Finite Group Databasebeta

← C5×#85480 #14511218

Subgroups — C5×#85480 #14511218

Order 1920.

798 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C22×C4×C5×A4C211yesmaximal, characteristic
2C2×C5×A4⋊Q8C211yesmaximal, characteristic, complemented
3C5×#5432C212yesmaximal
4C5×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C5×SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6C5×#2848—31nomaximal, Hall
7C23×C5×A4C2211yescharacteristic
8C2×C4×C5×A4—22no—
9C2×C4×C5×A4C2211yescharacteristic
10C2×C4×C5×A4C2211yescharacteristic
11C2×C5×SL(2,ℤ/4)C2212yes—
12C2×C5×SL(2,ℤ/4)C2212yes—
13C5×A4⋊Q8—22no—
14C5×#924—41nomaximal
15#85480C511yesmaximal, characteristic, direct product factor, Hall
16C24×C4×C5D311yescharacteristic
17C2×C5×C22⋊Q8—31no—
18C2×C5×C22⋊Q8—31no—
19C2×C5×C22⋊Q8—31no—
20C5×#385—32no—
21C5×#385—32no—
22C5×#385—32no—
23C5×#380—31no—
24C5×#380—31no—
25C5×#380—31no—
26C5×#379—32no—
27C22×C5×A4—21no—
28C22×C5×A4—21no—
29C22×C5×A4Q812yescomplemented
30C22×C5×A4C2311yescharacteristic
31C22×C3×C4×C5—41no—
32C2×C5×Dic6—41no—
33C4×C5×A4—21no—
34C4×C5×A4D412yes—
35C5×SL(2,ℤ/4)—22no—
36C5×SL(2,ℤ/4)—22no—
37C5×#217—42no—
38C5×C4⋊Dic3—41no—
39C5×Dic3⋊C4—42no—
40C22×C4×A4C2×C511yescharacteristic
41C2×A4⋊Q8C2×C511yescharacteristic, complemented
42#5432C2×C512yes—
43C4⋊SL(2,ℤ/4)C2×C511yescharacteristic, complemented
44SL(2,ℤ/4)⋊C4C2×C512yescomplemented
45C25×C5C2×D311yescharacteristic
46C22×C5×Q8—31no—
47C23×C4×C5—22no—
48C23×C4×C5—31no—
49C23×C4×C5—31no—
50C23×C4×C5—31no—
51C23×C4×C5—31no—
52C23×C4×C5—62no—
53C23×C4×C5C2×D311yescharacteristic
54C23×C4×C5C2×D311yescharacteristic
55C5×C22⋊Q8—62no—
56C5×C22⋊Q8—64no—
57C5×C22⋊Q8—61no—
58C5×C22⋊Q8—61no—
59C5×C22⋊Q8—61no—
60C5×C22⋊Q8—62no—
61C5×C22⋊Q8—61no—
62C2×C5×C4⋊C4—31no—
63C2×C5×C4⋊C4—31no—
64C2×C5×C4⋊C4—31no—
65C2×C5×C4⋊C4—32no—
66C2×C5×C4⋊C4—32no—
67C2×C5×C4⋊C4—32no—
68C2×C5×C22⋊C4—32no—
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—
74C5×#95—32no—
75C5×#95—32no—
76C5×#95—32no—
77#2848—31noSylow
78C23×C3×C5—41no—
79C2×C5×A4—22no—
80C2×C5×A4C2×D411yescharacteristic
81C2×C5×A4C2×Q811yescharacteristic
82C2×C5×A4#4111yescharacteristic
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—82no—
86C2×C5×Dic3—42no—
87C2×C5×Dic3—42no—
88C5×Dic6—82no—
89C23×A4C22×C511yescharacteristic
90C2×C4×A4—22no—
91C2×C4×A4C22×C511yescharacteristic
92C2×C4×A4C22×C511yescharacteristic
93C2×SL(2,ℤ/4)C22×C512yes—
94C2×SL(2,ℤ/4)C22×C512yes—
95A4⋊Q8—22no—
96#924—41no—
97C24×C5—21no—
98C24×C5—21no—
99C24×C5—31no—
100C24×C5—31no—
101C24×C5—32no—
102C24×C5—61no—
103C24×C5—61no—
104C24×C5Dic612yescomplemented
