Finite Group Databasebeta

← C2×C4×#4009 #13033413

Subgroups — C2×C4×#4009 #13033413

Order 1280.

1726 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D5C211yesmaximal, characteristic
2C22×#4009C212yesmaximal
3C2×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4C2×C4×D5⋊C8C211yesmaximal, characteristic
5C4×#4009C218yesmaximal, direct product factor
6C2×#64874C212yesmaximal
7C22×Q8×D5C2211yescharacteristic
8C22×C4⋊AGL(1,𝔽5)C2212yes—
9C22×C4×AGL(1,𝔽5)—21no—
10C22×D5⋊C8C2212yes—
11C2×C4×C5×Q8C411yescharacteristic, complemented
12C2×Q8×Dic5C411yescharacteristic, complemented
13C4×Q8×D5—22no—
14C4×Q8×D5—22no—
15C4×Q8×D5C414yescomplemented
16C4×Q8×D5C2214yes—
17C2×D5×C4⋊C4—21no—
18C2×D5×C4⋊C4C2211yescharacteristic
19C2×#4033—21no—
20C2×#4033C411yescharacteristic, complemented
21C2×C42×D5—21no—
22C2×C42×D5C2211yescharacteristic
23C2×C4×Dic10—21no—
24C2×C4×Dic10C411yescharacteristic, complemented
25C2×#4009C4116yesdirect product factor
26C2×#4009C2218yes—
27C2×(C2×AGL(1,𝔽5))⋊C4—21no—
28C2×C4×#170—21no—
29C2×C4×#170—21no—
30C4×C4⋊AGL(1,𝔽5)—22no—
31C4×C4⋊AGL(1,𝔽5)C2214yes—
32C4×D5⋊C8—22no—
33C4×D5⋊C8C2214yes—
34#64874C2218yes—
35C2×C4×#103—51nomaximal, Sylow
36C23×AGL(1,𝔽5)—22no—
37C22×C5×Q8C2×C411yescharacteristic
38C2×Q8×D5—21no—
39C2×Q8×D5—21no—
40C2×Q8×D5—24no—
41C2×Q8×D5C2×C418yes—
42C2×Q8×D5C2×C412yes—
43C2×Q8×D5C2312yes—
44C22×C4×D5—21no—
45C22×C4×D5—21no—
46C22×C4×D5D411yescharacteristic
47C22×C4×D5D411yescharacteristic
48C22×C4×D5C2311yescharacteristic
49C22×Dic10—21no—
50C22×Dic10C2×C411yescharacteristic
51C22×#170—22no—
52C2×C4⋊AGL(1,𝔽5)—22no—
53C2×C4⋊AGL(1,𝔽5)—24no—
54C2×C4⋊AGL(1,𝔽5)C2×C418yes—
55C2×C4⋊AGL(1,𝔽5)C2314yes—
56C2×C4×AGL(1,𝔽5)—24no—
57C2×C4×AGL(1,𝔽5)—44no—
58C2×D5⋊C8—22no—
59C2×D5⋊C8—24no—
60C2×D5⋊C8C2×C418yes—
61C2×D5⋊C8C2314yes—
62C4×C5×Q8—22no—
63C4×C5×Q8C2×C414yescomplemented
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4C2×C411yescharacteristic
66C2×C42×C5—21no—
67C2×C42×C5C2×C411yescharacteristic
68Q8×Dic5—22no—
69Q8×Dic5C2×C414yescomplemented
70C2×C4⋊Dic5—21no—
71C2×C4⋊Dic5C2×C411yescharacteristic
72C2×#683—21no—
73C2×#683—21no—
74C2×#683—21no—
75C2×C4×Dic5—21no—
76C2×C4×Dic5—21no—
77C2×C4×Dic5—21no—
78C2×C4×Dic5C2×C411yescharacteristic
79C2×C4×Dic5D411yescharacteristic
80C2×C4×Dic5D411yescharacteristic
81D5×C4⋊C4—24no—
82D5×C4⋊C4—24no—
83D5×C4⋊C4—21no—
84D5×C4⋊C4—21no—
85D5×C4⋊C4C2×C412yes—
86D5×C4⋊C4C2312yes—
87#4033—24no—
88#4033—22no—
89#4033—24no—
90#4033C2×C414yescomplemented
91C42×D5—24no—
92C42×D5—24no—
93C42×D5C2×C412yes—
94C42×D5D412yes—
95C42×D5D412yes—
96C42×D5C2312yes—
97C4×Dic10—22no—
98C4×Dic10—24no—
99C4×Dic10—24no—
100C4×Dic10C2×C414yescomplemented
