Finite Group Databasebeta

← #65987

Subgroups — #65987

Order 320.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×D4×D5C211yesmaximal, characteristic, complemented
2C22×D20C211yesmaximal, characteristic, complemented
3C5×#123C211yesmaximal, characteristic, complemented
4#4083C211yesmaximal, characteristic, complemented
5#4075C211yesmaximal, characteristic, complemented
6#4041C212yesmaximal, complemented
7#4040C212yesmaximal, complemented
8#4030C212yesmaximal, complemented
9#4028C212yesmaximal, complemented
10#4028C211yesmaximal, characteristic, complemented
11D5×C22⋊C4C211yesmaximal, characteristic, complemented
12C23×D5C2211yescharacteristic
13C23×D5C2211yescharacteristic, complemented
14C23×D5C2211yescharacteristic, complemented
15C2×C5×D4C2211yescharacteristic
16C22×C4×C5C2211yescharacteristic, complemented
17C2×C22⋊D5C2211yescharacteristic
18C2×C22⋊D5C2211yescharacteristic, complemented
19C2×C22⋊D5C2212yescomplemented
20D4×D5—24no—
21C2×D20—24no—
22C2×D20C2212yes—
23C2×D20C2212yescomplemented
24C2×D20C2211yescharacteristic, complemented
25C2×D20C2212yescomplemented
26C2×C4×D5C2211yescharacteristic, complemented
27C2×C4×D5C2212yescomplemented
28C5×C4⋊C4C2212yescomplemented
29C5×C22⋊C4C2211yescharacteristic, complemented
30C5×C22⋊C4C2212yescomplemented
31C22⋊Dic5C2211yescharacteristic, complemented
32#685C2212yescomplemented
33#685C2212yescomplemented
34#685C2212yescomplemented
35#685C2212yescomplemented
36#683C2212yescomplemented
37#534—51nomaximal, Sylow
38C23×C5C2311yescharacteristic
39C23×C5C2311yescharacteristic
40C22×D5—22no—
41C22×D5—22no—
42C22×D5—42no—
43C22×D5—42no—
44C22×D5—42no—
45C22×D5D414yescomplemented
46C22×D5C2312yes—
47C22×D5C2311yescharacteristic
48C22×D5C2311yescharacteristic
49C22×D5C2311yescharacteristic
50C5×D4—22no—
51C2×C4×C5—22no—
52C2×C4×C5C2312yes—
53C2×C4×C5C2312yes—
54C2×C4×C5C2311yescharacteristic
55C22⋊D5—22no—
56C22⋊D5—22no—
57C22⋊D5—42no—
58C2×Dic5C2311yescharacteristic
59C2×Dic5C2312yes—
60D20—24no—
61D20—24no—
62D20—22no—
63D20—42no—
64C4×D5—22no—
65C4×D5—42no—
66C22×D4—51no—
67C22×D4—51no—
68#123—51no—
69#123—52no—
70#123D511yescharacteristic, complemented
71#121—52no—
72#121—52no—
73#120—51no—
74#120—52no—
75#120—51no—
76C2×C22⋊C4—51no—
77C22×C5—22no—
78C22×C5—21no—
79C22×C5—22no—
80C22×C5C2×D412yes—
81C22×C5C2411yescharacteristic
82C2×D5—22no—
83C2×D5—22no—
84C2×D5—22no—
85C2×D5—21no—
86C2×D5—22no—
87C2×D5—22no—
88C2×D5—22no—
89C2×D5—44no—
90C2×D5—44no—
91C2×D5—44no—
92C2×D5C2×D412yes—
93C2×D5C2×D412yes—
94C4×C5—21no—
95C4×C5—22no—
96C4×C5—22no—
97Dic5—21no—
98Dic5—22no—
99C24—51no—
100C24—51no—
101C24—51no—
102C2×D4—51no—
103C2×D4—51no—
104C2×D4—52no—
105C2×D4—52no—
