Finite Group Databasebeta

← C5×#11760 #11934480

Subgroups — C5×#11760 #11934480

Order 1280.

436 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2611C211yesmaximal, characteristic
2C5×#1740C211yesmaximal, characteristic
3C5×#1464C211yesmaximal, characteristic
4C5×#1352C212yesmaximal
5C5×#1344C212yesmaximal
6C2×C5×#125C411yescharacteristic, complemented
7C2×C5×#117—21no—
8C2×C5×#117C2211yescharacteristic
9C5×#432—21no—
10C5×#400C411yescharacteristic, complemented
11C5×#399C2211yescharacteristic
12C5×#398C411yescharacteristic, complemented
13C5×#397C2211yescharacteristic
14C5×#389—21no—
15C5×#388—21no—
16C5×#387—21no—
17C5×#385—21no—
18C5×#376C411yescharacteristic, complemented
19C5×#342—21no—
20C5×#342C2212yes—
21C5×#324—21no—
22C5×#323C2212yes—
23#11760C511yesmaximal, characteristic, direct product factor, Sylow
24C23×C4×C5D411yescharacteristic
25C5×#125—41no—
26C5×#125—41no—
27C5×#117—24no—
28C5×#117—41no—
29C5×#117—41no—
30C5×#117—42no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—21no—
35C2×C5×C4⋊C4—41no—
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C4⋊C4C2×C411yescharacteristic
38C2×C5×C4⋊C4C2×C411yescharacteristic
39C2×C5×C4⋊C4D411yescharacteristic
40C2×C5×C22⋊C4—22no—
41C2×C5×C22⋊C4—21no—
42C2×C5×C22⋊C4—21no—
43C2×C5×C22⋊C4D411yescharacteristic
44C2×C5×C22⋊C4C2311yescharacteristic
45C2×C42×C5—21no—
46C2×C42×C5—41no—
47C2×C42×C5C2×C411yescharacteristic
48C2×C42×C5D411yescharacteristic
49C5×C4⋊C8—41no—
50C5×C22⋊C8—22no—
51C5×#97—41no—
52C5×#95—21no—
53C5×#95—21no—
54C5×#95—21no—
55C5×#95—21no—
56C5×#95—21no—
57C5×#95C2×C411yescharacteristic
58C5×#95C2×C411yescharacteristic
59#2611C2×C511yescharacteristic
60#1740C2×C511yescharacteristic
61#1464C2×C511yescharacteristic
62#1352C2×C512yes—
63#1344C2×C512yes—
64C24×C5C2×D411yescharacteristic
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—22no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5—42no—
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5C22⋊C411yescharacteristic
79C22×C4×C5C22×C411yescharacteristic
80C22×C4×C5C2×D411yescharacteristic
81C2×C5×C8—42no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—41no—
84C5×C4⋊C4—41no—
85C5×C4⋊C4—41no—
86C5×C4⋊C4—42no—
87C5×C4⋊C4—42no—
88C5×C4⋊C4—44no—
89C5×C4⋊C4—41no—
90C5×C4⋊C4—41no—
91C5×C4⋊C4—42no—
92C5×C22⋊C4—24no—
93C5×C22⋊C4—41no—
94C5×C22⋊C4—41no—
95C5×C22⋊C4—42no—
96C5×C22⋊C4—41no—
97C5×C22⋊C4—42no—
98C5×C22⋊C4—41no—
99C42×C5—21no—
100C42×C5—21no—
101C42×C5—21no—
102C42×C5—21no—
103C42×C5—42no—
104C42×C5—44no—
105C2×#125C4×C511yescharacteristic, complemented
106C2×#117—21no—
107C2×#117C22×C511yescharacteristic
108#432—21no—
109#400C4×C511yescharacteristic, complemented
110#399C22×C511yescharacteristic
111#398C4×C511yescharacteristic, complemented
112#397C22×C511yescharacteristic
113#389—21no—
114#388—21no—
115#387—21no—
116#385—21no—
117#376C4×C511yescharacteristic, complemented
118#342—21no—
119#342C22×C512yes—
120#324—21no—
121#323C22×C512yes—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5C2×C22⋊C411yescharacteristic
127C23×C5AGL(1,ℤ/8)11yescharacteristic
128C23×C5#13711yescharacteristic
129C2×C4×C5—22no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—22no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—41no—
148C2×C4×C5—44no—