101#4009C2×C4132yesdirect product factor
102(C2×AGL(1,𝔽5))⋊C4—24no—
103C4×#170—24no—
104C4×#170—24no—
105C22×C4×Q8—51no—
106C22×#103—52no—
107C22×C4×C8—51no—
108C2×C4×C4⋊C4—51no—
109C4×#103—58no—
110C2×#340—52no—
111C23×D5C2×D411yescharacteristic
112C22×AGL(1,𝔽5)—28no—
113C22×AGL(1,𝔽5)—24no—
114C22×AGL(1,𝔽5)—44no—
115C22×AGL(1,𝔽5)—44no—
116C2×C5×Q8—21no—
117C2×C5×Q8—22no—
118C2×C5×Q8C4214yescomplemented
119C2×C5×Q8C22×C412yes—
120C22×C4×C5—21no—
121C22×C4×C5—21no—
122C22×C4×C5C22⋊C411yescharacteristic
123C22×C4×C5C22⋊C411yescharacteristic
124C22×C4×C5C22×C411yescharacteristic
125C22×Dic5—21no—
126C22×Dic5—21no—
127C22×Dic5C22⋊C411yescharacteristic
128C22×Dic5C22⋊C411yescharacteristic
129C22×Dic5C2×D411yescharacteristic
130Q8×D5—24no—
131Q8×D5—28no—
132Q8×D5—28no—
133Q8×D5—24no—
134Q8×D5C4218yescomplemented
135Q8×D5C22×C418yes—
136C2×C4×D5—22no—
137C2×C4×D5—28no—
138C2×C4×D5—24no—
139C2×C4×D5—22no—
140C2×C4×D5—22no—
141C2×C4×D5—22no—
142C2×C4×D5—44no—
143C2×C4×D5C22⋊C412yes—
144C2×C4×D5C22⋊C412yes—
145C2×C4×D5SD1614yes—
146C2×C4×D5Q1614yes—
147C2×C4×D5C22×C411yescharacteristic
148C2×C4×D5C22×C414yes—
149C2×C4×D5C2×D412yes—
150C2×C4×D5C2×D411yescharacteristic
151C2×C4×D5C2×D411yescharacteristic
152C2×C4×D5C2×D412yes—
153C2×C4×D5#4114yes—
154C2×C4×D5C2411yescharacteristic
155C2×Dic10—21no—
156C2×Dic10—22no—
157C2×Dic10—22no—
158C2×Dic10—28no—
159C2×Dic10—22no—
160C2×Dic10C4214yescomplemented
161C2×Dic10C22×C412yes—
162C2×#170—28no—
163C2×#170—24no—
164C4⋊AGL(1,𝔽5)—28no—
165C4⋊AGL(1,𝔽5)—24no—
166C4⋊AGL(1,𝔽5)C22×C418yes—
167C4×AGL(1,𝔽5)—48no—
168D5⋊C8—28no—
169D5⋊C8—24no—
170D5⋊C8C22×C418yes—
171C5×C4⋊C4—24no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4C22×C412yes—
174C42×C5—24no—
175C42×C5C22⋊C412yes—
176C42×C5C22×C412yes—
177C4⋊Dic5—21no—
178C4⋊Dic5—24no—
179C4⋊Dic5C22×C412yes—
180#683—24no—
181#683—22no—
182#683—24no—
183#683—22no—
184C4×Dic5—24no—
185C4×Dic5—24no—
186C4×Dic5—24no—
187C4×Dic5C22⋊C412yes—
188C4×Dic5C22⋊C412yes—
189C4×Dic5C22⋊C412yes—
190C4×Dic5C22×C412yes—
191C4×Dic5C2×D412yes—
192C4×Dic5C2×D412yes—
193C23×Q8—51no—
194C23×C8—52no—
195C2×C4×Q8—51no—
196C2×C4×Q8—54no—
197C2×C4×Q8—54no—
198C2×C4×Q8—51no—
199C2×C4×Q8—51no—
200C2×C4×Q8—101no—
201C2×C4×Q8—101no—
202C2×C4×Q8—102no—
203C2×C4×Q8—102no—
204C2×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
205C22×C4⋊C4—52no—
206C22×C4⋊C4—51no—
207C22×C4⋊C4—101no—
208C22×C42—51no—
209C22×C42—101no—
210C22×C42—101no—
211C2×#103—58no—
212C2×#103—516no—
213C2×C4×C8—54no—
214C2×C4×C8—102no—
215C2×C4×C8—101no—
216C2×C4×C8—101no—
217C4×C4⋊C4—54no—
218C4×C4⋊C4—102no—
219C2×#95—101no—
220#340—58no—
221C23×C5C2×C22⋊C411yescharacteristic
222C22×D5C2×C22⋊C411yescharacteristic
