Finite Group Databasebeta

← #10604274

Subgroups — #10604274

Order 640.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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×#4039C211yesmaximal, characteristic
2#65759C211yesmaximal, characteristic, complemented
3C4×#704C211yesmaximal, characteristic, complemented
4#64871C214yesmaximal, complemented
5C22×D20C411yescharacteristic
6C22×C4×D5—21no—
7C2×#704—21no—
8C2×C5×C4⋊C4C411yescharacteristic
9C2×#685—21no—
10C2×C4×Dic5C2211yescharacteristic
11#4039—21no—
12#4039—21no—
13#4039C412yes—
14#4039C2212yes—
15C22⋊#170—21no—
16#4002—21no—
17#4002—21no—
18C4⋊#170C2212yescomplemented
19C4×#170C2212yescomplemented
20#1490—51nomaximal, Sylow
21C23×D5—21no—
22C22×C4×C5—21no—
23C22×C4×C5C2×C411yescharacteristic
24C22×Dic5D411yescharacteristic
25C22×Dic5D411yescharacteristic
26C2×D20—21no—
27C2×D20—24no—
28C2×D20C2×C412yes—
29C2×C4×D5—22no—
30C2×C4×D5—22no—
31C2×C4×D5—44no—
32#704—24no—
33C2×#170—22no—
34C2×#170—22no—
35C5×C4⋊C4—21no—
36C5×C4⋊C4C2×C412yes—
37#685—22no—
38#685—22no—
39C4×Dic5C2×C411yescharacteristic
40C4×Dic5D411yescharacteristic
41C4×Dic5D411yescharacteristic
42C4×Dic5C2311yescharacteristic
43C2×C4×D4—51no—
44#432—51no—
45C4×M16—51no—
46D4⋊C8—54no—
47C23×C5C22⋊C411yescharacteristic
48C22×D5—24no—
49C22×D5—22no—
50C22×D5—42no—
51C22×D5—42no—
52C2×C4×C5—22no—
53C2×C4×C5—22no—
54C2×C4×C5—42no—
55C2×C4×C5C22⋊C411yescharacteristic
56C2×C4×C5C22×C411yescharacteristic
57C2×Dic5—22no—
58C2×Dic5—22no—
59C2×Dic5C22⋊C411yescharacteristic
60C2×Dic5C22⋊C411yescharacteristic
61C2×Dic5C2×D411yescharacteristic
62C2×Dic5C2×D411yescharacteristic
63D20—24no—
64D20—22no—
65D20M1614yescomplemented
66C4×D5—44no—
67C4×D5—44no—
68#170—24no—
69#170—42no—
70C22×D4—51no—
71C23×C4—101no—
72C2×M16—101no—
73C4×D4—52no—
74C4×D4—52no—
75C4×D4—101no—
76C4×D4—101no—
77C2×C4⋊C4AGL(1,𝔽5)11yescharacteristic
78C2×C22⋊C4—101no—
79C2×C42—51no—
80C4⋊C8—52no—
81C22⋊C8—101no—
82#97—101no—
83#97—101no—
84C4×C8—52no—
85C22×C5—21no—
86C22×C5—21no—
87C22×C5#10412yes—
88C22×C5C2×C22⋊C411yescharacteristic
89C2×D5—24no—
90C2×D5—48no—
91C2×D5—42no—
92C2×D5—42no—
93C4×C5—21no—
94C4×C5—42no—
95C4×C5C2×M1612yes—
96Dic5—21no—
97Dic5—21no—
98Dic5—21no—
99Dic5AGL(1,ℤ/8)12yes—
100C24—101no—
101C2×D4—52no—
102C2×D4—104no—
103C2×D4—101no—
104C22×C4—21no—
105C22×C4—51no—
106C22×C4—51no—
107C22×C4—102no—
108C22×C4—102no—
109C22×C4—204no—
110C22×C4C2×AGL(1,𝔽5)11yescharacteristic
111M16—104no—
112C2×C8—102no—
113C2×C8—102no—
114C4⋊C4—21no—
115C4⋊C4C2×AGL(1,𝔽5)12yes—
116C22⋊C4—102no—
117C22⋊C4—102no—
118C42—51no—
119C42—51no—
120C42—51no—
121C42—51no—
122C2×C5—22no—
123C2×C5#40711yescharacteristic
124C2×C5#41811yescharacteristic
125C2×C5C2×#10411yescharacteristic
126D5—44no—
127C23—104no—
128C23—102no—
129C23—202no—
130C23—202no—
131C23C22⋊AGL(1,𝔽5)11yescharacteristic
132D4—54no—
133D4—104no—
134D4—102no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—51no—
139C2×C4—51no—
140C2×C4—51no—
141C2×C4—51no—
142C2×C4—102no—
143C2×C4—102no—
144C2×C4—204no—
145C2×C4—204no—
146C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
147C2×C4C22×AGL(1,𝔽5)11yescharacteristic
148C8—104no—
149C8—202no—
150C5#149011yescharacteristic, complemented, Sylow
151C22—21no—
152C22—21no—
153C22—104no—
154C22—208no—
155C22—202no—
156C22—202no—
157C22#401012yes—
158C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
159C4—21no—
160C4—42no—
161C4—52no—
162C4—101no—
163C4—101no—
164C4—101no—
165C4D5⋊M1612yes—
166C2—22no—
167C2—204no—
168C2#6575511yescharacteristic
169C2#6576711yescharacteristic
170C2C2×#401011yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊C4C4
socleC22×C5C2×C22⋊C4
prime radicalC22×D20C4
cosocleC4×Dic5C23
hypercenterC2×C4⋊C4AGL(1,𝔽5)
hypocenterC5#1490
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #65759C2semidirect
3. C4×#704C2semidirect
4. #64871C2semidirect
18. C4⋊#170C22semidirect
19. C4×#170C22semidirect
65. D20M16semidirect
150. C5#1490semidirect

Maximal subgroups

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

subgroupindex
C2×#40392
#657592
C4×#7042
#648712
#14905