106C2×D4—51no—
107C2×D4—52no—
108C2×D4—52no—
109C2×D4—104no—
110C2×D4—104no—
111C2×D4C2×D511yescharacteristic
112C22×C4—51no—
113C22×C4—52no—
114C22×C4C2×D511yescharacteristic, complemented
115C4⋊C4—52no—
116C4⋊C4C2×D512yescomplemented
117C22⋊C4—51no—
118C22⋊C4—52no—
119C22⋊C4—52no—
120C22⋊C4—52no—
121C22⋊C4—52no—
122C22⋊C4C2×D511yescharacteristic, complemented
123C22⋊C4C2×D512yescomplemented
124C2×C5—22no—
125C2×C5—41no—
126C2×C5C22×D411yescharacteristic
127C2×C5H(5,𝔽2)12yes—
128D5—22no—
129D5—22no—
130D5—42no—
131D5—41no—
132C23—51no—
133C23—51no—
134C23—54no—
135C23—51no—
136C23—52no—
137C23—102no—
138C23—102no—
139C23—202no—
140C23—202no—
141C23—202no—
142C23C22×D511yescharacteristic
143C23C22×D511yescharacteristic
144D4—22no—
145D4—104no—
146D4—104no—
147D4—102no—
148D4—102no—
149D4—102no—
150D4—202no—
151D4—202no—
152C2×C4—22no—
153C2×C4—52no—
154C2×C4—51no—
155C2×C4—102no—
156C2×C4—202no—
157C2×C4C22×D512yes—
158C2×C4C22×D511yescharacteristic
159C2×C4C22×D512yes—
160C5#53411yescharacteristic, complemented, Sylow
161C22—22no—
162C22—21no—
163C22—22no—
164C22—52no—
165C22—52no—
166C22—102no—
167C22—102no—
168C22—102no—
169C22—101no—
170C22—102no—
171C22—102no—
172C22—102no—
173C22—204no—
174C22—204no—
175C22—204no—
176C22D4×D512yes—
177C22C23×D511yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—101no—
182C4—102no—
183C2—22no—
184C2—41no—
185C2—102no—
186C2—102no—
187C2—202no—
188C2—201no—
189C2C2×D4×D511yescharacteristic
190C2#414912yes—
subgroupG/H
center, FrattiniC22C23×D5
commutator, socle, cosocleC22×C5C24
FittingC5×#123C2
hypercenter#123D5
hypocenterC5#534
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×D4×D5C2semidirect
2. C22×D20C2semidirect
3. C5×#123C2semidirect
4. #4083C2semidirect
5. #4075C2semidirect
6. #4041C2semidirect
7. #4040C2semidirect
8. #4030C2semidirect
9. #4028C2semidirect
10. #4028C2semidirect
11. D5×C22⋊C4C2semidirect
13. C23×D5C22semidirect
14. C23×D5C22semidirect
16. C22×C4×C5C22semidirect
18. C2×C22⋊D5C22semidirect
19. C2×C22⋊D5C22semidirect
23. C2×D20C22semidirect
24. C2×D20C22semidirect
25. C2×D20C22semidirect
26. C2×C4×D5C22semidirect
27. C2×C4×D5C22semidirect
28. C5×C4⋊C4C22semidirect
29. C5×C22⋊C4C22semidirect
30. C5×C22⋊C4C22semidirect
31. C22⋊Dic5C22semidirect
32. #685C22semidirect
33. #685C22semidirect
34. #685C22semidirect
35. #685C22semidirect
36. #683C22semidirect
45. C22×D5D4semidirect
70. #123D5semidirect
114. C22×C4C2×D5semidirect
116. C4⋊C4C2×D5semidirect
122. C22⋊C4C2×D5semidirect
123. C22⋊C4C2×D5semidirect
160. C5#534semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D52
C22×D202
C5×#1232
#40832
#40752
#40412
#40402
#40302
#40282
D5×C22⋊C42
#5345