Finite Group Databasebeta

← #13016199

Subgroups — #13016199

Order 1280.

698 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C2×#65682C212yesmaximal
2#10605879C211yesmaximal, characteristic
3#10598725C214yesmaximal, complemented
4C22×#700—22no—
5C2×#4038C412yes—
6C2×C42×D5C2211yescharacteristic
7C2×#4004—22no—
8C2×#4004C2214yes—
9#65682—24no—
10#65682C418yescomplemented
11C5×#382C411yescharacteristic
12#65265C2212yes—
13#65225C411yescharacteristic
14#64956—22no—
15#64938—22no—
16#8041—51nomaximal, Sylow
17C22×C4×D5C2×C411yescharacteristic
18C22×C4×D5D412yes—
19C22×#170—24no—
20C2×#700—28no—
21C2×#700—48no—
22C2×C5×C4⋊C4C2×C412yes—
23C2×C42×C5C2×C411yescharacteristic
24C2×#685—22no—
25C2×C4⋊Dic5C2×C412yes—
26C2×#683—24no—
27C2×C4×Dic5—22no—
28C2×C4×Dic5D412yes—
29C2×C4×Dic5C2311yescharacteristic
30#4038—44no—
31C42×D5C2×C414yes—
32C42×D5D412yes—
33C42×D5Q812yes—
34#4004—28no—
35#4004—24no—
36#4004C2×C418yes—
37C5×#95—22no—
38#3963—22no—
39C2×#423—52no—
40#2285—51no—
41#1296—54no—
42C23×D5C22⋊C411yescharacteristic
43C22×C4×C5—22no—
44C22×C4×C5C22⋊C412yes—
45C22×C4×C5C22×C411yescharacteristic
46C22×Dic5—22no—
47C22×Dic5C22⋊C411yescharacteristic
48C22×Dic5C2×D412yes—
49C2×C4×D5—22no—
50C2×C4×D5—22no—
51C2×C4×D5—24no—
52C2×C4×D5—24no—
53C2×C4×D5C4212yes—
54C2×C4×D5C22⋊C414yes—
55C2×C4×D5C4⋊C412yes—
56C2×C4×D5C4⋊C414yes—
57C2×#170—28no—
58C2×#170—48no—
59#700—416no—
60C5×C4⋊C4—42no—
61C42×C5C4212yes—
62C42×C5C4⋊C412yes—
63#685—44no—
64C4⋊Dic5—42no—
65#683—44no—
66#683—44no—
67C4×Dic5—44no—
68C4×Dic5C22⋊C414yes—
69C4×Dic5C4⋊C414yes—
70C4×Dic5C22×C412yes—
71C4×Dic5C2×D411yescharacteristic
72C4×Dic5C2×Q811yescharacteristic
73C22×M16—102no—
74C2×#117—52no—
75C22×C42—51no—
76#423—58no—
77#423—104no—
78C2×C4⋊C8—54no—
79C2×C4⋊C8—102no—
80#384—52no—
81#382—51no—
82#382AGL(1,𝔽5)11yescharacteristic
83#380—102no—
84C4×C4⋊C4—102no—
85C23×C5C2×C22⋊C411yescharacteristic
86C22×D5—21no—
87C22×D5—21no—
88C22×D5—24no—
89C22×D5#9512yes—
90C2×C4×C5—21no—
91C2×C4×C5—21no—
92C2×C4×C5—24no—
93C2×C4×C5—42no—
94C2×C4×C5—44no—
95C2×C4×C5#9514yes—
96C2×C4×C5C2×C4211yescharacteristic
97C2×C4×C5C2×C4⋊C411yescharacteristic
98C2×Dic5—42no—
99C2×Dic5—44no—
100C2×Dic5#9512yes—
101C2×Dic5#10012yes—
102C2×Dic5#10112yes—
103C2×Dic5C2×C22⋊C412yes—
104C2×Dic5C2×C4⋊C412yes—
105C2×Dic5AGL(1,ℤ/8)14yes—
106C2×Dic5#13714yes—
107C4×D5—24no—
108C4×D5—24no—
109C4×D5—24no—
110C4×D5—24no—
111C4×D5—28no—
112#170—48no—
113C23×C4—52no—
114C23×C4—51no—
115C2×M16—108no—
116C2×M16—208no—
117C22×C8—104no—
118#117—204no—
119C2×C4⋊C4—52no—
120C2×C4⋊C4—104no—
121C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
