Finite Group Databasebeta

← #13021451

Subgroups — #13021451

Order 1280.

518 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 132 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#10604187C211yesmaximal, characteristic
3C4×#4002C211yesmaximal, characteristic
4#10598721C214yesmaximal
5C2×C5×C4⋊Q8C411yescharacteristic
6C2×Q8×Dic5—21no—
7C2×#4002—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#9877—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—44no—
29C2×C4⋊Dic5—21no—
30C2×C4⋊Dic5—21no—
31C2×C4⋊Dic5—22no—
32C2×C4⋊Dic5C2×C412yes—
33C2×C4×Dic5—22no—
34C2×C4×Dic5C2×C411yescharacteristic
35C2×C4×Dic5D411yescharacteristic
36C2×C4×Dic5D411yescharacteristic
37C2×C4×Dic5D411yescharacteristic
38C2×C4×Dic5D411yescharacteristic
39C2×C4×Dic5C2311yescharacteristic
40#4002—24no—
41#4002—24no—
42C4⋊#170—24no—
43C4×#170—22no—
44C4×#170Q814yescomplemented
45#3963—22no—
46#3963—22no—
47C4×C4⋊Q8—51no—
48#1831—51no—
49C4×#97—51no—
50#1279—54no—
51C2×C5×Q8—24no—
52C2×C5×Q8—42no—
53C2×C5×Q8—42no—
54C22×C4×C5—22no—
55C22×C4×C5C22⋊C411yescharacteristic
56C22×C4×C5C22⋊C411yescharacteristic
57C22×C4×C5C22×C411yescharacteristic
58C22×Dic5—22no—
59C22×Dic5C22⋊C411yescharacteristic
60C22×Dic5C22⋊C411yescharacteristic
61C22×Dic5C2×D411yescharacteristic
62C22×Dic5C2×D411yescharacteristic
63C2×#170—24no—
64C2×#170—24no—
65C2×#170—24no—
66C2×#170—42no—
67C2×#170—42no—
68C2×#170—42no—
69C5×C4⋊C4—24no—
70C5×C4⋊C4—22no—
71C5×C4⋊C4M1614yes—
72C42×C5—22no—
73C4⋊Dic5—24no—
74C4⋊Dic5—44no—
75C4⋊Dic5—42no—
76C4⋊Dic5—42no—
77C4×Dic5—22no—
78C4×Dic5—22no—
79C4×Dic5—22no—
80C4×Dic5—22no—
81C4×Dic5—22no—
82C4×Dic5—44no—
83C4×Dic5C2×Q812yes—
84C4×Dic5#4112yes—
85C2×C4⋊Q8AGL(1,𝔽5)11yescharacteristic
86C2×C4×Q8—101no—
87C2×C4⋊C8—52no—
88C2×#97—101no—
89C2×#97—101no—
90C2×C4×C8—52no—
91#391—101no—
92#389—52no—
93#389—101no—
94#384—51no—
95C4×C4⋊C4—52no—
96C4×C4⋊C4—101no—
97C43—51no—
98#336—101no—
99#336—101no—
100C23×C5C2×C22⋊C411yescharacteristic
101C5×Q8—48no—
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5—21no—
105C2×C4×C5—21no—
106C2×C4×C5—24no—
107C2×C4×C5—42no—
108C2×C4×C5—42no—
109C2×C4×C5#10412yes—
110C2×C4×C5#10412yes—
111C2×C4×C5C4×Q812yes—
112C2×C4×C5C2×M1612yes—
113C2×C4×C5#13112yes—
114C2×Dic5—21no—
115C2×Dic5—21no—
116C2×Dic5—21no—
117C2×Dic5—21no—
118C2×Dic5—21no—
119C2×Dic5—21no—
120C2×Dic5—21no—
121C2×Dic5—21no—
122C2×Dic5—42no—
123C2×Dic5—42no—
124C2×Dic5—42no—
125C2×Dic5C22⋊Q812yes—
126C2×Dic5#12411yescharacteristic
127C2×Dic5C4⋊Q811yescharacteristic
128C2×Dic5AGL(1,ℤ/8)12yes—
129C2×Dic5#13712yes—
130#170—28no—
131#170—82no—
132C22×Q8—21no—
133C22×C8—102no—
134C22×C8—102no—
135C4⋊Q8—24no—
136C4×Q8—204no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—52no—
139C2×C4⋊C4—102no—
140C2×C4⋊C4—101no—
141C2×C4⋊C4—101no—
142C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
143C2×C42—51no—
144C2×C42—51no—
145C2×C42—51no—
