Finite Group Databasebeta

← #65954

Subgroups — #65954

Order 320.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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×#121C211yesmaximal, characteristic, complemented
2#4086C211yesmaximal, characteristic, complemented
3#4085C211yesmaximal, characteristic, complemented
4#4085C211yesmaximal, characteristic, complemented
5#4084C211yesmaximal, characteristic, complemented
6#4083C211yesmaximal, characteristic, complemented
7#4083C211yesmaximal, characteristic, complemented
8#4076C211yesmaximal, characteristic, complemented
9#4075C211yesmaximal, characteristic, complemented
10#4044C211yesmaximal, characteristic, complemented
11#4040C211yesmaximal, characteristic, complemented
12#4032C211yesmaximal, characteristic, complemented
13#4030C211yesmaximal, characteristic, complemented
14#4028C211yesmaximal, characteristic, complemented
15#4025C211yesmaximal, characteristic, complemented
16C23×D5C2211yescharacteristic
17C2×C5×D4C2211yescharacteristic
18C2×C5×D4C2211yescharacteristic
19C2×C5×D4C2211yescharacteristic, complemented
20C22×C4×C5C2211yescharacteristic, complemented
21C2×C22⋊D5C2211yescharacteristic
22C2×C22⋊D5C2211yescharacteristic
23C2×C22⋊D5C2211yescharacteristic
24C2×C22⋊D5C2211yescharacteristic
25C2×C22⋊D5C2211yescharacteristic
26C2×C22⋊D5C2211yescharacteristic, complemented
27C2×C22⋊D5C2211yescharacteristic, complemented
28C22×Dic5C2211yescharacteristic, complemented
29C2×D20C2211yescharacteristic
30C2×D20C2211yescharacteristic
31C2×C4×D5C2211yescharacteristic, complemented
32C5×C4⋊C4C2211yescharacteristic, complemented
33C5×C22⋊C4C2211yescharacteristic
34C5×C22⋊C4C2211yescharacteristic, complemented
35C22⋊Dic5C2211yescharacteristic
36C22⋊Dic5C2211yescharacteristic
37C22⋊Dic5C2211yescharacteristic, complemented
38#685C2211yescharacteristic
39#685C2211yescharacteristic
40#685C2211yescharacteristic
41#685C2211yescharacteristic
42#685C2211yescharacteristic
43#685C2211yescharacteristic
44#685C2211yescharacteristic, complemented
45C4⋊Dic5C2211yescharacteristic, complemented
46C4⋊Dic5C2211yescharacteristic, complemented
47#683C2211yescharacteristic, complemented
48#683C2211yescharacteristic, complemented
49#683C2211yescharacteristic, complemented
50C4×Dic5C2211yescharacteristic, complemented
51#560—51nomaximal, Sylow
52C23×C5C2311yescharacteristic
53C23×C5C2311yescharacteristic
54C23×C5C2311yescharacteristic
55C22×D5—41no—
56C22×D5—41no—
57C22×D5—41no—
58C22×D5C2311yescharacteristic
59C22×D5C2311yescharacteristic
60C22×D5C2311yescharacteristic
61C5×D4—41no—
62C5×D4—41no—
63C5×D4—41no—
64C2×C4×C5—41no—
65C2×C4×C5C2311yescharacteristic
66C2×C4×C5C2311yescharacteristic
67C2×C4×C5C2311yescharacteristic
68C2×C4×C5C2311yescharacteristic
69C22⋊D5—41no—
70C22⋊D5—41no—
71C22⋊D5—41no—
72C22⋊D5—41no—
73C22⋊D5—41no—
74C22⋊D5—41no—
75C22⋊D5—41no—
76C2×Dic5—41no—
77C2×Dic5C2311yescharacteristic
78C2×Dic5C2311yescharacteristic
79C2×Dic5C2311yescharacteristic
80C2×Dic5C2311yescharacteristic
81C2×Dic5C2311yescharacteristic
82D20—41no—
83D20—41no—
84C4×D5—41no—
85#127—51no—
86#126—51no—
87#126—51no—
88#123—51no—
89#123—51no—
90#123—51no—
91#121—51no—
92#121—51no—
93#121—51no—
94#121—51no—
95#121—51no—
96#121D511yescharacteristic, complemented
97#120—51no—
98#120—51no—
99#120—51no—
100C22×C5—21no—
101C22×C5—21no—
102C22×C5—21no—
103C22×C5—21no—
104C22×C5—21no—
105C22×C5—21no—
106C22×C5—21no—
107C22×C5—21no—
108C22×C5—21no—
109C22×C5C2411yescharacteristic
110C2×D5—21no—
111C2×D5—21no—
112C2×D5—21no—
