Finite Group Databasebeta

← C5×#70408 #14302875

Subgroups — C5×#70408 #14302875

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5830C211yesmaximal, characteristic
2C5×#5521C211yesmaximal, characteristic, complemented
3C3×C5×#421C211yesmaximal, characteristic, complemented
4C5×#5236C211yesmaximal, characteristic, complemented
5C2×C5×#823C211yesmaximal, characteristic, complemented
6C5×#4557C211yesmaximal, characteristic, complemented
7C5×#4531C211yesmaximal, characteristic, complemented
8C5×#1911—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C5×D12C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C5×#947C412yescomplemented
14C5×#940C412yescomplemented
15C5×#937—22no—
16C5×#930—22no—
17C5×#926C2211yescharacteristic
18C5×#926C2211yescharacteristic
19C2×C5×#208C2211yescharacteristic
20C5×#893—22no—
21C5×D3×C22⋊C4—22no—
22C4×C5×D12—22no—
23C3×C5×#104—22no—
24C5×#837—22no—
25C5×#823—22no—
26#70408C511yesmaximal, characteristic, direct product factor, Hall
27C5×#535—31no—
28C5×#518—31no—
29C5×#421—31no—
30C5×#421D311yescharacteristic, complemented
31C2×C5×#100—31no—
32C5×H(3,ℤ/4)—31no—
33C5×H(3,ℤ/4)—31no—
34C23×C5×D3D412yes—
35C3×C5×#41—21no—
36C3×C5×#41—21no—
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C2×C3×C5×Q8C2×C411yescharacteristic, complemented
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4C2×C411yescharacteristic, complemented
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2311yescharacteristic
44C2×C5×#67—24no—
45C2×C5×D12—21no—
46C2×C5×D12—21no—
47C2×C5×D12—21no—
48C2×C5×D12—21no—
49C2×C5×D12C2×C412yes—
50C2×C5×D12D412yes—
51C2×C4×C5×D3—24no—
52C3×C5×M16—21no—
53C3×C5×M16—21no—
54C2×C3×C5×C8—21no—
55C3×C5×C4⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C42×C5D412yescomplemented
58C5×#217—21no—
59C5×#217—21no—
60C5×(C2×D3)⋊C4—24no—
61C5×(C2×D3)⋊C4—24no—
62C5×C4⋊Dic3—21no—
63C5×C4⋊Dic3—21no—
64C4×C5×Dic3C2×C412yescomplemented
65C4×C5×Dic3D412yescomplemented
66C5×#208—21no—
67C5×#208—21no—
68C2×C5×C3⋊C8—21no—
69#5830C2×C511yescharacteristic
70#5521C2×C511yescharacteristic, complemented
71C3×#421C2×C511yescharacteristic, complemented
72#5236C2×C511yescharacteristic, complemented
73C2×#823C2×C511yescharacteristic, complemented
74#4557C2×C511yescharacteristic, complemented
75#4531C2×C511yescharacteristic, complemented
76C2×C5×#41C2×D311yescharacteristic
77C22×C5×D4—31no—
78C2×C5×M16—31no—
79C2×C5×M16C2×D311yescharacteristic
80C5×#127—32no—
81C5×#124—32no—
82C5×#121—62no—
83C5×#120—62no—
84C4×C5×D4—62no—
85C4×C5×D4—62no—
86C5×#117—31no—
87C5×#117—31no—
88C5×#117C2×D311yescharacteristic
89C2×C5×C22⋊C4—62no—
90C5×#104—22no—
91C5×#104—62no—
92C5×#100—62no—
93#1911—31noSylow
94C23×C3×C5—21no—
95C23×C3×C5C2×D411yescharacteristic
96C22×C5×D3—22no—
97C22×C5×D3—22no—
98C22×C5×D3—82no—
99C22×C5×D3C22⋊C414yes—
100C3×C5×Q8—21no—
101C3×C5×Q8—21no—
102C3×C5×D4—21no—
103C3×C5×D4—21no—
104C3×C5×D4—41no—
105C3×C5×D4—41no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5C22×C411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×C4×C5C2×D411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5#4111yescharacteristic
115C2×C3×C4×C5#4111yescharacteristic
116C5×#67—84no—
117C2×C5×Dic3—22no—
118C2×C5×Dic3—22no—
119C5×D12—22no—
120C5×D12—22no—
121C5×D12—22no—
122C5×D12—22no—
123C4×C5×D3—84no—
124C3×C5×C8—41no—
125C5×C3⋊C8—41no—
126C2×C3×#41C22×C511yescharacteristic
127C22×D12C22×C511yescharacteristic
128C2×C3×M16C22×C511yescharacteristic
129C3×#117C22×C511yescharacteristic
130#947C4×C512yescomplemented
131#940C4×C512yescomplemented
132#937—22no—
133#930—22no—
