Finite Group Databasebeta

← C5×#85313 #14511051

Subgroups — C5×#85313 #14511051

Order 1920.

544 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
2C5×(C2×S4)⋊C4C211yesmaximal, characteristic
3C5×(C2×S4)⋊C4C211yesmaximal, characteristic
4C5×(C2×S4)⋊C4C211yesmaximal, characteristic
5C5×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6C5×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
7C4×C5×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
8C5×#2481—31nomaximal, Hall
9C22×C5×S4C2211yescharacteristic
10C2×C4×C5×A4C2211yescharacteristic
11C2×C4×C5×A4C2211yescharacteristic
12C2×C4×C5×A4C2211yescharacteristic
13C2×C5×SL(2,ℤ/4)C2211yescharacteristic
14C2×C5×SL(2,ℤ/4)C2211yescharacteristic
15C2×C5×SL(2,ℤ/4)C2211yescharacteristic
16C5×#898—41nomaximal
17#85313C511yesmaximal, characteristic, direct product factor, Hall
18C2×C5×#126—31no—
19C22×C5×C4⋊C4D311yescharacteristic, complemented
20C5×#400—31no—
21C5×#400—31no—
22C5×#397—31no—
23C5×#396—31no—
24C5×#388—31no—
25C5×#387—31no—
26C5×#387—31no—
27C5×#386—31no—
28C5×#386—31no—
29C5×#386—31no—
30C5×#385—31no—
31C5×#380—31no—
32C4×C5×C22⋊C4—31no—
33C22×C5×A4C2311yescharacteristic
34C2×C5×S4—21no—
35C2×C5×S4—21no—
36C2×C5×S4—21no—
37C4×C5×A4—21no—
38C4×C5×A4—21no—
39C4×C5×A4—21no—
40C5×SL(2,ℤ/4)—21no—
41C5×SL(2,ℤ/4)—21no—
42C5×SL(2,ℤ/4)—21no—
43C3×C5×C4⋊C4—41no—
44C5×(C2×D3)⋊C4—41no—
45C5×(C2×D3)⋊C4—41no—
46C5×(C2×D3)⋊C4—41no—
47C5×C4⋊Dic3—41no—
48C5×Dic3⋊C4—41no—
49C4×C5×Dic3—41no—
50A4×C4⋊C4C2×C511yescharacteristic, complemented
51(C2×S4)⋊C4C2×C511yescharacteristic
52(C2×S4)⋊C4C2×C511yescharacteristic
53(C2×S4)⋊C4C2×C511yescharacteristic
54C4⋊SL(2,ℤ/4)C2×C511yescharacteristic, complemented
55SL(2,ℤ/4)⋊C4C2×C511yescharacteristic, complemented
56C4×SL(2,ℤ/4)C2×C511yescharacteristic, complemented
57C22×C5×D4—31no—
58C23×C4×C5C2×D311yescharacteristic
59C23×C4×C5C2×D311yescharacteristic
60C23×C4×C5C2×D311yescharacteristic
61C5×#126—61no—
62C5×#126—61no—
63C5×#126—61no—
64C5×#126—61no—
65C2×C5×C4⋊C4—31no—
66C2×C5×C4⋊C4—31no—
67C2×C5×C4⋊C4—31no—
68C2×C5×C4⋊C4—31no—
69C2×C5×C4⋊C4—31no—
70C2×C5×C4⋊C4—31no—
71C2×C5×C4⋊C4—31no—
72C2×C5×C4⋊C4—31no—
73C2×C5×C22⋊C4—31no—
74C2×C5×C22⋊C4—31no—
75C2×C5×C22⋊C4—31no—
76C2×C5×C22⋊C4—31no—
77C2×C5×C22⋊C4—31no—
78C2×C5×C22⋊C4—31no—
79C2×C5×C22⋊C4—31no—
80C2×C5×C22⋊C4—31no—
81C2×C5×C22⋊C4—31no—
82C2×C42×C5—31no—
83C2×C42×C5—31no—
84C5×#95—31no—
85C5×#95—31no—
86C5×#95—31no—
87C5×#95—31no—
88C5×#95—31no—
89C5×#95—31no—
90C5×#95—31no—
91C5×#95—31no—
92C5×#95—31no—
93C5×#95—31no—
94C5×#95—31no—
95C5×#95—31no—
96#2481—31noSylow
97C22×C5×D3—41no—
98C2×C5×A4#4111yescharacteristic
