Finite Group Databasebeta

← #10604632

Subgroups — #10604632

Order 640.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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#66062C211yesmaximal, characteristic
2#65689C211yesmaximal, characteristic
3#65678C211yesmaximal, characteristic
4#64926C214yesmaximal
5C2×C4⋊AGL(1,𝔽5)—21no—
6C5×C4⋊Q8C411yescharacteristic, complemented
7#4092C2212yes—
8Q8×Dic5C412yescomplemented
9#4043—21no—
10#4038—21no—
11#4036—21no—
12#4033—21no—
13C42×D5C2211yescharacteristic
14#4015C411yescharacteristic, complemented
15(C2×AGL(1,𝔽5))⋊C4—21no—
16(C2×AGL(1,𝔽5))⋊C4C2212yes—
17#4004—21no—
18#4003C2212yes—
19C4×#170—21no—
20#1518—51nomaximal, Sylow
21C22×AGL(1,𝔽5)—22no—
22C2×C5×Q8C2×C412yes—
23C2×C4×D5D411yescharacteristic
24C2×C4×D5D411yescharacteristic
25C2×C4×D5C2311yescharacteristic
26C2×Dic10—21no—
27C2×#170—22no—
28C4⋊AGL(1,𝔽5)—42no—
29C5×C4⋊C4—21no—
30C5×C4⋊C4—21no—
31C42×C5C2×C411yescharacteristic
32#685—22no—
33C4⋊Dic5—21no—
34C4⋊Dic5—22no—
35C4⋊Dic5C2×C412yes—
36#683—22no—
37C4×Dic5—22no—
38C4×Dic5C2×C411yescharacteristic
39C4×Dic5D411yescharacteristic
40C4×Dic5D411yescharacteristic
41#533—51no—
42#431—51no—
43#382—51no—
44#328—54no—
45C22×D5C2×D411yescharacteristic
46C2×AGL(1,𝔽5)—42no—
47C2×AGL(1,𝔽5)—42no—
48C2×AGL(1,𝔽5)—42no—
49C5×Q8—42no—
50C2×C4×C5—22no—
51C2×C4×C5C22⋊C411yescharacteristic
52C2×C4×C5C22⋊C411yescharacteristic
53C2×C4×C5C22×C411yescharacteristic
54C2×Dic5—22no—
55C2×Dic5C22⋊C411yescharacteristic
56C2×Dic5C22⋊C411yescharacteristic
57C2×Dic5C2×D411yescharacteristic
58C4×D5—41no—
59C4×D5—41no—
60C4×D5SD1612yes—
61C4×D5Q1612yes—
62Dic10—42no—
63#170—42no—
64C4⋊Q8—51no—
65C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
66#125—101no—
67C22⋊Q8—52no—
68C22⋊Q8—101no—
69C4×Q8—52no—
70C4×Q8—101no—
71#117—101no—
72C2×C4⋊C4—101no—
73C2×C42—51no—
74C4⋊C8—101no—
75C22⋊C8—52no—
76C4×C8—101no—
77#95—52no—
78#95—101no—
79C22×C5C2×C22⋊C411yescharacteristic
80C2×D5—21no—
81C2×D5—21no—
82C2×D5C2×SD1611yescharacteristic
83C2×D5C2×Q1611yescharacteristic
84AGL(1,𝔽5)—82no—
85C4×C5—21no—
86C4×C5—21no—
87C4×C5—42no—
88C4×C5#10312yes—
89Dic5—21no—
90Dic5—21no—
91Dic5—42no—
92Dic5#10312yes—
93C2×Q8—101no—
94C2×Q8C2×AGL(1,𝔽5)12yes—
95C22×C4—51no—
96C22×C4—51no—
97C22×C4—51no—
98C22×C4—102no—
99C2×C8—102no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—52no—
103C4⋊C4—102no—
104C4⋊C4—102no—
105C4⋊C4—101no—
106C4⋊C4—202no—
107C22⋊C4—102no—
108C42—51no—
109C42—51no—
110C42—51no—
111C42—102no—
112C42C2×AGL(1,𝔽5)11yescharacteristic
113C2×C5#41011yescharacteristic
114C2×C5C2×#10311yescharacteristic
115C2×C5#42111yescharacteristic
116D5—21no—
117D5—21no—
118C23—51no—
119Q8—42no—
120Q8—202no—
121C2×C4—22no—
122C2×C4—51no—
123C2×C4—52no—
124C2×C4—52no—
125C2×C4—51no—
126C2×C4—51no—
127C2×C4—102no—
128C2×C4—201no—
129C2×C4—201no—
130C2×C4—202no—
131C2×C4—202no—
132C2×C4—202no—
133C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
134C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
135C2×C4C22×AGL(1,𝔽5)11yescharacteristic
136C8—202no—
137C5#151811yescharacteristic, complemented, Sylow
138C22—51no—
139C22—51no—
140C22—101no—
141C22—101no—
142C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
143C4—21no—
144C4—21no—
145C4—42no—
146C4—52no—
147C4—101no—
148C4—101no—
149C4—202no—
150C4—402no—
151C4#400912yes—
152C2—101no—
153C2—101no—
154C2#6577111yescharacteristic
155C2C2×#400911yescharacteristic
156C2#6578411yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×C4⋊Q8C4
socleC22×C5C2×C22⋊C4
prime radicalC22×D5C2×D4
cosocleC2×C4×D5C23
hypercenterC4⋊Q8AGL(1,𝔽5)
hypocenterC5#1518
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
6. C5×C4⋊Q8C4semidirect
8. Q8×Dic5C4semidirect
14. #4015C4semidirect
65. C4⋊Q8AGL(1,𝔽5)semidirect
137. C5#1518semidirect

Maximal subgroups

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

subgroupindex
#660622
#656892
#656782
#649262
#15185