
230C23×C5—24no—
231C23×C5—22no—
232C23×C5—22no—
233C23×C5—24no—
234C23×C5—22no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—24no—
239C23×C5—24no—
240C23×C5—24no—
241C23×C5—24no—
242C23×C5—41no—
243C23×C5—42no—
244C23×C5—41no—
245C23×C5—41no—
246C23×C5—41no—
247C23×C5—42no—
248C23×C5—42no—
249C23×C5—42no—
250C23×C5—42no—
251C23×C5—42no—
252C23×C5C4×Q814yescomplemented
253C23×C5C23×C411yescharacteristic
254C23×C5C22×D412yes—
255C23×C5C22×D411yescharacteristic
256C23×C5C22×Q811yescharacteristic
257C23×C5C2×#4111yescharacteristic
258C23×C5C2×#4111yescharacteristic
259C23×C5C2×#4111yescharacteristic
260C23×C5C2×#4111yescharacteristic
261C23×C5C2×#4112yes—
262C23×C5H(5,𝔽2)12yes—
263C23×C52−1+412yes—
264C2×C4×C5—24no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—24no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—24no—
273C2×C4×C5—24no—
274C2×C4×C5—24no—
275C2×C4×C5—24no—
276C2×C4×C5—22no—
277C2×C4×C5—24no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—24no—
281C2×C4×C5—24no—
282C2×C4×C5—416no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—42no—
290C2×C4×C5—42no—
291C2×C4×C5—42no—
292C2×C4×C5C4×D414yes—
293C2×C4×C5#12014yes—
294C2×C4×C5C22⋊Q814yes—
295C2×Dic5—24no—
296C2×Dic5—24no—
297C2×Dic5—24no—
298C2×Dic5—24no—
299C2×Dic5—24no—
300C2×Dic5—24no—
301C2×Dic5—24no—
302C2×Dic5—24no—
303C2×Dic5—24no—
304C2×Dic5—24no—
305C2×Dic5—24no—
306C2×Dic5—24no—
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×Dic5—42no—
316C2×Dic5—42no—
317C2×Dic5—42no—
318C2×Dic5—42no—
319C2×Dic5—42no—
320C2×Dic5—42no—
321C2×Dic5—42no—
322C2×Dic5C4×D418yes—
323Dic10—232no—
324C25C22×D511yescharacteristic
325C22×Q8—54no—
326C22×Q8—104no—
327C22×Q8—104no—
328C22×Q8—102no—
329C22×Q8—102no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—24no—
334C23×C4—42no—
335C23×C4—52no—
336C23×C4—52no—
337C23×C4—52no—
338C23×C4—52no—
339C23×C4C22×D511yescharacteristic
340C23×C4C22×D511yescharacteristic
341C23×C4C22×D511yescharacteristic
342C23×C4C22×D511yescharacteristic
343C23×C4C22×D511yescharacteristic
344C23×C4C22×D511yescharacteristic
345C4×Q8—1032no—
346C2×C4⋊C4—58no—
347C2×C4⋊C4—104no—
348C2×C4⋊C4—104no—
349C2×C4⋊C4—104no—
350C2×C4⋊C4—102no—
351C2×C4⋊C4—104no—
352C2×C4⋊C4—104no—
353C2×C4⋊C4—202no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—58no—
359C2×C22⋊C4—104no—
360C2×C22⋊C4—104no—
361C2×C22⋊C4—104no—
362C2×C22⋊C4—104no—
363C2×C22⋊C4—202no—
364C2×C22⋊C4Dic1014yescomplemented
365C2×C22⋊C4C4×D514yes—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—42no—
369C2×C42—104no—
370C2×C42—104no—
371C2×C42—104no—
372C2×C42C22⋊D514yes—
373#95—22no—
374#95—22no—
375#95—58no—
376#95—54no—
377#95—104no—
378#95—104no—
379#95—104no—
380#95—102no—
381#95—104no—
382#95—202no—
383#95—202no—
384C22×C5—28no—
385C22×C5—24no—
386C22×C5—42no—
387C22×C5—42no—
388C22×C5—42no—
389C22×C5—44no—
390C22×C5—42no—
391C22×C5—42no—
392C22×C5—44no—
393C22×C5—44no—
394C22×C5—42no—
395C22×C5C2×C4×D411yescharacteristic
396C22×C5C2×C4×D412yes—
397C22×C5C2×C4×Q811yescharacteristic
398C22×C5C4×#4111yescharacteristic
399C22×C5#52012yes—
400C22×C5C2×#12011yescharacteristic
401C22×C5C2×C22⋊Q811yescharacteristic
402C22×C5#52511yescharacteristic
403C22×C5#52512yes—
404C22×C5#52511yescharacteristic
405C22×C5#52911yescharacteristic
406C22×C5#53311yescharacteristic
407C22×C5#53511yescharacteristic
408C22×C5#53611yescharacteristic
409C22×C5#54212yes—
410C22×C5#54612yes—
411C22×C5#54612yes—
412C22×C5#54812yes—
413C22×C5D4×Q812yes—
414C22×C5#55212yes—
415C22×C5#55412yes—
416C22×C5#55612yes—
417C22×C5#55912yes—
418C4×C5—24no—
419C4×C5—24no—
420C4×C5—24no—
421C4×C5—42no—
422C4×C5—42no—
423C4×C5—42no—
424Dic5—28no—
425Dic5—44no—
426Dic5—44no—
427Dic5—82no—
428C24—21no—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—21no—
436C24C2×Dic1012yes—
437C24C2×C4×D512yes—
438C24#70912yes—
439C24C23×D511yescharacteristic
440C2×Q8—1016no—
441C2×Q8—108no—
442C2×Q8—1016no—
443C2×Q8—108no—
444C2×Q8—108no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—54no—
