Finite Group Databasebeta

← #13034423

Subgroups — #13034423

Order 1280.

2038 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C42×D5C211yesmaximal, characteristic
2C2×#65680C212yesmaximal, complemented
3#10604899C212yesmaximal, complemented
4#10604898C212yesmaximal
5C4×C22⋊#170C212yesmaximal, complemented
6#10604184C212yesmaximal, complemented
7#10604182C212yesmaximal, complemented
8C4×#4003C212yesmaximal, complemented
9C23×C4×D5C411yescharacteristic
10C23×C4×D5C411yescharacteristic
11C23×C4×D5C2211yescharacteristic
12C22×C42×C5C411yescharacteristic
13C22×C4×Dic5C411yescharacteristic
14C22×C4×Dic5C2211yescharacteristic
15C22×C4×Dic5C2211yescharacteristic
16C2×C42×D5—28no—
17C2×C42×D5C411yescharacteristic
18C2×C42×D5C411yescharacteristic
19C2×C42×D5C411yescharacteristic
20C2×C42×D5C411yescharacteristic
21C2×C42×D5C2211yescharacteristic
22C2×C42×D5C2211yescharacteristic
23C2×C42×D5C2211yescharacteristic
24C2×C42×D5C2211yescharacteristic
25C2×C22⋊#170C2214yes—
26C2×#4003C2214yes—
27C2×#4003C2214yes—
28C2×C4⋊#170C2214yescomplemented
29C2×C4×#170C2214yescomplemented
30#65680—28no—
31#64918C2214yescomplemented
32#64918C2214yescomplemented
33#20714—51nomaximal, Sylow
34C24×D5C2×C411yescharacteristic
35C23×C4×C5C2×C411yescharacteristic
36C23×C4×C5C2×C411yescharacteristic
37C23×C4×C5C2×C411yescharacteristic
38C23×Dic5C2×C411yescharacteristic
39C23×Dic5C2×C411yescharacteristic
40C23×Dic5C2311yescharacteristic
41C22×C4×D5—24no—
42C22×C4×D5—24no—
43C22×C4×D5—24no—
44C22×C4×D5—24no—
45C22×C4×D5—24no—
46C22×C4×D5—24no—
47C22×C4×D5—24no—
48C22×C4×D5—24no—
49C22×C4×D5C818yescomplemented
50C22×C4×D5C2×C411yescharacteristic
51C22×C4×D5C2×C411yescharacteristic
52C22×C4×D5C2×C411yescharacteristic
53C22×C4×D5C2×C411yescharacteristic
54C22×C4×D5C2×C411yescharacteristic
55C22×C4×D5C2×C411yescharacteristic
56C22×C4×D5C2×C411yescharacteristic
57C22×C4×D5C2×C411yescharacteristic
58C22×C4×D5C2×C411yescharacteristic
59C22×C4×D5C2×C411yescharacteristic
60C22×C4×D5C2311yescharacteristic
61C22×C4×D5C2311yescharacteristic
62C22×#170C2318yes—
63C2×C42×C5—24no—
64C2×C42×C5C2×C411yescharacteristic
65C2×C42×C5C2×C411yescharacteristic
66C2×C42×C5C2×C411yescharacteristic
67C2×C42×C5C2×C411yescharacteristic
68C2×C4×Dic5—24no—
69C2×C4×Dic5—24no—
70C2×C4×Dic5—24no—
71C2×C4×Dic5C2×C411yescharacteristic
72C2×C4×Dic5C2×C411yescharacteristic
73C2×C4×Dic5C2×C411yescharacteristic
74C2×C4×Dic5C2×C411yescharacteristic
75C2×C4×Dic5C2×C411yescharacteristic
76C2×C4×Dic5C2×C411yescharacteristic
77C2×C4×Dic5C2×C411yescharacteristic
78C2×C4×Dic5C2×C411yescharacteristic
79C2×C4×Dic5C2311yescharacteristic
80C2×C4×Dic5C2311yescharacteristic
81C2×C4×Dic5C2311yescharacteristic
82C2×C4×Dic5C2311yescharacteristic
83C42×D5—24no—
84C42×D5—24no—
85C42×D5—24no—
86C42×D5—24no—
87C42×D5—24no—
88C42×D5—24no—
89C42×D5—24no—
