Finite Group Databasebeta

← #10604666

Subgroups — #10604666

Order 640.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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#65686C211yesmaximal, characteristic
3#65682C211yesmaximal, characteristic
4#64875C214yesmaximal
5C2×C4×AGL(1,𝔽5)C2212yes—
6C2×#700C2212yes—
7C5×C4⋊Q8C411yescharacteristic, complemented
8#4092—21no—
9Q8×Dic5—21no—
10#4043—21no—
11#4038C2212yes—
12#4036C412yescomplemented
13#4033—21no—
14C42×D5C2211yescharacteristic
15#4015C411yescharacteristic, complemented
16(C2×AGL(1,𝔽5))⋊C4—21no—
17(C2×AGL(1,𝔽5))⋊C4—21no—
18#4004—21no—
19#4004—21no—
20#1995—51nomaximal, Sylow
21C22×AGL(1,𝔽5)—22no—
22C2×C5×Q8—21no—
23C2×C4×D5D411yescharacteristic
24C2×C4×D5D411yescharacteristic
25C2×C4×D5C2311yescharacteristic
26C2×Dic10—21no—
27C2×#170—22no—
28C4×AGL(1,𝔽5)—22no—
29C4×AGL(1,𝔽5)Q814yescomplemented
30#700—24no—
31C5×C4⋊C4—21no—
32C5×C4⋊C4C2×C412yes—
33C42×C5C2×C411yescharacteristic
34#685—22no—
35C4⋊Dic5—21no—
36C4⋊Dic5—22no—
37C4⋊Dic5C2×C412yes—
38#683—22no—
39C4×Dic5—22no—
40C4×Dic5C2×C411yescharacteristic
41C4×Dic5D411yescharacteristic
42C4×Dic5D411yescharacteristic
43#533—51no—
44#423—51no—
45#376—51no—
46#339—54no—
47C22×D5C2×D411yescharacteristic
48C2×AGL(1,𝔽5)—24no—
49C2×AGL(1,𝔽5)—42no—
50C2×AGL(1,𝔽5)—42no—
51C5×Q8—42no—
52C2×C4×C5—22no—
53C2×C4×C5C22⋊C411yescharacteristic
54C2×C4×C5C22⋊C411yescharacteristic
55C2×C4×C5C22×C411yescharacteristic
56C2×Dic5—22no—
57C2×Dic5C22⋊C411yescharacteristic
58C2×Dic5C22⋊C411yescharacteristic
59C2×Dic5C2×D411yescharacteristic
60C4×D5—41no—
61C4×D5—41no—
62C4×D5C2×Q812yes—
63C4×D5#4112yes—
64Dic10—42no—
65#170—42no—
66C2×M16—52no—
67C4⋊Q8—51no—
68C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
69#125—52no—
70C22⋊Q8—101no—
71C22⋊Q8—101no—
72C4×Q8—101no—
73C4×Q8—101no—
74#117—52no—
75C2×C42—51no—
76C2×C42—52no—
77C4⋊C8—101no—
78C4⋊C8—101no—
79#95—101no—
80#95—101no—
81C22×C5C2×C22⋊C411yescharacteristic
82C2×D5—21no—
83C2×D5—21no—
84C2×D5C22⋊Q812yes—
85AGL(1,𝔽5)—44no—
86C4×C5—21no—
87C4×C5—21no—
88C4×C5—42no—
89C4×C5C4×Q812yes—
90Dic5—21no—
91Dic5—21no—
92Dic5—42no—
93Dic5#12411yescharacteristic
94Dic5C4⋊Q811yescharacteristic
95C2×Q8—21no—
96C2×Q8—101no—
97C22×C4—51no—
98C22×C4—51no—
99C22×C4—51no—
100C22×C4—102no—
101M16—104no—
102C2×C8—102no—
103C4⋊C4—21no—
104C4⋊C4—52no—
105C4⋊C4—102no—
106C4⋊C4—102no—
107C4⋊C4—101no—
108C4⋊C4C2×AGL(1,𝔽5)12yes—
109C22⋊C4—102no—
110C42—51no—
111C42—51no—
112C42—51no—
113C42—54no—
114C42—102no—
115C42—102no—
116C42C2×AGL(1,𝔽5)11yescharacteristic
117C2×C5#39111yescharacteristic
118C2×C5#42111yescharacteristic
119C2×C5#42111yescharacteristic
120D5—22no—
121C23—51no—
122Q8—42no—
123Q8—202no—
124C2×C4—22no—
125C2×C4—51no—
126C2×C4—52no—
127C2×C4—52no—
128C2×C4—51no—
129C2×C4—51no—
130C2×C4—104no—
131C2×C4—102no—
132C2×C4—201no—
133C2×C4—201no—
134C2×C4—202no—
135C2×C4—202no—
136C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
137C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
138C2×C4C22×AGL(1,𝔽5)11yescharacteristic
139C8—202no—
140C5#199511yescharacteristic, complemented, Sylow
141C22—52no—
142C22—101no—
143C22—101no—
144C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
145C4—21no—
146C4—21no—
147C4—42no—
148C4—51no—
149C4—51no—
150C4—101no—
151C4—101no—
152C4—204no—
153C4—202no—
154C4Q8×AGL(1,𝔽5)12yes—
155C2—102no—
156C2#6576911yescharacteristic
157C2#6578411yescharacteristic
158C2(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
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#1995
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
7. C5×C4⋊Q8C4semidirect
12. #4036C4semidirect
15. #4015C4semidirect
29. C4×AGL(1,𝔽5)Q8semidirect
68. C4⋊Q8AGL(1,𝔽5)semidirect
140. C5#1995semidirect

Maximal subgroups

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

subgroupindex
#660622
#656862
#656822
#648752
#19955