Finite Group Databasebeta

← C5×C22⋊C4×S4 #14511033

Subgroups — C5×C22⋊C4×S4 #14511033

Order 1920.

1314 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C5×S4C211yesmaximal, characteristic
2C2×C4×C5×S4C212yesmaximal, complemented
3C5×A4×C22⋊C4C211yesmaximal, characteristic, complemented
4C5×#5432C211yesmaximal, characteristic, complemented
5C5×(C2×S4)⋊C4C212yesmaximal, complemented
6C5×D4×C22⋊C4—31nomaximal, Hall
7C23×C5×A4C2211yescharacteristic
8C22×C5×S4—24no—
9C22×C5×S4C414yescomplemented
10C22×C5×S4C2211yescharacteristic
11C22×C5×S4C2211yescharacteristic
12C2×C4×C5×A4C2212yescomplemented
13C2×C5×SL(2,ℤ/4)C2212yescomplemented
14C4×C5×S4—24no—
15C5×D3×C22⋊C4—41nomaximal
16C22⋊C4×S4C511yesmaximal, characteristic, direct product factor, Hall
17C23×C5×D4—31no—
18C2×C4×C5×D4—32no—
19C22×C5×C22⋊C4—31no—
20C22×C5×C22⋊C4D311yescharacteristic, complemented
21C5×#390—32no—
22C5×#386—32no—
23C5×#386—32no—
24C5×#380—31no—
25C5×#379—31no—
26C5×#379—31no—
27C4×C5×C22⋊C4—31no—
28C23×C5×D3—41no—
29C22×C5×A4—22no—
30C22×C5×A4C2×C412yescomplemented
31C22×C5×A4C2311yescharacteristic
32C2×C5×S4—22no—
33C2×C5×S4—28no—
34C2×C5×S4C2×C412yes—
35C2×C5×S4C2×C412yescomplemented
36C2×C5×S4D414yes—
37C2×C4×C5×D3—42no—
38C4×C5×A4—22no—
39C5×SL(2,ℤ/4)—22no—
40C3×C5×C22⋊C4—41no—
41C5×#217—41no—
42C5×(C2×D3)⋊C4—42no—
43C23×S4C2×C511yescharacteristic
44C2×C4×S4C2×C512yescomplemented
45A4×C22⋊C4C2×C511yescharacteristic, complemented
46#5432C2×C511yescharacteristic, complemented
47(C2×S4)⋊C4C2×C512yescomplemented
48C25×C5—31no—
49C25×C5C2×D311yescharacteristic
50C22×C5×D4—31no—
51C22×C5×D4—31no—
52C22×C5×D4—31no—
53C22×C5×D4—34no—
54C22×C5×D4—34no—
55C22×C5×D4—31no—
56C22×C5×D4—64no—
57C22×C5×D4—64no—
58C23×C4×C5—31no—
59C23×C4×C5—32no—
60C23×C4×C5C2×D312yescomplemented
61C4×C5×D4—64no—
62C4×C5×D4—64no—
63C2×C5×C4⋊C4—32no—
64C2×C5×C22⋊C4—31no—
65C2×C5×C22⋊C4—31no—
66C2×C5×C22⋊C4—32no—
67C2×C5×C22⋊C4—31no—
68C2×C5×C22⋊C4—31no—
69C2×C5×C22⋊C4—32no—
70C2×C5×C22⋊C4—32no—
71C2×C5×C22⋊C4—32no—
72C2×C5×C22⋊C4—32no—
73C2×C5×C22⋊C4—32no—
74C2×C5×C22⋊C4—61no—
75C2×C5×C22⋊C4—62no—
76C2×C5×C22⋊C4—61no—
77C2×C42×C5—32no—
78C5×#95—32no—
79C5×#95—32no—
80D4×C22⋊C4—31noSylow
81C23×C3×C5—41no—
82C22×C5×D3—41no—
83C22×C5×D3—41no—
84C22×C5×D3—44no—
85C22×C5×D3—84no—
86C2×C5×A4—22no—
87C2×C5×A4C22×C411yescharacteristic
88C2×C5×A4C2×D412yes—
89C5×S4—22no—
90C5×S4C22⋊C414yesdirect product factor
91C2×C3×C4×C5—42no—
