Finite Group Databasebeta

← #65890

Subgroups — #65890

Order 320.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C4×C5×D4C211yesmaximal, characteristic, complemented
2#4083C211yesmaximal, characteristic
3#4081C211yesmaximal, characteristic, complemented
4#4075C211yesmaximal, characteristic, complemented
5C4×C22⋊D5C211yesmaximal, characteristic, complemented
6C2×#685C211yesmaximal, characteristic, complemented
7#4070C211yesmaximal, characteristic, complemented
8#4044C211yesmaximal, characteristic, complemented
9#4042C211yesmaximal, characteristic, complemented
10#4031C211yesmaximal, characteristic, complemented
11#4029C211yesmaximal, characteristic, complemented
12D5×C22⋊C4C211yesmaximal, characteristic, complemented
13#4023C211yesmaximal, characteristic, complemented
14#4022C211yesmaximal, characteristic, complemented
15#4018C211yesmaximal, characteristic, complemented
16C23×D5C2211yescharacteristic
17C2×C5×D4C2211yescharacteristic
18C22×C4×C5C2211yescharacteristic
19C22×C4×C5C2211yescharacteristic, complemented
20C2×C22⋊D5C2211yescharacteristic
21C2×C22⋊D5C2211yescharacteristic
22C22×Dic5C2211yescharacteristic
23C2×C4×D5C2211yescharacteristic
24C2×C4×D5C2211yescharacteristic
25C2×Dic10C2211yescharacteristic, complemented
26C5×C4⋊C4C2211yescharacteristic, complemented
27C5×C22⋊C4C2211yescharacteristic
28C5×C22⋊C4C2211yescharacteristic, complemented
29C42×C5C2211yescharacteristic, complemented
30C22⋊Dic5C2211yescharacteristic
31C22⋊Dic5C2211yescharacteristic
32C22⋊Dic5C2211yescharacteristic, complemented
33C22⋊Dic5C2211yescharacteristic, complemented
34#685C2211yescharacteristic
35#685C2211yescharacteristic
36#685C2211yescharacteristic
37#685C2211yescharacteristic
38#685C2211yescharacteristic
39#685C2211yescharacteristic
40#685C2211yescharacteristic
41#685C2211yescharacteristic
42C4⋊Dic5C2211yescharacteristic, complemented
43#683C2211yescharacteristic, complemented
44#683C2211yescharacteristic, complemented
45#683C2211yescharacteristic, complemented
46#683C2211yescharacteristic, complemented
47#683C2211yescharacteristic, complemented
48#683C2211yescharacteristic, complemented
49C4×Dic5C2211yescharacteristic, complemented
50C4×Dic5C2211yescharacteristic, complemented
51#551—51nomaximal, Sylow
52C23×C5C2311yescharacteristic
53C23×C5C2311yescharacteristic
54C22×D5—22no—
55C22×D5—22no—
56C22×D5—41no—
57C22×D5C2311yescharacteristic
58C22×D5C2311yescharacteristic
59C5×D4—22no—
60C2×C4×C5—41no—
61C2×C4×C5—41no—
62C2×C4×C5C2311yescharacteristic
63C2×C4×C5C2311yescharacteristic
64C2×C4×C5C2311yescharacteristic
65C2×C4×C5C2311yescharacteristic
66C2×C4×C5C2311yescharacteristic
67C22⋊D5—22no—
68C22⋊D5—41no—
69C2×Dic5—22no—
70C2×Dic5C2311yescharacteristic
71C2×Dic5C2311yescharacteristic
72C2×Dic5C2311yescharacteristic
73C2×Dic5C2311yescharacteristic
74C2×Dic5C2311yescharacteristic
75C2×Dic5C2311yescharacteristic
76C4×D5—22no—
77C4×D5—41no—
78Dic10—41no—
79#126—51no—
80#126—51no—
81#124—51no—
82#123—51no—
83#123—51no—
84#123—51no—
85C22⋊Q8—51no—
86C22⋊Q8—51no—
87#120—51no—
88C4×D4—51no—
89C4×D4D511yescharacteristic, complemented
90#117—51no—
91#117—51no—
92C2×C22⋊C4—51no—
93C2×C22⋊C4—51no—
94C22×C5—21no—
95C22×C5—21no—
96C22×C5—21no—
97C22×C5—21no—
98C22×C5—21no—
99C22×C5#4112yes—
100C22×C5C2411yescharacteristic
101C2×D5—21no—
102C2×D5—21no—
103C2×D5—21no—
104C2×D5—21no—
105C2×D5—21no—
