Finite Group Databasebeta

← C5×C4⋊C4×S4 #14511044

Subgroups — C5×C4⋊C4×S4 #14511044

Order 1920.

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

normal subgroups only

This group has 838 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
1C2×C4×C5×S4C212yesmaximal
2C2×C4×C5×S4C211yesmaximal, characteristic
3C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C5×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C5×SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6C5×D4×C4⋊C4—31nomaximal, Hall
7C22×C5×S4C2211yescharacteristic
8C2×C4×C5×A4C2212yes—
9C2×C4×C5×A4C2211yescharacteristic
10C2×C5×SL(2,ℤ/4)C2212yes—
11C2×C5×SL(2,ℤ/4)C2211yescharacteristic
12C4×C5×S4—24no—
13C4×C5×S4C414yescomplemented
14C5×D3×C4⋊C4—41nomaximal
15C4⋊C4×S4C511yesmaximal, characteristic, direct product factor, Hall
16C2×C4×C5×D4—31no—
17C2×C4×C5×D4—32no—
18C22×C5×C4⋊C4—31no—
19C22×C5×C4⋊C4D311yescharacteristic, complemented
20C5×#389—32no—
21C5×#385—32no—
22C5×#385—32no—
23C5×#384—31no—
24C5×#380—31no—
25C5×#380—31no—
26C4×C5×C4⋊C4—31no—
27C22×C5×A4C2311yescharacteristic
28C2×C5×S4C2×C412yes—
29C2×C5×S4D412yes—
30C2×C5×S4Q812yes—
31C2×C4×C5×D3—41no—
32C2×C4×C5×D3—42no—
33C4×C5×A4—22no—
34C4×C5×A4C2×C412yescomplemented
35C5×SL(2,ℤ/4)—22no—
36C5×SL(2,ℤ/4)C2×C412yescomplemented
37C3×C5×C4⋊C4—41no—
38C5×C4⋊Dic3—41no—
39C5×Dic3⋊C4—42no—
40C2×C4×S4C2×C512yes—
41C2×C4×S4C2×C511yescharacteristic
42A4×C4⋊C4C2×C511yescharacteristic, complemented
43C4⋊SL(2,ℤ/4)C2×C511yescharacteristic, complemented
44SL(2,ℤ/4)⋊C4C2×C512yescomplemented
45C22×C5×D4—31no—
46C23×C4×C5—31no—
47C23×C4×C5—32no—
48C23×C4×C5C2×D312yes—
49C23×C4×C5C2×D311yescharacteristic
50C4×C5×D4—34no—
51C4×C5×D4—34no—
52C4×C5×D4—64no—
53C4×C5×D4—64no—
54C2×C5×C4⋊C4—31no—
55C2×C5×C4⋊C4—31no—
56C2×C5×C4⋊C4—31no—
57C2×C5×C4⋊C4—32no—
58C2×C5×C4⋊C4—31no—
59C2×C5×C4⋊C4—31no—
60C2×C5×C4⋊C4—32no—
61C2×C5×C4⋊C4—32no—
62C2×C5×C4⋊C4—32no—
63C2×C5×C4⋊C4—62no—
64C2×C5×C4⋊C4—61no—
65C2×C5×C4⋊C4—61no—
66C2×C5×C22⋊C4—31no—
67C2×C5×C22⋊C4—31no—
68C2×C5×C22⋊C4—32no—
69C2×C5×C22⋊C4—32no—
70C2×C42×C5—31no—
71C2×C42×C5—32no—
72C5×#95—32no—
73C5×#95—32no—
74C5×#95—32no—
75D4×C4⋊C4—31noSylow
76C22×C5×D3—41no—
77C2×C5×A4C22×C411yescharacteristic
78C2×C5×A4C2×D411yescharacteristic
79C2×C5×A4C2×Q811yescharacteristic
80C5×S4C4⋊C414yesdirect product factor
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—42no—
83C2×C5×Dic3—41no—
84C2×C5×Dic3—42no—
85C4×C5×D3—44no—
86C4×C5×D3—84no—
87C22×S4C22×C511yescharacteristic
88C2×C4×A4C22×C512yes—
89C2×C4×A4C22×C511yescharacteristic
90C2×SL(2,ℤ/4)C22×C512yes—
