Finite Group Databasebeta

← C5×A4×C22⋊Q8 #14511323

Subgroups — C5×A4×C22⋊Q8 #14511323

Order 1920.

860 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×Q8×A4C211yesmaximal, characteristic, complemented
2C22×C4×C5×A4C211yesmaximal, characteristic
3C5×A4×C4⋊C4C212yesmaximal, complemented
4C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
5C5×A4×C22⋊C4C212yesmaximal
6C22×C5×C22⋊Q8C311yesmaximal, characteristic, complemented, Hall
7C23×C5×A4C2211yescharacteristic
8C5×Q8×A4—22no—
9C2×C4×C5×A4—22no—
10C2×C4×C5×A4C2211yescharacteristic
11C2×C4×C5×A4C2212yes—
12C2×C4×C5×A4C2212yes—
13C2×C4×C5×A4C2211yescharacteristic
14C3×C5×C22⋊Q8—41nomaximal
15A4×C22⋊Q8C511yesmaximal, characteristic, direct product factor, Hall
16C23×C5×Q8C2×C311yescharacteristic, complemented
17C24×C4×C5C2×C311yescharacteristic
18C2×C5×C22⋊Q8—31no—
19C2×C5×C22⋊Q8—32no—
20C2×C5×C22⋊Q8—31no—
21C2×C5×C22⋊Q8—32no—
22C2×C5×C22⋊Q8—31no—
23C2×C5×C22⋊Q8—31no—
24C22×C5×C4⋊C4C2×C311yescharacteristic, complemented
25C22×C5×C4⋊C4C2×C312yescomplemented
26C22×C5×C22⋊C4C2×C312yes—
27C22×C5×A4—21no—
28C22×C5×A4—21no—
29C22×C5×A4Q812yescomplemented
30C22×C5×A4C2311yescharacteristic
31C2×C3×C5×Q8—41no—
32C22×C3×C4×C5—41no—
33C4×C5×A4—21no—
34C4×C5×A4—22no—
35C4×C5×A4—22no—
36C4×C5×A4D412yes—
37C3×C5×C4⋊C4—41no—
38C3×C5×C4⋊C4—42no—
39C3×C5×C22⋊C4—42no—
40C2×Q8×A4C2×C511yescharacteristic, complemented
41C22×C4×A4C2×C511yescharacteristic
42A4×C4⋊C4C2×C512yescomplemented
43A4×C4⋊C4C2×C511yescharacteristic, complemented
44A4×C22⋊C4C2×C512yes—
45C25×C5C22×C311yescharacteristic
46C22×C5×Q8—22no—
47C22×C5×Q8—31no—
48C22×C5×Q8—32no—
49C22×C5×Q8—31no—
50C22×C5×Q8—62no—
51C23×C4×C5—22no—
52C23×C4×C5—31no—
53C23×C4×C5—31no—
54C23×C4×C5—31no—
55C23×C4×C5—31no—
56C23×C4×C5—62no—
57C23×C4×C5C22×C312yes—
58C23×C4×C5C22×C312yes—
59C23×C4×C5C22×C311yescharacteristic
60C23×C4×C5C22×C311yescharacteristic
61C5×C22⋊Q8—32no—
62C5×C22⋊Q8—32no—
63C5×C22⋊Q8—34no—
64C5×C22⋊Q8—34no—
65C5×C22⋊Q8—32no—
66C5×C22⋊Q8—31no—
67C5×C22⋊Q8—31no—
68C5×C22⋊Q8—32no—
69C5×C22⋊Q8—31no—
70C5×C22⋊Q8—32no—
71C5×C22⋊Q8A411yescharacteristic, direct product factor
72C2×C5×C4⋊C4—31no—
73C2×C5×C4⋊C4—32no—
74C2×C5×C4⋊C4—31no—
75C2×C5×C4⋊C4—32no—
76C2×C5×C4⋊C4—32no—
77C2×C5×C4⋊C4—32no—
78C2×C5×C4⋊C4—32no—
79C2×C5×C22⋊C4—32no—
80C2×C5×C22⋊C4—32no—
81C2×C5×C22⋊C4—32no—
82C2×C5×C22⋊C4—32no—
83C22×C22⋊Q8C3×C511yescharacteristic, complemented, Sylow
84C23×C3×C5—41no—
85C2×C5×A4—22no—
86C2×C5×A4C2×D411yescharacteristic
