Finite Group Databasebeta

← #66012

Subgroups — #66012

Order 320.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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×#124C211yesmaximal, characteristic, complemented
2#4093C211yesmaximal, characteristic, complemented
3#4092C211yesmaximal, characteristic, complemented
4#4085C211yesmaximal, characteristic, complemented
5#4083C211yesmaximal, characteristic, complemented
6#4032C211yesmaximal, characteristic, complemented
7#4030C211yesmaximal, characteristic, complemented
8#4030C211yesmaximal, characteristic, complemented
9#4029C211yesmaximal, characteristic, complemented
10#4028C211yesmaximal, characteristic, complemented
11#4027C211yesmaximal, characteristic, complemented
12D5×C22⋊C4C211yesmaximal, characteristic, complemented
13#4025C211yesmaximal, characteristic, complemented
14#4022C211yesmaximal, characteristic, complemented
15C4×D20C211yesmaximal, characteristic, complemented
16C23×D5C2211yescharacteristic
17C2×C5×Q8C2211yescharacteristic, complemented
18C2×C5×D4C2211yescharacteristic
19C2×C22⋊D5C2211yescharacteristic
20C2×C22⋊D5C2211yescharacteristic
21C2×C22⋊D5C2211yescharacteristic
22C2×C22⋊D5C2211yescharacteristic, complemented
23C22×Dic5C2211yescharacteristic, complemented
24C2×D20C2211yescharacteristic
25C2×D20C2211yescharacteristic
26C2×C4×D5C2211yescharacteristic
27C2×C4×D5C2211yescharacteristic
28C2×C4×D5C2211yescharacteristic, complemented
29C5×C22⋊C4C2211yescharacteristic
30C5×C22⋊C4C2211yescharacteristic
31C5×C22⋊C4C2211yescharacteristic, complemented
32C5×C22⋊C4C2211yescharacteristic, complemented
33C42×C5C2211yescharacteristic, complemented
34C22⋊Dic5C2211yescharacteristic
35C22⋊Dic5C2211yescharacteristic
36#685C2211yescharacteristic
37#685C2211yescharacteristic
38#685C2211yescharacteristic
39#685C2211yescharacteristic
40#685C2211yescharacteristic
41#685C2211yescharacteristic
42#685C2211yescharacteristic, complemented
43#685C2211yescharacteristic, complemented
44C4⋊Dic5C2211yescharacteristic, complemented
45C4⋊Dic5C2211yescharacteristic, complemented
46#683C2211yescharacteristic, complemented
47#683C2211yescharacteristic, complemented
48#683C2211yescharacteristic, complemented
49#683C2211yescharacteristic, complemented
50C4×Dic5C2211yescharacteristic, complemented
51#538—51nomaximal, Sylow
52C23×C5C2311yescharacteristic
53C23×C5C2311yescharacteristic
54C22×D5—22no—
55C22×D5—41no—
56C22×D5—41no—
57C22×D5C2311yescharacteristic
58C22×D5C2311yescharacteristic
59C22×D5C2311yescharacteristic
60C5×Q8—41no—
61C5×D4—41no—
62C2×C4×C5C2311yescharacteristic
63C2×C4×C5C2311yescharacteristic
64C2×C4×C5C2311yescharacteristic
65C2×C4×C5C2311yescharacteristic
66C2×C4×C5C2311yescharacteristic
67C22⋊D5—22no—
68C22⋊D5—41no—
69C22⋊D5—41no—
70C22⋊D5—41no—
71C2×Dic5—41no—
72C2×Dic5C2311yescharacteristic
73C2×Dic5C2311yescharacteristic
74C2×Dic5C2311yescharacteristic
75C2×Dic5C2311yescharacteristic
76C2×Dic5C2311yescharacteristic
77D20—22no—
78D20—41no—
79C4×D5—41no—
80C4×D5—41no—
81C4×D5—41no—
82#126—51no—
83#126—51no—
84#124—51no—
85#124D511yescharacteristic, complemented
86#123—51no—
87#123—51no—
88C22⋊Q8—51no—
89#121—51no—
90#121—51no—
91#121—51no—
92#120—51no—
93#120—51no—
94C4×D4—51no—
95C4×D4—51no—
96C2×C22⋊C4—51no—
97C22×C5—21no—
98C22×C5—21no—
99C22×C5—21no—
100C22×C5—21no—
101C22×C5—21no—
102C22×C5—21no—
103C22×C5C2411yescharacteristic
104C2×D5—21no—
105C2×D5—21no—
106C2×D5—21no—
107C2×D5—21no—
108C2×D5—21no—