91C2×SL(2,ℤ/4)C22×C511yescharacteristic
92C4×S4—24no—
93C4×S4C4×C514yescomplemented
94D3×C4⋊C4—41no—
95C24×C5—31no—
96C24×C5C22×D311yescharacteristic
97C2×C5×D4—32no—
98C2×C5×D4—32no—
99C2×C5×D4—32no—
100C2×C5×D4—32no—
101C2×C5×D4—32no—
102C2×C5×D4—32no—
103C22×C4×C5—22no—
104C22×C4×C5—31no—
105C22×C4×C5—31no—
106C22×C4×C5—31no—
107C22×C4×C5—32no—
108C22×C4×C5—31no—
109C22×C4×C5—31no—
110C22×C4×C5—34no—
111C22×C4×C5—32no—
112C22×C4×C5—32no—
113C22×C4×C5—32no—
114C22×C4×C5—32no—
115C22×C4×C5—61no—
116C22×C4×C5—62no—
117C22×C4×C5—62no—
118C22×C4×C5—61no—
119C22×C4×C5—62no—
120C22×C4×C5—64no—
121C22×C4×C5—62no—
122C22×C4×C5—122no—
123C22×C4×C5C4×D312yescomplemented
124C5×C4⋊C4—34no—
125C5×C4⋊C4—32no—
126C5×C4⋊C4—31no—
127C5×C4⋊C4—61no—
128C5×C4⋊C4—62no—
129C5×C4⋊C4—61no—
130C5×C4⋊C4—62no—
131C5×C4⋊C4—64no—
132C5×C4⋊C4—62no—
133C5×C4⋊C4—61no—
134C5×C4⋊C4—122no—
135C5×C4⋊C4S411yescharacteristic, direct product factor
136C5×C22⋊C4—34no—
137C5×C22⋊C4—32no—
138C5×C22⋊C4—32no—
139C5×C22⋊C4—64no—
140C5×C22⋊C4—62no—
141C5×C22⋊C4—62no—
142C42×C5—34no—
143C42×C5—64no—
144C2×C4×D4—31no—
145C2×C4×D4—32no—
146C22×C4⋊C4—31no—
147C22×C4⋊C4C5×D311yescharacteristic, complemented
148#389—32no—
149#385—32no—
150#385—32no—
151#384—31no—
152#380—31no—
153#380—31no—
154C4×C4⋊C4—31no—
155C22×C3×C5—41no—
156C2×C5×D3—42no—
157C2×C5×D3—42no—
158C2×C5×D3—42no—
159C5×A4C2×C4⋊C411yescharacteristic, complemented
160C3×C4×C5—42no—
161C3×C4×C5—82no—
162C5×Dic3—42no—
163C5×Dic3—82no—
164C22×A4C23×C511yescharacteristic
165C2×S4C2×C4×C512yes—
166C2×S4C5×D412yes—
167C2×S4C5×Q812yes—
168C2×C4×D3—41no—
169C2×C4×D3—42no—
170C4×A4—22no—
171C4×A4C2×C4×C512yescomplemented
172SL(2,ℤ/4)—22no—
173SL(2,ℤ/4)C2×C4×C512yescomplemented
174C3×C4⋊C4—41no—
175C4⋊Dic3—41no—
176Dic3⋊C4—42no—
177C23×C5—31no—
178C23×C5—31no—
179C23×C5—31no—
180C23×C5—31no—
181C23×C5—32no—
182C23×C5—32no—
183C23×C5—32no—
184C23×C5—61no—
185C23×C5—61no—
186C23×C5—61no—
187C23×C5—61no—
188C23×C5C2×C4×D311yescharacteristic
189C23×C5D3×D411yescharacteristic
190C23×C5D3×Q811yescharacteristic
191C5×D4—34no—
192C5×D4—34no—
193C5×D4—34no—
194C5×D4—34no—
195C2×C4×C5—32no—
196C2×C4×C5—32no—
197C2×C4×C5—32no—
198C2×C4×C5—31no—
199C2×C4×C5—32no—
200C2×C4×C5—32no—
201C2×C4×C5—32no—
202C2×C4×C5—32no—
203C2×C4×C5—32no—
204C2×C4×C5—32no—
205C2×C4×C5—62no—
206C2×C4×C5—62no—
207C2×C4×C5—61no—
208C2×C4×C5—64no—
209C2×C4×C5—61no—
210C2×C4×C5—61no—
211C2×C4×C5—61no—
212C2×C4×C5—61no—
213C2×C4×C5—64no—
214C2×C4×C5—62no—
215C2×C4×C5—62no—
216C2×C4×C5—62no—
217C2×C4×C5—62no—
218C2×C4×C5—62no—
219C2×C4×C5—62no—
