Finite Group Databasebeta

← C2×#10604167 #13034248

Subgroups — C2×#10604167 #13034248

Order 1280.

2066 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D20C211yesmaximal, characteristic
2C2×#65765C212yesmaximal, complemented
3C22×#3999C212yesmaximal
4C2×#64917C212yesmaximal, complemented
5#10604167C218yesmaximal, direct product factor
6C23×D20C2211yescharacteristic
7C23×C4×D5C411yescharacteristic, complemented
8C23×C4×D5C2211yescharacteristic
9C22×C22⋊AGL(1,𝔽5)C2214yes—
10C22×C4⋊AGL(1,𝔽5)—22no—
11C22×C42×C5C411yescharacteristic, complemented
12C22×#685C411yescharacteristic, complemented
13C22×#685C2211yescharacteristic
14C22×C4⋊Dic5C411yescharacteristic, complemented
15C2×C4×D20—22no—
16C2×C4×D20—22no—
17C2×C4×D20—24no—
18C2×C4×D20C414yescomplemented
19C2×C4×D20C2214yes—
20#65765—24no—
21#65765C2218yescomplemented
22C2×(C2×AGL(1,𝔽5))⋊C4—22no—
23C2×#3999—28no—
24C2×#3999C2218yes—
25#64917C2218yescomplemented
26C2×#1799—51nomaximal, Sylow
27C24×D5C2×C411yescharacteristic
28C24×D5C2311yescharacteristic
29C23×AGL(1,𝔽5)—24no—
30C23×C4×C5C2×C411yescharacteristic
31C23×C4×C5C2×C411yescharacteristic
32C23×C4×C5C2×C411yescharacteristic
33C23×Dic5C2×C411yescharacteristic
34C23×Dic5C2×C411yescharacteristic
35C22×D20—22no—
36C22×D20—22no—
37C22×D20—24no—
38C22×D20C2×C412yes—
39C22×D20D414yes—
40C22×D20Q814yes—
41C22×D20C2312yes—
42C22×C4×D5—22no—
43C22×C4×D5—21no—
44C22×C4×D5—21no—
45C22×C4×D5—44no—
46C22×C4×D5—44no—
47C22×C4×D5C2×C412yes—
48C22×C4×D5C2×C414yescomplemented
49C22×C4×D5D412yes—
50C22×C4×D5D412yes—
51C22×C4×D5D414yes—
52C22×C4×D5C2312yes—
53C2×C22⋊AGL(1,𝔽5)—28no—
54C2×C22⋊AGL(1,𝔽5)—28no—
55C2×C22⋊AGL(1,𝔽5)—24no—
56C2×C22⋊AGL(1,𝔽5)C2318yes—
57C2×C4⋊AGL(1,𝔽5)—28no—
58C2×C4⋊AGL(1,𝔽5)—48no—
59C2×C42×C5—22no—
60C2×C42×C5—21no—
61C2×C42×C5—21no—
62C2×C42×C5C2×C414yescomplemented
63C2×#685—22no—
64C2×#685—22no—
65C2×#685—22no—
66C2×#685—22no—
67C2×#685C2×C412yes—
68C2×#685C2×C414yescomplemented
69C2×#685C2312yes—
70C2×C4⋊Dic5—22no—
71C2×C4⋊Dic5—22no—
72C2×C4⋊Dic5C2×C414yescomplemented
73C4×D20—28no—
74C4×D20—28no—
75C4×D20—28no—
76C4×D20—28no—
77(C2×AGL(1,𝔽5))⋊C4—28no—
78#3999—216no—
79C22×C4×D4—51no—
80C22×#99—52no—
81C2×#380—52no—
82#1799—58no—
83C2×#342—52no—
84C24×C5C22×C411yescharacteristic
85C23×D5—21no—
86C23×D5—21no—
87C23×D5—22no—
88C23×D5—42no—
89C23×D5—42no—
90C23×D5—42no—
91C23×D5—42no—
92C23×D5C4⋊C414yescomplemented
93C23×D5C22×C412yes—
94C23×D5C22×C411yescharacteristic
95C23×D5C2×D412yes—
96C23×D5C2×D412yes—
97C23×D5C2×D412yes—
98C23×D5C2×Q812yes—