149C2×C4×C5—41no—
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—82no—
159C2×C4×C5—82no—
160C2×C4×C5#9911yescharacteristic
161C2×C4×C5#9911yescharacteristic
162C2×C4×C5#10412yes—
163C5×C8—82no—
164C23×C4C5×D411yescharacteristic
165#125—41no—
166#125—41no—
167#117—24no—
168#117—41no—
169#117—41no—
170#117—42no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4C2×C4×C511yescharacteristic
177C2×C4⋊C4C2×C4×C511yescharacteristic
178C2×C4⋊C4C2×C4×C511yescharacteristic
179C2×C4⋊C4C5×D411yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4C5×D411yescharacteristic
184C2×C22⋊C4C23×C511yescharacteristic
185C2×C42—21no—
186C2×C42—41no—
187C2×C42C2×C4×C511yescharacteristic
188C2×C42C5×D411yescharacteristic
189C4⋊C8—41no—
190C22⋊C8—22no—
191#97—41no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95C2×C4×C511yescharacteristic
198#95C2×C4×C511yescharacteristic
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—21no—
202C22×C5—21no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—81no—
208C22×C5C2×#9911yescharacteristic
209C22×C5#41711yescharacteristic
210C22×C5C2×#10411yescharacteristic
211C4×C5—21no—
212C4×C5—21no—
213C4×C5—22no—
214C4×C5—42no—
215C4×C5—41no—
216C4×C5—41no—
217C4×C5—41no—
218C4×C5—41no—
219C4×C5—81no—
220C4×C5—84no—
221C4×C5—81no—
222C4×C5—81no—
223C24C2×C5×D411yescharacteristic
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4C5×C22⋊C411yescharacteristic
235C22×C4C5×C22⋊C411yescharacteristic
236C22×C4C5×C22⋊C411yescharacteristic
237C22×C4C5×C22⋊C411yescharacteristic
238C22×C4C22×C4×C511yescharacteristic
239C22×C4C2×C5×D411yescharacteristic
240C2×C8—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C22⋊C4—24no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—42no—
263C42—44no—
264C2×C5—21no—
265C2×C5—21no—
266C2×C5—41no—
267C2×C5—41no—
268C2×C5#150411yescharacteristic
269C2×C5#212311yescharacteristic
270C2×C5#212611yescharacteristic
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23C2×C5×C22⋊C411yescharacteristic
276C23C5×AGL(1,ℤ/8)11yescharacteristic
277C23C5×#13711yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4C5×#9911yescharacteristic
310C2×C4C5×#9911yescharacteristic
311C2×C4C5×#10412yes—
312C8—82no—
313C5#1176011yescharacteristic, direct product factor, Sylow
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×C5×#9911yescharacteristic
324C22C5×#41711yescharacteristic
325C22C2×C5×#10411yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—42no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—84no—
336C4—81no—
337C4—81no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2C5×#150411yescharacteristic
343C2C5×#212311yescharacteristic
344C2C5×#212611yescharacteristic
subgroupG/H
center, socleC22×C5C2×#99
commutatorC22×C4C22×C4×C5
Frattini, cosocleC2×C22⋊C4C23×C5
prime radicalC24×C5C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#117 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C5×#125C4semidirect
10. C5×#400C4semidirect
12. C5×#398C4semidirect
18. C5×#376C4semidirect
23. #11760C5direct (N × H)
105. C2×#125C4×C5semidirect
109. #400C4×C5semidirect
111. #398C4×C5semidirect
117. #376C4×C5semidirect
313. C5#11760direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#26112
C5×#17402
C5×#14642
C5×#13522
C5×#13442
#117605