109C2×D5—21no—
110C2×D5—21no—
111C2×D5—21no—
112C2×D5—44no—
113C2×D5#4112yes—
114C4×C5—21no—
115C4×C5—21no—
116C4×C5—21no—
117C4×C5—21no—
118C4×C5—21no—
119Dic5—21no—
120Dic5—21no—
121Dic5—21no—
122Dic5—21no—
123Dic5—21no—
124C24—51no—
125C2×Q8C2×D511yescharacteristic, complemented
126C2×D4—51no—
127C2×D4—51no—
128C2×D4—51no—
129C2×D4—51no—
130C2×D4—51no—
131C2×D4—51no—
132C2×D4C2×D511yescharacteristic
133C22×C4—51no—
134C22×C4—51no—
135C22×C4—51no—
136C22×C4—51no—
137C4⋊C4—51no—
138C4⋊C4—51no—
139C4⋊C4—51no—
140C4⋊C4—51no—
141C4⋊C4—51no—
142C4⋊C4—51no—
143C22⋊C4—51no—
144C22⋊C4—51no—
145C22⋊C4—51no—
146C22⋊C4—51no—
147C22⋊C4—51no—
148C22⋊C4—51no—
149C22⋊C4—51no—
150C22⋊C4—51no—
151C22⋊C4—51no—
152C22⋊C4—51no—
153C22⋊C4C2×D511yescharacteristic
154C22⋊C4C2×D511yescharacteristic
155C22⋊C4C2×D511yescharacteristic, complemented
156C22⋊C4C2×D511yescharacteristic, complemented
157C42—51no—
158C42C2×D511yescharacteristic, complemented
159C2×C5—41no—
160C2×C5—41no—
161C2×C5C2×#4111yescharacteristic
162C2×C5H(5,𝔽2)11yescharacteristic
163C2×C5H(5,𝔽2)11yescharacteristic
164D5—22no—
165D5—41no—
166D5—41no—
167C23—51no—
168C23—51no—
169C23—51no—
170C23—102no—
171C23—201no—
172C23—201no—
173C23C22×D511yescharacteristic
174C23C22×D511yescharacteristic
175Q8—41no—
176D4—41no—
177D4—102no—
178D4—102no—
179D4—201no—
180D4—201no—
181D4—201no—
182D4—201no—
183C2×C4—51no—
184C2×C4—51no—
185C2×C4—51no—
186C2×C4—51no—
187C2×C4—51no—
188C2×C4—201no—
189C2×C4—201no—
190C2×C4—201no—
191C2×C4—201no—
192C2×C4C22×D511yescharacteristic
193C2×C4C22×D511yescharacteristic
194C2×C4C22×D511yescharacteristic
195C2×C4C22×D511yescharacteristic
196C2×C4C22×D511yescharacteristic
197C5#53811yescharacteristic, complemented, Sylow
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—52no—
205C22—101no—
206C22—101no—
207C22—101no—
208C22—101no—
209C22—101no—
210C22—101no—
211C22—101no—
212C22—101no—
213C22—204no—
214C22C23×D511yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—101no—
221C4—101no—
222C4—101no—
223C4—101no—
224C4—101no—
225C2—41no—
226C2—41no—
227C2—102no—
228C2—201no—
229C2—201no—
230C2#414411yescharacteristic
231C2D5×#4111yescharacteristic
232C2#414911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D5
commutator, socle, cosocleC22×C5C24
FittingC5×#124C2
hypercenter#124D5
hypocenterC5#538
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C5×#124C2semidirect
2. #4093C2semidirect
3. #4092C2semidirect
4. #4085C2semidirect
5. #4083C2semidirect
6. #4032C2semidirect
7. #4030C2semidirect
8. #4030C2semidirect
9. #4029C2semidirect
10. #4028C2semidirect
11. #4027C2semidirect
12. D5×C22⋊C4C2semidirect
13. #4025C2semidirect
14. #4022C2semidirect
15. C4×D20C2semidirect
17. C2×C5×Q8C22semidirect
22. C2×C22⋊D5C22semidirect
23. C22×Dic5C22semidirect
28. C2×C4×D5C22semidirect
31. C5×C22⋊C4C22semidirect
32. C5×C22⋊C4C22semidirect
33. C42×C5C22semidirect
42. #685C22semidirect
43. #685C22semidirect
44. C4⋊Dic5C22semidirect
45. C4⋊Dic5C22semidirect
46. #683C22semidirect
47. #683C22semidirect
48. #683C22semidirect
49. #683C22semidirect
50. C4×Dic5C22semidirect
85. #124D5semidirect
125. C2×Q8C2×D5semidirect
155. C22⋊C4C2×D5semidirect
156. C22⋊C4C2×D5semidirect
158. C42C2×D5semidirect
197. C5#538semidirect

Maximal subgroups

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

subgroupindex
C5×#1242
#40932
#40922
#40852
#40832
#40322
#40302
#40292
#40282
#40272
D5×C22⋊C42
#40252
#40222
C4×D202
#5385