Finite Group Databasebeta

← #13025020

Subgroups — #13025020

Order 1280.

360 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 64 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#10614934C211yesmaximal, characteristic
2#10604155C212yesmaximal, complemented
3#10598757C212yesmaximal, complemented
4#10598756C212yesmaximal, complemented
5C2×C5×C4⋊Q8C411yescharacteristic
6C2×#4088—22no—
7#65811C411yescharacteristic
8#65381C411yescharacteristic
9#65381C2211yescharacteristic
10#65375—22no—
11#65374C411yescharacteristic
12#65374C2211yescharacteristic
13C4×#681C2211yescharacteristic
14#64912—22no—
15#64861C2214yes—
16#64860—22no—
17#11993—51nomaximal, Sylow
18C22×C5×Q8—21no—
19C2×#709—21no—
20C5×C4⋊Q8—21no—
21C5×C4⋊Q8—41no—
22C5×C4⋊Q8C2×C411yescharacteristic
23C5×C4⋊Q8C2×C411yescharacteristic
24C2×C5×C4⋊C4—41no—
25C2×C42×C5C2×C411yescharacteristic
26C2×#689—22no—
27C2×#689—22no—
28#4088—48no—
29C2×#688—22no—
30C2×#688—22no—
31#4076—41no—
32C2×#681D412yes—
33#4021—21no—
34#4020C2×C411yescharacteristic
35C4×D20—41no—
36#4015C2×C411yescharacteristic
37#3998—24no—
38#3935D412yes—
39C4×#168C2×C411yescharacteristic
40C4×#168C2311yescharacteristic
41#3152—51no—
42#2165—52no—
43#1359—52no—
44#1358—52no—
45C2×C5×Q8—21no—
46C2×C5×Q8—21no—
47C2×C5×Q8—44no—
48C2×C5×Q8—41no—
49C22×C4×C5—21no—
50C22×C4×C5—41no—
51C22×C4×C5C22⋊C411yescharacteristic
52C2×C22⋊D5—41no—
53#709—44no—
54C2×D20—21no—
55C2×D20—41no—
56C2×C4×D5—41no—
57C2×Dic10—21no—
58C5×C4⋊C4—41no—
59C5×C4⋊C4—41no—
60C5×C4⋊C4—42no—
61C42×C5—21no—
62C42×C5C22⋊C411yescharacteristic
63C42×C5C22×C411yescharacteristic
64#689—44no—
65#689—44no—
66#688—44no—
67#688—44no—
68#685—42no—
69C4⋊Dic5—41no—
70#681—24no—
71C2×#168C22⋊C412yes—
72C2×#168C2×D412yes—
73#674—44no—
74C2×#137—102no—
75#532—51no—
76C2×C4⋊Q8AGL(1,𝔽5)11yescharacteristic
77#497—102no—
78#494—51no—
79#494—51no—
80#492—51no—
81#492—51no—
82C4×M16—51no—
83#364—54no—
84#349—102no—
85#347—102no—
86C23×C5#10011yescharacteristic
87C22×D5—41no—
88C5×Q8—42no—
89C5×Q8—42no—
90C5×Q8—82no—
91C2×C4×C5—21no—
92C2×C4×C5—22no—
93C2×C4×C5—21no—
94C2×C4×C5—42no—
95C2×C4×C5—41no—
96C2×C4×C5—41no—
97C2×C4×C5—82no—
98C2×C4×C5#10011yescharacteristic
99C2×C4×C5C2×C22⋊C411yescharacteristic
100C22⋊D5—82no—
101C2×Dic5—41no—
102D20—42no—
103D20—82no—
104C4×D5—82no—
105Dic10—42no—
106#168—22no—
107#168AGL(1,ℤ/8)14yes—
108C2×#41—101no—
109C22×Q8—21no—
110#137—208no—
111C2×Q16—102no—
112C2×Q16—102no—
113C2×SD16—102no—
114C2×SD16—102no—
115C2×M16—52no—
116C4⋊Q8—21no—
117C4⋊Q8—41no—
118C4⋊Q8—51no—
119C4⋊Q8C2×AGL(1,𝔽5)11yescharacteristic
120C4⋊Q8C2×AGL(1,𝔽5)11yescharacteristic
121#127—51no—
122#124—101no—
123#121—201no—
124C4×D4—201no—
125C2×C4⋊C4—41no—
126C2×C42C2×AGL(1,𝔽5)11yescharacteristic
127M32—104no—
128#97—52no—
129C4×C8—51no—
130C4×C8—51no—
131C22×C5—21no—
132C22×C5—41no—
133C22×C5C2×#10011yescharacteristic
134C2×D5—82no—
135C2×D5—81no—
136C4×C5—22no—
137C4×C5—21no—
138C4×C5—41no—
139C4×C5—81no—
140C4×C5—81no—
141C4×C5#41812yes—
142Dic5—81no—
143#41—204no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—44no—
147C2×Q8—41no—
148C2×Q8—101no—
149C2×D4—101no—
150C2×D4—201no—
151C2×D4—201no—
152C22×C4—21no—
153C22×C4—41no—
154C22×C4—201no—
155C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
156Q16—204no—
157Q16—204no—
158SD16—204no—
159SD16—204no—
160M16—104no—
161C2×C8—52no—
162C2×C8—52no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—201no—
167C22⋊C4—202no—
168C42—21no—
169C42C22⋊AGL(1,𝔽5)11yescharacteristic
170C42C22×AGL(1,𝔽5)11yescharacteristic
171C16—204no—
172C2×C5—21no—
173C2×C5—41no—
174C2×C5#155511yescharacteristic
175D5—161no—
176C23—201no—
177C23#401311yescharacteristic
178Q8—42no—
179Q8—42no—
180Q8—82no—
181Q8—202no—
182D4—202no—
183D4—402no—
184D4—402no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—82no—
192C2×C4—201no—
193C2×C4—402no—
194C2×C4#401311yescharacteristic
195C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
196C8—54no—
197C8—102no—
198C5#1199311yescharacteristic, complemented, Sylow
199C22—21no—
200C22—41no—
201C22—402no—
202C22—401no—
203C22C2×#401311yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—401no—
210C4#6576712yes—
211C2—21no—
212C2—41no—
213C2—801no—
214C2#1060492711yescharacteristic
subgroupG/H
centerC2#10604927
commutatorC42×C5C22×C4
FrattiniC42C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊Q8C4
socleC2×C5#1555
prime radical#65811C4
cosocleC4×#168C23
hypercenterC2×C4⋊Q8AGL(1,𝔽5)
hypocenterC5#11993
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
2. #10604155C2semidirect
3. #10598757C2semidirect
4. #10598756C2semidirect
198. C5#11993semidirect

Maximal subgroups

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

subgroupindex
#106149342
#106041552
#105987572
#105987562
#119935