Finite Group Databasebeta

← #13018282

Subgroups — #13018282

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#10604723C212yesmaximal, complemented
4#10598764C212yesmaximal, complemented
5C23×C5×C8C411yescharacteristic, complemented
6C2×C4×C22⋊D5C2211yescharacteristic
7C22×C8×D5C2211yescharacteristic
8C2×#4011—22no—
9#65765C2214yes—
10C2×#3994—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#8609—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#3994—44no—
44C22⋊#168—22no—
45#3952—22no—
46#3946—22no—
47C8×Dic5—21no—
48C8×Dic5—21no—
49C2×C8×D4—51no—
50#1808—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⋊D5D812yes—
65C2×C22⋊D5Q1612yes—
66C22×Dic5C4⋊C411yescharacteristic
67C22×Dic5C4⋊C411yescharacteristic
68C22×Dic5C2×Q811yescharacteristic
69C2×C4×D5—41no—
70C2×C4×D5—41no—
71C2×C4×D5C2×D412yes—
72C2×C4×D5#4112yes—
73C22⋊AGL(1,𝔽5)—44no—
74C22⋊AGL(1,𝔽5)—44no—
75C4⋊AGL(1,𝔽5)—44no—
76C4⋊AGL(1,𝔽5)—44no—
77C2×C5×C8—22no—
78C2×C5×C8—21no—
79C2×C5×C8—42no—
80C2×C5×C8—42no—
81C2×C5×C8—41no—
82C2×C5×C8C22⋊C411yescharacteristic
83C2×C5×C8C22⋊C411yescharacteristic
84C22⋊Dic5—22no—
85#685—22no—
86#683—22no—
87C4×Dic5—21no—
88C4×Dic5—21no—
89C2×#168—21no—
90C2×#168—41no—
91C2×#168—41no—
92C2×#168C22⋊C412yes—
93C8×D5—24no—
94C8×D5—42no—
95C23×C8—51no—
96C23×C8AGL(1,𝔽5)11yescharacteristic, complemented
97C2×C4×D4—51no—
98C8×D4—102no—
99C8×D4—102no—
100C2×#107—102no—
101C2×C4⋊C8—51no—
102C2×#99—102no—
103C2×C22⋊C8—51no—
104C2×C22⋊C8—51no—
105C2×C4×C8—51no—
106#380—54no—
107#342—102no—
108#340—102no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—41no—
112C23×C5—41no—
113C23×C5#10712yescomplemented
114C23×C5C2×C4⋊C411yescharacteristic
115C22×D5—41no—
116C22×D5—41no—
117C22×D5#12112yes—
118C22×D5C2×D811yescharacteristic
119C22×D5C2×Q1611yescharacteristic
120C22×D5#13512yes—
121C2×AGL(1,𝔽5)—84no—
122C2×AGL(1,𝔽5)—84no—
123C2×AGL(1,𝔽5)—84no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—42no—
128C2×C4×C5—41no—
129C2×C4×C5—41no—
130C2×C4×C5—41no—
131C2×C4×C5C2×C22⋊C411yescharacteristic
132C2×C4×C5#11711yescharacteristic
133C22⋊D5—24no—
134C22⋊D5—24no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—41no—
139C2×Dic5—41no—
140C2×Dic5#10712yescomplemented
141C2×Dic5C22⋊Q812yes—
142C4×D5—22no—
143C4×D5—22no—
144C4×D5—42no—
145C5×C8—21no—
146C5×C8—41no—
147C5×C8#9912yescomplemented
148#168—22no—
149#168—41no—
150C22×D4—51no—
151C23×C4—51no—
152C23×C4C2×AGL(1,𝔽5)11yescharacteristic
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—42no—
156C22×C8—51no—
157C22×C8—52no—
158C22×C8—52no—
159C22×C8—51no—
160C22×C8—202no—
161C22×C8C2×AGL(1,𝔽5)11yescharacteristic
162C22×C8C2×AGL(1,𝔽5)11yescharacteristic
163C4×D4—102no—
164C4×D4—102no—
165C2×C4⋊C4—51no—
166C2×C4⋊C4—104no—
167C2×C22⋊C4—51no—
168C2×C22⋊C4—51no—
169C2×C22⋊C4—104no—
170C2×C42—51no—
171#107—204no—
172C4⋊C8—102no—
173#99—204no—
174C22⋊C8—102no—
175C22⋊C8—102no—
176C4×C8—101no—
177C4×C8—101no—
178#95—104no—
179C22×C5—21no—
180C22×C5—21no—
181C22×C5—22no—
182C22×C5—42no—
183C22×C5—42no—
184C22×C5—41no—
185C22×C5—41no—
186C22×C5#38011yescharacteristic
