Finite Group Databasebeta

← #10604996

Subgroups — #10604996

Order 640.

258 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 54 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#66166C211yesmaximal, characteristic
2#65745C212yesmaximal, complemented
3#64928C212yesmaximal, complemented
4#64922C212yesmaximal, complemented
5C2×C5×#41C411yescharacteristic, complemented
6C2×#709C2211yescharacteristic
7#4130—22no—
8#4093C2211yescharacteristic
9#4090C411yescharacteristic, complemented
10#4086C2211yescharacteristic
11#4085—22no—
12#4082C411yescharacteristic, complemented
13C4×C22⋊D5—22no—
14C2×C4×Dic5C411yescharacteristic, complemented
15#4011—22no—
16#3999C2214yes—
17#2126—51nomaximal, Sylow
18C5×#41—41no—
19C5×#41—41no—
20C2×C5×Q8C2×C411yescharacteristic
21C2×C5×D4—21no—
22C2×C5×D4C2×C411yescharacteristic
23C22×C4×C5—21no—
24C22×C4×C5C2×C411yescharacteristic
25C2×C22⋊D5—22no—
26C2×C22⋊D5D412yes—
27C22×Dic5—21no—
28#709—41no—
29#709—41no—
30C2×D20C2311yescharacteristic
31C2×C4×D5D412yes—
32C2×Dic10C2×C411yescharacteristic
33C22⋊AGL(1,𝔽5)—24no—
34C4⋊AGL(1,𝔽5)—42no—
35C4×AGL(1,𝔽5)—42no—
36C22⋊Dic5—22no—
37#685—22no—
38#683—22no—
39C4×Dic5—21no—
40C4×Dic5C2×C411yescharacteristic
41C4×Dic5C2×C411yescharacteristic
42#532—51no—
43#410—52no—
44#354—52no—
45#353—52no—
46C23×C5—21no—
47C23×C5C22⋊C411yescharacteristic
48C22×D5C2×D412yes—
49C2×AGL(1,𝔽5)—44no—
50C5×Q8—41no—
51C5×D4—41no—
52C5×D4—42no—
53C2×C4×C5—21no—
54C2×C4×C5—21no—
55C2×C4×C5—21no—
56C2×C4×C5—41no—
57C2×C4×C5—41no—
58C2×C4×C5C22⋊C411yescharacteristic
59C2×C4×C5C22×C411yescharacteristic
60C22⋊D5—42no—
61C22⋊D5—44no—
62C2×Dic5—21no—
63C2×Dic5—21no—
64C2×Dic5—42no—
65C2×Dic5C22⋊C412yes—
66D20—41no—
67C4×D5—22no—
68C4×D5—22no—
69Dic10—41no—
70C2×#41—51no—
71C2×#41AGL(1,𝔽5)11yescharacteristic, complemented
72C4⋊Q8—51no—
73#127—51no—
74#124—51no—
75#124—51no—
76#121—102no—
77C4×D4—102no—
78#117—102no—
79C2×C42—51no—
80#99—54no—
81#99—102no—
82C22×C5—21no—
83C22×C5—41no—
84C22×C5—42no—
85C22×C5—41no—
86C22×C5C2×C22⋊C411yescharacteristic
87C2×D5—22no—
88C2×D5—42no—
89AGL(1,𝔽5)—84no—
90C4×C5—21no—
91C4×C5—41no—
92C4×C5#9911yescharacteristic
93C4×C5#9911yescharacteristic
94Dic5—22no—
95Dic5—22no—
96Dic5—41no—
97#41—41no—
98#41—41no—
99#41—201no—
100#41—201no—
101C2×Q8—51no—
102C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
103C2×D4—21no—
104C2×D4—52no—
105C2×D4—51no—
106C2×D4—102no—
107C2×D4C2×AGL(1,𝔽5)11yescharacteristic
108C22×C4—21no—
109C22×C4—52no—
110C22×C4—101no—
111C22×C4C2×AGL(1,𝔽5)11yescharacteristic
112C4⋊C4—102no—
113C4⋊C4—202no—
114C22⋊C4—102no—
115C22⋊C4—102no—
116C22⋊C4—104no—
117C42—51no—
118C42—51no—
119C42—101no—
120C42—202no—
121C2×C5—21no—
122C2×C5—41no—
123C2×C5—81no—
124C2×C5C2×#9911yescharacteristic
125D5—42no—
126C23—21no—
127C23—52no—
128C23C22⋊AGL(1,𝔽5)11yescharacteristic
129Q8—41no—
130Q8—201no—
131D4—41no—
132D4—42no—
133D4—201no—
134D4—202no—
135D4—204no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—52no—
142C2×C4—101no—
143C2×C4—102no—
144C2×C4—102no—
145C2×C4—101no—
146C2×C4—202no—
147C2×C4—204no—
148C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
149C2×C4C22×AGL(1,𝔽5)11yescharacteristic
150C5#212611yescharacteristic, complemented, Sylow
151C22—21no—
152C22—41no—
153C22—42no—
154C22—41no—
155C22—102no—
156C22—202no—
157C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
158C4—21no—
159C4—41no—
160C4—102no—
161C4—102no—
162C4—201no—
163C4—404no—
164C4#401111yescharacteristic
165C4#401111yescharacteristic
166C2—21no—
167C2—41no—
168C2—81no—
169C2—202no—
170C2C2×#401111yescharacteristic
subgroupG/H
centerC4#4011
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC2×C5×#41C4
socleC2×C5C2×#99
prime radical#66166C2
cosocleC2×D20C23
hypercenterC2×#41AGL(1,𝔽5)
hypocenterC5#2126
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #65745C2semidirect
3. #64928C2semidirect
4. #64922C2semidirect
5. C2×C5×#41C4semidirect
9. #4090C4semidirect
12. #4082C4semidirect
14. C2×C4×Dic5C4semidirect
71. C2×#41AGL(1,𝔽5)semidirect
150. C5#2126semidirect

Maximal subgroups

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

subgroupindex
#661662
#657452
#649282
#649222
#21265