99C23×D5#4114yes—
100C23×D5C2411yescharacteristic
101C22×AGL(1,𝔽5)—28no—
102C22×AGL(1,𝔽5)—48no—
103C22×AGL(1,𝔽5)—48no—
104C22×AGL(1,𝔽5)—48no—
105C22×C4×C5—21no—
106C22×C4×C5—21no—
107C22×C4×C5—22no—
108C22×C4×C5—24no—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—42no—
112C22×C4×C5C22⋊C412yes—
113C22×C4×C5C22⋊C411yescharacteristic
114C22×C4×C5C22⋊C411yescharacteristic
115C22×C4×C5C4⋊C414yescomplemented
116C22×C4×C5C22×C412yes—
117C22×C4×C5C22×C412yes—
118C22×C4×C5C22×C412yes—
119C22×Dic5—21no—
120C22×Dic5—21no—
121C22×Dic5—42no—
122C22×Dic5—42no—
123C22×Dic5C22⋊C412yes—
124C22×Dic5C22⋊C412yes—
125C22×Dic5C22×C412yes—
126C22×Dic5C22×C412yes—
127C2×D20—24no—
128C2×D20—24no—
129C2×D20—28no—
130C2×D20—28no—
131C2×D20—28no—
132C2×D20—28no—
133C2×D20—24no—
134C2×D20—24no—
135C2×D20C4⋊C418yescomplemented
136C2×D20C2×D414yes—
137C2×D20C2×Q814yes—
138C2×C4×D5—28no—
139C2×C4×D5—24no—
140C2×C4×D5—24no—
141C2×C4×D5—44no—
142C2×C4×D5—48no—
143C2×C4×D5—44no—
144C2×C4×D5—44no—
145C2×C4×D5—44no—
146C2×C4×D5—48no—
147C2×C4×D5—44no—
148C2×C4×D5—48no—
149C2×C4×D5C22⋊C414yes—
150C2×C4×D5C22⋊C414yes—
151C2×C4×D5C2×D414yes—
152C2×C4×D5C2×D414yes—
153C22⋊AGL(1,𝔽5)—28no—
154C22⋊AGL(1,𝔽5)—28no—
155C22⋊AGL(1,𝔽5)—48no—
156C4⋊AGL(1,𝔽5)—416no—
157C42×C5—24no—
158C42×C5—24no—
159#685—24no—
160#685—24no—
161#685—28no—
162C4⋊Dic5—28no—
163C23×D4—51no—
164C24×C4—51no—
165C24×C4—51no—
166C2×C4×D4—54no—
167C2×C4×D4—54no—
168C2×C4×D4—102no—
169C2×C4×D4—102no—
170C2×C4×D4—104no—
171C22×C4⋊C4—51no—
172C22×C4⋊C4—102no—
173C22×C22⋊C4—51no—
174C22×C22⋊C4—51no—
175C22×C22⋊C4—54no—
176C22×C42AGL(1,𝔽5)11yescharacteristic, complemented
177C2×#99—58no—
178C2×#99—108no—
179#380—58no—
180#380—104no—
181C2×#95—102no—
182#342—58no—
183C23×C5—22no—
184C23×C5—22no—
185C23×C5C2×C22⋊C411yescharacteristic
186C23×C5C2×C22⋊C411yescharacteristic
187C23×C5C2×C4⋊C412yes—
188C23×C5#11712yes—
189C23×C5C23×C411yescharacteristic
190C22×D5—24no—
191C22×D5—24no—
192C22×D5—22no—
193C22×D5—22no—
194C22×D5—24no—
195C22×D5—24no—
196C22×D5—41no—
197C22×D5—42no—
198C22×D5—41no—
199C22×D5—41no—
200C22×D5—44no—
201C22×D5—42no—
202C22×D5—44no—
203C22×D5—48no—
204C22×D5—41no—
205C22×D5—42no—
206C22×D5—48no—
207C22×D5—42no—
208C22×D5—42no—
209C22×D5—42no—
210C22×D5—42no—
211C22×D5—48no—
212C22×D5—42no—
213C22×D5C2×C22⋊C412yes—
214C22×D5C2×C4⋊C412yes—
215C22×D5C2×C4⋊C414yescomplemented
216C22×D5#11712yes—
217C22×D5#12114yes—
218C22×D5#12114yes—
219C22×D5C22⋊Q814yes—
220C22×D5C22⋊Q814yes—
221C22×D5C22×D412yes—
222C22×D5C22×D411yescharacteristic