99C2×C5×A4#4111yescharacteristic
100C2×C5×A4#4111yescharacteristic
101C5×S4—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C5×Dic3—41no—
106C2×C5×Dic3—41no—
107C2×C5×Dic3—41no—
108C22×S4C22×C511yescharacteristic
109C2×C4×A4C22×C511yescharacteristic
110C2×C4×A4C22×C511yescharacteristic
111C2×C4×A4C22×C511yescharacteristic
112C2×SL(2,ℤ/4)C22×C511yescharacteristic
113C2×SL(2,ℤ/4)C22×C511yescharacteristic
114C2×SL(2,ℤ/4)C22×C511yescharacteristic
115#898—41no—
116C24×C5—31no—
117C24×C5C22×D311yescharacteristic
118C2×C5×D4—61no—
119C2×C5×D4—61no—
120C2×C5×D4—61no—
121C2×C5×D4—61no—
122C2×C5×D4—61no—
123C2×C5×D4—61no—
124C22×C4×C5—21no—
125C22×C4×C5—21no—
126C22×C4×C5—21no—
127C22×C4×C5—31no—
128C22×C4×C5—31no—
129C22×C4×C5—31no—
130C22×C4×C5—31no—
131C22×C4×C5—31no—
132C22×C4×C5—31no—
133C22×C4×C5—31no—
134C22×C4×C5—31no—
135C22×C4×C5—31no—
136C22×C4×C5—31no—
137C22×C4×C5—31no—
138C22×C4×C5—31no—
139C22×C4×C5—31no—
140C22×C4×C5—61no—
141C22×C4×C5—61no—
142C22×C4×C5—61no—
143C5×C4⋊C4—31no—
144C5×C4⋊C4—31no—
145C5×C4⋊C4—31no—
146C5×C4⋊C4—61no—
147C5×C4⋊C4—62no—
148C5×C4⋊C4—61no—
149C5×C4⋊C4—61no—
150C5×C4⋊C4—61no—
151C5×C4⋊C4—61no—
152C5×C4⋊C4—121no—
153C5×C4⋊C4S411yescharacteristic, complemented
154C5×C22⋊C4—62no—
155C5×C22⋊C4—61no—
156C5×C22⋊C4—61no—
157C5×C22⋊C4—61no—
158C5×C22⋊C4—61no—
159C5×C22⋊C4—61no—
160C5×C22⋊C4—61no—
161C5×C22⋊C4—61no—
162C5×C22⋊C4—61no—
163C5×C22⋊C4—61no—
164C5×C22⋊C4—61no—
165C5×C22⋊C4—61no—
166C5×C22⋊C4—61no—
167C5×C22⋊C4—121no—
168C5×C22⋊C4—121no—
169C42×C5—61no—
170C42×C5—61no—
171C42×C5—121no—
172C2×#126—31no—
173C22×C4⋊C4C5×D311yescharacteristic, complemented
174#400—31no—
175#400—31no—
176#397—31no—
177#396—31no—
178#388—31no—
179#387—31no—
180#387—31no—
181#386—31no—
182#386—31no—
183#386—31no—
184#385—31no—
185#380—31no—
186C4×C22⋊C4—31no—
187C22×C3×C5—41no—
188C2×C5×D3—81no—
189C2×C5×D3—81no—
190C2×C5×D3—81no—
191C5×A4#12611yescharacteristic, complemented
192C3×C4×C5—81no—
193C3×C4×C5—81no—
194C3×C4×C5—81no—
195C5×Dic3—81no—
196C5×Dic3—81no—
197C5×Dic3—81no—
198C22×A4C23×C511yescharacteristic
199C2×S4—21no—
200C2×S4—21no—
201C2×S4—21no—
202C4×A4—21no—
203C4×A4—21no—
204C4×A4—21no—
205SL(2,ℤ/4)—21no—
206SL(2,ℤ/4)—21no—
207SL(2,ℤ/4)—21no—
208C3×C4⋊C4—41no—
209(C2×D3)⋊C4—41no—
210(C2×D3)⋊C4—41no—
211(C2×D3)⋊C4—41no—
212C4⋊Dic3—41no—
213Dic3⋊C4—41no—
214C4×Dic3—41no—
215C23×C5—31no—
216C23×C5—31no—
217C23×C5—31no—
218C23×C5—31no—
219C23×C5—61no—
220C23×C5—61no—
221C23×C5—61no—
222C23×C5—61no—
223C23×C5—61no—
224C23×C5—61no—
225C23×C5—61no—
226C23×C5#23511yescharacteristic
227C23×C5#23711yescharacteristic
228C23×C5#23911yescharacteristic
229C5×D4—121no—
