Finite Group Databasebeta

← C5×#11727 #11934447

Subgroups — C5×#11727 #11934447

Order 1280.

456 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2395C211yesmaximal, characteristic
2C4×C5×#99C211yesmaximal, characteristic
3C5×#1460C211yesmaximal, characteristic
4C5×#1352C212yesmaximal, complemented
5C5×#1344C212yesmaximal, complemented
6C2×C4×C5×D4C2211yescharacteristic
7C2×C5×#117C411yescharacteristic, complemented
8C5×#431—21no—
9C2×C5×#99—21no—
10C5×#400C411yescharacteristic, complemented
11C5×#399C2211yescharacteristic
12C5×#398C411yescharacteristic, complemented
13C5×#397—21no—
14C5×#397C2211yescharacteristic
15C5×#394—21no—
16C4×C5×C4⋊C4—21no—
17C4×C5×C22⋊C4—21no—
18C4×C5×C22⋊C4—21no—
19C4×C5×C22⋊C4C411yescharacteristic, complemented
20C5×#342C2212yes—
21C5×#323—21no—
22C5×#323C2212yes—
23#11727C511yesmaximal, characteristic, direct product factor, Sylow
24C22×C5×D4D411yescharacteristic
25C23×C4×C5C2×C411yescharacteristic
26C23×C4×C5D411yescharacteristic
27C4×C5×D4—41no—
28C4×C5×D4—41no—
29C5×#117—41no—
30C5×#117—41no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C4⋊C4C2×C411yescharacteristic
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—21no—
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4D411yescharacteristic
42C2×C5×C22⋊C4C2311yescharacteristic
43C2×C42×C5—21no—
44C2×C42×C5—21no—
45C2×C42×C5—41no—
46C2×C42×C5D411yescharacteristic
47C5×C4⋊C8—41no—
48C5×#99—24no—
49C5×C22⋊C8—22no—
50C4×C5×C8—41no—
51C5×#95—21no—
52C5×#95—21no—
53C5×#95—21no—
54C5×#95—21no—
55C5×#95—41no—
56C5×#95C2×C411yescharacteristic
57C5×#95C2×C411yescharacteristic
58#2395C2×C511yescharacteristic
59C4×#99C2×C511yescharacteristic
60#1460C2×C511yescharacteristic
61#1352C2×C512yescomplemented
62#1344C2×C512yescomplemented
63C24×C5C22⋊C411yescharacteristic
64C24×C5C2×D411yescharacteristic
65C2×C5×D4—22no—
66C2×C5×D4—22no—
67C2×C5×D4—41no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—42no—
77C22×C4×C5—42no—
78C22×C4×C5—42no—
79C22×C4×C5C22⋊C411yescharacteristic
80C22×C4×C5C22⋊C411yescharacteristic
81C22×C4×C5C22⋊C411yescharacteristic
82C22×C4×C5C22×C411yescharacteristic
83C22×C4×C5C2×D411yescharacteristic
84C2×C5×C8—42no—
85C5×C4⋊C4—41no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—42no—
89C5×C4⋊C4—42no—
90C5×C22⋊C4—24no—
91C5×C22⋊C4—41no—
92C5×C22⋊C4—41no—
93C5×C22⋊C4—42no—
94C5×C22⋊C4—42no—
95C5×C22⋊C4—42no—
96C5×C22⋊C4—42no—
97C5×C22⋊C4—42no—
98C42×C5—21no—
99C42×C5—21no—
100C42×C5—42no—
101C42×C5—41no—
102C42×C5—41no—
103C42×C5—82no—
104C2×C4×D4C22×C511yescharacteristic
105C2×#117C4×C511yescharacteristic, complemented
106#431—21no—
107C2×#99—21no—
108#400C4×C511yescharacteristic, complemented
109#399C22×C511yescharacteristic
110#398C4×C511yescharacteristic, complemented
111#397—21no—
112#397C22×C511yescharacteristic
113#394—21no—
114C4×C4⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4C4×C511yescharacteristic, complemented
118#342C22×C512yes—
119#323—21no—
120#323C22×C512yes—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—41no—
127C23×C5—41no—
128C23×C5—41no—
129C23×C5—41no—
130C23×C5C2×C22⋊C411yescharacteristic
131C23×C5#13512yes—
132C5×D4—44no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—22no—
136C2×C4×C5—22no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—42no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—42no—
144C2×C4×C5—42no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5—44no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—41no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5—41no—
