Finite Group Databasebeta

← #13018278

Subgroups — #13018278

Order 1280.

550 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 108 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×C8×C22⋊D5C211yesmaximal, characteristic
2#10604892C212yesmaximal
3#10604722C212yesmaximal, complemented
4#10598764C212yesmaximal, complemented
5C23×C5×C8C411yescharacteristic, complemented
6C2×C4×C22⋊D5C2211yescharacteristic
7C22×C8×D5C2211yescharacteristic
8C2×#4011—22no—
9#65765C2214yes—
10C2×#3993—22no—
11C2×C22⋊#168C411yescharacteristic, complemented
12C8×C22⋊D5—22no—
13C8×C22⋊D5—22no—
14C2×#3952C2211yescharacteristic
15C2×#3946C411yescharacteristic, complemented
16C2×C8×Dic5C411yescharacteristic, complemented
17#64917—22no—
18#64893—22no—
19#8608—51nomaximal, Sylow
20C23×C4×C5C2×C411yescharacteristic
21C22×C22⋊D5D411yescharacteristic
22C22×C4×D5C2311yescharacteristic
23C2×C22⋊AGL(1,𝔽5)—24no—
24C2×C4⋊AGL(1,𝔽5)—24no—
25C22×C5×C8—21no—
26C22×C5×C8—21no—
27C22×C5×C8—42no—
28C22×C5×C8C2×C411yescharacteristic
29C22×C5×C8C2×C411yescharacteristic
30C2×C22⋊Dic5Q811yescharacteristic
31C4×C22⋊D5—22no—
32C4×C22⋊D5—22no—
33C2×#685D411yescharacteristic
34C2×#683Q811yescharacteristic
35C2×C4×Dic5C2×C411yescharacteristic
36C22×#168C2×C411yescharacteristic
37C22×#168C2×C411yescharacteristic
38C2×C8×D5—42no—
39C2×C8×D5D412yes—
40C2×C8×D5D412yes—
41#4011—44no—
42(C2×AGL(1,𝔽5))⋊C4—24no—
43#3993—44no—
44C22⋊#168—22no—
45#3952—22no—
46#3946—22no—
47C8×Dic5—21no—
48C8×Dic5—21no—
49C2×C8×D4—51no—
50#1807—52no—
51#1799—52no—
52#1289—52no—
53C24×C5C4⋊C411yescharacteristic
54C23×D5C2×D411yescharacteristic
55C22×AGL(1,𝔽5)—44no—
56C22×C4×C5—21no—
57C22×C4×C5—21no—
58C22×C4×C5—41no—
59C22×C4×C5—41no—
60C22×C4×C5C4⋊C411yescharacteristic
61C22×C4×C5C22×C411yescharacteristic
62C2×C22⋊D5—22no—
63C2×C22⋊D5—22no—
64C2×C22⋊D5SD1614yes—
65C22×Dic5C4⋊C411yescharacteristic
66C22×Dic5C4⋊C411yescharacteristic
67C22×Dic5C2×Q811yescharacteristic
68C2×C4×D5—41no—
69C2×C4×D5—41no—
70C2×C4×D5C2×D412yes—
71C2×C4×D5#4112yes—
72C22⋊AGL(1,𝔽5)—44no—
73C22⋊AGL(1,𝔽5)—44no—
74C4⋊AGL(1,𝔽5)—44no—
75C4⋊AGL(1,𝔽5)—44no—
76C2×C5×C8—22no—
77C2×C5×C8—21no—
78C2×C5×C8—42no—
79C2×C5×C8—42no—
80C2×C5×C8—41no—
81C2×C5×C8C22⋊C411yescharacteristic
82C2×C5×C8C22⋊C411yescharacteristic
83C22⋊Dic5—22no—
84#685—22no—
85#683—22no—
86C4×Dic5—21no—
87C4×Dic5—21no—
88C2×#168—21no—
89C2×#168—41no—
90C2×#168—41no—
91C2×#168C22⋊C412yes—
92C8×D5—24no—
93C8×D5—42no—
94C23×C8—51no—
95C23×C8AGL(1,𝔽5)11yescharacteristic, complemented
96C2×C4×D4—51no—
97C8×D4—102no—
98C8×D4—102no—
99C2×#106—102no—
100C2×C4⋊C8—51no—
101C2×#99—102no—
102C2×C22⋊C8—51no—
103C2×C22⋊C8—51no—
104C2×C4×C8—51no—
105#380—54no—
106#342—102no—
107#340—102no—
108C23×C5—21no—
109C23×C5—21no—
110C23×C5—41no—
111C23×C5—41no—
112C23×C5#10612yescomplemented
113C23×C5C2×C4⋊C411yescharacteristic
114C22×D5—41no—
115C22×D5—41no—
116C22×D5#12112yes—
117C22×D5C2×SD1612yes—
118C22×D5#13512yes—
119C2×AGL(1,𝔽5)—84no—
120C2×AGL(1,𝔽5)—84no—
121C2×AGL(1,𝔽5)—84no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—42no—
126C2×C4×C5—41no—
127C2×C4×C5—41no—
128C2×C4×C5—41no—
129C2×C4×C5C2×C22⋊C411yescharacteristic
130C2×C4×C5#11711yescharacteristic
131C22⋊D5—28no—
132C2×Dic5—21no—
133C2×Dic5—21no—
134C2×Dic5—21no—
135C2×Dic5—41no—
136C2×Dic5—41no—
137C2×Dic5#10612yescomplemented
138C2×Dic5C22⋊Q812yes—
139C4×D5—22no—
140C4×D5—22no—
141C4×D5—42no—
142C5×C8—21no—
143C5×C8—41no—
144C5×C8#9912yescomplemented
145#168—22no—
146#168—41no—
147C22×D4—51no—
148C23×C4—51no—
149C23×C4C2×AGL(1,𝔽5)11yescharacteristic
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—42no—
153C22×C8—51no—
154C22×C8—52no—
155C22×C8—52no—
156C22×C8—51no—