92C2×C5×Dic3—42no—
93C4×C5×D3—84no—
94C23×A4C22×C511yescharacteristic
95C22×S4—24no—
96C22×S4C4×C514yescomplemented
97C22×S4C22×C511yescharacteristic
98C22×S4C22×C511yescharacteristic
99C2×C4×A4C22×C512yescomplemented
100C2×SL(2,ℤ/4)C22×C512yescomplemented
101C4×S4—24no—
102D3×C22⋊C4—41no—
103C24×C5—22no—
104C24×C5—31no—
105C24×C5—31no—
106C24×C5—31no—
107C24×C5—32no—
108C24×C5—31no—
109C24×C5—34no—
110C24×C5—62no—
111C24×C5—61no—
112C24×C5—64no—
113C24×C5—62no—
114C24×C5—61no—
115C24×C5—122no—
116C24×C5C4×D312yescomplemented
117C24×C5C22×D311yescharacteristic
118C2×C5×D4—34no—
119C2×C5×D4—32no—
120C2×C5×D4—32no—
121C2×C5×D4—34no—
122C2×C5×D4—34no—
123C2×C5×D4—32no—
124C2×C5×D4—34no—
125C2×C5×D4—32no—
126C2×C5×D4—64no—
127C2×C5×D4—68no—
128C2×C5×D4—68no—
129C2×C5×D4—616no—
130C2×C5×D4—62no—
131C2×C5×D4—64no—
132C2×C5×D4—62no—
133C22×C4×C5—22no—
134C22×C4×C5—31no—
135C22×C4×C5—34no—
136C22×C4×C5—31no—
137C22×C4×C5—32no—
138C22×C4×C5—32no—
139C22×C4×C5—32no—
140C22×C4×C5—32no—
141C22×C4×C5—64no—
142C22×C4×C5—62no—
143C22×C4×C5—62no—
144C22×C4×C5—64no—
145C22×C4×C5—62no—
146C22×C4×C5—122no—
147C5×C4⋊C4—64no—
148C5×C22⋊C4—31no—
149C5×C22⋊C4—32no—
150C5×C22⋊C4—61no—
151C5×C22⋊C4—62no—
152C5×C22⋊C4—62no—
153C5×C22⋊C4—61no—
154C5×C22⋊C4—64no—
155C5×C22⋊C4—62no—
156C5×C22⋊C4—62no—
157C5×C22⋊C4—61no—
158C5×C22⋊C4—62no—
159C5×C22⋊C4—122no—
160C5×C22⋊C4—122no—
161C5×C22⋊C4S411yescharacteristic, direct product factor
162C42×C5—64no—
163C23×D4—31no—
164C2×C4×D4—32no—
165C22×C22⋊C4—31no—
166C22×C22⋊C4C5×D311yescharacteristic, complemented
167#390—32no—
168#386—32no—
169#386—32no—
170#380—31no—
171#379—31no—
172#379—31no—
173C4×C22⋊C4—31no—
174C22×C3×C5—42no—
175C22×C3×C5—41no—
176C22×C3×C5—82no—
177C2×C5×D3—42no—
178C2×C5×D3—44no—
179C2×C5×D3—42no—
180C2×C5×D3—88no—
181C2×C5×D3—82no—
182C5×A4C2×C22⋊C411yescharacteristic, complemented
183C3×C4×C5—82no—
184C5×Dic3—82no—
185C23×D3—41no—
186C22×A4—22no—
187C22×A4C2×C4×C512yescomplemented
188C22×A4C23×C511yescharacteristic
189C2×S4—22no—
190C2×S4—28no—
191C2×S4C2×C4×C512yes—
192C2×S4C2×C4×C512yescomplemented
193C2×S4C5×D414yes—
194C2×C4×D3—42no—
195C4×A4—22no—
196SL(2,ℤ/4)—22no—
197C3×C22⋊C4—41no—
198#217—41no—
199(C2×D3)⋊C4—42no—
200C23×C5—22no—
201C23×C5—31no—
202C23×C5—32no—
203C23×C5—31no—
204C23×C5—34no—
205C23×C5—32no—
206C23×C5—32no—
207C23×C5—32no—
208C23×C5—32no—
209C23×C5—31no—
210C23×C5—32no—
211C23×C5—62no—
212C23×C5—62no—