158C2×C4×C5—81no—
159C2×C4×C5—82no—
160C2×C4×C5—82no—
161C2×C4×C5#9911yescharacteristic
162C2×C4×C5#9911yescharacteristic
163C2×C4×C5#10412yes—
164C5×C8—82no—
165C22×D4C5×D411yescharacteristic
166C23×C4C2×C4×C511yescharacteristic
167C23×C4C5×D411yescharacteristic
168C4×D4—41no—
169C4×D4—41no—
170#117—41no—
171#117—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C2×C4×C511yescharacteristic
175C2×C4⋊C4C2×C4×C511yescharacteristic
176C2×C4⋊C4C2×C4×C511yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4C5×D411yescharacteristic
183C2×C22⋊C4C23×C511yescharacteristic
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—41no—
187C2×C42C5×D411yescharacteristic
188C4⋊C8—41no—
189#99—24no—
190C22⋊C8—22no—
191C4×C8—41no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—41no—
197#95C2×C4×C511yescharacteristic
198#95C2×C4×C511yescharacteristic
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—22no—
202C22×C5—42no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5—81no—
209C22×C5—81no—
210C22×C5—81no—
211C22×C5C2×#9911yescharacteristic
212C22×C5#41611yescharacteristic
213C22×C5C2×#10411yescharacteristic
214C4×C5—21no—
215C4×C5—22no—
216C4×C5—42no—
217C4×C5—41no—
218C4×C5—41no—
219C4×C5—41no—
220C4×C5—81no—
221C4×C5—84no—
222C4×C5—81no—
223C4×C5—81no—
224C4×C5#35212yes—
225C24C5×C22⋊C411yescharacteristic
226C24C2×C5×D411yescharacteristic
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C5×C22⋊C411yescharacteristic
242C22×C4C5×C22⋊C411yescharacteristic
243C22×C4C5×C22⋊C411yescharacteristic
244C22×C4C22×C4×C511yescharacteristic
245C22×C4C2×C5×D411yescharacteristic
246C2×C8—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C22⋊C4—24no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C42—21no—
261C42—21no—
262C42—42no—
263C42—41no—
264C42—41no—
265C42—82no—
266C2×C5—21no—
267C2×C5—21no—
268C2×C5—42no—
269C2×C5—81no—
270C2×C5#150111yescharacteristic
271C2×C5C2×#35211yescharacteristic
272C2×C5#212611yescharacteristic
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23C2×C5×C22⋊C411yescharacteristic
283C23C5×#13512yes—
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4C5×#9911yescharacteristic
314C2×C4C5×#9911yescharacteristic
315C2×C4C5×#10412yes—
316C8—82no—
317C5#1172711yescharacteristic, direct product factor, Sylow
318C22—21no—
319C22—21no—
320C22—22no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C2×C5×#9911yescharacteristic
331C22C5×#41611yescharacteristic
332C22C2×C5×#10411yescharacteristic
333C4—22no—
334C4—21no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—84no—
341C4—81no—
342C4—81no—
343C4C5×#35212yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2C5×#150111yescharacteristic
349C2C2×C5×#35211yescharacteristic
350C2C5×#212611yescharacteristic
subgroupG/H
centerC2×C4×C5#99
commutatorC22×C4C22×C4×C5
Frattini, cosocleC2×C22⋊C4C23×C5
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C5×#1352C2semidirect
5. C5×#1344C2semidirect
7. C2×C5×#117C4semidirect
10. C5×#400C4semidirect
12. C5×#398C4semidirect
19. C4×C5×C22⋊C4C4semidirect
23. #11727C5direct (N × H)
61. #1352C2×C5semidirect
62. #1344C2×C5semidirect
105. C2×#117C4×C5semidirect
108. #400C4×C5semidirect
110. #398C4×C5semidirect
117. C4×C22⋊C4C4×C5semidirect
317. C5#11727direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#23952
C4×C5×#992
C5×#14602
C5×#13522
C5×#13442
#117275