Finite Group Databasebeta

← C2×#10604892 #13034246

Subgroups — C2×#10604892 #13034246

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×C22⋊D5C211yesmaximal, characteristic
2C22×#4011C212yesmaximal
3C2×#65765C212yesmaximal, complemented
4#10604892C218yesmaximal, direct product factor
5C2×#64917C212yesmaximal, complemented
6C24×C4×C5C411yescharacteristic, complemented
7C23×C22⋊D5C2211yescharacteristic
8C23×C4×D5C2211yescharacteristic
9C22×C22⋊AGL(1,𝔽5)C2214yes—
10C22×C4⋊AGL(1,𝔽5)—22no—
11C22×C22⋊Dic5C411yescharacteristic, complemented
12C2×C4×C22⋊D5—22no—
13C2×C4×C22⋊D5—22no—
14C2×C4×C22⋊D5—24no—
15C2×C4×C22⋊D5C414yescomplemented
16C2×C4×C22⋊D5C2214yes—
17C22×#685C2211yescharacteristic
18C22×#683C411yescharacteristic, complemented
19C22×C4×Dic5C411yescharacteristic, complemented
20C2×#4011—28no—
21C2×#4011C2218yes—
22#65765—24no—
23#65765C2218yescomplemented
24C2×(C2×AGL(1,𝔽5))⋊C4—22no—
25#64917C2218yescomplemented
26C2×#1799—51nomaximal, Sylow
27C25×C5C2×C411yescharacteristic
28C24×D5C2311yescharacteristic
29C23×AGL(1,𝔽5)—24no—
30C23×C4×C5—22no—
31C23×C4×C5—21no—
32C23×C4×C5—21no—
33C23×C4×C5—44no—
34C23×C4×C5C2×C411yescharacteristic
35C23×C4×C5C2×C411yescharacteristic
36C23×C4×C5C2×C414yescomplemented
37C22×C22⋊D5—22no—
38C22×C22⋊D5—22no—
39C22×C22⋊D5—24no—
40C22×C22⋊D5C2×C412yes—
41C22×C22⋊D5D414yes—
42C22×C22⋊D5Q814yes—
43C22×C22⋊D5C2312yes—
44C23×Dic5C2×C411yescharacteristic
45C23×Dic5C2×C411yescharacteristic
46C23×Dic5C2×C411yescharacteristic
47C22×C4×D5—44no—
48C22×C4×D5C2×C412yes—
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×C22⋊Dic5—22no—
60C2×C22⋊Dic5—22no—
61C2×C22⋊Dic5C2×C414yescomplemented
62C4×C22⋊D5—28no—
63C4×C22⋊D5—28no—
64C4×C22⋊D5—28no—
65C4×C22⋊D5—28no—
66C2×#685—22no—
67C2×#685—22no—
68C2×#685C2×C412yes—
69C2×#685C2312yes—
70C2×#683—22no—
71C2×#683—22no—
72C2×#683C2×C414yescomplemented
73C2×C4×Dic5—22no—
74C2×C4×Dic5—21no—
75C2×C4×Dic5—21no—
76C2×C4×Dic5C2×C414yescomplemented
77#4011—216no—
78(C2×AGL(1,𝔽5))⋊C4—28no—
79C22×C4×D4—51no—
80C22×#99—52no—
81C2×#380—52no—
82#1799—58no—
83C2×#342—52no—
84C24×C5—21no—
85C24×C5—21no—
86C24×C5—22no—
87C24×C5—42no—
88C24×C5—42no—
89C24×C5C4⋊C414yescomplemented
90C24×C5C22×C411yescharacteristic
91C24×C5C22×C412yes—
92C23×D5—42no—
93C23×D5—42no—
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—24no—
106C22×C4×C5—21no—
107C22×C4×C5—21no—
108C22×C4×C5—24no—
109C22×C4×C5—24no—
110C22×C4×C5—44no—
111C22×C4×C5—48no—
112C22×C4×C5—48no—
113C22×C4×C5—44no—
114C22×C4×C5—42no—
115C22×C4×C5C22⋊C412yes—
116C22×C4×C5C22⋊C411yescharacteristic
117C22×C4×C5C22⋊C411yescharacteristic
118C22×C4×C5C22×C412yes—
119C22×C4×C5C22×C412yes—
120C2×C22⋊D5—24no—
121C2×C22⋊D5—24no—
122C2×C22⋊D5—28no—
123C2×C22⋊D5—28no—
124C2×C22⋊D5—28no—
125C2×C22⋊D5—28no—