230C5×D4—121no—
231C5×D4—121no—
232C5×D4—121no—
233C2×C4×C5—32no—
234C2×C4×C5—31no—
235C2×C4×C5—31no—
236C2×C4×C5—31no—
237C2×C4×C5—61no—
238C2×C4×C5—61no—
239C2×C4×C5—61no—
240C2×C4×C5—61no—
241C2×C4×C5—61no—
242C2×C4×C5—61no—
243C2×C4×C5—61no—
244C2×C4×C5—61no—
245C2×C4×C5—61no—
246C2×C4×C5—61no—
247C2×C4×C5—61no—
248C2×C4×C5—61no—
249C2×C4×C5—61no—
250C2×C4×C5—61no—
251C2×C4×C5—61no—
252C2×C4×C5—61no—
253C2×C4×C5—61no—
254C2×C4×C5—61no—
255C2×C4×C5—61no—
256C2×C4×C5—61no—
257C2×C4×C5—61no—
258C2×C4×C5—61no—
259C2×C4×C5—61no—
260C2×C4×C5—61no—
261C2×C4×C5—61no—
262C2×C4×C5—61no—
263C2×C4×C5—61no—
264C2×C4×C5—61no—
265C2×C4×C5—61no—
266C2×C4×C5—61no—
267C2×C4×C5—61no—
268C2×C4×C5—61no—
269C2×C4×C5C2×S411yescharacteristic
270C2×C4×C5C2×S411yescharacteristic
271C2×C4×C5C2×S411yescharacteristic
272C22×D4—31no—
273C23×C4C2×C5×D311yescharacteristic
274C23×C4C2×C5×D311yescharacteristic
275C23×C4C2×C5×D311yescharacteristic
276#126—61no—
277#126—61no—
278#126—61no—
279#126—61no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—31no—
290C2×C22⋊C4—31no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—31no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C42—31no—
298C2×C42—31no—
299#95—31no—
300#95—31no—
301#95—31no—
302#95—31no—
303#95—31no—
304#95—31no—
305#95—31no—
306#95—31no—
307#95—31no—
308#95—31no—
309#95—31no—
310#95—31no—
311C2×C3×C5—41no—
312C2×C3×C5—41no—
313C2×C3×C5—41no—
314C5×D3—161no—
315C22×D3—41no—
316C2×A4C5×#4111yescharacteristic
317C2×A4C5×#4111yescharacteristic
318C2×A4C5×#4111yescharacteristic
319S4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C22×C5—31no—
327C22×C5—31no—
328C22×C5—31no—
329C22×C5—31no—
330C22×C5—31no—
331C22×C5—31no—
332C22×C5—31no—
333C22×C5—31no—
334C22×C5—31no—
335C22×C5—61no—
336C22×C5—121no—
337C22×C5—121no—
338C22×C5—121no—
339C22×C5—121no—
340C22×C5—121no—
341C22×C5—121no—
342C22×C5—121no—
343C22×C5#89811yescharacteristic, complemented
344C22×C5C22×S411yescharacteristic
345C4×C5—21no—
346C4×C5—21no—
347C4×C5—21no—
348C4×C5—61no—
349C4×C5—61no—
350C4×C5—61no—
351C4×C5—121no—
352C4×C5—121no—
353C4×C5—121no—
354C4×C5—121no—
355C4×C5—121no—
356C4×C5—121no—
357C4×C5—121no—
358C24—31no—
359C24C22×C5×D311yescharacteristic
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—61no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C4⋊C4—31no—
386C4⋊C4—31no—
387C4⋊C4—31no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—121no—
395C4⋊C4C5×S411yescharacteristic, complemented
396C22⋊C4—61no—
397C22⋊C4—61no—
398C22⋊C4—61no—
399C22⋊C4—62no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—61no—
403C22⋊C4—61no—
404C22⋊C4—61no—
