Finite Group Databasebeta

← #66140

Subgroups — #66140

Order 320.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×C5×D4C211yesmaximal, characteristic, complemented
2C2×#711C211yesmaximal, characteristic, complemented
3#4099C212yesmaximal, complemented
4C2×C22⋊Dic5C212yesmaximal, complemented
5#4085C212yesmaximal, complemented
6#4084C211yesmaximal, characteristic, complemented
7#4082C211yesmaximal, characteristic, complemented
8#4081C212yesmaximal, complemented
9D4×Dic5C211yesmaximal, characteristic, complemented
10C4×C22⋊D5C211yesmaximal, characteristic, complemented
11#4070C211yesmaximal, characteristic, complemented
12C24×C5C2212yes—
13C2×C5×D4—24no—
14C2×C5×D4C2212yes—
15C2×C5×D4C2211yescharacteristic
16C2×C5×D4C2211yescharacteristic
17C22×C4×C5C2211yescharacteristic
18C2×C22⋊D5C2211yescharacteristic
19C2×C22⋊D5C2212yescomplemented
20C2×C22⋊D5C2212yescomplemented
21C22×Dic5C2212yescomplemented
22C22×Dic5C2212yescomplemented
23#711—24no—
24C2×C4×D5C2211yescharacteristic, complemented
25C2×Dic10C2211yescharacteristic, complemented
26C22⋊Dic5C2211yescharacteristic
27C22⋊Dic5C2212yes—
28C22⋊Dic5C2212yescomplemented
29C22⋊Dic5C2212yescomplemented
30C22⋊Dic5C2212yescomplemented
31C22⋊Dic5C2212yescomplemented
32#685C2211yescharacteristic, complemented
33C4⋊Dic5C2211yescharacteristic, complemented
34#683C2212yescomplemented
35#683C2211yescharacteristic, complemented
36C4×Dic5C2211yescharacteristic, complemented
37#546—51nomaximal, Sylow
38C23×C5—24no—
39C23×C5—24no—
40C23×C5—42no—
41C23×C5C2312yes—
42C23×C5C2312yes—
43C23×C5C2311yescharacteristic
44C22×D5C2311yescharacteristic
45C5×D4—24no—
46C5×D4—22no—
47C5×D4D414yescomplemented
48C2×C4×C5—22no—
49C2×C4×C5C2311yescharacteristic
50C2×C4×C5C2311yescharacteristic
51C22⋊D5—24no—
52C22⋊D5—22no—
53C22⋊D5—42no—
54C2×Dic5—24no—
55C2×Dic5—42no—
56C2×Dic5C2311yescharacteristic
57C2×Dic5C2311yescharacteristic
58C2×Dic5C2312yes—
59C2×Dic5C2312yes—
60C2×Dic5C2311yescharacteristic
61C4×D5—22no—
62Dic10—22no—
63C2×#41—51no—
64C22×D4D511yescharacteristic, complemented
65#124—51no—
66#123—52no—
67C22⋊Q8—51no—
68#121—51no—
69#121—52no—
70#120—52no—
71C4×D4—51no—
72C4×D4—51no—
73C2×C22⋊C4—52no—
74C22×C5—21no—
75C22×C5—22no—
76C22×C5—22no—
77C22×C5—21no—
78C22×C5—22no—
79C22×C5—22no—
80C22×C5—22no—
81C22×C5—48no—
82C22×C5C2×D414yes—
83C22×C5#4112yescomplemented
84C22×C5C2411yescharacteristic
85C2×D5—21no—
86C2×D5—21no—
87C2×D5—21no—
88C4×C5—21no—
89C4×C5C2×D412yes—
90Dic5—22no—
91Dic5—21no—
92Dic5—21no—
93Dic5—21no—
94Dic5—22no—
95C24C2×D512yes—
96#41—104no—
97C2×Q8—51no—
98C2×D4—24no—
99C2×D4—51no—
100C2×D4—52no—
101C2×D4—52no—
102C2×D4C2×D512yes—
103C2×D4C2×D511yescharacteristic
104C2×D4C2×D511yescharacteristic
105C22×C4—52no—
106C22×C4—52no—
107C22×C4—51no—
108C22×C4C2×D511yescharacteristic
109C4⋊C4—51no—
110C4⋊C4—51no—
111C4⋊C4—52no—
112C22⋊C4—51no—
113C22⋊C4—52no—
114C22⋊C4—52no—
115C22⋊C4—52no—
116C22⋊C4—52no—
117C22⋊C4—52no—
118C22⋊C4—51no—
119C42—51no—
120C2×C5—22no—
121C2×C5—24no—
122C2×C5—42no—
123C2×C5C22×D411yescharacteristic
124C2×C5C2×#4111yescharacteristic
125C2×C5H(5,𝔽2)11yescharacteristic
126D5—41no—
127C23—24no—
128C23—24no—
129C23—42no—
130C23—51no—
131C23C22×D512yes—
132C23C22×D511yescharacteristic
133C23C22×D512yes—
134Q8—102no—
135D4—24no—
136D4—22no—
137D4—102no—
138D4—104no—
139D4—202no—
140D4C22⋊D514yescomplemented
141C2×C4—22no—
142C2×C4—51no—
143C2×C4—51no—
144C2×C4—52no—
145C2×C4—51no—
146C2×C4—52no—
147C2×C4—104no—
148C2×C4—102no—
149C2×C4—202no—
150C2×C4C22×D511yescharacteristic
151C2×C4C22×D511yescharacteristic
152C5#54611yescharacteristic, complemented, Sylow
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—101no—
162C22—101no—
163C22—101no—
164C22#71112yescomplemented
165C22C2×C22⋊D514yes—
166C22C23×D511yescharacteristic
167C4—21no—
168C4—101no—
169C4—101no—
170C4—102no—
171C4—101no—
172C4—102no—
173C4C2×C22⋊D512yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—201no—
178C2C2×#71111yescharacteristic
179C2#414411yescharacteristic
180C2C22×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D5
commutator, socle, cosocleC22×C5C24
FittingC22×C5×D4C2
hypercenterC22×D4D5
hypocenterC5#546
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C5×D4C2semidirect
2. C2×#711C2semidirect
3. #4099C2semidirect
4. C2×C22⋊Dic5C2semidirect
5. #4085C2semidirect
6. #4084C2semidirect
7. #4082C2semidirect
8. #4081C2semidirect
9. D4×Dic5C2semidirect
10. C4×C22⋊D5C2semidirect
11. #4070C2semidirect
19. C2×C22⋊D5C22semidirect
20. C2×C22⋊D5C22semidirect
21. C22×Dic5C22semidirect
22. C22×Dic5C22semidirect
24. C2×C4×D5C22semidirect
25. C2×Dic10C22semidirect
28. C22⋊Dic5C22semidirect
29. C22⋊Dic5C22semidirect
30. C22⋊Dic5C22semidirect
31. C22⋊Dic5C22semidirect
32. #685C22semidirect
33. C4⋊Dic5C22semidirect
34. #683C22semidirect
35. #683C22semidirect
36. C4×Dic5C22semidirect
47. C5×D4D4semidirect
64. C22×D4D5semidirect
83. C22×C5#41semidirect
140. D4C22⋊D5semidirect
152. C5#546semidirect
164. C22#711semidirect

Maximal subgroups

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

subgroupindex
C22×C5×D42
C2×#7112
#40992
C2×C22⋊Dic52
#40852
#40842
#40822
#40812
D4×Dic52
C4×C22⋊D52
#40702
#5465