126C2×C22⋊D5—24no—
127C2×C22⋊D5—24no—
128C2×C22⋊D5C4⋊C418yescomplemented
129C2×C22⋊D5C2×D414yes—
130C2×C22⋊D5C2×Q814yes—
131C22×Dic5—21no—
132C22×Dic5—21no—
133C22×Dic5—22no—
134C22×Dic5—21no—
135C22×Dic5—21no—
136C22×Dic5—42no—
137C22×Dic5—42no—
138C22×Dic5C22⋊C412yes—
139C22×Dic5C22⋊C412yes—
140C22×Dic5C4⋊C414yescomplemented
141C22×Dic5C22×C412yes—
142C22×Dic5C22×C412yes—
143C22×Dic5C22×C412yes—
144C2×C4×D5—28no—
145C2×C4×D5—44no—
146C2×C4×D5—48no—
147C2×C4×D5—44no—
148C2×C4×D5—44no—
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—
157C22⋊Dic5—28no—
158#685—24no—
159#685—24no—
160#683—28no—
161C4×Dic5—24no—
162C4×Dic5—24no—
163C23×D4—51no—
164C24×C4—51no—
165C24×C4AGL(1,𝔽5)11yescharacteristic, complemented
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×C42—51no—
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×C5—22no—
186C23×C5—22no—
187C23×C5—24no—
188C23×C5—24no—
189C23×C5—42no—
190C23×C5—48no—
191C23×C5—42no—
192C23×C5—42no—
193C23×C5—48no—
194C23×C5—42no—
195C23×C5—42no—
196C23×C5—42no—
197C23×C5C2×C22⋊C411yescharacteristic
198C23×C5C2×C22⋊C411yescharacteristic
199C23×C5C2×C4⋊C412yes—
200C23×C5C2×C4⋊C414yescomplemented
201C23×C5#11712yes—
202C23×C5C23×C411yescharacteristic
203C22×D5—24no—
204C22×D5—24no—
205C22×D5—41no—
206C22×D5—42no—
207C22×D5—41no—
208C22×D5—41no—
209C22×D5—44no—
210C22×D5—42no—
211C22×D5—44no—
212C22×D5—48no—
213C22×D5—41no—
214C22×D5C2×C22⋊C412yes—
215C22×D5C2×C4⋊C412yes—
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—24no—
231C2×C4×C5—24no—
232C2×C4×C5—22no—
233C2×C4×C5—48no—
234C2×C4×C5—42no—
235C2×C4×C5—42no—
236C2×C4×C5—48no—
237C2×C4×C5—42no—
238C2×C4×C5#9914yescomplemented
239C2×C4×C5C2×C22⋊C412yes—
240C2×C4×C5C2×C22⋊C412yes—
241C22⋊D5—28no—
242C22⋊D5—28no—
243C22⋊D5—216no—
244C2×Dic5—22no—
245C2×Dic5—22no—
246C2×Dic5—24no—
247C2×Dic5—24no—
248C2×Dic5—24no—
249C2×Dic5—22no—
250C2×Dic5—42no—
251C2×Dic5—42no—
252C2×Dic5—42no—
253C2×Dic5—42no—
254C2×Dic5—42no—
255C2×Dic5C2×C22⋊C414yes—
256C2×Dic5C2×C4⋊C414yescomplemented
257C4×D5—28no—
258C4×D5—28no—
259C4×D5—44no—
260C4×D5—44no—
261C25—51no—
262C25C2×AGL(1,𝔽5)11yescharacteristic
263C22×D4—54no—
264C22×D4—54no—
265C22×D4—52no—
266C22×D4—52no—
267C22×D4—102no—
268C22×D4—102no—
269C22×D4—104no—
270C23×C4—22no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—44no—
274C23×C4—51no—
275C23×C4—51no—
276C23×C4—52no—
277C23×C4—52no—
278C23×C4—54no—
279C23×C4—52no—
280C23×C4—52no—
281C23×C4—51no—
282C23×C4—104no—
283C23×C4—204no—
284C23×C4C2×AGL(1,𝔽5)11yescharacteristic
285C23×C4C2×AGL(1,𝔽5)11yescharacteristic
286C23×C4C2×AGL(1,𝔽5)14yescomplemented
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—54no—
297C2×C22⋊C4—52no—
298C2×C22⋊C4—52no—
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—54no—
