Finite Group Databasebeta

← #10604219

Subgroups — #10604219

Order 640.

296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 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
1#65820C211yesmaximal, characteristic
2C2×#4011C212yesmaximal
3#64866C214yesmaximal, complemented
4C22×C22⋊D5C2211yescharacteristic
5C2×C22⋊AGL(1,𝔽5)C2214yes—
6C2×C5×C22⋊C4C411yescharacteristic, complemented
7C2×C22⋊Dic5C411yescharacteristic, complemented
8C4×C22⋊D5—22no—
9#4027—22no—
10D5×C22⋊C4C2212yes—
11#4023C412yescomplemented
12#4011—22no—
13#4011—22no—
14#1828—51nomaximal, Sylow
15C24×C5C2×C411yescharacteristic
16C23×D5C2311yescharacteristic
17C22×AGL(1,𝔽5)—24no—
18C22×C4×C5—21no—
19C2×C22⋊D5—24no—
20C2×C22⋊D5D412yes—
21C2×C22⋊D5D412yes—
22C22×Dic5—21no—
23C22×Dic5C2×C411yescharacteristic
24C2×C4×D5—22no—
25C22⋊AGL(1,𝔽5)—24no—
26C22⋊AGL(1,𝔽5)—24no—
27C5×C22⋊C4—21no—
28C5×C22⋊C4C2×C412yes—
29C22⋊Dic5—21no—
30C22⋊Dic5C2×C412yes—
31#685—22no—
32#683—22no—
33C4×Dic5—22no—
34#518—51no—
35C2×#99—52no—
36#342—54no—
37C23×C5—22no—
38C23×C5—81no—
39C23×C5C22⋊C411yescharacteristic
40C23×C5C22⋊C411yescharacteristic
41C23×C5C22×C411yescharacteristic
42C22×D5—81no—
43C22×D5C2×D412yes—
44C22×D5#4114yes—
45C2×AGL(1,𝔽5)—44no—
46C2×AGL(1,𝔽5)—84no—
47C2×C4×C5—22no—
48C2×C4×C5—81no—
49C22⋊D5—24no—
50C22⋊D5—24no—
51C2×Dic5—22no—
52C2×Dic5—22no—
53C2×Dic5—81no—
54C2×Dic5C22⋊C412yes—
55C4×D5—82no—
56C22×D4—51no—
57C4×D4—102no—
58C4×D4—102no—
59#117—52no—
60C2×C22⋊C4—51no—
61C2×C22⋊C4—52no—
62C2×C22⋊C4—54no—
63C2×C22⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
64#99—102no—
65#99—102no—
66C22×C5—21no—
67C22×C5—21no—
68C22×C5—41no—
69C22×C5—41no—
70C22×C5—41no—
71C22×C5—82no—
72C22×C5C2×C22⋊C411yescharacteristic
73C22×C5#11712yes—
74C2×D5—42no—
75C2×D5—44no—
76C2×D5#12314yes—
77AGL(1,𝔽5)—84no—
78C4×C5—42no—
79Dic5—22no—
80Dic5—42no—
81C24—51no—
82C24C2×AGL(1,𝔽5)11yescharacteristic
83C2×D4—52no—
84C2×D4—52no—
85C2×D4—104no—
86C22×C4—21no—
87C22×C4—51no—
88C22×C4—102no—
89C22×C4—101no—
90C22×C4—104no—
91C4⋊C4—102no—
92C22⋊C4—21no—
93C22⋊C4—52no—
94C22⋊C4—102no—
95C22⋊C4—101no—
96C22⋊C4—104no—
97C22⋊C4—104no—
98C22⋊C4C2×AGL(1,𝔽5)12yes—
99C42—102no—
100C2×C5—21no—
101C2×C5—22no—
102C2×C5—41no—
103C2×C5—41no—
104C2×C5#38111yescharacteristic
105D5—24no—
106C23—22no—
107C23—54no—
108C23—52no—
109C23—81no—
110C23—401no—
111C23C22⋊AGL(1,𝔽5)11yescharacteristic
112C23C22⋊AGL(1,𝔽5)11yescharacteristic
113C23C22×AGL(1,𝔽5)11yescharacteristic
114D4—104no—
115D4—104no—
116C2×C4—22no—
117C2×C4—52no—
118C2×C4—81no—
119C2×C4—102no—
120C2×C4—102no—
121C2×C4—204no—
122C2×C4—402no—
123C2×C4—401no—
124C2×C4—404no—
125C5#182811yescharacteristic, complemented, Sylow
126C22—21no—
127C22—21no—
128C22—41no—
129C22—41no—
130C22—41no—
131C22—54no—
132C22—82no—
133C22—202no—
134C22—204no—
135C22#413012yes—
136C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
137C4—42no—
138C4—102no—
139C4—202no—
140C4—404no—
141C2—21no—
142C2—22no—
143C2—41no—
144C2—41no—
145C2—104no—
146C2#6576411yescharacteristic
subgroupG/H
centerC2#65764
commutatorC23×C5C22×C4
FrattiniC23C22×AGL(1,𝔽5)
FittingC2×C5×C22⋊C4C4
socleC2×C5#381
prime radicalC22×C22⋊D5C22
cosocleC23×D5C23
hypercenterC2×C22⋊C4AGL(1,𝔽5)
hypocenterC5#1828
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #64866C2semidirect
6. C2×C5×C22⋊C4C4semidirect
7. C2×C22⋊Dic5C4semidirect
11. #4023C4semidirect
63. C2×C22⋊C4AGL(1,𝔽5)semidirect
125. C5#1828semidirect

Maximal subgroups

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

subgroupindex
#658202
C2×#40112
#648662
#18285