105C24×C5C22×D311yescharacteristic
106C2×C5×Q8—61no—
107C2×C5×Q8—62no—
108C2×C5×Q8—62no—
109C2×C5×Q8—61no—
110C22×C4×C5—21no—
111C22×C4×C5—31no—
112C22×C4×C5—31no—
113C22×C4×C5—31no—
114C22×C4×C5—32no—
115C22×C4×C5—34no—
116C22×C4×C5—31no—
117C22×C4×C5—31no—
118C22×C4×C5—31no—
119C22×C4×C5—31no—
120C22×C4×C5—32no—
121C22×C4×C5—32no—
122C22×C4×C5—32no—
123C22×C4×C5—32no—
124C22×C4×C5—62no—
125C22×C4×C5—62no—
126C22×C4×C5—61no—
127C22×C4×C5—61no—
128C22×C4×C5—62no—
129C22×C4×C5—62no—
130C22×C4×C5—62no—
131C22×C4×C5—62no—
132C22×C4×C5—62no—
133C22×C4×C5#6712yes—
134C22×C4×C5S411yescharacteristic
135C5×C4⋊C4—61no—
136C5×C4⋊C4—62no—
137C5×C4⋊C4—62no—
138C5×C4⋊C4—64no—
139C5×C4⋊C4—64no—
140C5×C4⋊C4—62no—
141C5×C4⋊C4—62no—
142C5×C4⋊C4—61no—
143C5×C22⋊C4—62no—
144C5×C22⋊C4—64no—
145C5×C22⋊C4—64no—
146C5×C22⋊C4—62no—
147C5×C22⋊C4—64no—
148C5×C22⋊C4—62no—
149C5×C22⋊C4—62no—
150C5×C22⋊C4—62no—
151C5×C22⋊C4—62no—
152C24×C4C5×D311yescharacteristic
153C2×C22⋊Q8—31no—
154C2×C22⋊Q8—31no—
155C2×C22⋊Q8—31no—
156#385—32no—
157#385—32no—
158#385—32no—
159#380—31no—
160#380—31no—
161#380—31no—
162#379—32no—
163C22×C3×C5—42no—
164C22×C3×C5—41no—
165C22×C3×C5—81no—
166C22×C3×C5—81no—
167C5×A4C22⋊Q811yescharacteristic, complemented
168C3×C4×C5—42no—
169C3×C4×C5—81no—
170C5×Dic3—82no—
171C5×Dic3—82no—
172C22×A4—21no—
173C22×A4—21no—
174C22×A4C5×Q812yescomplemented
175C22×A4C23×C511yescharacteristic
176C22×C3×C4—41no—
177C2×Dic6—41no—
178C4×A4—21no—
179C4×A4C5×D412yes—
180SL(2,ℤ/4)—22no—
181SL(2,ℤ/4)—22no—
182#217—42no—
183C4⋊Dic3—41no—
184Dic3⋊C4—42no—
185C23×C5—22no—
186C23×C5—31no—
187C23×C5—31no—
188C23×C5—31no—
189C23×C5—32no—
190C23×C5—31no—
191C23×C5—32no—
192C23×C5—32no—
193C23×C5—32no—
194C23×C5—32no—
195C23×C5—31no—
196C23×C5—61no—
197C23×C5—61no—
198C23×C5—61no—
199C23×C5—61no—
200C23×C5—62no—
201C23×C5—61no—
202C23×C5—61no—
203C23×C5—62no—
204C23×C5—62no—
205C23×C5—61no—
206C23×C5—64no—
207C23×C5C2×Dic611yescharacteristic
208C23×C5#23511yescharacteristic
209C23×C5C2×#6711yescharacteristic
210C23×C5C2×S411yescharacteristic
211C5×Q8—122no—
212C5×Q8—122no—
213C2×C4×C5—22no—
214C2×C4×C5—31no—
215C2×C4×C5—32no—
216C2×C4×C5—32no—
217C2×C4×C5—32no—
218C2×C4×C5—32no—
219C2×C4×C5—31no—
220C2×C4×C5—62no—
221C2×C4×C5—61no—
222C2×C4×C5—61no—
223C2×C4×C5—61no—
224C2×C4×C5—62no—
225C2×C4×C5—64no—
226C2×C4×C5—62no—
227C2×C4×C5—61no—
228C2×C4×C5—62no—
229C2×C4×C5—62no—
230C2×C4×C5—62no—
231C2×C4×C5—62no—
232C2×C4×C5—62no—
233C2×C4×C5—62no—
234C2×C4×C5—62no—
235C2×C4×C5—62no—
236C2×C4×C5—62no—
237C2×C4×C5—62no—
238C2×C4×C5—62no—
239C2×C4×C5—62no—
240C2×C4×C5C2×S411yescharacteristic