90C42×D5D418yes—
91C22⋊#170—28no—
92#4003—28no—
93#4003—28no—
94C4⋊#170—28no—
95C4×#170—28no—
96C23×C42—51no—
97C2×#431—52no—
98#1817—52no—
99#1797—52no—
100#1797—52no—
101#1779—52no—
102C4×C22⋊C8—52no—
103C4×C22⋊C8—52no—
104C24×C5C22×C411yescharacteristic
105C23×D5—22no—
106C23×D5—22no—
107C23×D5—22no—
108C23×D5—22no—
109C23×D5—22no—
110C23×D5—22no—
111C23×D5C2×C814yes—
112C23×D5C22×C411yescharacteristic
113C23×D5C22×C411yescharacteristic
114C22×C4×C5—22no—
115C22×C4×C5—22no—
116C22×C4×C5—22no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5C2×C814yes—
122C22×C4×C5M1614yes—
123C22×C4×C5C22×C411yescharacteristic
124C22×C4×C5C22×C411yescharacteristic
125C22×C4×C5C22×C411yescharacteristic
126C22×C4×C5C22×C411yescharacteristic
127C22×C4×C5C22×C411yescharacteristic
128C22×C4×C5C22×C411yescharacteristic
129C22×Dic5—22no—
130C22×Dic5—22no—
131C22×Dic5—22no—
132C22×Dic5—22no—
133C22×Dic5—22no—
134C22×Dic5—22no—
135C22×Dic5—22no—
136C22×Dic5C2×C814yes—
137C22×Dic5C22×C411yescharacteristic
138C22×Dic5C22×C411yescharacteristic
139C22×Dic5C22×C411yescharacteristic
140C22×Dic5C22×C411yescharacteristic
141C22×Dic5C22×C411yescharacteristic
142C22×Dic5#4114yes—
143C22×Dic5C2411yescharacteristic
144C2×C4×D5—232no—
145C2×C4×D5—232no—
146C2×C4×D5—22no—
147C2×C4×D5—22no—
148C2×C4×D5—22no—
149C2×C4×D5—22no—
150C2×C4×D5—22no—
151C2×C4×D5—22no—
152C2×C4×D5—22no—
153C2×C4×D5—22no—
154C2×C4×D5—22no—
155C2×C4×D5—22no—
156C2×C4×D5—22no—
157C2×C4×D5—22no—
158C2×C4×D5—22no—
159C2×C4×D5—22no—
160C2×C4×D5—22no—
161C2×C4×D5—22no—
162C2×C4×D5—22no—
163C2×C4×D5—22no—
164C2×C4×D5—22no—
165C2×C4×D5—22no—
166C2×C4×D5—22no—
167C2×C4×D5—22no—
168C2×C4×D5—22no—
169C2×C4×D5—22no—
170C2×C4×D5—22no—
171C2×C4×D5—22no—
172C2×C4×D5—232no—
173C2×C4×D5C22⋊C414yes—
174C2×C4×D5C22⋊C414yes—
175C2×C4×D5C2×C814yes—
176C2×C4×D5C2×C814yes—
177C2×C4×D5C2×C814yes—
178C2×C4×D5C2×C814yes—
179C2×C4×D5M1614yes—
180C2×C4×D5M1614yes—
181C2×C4×D5C2×D414yes—
182C2×C4×D5#4114yes—
183C2×#170—28no—
184C2×#170—28no—
185C2×#170—28no—
186C42×C5—22no—
187C42×C5—22no—
188C42×C5—22no—
189C42×C5C22⋊C414yes—
190C4×Dic5—22no—
191C4×Dic5—22no—
192C4×Dic5—22no—
193C4×Dic5—22no—
194C4×Dic5—22no—
195C4×Dic5—22no—
196C4×Dic5—22no—
197C4×Dic5—22no—
198C4×Dic5—22no—
199C4×Dic5C22⋊C414yes—
200C4×Dic5C2×D414yes—
201C4×Dic5C2×D414yes—
202C24×C4—51no—
203C24×C4—51no—
204C24×C4—51no—
205C22×C42—51no—
206C22×C42—51no—
207C22×C42—51no—
208C22×C42—51no—
209C22×C42—51no—
210C22×C42—51no—
211C22×C42—51no—
212C22×C42—51no—
213C22×C42—51no—
214C22×C42—51no—
215C22×C42—51no—
216C22×C42—108no—
217C22×C42AGL(1,𝔽5)11yescharacteristic
218#431—108no—
219C2×C4⋊C8—54no—
220C2×C22⋊C8—54no—
221C2×C22⋊C8—54no—
222C2×C22⋊C8—54no—
223C2×C4×C8—54no—
224#336—54no—
225#336—54no—
226C23×C5—21no—
227C23×C5—21no—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5#11712yes—