308C2×C42—102no—
309C2×C42—101no—
310C2×C42—101no—
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—24no—
320C22×C5—44no—
321C22×C5—44no—
322C22×C5—48no—
323C22×C5—48no—
324C22×C5—42no—
325C22×C5—42no—
326C22×C5#38012yes—
327C22×C5#38012yes—
328C22×C5C2×#9912yes—
329C22×C5#41012yes—
330C22×C5C22×C22⋊C411yescharacteristic
331C22×C5C22×C4⋊C411yescharacteristic
332C22×C5C2×#11711yescharacteristic
333C2×D5—28no—
334C2×D5—28no—
335C2×D5—22no—
336C2×D5—22no—
337C2×D5—22no—
338C2×D5—22no—
339C2×D5—42no—
340C2×D5—42no—
341C2×D5—44no—
342C2×D5—44no—
343C2×D5—41no—
344C2×D5—41no—
345C2×D5—42no—
346C2×D5#38014yes—
347C2×D5C2×#12112yes—
348C2×D5C2×C22⋊Q812yes—
349AGL(1,𝔽5)—88no—
350C4×C5—22no—
351C4×C5—42no—
352C4×C5C2×#9914yescomplemented
353Dic5—24no—
354Dic5—24no—
355Dic5—42no—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—42no—
360C24—42no—
361C24—52no—
362C24—52no—
363C24—52no—
364C24—54no—
365C24—52no—
366C24—51no—
367C24—51no—
368C24—202no—
369C24—202no—
370C24C4⋊AGL(1,𝔽5)14yescomplemented
371C24C22×AGL(1,𝔽5)11yescharacteristic
372C24C22×AGL(1,𝔽5)12yes—
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—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—54no—
395C22×C4—52no—
396C22×C4—52no—
397C22×C4—52no—
398C22×C4—52no—
399C22×C4—54no—
400C22×C4—54no—
401C22×C4—54no—
402C22×C4—54no—
403C22×C4—52no—
404C22×C4—102no—
405C22×C4—101no—
406C22×C4—101no—
407C22×C4—108no—
408C22×C4—101no—
409C22×C4—101no—
410C22×C4—108no—
411C22×C4—208no—
412C22×C4—204no—
413C22×C4—204no—
414C22×C4—204no—
415C22×C4—202no—
416C22×C4—202no—
417C22×C4—208no—
418C22×C4—208no—
419C22×C4—208no—
420C22×C4C22⋊AGL(1,𝔽5)12yes—
421C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
422C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
423C22×C4C22×AGL(1,𝔽5)12yes—
424C22×C4C22×AGL(1,𝔽5)12yes—
425C4⋊C4—108no—
426C4⋊C4—2016no—
427C22⋊C4—104no—
428C22⋊C4—104no—
429C22⋊C4—108no—
430C22⋊C4—108no—
431C22⋊C4—108no—
432C22⋊C4—208no—
433C42—104no—
434C42—104no—
435C2×C5—21no—
436C2×C5—21no—
437C2×C5—22no—
438C2×C5—44no—
439C2×C5#179914yescomplemented
440C2×C5C2×#38011yescharacteristic
441C2×C5C22×#9911yescharacteristic
442C2×C5C2×#41011yescharacteristic
443D5—22no—
444D5—22no—
445D5—24no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—42no—
453C23—48no—
454C23—42no—
455C23—42no—
456C23—48no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—52no—
461C23—52no—
462C23—54no—
463C23—54no—
464C23—51no—
465C23—52no—
466C23—51no—
467C23—52no—
468C23—52no—
469C23—54no—
470C23—54no—
471C23—104no—
472C23—104no—
473C23—202no—
474C23—201no—
475C23—201no—
476C23—204no—
477C23—204no—
478C23—208no—
479C23—202no—
480C23—201no—
481C23—201no—
482C23C2×C4⋊AGL(1,𝔽5)12yes—
483C23C2×C4⋊AGL(1,𝔽5)14yescomplemented
