Finite Group Databasebeta

← C5×#85308 #14511046

Subgroups — C5×#85308 #14511046

Order 1920.

854 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 108 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×C5×C4⋊S4C211yesmaximal, characteristic
2C2×C4×C5×S4C212yesmaximal, complemented
3C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C5×(C2×S4)⋊C4C212yesmaximal, complemented
5C4×C5×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6C5×#2361—31nomaximal, Hall
7C22×C5×S4C2212yes—
8C2×C4×C5×A4C2212yes—
9C2×C4×C5×A4C2211yescharacteristic
10C2×C5×SL(2,ℤ/4)C2211yescharacteristic
11C2×C5×SL(2,ℤ/4)C2211yescharacteristic
12C5×C4⋊S4C414yescomplemented
13C4×C5×S4—24no—
14C5×#893—41nomaximal
15#85308C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×#121—31no—
17C2×C4×C5×D4—31no—
18C2×C4×C5×D4—32no—
19C22×C5×C4⋊C4D311yescharacteristic, complemented
20C5×#390—31no—
21C5×#389—31no—
22C5×#388—32no—
23C5×#386—32no—
24C5×#385—32no—
25C4×C5×C4⋊C4—31no—
26C4×C5×C22⋊C4—31no—
27C22×C5×A4C2311yescharacteristic
28C2×C5×S4—22no—
29C2×C5×S4—22no—
30C2×C5×S4C2×C414yescomplemented
31C2×C5×D12—41no—
32C2×C4×C5×D3—42no—
33C4×C5×A4—22no—
34C4×C5×A4C2×C412yescomplemented
35C5×SL(2,ℤ/4)—21no—
36C5×SL(2,ℤ/4)D412yescomplemented
37C3×C5×C4⋊C4—41no—
38C5×(C2×D3)⋊C4—42no—
39C4×C5×Dic3—41no—
40C2×C4⋊S4C2×C511yescharacteristic
41C2×C4×S4C2×C512yescomplemented
42A4×C4⋊C4C2×C511yescharacteristic, complemented
43(C2×S4)⋊C4C2×C512yescomplemented
44C4×SL(2,ℤ/4)C2×C511yescharacteristic, complemented
45C22×C5×D4—31no—
46C22×C5×D4—32no—
47C23×C4×C5—32no—
48C23×C4×C5C2×D312yes—
49C23×C4×C5C2×D311yescharacteristic
50C5×#121—34no—
51C5×#121—34no—
52C4×C5×D4—64no—
53C4×C5×D4—64no—
54C4×C5×D4—62no—
55C4×C5×D4—61no—
56C4×C5×D4—61no—
57C2×C5×C4⋊C4—31no—
58C2×C5×C4⋊C4—31no—
59C2×C5×C4⋊C4—32no—
60C2×C5×C4⋊C4—31no—
61C2×C5×C4⋊C4—32no—
62C2×C5×C22⋊C4—32no—
63C2×C5×C22⋊C4—31no—
64C2×C5×C22⋊C4—32no—
65C2×C5×C22⋊C4—32no—
66C2×C5×C22⋊C4—31no—
67C2×C5×C22⋊C4—32no—
68C2×C42×C5—31no—
69C2×C42×C5—31no—
70C2×C42×C5—32no—
71C5×#95—31no—
72C5×#95—31no—
73C5×#95—32no—
74C5×#95—32no—
75#2361—31noSylow
76C22×C5×D3—42no—
77C2×C5×A4C22×C411yescharacteristic
78C2×C5×A4C2×D411yescharacteristic
79C2×C5×A4#4111yescharacteristic
80C5×S4—24no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—42no—
83C2×C5×Dic3—41no—
84C2×C5×Dic3—41no—
85C5×D12—44no—
86C4×C5×D3—84no—
87C22×S4C22×C512yes—
88C2×C4×A4C22×C512yes—
89C2×C4×A4C22×C511yescharacteristic
90C2×SL(2,ℤ/4)C22×C511yescharacteristic
91C2×SL(2,ℤ/4)C22×C511yescharacteristic