223C22×D5C22×Q811yescharacteristic
224C22×D5C2×#4112yes—
225C2×AGL(1,𝔽5)—48no—
226C2×AGL(1,𝔽5)—48no—
227C2×AGL(1,𝔽5)—48no—
228C2×AGL(1,𝔽5)—816no—
229C2×C4×C5—22no—
230C2×C4×C5—22no—
231C2×C4×C5—24no—
232C2×C4×C5—24no—
233C2×C4×C5—22no—
234C2×C4×C5—24no—
235C2×C4×C5—24no—
236C2×C4×C5—22no—
237C2×C4×C5—42no—
238C2×C4×C5—42no—
239C2×C4×C5—42no—
240C2×C4×C5#9914yescomplemented
241C2×C4×C5C2×C22⋊C412yes—
242C2×C4×C5C2×C22⋊C412yes—
243C2×C4×C5C2×C4⋊C414yescomplemented
244C2×Dic5—24no—
245C2×Dic5—22no—
246C2×Dic5—42no—
247C2×Dic5—42no—
248C2×Dic5—42no—
249C2×Dic5—42no—
250C2×Dic5—42no—
251C2×Dic5C2×C22⋊C414yes—
252D20—28no—
253D20—28no—
254D20—216no—
255C4×D5—28no—
256C4×D5—28no—
257C4×D5—44no—
258C4×D5—44no—
259C4×D5—48no—
260C4×D5—48no—
261C25—51no—
262C25—51no—
263C22×D4—54no—
264C22×D4—54no—
265C22×D4—52no—
266C22×D4—52no—
267C22×D4—102no—
268C22×D4—102no—
269C22×D4—104no—
270C23×C4—51no—
271C23×C4—52no—
272C23×C4—52no—
273C23×C4—54no—
274C23×C4—52no—
275C23×C4—52no—
276C23×C4—51no—
277C23×C4—54no—
278C23×C4—102no—
279C23×C4—101no—
280C23×C4—101no—
281C23×C4—104no—
282C23×C4—204no—
283C23×C4—204no—
284C23×C4C2×AGL(1,𝔽5)11yescharacteristic
285C23×C4C2×AGL(1,𝔽5)11yescharacteristic
286C23×C4C2×AGL(1,𝔽5)11yescharacteristic
287C4×D4—108no—
288C4×D4—108no—
289C4×D4—108no—
290C4×D4—108no—
291C2×C4⋊C4—54no—
292C2×C4⋊C4—102no—
293C2×C4⋊C4—102no—
294C2×C4⋊C4—108no—
295C2×C4⋊C4—208no—
296C2×C22⋊C4—52no—
297C2×C22⋊C4—52no—
298C2×C22⋊C4—54no—
299C2×C22⋊C4—58no—
300C2×C22⋊C4—102no—
301C2×C22⋊C4—102no—
302C2×C22⋊C4—102no—
303C2×C22⋊C4—102no—
304C2×C22⋊C4—108no—
305C2×C22⋊C4—108no—
306C2×C22⋊C4—104no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42C2×AGL(1,𝔽5)14yescomplemented
311#99—1016no—
312#95—108no—
313C22×C5—24no—
314C22×C5—24no—
315C22×C5—21no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5—21no—
319C22×C5#38012yes—
320C22×C5#38012yes—
321C22×C5C2×#9912yes—
322C22×C5#41012yes—
323C22×C5C22×C22⋊C411yescharacteristic
324C22×C5C22×C4⋊C411yescharacteristic
325C22×C5C2×#11711yescharacteristic
326C2×D5—28no—
327C2×D5—28no—
328C2×D5—22no—
329C2×D5—22no—
330C2×D5—22no—
331C2×D5—22no—
332C2×D5—24no—
333C2×D5—42no—
334C2×D5—42no—
335C2×D5—44no—
336C2×D5—44no—
337C2×D5—41no—
338C2×D5—41no—
339C2×D5—42no—
340C2×D5—44no—
341C2×D5—48no—
342C2×D5—48no—
343C2×D5—42no—
344C2×D5—42no—
345C2×D5—44no—
346C2×D5#38014yes—
347C2×D5C2×#12112yes—
348C2×D5C2×C22⋊Q812yes—
349AGL(1,𝔽5)—88no—
350C4×C5—22no—
351C4×C5—24no—
352C4×C5—42no—
353C4×C5C2×#9914yescomplemented
354Dic5—24no—
355Dic5—42no—
356C24—52no—
