Finite Group Databasebeta

← #13025804

Subgroups — #13025804

Order 1280.

470 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 98 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#10615465C211yesmaximal, characteristic
2#10604887C212yesmaximal
3#10604726C212yesmaximal, complemented
4#10598770C211yesmaximal, characteristic, complemented
5#10598768C211yesmaximal, characteristic, complemented
6C23×C5×C8C411yescharacteristic, complemented
7#66125C2211yescharacteristic
8C2×#4047C2211yescharacteristic
9#65761C2212yes—
10#65745C2212yes—
11#65718—22no—
12#65496C411yescharacteristic, complemented
13#65400—22no—
14#65399C2211yescharacteristic
15C8×C22⋊D5—22no—
16#65391C411yescharacteristic, complemented
17#65390C411yescharacteristic, complemented
18#64934C414yescomplemented
19#64916C414yescomplemented
20#64894—22no—
21#12623—51nomaximal, Sylow
22C23×C4×C5C2×C411yescharacteristic
23C2×#709C2311yescharacteristic
24#4130—22no—
25D5⋊M16—22no—
26C22×C5×C8—21no—
27C22×C5×C8—21no—
28C22×C5×C8—41no—
29C22×C5×C8—41no—
30C22×C5×C8C2×C411yescharacteristic
31C22×C5×C8C2×C411yescharacteristic
32#4099C2×C412yes—
33#4076—21no—
34#4075C2×C412yes—
35C4×C22⋊D5—22no—
36#4072C2×C411yescharacteristic
37#4070—21no—
38C2×#681C2×C411yescharacteristic
39C2×#681C2×C411yescharacteristic
40#4047—41no—
41#4047—41no—
42C2×#676D412yes—
43C2×C8×D5D412yes—
44#4013C2×C414yes—
45#4011C2×C414yes—
46#4005—22no—
47#3999—22no—
48#3992—42no—
49C8×AGL(1,𝔽5)—42no—
50C22⋊#168—22no—
51#3952—22no—
52#3947—21no—
53#3946—22no—
54C8×Dic5—21no—
55#2929—51no—
56#2120—52no—
57#1757—52no—
58#1392—51no—
59#1391—51no—
60C24×C5C4211yescharacteristic
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—41no—
64C22×C4×C5—41no—
65C22×C4×C5C4211yescharacteristic
66C22×C4×C5C22×C411yescharacteristic
67C2×C22⋊D5—22no—
68C2×C22⋊D5C22×C412yes—
69#709—41no—
70#709—41no—
71C2×D20#4111yescharacteristic
72C2×C4×D5C2×D412yes—
73C2×Dic10#4111yescharacteristic
74C22⋊AGL(1,𝔽5)—24no—
75#704—24no—
76C4⋊AGL(1,𝔽5)—42no—
77C4×AGL(1,𝔽5)—42no—
78#700—42no—
79D5⋊C8—42no—
80C2×C5×C8—21no—
81C2×C5×C8—21no—
82C2×C5×C8—21no—
83C2×C5×C8—42no—
84C2×C5×C8—41no—
85C2×C5×C8—41no—
86C2×C5×C8—41no—
87C2×C5×C8C22⋊C411yescharacteristic
88C2×C5×C8C22⋊C411yescharacteristic
89C22⋊Dic5—22no—
90C22⋊Dic5C4212yes—
91#685—22no—
92C4⋊Dic5—21no—
93#683—22no—
94C4×Dic5—21no—
95#681—41no—
96#681—41no—
97C2×#168—21no—
98C2×#168C22⋊C412yes—
99#676—42no—
100C8×D5—22no—
101C8×D5—22no—
102C2×#131—51no—
103C23×C8AGL(1,𝔽5)11yescharacteristic, complemented
104#525—51no—
105#435—102no—
106C8×D4—102no—
107#424—51no—
108#413—52no—
109#410—52no—
110#408—51no—
111#407—51no—
112#405—51no—
113#405—102no—
114#352—54no—
115AΣL(1,𝔽16)—54no—
116#344—102no—
117C23×C5—21no—
118C23×C5—41no—
119C23×C5—42no—
120C23×C5C2×C4211yescharacteristic
121C22×D5C4×D412yes—
122C2×AGL(1,𝔽5)—44no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—42no—
127C2×C4×C5—41no—
128C2×C4×C5—41no—
129C2×C4×C5—41no—
130C2×C4×C5C2×C22⋊C411yescharacteristic
131C2×C4×C5#11711yescharacteristic
132C22⋊D5—42no—
133C22⋊D5—44no—
134C2×Dic5—22no—
135C2×Dic5C4×D412yes—
136D20—41no—
137C4×D5—22no—
138C4×D5—22no—
139Dic10—41no—
140#170—44no—
141C5×C8—21no—
142C5×C8—41no—
143C5×C8#9911yescharacteristic
144C5×C8#9911yescharacteristic
145#168—22no—
146#168—41no—
147C2×#41—51no—
