Finite Group Databasebeta

← #13018215

Subgroups — #13018215

Order 1280.

516 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 140 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#10608212C211yesmaximal, characteristic
2C2×#64931C212yesmaximal
3#10604191C212yesmaximal
4#10598725C212yesmaximal
5C2×Q8×Dic5C411yescharacteristic
6C2×#4090C2211yescharacteristic
7C2×#4033C411yescharacteristic
8C2×#4004—22no—
9C2×#4004C2214yes—
10C2×C4⋊#170—22no—
11C5×#391C411yescharacteristic
12#65265C2211yescharacteristic
13C42×Dic5C2211yescharacteristic
14#65218C411yescharacteristic
15#64938—22no—
16#64936—22no—
17#64931—24no—
18#8587—51nomaximal, Sylow
19C22×C5×Q8C2×C411yescharacteristic
20C22×Dic10C2×C411yescharacteristic
21C22×#170—24no—
22C2×C5×C4⋊C4C2×C411yescharacteristic
23C2×C42×C5C2×C411yescharacteristic
24Q8×Dic5—42no—
25#4090—22no—
26#4090D412yes—
27#4090Q812yes—
28C2×C4⋊Dic5—21no—
29C2×C4⋊Dic5C2×C411yescharacteristic
30C2×#683—22no—
31C2×#683—21no—
32C2×#683—22no—
33C2×C4×Dic5—21no—
34C2×C4×Dic5—21no—
35C2×C4×Dic5C2×C411yescharacteristic
36C2×C4×Dic5D412yes—
37C2×C4×Dic5D412yes—
38C2×C4×Dic5C2311yescharacteristic
39#4033—42no—
40#4004—24no—
41#4004—24no—
42#4004—44no—
43C4⋊#170—44no—
44C5×#95—22no—
45#3963—22no—
46C4×C4⋊Q8—51no—
47#1842—52no—
48C2×#333—52no—
49#1296—52no—
50C2×C5×Q8—21no—
51C2×C5×Q8—41no—
52C2×C5×Q8—41no—
53C2×C5×Q8C4⋊C412yes—
54C22×C4×C5—21no—
55C22×C4×C5—21no—
56C22×C4×C5C22⋊C412yes—
57C22×C4×C5C22×C411yescharacteristic
58C22×Dic5—21no—
59C22×Dic5—21no—
60C22×Dic5C22⋊C411yescharacteristic
61C22×Dic5C22⋊C411yescharacteristic
62C22×Dic5C2×D412yes—
63C2×Dic10—21no—
64C2×Dic10—41no—
65C2×Dic10—41no—
66C2×Dic10C4⋊C412yes—
67C2×#170—44no—
68C2×#170—44no—
69C2×#170—44no—
70C5×C4⋊C4—41no—
71C42×C5—41no—
72C4⋊Dic5—42no—
73C4⋊Dic5—41no—
74#683—24no—
75#683—24no—
76#683—42no—
77#683—42no—
78#683—42no—
79C4×Dic5—22no—
80C4×Dic5—22no—
81C4×Dic5—22no—
82C4×Dic5—42no—
83C4×Dic5—42no—
84C4×Dic5SD1614yes—
85C4×Dic5Q1614yes—
86C4×Dic5C2×D411yescharacteristic
87C4×Dic5C2×Q811yescharacteristic
88C4×Dic5#4112yes—
89C2×C4⋊Q8—51no—
90C2×C4×Q8—51no—
91C2×C4×Q8—51no—
92C2×C4⋊C8—54no—
93C2×C4⋊C8—102no—
94C2×C4⋊C8—102no—
95#391—51no—
96#391AGL(1,𝔽5)11yescharacteristic
97#389—102no—
98#384—51no—
99C4×C4⋊C4—102no—
100C43—51no—
101#333—104no—
102C23×C5C2×C22⋊C411yescharacteristic
103C5×Q8—42no—
104C5×Q8—42no—
105C2×C4×C5—21no—
106C2×C4×C5—21no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—41no—
111C2×C4×C5—41no—
112C2×C4×C5—41no—
113C2×C4×C5—42no—
114C2×C4×C5#10314yes—
115C2×C4×C5C2×C4⋊C411yescharacteristic
116C2×C4×C5#11711yescharacteristic
117C2×Dic5—22no—
118C2×Dic5—21no—
119C2×Dic5—22no—
120C2×Dic5—41no—
121C2×Dic5—41no—
122C2×Dic5—41no—
123C2×Dic5—42no—
124C2×Dic5#10012yes—
125C2×Dic5#10314yes—
126C2×Dic5#12112yes—
127C2×Dic5C22⋊Q812yes—
128C2×Dic5C2×SD1612yes—
129C2×Dic5C2×Q1612yes—
130C2×Dic5AGL(1,ℤ/8)12yes—
131C2×Dic5#13712yes—
132Dic10—42no—
133Dic10—42no—
134#170—84no—
135C22×Q8—51no—
136C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