92C4⋊S4C4×C514yescomplemented
93C4×S4—24no—
94#893—41no—
95C24×C5—32no—
96C24×C5C22×D311yescharacteristic
97C2×C5×D4—34no—
98C2×C5×D4—34no—
99C2×C5×D4—34no—
100C2×C5×D4—61no—
101C2×C5×D4—62no—
102C2×C5×D4—62no—
103C2×C5×D4—62no—
104C2×C5×D4—62no—
105C2×C5×D4—62no—
106C2×C5×D4—61no—
107C22×C4×C5—22no—
108C22×C4×C5—31no—
109C22×C4×C5—31no—
110C22×C4×C5—32no—
111C22×C4×C5—31no—
112C22×C4×C5—31no—
113C22×C4×C5—32no—
114C22×C4×C5—32no—
115C22×C4×C5—32no—
116C22×C4×C5—31no—
117C22×C4×C5—31no—
118C22×C4×C5—64no—
119C22×C4×C5—62no—
120C22×C4×C5—62no—
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—64no—
129C5×C4⋊C4S411yescharacteristic, complemented
130C5×C22⋊C4—32no—
131C5×C22⋊C4—38no—
132C5×C22⋊C4—32no—
133C5×C22⋊C4—64no—
134C5×C22⋊C4—62no—
135C5×C22⋊C4—61no—
136C5×C22⋊C4—61no—
137C5×C22⋊C4—62no—
138C5×C22⋊C4—122no—
139C42×C5—62no—
140C42×C5—61no—
141C42×C5—64no—
142C42×C5—61no—
143C2×#121—31no—
144C2×C4×D4—31no—
145C2×C4×D4—32no—
146C22×C4⋊C4C5×D311yescharacteristic, complemented
147#390—31no—
148#389—31no—
149#388—32no—
150#386—32no—
151#385—32no—
152C4×C4⋊C4—31no—
153C4×C22⋊C4—31no—
154C22×C3×C5—41no—
155C2×C5×D3—44no—
156C2×C5×D3—82no—
157C2×C5×D3—82no—
158C5×A4C4×D411yescharacteristic, complemented
159C3×C4×C5—42no—
160C3×C4×C5—82no—
161C5×Dic3—42no—
162C5×Dic3—81no—
163C22×A4C23×C511yescharacteristic
164C2×S4—22no—
165C2×S4—22no—
166C2×S4C2×C4×C514yescomplemented
167C2×D12—41no—
168C2×C4×D3—42no—
169C4×A4—22no—
170C4×A4C2×C4×C512yescomplemented
171SL(2,ℤ/4)—21no—
172SL(2,ℤ/4)C5×D412yescomplemented
173C3×C4⋊C4—41no—
174(C2×D3)⋊C4—42no—
175C4×Dic3—41no—
176C23×C5—31no—
177C23×C5—31no—
178C23×C5—31no—
179C23×C5—31no—
180C23×C5—34no—
181C23×C5—62no—
182C23×C5—62no—
183C23×C5—62no—
184C23×C5—62no—
185C23×C5—62no—
186C23×C5—62no—
187C23×C5C2×C4×D311yescharacteristic
188C23×C5D3×D411yescharacteristic
189C23×C5#23911yescharacteristic
190C5×D4—64no—
191C5×D4—64no—
192C5×D4—64no—
193C5×D4—64no—
194C5×D4—64no—
195C5×D4—64no—
196C2×C4×C5—31no—
197C2×C4×C5—32no—
198C2×C4×C5—32no—
199C2×C4×C5—32no—
200C2×C4×C5—32no—
201C2×C4×C5—34no—
202C2×C4×C5—32no—
203C2×C4×C5—32no—
204C2×C4×C5—62no—
205C2×C4×C5—62no—
206C2×C4×C5—61no—
207C2×C4×C5—62no—
208C2×C4×C5—61no—
209C2×C4×C5—61no—
210C2×C4×C5—61no—
211C2×C4×C5—62no—
212C2×C4×C5—61no—
213C2×C4×C5—62no—
214C2×C4×C5—62no—
215C2×C4×C5—61no—
216C2×C4×C5—61no—
217C2×C4×C5—61no—
218C2×C4×C5—61no—
219C2×C4×C5—61no—
220C2×C4×C5—61no—
221C2×C4×C5—124no—
222C2×C4×C5—124no—
223C2×C4×C5C2×S412yes—
224C2×C4×C5C2×S411yescharacteristic
225C22×D4—31no—