405C22⋊C4—61no—
406C22⋊C4—61no—
407C22⋊C4—61no—
408C22⋊C4—61no—
409C22⋊C4—121no—
410C22⋊C4—121no—
411C42—61no—
412C42—61no—
413C42—121no—
414C3×C5—41noHall
415C22×C3—41no—
416C2×D3—81no—
417C2×D3—81no—
418C2×D3—81no—
419A4C5×#12611yescharacteristic, complemented
420C3×C4—81no—
421C3×C4—81no—
422C3×C4—81no—
423Dic3—81no—
424Dic3—81no—
425Dic3—81no—
426C2×C5—31no—
427C2×C5—31no—
428C2×C5—31no—
429C2×C5—31no—
430C2×C5—241no—
431C2×C5#591211yescharacteristic
432C2×C5#591411yescharacteristic
433C2×C5#591911yescharacteristic
434C23—31no—
435C23—31no—
436C23—31no—
437C23—31no—
438C23—61no—
439C23—61no—
440C23—61no—
441C23—61no—
442C23—61no—
443C23—61no—
444C23—61no—
445C23C5×#23511yescharacteristic
446C23C5×#23711yescharacteristic
447C23C5×#23911yescharacteristic
448D4—121no—
449D4—121no—
450D4—121no—
451D4—121no—
452C2×C4—32no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4C2×C5×S411yescharacteristic
489C2×C4C2×C5×S411yescharacteristic
490C2×C4C2×C5×S411yescharacteristic
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494D3—161no—
495C5#8531311yescharacteristic, direct product factor, Sylow
496C22—31no—
497C22—31no—
498C22—31no—
499C22—31no—
500C22—31no—
501C22—31no—
502C22—31no—
503C22—31no—
504C22—31no—
505C22—61no—
506C22—121no—
507C22—121no—
508C22—121no—
509C22—121no—
510C22—121no—
511C22—121no—
512C22—121no—
513C22C5×#89811yescharacteristic, complemented
514C22C22×C5×S411yescharacteristic
515C4—21no—
516C4—21no—
517C4—21no—
518C4—61no—
519C4—61no—
520C4—61no—
521C4—121no—
522C4—121no—
523C4—121no—
524C4—121no—
525C4—121no—
526C4—121no—
527C4—121no—
528C3—41noSylow
529C2—31no—
530C2—31no—
531C2—31no—
532C2—31no—
533C2—241no—
534C2C5×#591211yescharacteristic
535C2C5×#591411yescharacteristic
536C2C5×#591911yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5C22×D3
prime radicalC22×C5×S4C22
hypercenterC5×C4⋊C4S4
hypocenterA4C5×#126
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C5×A4×C4⋊C4C2semidirect
5. C5×C4⋊SL(2,ℤ/4)C2semidirect
6. C5×SL(2,ℤ/4)⋊C4C2semidirect
7. C4×C5×SL(2,ℤ/4)C2semidirect
17. #85313C5direct (N × H)
19. C22×C5×C4⋊C4D3semidirect
50. A4×C4⋊C4C2×C5semidirect
54. C4⋊SL(2,ℤ/4)C2×C5semidirect
55. SL(2,ℤ/4)⋊C4C2×C5semidirect
56. C4×SL(2,ℤ/4)C2×C5semidirect
153. C5×C4⋊C4S4semidirect
173. C22×C4⋊C4C5×D3semidirect
191. C5×A4#126semidirect
343. C22×C5#898semidirect
395. C4⋊C4C5×S4semidirect
419. A4C5×#126semidirect
495. C5#85313direct (N × H)
513. C22C5×#898semidirect

Maximal subgroups

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

subgroupindex
C5×A4×C4⋊C42
C5×(C2×S4)⋊C42
C5×C4⋊SL(2,ℤ/4)2
C5×SL(2,ℤ/4)⋊C42
C4×C5×SL(2,ℤ/4)2
C5×#24813
C5×#8984
#853135