231C23×C5C22×C812yes—
232C23×C5C2×M1612yes—
233C23×C5C23×C411yescharacteristic
234C22×D5—28no—
235C22×D5—28no—
236C22×D5—28no—
237C22×D5—28no—
238C22×D5—21no—
239C22×D5—21no—
240C22×D5—21no—
241C22×D5—21no—
242C22×D5—21no—
243C22×D5—21no—
244C22×D5—21no—
245C22×D5—21no—
246C22×D5—28no—
247C22×D5—28no—
248C22×D5—28no—
249C22×D5C2×C22⋊C412yes—
250C22×D5#11712yes—
251C22×D5C22×C812yes—
252C22×D5C22×C812yes—
253C22×D5C2×M1612yes—
254C22×D5C2×M1612yes—
255C2×C4×C5—28no—
256C2×C4×C5—21no—
257C2×C4×C5—21no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—28no—
265C2×C4×C5—28no—
266C2×C4×C5C2×C22⋊C412yes—
267C2×C4×C5C2×C22⋊C412yes—
268C2×C4×C5C2×C22⋊C412yes—
269C2×C4×C5#11712yes—
270C2×C4×C5C22×C812yes—
271C2×C4×C5C22×C812yes—
272C2×C4×C5C2×M1612yes—
273C2×C4×C5C2×M1612yes—
274C2×C4×C5C2×M1612yes—
275C2×C4×C5C2×M1612yes—
276C2×Dic5—28no—
277C2×Dic5—21no—
278C2×Dic5—21no—
279C2×Dic5—21no—
280C2×Dic5—21no—
281C2×Dic5—21no—
282C2×Dic5—21no—
283C2×Dic5—21no—
284C2×Dic5—21no—
285C2×Dic5—28no—
286C2×Dic5—28no—
287C2×Dic5C2×C22⋊C412yes—
288C2×Dic5C2×C22⋊C412yes—
289C2×Dic5#11712yes—
290C2×Dic5C22×C812yes—
291C2×Dic5C22×C812yes—
292C2×Dic5C2×M1612yes—
293C2×Dic5C2×M1612yes—
294C2×Dic5C22×D412yes—
295C2×Dic5C2×#4112yes—
296C2×Dic5C2×#4112yes—
297C4×D5—28no—
298C4×D5—28no—
299C4×D5—28no—
300C4×D5—28no—
301C4×D5—28no—
302C4×D5—28no—
303C4×D5—28no—
304C4×D5—28no—
305C4×D5—28no—
306C4×D5—28no—
307C4×D5—28no—
308C4×D5C22⋊C8116yescomplemented
309#170—48no—
310C25—51no—
311C23×C4—51no—
312C23×C4—51no—
313C23×C4—51no—
314C23×C4—58no—
315C23×C4—51no—
316C23×C4—51no—
317C23×C4—51no—
318C23×C4—51no—
319C23×C4—51no—
320C23×C4—51no—
321C23×C4—51no—
322C23×C4—51no—
323C23×C4—51no—
324C23×C4—51no—
325C23×C4—51no—
326C23×C4—51no—
327C23×C4—104no—
328C23×C4—104no—
329C23×C4—104no—
330C23×C4—104no—
331C23×C4—104no—
332C23×C4—104no—
333C23×C4—104no—
334C23×C4—104no—
335C23×C4C2×AGL(1,𝔽5)11yescharacteristic
336C23×C4C2×AGL(1,𝔽5)11yescharacteristic
337C23×C4C2×AGL(1,𝔽5)11yescharacteristic
338C22×C8—58no—
339C2×C42—24no—
340C2×C42—51no—
341C2×C42—51no—
342C2×C42—51no—
343C2×C42—58no—
344C2×C42—51no—
345C2×C42—51no—
346C2×C42—51no—
347C2×C42—51no—
348C2×C42—51no—
349C2×C42—51no—
350C2×C42—51no—
351C2×C42—51no—
352C2×C42—51no—
353C2×C42—104no—
354C2×C42—104no—
355C2×C42—104no—
356C2×C42—104no—
357C2×C42—104no—
358C2×C42—104no—
359C2×C42—104no—
360C2×C42—104no—
361C2×C42—104no—
362C2×C42—104no—
363C2×C42C2×AGL(1,𝔽5)11yescharacteristic
364C2×C42C2×AGL(1,𝔽5)11yescharacteristic
365C2×C42C2×AGL(1,𝔽5)11yescharacteristic
366C2×C42C2×AGL(1,𝔽5)11yescharacteristic
367C4⋊C8—108no—
368C22⋊C8—108no—
369C22⋊C8—108no—
370C22⋊C8—108no—
371C4×C8—108no—
372C22×C5—22no—
373C22×C5—22no—
374C22×C5—22no—
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5#43114yescomplemented
379C22×C5C22×C22⋊C411yescharacteristic