226C22×D4—32no—
227C23×C4—32no—
228C23×C4C2×C5×D312yes—
229C23×C4C2×C5×D311yescharacteristic
230#121—34no—
231#121—34no—
232C4×D4—64no—
233C4×D4—64no—
234C4×D4—62no—
235C4×D4—61no—
236C4×D4—61no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—32no—
242C2×C22⋊C4—32no—
243C2×C22⋊C4—31no—
244C2×C22⋊C4—32no—
245C2×C22⋊C4—32no—
246C2×C22⋊C4—31no—
247C2×C22⋊C4—32no—
248C2×C42—31no—
249C2×C42—32no—
250C2×C42—31no—
251#95—31no—
252#95—31no—
253#95—32no—
254#95—32no—
255C2×C3×C5—41no—
256C2×C3×C5—41no—
257C2×C3×C5—41no—
258C5×D3—84no—
259C22×D3—42no—
260C2×A4C22×C4×C511yescharacteristic
261C2×A4C2×C5×D411yescharacteristic
262C2×A4C5×#4111yescharacteristic
263S4—24no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268D12—44no—
269C4×D3—84no—
270C22×C5—31no—
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—61no—
280C22×C5—64no—
281C22×C5—64no—
282C22×C5—64no—
283C22×C5—122no—
284C22×C5—122no—
285C22×C5—122no—
286C22×C5—122no—
287C22×C5#89311yescharacteristic, complemented
288C22×C5C22×S411yescharacteristic
289C4×C5—22no—
290C4×C5—32no—
291C4×C5—62no—
292C4×C5—64no—
293C4×C5—62no—
294C4×C5—62no—
295C4×C5—121no—
296C4×C5—121no—
297C4×C5C4×S412yescomplemented
298C24—32no—
299C24C22×C5×D311yescharacteristic
300C2×D4—34no—
301C2×D4—34no—
302C2×D4—34no—
303C2×D4—61no—
304C2×D4—62no—
305C2×D4—62no—
306C2×D4—62no—
307C2×D4—62no—
308C2×D4—62no—
309C2×D4—61no—
310C22×C4—22no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—32no—
314C22×C4—32no—
315C22×C4—31no—
316C22×C4—31no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—31no—
320C22×C4—31no—
321C22×C4—64no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—122no—
326C22×C4C4×C5×D312yescomplemented
327C4⋊C4—34no—
328C4⋊C4—32no—
329C4⋊C4—31no—
330C4⋊C4—61no—
331C4⋊C4—64no—
332C4⋊C4C5×S411yescharacteristic, complemented
333C22⋊C4—32no—
334C22⋊C4—38no—
335C22⋊C4—32no—
336C22⋊C4—64no—
337C22⋊C4—62no—
338C22⋊C4—61no—
339C22⋊C4—61no—
340C22⋊C4—62no—
341C22⋊C4—122no—
342C42—61no—
343C42—64no—
344C42—62no—
345C42—61no—
346C3×C5—41noHall
347C22×C3—41no—
348C2×D3—44no—
349C2×D3—82no—
350C2×D3—82no—
351A4C4×C5×D411yescharacteristic, complemented
352C3×C4—42no—
353C3×C4—82no—
354Dic3—42no—
355Dic3—81no—
356C2×C5—31no—
357C2×C5—31no—
358C2×C5—31no—
359C2×C5—31no—
360C2×C5—124no—
361C2×C5C2×C4×S411yescharacteristic
362C2×C5D4×S411yescharacteristic
363C2×C5#591911yescharacteristic
364C23—31no—
365C23—31no—
366C23—31no—
367C23—31no—
368C23—34no—
369C23—62no—
370C23—62no—
371C23—62no—
372C23—62no—
373C23—62no—
374C23—62no—