357C24—52no—
358C24—52no—
359C24—54no—
360C24—52no—
361C24—51no—
362C24—51no—
363C24—54no—
364C24—52no—
365C24—102no—
366C24—101no—
367C24—101no—
368C24—202no—
369C24—202no—
370C24—202no—
371C24—202no—
372C24C22×AGL(1,𝔽5)11yescharacteristic
373C2×D4—54no—
374C2×D4—58no—
375C2×D4—54no—
376C2×D4—104no—
377C2×D4—104no—
378C2×D4—104no—
379C2×D4—104no—
380C2×D4—108no—
381C2×D4—108no—
382C2×D4—108no—
383C2×D4—108no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—42no—
391C22×C4—52no—
392C22×C4—52no—
393C22×C4—52no—
394C22×C4—54no—
395C22×C4—54no—
396C22×C4—54no—
397C22×C4—54no—
398C22×C4—52no—
399C22×C4—108no—
400C22×C4—101no—
401C22×C4—101no—
402C22×C4—104no—
403C22×C4—104no—
404C22×C4—108no—
405C22×C4—208no—
406C22×C4—204no—
407C22×C4—204no—
408C22×C4—204no—
409C22×C4—204no—
410C22×C4—208no—
411C22×C4—208no—
412C22×C4—202no—
413C22×C4—204no—
414C22×C4—202no—
415C22×C4—208no—
416C22×C4—208no—
417C22×C4—208no—
418C22×C4C4⋊AGL(1,𝔽5)14yescomplemented
419C22×C4C22⋊AGL(1,𝔽5)12yes—
420C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
421C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
422C22×C4C22×AGL(1,𝔽5)12yes—
423C22×C4C22×AGL(1,𝔽5)12yes—
424C22×C4C22×AGL(1,𝔽5)12yes—
425C4⋊C4—108no—
426C4⋊C4—2016no—
427C22⋊C4—108no—
428C22⋊C4—104no—
429C22⋊C4—104no—
430C22⋊C4—108no—
431C22⋊C4—108no—
432C22⋊C4—208no—
433C42—24no—
434C42—24no—
435C2×C5—21no—
436C2×C5—21no—
437C2×C5—22no—
438C2×C5#179914yescomplemented
439C2×C5C2×#38011yescharacteristic
440C2×C5C22×#9911yescharacteristic
441C2×C5C2×#41011yescharacteristic
442D5—22no—
443D5—22no—
444D5—24no—
445D5—44no—
446C23—22no—
447C23—22no—
448C23—52no—
449C23—52no—
450C23—54no—
451C23—54no—
452C23—51no—
453C23—52no—
454C23—51no—
455C23—52no—
456C23—52no—
457C23—54no—
458C23—54no—
459C23—54no—
460C23—104no—
461C23—104no—
462C23—102no—
463C23—102no—
464C23—104no—
465C23—104no—
466C23—202no—
467C23—201no—
468C23—201no—
469C23—202no—
470C23—201no—
471C23—201no—
472C23—204no—
473C23—204no—
474C23—208no—
475C23—202no—
476C23—202no—
477C23—208no—
478C23—202no—
479C23—202no—
480C23—208no—
481C23—202no—
482C23—202no—
483C23C2×C4⋊AGL(1,𝔽5)12yes—
484C23#413012yes—
485C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
486C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
487C23C23×AGL(1,𝔽5)11yescharacteristic
488D4—108no—
489D4—108no—
490D4—1016no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—54no—
503C2×C4—104no—
504C2×C4—108no—
505C2×C4—108no—
506C2×C4—102no—
507C2×C4—204no—
508C2×C4—204no—
509C2×C4—208no—
510C2×C4—202no—