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—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—54no—
501C2×C4—54no—
502C2×C4—104no—
503C2×C4—102no—
504C2×C4—102no—
505C2×C4—104no—
506C2×C4—104no—
507C2×C4—108no—
508C2×C4—108no—
509C2×C4—102no—
510C2×C4—204no—
511C2×C4—204no—
512C2×C4—202no—
513C2×C4—202no—
514C2×C4—202no—
515C2×C4—202no—
516C2×C4—202no—
517C2×C4—208no—
518C2×C4—208no—
519C2×C4—208no—
520C2×C4—4016no—
521C2×C4#401114yescomplemented
522C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
523C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
524C5C2×#179911yescharacteristic, complemented, Sylow
525C22—24no—
526C22—24no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—24no—
532C22—44no—
533C22—44no—
534C22—48no—
535C22—48no—
536C22—42no—
537C22—42no—
538C22—52no—
539C22—52no—
540C22—54no—
541C22—108no—
542C22—108no—
543C22—102no—
544C22—102no—
545C22—102no—
546C22—102no—
547C22—201no—
548C22—202no—
549C22—201no—
550C22—202no—
551C22—202no—
552C22—204no—
553C22—204no—
554C22#6574512yes—
555C22#6576512yes—
556C22#6576512yes—
557C22C2×#401112yes—
558C22C22×C4⋊AGL(1,𝔽5)11yescharacteristic
559C22C2×#413011yescharacteristic
560C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
561C4—22no—
562C4—42no—
563C4—104no—
564C4—104no—
565C4—202no—
566C4—408no—
567C4C2×#401114yescomplemented
568C2—21no—
569C2—21no—
570C2—22no—
571C2—44no—
572C2—102no—
573C2—102no—
574C2—104no—
575C2#1060489214yesdirect product factor
576C2C2×#6574511yescharacteristic
577C2C2×#6576511yescharacteristic
578C2C22×#401111yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C23×C4
FrattiniC23C23×AGL(1,𝔽5)
FittingC24×C4×C5C4
socleC23×C5C2×C22⋊C4
prime radicalC23×C22⋊D5C22
cosocleC23×D5C24
hypercenterC24×C4AGL(1,𝔽5)
hypocenterC5C2×#1799
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
3. C2×#65765C2semidirect
4. #10604892C2direct (N × H)
5. C2×#64917C2semidirect
6. C24×C4×C5C4semidirect
11. C22×C22⋊Dic5C4semidirect
15. C2×C4×C22⋊D5C4semidirect
18. C22×#683C4semidirect
19. C22×C4×Dic5C4semidirect
23. #65765C22semidirect
25. #64917C22semidirect
36. C23×C4×C5C2×C4semidirect
61. C2×C22⋊Dic5C2×C4semidirect
72. C2×#683C2×C4semidirect
76. C2×C4×Dic5C2×C4semidirect
89. C24×C5C4⋊C4semidirect
128. C2×C22⋊D5C4⋊C4semidirect
140. C22×Dic5C4⋊C4semidirect
165. C24×C4AGL(1,𝔽5)semidirect
200. C23×C5C2×C4⋊C4semidirect
238. C2×C4×C5#99semidirect
256. C2×Dic5C2×C4⋊C4semidirect
286. C23×C4C2×AGL(1,𝔽5)semidirect
352. C4×C5C2×#99semidirect
370. C24C4⋊AGL(1,𝔽5)semidirect
439. C2×C5#1799semidirect
483. C23C2×C4⋊AGL(1,𝔽5)semidirect
521. C2×C4#4011semidirect
524. C5C2×#1799semidirect
567. C4C2×#4011semidirect
575. C2#10604892direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D52
C22×#40112
C2×#657652
#106048922
C2×#649172
C2×#17995