Finite Group Databasebeta

← #11997269

Subgroups — #11997269

Order 1280.

524 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C5×#2973C211yesmaximal, characteristic
2#10614105C211yesmaximal, characteristic
3#10602395C211yesmaximal, characteristic
4#10600766C212yesmaximal, complemented
5#10598175C212yesmaximal, complemented
6C2×C4×C5×D4C2211yescharacteristic
7C22×C4⋊Dic5C2211yescharacteristic
8C5×#435—24no—
9C5×#432—21no—
10C5×#407C2212yes—
11C2×C5×C22⋊C8—21no—
12#65509C412yes—
13#65501—21no—
14#65501C2211yescharacteristic
15#65496C2212yes—
16#65275—21no—
17#65238—21no—
18#65235C412yes—
19#65217—21no—
20#64774—24no—
21#64748—21no—
22#64745—21no—
23#9509—51nomaximal, Sylow
24C22×C5×D4C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5C2311yescharacteristic
27C23×Dic5C2×C412yes—
28C2×C5×M16D414yes—
29C22×C5×C8—22no—
30C4×C5×D4—22no—
31C4×C5×D4—22no—
32C2×C5×C4⋊C4D411yescharacteristic
33C2×C5×C22⋊C4C2×C411yescharacteristic
34C2×C5×C22⋊C4D411yescharacteristic
35C2×C42×C5D411yescharacteristic
36C2×C22⋊Dic5—22no—
37C2×C22⋊Dic5C2×C412yes—
38C2×C4⋊Dic5—22no—
39C2×C4⋊Dic5—21no—
40C2×C4⋊Dic5—21no—
41C2×C4⋊Dic5—21no—
42C2×C4⋊Dic5—22no—
43C2×#683—22no—
44C2×#681—22no—
45C5×C4⋊C8—22no—
46C5×C22⋊C8—22no—
47C5×C22⋊C8—22no—
48C5×C22⋊C8—22no—
49C5×#97—22no—
50#3963—22no—
51#3963—22no—
52#3963—21no—
53C22⋊#168—22no—
54C22⋊#168—22no—
55#2973D511yescharacteristic
56#2487—51no—
57#1821—52no—
58#1465—51no—
59#1305—52no—
60C24×C5C22⋊C411yescharacteristic
61C24×C5C22×C411yescharacteristic
62C2×C5×D4—22no—
63C2×C5×D4—22no—
64C2×C5×D4—41no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5C2×D411yescharacteristic
73C22×C4×C5C2×D412yes—
74C22×C4×C5C2×D411yescharacteristic
75C22×C4×C5#4112yes—
76C22×Dic5—22no—
77C22×Dic5—22no—
78C22×Dic5—24no—
79C22×Dic5—82no—
80C5×M16—44no—
81C2×C5×C8—24no—
82C2×C5×C8—82no—
83C5×C4⋊C4—22no—
84C5×C22⋊C4—22no—
85C5×C22⋊C4—22no—
86C42×C5—21no—
87C42×C5—21no—
88C22⋊Dic5—42no—
89C22⋊Dic5—82no—
90C4⋊Dic5—42no—
91C4⋊Dic5—41no—
92C4⋊Dic5—42no—
93C4⋊Dic5—41no—
94#683—82no—
95#681—44no—
96C2×#168—42no—
97C2×C4×D4C2×D511yescharacteristic
98C22×C4⋊C4—51no—
99#435—24no—
100#432—21no—
101#407—52no—
102#407C2×D512yes—
103C2×C22⋊C8—21no—
104#400—101no—
105#399—101no—
106#389—101no—
107#386—52no—
108#385—52no—
109#380—51no—
110#380—101no—
111#343—104no—
112#324—101no—
113#323—101no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—41no—
119C23×C5—41no—
120C23×C5—41no—
121C23×C5—41no—
122C23×C5#10112yes—
123C23×C5C2×C22⋊C411yescharacteristic
124C23×C5C4×D412yes—
125C5×D4—44no—
126C2×C4×C5—24no—
127C2×C4×C5—22no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—41no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—41no—
137C2×C4×C5—41no—
138C2×C4×C5—81no—
139C2×C4×C5#12011yescharacteristic
140C2×C4×C5#12112yes—
141C2×C4×C5#12311yescharacteristic
142C2×Dic5—42no—
143C2×Dic5—42no—
144C2×Dic5—42no—
145C2×Dic5—42no—
146C2×Dic5—42no—
147C2×Dic5—42no—
148C2×Dic5—84no—
149C5×C8—44no—
150#168—82no—
151C22×D4C4×D511yescharacteristic
152C23×C4—52no—
153C23×C4C22⋊D511yescharacteristic
154C23×C4C22×D511yescharacteristic
155C2×M16—102no—
156C2×M16C22⋊D514yes—