106C2×D5—44no—
107C2×D5#4112yes—
108C4×C5—21no—
109C4×C5—21no—
110C4×C5—21no—
111C4×C5—21no—
112C4×C5—21no—
113Dic5—21no—
114Dic5—21no—
115Dic5—21no—
116Dic5—21no—
117Dic5—21no—
118Dic5—21no—
119C24—51no—
120C2×Q8—51no—
121C2×D4—51no—
122C2×D4—51no—
123C2×D4C2×D511yescharacteristic
124C22×C4—51no—
125C22×C4—51no—
126C22×C4—51no—
127C22×C4C2×D511yescharacteristic
128C22×C4C2×D511yescharacteristic, complemented
129C4⋊C4—51no—
130C4⋊C4—51no—
131C4⋊C4—51no—
132C4⋊C4—51no—
133C4⋊C4—51no—
134C4⋊C4—51no—
135C4⋊C4—51no—
136C4⋊C4C2×D511yescharacteristic, complemented
137C22⋊C4—51no—
138C22⋊C4—51no—
139C22⋊C4—51no—
140C22⋊C4—51no—
141C22⋊C4—51no—
142C22⋊C4—51no—
143C22⋊C4—51no—
144C22⋊C4—51no—
145C22⋊C4—51no—
146C22⋊C4—51no—
147C22⋊C4—51no—
148C22⋊C4—51no—
149C22⋊C4C2×D511yescharacteristic
150C22⋊C4C2×D511yescharacteristic, complemented
151C42—51no—
152C42—51no—
153C42C2×D511yescharacteristic, complemented
154C2×C5—22no—
155C2×C5—41no—
156C2×C5C2×#4111yescharacteristic
157C2×C5C2×#4111yescharacteristic
158C2×C5H(5,𝔽2)11yescharacteristic
159D5—22no—
160D5—41no—
161C23—51no—
162C23—51no—
163C23—102no—
164C23—102no—
165C23—201no—
166C23C22×D511yescharacteristic
167C23C22×D511yescharacteristic
168Q8—201no—
169D4—22no—
170D4—102no—
171D4—201no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—51no—
175C2×C4—51no—
176C2×C4—51no—
177C2×C4—51no—
178C2×C4—51no—
179C2×C4—51no—
180C2×C4—102no—
181C2×C4—102no—
182C2×C4—201no—
183C2×C4C22×D511yescharacteristic
184C2×C4C22×D511yescharacteristic
185C2×C4C22×D511yescharacteristic
186C2×C4C22×D511yescharacteristic
187C2×C4C22×D511yescharacteristic
188C5#55111yescharacteristic, complemented, Sylow
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—52no—
195C22—101no—
196C22—101no—
197C22—101no—
198C22—101no—
199C22—101no—
200C22—204no—
201C22#70912yes—
202C22C23×D511yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—101no—
209C4—101no—
210C4—101no—
211C4—101no—
212C4—101no—
213C4—101no—
214C2—22no—
215C2—41no—
216C2—102no—
217C2—201no—
218C2C2×#70911yescharacteristic
219C2#414411yescharacteristic
220C2D5×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D5
commutator, socle, cosocleC22×C5C24
FittingC4×C5×D4C2
prime radical#4083C2
hypercenterC4×D4D5
hypocenterC5#551
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×C5×D4C2semidirect
3. #4081C2semidirect
4. #4075C2semidirect
5. C4×C22⋊D5C2semidirect
6. C2×#685C2semidirect
7. #4070C2semidirect
8. #4044C2semidirect
9. #4042C2semidirect
10. #4031C2semidirect
11. #4029C2semidirect
12. D5×C22⋊C4C2semidirect
13. #4023C2semidirect
14. #4022C2semidirect
15. #4018C2semidirect
19. C22×C4×C5C22semidirect
25. C2×Dic10C22semidirect
26. C5×C4⋊C4C22semidirect
28. C5×C22⋊C4C22semidirect
29. C42×C5C22semidirect
32. C22⋊Dic5C22semidirect
33. C22⋊Dic5C22semidirect
42. C4⋊Dic5C22semidirect
43. #683C22semidirect
44. #683C22semidirect
45. #683C22semidirect
46. #683C22semidirect
47. #683C22semidirect
48. #683C22semidirect
49. C4×Dic5C22semidirect
50. C4×Dic5C22semidirect
89. C4×D4D5semidirect
128. C22×C4C2×D5semidirect
136. C4⋊C4C2×D5semidirect
150. C22⋊C4C2×D5semidirect
153. C42C2×D5semidirect
188. C5#551semidirect

Maximal subgroups

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

subgroupindex
C4×C5×D42
#40832
#40812
#40752
C4×C22⋊D52
C2×#6852
#40702
#40442
#40422
#40312
#40292
D5×C22⋊C42
#40232
#40222
#40182
#5515