Finite Group Databasebeta

← C5×D5⋊#104 #13104552

Subgroups — C5×D5⋊#104 #13104552

Order 1600.

360 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 100 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×D5×#41C211yesmaximal, characteristic
2C2×C4×C5×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3C5×D5⋊M16C211yesmaximal, characteristic, complemented
4C5×#4010C212yesmaximal, complemented
5C5×#4008C212yesmaximal, complemented
6C22×C5×AGL(1,𝔽5)—21no—
7C52×#41C411yescharacteristic, complemented
8C5×#713C2211yescharacteristic
9C5×Q8×D5C411yescharacteristic, complemented
10C5×#711—21no—
11C5×#711C2211yescharacteristic
12C5×D4×D5—21no—
13C5×D4×D5C411yescharacteristic, complemented
14C5×#709—21no—
15C5×#709C411yescharacteristic, complemented
16C2×C4×C5×D5—21no—
17C2×C4×C5×D5C2211yescharacteristic
18C5×#704—21no—
19C4×C5×AGL(1,𝔽5)—21no—
20C4×C5×AGL(1,𝔽5)C2212yescomplemented
21C5×#700C2212yescomplemented
22C5×D5⋊C8—21no—
23D5⋊#104C511yesmaximal, characteristic, direct product factor
24C2×C5×#104—51nomaximal
25C22×C5×D5—21no—
26C22×C5×D5D411yescharacteristic
27C2×C5×AGL(1,𝔽5)—24no—
28C2×C5×AGL(1,𝔽5)—22no—
29C52×Q8C2×C411yescharacteristic, complemented
30C52×D4—21no—
31C52×D4C2×C411yescharacteristic, complemented
32C2×C4×C52—21no—
33C2×C4×C52C2×C411yescharacteristic
34C5×C22⋊D5—21no—
35C5×C22⋊D5—21no—
36C5×C22⋊D5—21no—
37C2×C5×Dic5—21no—
38C2×C5×Dic5D411yescharacteristic
39C5×D20—21no—
40C5×D20C2×C411yescharacteristic, complemented
41C4×C5×D5—21no—
42C4×C5×D5—21no—
43C4×C5×D5—21no—
44C4×C5×D5C2×C411yescharacteristic
45C4×C5×D5D411yescharacteristic
46C4×C5×D5D411yescharacteristic
47C4×C5×D5C2311yescharacteristic
48C5×Dic10—21no—
49C5×Dic10C2×C411yescharacteristic, complemented
50C5×#170—22no—
51C2×C5×#41—51no—
52D5×#41C2×C511yescharacteristic
53C2×C4×AGL(1,𝔽5)C2×C511yescharacteristic, complemented
54D5⋊M16C2×C511yescharacteristic, complemented
55C2×C5×M16—51no—
56C2×C42×C5—51no—
57#4010C2×C512yescomplemented
58#4008C2×C512yescomplemented
59C5×#104—52no—
60C5×#104—52no—
61C22×C52—21no—
62C22×C52C22⋊C411yescharacteristic
63C2×C5×D5—22no—
64C2×C5×D5—21no—
65C2×C5×D5—42no—
66C2×C5×D5C22⋊C411yescharacteristic
67C2×C5×D5C2×D411yescharacteristic
68C5×AGL(1,𝔽5)—24no—
69C4×C52—21no—
70C4×C52C22⋊C411yescharacteristic
71C4×C52C22×C411yescharacteristic
72C5×Dic5—21no—
73C5×Dic5C22⋊C411yescharacteristic
74C5×Dic5C2×D411yescharacteristic
75C22×AGL(1,𝔽5)—21no—
76C5×#41—41no—
77C5×#41—51no—
78C5×#41—51no—
79C5×#41—51no—
80C5×#41—101no—
81C5×#41—101no—
82C5×#41C4×C511yescharacteristic, complemented
83C5×#41AGL(1,𝔽5)11yescharacteristic, complemented
84C2×C5×Q8—51no—
85C2×C5×D4—51no—
86C2×C5×D4—101no—
87C22×C4×C5—51no—
88C22×C4×C5—101no—
89C22×C4×C5—101no—
90#713C22×C511yescharacteristic
91Q8×D5C4×C511yescharacteristic, complemented
92#711—21no—
93#711C22×C511yescharacteristic
94D4×D5—21no—
95D4×D5C4×C511yescharacteristic, complemented
96#709—21no—
97#709C4×C511yescharacteristic, complemented
98C2×C4×D5—21no—
99C2×C4×D5C22×C511yescharacteristic
100#704—21no—
101C4×AGL(1,𝔽5)—21no—
102C4×AGL(1,𝔽5)C22×C512yescomplemented
103#700C22×C512yescomplemented
104D5⋊C8—21no—
105C5×M16—52no—
106C5×M16—101no—
107C2×C5×C8—101no—
108C42×C5—52no—
109C42×C5—101no—
110C2×#104—51noSylow
111C2×C52—21no—
112C2×C52—41no—
113C2×C52C2×C22⋊C411yescharacteristic
114C5×D5—21no—
115C5×D5—41no—
116C5×D5#10412yescomplemented
117C23×C5—51no—
118C23×C5—101no—
119C22×D5—21no—
120C22×D5C5×D411yescharacteristic
121C2×AGL(1,𝔽5)—24no—
122C2×AGL(1,𝔽5)—22no—
123C5×Q8—41no—
124C5×Q8—51no—
125C5×Q8—101no—
126C5×Q8C2×C4×C511yescharacteristic, complemented
127C5×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
128C5×D4—21no—
129C5×D4—21no—
130C5×D4—42no—
131C5×D4—41no—
