Finite Group Databasebeta

← C5×#11508 #11934228

Subgroups — C5×#11508 #11934228

Order 1280.

574 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2311C211yesmaximal, characteristic
2C5×#1463C212yesmaximal, complemented
3C5×#1319C212yesmaximal, complemented
4C5×#1318C212yesmaximal, complemented
5C22×C5×#41C411yescharacteristic
6C2×C5×#117C411yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C5×#407—22no—
9C5×#391C411yescharacteristic
10C5×#391C2211yescharacteristic
11C5×#390C411yescharacteristic
12C5×#390C2211yescharacteristic
13C5×#386—22no—
14C5×#380—22no—
15C5×#324—22no—
16C5×#324C2214yes—
17#11508C511yesmaximal, characteristic, direct product factor, Sylow
18C2×C5×#41—22no—
19C2×C5×#41—22no—
20C2×C5×#41—42no—
21C2×C5×#41—42no—
22C22×C5×Q8C2×C411yescharacteristic
23C22×C5×D4—21no—
24C22×C5×D4C2×C411yescharacteristic
25C23×C4×C5—21no—
26C23×C4×C5C2×C411yescharacteristic
27C23×C4×C5D412yes—
28C2×C5×M16—42no—
29C5×#117—42no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—42no—
32C2×C5×C4⋊C4C2×C411yescharacteristic
33C2×C5×C4⋊C4D412yes—
34C2×C5×C4⋊C4C2311yescharacteristic
35C2×C5×C22⋊C4—21no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—22no—
38C2×C42×C5C2×C411yescharacteristic
39C2×C42×C5C2×C411yescharacteristic
40C5×C22⋊C8—24no—
41C5×C22⋊C8—42no—
42C5×#95—22no—
43C5×#95—22no—
44#2311C2×C511yescharacteristic
45#1463C2×C512yescomplemented
46#1319C2×C512yescomplemented
47#1318C2×C512yescomplemented
48C24×C5—21no—
49C24×C5C22⋊C411yescharacteristic
50C5×#41—48no—
51C5×#41—48no—
52C2×C5×Q8—21no—
53C2×C5×Q8—41no—
54C2×C5×Q8—41no—
55C2×C5×Q8M1612yescomplemented
56C2×C5×D4—24no—
57C2×C5×D4—21no—
58C2×C5×D4—41no—
59C2×C5×D4—41no—
60C2×C5×D4—42no—
61C2×C5×D4—42no—
62C2×C5×D4M1612yescomplemented
63C22×C4×C5—22no—
64C22×C4×C5—22no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—24no—
68C22×C4×C5—22no—
69C22×C4×C5—21no—
70C22×C4×C5—22no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—42no—
74C22×C4×C5—42no—
75C22×C4×C5—82no—
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C22⋊C412yes—
78C22×C4×C5C22×C411yescharacteristic
79C22×C4×C5C2×D412yes—
80C5×M16—84no—
81C2×C5×C8—44no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—41no—
85C5×C4⋊C4—42no—
86C5×C22⋊C4—42no—
87C5×C22⋊C4—82no—
88C5×C22⋊C4—82no—
89C42×C5—41no—
90C42×C5—41no—
91C22×#41C4×C511yescharacteristic
92C2×#117C4×C511yescharacteristic
93C22×C4⋊C4C22×C511yescharacteristic
94#407—22no—
95#391C4×C511yescharacteristic
96#391C22×C511yescharacteristic
97#390C4×C511yescharacteristic
98#390C22×C511yescharacteristic
99#386—22no—
100#380—22no—
101#324—22no—
102#324C22×C514yes—
103C23×C5—21no—
104C23×C5—21no—
105C23×C5—22no—
106C23×C5—41no—
107C23×C5—41no—
108C23×C5—42no—
109C23×C5—42no—
110C23×C5—41no—
111C23×C5—41no—
112C23×C5#9912yes—
113C23×C5C2×C22⋊C411yescharacteristic
114C5×Q8—42no—
115C5×Q8—42no—
116C5×D4—42no—
117C5×D4—42no—
118C5×D4—48no—
119C2×C4×C5—24no—
120C2×C4×C5—24no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—22no—
126C2×C4×C5—22no—
127C2×C4×C5—22no—
128C2×C4×C5—48no—
129C2×C4×C5—44no—
130C2×C4×C5—41no—
131C2×C4×C5—41no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—41no—
135C2×C4×C5—41no—
136C2×C4×C5—41no—
137C2×C4×C5—41no—
138C2×C4×C5—41no—
139C2×C4×C5—41no—
140C2×C4×C5—82no—
141C2×C4×C5#10012yes—