146C2×C42—51no—
147C2×C42—51no—
148C2×C42—51no—
149C2×C42—102no—
150C2×C42C2×AGL(1,𝔽5)11yescharacteristic
151C4⋊C8—104no—
152#97—104no—
153#97—104no—
154C4×C8—54no—
155C4×C8—102no—
156#95—102no—
157#95—102no—
158C22×C5#39111yescharacteristic
159C22×C5#40711yescharacteristic
160C22×C5#40811yescharacteristic
161C22×C5#41811yescharacteristic
162C22×C5#41911yescharacteristic
163C22×C5C2×#10411yescharacteristic
164C22×C5C2×#10411yescharacteristic
165C4×C5—22no—
166C4×C5—22no—
167C4×C5—44no—
168C4×C5#44614yes—
169Dic5—22no—
170Dic5—41no—
171Dic5—41no—
172Dic5—82no—
173Dic5#48912yes—
174Dic5#50112yes—
175C2×Q8—24no—
176C2×Q8—42no—
177C2×Q8—42no—
178C22×C4—22no—
179C22×C4—51no—
180C22×C4—51no—
181C22×C4—51no—
182C22×C4—51no—
183C22×C4—102no—
184C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
185C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
186C22×C4C22×AGL(1,𝔽5)11yescharacteristic
187C2×C8—104no—
188C2×C8—104no—
189C2×C8—104no—
190C2×C8—202no—
191C2×C8—202no—
192C2×C8—202no—
193C4⋊C4—24no—
194C4⋊C4—22no—
195C4⋊C4—104no—
196C4⋊C4—202no—
197C4⋊C4—204no—
198C4⋊C4—202no—
199C4⋊C4#70414yes—
200C42—22no—
201C42—52no—
202C42—52no—
203C42—102no—
204C42—102no—
205C42—102no—
206C42—102no—
207C42—102no—
208C42—204no—
209C2×C5#148411yescharacteristic
210C2×C5#148511yescharacteristic
211C2×C5#149011yescharacteristic
212C2×C5#149111yescharacteristic
213C2×C5#198511yescharacteristic
214C2×C5#199311yescharacteristic
215C2×C5#199411yescharacteristic
216C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
217Q8—48no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—52no—
226C2×C4—52no—
227C2×C4—51no—
228C2×C4—51no—
229C2×C4—52no—
230C2×C4—101no—
231C2×C4—101no—
232C2×C4—101no—
233C2×C4—101no—
234C2×C4—101no—
235C2×C4—101no—
236C2×C4—101no—
237C2×C4—101no—
238C2×C4—202no—
239C2×C4—202no—
240C2×C4—202no—
241C2×C4#400812yes—
242C2×C4#401012yes—
243C2×C4#413312yes—
244C2×C4Q8×AGL(1,𝔽5)12yes—
245C2×C4C2×#70412yes—
246C8—108no—
247C8—402no—
248C5#987711yescharacteristic, complemented, Sylow
249C22#6576711yescharacteristic
250C22C2×#400811yescharacteristic
251C22#6578211yescharacteristic
252C22C2×#401011yescharacteristic
253C22#6578711yescharacteristic
254C22(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
255C22#6579811yescharacteristic
256C4—22no—
257C4—22no—
258C4—44no—
259C4—52no—
260C4—52no—
261C4—102no—
262C4—201no—
263C4—201no—
264C4—402no—
265C4#6578814yes—
266C2#1060464711yescharacteristic
267C2#1060464911yescharacteristic
268C2#1060466511yescharacteristic
269C2#1060467511yescharacteristic
270C2#1060492411yescharacteristic
271C2#1060496911yescharacteristic
272C2#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#9877
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
248. C5#9877semidirect

Maximal subgroups

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

subgroupindex
Dic5×C4⋊Q82
#106041872
C4×#40022
#105987212
#98775