132C5×D4—51no—
133C5×D4—101no—
134C5×D4—101no—
135C5×D4—101no—
136C5×D4—101no—
137C5×D4C2×C4×C511yescharacteristic, complemented
138C5×D4C2×AGL(1,𝔽5)11yescharacteristic, complemented
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—42no—
142C2×C4×C5—41no—
143C2×C4×C5—51no—
144C2×C4×C5—51no—
145C2×C4×C5—51no—
146C2×C4×C5—51no—
147C2×C4×C5—51no—
148C2×C4×C5—101no—
149C2×C4×C5—101no—
150C2×C4×C5—101no—
151C2×C4×C5—102no—
152C2×C4×C5—104no—
153C2×C4×C5—101no—
154C2×C4×C5C2×C4×C511yescharacteristic
155C2×C4×C5C2×AGL(1,𝔽5)11yescharacteristic
156C22⋊D5—21no—
157C22⋊D5—21no—
158C22⋊D5—21no—
159C2×Dic5—21no—
160C2×Dic5C5×D411yescharacteristic
161D20—21no—
162D20C2×C4×C511yescharacteristic, complemented
163C4×D5—21no—
164C4×D5—21no—
165C4×D5—21no—
166C4×D5C2×C4×C511yescharacteristic
167C4×D5C5×D411yescharacteristic
168C4×D5C5×D411yescharacteristic
169C4×D5C23×C511yescharacteristic
170Dic10—21no—
171Dic10C2×C4×C511yescharacteristic, complemented
172#170—22no—
173C5×C8—102no—
174C2×#41—51no—
175C2×M16—51no—
176C2×C42—51no—
177#104—52no—
178#104—52no—
179C52C2×#10411yescharacteristic, complemented, Sylow
180C22×C5—21no—
181C22×C5—21no—
182C22×C5—42no—
183C22×C5—41no—
184C22×C5—51no—
185C22×C5—51no—
186C22×C5—102no—
187C22×C5—101no—
188C22×C5—202no—
189C22×C5C5×C22⋊C411yescharacteristic
190C22×C5C22⋊AGL(1,𝔽5)11yescharacteristic
191C2×D5—22no—
192C2×D5—21no—
193C2×D5—42no—
194C2×D5C5×C22⋊C411yescharacteristic
195C2×D5C2×C5×D411yescharacteristic
196AGL(1,𝔽5)—24no—
197C4×C5—21no—
198C4×C5—21no—
199C4×C5—42no—
200C4×C5—41no—
201C4×C5—41no—
202C4×C5—51no—
203C4×C5—51no—
204C4×C5—104no—
205C4×C5—101no—
206C4×C5C5×C22⋊C411yescharacteristic
207C4×C5C22⋊AGL(1,𝔽5)11yescharacteristic
208C4×C5C22×C4×C511yescharacteristic
209C4×C5C22×AGL(1,𝔽5)11yescharacteristic
210Dic5—21no—
211Dic5C5×C22⋊C411yescharacteristic
212Dic5C2×C5×D411yescharacteristic
213#41—51no—
214#41—51no—
215#41—51no—
216#41—101no—
217#41—101no—
218#41C5×AGL(1,𝔽5)11yescharacteristic, complemented
219C2×Q8—51no—
220C2×D4—51no—
221C2×D4—101no—
222C22×C4—51no—
223C22×C4—101no—
224C22×C4—101no—
225M16—52no—
226M16—101no—
227C2×C8—101no—
228C42—52no—
229C42—101no—
230C2×C5—21no—
231C2×C5—21no—
232C2×C5—41no—
233C2×C5—41no—
234C2×C5—41no—
235C2×C5—52no—
236C2×C5—81no—
237C2×C5—82no—
238C2×C5—101no—
239C2×C5—201no—
240C2×C5C2×C5×C22⋊C411yescharacteristic
241C2×C5C2×C22⋊AGL(1,𝔽5)11yescharacteristic
242D5—21no—
243D5—41no—
244D5C5×#10412yescomplemented
245C23—51no—
246C23—101no—
247Q8—51no—
248Q8—101no—
249Q8C2×C5×AGL(1,𝔽5)11yescharacteristic, complemented
250D4—21no—
251D4—51no—
252D4—101no—
253D4—101no—
254D4—101no—
255D4—101no—
256D4C2×C5×AGL(1,𝔽5)11yescharacteristic, complemented
257C2×C4—21no—
258C2×C4—51no—
259C2×C4—51no—
260C2×C4—51no—
261C2×C4—51no—
262C2×C4—51no—
263C2×C4—101no—
264C2×C4—101no—
265C2×C4—101no—
266C2×C4—102no—
267C2×C4—104no—
268C2×C4—101no—
269C2×C4C2×C5×AGL(1,𝔽5)11yescharacteristic
270C8—102no—
271C5—41no—
272C5C2×C5×#10411yescharacteristic, complemented
273C5D5⋊#10411yescharacteristic, direct product factor
274C22—21no—
275C22—51no—
276C22—51no—
277C22—102no—
278C22—101no—
279C22—202no—
280C22C5×C22⋊AGL(1,𝔽5)11yescharacteristic
281C4—21no—
282C4—51no—
283C4—51no—
284C4—104no—
285C4—101no—
286C4C5×C22⋊AGL(1,𝔽5)11yescharacteristic
287C4C22×C5×AGL(1,𝔽5)11yescharacteristic
288C2—21no—
289C2—41no—
290C2—52no—
291C2—101no—
292C2—201no—
293C2C2×C5×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC4×C5C22⋊AGL(1,𝔽5)
commutatorC4×C5C22×C4×C5
FrattiniC4C22×C5×AGL(1,𝔽5)
FittingC52×#41C4
socleC2×C52C2×C22⋊C4
prime radicalC5×D5×#41C2
cosocleC4×D5C23×C5
hypercenterC5×#41AGL(1,𝔽5)
hypocenterC5C2×C5×#104
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C5×#41