213C23×C5—61no—
214C23×C5—62no—
215C23×C5—62no—
216C23×C5—61no—
217C23×C5—62no—
218C23×C5—64no—
219C23×C5—62no—
220C23×C5—62no—
221C23×C5—62no—
222C23×C5—68no—
223C23×C5—61no—
224C23×C5—64no—
225C23×C5—61no—
226C23×C5—64no—
227C23×C5—61no—
228C23×C5—124no—
229C23×C5—122no—
230C23×C5—124no—
231C23×C5—128no—
232C23×C5—124no—
233C23×C5C2×C4×D311yescharacteristic
234C23×C5D3×D412yes—
235C23×C5C2×S411yescharacteristic
236C5×D4—34no—
237C5×D4—38no—
238C5×D4—34no—
239C5×D4—68no—
240C5×D4—62no—
241C5×D4—64no—
242C5×D4—68no—
243C5×D4—62no—
244C2×C4×C5—34no—
245C2×C4×C5—32no—
246C2×C4×C5—32no—
247C2×C4×C5—32no—
248C2×C4×C5—62no—
249C2×C4×C5—62no—
250C2×C4×C5—68no—
251C2×C4×C5—62no—
252C2×C4×C5—62no—
253C2×C4×C5—62no—
254C2×C4×C5—62no—
255C2×C4×C5—62no—
256C2×C4×C5—62no—
257C2×C4×C5—62no—
258C2×C4×C5—62no—
259C2×C4×C5—62no—
260C2×C4×C5—124no—
261C2×C4×C5—124no—
262C2×C4×C5C2×S412yescomplemented
263C25—31no—
264C25C2×C5×D311yescharacteristic
265C22×D4—31no—
266C22×D4—34no—
267C22×D4—31no—
268C22×D4—31no—
269C22×D4—34no—
270C22×D4—31no—
271C22×D4—64no—
272C22×D4—64no—
273C23×C4—31no—
274C23×C4—32no—
275C23×C4C2×C5×D312yescomplemented
276C4×D4—64no—
277C4×D4—64no—
278C2×C4⋊C4—32no—
279C2×C22⋊C4—31no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—32no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—32no—
284C2×C22⋊C4—32no—
285C2×C22⋊C4—31no—
286C2×C22⋊C4—32no—
287C2×C22⋊C4—32no—
288C2×C22⋊C4—32no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—61no—
292C2×C42—32no—
293#95—32no—
294#95—32no—
295C2×C3×C5—41no—
296C2×C3×C5—42no—
297C2×C3×C5—82no—
298C5×D3—44no—
299C5×D3—82no—
300C23×C3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—44no—
304C22×D3—84no—
305C2×A4—22no—
306C2×A4C22×C4×C511yescharacteristic
307C2×A4C2×C5×D412yes—
308S4—22no—
309S4C5×C22⋊C414yesdirect product factor
310C2×C3×C4—42no—
311C2×Dic3—42no—
312C4×D3—84no—
313C22×C5—22no—
314C22×C5—31no—
315C22×C5—32no—
316C22×C5—31no—
317C22×C5—31no—
318C22×C5—32no—
319C22×C5—32no—
320C22×C5—34no—
321C22×C5—32no—
322C22×C5—62no—
323C22×C5—64no—
324C22×C5—62no—
325C22×C5—61no—
326C22×C5—62no—
327C22×C5—62no—
328C22×C5—64no—
329C22×C5—62no—
330C22×C5—62no—
331C22×C5—62no—
332C22×C5—62no—
333C22×C5—64no—
334C22×C5—62no—
335C22×C5—62no—
336C22×C5—128no—
337C22×C5—122no—
338C22×C5—122no—
339C22×C5—128no—
340C22×C5—122no—
341C22×C5D3×C22⋊C411yescharacteristic, complemented
342C22×C5C4×S412yescomplemented