375C23C2×C4×C5×D311yescharacteristic
376C23C5×D3×D411yescharacteristic
377C23C5×#23911yescharacteristic
378D4—64no—
379D4—64no—
380D4—64no—
381D4—64no—
382D4—64no—
383D4—64no—
384C2×C4—31no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—32no—
389C2×C4—34no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—61no—
395C2×C4—62no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—62no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—62no—
402C2×C4—62no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—124no—
410C2×C4—124no—
411C2×C4C2×C5×S412yes—
412C2×C4C2×C5×S411yescharacteristic
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—41no—
416D3—84no—
417C5#8530811yescharacteristic, direct product factor, Sylow
418C22—31no—
419C22—31no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—31no—
427C22—61no—
428C22—64no—
429C22—64no—
430C22—64no—
431C22—122no—
432C22—122no—
433C22—122no—
434C22—122no—
435C22C5×#89311yescharacteristic, complemented
436C22C22×C5×S411yescharacteristic
437C4—22no—
438C4—32no—
439C4—64no—
440C4—62no—
441C4—62no—
442C4—62no—
443C4—121no—
444C4—121no—
445C4C4×C5×S412yescomplemented
446C3—41noSylow
447C2—31no—
448C2—31no—
449C2—31no—
450C2—31no—
451C2—124no—
452C2C2×C4×C5×S411yescharacteristic
453C2C5×D4×S411yescharacteristic
454C2C5×#591911yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutatorC2×A4C22×C4×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5C22×D3
prime radicalC2×C5×C4⋊S4C2
cosocleC22×A4C23×C5
hypercenterC5×C4⋊C4S4
hypocenterA4C4×C5×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C5×S4C2semidirect
3. C5×A4×C4⋊C4C2semidirect
4. C5×(C2×S4)⋊C4C2semidirect
5. C4×C5×SL(2,ℤ/4)C2semidirect
12. C5×C4⋊S4C4semidirect
15. #85308C5direct (N × H)
19. C22×C5×C4⋊C4D3semidirect
30. C2×C5×S4C2×C4semidirect
34. C4×C5×A4C2×C4semidirect
36. C5×SL(2,ℤ/4)D4semidirect
41. C2×C4×S4C2×C5semidirect
42. A4×C4⋊C4C2×C5semidirect
43. (C2×S4)⋊C4C2×C5semidirect
44. C4×SL(2,ℤ/4)C2×C5semidirect
92. C4⋊S4C4×C5semidirect
123. C22×C4×C5C4×D3semidirect
129. C5×C4⋊C4S4semidirect
146. C22×C4⋊C4C5×D3semidirect
158. C5×A4C4×D4semidirect
166. C2×S4C2×C4×C5semidirect
170. C4×A4C2×C4×C5semidirect
172. SL(2,ℤ/4)C5×D4semidirect
287. C22×C5#893semidirect
297. C4×C5C4×S4semidirect
326. C22×C4C4×C5×D3semidirect
332. C4⋊C4C5×S4semidirect
351. A4C4×C5×D4semidirect
417. C5#85308direct (N × H)
435. C22C5×#893semidirect
445. C4C4×C5×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C4⋊S42
C2×C4×C5×S42
C5×A4×C4⋊C42
C5×(C2×S4)⋊C42
C4×C5×SL(2,ℤ/4)2
C5×#23613
C5×#8934
#853085