148C23×C4C2×AGL(1,𝔽5)11yescharacteristic
149#131—201no—
150#131—201no—
151C2×M16—51no—
152C2×M16—52no—
153C2×M16—51no—
154C2×M16—102no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—41no—
158C22×C8—41no—
159C22×C8—52no—
160C22×C8C2×AGL(1,𝔽5)11yescharacteristic
161C22×C8C2×AGL(1,𝔽5)11yescharacteristic
162#123—52no—
163C22⋊Q8—101no—
164#121—101no—
165#120—52no—
166C4×D4—102no—
167#117—51no—
168#117—102no—
169C4⋊C8—102no—
170#101—102no—
171#100—54no—
172#99—54no—
173#99—102no—
174C22⋊C8—102no—
175C22⋊C8—102no—
176#97—101no—
177#97—202no—
178C4×C8—101no—
179C4×C8—202no—
180C22×C5—22no—
181C22×C5—21no—
182C22×C5—41no—
183C22×C5—41no—
184C22×C5—42no—
185C22×C5—42no—
186C22×C5—41no—
187C22×C5C4×C22⋊C411yescharacteristic
188C2×D5—22no—
189C2×D5—42no—
190AGL(1,𝔽5)—84no—
191C4×C5—21no—
192C4×C5—41no—
193C4×C5C2×#9911yescharacteristic
194C4×C5#41011yescharacteristic
195Dic5—22no—
196Dic5—42no—
197C24C4×AGL(1,𝔽5)11yescharacteristic
198#41—201no—
199#41—201no—
200C2×Q8—51no—
201C2×D4—52no—
202C2×D4—51no—
203C2×D4—102no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—52no—
209C22×C4C4×AGL(1,𝔽5)11yescharacteristic
210C22×C4C22×AGL(1,𝔽5)11yescharacteristic
211M16—104no—
212M16—202no—
213M16—201no—
214M16—202no—
215M16—201no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—52no—
224C2×C8—102no—
225C2×C8—102no—
226C2×C8—101no—
227C2×C8—202no—
228C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
229C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
230C4⋊C4—102no—
231C4⋊C4—101no—
232C4⋊C4—202no—
233C22⋊C4—52no—
234C22⋊C4—102no—
235C22⋊C4—102no—
236C22⋊C4—104no—
237C42—101no—
238C42—202no—
239C2×C5—21no—
240C2×C5—42no—
241C2×C5—41no—
242C2×C5C4×#9911yescharacteristic
243D5—42no—
244C23—21no—
245C23—41no—
246C23—42no—
247C23—52no—
248C23C2×C4×AGL(1,𝔽5)11yescharacteristic
249Q8—201no—
250D4—201no—
251D4—202no—
252D4—204no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—52no—
261C2×C4—102no—
262C2×C4—102no—
263C2×C4—102no—
264C2×C4—204no—
265C2×C4#413011yescharacteristic
266C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
267C8—21no—
268C8—41no—
269C8—102no—
270C8—204no—
271C8—201no—
272C8#401111yescharacteristic
273C8#401111yescharacteristic
274C5#1262311yescharacteristic, complemented, Sylow
275C22—22no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—102no—
283C22—202no—
284C22C4×C22⋊AGL(1,𝔽5)11yescharacteristic
285C4—21no—
286C4—41no—
287C4—102no—
288C4—202no—
289C4—404no—
290C4#6574511yescharacteristic
291C4C2×#401111yescharacteristic
292C2—21no—
293C2—42no—
294C2—41no—
295C2—202no—
296C2C4×#401111yescharacteristic
subgroupG/H
centerC8#4011
commutatorC23×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC23×C5×C8C4
socleC2×C5C4×#99
prime radical#66125C22
cosocleC2×#709C23
hypercenterC23×C8AGL(1,𝔽5)
hypocenterC5#12623
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
3. #10604726C2semidirect
4. #10598770C2semidirect
5. #10598768C2semidirect
6. C23×C5×C8C4semidirect
12. #65496C4semidirect
16. #65391C4semidirect
17. #65390C4semidirect
18. #64934C4semidirect
19. #64916C4semidirect
103. C23×C8AGL(1,𝔽5)semidirect
274. C5#12623semidirect

Maximal subgroups

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

subgroupindex
#106154652
#106048872
#106047262
#105987702
#105987682
#126235