157C22×C8—202no—
158C22×C8C2×AGL(1,𝔽5)11yescharacteristic
159C22×C8C2×AGL(1,𝔽5)11yescharacteristic
160C4×D4—102no—
161C4×D4—102no—
162C2×C4⋊C4—51no—
163C2×C4⋊C4—104no—
164C2×C22⋊C4—51no—
165C2×C22⋊C4—51no—
166C2×C22⋊C4—104no—
167C2×C42—51no—
168#106—204no—
169C4⋊C8—102no—
170#99—204no—
171C22⋊C8—102no—
172C22⋊C8—102no—
173C4×C8—101no—
174C4×C8—101no—
175#95—104no—
176C22×C5—21no—
177C22×C5—21no—
178C22×C5—22no—
179C22×C5—42no—
180C22×C5—42no—
181C22×C5—41no—
182C22×C5—41no—
183C22×C5#38011yescharacteristic
184C22×C5C2×#10611yescharacteristic
185C22×C5#42711yescharacteristic
186C2×D5—22no—
187C2×D5—22no—
188C2×D5—42no—
189C2×D5—42no—
190C2×D5#46514yes—
191AGL(1,𝔽5)—164no—
192C4×C5—21no—
193C4×C5—41no—
194C4×C5C2×#9911yescharacteristic
195C4×C5#41011yescharacteristic
196Dic5—22no—
197Dic5—22no—
198Dic5—41no—
199C24—51no—
200C24C4⋊AGL(1,𝔽5)11yescharacteristic
201C2×D4—54no—
202C2×D4—102no—
203C2×D4—102no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—51no—
209C22×C4—51no—
210C22×C4—52no—
211C22×C4—52no—
212C22×C4—51no—
213C22×C4—201no—
214C22×C4—201no—
215C22×C4—204no—
216C22×C4C4⋊AGL(1,𝔽5)11yescharacteristic
217C22×C4C22×AGL(1,𝔽5)11yescharacteristic
218C2×C8—22no—
219C2×C8—21no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8—52no—
224C2×C8—104no—
225C2×C8—101no—
226C2×C8—202no—
227C2×C8—201no—
228C2×C8—201no—
229C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
230C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
231C4⋊C4—102no—
232C4⋊C4—204no—
233C4⋊C4—204no—
234C22⋊C4—102no—
235C22⋊C4—102no—
236C22⋊C4—204no—
237C22⋊C4—204no—
238C42—101no—
239C42—101no—
240C2×C5—21no—
241C2×C5—21no—
242C2×C5—42no—
243C2×C5#179911yescharacteristic
244C2×C5#180711yescharacteristic
245C2×C5#181211yescharacteristic
246D5—24no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—52no—
252C23—52no—
253C23—52no—
254C23—201no—
255C23—201no—
256C23#399312yescomplemented
257C23C2×C4⋊AGL(1,𝔽5)11yescharacteristic
258D4—108no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—52no—
267C2×C4—52no—
268C2×C4—101no—
269C2×C4—101no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4—101no—
273C2×C4—202no—
274C2×C4—201no—
275C2×C4—201no—
276C2×C4—404no—
277C2×C4—404no—
278C2×C4—404no—
279C2×C4#413011yescharacteristic
280C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
281C8—21no—
282C8—41no—
283C8—102no—
284C8—201no—
285C8#401112yescomplemented
286C5#860811yescharacteristic, complemented, Sylow
287C22—21no—
288C22—21no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—54no—
295C22—102no—
296C22—102no—
297C22—202no—
298C22—202no—
299C22C2×#399311yescharacteristic
300C22#6572111yescharacteristic
301C22#6576511yescharacteristic
302C4—21no—
303C4—41no—
304C4—102no—
305C4—102no—
306C4—201no—
307C4—804no—
308C4#6574511yescharacteristic
309C4C2×#401111yescharacteristic
310C2—21no—
311C2—21no—
312C2—42no—
313C2—104no—
314C2#1060472211yescharacteristic
315C2#1060472711yescharacteristic
316C2#1060489211yescharacteristic
subgroupG/H
centerC22#65765
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC23×C5×C8C4
socleC22×C5#380
prime radicalC22×C22⋊D5D4
cosocleC22×C4×D5C23
hypercenterC23×C8AGL(1,𝔽5)
hypocenterC5#8608
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
3. #10604722C2semidirect
4. #10598764C2semidirect
5. C23×C5×C8C4semidirect
11. C2×C22⋊#168C4semidirect
15. C2×#3946C4semidirect
16. C2×C8×Dic5C4semidirect
95. C23×C8AGL(1,𝔽5)semidirect
112. C23×C5#106semidirect
137. C2×Dic5#106semidirect
144. C5×C8#99semidirect
256. C23#3993semidirect
285. C8#4011semidirect
286. C5#8608semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C22⋊D52
#106048922
#106047222
#105987642
#86085