142C2×C4×C5C2×M1611yescharacteristic
143C2×C4×C5C2×M1611yescharacteristic
144C5×C8—84no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—42no—
148C2×#41—42no—
149C22×Q8C2×C4×C511yescharacteristic
150C22×D4—21no—
151C22×D4C2×C4×C511yescharacteristic
152C23×C4—21no—
153C23×C4C2×C4×C511yescharacteristic
154C23×C4C5×D412yes—
155C2×M16—42no—
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C2×C4×C511yescharacteristic
160C2×C4⋊C4C5×D412yes—
161C2×C4⋊C4C23×C511yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C42C2×C4×C511yescharacteristic
166C2×C42C2×C4×C511yescharacteristic
167C22⋊C8—24no—
168C22⋊C8—42no—
169#95—22no—
170#95—22no—
171C22×C5—21no—
172C22×C5—21no—
173C22×C5—22no—
174C22×C5—22no—
175C22×C5—22no—
176C22×C5—42no—
177C22×C5—42no—
178C22×C5—44no—
179C22×C5—42no—
180C22×C5—82no—
181C22×C5—81no—
182C22×C5—81no—
183C22×C5#40711yescharacteristic
184C22×C5C2×#9911yescharacteristic
185C22×C5C2×#10011yescharacteristic
186C4×C5—22no—
187C4×C5—21no—
188C4×C5—21no—
189C4×C5—42no—
190C4×C5—42no—
191C4×C5—42no—
192C4×C5—81no—
193C4×C5—81no—
194C24—21no—
195C24C5×C22⋊C411yescharacteristic
196#41—48no—
197#41—48no—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8C5×M1612yescomplemented
202C2×D4—24no—
203C2×D4—21no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4C5×M1612yescomplemented
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—82no—
222C22×C4C5×C22⋊C411yescharacteristic
223C22×C4C5×C22⋊C412yes—
224C22×C4C22×C4×C511yescharacteristic
225C22×C4C2×C5×D412yes—
226M16—84no—
227C2×C8—44no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C42—41no—
236C42—41no—
237C2×C5—21no—
238C2×C5—21no—
239C2×C5—42no—
240C2×C5—82no—
241C2×C5#146211yescharacteristic
242C2×C5#212711yescharacteristic
243C2×C5#213311yescharacteristic
244C23—21no—
245C23—21no—
246C23—22no—
247C23—41no—
248C23—41no—
249C23—42no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23C5×#9912yes—
254C23C2×C5×C22⋊C411yescharacteristic
255Q8—42no—
256Q8—42no—
257D4—42no—
258D4—42no—
259D4—48no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—48no—
270C2×C4—44no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—82no—
282C2×C4C5×#10012yes—
283C2×C4C2×C5×M1611yescharacteristic
284C2×C4C2×C5×M1611yescharacteristic
285C8—84no—
286C5#1150811yescharacteristic, direct product factor, Sylow
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—44no—
295C22—42no—
296C22—82no—
297C22—81no—
298C22—81no—
299C22C5×#40711yescharacteristic
300C22C2×C5×#9911yescharacteristic
301C22C2×C5×#10011yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—82no—
314C2C5×#146211yescharacteristic
315C2C5×#212711yescharacteristic
316C2C5×#213311yescharacteristic
subgroupG/H
center, socleC22×C5C2×#99
commutatorC22×C4C22×C4×C5
Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC22×C5×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C24×C5 ◁ C22×C5×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1463C2semidirect
3. C5×#1319C2semidirect
4. C5×#1318C2semidirect
17. #11508C5direct (N × H)
45. #1463C2×C5semidirect
46. #1319C2×C5semidirect
47. #1318C2×C5semidirect
55. C2×C5×Q8M16semidirect
62. C2×C5×D4M16semidirect
201. C2×Q8C5×M16semidirect
208. C2×D4C5×M16semidirect
286. C5#11508direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#23112
C5×#14632
C5×#13192
C5×#13182
#115085