113C2×D5—21no—
114C2×D5—21no—
115C2×D5—21no—
116C2×D5—21no—
117C2×D5—21no—
118C2×D5—21no—
119C2×D5—44no—
120C4×C5—21no—
121C4×C5—21no—
122C4×C5—21no—
123C4×C5—21no—
124Dic5—21no—
125Dic5—21no—
126Dic5—21no—
127Dic5—21no—
128Dic5—21no—
129C24—51no—
130C2×D4—51no—
131C2×D4—51no—
132C2×D4—51no—
133C2×D4—51no—
134C2×D4—51no—
135C2×D4—51no—
136C2×D4—51no—
137C2×D4—51no—
138C2×D4—51no—
139C2×D4C2×D511yescharacteristic
140C2×D4C2×D511yescharacteristic
141C2×D4C2×D511yescharacteristic, complemented
142C22×C4—51no—
143C22×C4—51no—
144C22×C4C2×D511yescharacteristic, complemented
145C4⋊C4—51no—
146C4⋊C4—51no—
147C4⋊C4—51no—
148C4⋊C4—51no—
149C4⋊C4—51no—
150C4⋊C4C2×D511yescharacteristic, complemented
151C22⋊C4—51no—
152C22⋊C4—51no—
153C22⋊C4—51no—
154C22⋊C4—51no—
155C22⋊C4—51no—
156C22⋊C4—51no—
157C22⋊C4—51no—
158C22⋊C4—51no—
159C22⋊C4—51no—
160C22⋊C4—51no—
161C22⋊C4C2×D511yescharacteristic
162C22⋊C4C2×D511yescharacteristic, complemented
163C42—51no—
164C2×C5—41no—
165C2×C5—41no—
166C2×C5—41no—
167C2×C5H(5,𝔽2)11yescharacteristic
168C2×C5H(5,𝔽2)11yescharacteristic
169C2×C5H(5,𝔽2)11yescharacteristic
170D5—41no—
171D5—41no—
172D5—41no—
173C23—51no—
174C23—51no—
175C23—51no—
176C23—201no—
177C23—201no—
178C23—201no—
179C23C22×D511yescharacteristic
180C23C22×D511yescharacteristic
181C23C22×D511yescharacteristic
182D4—41no—
183D4—41no—
184D4—41no—
185D4—201no—
186D4—201no—
187D4—201no—
188D4—201no—
189D4—201no—
190D4—201no—
191D4—201no—
192D4—201no—
193D4—201no—
194C2×C4—41no—
195C2×C4—51no—
196C2×C4—51no—
197C2×C4—51no—
198C2×C4—51no—
199C2×C4—51no—
200C2×C4—201no—
201C2×C4—201no—
202C2×C4C22×D511yescharacteristic
203C2×C4C22×D511yescharacteristic
204C2×C4C22×D511yescharacteristic
205C2×C4C22×D511yescharacteristic
206C5#56011yescharacteristic, complemented, Sylow
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—101no—
217C22—101no—
218C22—101no—
219C22—101no—
220C22—101no—
221C22—101no—
222C22—101no—
223C22—101no—
224C22—101no—
225C22—204no—
226C22C23×D511yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—101no—
232C4—101no—
233C4—101no—
234C4—101no—
235C4—101no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—201no—
240C2—201no—
241C2—201no—
242C2#414411yescharacteristic
243C2#414411yescharacteristic
244C2#414911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D5
commutator, socle, cosocleC22×C5C24
FittingC5×#121C2
hypercenter#121D5
hypocenterC5#560
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C5×#121C2semidirect
2. #4086C2semidirect
3. #4085C2semidirect
4. #4085C2semidirect
5. #4084C2semidirect
6. #4083C2semidirect
7. #4083C2semidirect
8. #4076C2semidirect
9. #4075C2semidirect
10. #4044C2semidirect
11. #4040C2semidirect
12. #4032C2semidirect
13. #4030C2semidirect
14. #4028C2semidirect
15. #4025C2semidirect
19. C2×C5×D4C22semidirect
20. C22×C4×C5C22semidirect
26. C2×C22⋊D5C22semidirect
27. C2×C22⋊D5C22semidirect
28. C22×Dic5C22semidirect
31. C2×C4×D5C22semidirect
32. C5×C4⋊C4C22semidirect
34. C5×C22⋊C4C22semidirect
37. C22⋊Dic5C22semidirect
44. #685C22semidirect
45. C4⋊Dic5C22semidirect
46. C4⋊Dic5C22semidirect
47. #683C22semidirect
48. #683C22semidirect
49. #683C22semidirect
50. C4×Dic5C22semidirect
96. #121D5semidirect
141. C2×D4C2×D5semidirect
144. C22×C4C2×D5semidirect
150. C4⋊C4C2×D5semidirect
162. C22⋊C4C2×D5semidirect
206. C5#560semidirect

Maximal subgroups

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

subgroupindex
C5×#1212
#40862
#40852
#40842
#40832
#40762
#40752
#40442
#40402
#40322
#40302
#40282
#40252
#5605