223C22×D5C4×D414yes—
224C22×D5C2×SD1612yes—
225C22×D5C2×Q1612yes—
226C22×D5#13514yes—
227C22×D5C22×D411yescharacteristic
228C2×AGL(1,𝔽5)—28no—
229C2×AGL(1,𝔽5)—44no—
230C2×AGL(1,𝔽5)—44no—
231C2×AGL(1,𝔽5)—416no—
232C5×Q8—22no—
233C5×Q8—24no—
234C5×Q8C2×C4214yescomplemented
235C2×C4×C5—22no—
236C2×C4×C5—24no—
237C2×C4×C5—22no—
238C2×C4×C5—22no—
239C2×C4×C5—42no—
240C2×C4×C5#10314yescomplemented
241C2×C4×C5C2×C4212yes—
242C2×C4×C5C2×C22⋊C412yes—
243C2×C4×C5C2×C22⋊C411yescharacteristic
244C2×C4×C5C2×C22⋊C412yes—
245C2×C4×C5#11712yes—
246C2×C4×C5C23×C411yescharacteristic
247C2×Dic5—22no—
248C2×Dic5—24no—
249C2×Dic5—22no—
250C2×Dic5—22no—
251C2×Dic5—42no—
252C2×Dic5#10314yescomplemented
253C2×Dic5C2×C22⋊C412yes—
254C2×Dic5C2×C22⋊C411yescharacteristic
255C2×Dic5C2×C22⋊C412yes—
256C2×Dic5C4×D414yes—
257C2×Dic5C22×D411yescharacteristic
258C4×D5—28no—
259C4×D5—24no—
260C4×D5—28no—
261C4×D5—24no—
262C4×D5—44no—
263C4×D5—44no—
264C4×D5#10318yescomplemented
265C4×D5C2×C4212yes—
266C4×D5#11712yes—
267C4×D5C4×D414yes—
268C4×D5C4×D414yes—
269C4×D5C2×SD1614yes—
270C4×D5C2×Q1614yes—
271C4×D5C23×C412yes—
272C4×D5C2×#4112yes—
273Dic10—22no—
274Dic10—24no—
275Dic10—28no—
276Dic10—28no—
277Dic10C2×C4214yescomplemented
278#170—28no—
279C22×Q8—58no—
280C22×Q8—52no—
281C22×Q8—52no—
282C22×Q8—51no—
283C22×Q8—101no—
284C22×Q8—101no—
285C22×Q8—101no—
286C22×Q8—104no—
287C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
288C23×C4—51no—
289C23×C4—51no—
290C23×C4—51no—
291C23×C4—101no—
292C23×C4—101no—
293C23×C4—102no—
294C22×C8—58no—
295C22×C8—54no—
296C22×C8—104no—
297C22×C8—102no—
298C22×C8—102no—
299C4×Q8—22no—
300C4×Q8—54no—
301C4×Q8—54no—
302C4×Q8—54no—
303C4×Q8—104no—
304C4×Q8—104no—
305C4×Q8—104no—
306C4×Q8—104no—
307C4×Q8—102no—
308C4×Q8—102no—
309C4×Q8—102no—
310C4×Q8C2×AGL(1,𝔽5)14yescomplemented
311C2×C4⋊C4—21no—
312C2×C4⋊C4—58no—
313C2×C4⋊C4—54no—
314C2×C4⋊C4—51no—
315C2×C4⋊C4—52no—
316C2×C4⋊C4—52no—
317C2×C4⋊C4—101no—
318C2×C4⋊C4—101no—
319C2×C4⋊C4—104no—
320C2×C4⋊C4—104no—
321C2×C4⋊C4—101no—
322C2×C4⋊C4—101no—
323C2×C4⋊C4—101no—
324C2×C4⋊C4—104no—
325C2×C4⋊C4—102no—
326C2×C4⋊C4—101no—
327C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
328C2×C42—21no—
329C2×C42—51no—
330C2×C42—51no—
331C2×C42—51no—
332C2×C42—52no—
333C2×C42—52no—
334C2×C42—52no—
335C2×C42—52no—
336C2×C42—104no—
337C2×C42—104no—
338C2×C42—101no—
339C2×C42—101no—
340C2×C42—101no—
341C2×C42—104no—
342C2×C42—204no—
343C2×C42C2×AGL(1,𝔽5)11yescharacteristic
344#103—532no—
345C4×C8—104no—
346C4×C8—104no—
347#95—104no—
348C22×C5C4×C22⋊C412yes—
349C22×C5C2×#10312yes—
350C22×C5#41612yes—
351C22×C5C22×C22⋊C411yescharacteristic
352C2×D5C4×C22⋊C412yes—
353C2×D5C2×#10312yes—
354C2×D5#41612yes—