122C2×C22⋊C4—102no—
123C2×C42—52no—
124C2×C42—51no—
125C2×C42—54no—
126C2×C42—52no—
127C2×C42—52no—
128C2×C42—102no—
129C2×C42C2×AGL(1,𝔽5)11yescharacteristic
130C4⋊C8—58no—
131C4⋊C8—104no—
132C4⋊C8—108no—
133#95—22no—
134#95—102no—
135C22×C5C2×#9511yescharacteristic
136C22×C5C2×#10011yescharacteristic
137C22×C5C2×#10111yescharacteristic
138C22×C5#41714yes—
139C2×D5—24no—
140C2×D5—22no—
141C2×D5—42no—
142C2×D5—42no—
143C4×C5—24no—
144C4×C5—21no—
145C4×C5—21no—
146C4×C5—82no—
147Dic5—82no—
148Dic5#34314yes—
149Dic5#41812yes—
150Dic5#41912yes—
151Dic5#42814yes—
152C24—51no—
153C22×C4—22no—
154C22×C4—51no—
155C22×C4—52no—
156C22×C4—52no—
157C22×C4—52no—
158C22×C4—54no—
159C22×C4—54no—
160C22×C4—102no—
161C22×C4—104no—
162C22×C4—104no—
163C22×C4—102no—
164C22×C4—102no—
165C22×C4C22⋊AGL(1,𝔽5)12yes—
166C22×C4C22×AGL(1,𝔽5)11yescharacteristic
167M16—2016no—
168C2×C8—108no—
169C2×C8—208no—
170C4⋊C4—42no—
171C4⋊C4—202no—
172C4⋊C4—204no—
173C4⋊C4—204no—
174C22⋊C4—204no—
175C42—52no—
176C42—51no—
177C42—51no—
178C42—54no—
179C42—54no—
180C42—204no—
181C42C4×AGL(1,𝔽5)12yes—
182C42C4⋊AGL(1,𝔽5)12yes—
183C2×C5#156012yes—
184C2×C5#156112yes—
185C2×C5#173612yes—
186C2×C5C2×#34311yescharacteristic
187D5—42no—
188C23—52no—
189C23—104no—
190C23—101no—
191C23—101no—
192C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—54no—
199C2×C4—54no—
200C2×C4—52no—
201C2×C4—52no—
202C2×C4—52no—
203C2×C4—52no—
204C2×C4—52no—
205C2×C4—104no—
206C2×C4—104no—
207C2×C4—104no—
208C2×C4—104no—
209C2×C4—108no—
210C2×C4—204no—
211C2×C4—202no—
212C2×C4(C2×AGL(1,𝔽5))⋊C414yes—
213C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
214C2×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
215C8—208no—
216C5#804111yescharacteristic, complemented, Sylow
217C22—102no—
218C22—104no—
219C22—202no—
220C22—202no—
221C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
222C22D5⋊#10011yescharacteristic
223C22D5⋊#10111yescharacteristic
224C22#6579314yes—
225C4—21no—
226C4—21no—
227C4—24no—
228C4—54no—
229C4—52no—
230C4—52no—
231C4—54no—
232C4—82no—
233C4—402no—
234C2—202no—
235C2#1060433411yescharacteristic
236C2#1060437912yes—
237C2#1060455812yes—
238C2#1060458612yes—
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#382C4
socleC23×C5C2×C22⋊C4
prime radicalC23×D5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenter#382AGL(1,𝔽5)
hypocenterC5#8041
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
3. #10598725C2semidirect
10. #65682C4semidirect
216. C5#8041semidirect

Maximal subgroups

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

subgroupindex
C2×#656822
#106058792
#105987252
#80415