511C2×C4—202no—
512C2×C4—202no—
513C2×C4—202no—
514C2×C4—208no—
515C2×C4—202no—
516C2×C4—208no—
517C2×C4—208no—
518C2×C4—208no—
519C2×C4—4016no—
520C2×C4#399914yescomplemented
521C2×C4C2×C4⋊AGL(1,𝔽5)14yescomplemented
522C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
523C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
524C5C2×#179911yescharacteristic, complemented, Sylow
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—24no—
530C22—24no—
531C22—52no—
532C22—52no—
533C22—54no—
534C22—102no—
535C22—102no—
536C22—102no—
537C22—102no—
538C22—108no—
539C22—108no—
540C22—104no—
541C22—201no—
542C22—202no—
543C22—201no—
544C22—202no—
545C22—202no—
546C22—204no—
547C22—204no—
548C22—204no—
549C22—204no—
550C22—208no—
551C22—202no—
552C22—202no—
553C22—208no—
554C22C2×#399912yes—
555C22#6574512yes—
556C22#6576512yes—
557C22#6576512yes—
558C22C22×C4⋊AGL(1,𝔽5)11yescharacteristic
559C22C2×#413011yescharacteristic
560C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
561C4—22no—
562C4—24no—
563C4—42no—
564C4—104no—
565C4—202no—
566C4—408no—
567C4C2×#399914yescomplemented
568C2—21no—
569C2—21no—
570C2—22no—
571C2—102no—
572C2—102no—
573C2—104no—
574C2—204no—
575C2#1060416714yesdirect product factor
576C2C22×#399911yescharacteristic
577C2C2×#6574511yescharacteristic
578C2C2×#6576511yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C23×C4
FrattiniC23C23×AGL(1,𝔽5)
FittingC22×C42×C5C4
socleC23×C5C2×C22⋊C4
prime radicalC23×D20C22
cosocleC23×D5C24
hypercenterC22×C42AGL(1,𝔽5)
hypocenterC5C2×#1799
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#65765C2semidirect
4. C2×#64917C2semidirect
5. #10604167C2direct (N × H)
7. C23×C4×D5C4semidirect
11. C22×C42×C5C4semidirect
12. C22×#685C4semidirect
14. C22×C4⋊Dic5C4semidirect
18. C2×C4×D20C4semidirect
21. #65765C22semidirect
25. #64917C22semidirect
48. C22×C4×D5C2×C4semidirect
62. C2×C42×C5C2×C4semidirect
68. C2×#685C2×C4semidirect
72. C2×C4⋊Dic5C2×C4semidirect
92. C23×D5C4⋊C4semidirect
115. C22×C4×C5C4⋊C4semidirect
135. C2×D20C4⋊C4semidirect
176. C22×C42AGL(1,𝔽5)semidirect
215. C22×D5C2×C4⋊C4semidirect
240. C2×C4×C5#99semidirect
243. C2×C4×C5C2×C4⋊C4semidirect
310. C2×C42C2×AGL(1,𝔽5)semidirect
353. C4×C5C2×#99semidirect
418. C22×C4C4⋊AGL(1,𝔽5)semidirect
438. C2×C5#1799semidirect
520. C2×C4#3999semidirect
521. C2×C4C2×C4⋊AGL(1,𝔽5)semidirect
524. C5C2×#1799semidirect
567. C4C2×#3999semidirect
575. C2#10604167direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D202
C2×#657652
C22×#39992
C2×#649172
#106041672
C2×#17995