87C2×C5×A4C2×Q811yescharacteristic
88C2×C5×A4#4111yescharacteristic
89C3×C5×Q8—82no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5—42no—
92C2×C3×C4×C5—42no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—82no—
95C23×A4C22×C511yescharacteristic
96Q8×A4—22no—
97C2×C4×A4—22no—
98C2×C4×A4C22×C511yescharacteristic
99C2×C4×A4C22×C512yes—
100C2×C4×A4C22×C512yes—
101C2×C4×A4C22×C511yescharacteristic
102C3×C22⋊Q8—41no—
103C24×C5—21no—
104C24×C5—21no—
105C24×C5—31no—
106C24×C5—32no—
107C24×C5—31no—
108C24×C5—61no—
109C24×C5—61no—
110C24×C5C3×Q812yescomplemented
111C24×C5C23×C311yescharacteristic
112C2×C5×Q8—32no—
113C2×C5×Q8—32no—
114C2×C5×Q8—31no—
115C2×C5×Q8—62no—
116C2×C5×Q8—62no—
117C2×C5×Q8—62no—
118C2×C5×Q8—64no—
119C2×C5×Q8—62no—
120C2×C5×Q8—64no—
121C2×C5×Q8C2×A411yescharacteristic, complemented
122C22×C4×C5—21no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—31no—
126C22×C4×C5—31no—
127C22×C4×C5—32no—
128C22×C4×C5—32no—
129C22×C4×C5—31no—
130C22×C4×C5—32no—
131C22×C4×C5—32no—
132C22×C4×C5—31no—
133C22×C4×C5—31no—
134C22×C4×C5—31no—
135C22×C4×C5—32no—
136C22×C4×C5—32no—
137C22×C4×C5—31no—
138C22×C4×C5—62no—
139C22×C4×C5—62no—
140C22×C4×C5—62no—
141C22×C4×C5—62no—
142C22×C4×C5—62no—
143C22×C4×C5—62no—
144C22×C4×C5—62no—
145C22×C4×C5—62no—
146C22×C4×C5—61no—
147C22×C4×C5—62no—
148C22×C4×C5—62no—
149C22×C4×C5C3×D412yes—
150C22×C4×C5C2×A411yescharacteristic
151C5×C4⋊C4—32no—
152C5×C4⋊C4—34no—
153C5×C4⋊C4—32no—
154C5×C4⋊C4—31no—
155C5×C4⋊C4—32no—
156C5×C4⋊C4—32no—
157C5×C4⋊C4—32no—
158C5×C4⋊C4C2×A411yescharacteristic, complemented
159C5×C4⋊C4C2×A412yescomplemented
160C5×C22⋊C4—34no—
161C5×C22⋊C4—32no—
162C5×C22⋊C4—32no—
163C5×C22⋊C4—32no—
164C5×C22⋊C4C2×A412yes—
165C23×Q8C2×C3×C511yescharacteristic, complemented
166C24×C4C2×C3×C511yescharacteristic
167C2×C22⋊Q8—31no—
168C2×C22⋊Q8—32no—
169C2×C22⋊Q8—31no—
170C2×C22⋊Q8—32no—
171C2×C22⋊Q8—31no—
172C2×C22⋊Q8—31no—
173C22×C4⋊C4C2×C3×C511yescharacteristic, complemented
174C22×C4⋊C4C2×C3×C512yescomplemented
175C22×C22⋊C4C2×C3×C512yes—
176C22×C3×C5—42no—
177C22×C3×C5—41no—
178C22×C3×C5—81no—
179C22×C3×C5—81no—
180C5×A4C22⋊Q811yescharacteristic, direct product factor
181C3×C4×C5—42no—
182C3×C4×C5—82no—
183C3×C4×C5—81no—
184C3×C4×C5—82no—
185C22×A4—21no—
186C22×A4—21no—
187C22×A4C5×Q812yescomplemented
188C22×A4C23×C511yescharacteristic
189C2×C3×Q8—41no—
190C22×C3×C4—41no—
191C4×A4—21no—
192C4×A4—22no—
193C4×A4—22no—
194C4×A4C5×D412yes—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—42no—