380C22×C5C2×#11711yescharacteristic
381C22×C5C2×#11711yescharacteristic
382C22×C5C23×C811yescharacteristic
383C22×C5C22×M1611yescharacteristic
384C22×C5C22×M1611yescharacteristic
385C22×C5C22×M1611yescharacteristic
386C2×D5—232no—
387C2×D5—22no—
388C2×D5—22no—
389C2×D5—22no—
390C2×D5—22no—
391C2×D5—22no—
392C2×D5—22no—
393C2×D5—22no—
394C2×D5—22no—
395C2×D5—22no—
396C2×D5—22no—
397C2×D5—22no—
398C2×D5—22no—
399C2×D5C2×C22⋊C814yes—
400C2×D5#43114yescomplemented
401C4×C5—22no—
402C4×C5—22no—
403C4×C5—22no—
404C4×C5—22no—
405C4×C5C2×C22⋊C814yes—
406C4×C5#40714yes—
407Dic5—22no—
408Dic5—22no—
409Dic5—22no—
410Dic5—22no—
411Dic5C2×C22⋊C814yes—
412Dic5#52514yes—
413C24—51no—
414C24—51no—
415C24—54no—
416C24—102no—
417C24—102no—
418C24—102no—
419C24—102no—
420C24—102no—
421C24—102no—
422C24C22×AGL(1,𝔽5)11yescharacteristic
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—51no—
431C22×C4—51no—
432C22×C4—54no—
433C22×C4—54no—
434C22×C4—54no—
435C22×C4—54no—
436C22×C4—54no—
437C22×C4—51no—
438C22×C4—51no—
439C22×C4—51no—
440C22×C4—51no—
441C22×C4—54no—
442C22×C4—54no—
443C22×C4—54no—
444C22×C4—54no—
445C22×C4—54no—
446C22×C4—54no—
447C22×C4—54no—
448C22×C4—102no—
449C22×C4—102no—
450C22×C4—102no—
451C22×C4—102no—
452C22×C4—102no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—1032no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—102no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—102no—
465C22×C4—1032no—
466C22×C4—1032no—
467C22×C4—102no—
468C22×C4—102no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4—102no—
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×C4D5⋊C814yes—
485C22×C4#70014yes—
486C22×C4C22×AGL(1,𝔽5)11yescharacteristic
487C22×C4C22×AGL(1,𝔽5)11yescharacteristic
488C22×C4C22×AGL(1,𝔽5)11yescharacteristic
489C22×C4C22×AGL(1,𝔽5)11yescharacteristic
490C22×C4C22×AGL(1,𝔽5)11yescharacteristic
491C22×C4C22×AGL(1,𝔽5)11yescharacteristic
492C2×C8—108no—
493C2×C8—108no—
494C2×C8—108no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—54no—
499C42—54no—
500C42—54no—
501C42—102no—
502C42—102no—
503C42—102no—
504C42—102no—
505C42—102no—
506C42—102no—
507C42—102no—
508C42—102no—
509C42—102no—
510C42C22⋊AGL(1,𝔽5)14yes—
511C2×C5—24no—
512C2×C5#228011yescharacteristic
513C2×C5C22×C22⋊C811yescharacteristic
514C2×C5C2×#40711yescharacteristic
515C2×C5C2×#43111yescharacteristic
516C2×C5C2×#43111yescharacteristic
517C2×C5#292011yescharacteristic
518C2×C5#292011yescharacteristic
519D5—24no—
520D5—24no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—52no—
526C23—52no—
527C23—52no—
528C23—52no—
529C23—52no—
530C23—52no—
531C23—108no—
532C23—108no—
533C23—108no—
534C23—108no—
535C23—108no—
536C23—108no—
537C23—108no—
538C23—101no—
539C23—101no—
540C23—101no—
541C23—101no—
542C23—101no—
543C23—101no—
544C23—101no—
545C23—101no—
546C23C2×D5⋊C812yes—
547C23C2×#70012yes—