134#926C22×C511yescharacteristic
135#926C22×C511yescharacteristic
136C2×#208C22×C511yescharacteristic
137#893—22no—
138D3×C22⋊C4—22no—
139C4×D12—22no—
140C3×#104—22no—
141#837—22no—
142#823—22no—
143C24×C5—32no—
144C5×#41—21no—
145C5×#41—21no—
146C5×#41—21no—
147C5×#41—21no—
148C2×C5×Q8C4×D311yescharacteristic, complemented
149C2×C5×D4—21no—
150C2×C5×D4—32no—
151C2×C5×D4—32no—
152C2×C5×D4—64no—
153C2×C5×D4—61no—
154C2×C5×D4—61no—
155C2×C5×D4—61no—
156C2×C5×D4—61no—
157C2×C5×D4C4×D311yescharacteristic, complemented
158C22×C4×C5—21no—
159C22×C4×C5—64no—
160C22×C4×C5C22×D311yescharacteristic
161C5×M16—21no—
162C5×M16—21no—
163C5×M16—61no—
164C5×M16—61no—
165C2×C5×C8—21no—
166C2×C5×C8—61no—
167C5×C4⋊C4—21no—
168C5×C4⋊C4—61no—
169C5×C4⋊C4—61no—
170C5×C22⋊C4—21no—
171C5×C22⋊C4—64no—
172C5×C22⋊C4—64no—
173C5×C22⋊C4—61no—
174C5×C22⋊C4—61no—
175C42×C5—32no—
176C42×C5—32no—
177C42×C5#6712yescomplemented
178#535—31no—
179#518—31no—
180#421—31no—
181#421C5×D311yescharacteristic, complemented
182C2×#100—31no—
183H(3,ℤ/4)—31no—
184H(3,ℤ/4)—31no—
185C22×C3×C5—41no—
186C22×C3×C5—42no—
187C22×C3×C5—41no—
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C22×C3×C5#12011yescharacteristic
190C22×C3×C5#12311yescharacteristic
191C2×C5×D3—24no—
192C2×C5×D3—44no—
193C2×C5×D3—44no—
194C2×C5×D3—82no—
195C3×C4×C5—42no—
196C3×C4×C5—41no—
197C3×C4×C5C4×D411yescharacteristic
198C3×C4×C5C4×D411yescharacteristic
199C3×C4×C5#12111yescharacteristic
200C3×C4×C5#12111yescharacteristic
201C5×Dic3—42no—
202C5×Dic3—42no—
203C23×D3C5×D412yes—
204C3×#41—21no—
205C3×#41—21no—
206C3×#41—21no—
207C3×#41—21no—
208C2×C3×Q8C2×C4×C511yescharacteristic, complemented
209C2×C3×D4—21no—
210C2×C3×D4C2×C4×C511yescharacteristic, complemented
211C22×C3×C4—21no—
212C22×C3×C4C23×C511yescharacteristic
213C2×#67—24no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12C2×C4×C512yes—
219C2×D12C5×D412yes—
220C2×C4×D3—24no—
221C3×M16—21no—
222C3×M16—21no—
223C2×C3×C8—21no—
224C3×C4⋊C4—21no—
225C3×C22⋊C4—21no—
226C3×C42C5×D412yescomplemented
227#217—21no—
228#217—21no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—24no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4×Dic3C2×C4×C512yescomplemented
234C4×Dic3C5×D412yescomplemented
235#208—21no—
236#208—21no—
237C2×C3⋊C8—21no—
238C23×C5—21no—
239C23×C5—34no—
240C23×C5—62no—
241C23×C5—62no—
242C23×C5—242no—
243C23×C5D3×D411yescharacteristic
244C5×Q8—21no—
245C5×Q8—21no—
246C5×D4—21no—
247C5×D4—21no—
248C5×D4—41no—
249C5×D4—41no—
250C5×D4—62no—
251C5×D4—62no—
252C5×D4—62no—
253C5×D4—62no—
254C5×D4—244no—
255C2×C4×C5—21no—
256C2×C4×C5—22no—
257C2×C4×C5—41no—
258C2×C4×C5—41no—
259C2×C4×C5—62no—
260C2×C4×C5—62no—
261C2×C4×C5—244no—
262C2×C4×C5C2×C4×D311yescharacteristic
263C2×C4×C5#23511yescharacteristic
264C2×C4×C5D3×D411yescharacteristic
265C2×C4×C5D3×D411yescharacteristic
266C2×C4×C5#23711yescharacteristic
267C2×C4×C5C2×#6711yescharacteristic
268C5×C8—41no—
269C5×C8—121no—
270C2×#41C2×C5×D311yescharacteristic
271C22×D4—31no—
272C2×M16—31no—
273C2×M16C2×C5×D311yescharacteristic
274#127—32no—
275#124—32no—
276#121—62no—
277#120—62no—
278C4×D4—62no—
279C4×D4—62no—
280#117—31no—
281#117—31no—
282#117C2×C5×D311yescharacteristic
283C2×C22⋊C4—62no—
284#104—22no—
285#104—62no—
286#100—62no—
287C2×C3×C5—21no—
288C2×C3×C5—21no—
289C2×C3×C5—21no—
290C2×C3×C5—81no—
291C2×C3×C5#38611yescharacteristic
292C5×D3—44no—
293C23×C3—21no—
294C23×C3C2×C5×D411yescharacteristic
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—82no—
298C22×D3C5×C22⋊C414yes—
299C3×Q8—21no—
300C3×Q8—21no—
301C3×D4—21no—
302C3×D4—21no—
