Finite Group Databasebeta

← #10604639

Subgroups — #10604639

Order 640.

194 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 56 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#65684C211yesmaximal, characteristic
3#65684C211yesmaximal, characteristic
4#64870C214yesmaximal, complemented
5C5×C4⋊Q8C411yescharacteristic
6#4092—21no—
7Q8×Dic5—21no—
8#4043—21no—
9#4038C412yes—
10#4036C2212yes—
11#4033—21no—
12C42×D5C2211yescharacteristic
13#4015C411yescharacteristic
14#4004—21no—
15#4004—21no—
16#4003—21no—
17#4003—21no—
18#4002C2212yes—
19#4002C2212yes—
20#1537—51nomaximal, Sylow
21C2×C5×Q8—21no—
22C2×C4×D5C2×C411yescharacteristic
23C2×C4×D5D411yescharacteristic
24C2×C4×D5D411yescharacteristic
25C2×Dic10—21no—
26C2×#170—22no—
27C2×#170—22no—
28C5×C4⋊C4—21no—
29C5×C4⋊C4C2×C412yes—
30C42×C5C2×C411yescharacteristic
31#685—22no—
32C4⋊Dic5—21no—
33C4⋊Dic5—22no—
34C4⋊Dic5C2×C412yes—
35#683—22no—
36C4×Dic5—22no—
37C4×Dic5D411yescharacteristic
38C4×Dic5D411yescharacteristic
39C4×Dic5C2311yescharacteristic
40#533—51no—
41#432—51no—
42#432—51no—
43#330—54no—
44C22×D5C22⋊C411yescharacteristic
45C5×Q8—42no—
46C2×C4×C5—22no—
47C2×C4×C5C22⋊C411yescharacteristic
48C2×C4×C5C22⋊C411yescharacteristic
49C2×C4×C5C22×C411yescharacteristic
50C2×Dic5—22no—
51C2×Dic5C22⋊C411yescharacteristic
52C2×Dic5C2×D411yescharacteristic
53C2×Dic5C2×D411yescharacteristic
54C4×D5—22no—
55C4×D5—41no—
56C4×D5—41no—
57Dic10—42no—
58#170—42no—
59#170—42no—
60C4⋊Q8—51no—
61C4⋊Q8AGL(1,𝔽5)11yescharacteristic
62#125—52no—
63C22⋊Q8—101no—
64C22⋊Q8—101no—
65C4×Q8—101no—
66C4×Q8—101no—
67#117—52no—
68C2×C42—51no—
69C4⋊C8—101no—
70C4⋊C8—101no—
71C22⋊C8—101no—
72C22⋊C8—101no—
73#97—52no—
74#97—52no—
75C22×C5C2×C22⋊C411yescharacteristic
76C2×D5—21no—
77C2×D5—21no—
78C2×D5—21no—
79C4×C5—21no—
80C4×C5—21no—
81C4×C5—21no—
82C4×C5—42no—
83Dic5—21no—
84Dic5—21no—
85Dic5—42no—
86Dic5#10112yes—
87C2×Q8—21no—
88C2×Q8—101no—
89C22×C4—51no—
90C22×C4—51no—
91C22×C4—51no—
92C2×C8—102no—
93C2×C8—102no—
94C4⋊C4—21no—
95C4⋊C4—52no—
96C4⋊C4—102no—
97C4⋊C4—102no—
98C4⋊C4—101no—
99C4⋊C4C2×AGL(1,𝔽5)12yes—
100C22⋊C4—102no—
101C42—51no—
102C42—51no—
103C42—51no—
104C42—102no—
105C42C2×AGL(1,𝔽5)11yescharacteristic
106C2×C5C2×#10111yescharacteristic
107C2×C5#42111yescharacteristic
108C2×C5#42111yescharacteristic
109D5—41no—
110C23—51no—
111Q8—42no—
112Q8—202no—
113C2×C4—22no—
114C2×C4—51no—
115C2×C4—51no—
116C2×C4—51no—
117C2×C4—102no—
118C2×C4—102no—
119C2×C4—201no—
120C2×C4—201no—
121C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
122C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
123C2×C4C22×AGL(1,𝔽5)11yescharacteristic
124C8—202no—
125C8—202no—
126C5#153711yescharacteristic, complemented, Sylow
127C22—101no—
128C22—101no—
129C22—101no—
130C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
131C4—21no—
132C4—21no—
133C4—21no—
134C4—42no—
135C4—52no—
136C4—101no—
137C4—101no—
138C4—202no—
139C2—201no—
140C2#6576911yescharacteristic
141C2#6578411yescharacteristic
142C2D5⋊#10111yescharacteristic
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×D5C22⋊C4
cosocleC4×Dic5C23
hypercenterC4⋊Q8AGL(1,𝔽5)
hypocenterC5#1537
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
4. #64870C2semidirect
126. C5#1537semidirect

Maximal subgroups

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

subgroupindex
#660622
#656842
#648702
#15375