197C3×C22⋊C4—42no—
198C23×C5—22no—
199C23×C5—31no—
200C23×C5—31no—
201C23×C5—31no—
202C23×C5—31no—
203C23×C5—32no—
204C23×C5—32no—
205C23×C5—32no—
206C23×C5—32no—
207C23×C5—31no—
208C23×C5—61no—
209C23×C5—61no—
210C23×C5—61no—
211C23×C5—61no—
212C23×C5—62no—
213C23×C5—61no—
214C23×C5—61no—
215C23×C5—62no—
216C23×C5—62no—
217C23×C5—61no—
218C23×C5—61no—
219C23×C5—64no—
220C23×C5C2×C3×D411yescharacteristic
221C23×C5C2×C3×Q811yescharacteristic
222C23×C5C3×#4111yescharacteristic
223C23×C5C22×A411yescharacteristic
224C5×Q8—22no—
225C5×Q8—64no—
226C5×Q8—64no—
227C5×Q8—62no—
228C2×C4×C5—22no—
229C2×C4×C5—32no—
230C2×C4×C5—31no—
231C2×C4×C5—32no—
232C2×C4×C5—32no—
233C2×C4×C5—32no—
234C2×C4×C5—32no—
235C2×C4×C5—32no—
236C2×C4×C5—31no—
237C2×C4×C5—62no—
238C2×C4×C5—62no—
239C2×C4×C5—61no—
240C2×C4×C5—62no—
241C2×C4×C5—62no—
242C2×C4×C5—61no—
243C2×C4×C5—62no—
244C2×C4×C5—62no—
245C2×C4×C5—64no—
246C2×C4×C5—62no—
247C2×C4×C5—62no—
248C2×C4×C5—61no—
249C2×C4×C5—62no—
250C2×C4×C5—62no—
251C2×C4×C5—61no—
252C2×C4×C5—62no—
253C2×C4×C5C22×A412yes—
254C2×C4×C5C22×A412yes—
255C2×C4×C5C22×A411yescharacteristic
256C2×C4×C5C22×A411yescharacteristic
257C25C22×C3×C511yescharacteristic
258C22×Q8—22no—
259C22×Q8—31no—
260C22×Q8—32no—
261C22×Q8—31no—
262C22×Q8—62no—
263C23×C4—22no—
264C23×C4—31no—
265C23×C4—31no—
266C23×C4—31no—
267C23×C4—31no—
268C23×C4—62no—
269C23×C4C22×C3×C512yes—
270C23×C4C22×C3×C512yes—
271C23×C4C22×C3×C511yescharacteristic
272C23×C4C22×C3×C511yescharacteristic
273C22⋊Q8—32no—
274C22⋊Q8—32no—
275C22⋊Q8—34no—
276C22⋊Q8—34no—
277C22⋊Q8—32no—
278C22⋊Q8—31no—
279C22⋊Q8—31no—
280C22⋊Q8—32no—
281C22⋊Q8—31no—
282C22⋊Q8—32no—
283C22⋊Q8C5×A411yescharacteristic, direct product factor
284C2×C4⋊C4—31no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—32no—
291C2×C22⋊C4—32no—
292C2×C22⋊C4—32no—
293C2×C22⋊C4—32no—
294C2×C22⋊C4—32no—
295C2×C3×C5—41no—
296C2×C3×C5—41no—
297C2×C3×C5—41no—
298C2×C3×C5—82no—
299C23×C3—41no—
300C2×A4—22no—
301C2×A4C2×C5×D411yescharacteristic
302C2×A4C2×C5×Q811yescharacteristic
303C2×A4C5×#4111yescharacteristic
304C3×Q8—82no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—82no—
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—31no—
313C22×C5—31no—
314C22×C5—31no—
315C22×C5—32no—
316C22×C5—31no—
317C22×C5—31no—
318C22×C5—31no—
319C22×C5—31no—
320C22×C5—31no—
321C22×C5—31no—
322C22×C5—32no—
323C22×C5—62no—
324C22×C5—62no—
325C22×C5—62no—
326C22×C5—62no—
327C22×C5—62no—