548C23#413012yes—
549C23C23×AGL(1,𝔽5)11yescharacteristic
550C2×C4—28no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—52no—
562C2×C4—52no—
563C2×C4—52no—
564C2×C4—52no—
565C2×C4—52no—
566C2×C4—52no—
567C2×C4—516no—
568C2×C4—52no—
569C2×C4—52no—
570C2×C4—52no—
571C2×C4—52no—
572C2×C4—108no—
573C2×C4—108no—
574C2×C4—108no—
575C2×C4—108no—
576C2×C4—108no—
577C2×C4—108no—
578C2×C4—108no—
579C2×C4—108no—
580C2×C4—108no—
581C2×C4—108no—
582C2×C4—108no—
583C2×C4—101no—
584C2×C4—101no—
585C2×C4—101no—
586C2×C4—101no—
587C2×C4—101no—
588C2×C4—101no—
589C2×C4—101no—
590C2×C4—101no—
591C2×C4—108no—
592C2×C4—108no—
593C2×C4—108no—
594C2×C4C2×D5⋊C812yes—
595C2×C4C2×D5⋊C812yes—
596C2×C4C2×#70012yes—
597C2×C4C2×#70012yes—
598C2×C4D5⋊M1612yes—
599C2×C4D5⋊M1612yes—
600C2×C4#413012yes—
601C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
602C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
603C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
604C8—208no—
605C5#2071411yescharacteristic, complemented, Sylow
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—54no—
613C22—54no—
614C22—1032no—
615C22—102no—
616C22—102no—
617C22—102no—
618C22—102no—
619C22—102no—
620C22—102no—
621C22—102no—
622C22—102no—
623C22—102no—
624C22—102no—
625C22—102no—
626C22—102no—
627C22#6568014yescomplemented
628C22C22×D5⋊C811yescharacteristic
629C22C22×#70011yescharacteristic
630C22C2×D5⋊M1611yescharacteristic
631C22C2×D5⋊M1611yescharacteristic
632C22C2×#413011yescharacteristic
633C22C2×#413011yescharacteristic
634C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—54no—
640C4—54no—
641C4—102no—
642C4—102no—
643C4—102no—
644C4—102no—
645C4D5⋊(C22⋊C8)14yes—
646C4#6575614yes—
647C2—24no—
648C2—104no—
649C2—104no—
650C2C2×#6568011yescharacteristic
651C2D5⋊#43111yescharacteristic
652C2#1061703711yescharacteristic
653C2#1061703811yescharacteristic
654C2C2×D5⋊(C22⋊C8)11yescharacteristic
655C2C2×#6575611yescharacteristic
656C2#1061748011yescharacteristic
subgroupG/H
centerC22×C4C22×AGL(1,𝔽5)
commutatorC22×C5C23×C8
FrattiniC23C23×AGL(1,𝔽5)
FittingC22×C42×C5C4
socleC23×C5C23×C4
prime radicalC24×D5C2×C4
cosocleC22×Dic5C24
hypercenterC22×C42AGL(1,𝔽5)
hypocenterC5#20714
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#65680C2semidirect
3. #10604899C2semidirect
5. C4×C22⋊#170C2semidirect
6. #10604184C2semidirect
7. #10604182C2semidirect
8. C4×#4003C2semidirect
28. C2×C4⋊#170C22semidirect
29. C2×C4×#170C22semidirect
31. #64918C22semidirect
32. #64918C22semidirect
49. C22×C4×D5C8semidirect
308. C4×D5C22⋊C8semidirect
378. C22×C5#431semidirect
400. C2×D5#431semidirect
605. C5#20714semidirect
627. C22#65680semidirect

Maximal subgroups

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

subgroupindex
C22×C42×D52
C2×#656802
#106048992
#106048982
C4×C22⋊#1702
#106041842
#106041822
C4×#40032
#207145