Finite Group Databasebeta

← #10604556

Subgroups — #10604556

Order 640.

246 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 60 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#66002C211yesmaximal, characteristic
2#65678C212yesmaximal, complemented
3#64931C212yesmaximal, complemented
4#64925C212yesmaximal, complemented
5C2×#713C411yescharacteristic
6C2×#711C411yescharacteristic
7C5×#124C411yescharacteristic
8#4090C2211yescharacteristic
9#4086C2211yescharacteristic
10#4030—22no—
11#4027—22no—
12#4021C411yescharacteristic
13C42×D5C2211yescharacteristic
14#4004C2214yes—
15#4003—22no—
16C4×#170—22no—
17#1543—51nomaximal, Sylow
18C2×C5×Q8C2×C411yescharacteristic
19C2×C5×D4C2×C411yescharacteristic
20C2×C22⋊D5—21no—
21C2×C22⋊D5—22no—
22C22×Dic5—21no—
23#713—42no—
24#711—42no—
25C2×D20—21no—
26C2×D20C2×C411yescharacteristic
27C2×C4×D5—21no—
28C2×C4×D5C2×C411yescharacteristic
29C2×C4×D5D412yes—
30C2×Dic10C2×C411yescharacteristic
31C2×#170—24no—
32C5×C22⋊C4—22no—
33C42×C5C2×C411yescharacteristic
34#685—22no—
35#683—22no—
36C4×Dic5D412yes—
37C4×Dic5C2311yescharacteristic
38#532—51no—
39#431—52no—
40#333—52no—
41#331—52no—
42C23×C5—21no—
43C22×D5—21no—
44C22×D5C22⋊C411yescharacteristic
45C5×Q8—41no—
46C5×D4—41no—
47C2×C4×C5—21no—
48C2×C4×C5C22⋊C412yes—
49C2×C4×C5C22×C411yescharacteristic
50C22⋊D5—42no—
51C22⋊D5—44no—
52C2×Dic5—21no—
53C2×Dic5—42no—
54C2×Dic5C22⋊C411yescharacteristic
55C2×Dic5C2×D412yes—
56D20—41no—
57D20—42no—
58C4×D5—22no—
59C4×D5—24no—
60C4×D5—42no—
61Dic10—41no—
62#170—44no—
63C2×#41—51no—
64C2×#41—51no—
65C4⋊Q8—51no—
66#127—51no—
67#124—51no—
68#124AGL(1,𝔽5)11yescharacteristic
69#121—102no—
70C4×D4—102no—
71C2×C42—51no—
72C4⋊C8—54no—
73C22⋊C8—102no—
74C4×C8—102no—
75C22×C5—41no—
76C22×C5—42no—
77C22×C5C2×C22⋊C411yescharacteristic
78C2×D5—21no—
79C2×D5—22no—
80C2×D5—41no—
81C2×D5—42no—
82C4×C5—21no—
83C4×C5—22no—
84C4×C5—41no—
85Dic5—41no—
86Dic5#10012yes—
87Dic5#13514yes—
88#41—202no—
89#41—202no—
90C2×Q8—51no—
91C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
92C2×D4—51no—
93C2×D4—101no—
94C2×D4—102no—
95C2×D4—101no—
96C2×D4C2×AGL(1,𝔽5)11yescharacteristic
97C22×C4—52no—
98C22×C4—51no—
99C22×C4—101no—
100C22×C4—101no—
101C2×C8—104no—
102C4⋊C4—102no—
103C22⋊C4—22no—
104C22⋊C4—102no—
105C42—52no—
106C42—51no—
107C42C2×AGL(1,𝔽5)11yescharacteristic
108C2×C5—81no—
109C2×C5C2×#10011yescharacteristic
110C2×C5#41612yes—
111D5—41no—
112D5—81no—
113C23—21no—
114C23—51no—
115C23—101no—
116Q8—41no—
117Q8—201no—
118D4—41no—
119D4—201no—
120D4—202no—
121D4—204no—
122D4—202no—
123C2×C4—21no—
124C2×C4—52no—
125C2×C4—51no—
126C2×C4—102no—
127C2×C4—104no—
128C2×C4—101no—
129C2×C4—202no—
130C2×C4—202no—
131C2×C4C22⋊AGL(1,𝔽5)12yes—
132C2×C4C22×AGL(1,𝔽5)11yescharacteristic
133C8—204no—
134C5#154311yescharacteristic, complemented, Sylow
135C22—42no—
136C22—41no—
137C22—102no—
138C22—101no—
139C22—202no—
140C22—201no—
141C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
142C4—22no—
143C4—21no—
144C4—41no—
145C4—54no—
146C4—52no—
147C4—201no—
148C2—81no—
149C2—201no—
150C2—401no—
151C2D5⋊#10011yescharacteristic
152C2#6579412yes—
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#124C4
socleC22×C5C2×C22⋊C4
prime radical#66002C2
cosocleC4×Dic5C23
hypercenter#124AGL(1,𝔽5)
hypocenterC5#1543
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #65678C2semidirect
3. #64931C2semidirect
4. #64925C2semidirect
134. C5#1543semidirect

Maximal subgroups

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

subgroupindex
#660022
#656782
#649312
#649252
#15435