Complements

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

NH ≅ G/Ntype
2. C2×C4×C5×AGL(1,𝔽5)C2semidirect
3. C5×D5⋊M16C2semidirect
4. C5×#4010C2semidirect
5. C5×#4008C2semidirect
7. C52×#41C4semidirect
9. C5×Q8×D5C4semidirect
13. C5×D4×D5C4semidirect
15. C5×#709C4semidirect
20. C4×C5×AGL(1,𝔽5)C22semidirect
21. C5×#700C22semidirect
23. D5⋊#104C5direct (N × H)
29. C52×Q8C2×C4semidirect
31. C52×D4C2×C4semidirect
40. C5×D20C2×C4semidirect
49. C5×Dic10C2×C4semidirect
53. C2×C4×AGL(1,𝔽5)C2×C5semidirect
54. D5⋊M16C2×C5semidirect
57. #4010C2×C5semidirect
58. #4008C2×C5semidirect
82. C5×#41C4×C5semidirect
83. C5×#41AGL(1,𝔽5)semidirect
91. Q8×D5C4×C5semidirect
95. D4×D5C4×C5semidirect
97. #709C4×C5semidirect
102. C4×AGL(1,𝔽5)C22×C5semidirect
103. #700C22×C5semidirect
116. C5×D5#104semidirect
126. C5×Q8C2×C4×C5semidirect
127. C5×Q8C2×AGL(1,𝔽5)semidirect
137. C5×D4C2×C4×C5semidirect
138. C5×D4C2×AGL(1,𝔽5)semidirect
162. D20C2×C4×C5semidirect
171. Dic10C2×C4×C5semidirect
179. C52C2×#104semidirect
218. #41C5×AGL(1,𝔽5)semidirect
244. D5C5×#104semidirect
249. Q8C2×C5×AGL(1,𝔽5)semidirect
256. D4C2×C5×AGL(1,𝔽5)semidirect
272. C5C2×C5×#104semidirect
273. C5D5⋊#104direct (N × H)

Maximal subgroups

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

subgroupindex
C5×D5×#412
C2×C4×C5×AGL(1,𝔽5)2
C5×D5⋊M162
C5×#40102
C5×#40082
D5⋊#1045
C2×C5×#1045