Finite Group Databasebeta

← #13021427

Subgroups — #13021427

Order 1280.

512 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 138 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
1Dic5×C4⋊Q8C211yesmaximal, characteristic
2#10604181C211yesmaximal, characteristic
3C4×#4004C211yesmaximal, characteristic
4#10598721C214yesmaximal
5C2×C5×C4⋊Q8C411yescharacteristic
6C2×Q8×Dic5—21no—
7C2×#4004—21no—
8C2×#4002—21no—
9C2×C4⋊#170C2212yes—
10C2×C4×#170C2212yes—
11#65519—21no—
12#65274—21no—
13#65274C412yes—
14C4⋊C4×Dic5—21no—
15C4⋊C4×Dic5C2212yes—
16#65224C411yescharacteristic
17C42×Dic5C2211yescharacteristic
18#64918—21no—
19#64918—21no—
20#9869—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×#170—22no—
23C22×#170—22no—
24C5×C4⋊Q8—24no—
25C2×C5×C4⋊C4—21no—
26C2×C5×C4⋊C4C2×C412yes—
27C2×C42×C5C2×C411yescharacteristic
28Q8×Dic5—42no—
29Q8×Dic5—42no—
30C2×C4⋊Dic5—21no—
31C2×C4⋊Dic5—21no—
32C2×C4⋊Dic5—22no—
33C2×C4⋊Dic5C2×C412yes—
34C2×C4×Dic5—22no—
35C2×C4×Dic5C2×C411yescharacteristic
36C2×C4×Dic5D411yescharacteristic
37C2×C4×Dic5D411yescharacteristic
38C2×C4×Dic5D411yescharacteristic
39C2×C4×Dic5D411yescharacteristic
40C2×C4×Dic5C2311yescharacteristic
41#4004—24no—
42#4002—42no—
43C4⋊#170—22no—
44C4⋊#170Q814yescomplemented
45C4×#170—24no—
46#3963—22no—
47#3963—22no—
48C4×C4⋊Q8—51no—
49#1767—51no—
50C4×C4⋊C8—51no—
51#1279—54no—
52C2×C5×Q8—24no—
53C2×C5×Q8—42no—
54C2×C5×Q8—42no—
55C22×C4×C5—22no—
56C22×C4×C5C22⋊C411yescharacteristic
57C22×C4×C5C22⋊C411yescharacteristic
58C22×C4×C5C22×C411yescharacteristic
59C22×Dic5—22no—
60C22×Dic5C22⋊C411yescharacteristic
61C22×Dic5C22⋊C411yescharacteristic
62C22×Dic5C2×D411yescharacteristic
63C22×Dic5C2×D411yescharacteristic
64C2×#170—24no—
65C2×#170—24no—
66C2×#170—42no—
67C2×#170—42no—
68C2×#170—42no—
69C2×#170—42no—
70C5×C4⋊C4—24no—
71C5×C4⋊C4—22no—
72C5×C4⋊C4M1614yes—
73C42×C5—22no—
74C4⋊Dic5—24no—
75C4⋊Dic5—42no—
76C4⋊Dic5—42no—
77C4⋊Dic5—44no—
78C4×Dic5—22no—
79C4×Dic5—22no—
80C4×Dic5—22no—
81C4×Dic5—41no—
82C4×Dic5—42no—
83C4×Dic5—42no—
84C4×Dic5D812yes—
85C4×Dic5SD1612yes—
86C4×Dic5C2×Q812yes—
87C4×Dic5#4112yes—
88C2×C4⋊Q8AGL(1,𝔽5)11yescharacteristic
89C2×C4×Q8—101no—
90C2×C4⋊C8—52no—
91C2×C4⋊C8—101no—
92C2×#97—101no—
93C2×C4×C8—52no—
94#391—101no—
95#389—52no—
96#389—101no—
97#384—51no—
98C4×C4⋊C4—52no—
99C4×C4⋊C4—101no—
100C43—51no—
101#336—101no—
102#336—101no—
103C23×C5C2×C22⋊C411yescharacteristic
104C5×Q8—48no—
105C2×C4×C5—21no—
106C2×C4×C5—21no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5—24no—
110C2×C4×C5—42no—
111C2×C4×C5—42no—
112C2×C4×C5#10212yes—
113C2×C4×C5#10412yes—
114C2×C4×C5C4×Q812yes—
115C2×C4×C5C2×M1612yes—
116C2×C4×C5#13112yes—
117C2×Dic5—21no—
118C2×Dic5—21no—
119C2×Dic5—21no—
120C2×Dic5—21no—
121C2×Dic5—21no—
122C2×Dic5—21no—
123C2×Dic5—21no—
124C2×Dic5—42no—
125C2×Dic5—42no—
126C2×Dic5—42no—
127C2×Dic5#10212yes—
128C2×Dic5C22⋊Q812yes—
129C2×Dic5#12411yescharacteristic
130C2×Dic5C4⋊Q811yescharacteristic
131C2×Dic5C2×D811yescharacteristic
132C2×Dic5C2×SD1611yescharacteristic
133C2×Dic5#13712yes—
134#170—44no—
135#170—44no—
136C22×Q8—21no—
137C22×C8—102no—
138C22×C8—102no—
139C4⋊Q8—24no—
140C4×Q8—202no—
141C4×Q8—202no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—52no—
144C2×C4⋊C4—102no—
145C2×C4⋊C4—101no—