470C22×C4—54no—
471C22×C4—54no—
472C22×C4—54no—
473C22×C4—54no—
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×C4—102no—
488C22×C4—102no—
489C22×C4—102no—
490C22×C4—102no—
491C22×C4—102no—
492C22×C4—102no—
493C22×C4—102no—
494C22×C4—102no—
495C22×C4—102no—
496C22×C4—102no—
497C22×C4C2×Dic1012yes—
498C22×C4C2×Dic1012yes—
499C22×C4C2×C4×D512yes—
500C22×C4C2×C4×D512yes—
501C22×C4#70912yes—
502C22×C4#70912yes—
503C22×C4#70912yes—
504C22×C4D4×D514yes—
505C22×C4C2×C22⋊D512yes—
506C22×C4C2×C22⋊D512yes—
507C22×C4C2×C22⋊D512yes—
508C4⋊C4—108no—
509C4⋊C4—1016no—
510C4⋊C4—208no—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—1016no—
514C22⋊C4—208no—
515C42—28no—
516C42—44no—
517C42—1016no—
518C2×C5—44no—
519C2×C5C4×#12011yescharacteristic
520C2×C5C4×C22⋊Q811yescharacteristic
521C2×C5#232611yescharacteristic
522C2×C5#236012yes—
523C2×C5#236712yes—
524C2×C5#240411yescharacteristic
525C2×C5#240911yescharacteristic
526C2×C5#245812yes—
527C2×C5#246512yes—
528C2×C5#258011yescharacteristic
529C2×C5#259911yescharacteristic
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23C4×Dic1014yescomplemented
553C23C22×Dic1011yescharacteristic
554C23C22×C4×D511yescharacteristic
555C23C2×#70911yescharacteristic
556C23C2×#70911yescharacteristic
557C23C2×#70911yescharacteristic
558C23C2×#70911yescharacteristic
559C23C2×D4×D512yes—
560C23#414412yes—
561C23D5×#4112yes—
562C23#415012yes—
563C23C22×C22⋊D511yescharacteristic
564Q8—1032no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—58no—
594C2×C4—104no—
595C2×C4—104no—
596C2×C4—104no—
597C2×C4—104no—
598C2×C4—104no—
599C2×C4—104no—
600C2×C4—104no—
601C2×C4—104no—
602C2×C4—104no—
603C2×C4—104no—
604C2×C4—104no—
605C2×C4—104no—
606C2×C4—202no—
607C2×C4—202no—
608C2×C4—202no—
609C2×C4—202no—
610C2×C4—202no—
611C2×C4—202no—
612C2×C4—202no—
613C2×C4—202no—
614C2×C4—202no—
615C2×C4—202no—
616C2×C4—202no—
617C2×C4—202no—
618C2×C4—202no—
619C2×C4—202no—
620C2×C4—202no—
621C2×C4#407014yes—
622C2×C4C4×C22⋊D514yes—
623C2×C4#408314yes—
624C5#1417211yescharacteristic, complemented, Sylow
625C22—24no—
626C22—28no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22C2×C4×Dic1011yescharacteristic
637C22#6580411yescharacteristic
638C22C4×#70911yescharacteristic
639C22#6581311yescharacteristic
640C22#6582412yes—
641C22#6582412yes—
642C22#6586512yes—
643C22#6586712yes—
644C22D4×Dic1012yes—
645C22#6587312yes—
646C22C4×D4×D512yes—
647C22#6588012yes—
648C22#6588112yes—
649C22#6588612yes—
650C22#6589312yes—
651C22#6589512yes—
652C22C2×#407011yescharacteristic
653C22C2×C4×C22⋊D511yescharacteristic
654C22#6612511yescharacteristic
655C22C2×#408311yescharacteristic
656C22#6613911yescharacteristic
657C22#6615911yescharacteristic
658C22#6616211yescharacteristic
659C4—24no—
660C4—24no—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—108no—
666C4—204no—
667C4—204no—
668C4—402no—
669C2—44no—
670C2#1060519712yes—
671C2#1060520312yes—
672C2#1060522712yes—
673C2#1060522912yes—
674C2C4×#407011yescharacteristic
675C2#1061385611yescharacteristic
676C2#1061398911yescharacteristic
677C2#1061405911yescharacteristic
678C2#1061408011yescharacteristic
679C2#1061411911yescharacteristic
680C2C4×#408311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C4×C5×C22⋊C4C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×C4×C22⋊C4D5
hypocenterC5#14172
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic10C2semidirect
7. #10602329C2semidirect
8. #10602327C2semidirect
9. #10600217C2semidirect
10. #10600213C2semidirect
11. #10600199C2semidirect
39. #65236C4semidirect
61. C22×Dic10C2×C4semidirect
67. C2×C5×C22⋊C4Q8semidirect
99. #3963C2×C4semidirect
100. #3963C2×C4semidirect
252. C23×C5C4×Q8semidirect
364. C2×C22⋊C4Dic10semidirect
552. C23C4×Dic10semidirect
624. C5#14172semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic102
C2×C4×C5×C22⋊C42
C2×#654992
C2×#652362
#106023392
#106023382
#106023292
#106023272
#106002172
#106002132
#106001992
#141725