343C22×C5C22×S411yescharacteristic
344C4×C5—22no—
345C4×C5—34no—
346C4×C5—62no—
347C4×C5—62no—
348C4×C5—122no—
349C4×C5—122no—
350C24—22no—
351C24—31no—
352C24—31no—
353C24—31no—
354C24—32no—
355C24—31no—
356C24—34no—
357C24—62no—
358C24—61no—
359C24—64no—
360C24—62no—
361C24—61no—
362C24—122no—
363C24C4×C5×D312yescomplemented
364C24C22×C5×D311yescharacteristic
365C2×D4—34no—
366C2×D4—32no—
367C2×D4—32no—
368C2×D4—34no—
369C2×D4—34no—
370C2×D4—32no—
371C2×D4—34no—
372C2×D4—32no—
373C2×D4—64no—
374C2×D4—68no—
375C2×D4—68no—
376C2×D4—616no—
377C2×D4—62no—
378C2×D4—64no—
379C2×D4—62no—
380C22×C4—22no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—34no—
384C22×C4—32no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—32no—
388C22×C4—64no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—64no—
392C22×C4—62no—
393C22×C4—122no—
394C4⋊C4—64no—
395C22⋊C4—31no—
396C22⋊C4—32no—
397C22⋊C4—61no—
398C22⋊C4—62no—
399C22⋊C4—61no—
400C22⋊C4—64no—
401C22⋊C4—62no—
402C22⋊C4—62no—
403C22⋊C4—62no—
404C22⋊C4—61no—
405C22⋊C4—62no—
406C22⋊C4—122no—
407C22⋊C4—122no—
408C22⋊C4C5×S411yescharacteristic, direct product factor
409C42—64no—
410C3×C5—41noHall
411C22×C3—42no—
412C22×C3—41no—
413C22×C3—82no—
414C2×D3—44no—
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—88no—
418C2×D3—82no—
419A4C2×C5×C22⋊C411yescharacteristic, complemented
420C3×C4—82no—
421Dic3—82no—
422C2×C5—22no—
423C2×C5—31no—
424C2×C5—32no—
425C2×C5—31no—
426C2×C5—62no—
427C2×C5—64no—
428C2×C5—122no—
429C2×C5C2×C4×S411yescharacteristic
430C2×C5D4×S412yes—
431C23—22no—
432C23—31no—
433C23—32no—
434C23—31no—
435C23—34no—
436C23—32no—
437C23—32no—
438C23—32no—
439C23—32no—
440C23—31no—
441C23—32no—
442C23—62no—
443C23—62no—
444C23—61no—
445C23—62no—
446C23—62no—
447C23—61no—
448C23—62no—
449C23—64no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—68no—
454C23—61no—
455C23—64no—
456C23—61no—
457C23—64no—
458C23—61no—
459C23—124no—
460C23—122no—
461C23—124no—
462C23—128no—
463C23—124no—
464C23C2×C4×C5×D311yescharacteristic
465C23C5×D3×D412yes—
466C23C2×C5×S411yescharacteristic
467D4—34no—
468D4—38no—
469D4—34no—
470D4—68no—
471D4—62no—
472D4—64no—
473D4—68no—
474D4—62no—
475C2×C4—34no—
476C2×C4—32no—
477C2×C4—32no—
478C2×C4—32no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—68no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—124no—
492C2×C4—124no—
493C2×C4C2×C5×S412yescomplemented
494C2×C3—41no—