157C22×C8—22no—
158C4×D4—22no—
159C4×D4—22no—
160C2×C4⋊C4—101no—
161C2×C4⋊C4—102no—
162C2×C4⋊C4—102no—
163C2×C4⋊C4—101no—
164C2×C4⋊C4—101no—
165C2×C4⋊C4—102no—
166C2×C4⋊C4D2011yescharacteristic
167C2×C22⋊C4—52no—
168C2×C22⋊C4—102no—
169C2×C22⋊C4C4×D511yescharacteristic
170C2×C22⋊C4D2011yescharacteristic
171C2×C42C22⋊D511yescharacteristic
172C4⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—102no—
177C22⋊C8—102no—
178#97—22no—
179#95—102no—
180#95—102no—
181#95—101no—
182C22×C5—21no—
183C22×C5—21no—
184C22×C5—22no—
185C22×C5—22no—
186C22×C5—22no—
187C22×C5—22no—
188C22×C5—42no—
189C22×C5—42no—
190C22×C5—42no—
191C22×C5—41no—
192C22×C5—41no—
193C22×C5—81no—
194C22×C5#38611yescharacteristic
195C22×C5C2×#10111yescharacteristic
196C22×C5#41311yescharacteristic
197C4×C5—22no—
198C4×C5—21no—
199C4×C5—41no—
200C4×C5—41no—
201C4×C5—41no—
202C4×C5#45812yes—
203Dic5—82no—
204Dic5—82no—
205C24#68511yescharacteristic
206C24C2×C4×D511yescharacteristic
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—41no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—102no—
215C22×C4—102no—
216C22×C4—104no—
217C22×C4—402no—
218C22×C4#68511yescharacteristic
219C22×C4#68511yescharacteristic
220C22×C4#68511yescharacteristic
221C22×C4C2×D2011yescharacteristic
222C22×C4#70912yes—
223C22×C4C2×C22⋊D511yescharacteristic
224C22×C4C2×C22⋊D512yes—
225M16—44no—
226M16—204no—
227C2×C8—24no—
228C2×C8—82no—
229C2×C8—202no—
230C4⋊C4—22no—
231C4⋊C4—202no—
232C4⋊C4—201no—
233C4⋊C4—201no—
234C4⋊C4—202no—
235C4⋊C4—402no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—202no—
239C22⋊C4—402no—
240C42—21no—
241C42—21no—
242C2×C5—21no—
243C2×C5—21no—
244C2×C5—42no—
245C2×C5—42no—
246C2×C5#189011yescharacteristic
247C2×C5#190111yescharacteristic
248C2×C5#191411yescharacteristic
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#395612yes—
258C23C2×#68511yescharacteristic
259C23C4×C22⋊D512yes—
260D4—44no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—202no—
275C2×C4—202no—
276C2×C4—202no—
277C2×C4—202no—
278C2×C4—202no—
279C2×C4—202no—
280C2×C4—404no—
281C2×C4#407511yescharacteristic
282C2×C4#407612yes—
283C2×C4#409911yescharacteristic
284C8—44no—
285C8—402no—
286C5#950911yescharacteristic, complemented, Sylow
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#6542111yescharacteristic
300C22C2×#395611yescharacteristic
301C22#6550211yescharacteristic
302C4—22no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—402no—
308C4—402no—
309C4#6553212yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—42no—
314C2#1060244811yescharacteristic
315C2#1060279611yescharacteristic
316C2#1060282311yescharacteristic
subgroupG/H
centerC22#65502
commutatorC24×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2973C2
socleC22×C5#386
prime radicalC22×C5×D4C2×C4
cosocleC23×C4×C5C23
hypercenter#2973D5
hypocenterC5#9509
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #10600766C2semidirect
5. #10598175C2semidirect
286. C5#9509semidirect

Maximal subgroups

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

subgroupindex
C5×#29732
#106141052
#106023952
#106007662
#105981752
#95095