328C22×C5—62no—
329C22×C5—62no—
330C22×C5—62no—
331C22×C5—61no—
332C22×C5—61no—
333C22×C5C3×C22⋊Q811yescharacteristic, complemented
334C22×C5Q8×A412yescomplemented
335C22×C5C23×A411yescharacteristic
336C4×C5—21no—
337C4×C5—22no—
338C4×C5—22no—
339C4×C5—32no—
340C4×C5—61no—
341C4×C5—62no—
342C4×C5—62no—
343C4×C5D4×A412yes—
344C24—21no—
345C24—21no—
346C24—31no—
347C24—32no—
348C24—31no—
349C24—61no—
350C24—61no—
351C24C3×C5×Q812yescomplemented
352C24C23×C3×C511yescharacteristic
353C2×Q8—32no—
354C2×Q8—32no—
355C2×Q8—31no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—64no—
360C2×Q8—62no—
361C2×Q8—64no—
362C2×Q8C2×C5×A411yescharacteristic, complemented
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—31no—
371C22×C4—32no—
372C22×C4—32no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—31no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4C3×C5×D412yes—
391C22×C4C2×C5×A411yescharacteristic
392C4⋊C4—32no—
393C4⋊C4—34no—
394C4⋊C4—32no—
395C4⋊C4—31no—
396C4⋊C4—32no—
397C4⋊C4—32no—
398C4⋊C4—32no—
399C4⋊C4C2×C5×A411yescharacteristic, complemented
400C4⋊C4C2×C5×A412yescomplemented
401C22⋊C4—34no—
402C22⋊C4—32no—
403C22⋊C4—32no—
404C22⋊C4—32no—
405C22⋊C4C2×C5×A412yes—
406C3×C5—41noHall
407C22×C3—42no—
408C22×C3—41no—
409C22×C3—81no—
410C22×C3—81no—
411A4C5×C22⋊Q811yescharacteristic, direct product factor
412C3×C4—42no—
413C3×C4—81no—
414C3×C4—82no—
415C3×C4—82no—
416C2×C5—22no—
417C2×C5—31no—
418C2×C5—31no—
419C2×C5—31no—
420C2×C5—31no—
421C2×C5—62no—
422C2×C5C2×D4×A411yescharacteristic
423C2×C5C2×Q8×A411yescharacteristic
424C2×C5A4×#4111yescharacteristic
425C23—22no—
426C23—31no—
427C23—31no—
428C23—31no—
429C23—31no—
430C23—32no—
431C23—32no—
432C23—32no—
433C23—32no—
434C23—31no—
435C23—61no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—62no—
440C23—61no—
441C23—61no—
442C23—62no—
443C23—62no—
444C23—61no—
445C23—61no—
446C23—64no—
447C23C2×C3×C5×D411yescharacteristic
448C23C2×C3×C5×Q811yescharacteristic
449C23C3×C5×#4111yescharacteristic
450C23C22×C5×A411yescharacteristic
451Q8—22no—
452Q8—64no—
453Q8—64no—
454Q8—62no—
455C2×C4—22no—
456C2×C4—32no—
457C2×C4—31no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—32no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—31no—
464C2×C4—62no—
465C2×C4—61no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—61no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—61no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—61no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4C22×C5×A412yes—