495C2×C3—42no—
496C2×C3—82no—
497D3—44no—
498D3—82no—
499C5C22⋊C4×S411yescharacteristic, direct product factor, Sylow
500C22—22no—
501C22—31no—
502C22—32no—
503C22—31no—
504C22—31no—
505C22—32no—
506C22—32no—
507C22—34no—
508C22—32no—
509C22—62no—
510C22—64no—
511C22—62no—
512C22—61no—
513C22—62no—
514C22—62no—
515C22—64no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—62no—
520C22—64no—
521C22—62no—
522C22—62no—
523C22—128no—
524C22—122no—
525C22—122no—
526C22—128no—
527C22—122no—
528C22C5×D3×C22⋊C411yescharacteristic, complemented
529C22C4×C5×S412yescomplemented
530C22C22×C5×S411yescharacteristic
531C4—22no—
532C4—34no—
533C4—62no—
534C4—62no—
535C4—122no—
536C4—122no—
537C3—41noSylow
538C2—22no—
539C2—31no—
540C2—32no—
541C2—31no—
542C2—62no—
543C2—64no—
544C2—122no—
545C2C2×C4×C5×S411yescharacteristic
546C2C5×D4×S412yes—
subgroupG/H
centerC22×C5C22×S4
commutatorC2×A4C22×C4×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C22⋊C4D3
socleC24×C5C22×D3
prime radicalC23×C5×S4C2
cosocleC22×A4C23×C5
hypercenterC5×C22⋊C4S4
hypocenterA4C2×C5×C22⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C5×S4C2semidirect
3. C5×A4×C22⋊C4C2semidirect
4. C5×#5432C2semidirect
5. C5×(C2×S4)⋊C4C2semidirect
9. C22×C5×S4C4semidirect
12. C2×C4×C5×A4C22semidirect
13. C2×C5×SL(2,ℤ/4)C22semidirect
16. C22⋊C4×S4C5direct (N × H)
20. C22×C5×C22⋊C4D3semidirect
30. C22×C5×A4C2×C4semidirect
35. C2×C5×S4C2×C4semidirect
44. C2×C4×S4C2×C5semidirect
45. A4×C22⋊C4C2×C5semidirect
46. #5432C2×C5semidirect
47. (C2×S4)⋊C4C2×C5semidirect
60. C23×C4×C5C2×D3semidirect
90. C5×S4C22⋊C4direct (N × H)
96. C22×S4C4×C5semidirect
99. C2×C4×A4C22×C5semidirect
100. C2×SL(2,ℤ/4)C22×C5semidirect
116. C24×C5C4×D3semidirect
161. C5×C22⋊C4S4direct (N × H)
166. C22×C22⋊C4C5×D3semidirect
182. C5×A4C2×C22⋊C4semidirect
187. C22×A4C2×C4×C5semidirect
192. C2×S4C2×C4×C5semidirect
262. C2×C4×C5C2×S4semidirect
275. C23×C4C2×C5×D3semidirect
309. S4C5×C22⋊C4direct (N × H)
341. C22×C5D3×C22⋊C4semidirect
342. C22×C5C4×S4semidirect
363. C24C4×C5×D3semidirect
408. C22⋊C4C5×S4direct (N × H)
419. A4C2×C5×C22⋊C4semidirect
493. C2×C4C2×C5×S4semidirect
499. C5C22⋊C4×S4direct (N × H)
528. C22C5×D3×C22⋊C4semidirect
529. C22C4×C5×S4semidirect

Maximal subgroups

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

subgroupindex
C23×C5×S42
C2×C4×C5×S42
C5×A4×C22⋊C42
C5×#54322
C5×(C2×S4)⋊C42
C5×D4×C22⋊C43
C5×D3×C22⋊C44
C22⋊C4×S45