187C22×C5C2×#10711yescharacteristic
188C22×C5#42711yescharacteristic
189C2×D5—22no—
190C2×D5—22no—
191C2×D5—41no—
192C2×D5—41no—
193C2×D5—42no—
194C2×D5#46612yes—
195C2×D5#46712yes—
196AGL(1,𝔽5)—164no—
197C4×C5—21no—
198C4×C5—41no—
199C4×C5C2×#9911yescharacteristic
200C4×C5#41011yescharacteristic
201Dic5—22no—
202Dic5—22no—
203Dic5—41no—
204C24—51no—
205C24C4⋊AGL(1,𝔽5)11yescharacteristic
206C2×D4—52no—
207C2×D4—52no—
208C2×D4—102no—
209C2×D4—102no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—51no—
215C22×C4—51no—
216C22×C4—52no—
217C22×C4—52no—
218C22×C4—51no—
219C22×C4—201no—
220C22×C4—201no—
221C22×C4—204no—
222C22×C4C4⋊AGL(1,𝔽5)11yescharacteristic
223C22×C4C22×AGL(1,𝔽5)11yescharacteristic
224C2×C8—22no—
225C2×C8—21no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—41no—
229C2×C8—52no—
230C2×C8—104no—
231C2×C8—101no—
232C2×C8—202no—
233C2×C8—201no—
234C2×C8—201no—
235C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
236C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
237C4⋊C4—102no—
238C4⋊C4—204no—
239C4⋊C4—204no—
240C22⋊C4—102no—
241C22⋊C4—102no—
242C22⋊C4—204no—
243C22⋊C4—204no—
244C42—101no—
245C42—101no—
246C2×C5—21no—
247C2×C5—21no—
248C2×C5—42no—
249C2×C5#179911yescharacteristic
250C2×C5#180811yescharacteristic
251C2×C5#181211yescharacteristic
252D5—22no—
253D5—22no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—52no—
259C23—51no—
260C23—51no—
261C23—52no—
262C23—201no—
263C23—201no—
264C23#399412yescomplemented
265C23C2×C4⋊AGL(1,𝔽5)11yescharacteristic
266D4—104no—
267D4—104no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—52no—
276C2×C4—52no—
277C2×C4—101no—
278C2×C4—101no—
279C2×C4—102no—
280C2×C4—102no—
281C2×C4—101no—
282C2×C4—202no—
283C2×C4—201no—
284C2×C4—201no—
285C2×C4—404no—
286C2×C4—404no—
287C2×C4—404no—
288C2×C4#413011yescharacteristic
289C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
290C8—21no—
291C8—41no—
292C8—102no—
293C8—201no—
294C8#401112yescomplemented
295C5#860911yescharacteristic, complemented, Sylow
296C22—21no—
297C22—21no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—52no—
304C22—52no—
305C22—102no—
306C22—102no—
307C22—201no—
308C22—201no—
309C22—202no—
310C22C2×#399411yescharacteristic
311C22#6572111yescharacteristic
312C22#6576511yescharacteristic
313C4—21no—
314C4—41no—
315C4—102no—
316C4—102no—
317C4—201no—
318C4—804no—
319C4#6574511yescharacteristic
320C4C2×#401111yescharacteristic
321C2—21no—
322C2—21no—
323C2—42no—
324C2—102no—
325C2—102no—
326C2#1060472311yescharacteristic
327C2#1060472711yescharacteristic
328C2#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#8609
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. #10604723C2semidirect
4. #10598764C2semidirect
5. C23×C5×C8C4semidirect
11. C2×C22⋊#168C4semidirect
15. C2×#3946C4semidirect
16. C2×C8×Dic5C4semidirect
96. C23×C8AGL(1,𝔽5)semidirect
113. C23×C5#107semidirect
140. C2×Dic5#107semidirect
147. C5×C8#99semidirect
264. C23#3994semidirect
294. C8#4011semidirect
295. C5#8609semidirect

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
#106047232
#105987642
#86095