481C2×C4C22×C5×A412yes—
482C2×C4C22×C5×A411yescharacteristic
483C2×C4C22×C5×A411yescharacteristic
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—41no—
487C2×C3—82no—
488C5A4×C22⋊Q811yescharacteristic, direct product factor, Sylow
489C22—21no—
490C22—21no—
491C22—31no—
492C22—31no—
493C22—31no—
494C22—32no—
495C22—31no—
496C22—31no—
497C22—31no—
498C22—31no—
499C22—31no—
500C22—31no—
501C22—32no—
502C22—62no—
503C22—62no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—62no—
509C22—62no—
510C22—61no—
511C22—61no—
512C22C3×C5×C22⋊Q811yescharacteristic, complemented
513C22C5×Q8×A412yescomplemented
514C22C23×C5×A411yescharacteristic
515C4—21no—
516C4—22no—
517C4—22no—
518C4—32no—
519C4—61no—
520C4—62no—
521C4—62no—
522C4C5×D4×A412yes—
523C3—41noSylow
524C2—22no—
525C2—31no—
526C2—31no—
527C2—31no—
528C2—31no—
529C2—62no—
530C2C2×C5×D4×A411yescharacteristic
531C2C2×C5×Q8×A411yescharacteristic
532C2C5×A4×#4111yescharacteristic
subgroupG/H
centerC22×C5C23×A4
commutator, cosocleC24C23×C3×C5
FrattiniC22C23×C5×A4
FittingC22×C5×C22⋊Q8C3
socleC24×C5C23×C3
prime radicalC23×C5×A4C22
hypercenterC5×C22⋊Q8A4
hypocenterC22C3×C5×C22⋊Q8
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C5 ◁ C5×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×Q8×A4C2semidirect
3. C5×A4×C4⋊C4C2semidirect
4. C5×A4×C4⋊C4C2semidirect
6. C22×C5×C22⋊Q8C3semidirect
15. A4×C22⋊Q8C5direct (N × H)
16. C23×C5×Q8C2×C3semidirect
24. C22×C5×C4⋊C4C2×C3semidirect
25. C22×C5×C4⋊C4C2×C3semidirect
29. C22×C5×A4Q8semidirect
40. C2×Q8×A4C2×C5semidirect
42. A4×C4⋊C4C2×C5semidirect
43. A4×C4⋊C4C2×C5semidirect
71. C5×C22⋊Q8A4direct (N × H)
83. C22×C22⋊Q8C3×C5semidirect
110. C24×C5C3×Q8semidirect
121. C2×C5×Q8C2×A4semidirect
158. C5×C4⋊C4C2×A4semidirect
159. C5×C4⋊C4C2×A4semidirect
165. C23×Q8C2×C3×C5semidirect
173. C22×C4⋊C4C2×C3×C5semidirect
174. C22×C4⋊C4C2×C3×C5semidirect
180. C5×A4C22⋊Q8direct (N × H)
187. C22×A4C5×Q8semidirect
283. C22⋊Q8C5×A4direct (N × H)
333. C22×C5C3×C22⋊Q8semidirect
334. C22×C5Q8×A4semidirect
351. C24C3×C5×Q8semidirect
362. C2×Q8C2×C5×A4semidirect
399. C4⋊C4C2×C5×A4semidirect
400. C4⋊C4C2×C5×A4semidirect
411. A4C5×C22⋊Q8direct (N × H)
488. C5A4×C22⋊Q8direct (N × H)
512. C22C3×C5×C22⋊Q8semidirect
513. C22C5×Q8×A4semidirect

Maximal subgroups

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

subgroupindex
C2×C5×Q8×A42
C22×C4×C5×A42
C5×A4×C4⋊C42
C5×A4×C22⋊C42
C22×C5×C22⋊Q83
C3×C5×C22⋊Q84
A4×C22⋊Q85