355C2×D5C4×SD1618yes—
356C2×D5C4×Q1618yes—
357C2×D5C2×C4×D412yes—
358C2×D5C22×SD1611yescharacteristic
359C2×D5C22×Q1611yescharacteristic
360C2×D5C2×#13512yes—
361AGL(1,𝔽5)—48no—
362C4×C5—22no—
363C4×C5—24no—
364C4×C5—42no—
365C4×C5C4×C22⋊C412yes—
366C4×C5C2×#10314yescomplemented
367C4×C5C22×C4211yescharacteristic
368C4×C5C2×#11711yescharacteristic
369Dic5—22no—
370Dic5—24no—
371Dic5—42no—
372Dic5C4×C22⋊C412yes—
373Dic5C2×#10314yescomplemented
374Dic5C2×C4×D412yes—
375C24—51no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—58no—
379C2×Q8—58no—
380C2×Q8—54no—
381C2×Q8—52no—
382C2×Q8—102no—
383C2×Q8—102no—
384C2×Q8—108no—
385C2×Q8—104no—
386C2×Q8—104no—
387C2×Q8—108no—
388C2×Q8—108no—
389C2×Q8—102no—
390C2×Q8—101no—
391C2×Q8C4×AGL(1,𝔽5)14yescomplemented
392C2×Q8C22×AGL(1,𝔽5)12yes—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—51no—
396C22×C4—51no—
397C22×C4—54no—
398C22×C4—54no—
399C22×C4—52no—
400C22×C4—52no—
401C22×C4—51no—
402C22×C4—51no—
403C22×C4—51no—
404C22×C4—51no—
405C22×C4—54no—
406C22×C4—54no—
407C22×C4—51no—
408C22×C4—52no—
409C22×C4—52no—
410C22×C4—102no—
411C22×C4—102no—
412C22×C4—104no—
413C22×C4—108no—
414C22×C4—102no—
415C22×C4—102no—
416C22×C4—101no—
417C22×C4—104no—
418C22×C4—108no—
419C22×C4—101no—
420C22×C4—204no—
421C22×C4—204no—
422C22×C4—204no—
423C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
424C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
425C22×C4C22×AGL(1,𝔽5)11yescharacteristic
426C2×C8—58no—
427C2×C8—104no—
428C2×C8—104no—
429C2×C8—108no—
430C2×C8—108no—
431C4⋊C4—24no—
432C4⋊C4—21no—
433C4⋊C4—58no—
434C4⋊C4—52no—
435C4⋊C4—102no—
436C4⋊C4—102no—
437C4⋊C4—104no—
438C4⋊C4—104no—
439C4⋊C4—104no—
440C4⋊C4—108no—
441C4⋊C4—101no—
442C4⋊C4—104no—
443C4⋊C4C22×AGL(1,𝔽5)12yes—
444C42—24no—
445C42—52no—
446C42—52no—
447C42—52no—
448C42—52no—
449C42—52no—
450C42—52no—
451C42—104no—
452C42—104no—
453C42—104no—
454C42—208no—
455C42C22⋊AGL(1,𝔽5)12yes—
456C42C22×AGL(1,𝔽5)12yes—
457C2×C5C4×#10314yescomplemented
458C2×C5C2×C4×C22⋊C411yescharacteristic
459C2×C5C22×#10311yescharacteristic
460C2×C5C2×#41611yescharacteristic
461D5C4×#10314yescomplemented
462D5C2×C4×SD1612yes—
463D5C2×C4×Q1612yes—
464C23—54no—
465C23—51no—
466C23—51no—
467C23—54no—
468C23—52no—
469C23—52no—
470C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
471Q8—22no—
472Q8—24no—
473Q8—54no—
474Q8—108no—
475Q8—108no—
476Q8—102no—
477Q8—104no—
478Q8C2×C4×AGL(1,𝔽5)14yescomplemented
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—52no—
485C2×C4—51no—
486C2×C4—51no—
487C2×C4—54no—
488C2×C4—54no—
489C2×C4—52no—
490C2×C4—52no—
491C2×C4—52no—
492C2×C4—52no—
493C2×C4—54no—
494C2×C4—54no—
495C2×C4—54no—