220C2×C4×C5—62no—
221C2×C4×C5—62no—
222C2×C4×C5—124no—
223C2×C4×C5—124no—
224C2×C4×C5C2×S412yes—
225C2×C4×C5C2×S411yescharacteristic
226C22×D4—31no—
227C23×C4—31no—
228C23×C4—32no—
229C23×C4C2×C5×D312yes—
230C23×C4C2×C5×D311yescharacteristic
231C4×D4—34no—
232C4×D4—34no—
233C4×D4—64no—
234C4×D4—64no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—31no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4—61no—
247C2×C22⋊C4—31no—
248C2×C22⋊C4—31no—
249C2×C22⋊C4—32no—
250C2×C22⋊C4—32no—
251C2×C42—31no—
252C2×C42—32no—
253#95—32no—
254#95—32no—
255#95—32no—
256C2×C3×C5—41no—
257C2×C3×C5—41no—
258C2×C3×C5—41no—
259C5×D3—44no—
260C22×D3—41no—
261C2×A4C22×C4×C511yescharacteristic
262C2×A4C2×C5×D411yescharacteristic
263C2×A4C2×C5×Q811yescharacteristic
264S4C5×C4⋊C414yesdirect product factor
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269C4×D3—44no—
270C4×D3—84no—
271C22×C5—31no—
272C22×C5—31no—
273C22×C5—31no—
274C22×C5—31no—
275C22×C5—31no—
276C22×C5—31no—
277C22×C5—31no—
278C22×C5—31no—
279C22×C5—31no—
280C22×C5—34no—
281C22×C5—61no—
282C22×C5—62no—
283C22×C5—62no—
284C22×C5—62no—
285C22×C5—62no—
286C22×C5—62no—
287C22×C5—62no—
288C22×C5D3×C4⋊C411yescharacteristic, complemented
289C22×C5C22×S411yescharacteristic
290C4×C5—22no—
291C4×C5—34no—
292C4×C5—32no—
293C4×C5—62no—
294C4×C5—62no—
295C4×C5—62no—
296C4×C5—122no—
297C4×C5—122no—
298C4×C5C4×S412yescomplemented
299C24—31no—
300C24C22×C5×D311yescharacteristic
301C2×D4—32no—
302C2×D4—32no—
303C2×D4—32no—
304C2×D4—32no—
305C2×D4—32no—
306C2×D4—32no—
307C22×C4—22no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4—32no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—34no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—61no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—64no—
325C22×C4—62no—
326C22×C4—122no—
327C22×C4C4×C5×D312yescomplemented
328C4⋊C4—34no—
329C4⋊C4—32no—
330C4⋊C4—31no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—62no—
334C4⋊C4—64no—
335C4⋊C4—62no—
336C4⋊C4—62no—
337C4⋊C4—61no—
338C4⋊C4—122no—
339C4⋊C4C5×S411yescharacteristic, direct product factor
340C22⋊C4—34no—
341C22⋊C4—32no—
342C22⋊C4—32no—
343C22⋊C4—64no—
344C22⋊C4—62no—
345C22⋊C4—62no—
346C42—34no—
347C42—64no—
348C3×C5—41noHall
349C22×C3—41no—
350C2×D3—42no—
351C2×D3—42no—
352C2×D3—42no—
353A4C2×C5×C4⋊C411yescharacteristic, complemented
354C3×C4—42no—
355C3×C4—82no—
356Dic3—42no—
357Dic3—82no—
358C2×C5—31no—
359C2×C5—31no—
360C2×C5—31no—
361C2×C5—31no—
362C2×C5—64no—
363C2×C5C2×C4×S411yescharacteristic
364C2×C5D4×S411yescharacteristic
