Finite Group Databasebeta

← #13024819

Subgroups — #13024819

Order 1280.

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

normal subgroups only

This group has 274 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#10614929C211yesmaximal, characteristic
2#10604155C212yesmaximal
3#10598752C214yesmaximal, complemented
4C2×C5×C4⋊Q8C411yescharacteristic
5#65804C411yescharacteristic
6#65514—22no—
7#65371C412yes—
8#65371C2212yes—
9#65370—22no—
10C4×#681C2211yescharacteristic
11#64912—22no—
12#64861—22no—
13#64860C2214yes—
14#11916—51nomaximal, Sylow
15C22×C5×Q8—21no—
16C5×C4⋊Q8—21no—
17C5×C4⋊Q8—41no—
18C5×C4⋊Q8C2×C412yes—
19C2×C5×C4⋊C4—41no—
20C2×C42×C5C2×C411yescharacteristic
21#4072—21no—
22#4070—41no—
23C2×#681D412yes—
24#4016C2×C412yes—
25#4015—21no—
26C4×Dic10—41no—
27#3998—24no—
28#3967—24no—
29#3967—24no—
30#3935C2×C411yescharacteristic
31#3935C2311yescharacteristic
32C4×#168D412yes—
33#3139—51no—
34#2165—52no—
35#1356—54no—
36C2×C5×Q8—22no—
37C2×C5×Q8—44no—
38C2×C5×Q8—41no—
39C22×C4×C5—21no—
40C22×C4×C5—41no—
41C22×C4×C5C22⋊C411yescharacteristic
42C2×Dic10—41no—
43C5×C4⋊C4—42no—
44C5×C4⋊C4—42no—
45C42×C5—21no—
46C42×C5C22⋊C411yescharacteristic
47C42×C5C22×C411yescharacteristic
48C22⋊Dic5—41no—
49C4⋊Dic5—22no—
50C4⋊Dic5—41no—
51#683—42no—
52C4×Dic5—41no—
53#681—42no—
54C2×#168C22⋊C412yes—
55C2×#168C2×D412yes—
56#674—44no—
57#533—51no—
58C2×C4⋊Q8AGL(1,𝔽5)11yescharacteristic
59#491—52no—
60#491—52no—
61#487—102no—
62#419—102no—
63C4×M16—51no—
64#364—102no—
65#349—102no—
66#347—54no—
67C23×C5#10011yescharacteristic
68C5×Q8—44no—
69C5×Q8—82no—
70C2×C4×C5—21no—
71C2×C4×C5—22no—
72C2×C4×C5—21no—
73C2×C4×C5—42no—
74C2×C4×C5—42no—
75C2×C4×C5—82no—
76C2×C4×C5#10011yescharacteristic
77C2×C4×C5C2×C22⋊C411yescharacteristic
78C2×Dic5—42no—
79Dic10—82no—
80#168—22no—
81#168—22no—
82C22×Q8—21no—
83C2×M16—52no—
84C4⋊Q8—21no—
85C4⋊Q8—41no—
86C4⋊Q8—101no—
87C4⋊Q8C2×AGL(1,𝔽5)12yes—
88#125—52no—
89C22⋊Q8—201no—
90C4×Q8—201no—
91#117—101no—
92C2×C4⋊C4—41no—
93C2×C42C2×AGL(1,𝔽5)11yescharacteristic
94M32—104no—
95#103—104no—
96#103—104no—
97#97—51no—
98#97—51no—
99C4×C8—52no—
100C22×C5—21no—
101C22×C5—41no—
102C22×C5C2×#10011yescharacteristic
103C4×C5—22no—
104C4×C5—21no—
105C4×C5—41no—
106C4×C5—82no—
107C4×C5#42112yes—
108Dic5—82no—
109C2×Q8—22no—
110C2×Q8—44no—
111C2×Q8—41no—
112C2×Q8—201no—
113C22×C4—21no—
114C22×C4—41no—
115C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
116M16—202no—
117C2×C8—52no—
118C2×C8—52no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—102no—
122C4⋊C4—202no—
123C4⋊C4—201no—
124C22⋊C4—201no—
125C42—21no—
126C42—201no—
127C42C22⋊AGL(1,𝔽5)11yescharacteristic
128C42C22×AGL(1,𝔽5)11yescharacteristic
129C16—204no—
130C2×C5—21no—
131C2×C5—41no—
132C2×C5#153311yescharacteristic
133C23#401311yescharacteristic
134Q8—44no—
135Q8—82no—
136Q8—402no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—82no—
143C2×C4—202no—
144C2×C4#401311yescharacteristic
145C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
146C8—102no—
147C8—102no—
148C5#1191611yescharacteristic, complemented, Sylow
149C22—21no—
150C22—41no—
151C22C2×#401311yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—82no—
156C4—402no—
157C4#6576912yes—
158C2—21no—
159C2—41no—
160C2#1060491611yescharacteristic
subgroupG/H
centerC2#10604916
commutatorC42×C5C22×C4
FrattiniC42C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊Q8C4
socleC2×C5#1533
prime radicalC23×C5#100
cosocle#3935C23
hypercenterC2×C4⋊Q8AGL(1,𝔽5)
hypocenterC5#11916
LCSG ▷ C42×C5 ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #10598752C2semidirect
148. C5#11916semidirect

Maximal subgroups

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

subgroupindex
#106149292
#106041552
#105987522
#119165