496C2×C4—58no—
497C2×C4—54no—
498C2×C4—52no—
499C2×C4—104no—
500C2×C4—104no—
501C2×C4—108no—
502C2×C4—108no—
503C2×C4—102no—
504C2×C4—102no—
505C2×C4—104no—
506C2×C4—108no—
507C2×C4—102no—
508C2×C4—204no—
509C2×C4—204no—
510C2×C4—204no—
511C2×C4—2016no—
512C2×C4—204no—
513C2×C4—202no—
514C2×C4#400914yesdirect product factor
515C2×C4C2×C4×AGL(1,𝔽5)12yes—
516C2×C4#413012yes—
517C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
518C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
519C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
520C2×C4C23×AGL(1,𝔽5)11yescharacteristic
521C8—108no—
522C5C2×C4×#10311yescharacteristic, complemented, Sylow
523C22—52no—
524C22—52no—
525C22—51no—
526C22—51no—
527C22—52no—
528C22—58no—
529C22—58no—
530C22—52no—
531C22—52no—
532C22C4×C22⋊AGL(1,𝔽5)12yes—
533C22C2×#400912yes—
534C22#6579412yes—
535C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
536C4—24no—
537C4—22no—
538C4—42no—
539C4—52no—
540C4—52no—
541C4—54no—
542C4—104no—
543C4—102no—
544C4—208no—
545C4—202no—
546C4C4×C22⋊AGL(1,𝔽5)12yes—
547C4C2×#400914yesdirect product factor
548C4C22×C4×AGL(1,𝔽5)11yescharacteristic
549C4C2×#413011yescharacteristic
550C2—52no—
551C2—52no—
552C2—54no—
553C2C4×#400914yesdirect product factor
554C2C2×C4×C22⋊AGL(1,𝔽5)11yescharacteristic
555C2C22×#400911yescharacteristic
556C2C2×#6579411yescharacteristic
subgroupG/H
centerC22×C4C22⋊AGL(1,𝔽5)
commutatorC4×C5C22×C42
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC2×C4×C5×Q8C4
socleC23×C5C2×C22⋊C4
prime radicalC23×D5C2×D4
cosocleC2×C4×D5C24
hypercenterC2×C4×Q8AGL(1,𝔽5)
hypocenterC5C2×C4×#103
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
5. C4×#4009C2direct (N × H)
11. C2×C4×C5×Q8C4semidirect
12. C2×Q8×Dic5C4semidirect
15. C4×Q8×D5C4semidirect
20. C2×#4033C4semidirect
24. C2×C4×Dic10C4semidirect
25. C2×#4009C4direct (N × H)
63. C4×C5×Q8C2×C4semidirect
69. Q8×Dic5C2×C4semidirect
90. #4033C2×C4semidirect
100. C4×Dic10C2×C4semidirect
101. #4009C2×C4direct (N × H)
118. C2×C5×Q8C42semidirect
134. Q8×D5C42semidirect
160. C2×Dic10C42semidirect
204. C2×C4×Q8AGL(1,𝔽5)semidirect
234. C5×Q8C2×C42semidirect
240. C2×C4×C5#103semidirect
252. C2×Dic5#103semidirect
264. C4×D5#103semidirect
277. Dic10C2×C42semidirect
310. C4×Q8C2×AGL(1,𝔽5)semidirect
366. C4×C5C2×#103semidirect
373. Dic5C2×#103semidirect
391. C2×Q8C4×AGL(1,𝔽5)semidirect
457. C2×C5C4×#103semidirect
461. D5C4×#103semidirect
478. Q8C2×C4×AGL(1,𝔽5)semidirect
514. C2×C4#4009direct (N × H)
522. C5C2×C4×#103semidirect
547. C4C2×#4009direct (N × H)
553. C2C4×#4009direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D52
C22×#40092
C2×C4×C4⋊AGL(1,𝔽5)2
C2×C4×D5⋊C82
C4×#40092
C2×#648742
C2×C4×#1035