303C3×D4—41no—
304C3×D4—41no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4C22×C4×C511yescharacteristic
310C2×C3×C4C2×C5×D411yescharacteristic
311C2×C3×C4C2×C5×D411yescharacteristic
312C2×C3×C4C2×C5×D411yescharacteristic
313C2×C3×C4C5×#4111yescharacteristic
314C2×C3×C4C5×#4111yescharacteristic
315#67—84no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318D12—22no—
319D12—22no—
320D12—22no—
321D12—22no—
322C4×D3—84no—
323C3×C8—41no—
324C3⋊C8—41no—
325C22×C5—41no—
326C22×C5—42no—
327C22×C5—41no—
328C22×C5—64no—
329C22×C5—124no—
330C22×C5—124no—
331C22×C5—242no—
332C22×C5D3×C22⋊C411yescharacteristic
333C22×C5#88311yescharacteristic
334C22×C5#93711yescharacteristic
335C4×C5—42no—
336C4×C5—41no—
337C4×C5—122no—
338C4×C5—122no—
339C4×C5#67⋊C411yescharacteristic
340C4×C5#88411yescharacteristic
341C4×C5C4×#6711yescharacteristic
342C4×C5#93811yescharacteristic
343C24—32no—
344#41—21no—
345#41—21no—
346#41—21no—
347#41—21no—
348C2×Q8C4×C5×D311yescharacteristic, complemented
349C2×D4—21no—
350C2×D4—32no—
351C2×D4—32no—
352C2×D4—64no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4C4×C5×D311yescharacteristic, complemented
358C22×C4—21no—
359C22×C4—64no—
360C22×C4C22×C5×D311yescharacteristic
361M16—21no—
362M16—21no—
363M16—61no—
364M16—61no—
365C2×C8—21no—
366C2×C8—61no—
367C4⋊C4—21no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C22⋊C4—21no—
371C22⋊C4—64no—
372C22⋊C4—64no—
373C22⋊C4—61no—
374C22⋊C4—61no—
375C42—32no—
376C42—32no—
377C42C5×#6712yescomplemented
378C3×C5#191111yescharacteristic, complemented, Hall
379C22×C3—41no—
380C22×C3—42no—
381C22×C3—41no—
382C22×C3C2×C5×C22⋊C411yescharacteristic
383C22×C3C5×#12011yescharacteristic
384C22×C3C5×#12311yescharacteristic
385C2×D3—24no—
386C2×D3—44no—
387C2×D3—44no—
388C2×D3—82no—
389C3×C4—42no—
390C3×C4—41no—
391C3×C4C4×C5×D411yescharacteristic
392C3×C4C4×C5×D411yescharacteristic
393C3×C4C5×#12111yescharacteristic
394C3×C4C5×#12111yescharacteristic
395Dic3—42no—
396Dic3—42no—
397C2×C5—21no—
398C2×C5—21no—
399C2×C5—21no—
400C2×C5—81no—
401C2×C5—124no—
402C2×C5#495611yescharacteristic
403C23—21no—
404C23—34no—
405C23—62no—
406C23—62no—
407C23—242no—
408C23C5×D3×D411yescharacteristic
409Q8—21no—
410Q8—21no—
411D4—21no—
412D4—21no—
413D4—41no—
414D4—41no—
415D4—62no—
416D4—62no—
417D4—62no—
418D4—62no—
419D4—244no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—244no—
427C2×C4C2×C4×C5×D311yescharacteristic
428C2×C4C5×#23511yescharacteristic
429C2×C4C5×D3×D411yescharacteristic
430C2×C4C5×D3×D411yescharacteristic
431C2×C4C5×#23711yescharacteristic
432C2×C4C2×C5×#6711yescharacteristic
433C8—41no—
434C8—121no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—81no—
439C2×C3C5×#38611yescharacteristic
440D3—44no—
441C5#7040811yescharacteristic, direct product factor, Sylow
442C22—41no—
443C22—42no—
444C22—41no—
445C22—64no—
446C22—124no—
447C22—124no—
448C22—242no—
449C22C5×D3×C22⋊C411yescharacteristic
450C22C5×#88311yescharacteristic
451C22C5×#93711yescharacteristic
452C4—42no—
453C4—41no—
454C4—122no—
455C4—122no—
456C4C5×#67⋊C411yescharacteristic
457C4C5×#88411yescharacteristic
458C4C4×C5×#6711yescharacteristic
459C4C5×#93811yescharacteristic
460C3C5×#191111yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—21no—
464C2—81no—
465C2—124no—
466C2C5×#495611yescharacteristic
subgroupG/H
centerC2×C5#4956
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#421C2
socleC2×C3×C5#386
prime radicalC5×#5830C2
cosocleC22×C3×C4C23×C5
hypercenterC5×#421D3
hypocenterC3C5×#1911
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#421