241C2×C4×C5C2×S411yescharacteristic
242C25C2×C5×D311yescharacteristic
243C22×Q8—31no—
244C23×C4—22no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—62no—
250C23×C4C2×C5×D311yescharacteristic
251C23×C4C2×C5×D311yescharacteristic
252C22⋊Q8—62no—
253C22⋊Q8—64no—
254C22⋊Q8—61no—
255C22⋊Q8—61no—
256C22⋊Q8—61no—
257C22⋊Q8—62no—
258C22⋊Q8—61no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—32no—
265C2×C22⋊C4—32no—
266C2×C22⋊C4—32no—
267C2×C22⋊C4—32no—
268C2×C22⋊C4—32no—
269C2×C22⋊C4—32no—
270C2×C22⋊C4—32no—
271#95—32no—
272#95—32no—
273#95—32no—
274C2×C3×C5—41no—
275C2×C3×C5—41no—
276C2×C3×C5—41no—
277C2×C3×C5—82no—
278C23×C3—41no—
279C2×A4—22no—
280C2×A4C2×C5×D411yescharacteristic
281C2×A4C2×C5×Q811yescharacteristic
282C2×A4C5×#4111yescharacteristic
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—82no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288Dic6—82no—
289C22×C5—21no—
290C22×C5—21no—
291C22×C5—31no—
292C22×C5—31no—
293C22×C5—31no—
294C22×C5—31no—
295C22×C5—31no—
296C22×C5—31no—
297C22×C5—31no—
298C22×C5—31no—
299C22×C5—31no—
300C22×C5—34no—
301C22×C5—32no—
302C22×C5—62no—
303C22×C5—62no—
304C22×C5—62no—
305C22×C5—62no—
306C22×C5—61no—
307C22×C5—62no—
308C22×C5—62no—
309C22×C5—62no—
310C22×C5—61no—
311C22×C5—61no—
312C22×C5#92411yescharacteristic, complemented
313C22×C5A4⋊Q812yescomplemented
314C22×C5C22×S411yescharacteristic
315C4×C5—21no—
316C4×C5—32no—
317C4×C5—61no—
318C4×C5—122no—
319C4×C5—122no—
320C4×C5—122no—
321C4×C5—122no—
322C4×C5GL(2,ℤ/4)12yes—
323C24—21no—
324C24—21no—
325C24—31no—
326C24—31no—
327C24—32no—
328C24—61no—
329C24—61no—
330C24C5×Dic612yescomplemented
331C24C22×C5×D311yescharacteristic
332C2×Q8—61no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—61no—
336C22×C4—21no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—32no—
341C22×C4—34no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—32no—
347C22×C4—32no—
348C22×C4—32no—
349C22×C4—32no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4C5×#6712yes—
360C22×C4C5×S411yescharacteristic
361C4⋊C4—61no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—64no—
365C4⋊C4—64no—
366C4⋊C4—62no—
367C4⋊C4—62no—
368C4⋊C4—61no—
369C22⋊C4—62no—
370C22⋊C4—64no—
371C22⋊C4—62no—
372C22⋊C4—62no—
373C22⋊C4—64no—
374C22⋊C4—64no—
375C22⋊C4—62no—
376C22⋊C4—62no—
377C22⋊C4—62no—
378C3×C5—41noHall
379C22×C3—42no—
380C22×C3—41no—
381C22×C3—81no—
382C22×C3—81no—
383A4C5×C22⋊Q811yescharacteristic, complemented
384C3×C4—42no—
385C3×C4—81no—
386Dic3—82no—
387Dic3—82no—
388C2×C5—22no—
389C2×C5—31no—
390C2×C5—31no—
391C2×C5—31no—
392C2×C5—31no—