365C2×C5Q8×S411yescharacteristic
366C23—31no—
367C23—31no—
368C23—31no—
369C23—31no—
370C23—32no—
371C23—32no—
372C23—32no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23C2×C4×C5×D311yescharacteristic
378C23C5×D3×D411yescharacteristic
379C23C5×D3×Q811yescharacteristic
380D4—34no—
381D4—34no—
382D4—34no—
383D4—34no—
384C2×C4—32no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—31no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—61no—
397C2×C4—64no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—64no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—124no—
412C2×C4—124no—
413C2×C4C2×C5×S412yes—
414C2×C4C2×C5×S411yescharacteristic
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418D3—44no—
419C5C4⋊C4×S411yescharacteristic, direct product factor, Sylow
420C22—31no—
421C22—31no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—31no—
427C22—31no—
428C22—31no—
429C22—34no—
430C22—61no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22C5×D3×C4⋊C411yescharacteristic, complemented
438C22C22×C5×S411yescharacteristic
439C4—22no—
440C4—34no—
441C4—32no—
442C4—62no—
443C4—62no—
444C4—62no—
445C4—122no—
446C4—122no—
447C4C4×C5×S412yescomplemented
448C3—41noSylow
449C2—31no—
450C2—31no—
451C2—31no—
452C2—31no—
453C2—64no—
454C2C2×C4×C5×S411yescharacteristic
455C2C5×D4×S411yescharacteristic
456C2C5×Q8×S411yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutatorC2×A4C22×C4×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5C22×D3
prime radicalC22×C5×S4C22
cosocleC22×A4C23×C5
hypercenterC5×C4⋊C4S4
hypocenterA4C2×C5×C4⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C5×A4×C4⋊C4C2semidirect
4. C5×C4⋊SL(2,ℤ/4)C2semidirect
5. C5×SL(2,ℤ/4)⋊C4C2semidirect
13. C4×C5×S4C4semidirect
15. C4⋊C4×S4C5direct (N × H)
19. C22×C5×C4⋊C4D3semidirect
34. C4×C5×A4C2×C4semidirect
36. C5×SL(2,ℤ/4)C2×C4semidirect
42. A4×C4⋊C4C2×C5semidirect
43. C4⋊SL(2,ℤ/4)C2×C5semidirect
44. SL(2,ℤ/4)⋊C4C2×C5semidirect
80. C5×S4C4⋊C4direct (N × H)
93. C4×S4C4×C5semidirect
123. C22×C4×C5C4×D3semidirect
135. C5×C4⋊C4S4direct (N × H)
147. C22×C4⋊C4C5×D3semidirect
159. C5×A4C2×C4⋊C4semidirect
171. C4×A4C2×C4×C5semidirect
173. SL(2,ℤ/4)C2×C4×C5semidirect
264. S4C5×C4⋊C4direct (N × H)
288. C22×C5D3×C4⋊C4semidirect
298. C4×C5C4×S4semidirect
327. C22×C4C4×C5×D3semidirect
339. C4⋊C4C5×S4direct (N × H)
353. A4C2×C5×C4⋊C4semidirect
419. C5C4⋊C4×S4direct (N × H)
437. C22C5×D3×C4⋊C4semidirect
447. C4C4×C5×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×S42
C5×A4×C4⋊C42
C5×C4⋊SL(2,ℤ/4)2
C5×SL(2,ℤ/4)⋊C42
C5×D4×C4⋊C43
C5×D3×C4⋊C44
C4⋊C4×S45