Complements

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

NH ≅ G/Ntype
2. C5×#5521C2semidirect
3. C3×C5×#421C2semidirect
4. C5×#5236C2semidirect
5. C2×C5×#823C2semidirect
6. C5×#4557C2semidirect
7. C5×#4531C2semidirect
13. C5×#947C4semidirect
14. C5×#940C4semidirect
26. #70408C5direct (N × H)
30. C5×#421D3semidirect
39. C2×C3×C5×Q8C2×C4semidirect
41. C2×C3×C5×D4C2×C4semidirect
57. C3×C42×C5D4semidirect
64. C4×C5×Dic3C2×C4semidirect
65. C4×C5×Dic3D4semidirect
70. #5521C2×C5semidirect
71. C3×#421C2×C5semidirect
72. #5236C2×C5semidirect
73. C2×#823C2×C5semidirect
74. #4557C2×C5semidirect
75. #4531C2×C5semidirect
130. #947C4×C5semidirect
131. #940C4×C5semidirect
148. C2×C5×Q8C4×D3semidirect
157. C2×C5×D4C4×D3semidirect
177. C42×C5#67semidirect
181. #421C5×D3semidirect
208. C2×C3×Q8C2×C4×C5semidirect
210. C2×C3×D4C2×C4×C5semidirect
226. C3×C42C5×D4semidirect
233. C4×Dic3C2×C4×C5semidirect
234. C4×Dic3C5×D4semidirect
348. C2×Q8C4×C5×D3semidirect
357. C2×D4C4×C5×D3semidirect
377. C42C5×#67semidirect
378. C3×C5#1911semidirect
441. C5#70408direct (N × H)
460. C3C5×#1911semidirect

Maximal subgroups

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

subgroupindex
C5×#58302
C5×#55212
C3×C5×#4212
C5×#52362
C2×C5×#8232
C5×#45572
C5×#45312
C5×#19113
#704085