146C2×C4⋊C4—101no—
147C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
148C2×C42—51no—
149C2×C42—51no—
150C2×C42—51no—
151C2×C42—51no—
152C2×C42—51no—
153C2×C42—51no—
154C2×C42—102no—
155C2×C42C2×AGL(1,𝔽5)11yescharacteristic
156C4⋊C8—54no—
157C4⋊C8—102no—
158C4⋊C8—104no—
159#97—202no—
160C4×C8—104no—
161#95—102no—
162#95—102no—
163C22×C5#39111yescharacteristic
164C22×C5#40711yescharacteristic
165C22×C5#40811yescharacteristic
166C22×C5C2×#10211yescharacteristic
167C22×C5#41911yescharacteristic
168C22×C5C2×#10411yescharacteristic
169C22×C5#42111yescharacteristic
170C4×C5—22no—
171C4×C5—22no—
172C4×C5—44no—
173C4×C5#44614yes—
174Dic5—21no—
175Dic5—21no—
176Dic5—22no—
177Dic5—41no—
178Dic5—82no—
179Dic5#47412yes—
180Dic5#47512yes—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8—42no—
184C22×C4—22no—
185C22×C4—51no—
186C22×C4—51no—
187C22×C4—51no—
188C22×C4—51no—
189C22×C4—102no—
190C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
191C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
192C22×C4C22×AGL(1,𝔽5)11yescharacteristic
193C2×C8—104no—
194C2×C8—104no—
195C2×C8—202no—
196C2×C8—202no—
197C2×C8—202no—
198C2×C8—202no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—104no—
202C4⋊C4—204no—
203C4⋊C4—202no—
204C4⋊C4—202no—
205C4⋊C4#70414yes—
206C42—22no—
207C42—52no—
208C42—52no—
209C42—52no—
210C42—52no—
211C42—102no—
212C42—102no—
213C42—102no—
214C42—201no—
215C42—202no—
216C42—202no—
217C2×C5#148611yescharacteristic
218C2×C5#149111yescharacteristic
219C2×C5#149211yescharacteristic
220C2×C5#149711yescharacteristic
221C2×C5#198511yescharacteristic
222C2×C5#198611yescharacteristic
223C2×C5#198811yescharacteristic
224C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
225Q8—48no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—51no—
234C2×C4—51no—
235C2×C4—52no—
236C2×C4—52no—
237C2×C4—52no—
238C2×C4—51no—
239C2×C4—51no—
240C2×C4—101no—
241C2×C4—101no—
242C2×C4—101no—
243C2×C4—101no—
244C2×C4—101no—
245C2×C4—101no—
246C2×C4—101no—
247C2×C4—202no—
248C2×C4—202no—
249C2×C4—202no—
250C2×C4#400712yes—
251C2×C4#401012yes—
252C2×C4#413312yes—
253C2×C4Q8×AGL(1,𝔽5)12yes—
254C2×C4C2×#70412yes—
255C8—204no—
256C8—204no—
257C5#986911yescharacteristic, complemented, Sylow
258C22C2×#400711yescharacteristic
259C22#6576911yescharacteristic
260C22#6578211yescharacteristic
261C22C2×#401011yescharacteristic
262C22#6578711yescharacteristic
263C22(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
264C22#6579811yescharacteristic
265C4—22no—
266C4—22no—
267C4—44no—
268C4—52no—
269C4—52no—
270C4—102no—
271C4—101no—
272C4—101no—
273C4—201no—
274C4—402no—
275C4#6578814yes—
276C2#1060465011yescharacteristic
277C2#1060465511yescharacteristic
278C2#1060466911yescharacteristic
279C2#1060468011yescharacteristic
280C2#1060492311yescharacteristic
281C2#1060496911yescharacteristic
282C2#1060498511yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊Q8C4
socle, prime radicalC23×C5C2×C22⋊C4
cosocleC2×C4×Dic5C23
hypercenterC2×C4⋊Q8AGL(1,𝔽5)
hypocenterC5#9869
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
44. C4⋊#170Q8semidirect
257. C5#9869semidirect

Maximal subgroups

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

subgroupindex
Dic5×C4⋊Q82
#106041812
C4×#40042
#105987212
#98695