393C2×C5—62no—
394C2×C5C2×A4⋊Q811yescharacteristic
395C2×C5#591211yescharacteristic
396C2×C5C2×GL(2,ℤ/4)11yescharacteristic
397C23—22no—
398C23—31no—
399C23—31no—
400C23—31no—
401C23—32no—
402C23—31no—
403C23—32no—
404C23—32no—
405C23—32no—
406C23—32no—
407C23—31no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23—61no—
412C23—62no—
413C23—61no—
414C23—61no—
415C23—62no—
416C23—62no—
417C23—61no—
418C23—64no—
419C23C2×C5×Dic611yescharacteristic
420C23C5×#23511yescharacteristic
421C23C2×C5×#6711yescharacteristic
422C23C2×C5×S411yescharacteristic
423Q8—122no—
424Q8—122no—
425C2×C4—22no—
426C2×C4—31no—
427C2×C4—32no—
428C2×C4—32no—
429C2×C4—32no—
430C2×C4—32no—
431C2×C4—31no—
432C2×C4—62no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—62no—
437C2×C4—64no—
438C2×C4—62no—
439C2×C4—61no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4C2×C5×S411yescharacteristic
453C2×C4C2×C5×S411yescharacteristic
454C2×C3—41no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—82no—
458C5#8548011yescharacteristic, direct product factor, Sylow
459C22—21no—
460C22—21no—
461C22—31no—
462C22—31no—
463C22—31no—
464C22—31no—
465C22—31no—
466C22—31no—
467C22—31no—
468C22—31no—
469C22—31no—
470C22—34no—
471C22—32no—
472C22—62no—
473C22—62no—
474C22—62no—
475C22—62no—
476C22—61no—
477C22—62no—
478C22—62no—
479C22—62no—
480C22—61no—
481C22—61no—
482C22C5×#92411yescharacteristic, complemented
483C22C5×A4⋊Q812yescomplemented
484C22C22×C5×S411yescharacteristic
485C4—21no—
486C4—32no—
487C4—61no—
488C4—122no—
489C4—122no—
490C4—122no—
491C4—122no—
492C4C5×GL(2,ℤ/4)12yes—
493C3—41noSylow
494C2—22no—
495C2—31no—
496C2—31no—
497C2—31no—
498C2—31no—
499C2—62no—
500C2C2×C5×A4⋊Q811yescharacteristic
501C2C5×#591211yescharacteristic
502C2C2×C5×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC24×C4×C5D3
socleC24×C5C22×D3
prime radicalC23×C5×A4C22
hypercenterC22×C4×C5S4
hypocenterA4C5×C22⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
2. C2×C5×A4⋊Q8C2semidirect
4. C5×C4⋊SL(2,ℤ/4)C2semidirect
5. C5×SL(2,ℤ/4)⋊C4C2semidirect
15. #85480C5direct (N × H)
29. C22×C5×A4Q8semidirect
41. C2×A4⋊Q8C2×C5semidirect
43. C4⋊SL(2,ℤ/4)C2×C5semidirect
44. SL(2,ℤ/4)⋊C4C2×C5semidirect
104. C24×C5Dic6semidirect
167. C5×A4C22⋊Q8semidirect
174. C22×A4C5×Q8semidirect
312. C22×C5#924semidirect
313. C22×C5A4⋊Q8semidirect
330. C24C5×Dic6semidirect
383. A4C5×C22⋊Q8semidirect
458. C5#85480direct (N × H)
482. C22C5×#924semidirect
483. C22C5×A4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C5×A42
C2×C5×A4⋊Q82
C5×#54322
C5×C4⋊SL(2,ℤ/4)2
C5×SL(2,ℤ/4)⋊C42
C5×#28483
C5×#9244
#854805