137C22×C8—104no—
138C4⋊Q8—52no—
139C4⋊Q8—52no—
140C4⋊Q8—102no—
141C4×Q8—202no—
142C4×Q8—202no—
143C2×C4⋊C4—51no—
144C2×C4⋊C4—102no—
145C2×C4⋊C4—101no—
146C2×C4⋊C4—101no—
147C2×C4⋊C4—102no—
148C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
149C2×C42—52no—
150C2×C42—51no—
151C2×C42—52no—
152C2×C42—51no—
153C2×C42—101no—
154C2×C42—101no—
155C2×C42C2×AGL(1,𝔽5)11yescharacteristic
156C4⋊C8—104no—
157C4⋊C8—104no—
158C4⋊C8—204no—
159C4⋊C8—204no—
160#95—22no—
161#95—102no—
162C22×C5#38011yescharacteristic
163C22×C5C2×#10011yescharacteristic
164C22×C5#41311yescharacteristic
165C22×C5C2×#10312yes—
166C22×C5#41712yes—
167C4×C5—21no—
168C4×C5—21no—
169C4×C5—24no—
170C4×C5—42no—
171C4×C5—81no—
172Dic5—42no—
173Dic5—41no—
174Dic5—81no—
175Dic5#33314yescomplemented
176Dic5#46012yes—
177Dic5#46212yes—
178Dic5#47512yes—
179Dic5#47712yes—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—52no—
184C2×Q8—101no—
185C2×Q8—201no—
186C2×Q8—201no—
187C2×Q8C4⋊AGL(1,𝔽5)12yes—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—51no—
191C22×C4—52no—
192C22×C4—51no—
193C22×C4—101no—
194C22×C4—101no—
195C22×C4C22⋊AGL(1,𝔽5)12yes—
196C22×C4C22×AGL(1,𝔽5)11yescharacteristic
197C2×C8—204no—
198C2×C8—204no—
199C2×C8—204no—
200C4⋊C4—41no—
201C4⋊C4—104no—
202C4⋊C4—104no—
203C4⋊C4—201no—
204C4⋊C4—202no—
205C4⋊C4—202no—
206C4⋊C4—202no—
207C4⋊C4—202no—
208C42—41no—
209C42—51no—
210C42—51no—
211C42—52no—
212C42—54no—
213C42—54no—
214C42—102no—
215C42—102no—
216C42—102no—
217C42—202no—
218C42—202no—
219C2×C5C2×#33311yescharacteristic
220C2×C5#155012yes—
221C2×C5#156011yescharacteristic
222C2×C5#180011yescharacteristic
223C2×C5#180312yes—
224C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
225Q8—42no—
226Q8—42no—
227Q8—202no—
228Q8—202no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—52no—
239C2×C4—52no—
240C2×C4—52no—
241C2×C4—52no—
242C2×C4—52no—
243C2×C4—52no—
244C2×C4—54no—
245C2×C4—52no—
246C2×C4—102no—
247C2×C4—102no—
248C2×C4—101no—
249C2×C4—201no—
250C2×C4—201no—
251C2×C4—202no—
252C2×C4—201no—
253C2×C4#400914yes—
254C2×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
255C2×C4#413011yescharacteristic
256C8—404no—
257C5#858711yescharacteristic, complemented, Sylow
258C22#6576511yescharacteristic
259C22#6577711yescharacteristic
260C22D5⋊#10011yescharacteristic
261C22C2×#400912yes—
262C22#6579312yes—
263C4—21no—
264C4—21no—
265C4—24no—
266C4—42no—
267C4—54no—
268C4—52no—
269C4—52no—
270C4—52no—
271C4—52no—
272C4—81no—
273C4—201no—
274C4—202no—
275C4—401no—
276C2#1060424611yescharacteristic
277C2#1060445112yes—
278C2#1060451812yes—
279C2#1060455811yescharacteristic
280C2D5⋊#33311yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#391C4
socle, prime radicalC23×C5C2×C22⋊C4
cosocleC2×C4×Dic5C23
hypercenter#391AGL(1,𝔽5)
hypocenterC5#8587
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
175. Dic5#333semidirect
257. C5#8587semidirect

Maximal subgroups

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

subgroupindex
#106082122
C2×#649312
#106041912
#105987252
#85875