Finite Group Databasebeta

← #11996979

Subgroups — #11996979

Order 1280.

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

normal subgroups only

This group has 518 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#10614113C211yesmaximal, characteristic
3#10602396C211yesmaximal, characteristic
4#10601617C212yesmaximal, complemented
5#10598241C212yesmaximal, complemented
6C2×C4×C5×D4C2211yescharacteristic
7C2×#4082C412yes—
8C2×#4015C2211yescharacteristic
9C5×#435—24no—
10C5×#432C2212yes—
11C5×#407—21no—
12C2×C5×C22⋊C8—21no—
13#65501—21no—
14#65501—21no—
15#65263C412yes—
16#65242—21no—
17#65242—21no—
18C4×C4⋊Dic5C2211yescharacteristic
19#65215C2212yes—
20#64745—21no—
21#64745—21no—
22#64701—24no—
23#9451—51nomaximal, Sylow
24C22×C5×D4C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5D411yescharacteristic
27C22×Dic10C2×C412yes—
28C2×C5×M16—22no—
29C22×C5×C8—22no—
30C4×C5×D4—22no—
31C4×C5×D4—22no—
32C2×C5×C4⋊C4C2×C411yescharacteristic
33C2×C5×C22⋊C4D411yescharacteristic
34C2×C5×C22⋊C4D411yescharacteristic
35C2×C42×C5C2311yescharacteristic
36C2×C22⋊Dic5—22no—
37C2×C22⋊Dic5—22no—
38#4082—44no—
39C2×C4⋊Dic5—21no—
40C2×C4⋊Dic5—21no—
41C2×C4⋊Dic5—21no—
42C2×C4×Dic5C2×C412yes—
43#4015—22no—
44#4015—22no—
45C5×C4⋊C8D414yes—
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—
54C4⋊#168—22no—
55#3935—22no—
56#2973D511yescharacteristic
57#2547—51no—
58#1558—52no—
59#1466—51no—
60#1285—52no—
61C24×C5C22⋊C411yescharacteristic
62C24×C5C22⋊C411yescharacteristic
63C2×C5×D4—22no—
64C2×C5×D4—22no—
65C2×C5×D4—41no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—42no—
71C22×C4×C5—42no—
72C22×C4×C5C22⋊C411yescharacteristic
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22×C411yescharacteristic
75C22×C4×C5C2×D411yescharacteristic
76C22×C4×C5C2×D411yescharacteristic
77C22×Dic5—22no—
78C22×Dic5—22no—
79C2×Dic10—24no—
80C2×Dic10—42no—
81C2×Dic10—42no—
82C5×M16—82no—
83C2×C5×C8—24no—
84C2×C5×C8—82no—
85C5×C4⋊C4—22no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4—22no—
88C42×C5C2×D412yes—
89C42×C5#4112yes—
90C22⋊Dic5—44no—
91C22⋊Dic5—44no—
92C4⋊Dic5—42no—
93C4⋊Dic5—41no—
94C4⋊Dic5—42no—
95C4⋊Dic5—41no—
96C4×Dic5—42no—
97C2×#168—42no—
98C2×C4⋊Q8—51no—
99C2×#124—52no—
100C2×C4×D4C2×D511yescharacteristic
101#435—24no—
102#432—52no—
103#432C2×D512yes—
104#407—21no—
105C2×C22⋊C8—21no—
106#397—101no—
107#397—101no—
108#391—52no—
109#380—101no—
110#380—101no—
111C4×C4⋊C4—51no—
112#333—104no—
113#323—101no—
114#323—101no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—41no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5—41no—
124C23×C5#10012yes—
125C23×C5C2×C22⋊C411yescharacteristic
126C5×D4—44no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—22no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—41no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—41no—
136C2×C4×C5—41no—
137C2×C4×C5—81no—
138C2×C4×C5—81no—
139C2×C4×C5C4×D412yes—
140C2×C4×C5#12011yescharacteristic
141C2×C4×C5#12112yes—
142C2×C4×C5#12311yescharacteristic
143C2×Dic5—42no—
144C2×Dic5—42no—
145C2×Dic5—42no—
146C2×Dic5—42no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149Dic10—84no—
150C5×C8—44no—
151#168—82no—
152C22×Q8—52no—
153C22×D4C4×D511yescharacteristic
154C23×C4C22⋊D511yescharacteristic
155C23×C4C22⋊D511yescharacteristic
156C2×M16—22no—
157C22×C8—22no—
158C4⋊Q8—102no—
159C4⋊Q8—102no—
160#124—204no—
161C4×D4—22no—
162C4×D4—22no—
163C2×C4⋊C4—101no—
164C2×C4⋊C4—101no—
165C2×C4⋊C4—101no—
166C2×C4⋊C4C4×D511yescharacteristic
167C2×C22⋊C4—102no—
168C2×C22⋊C4—102no—
169C2×C22⋊C4D2011yescharacteristic
170C2×C22⋊C4D2011yescharacteristic
171C2×C42—52no—
172C2×C42C22×D511yescharacteristic
173C4⋊C8—102no—
174C4⋊C8C22⋊D514yes—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—102no—
179#97—22no—
180#97—102no—
181#95—102no—
182#95—102no—
183#95—101no—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—22no—
187C22×C5—42no—
188C22×C5—42no—
189C22×C5—41no—
190C22×C5—41no—
191C22×C5—41no—
192C22×C5—41no—
193C22×C5—41no—
194C22×C5—81no—
195C22×C5—81no—
196C22×C5#38611yescharacteristic
197C22×C5C2×#10011yescharacteristic
198C22×C5#41311yescharacteristic
199C4×C5—21no—
200C4×C5—22no—
201C4×C5—22no—
202C4×C5—41no—
203C4×C5—41no—
204C4×C5#45512yes—
205Dic5—82no—
206Dic5—82no—
207C24#68511yescharacteristic
208C24#68511yescharacteristic
209C2×Q8—104no—
210C2×Q8—202no—
211C2×Q8—202no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—102no—
222C22×C4—102no—
223C22×C4#68511yescharacteristic
224C22×C4#68511yescharacteristic
225C22×C4C2×C4×D511yescharacteristic
226C22×C4C2×D2011yescharacteristic
227C22×C4C2×C22⋊D511yescharacteristic
228M16—82no—
229C2×C8—24no—
230C2×C8—82no—
231C2×C8—202no—
232C4⋊C4—22no—
233C4⋊C4—202no—
234C4⋊C4—201no—
235C4⋊C4—201no—
236C4⋊C4—202no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—204no—
240C22⋊C4—204no—
241C42—202no—
242C42#70912yes—
243C42C2×C22⋊D512yes—
244C2×C5—21no—
245C2×C5—21no—
246C2×C5—42no—
247C2×C5—81no—
248C2×C5#188511yescharacteristic
249C2×C5#190011yescharacteristic
250C2×C5#190911yescharacteristic
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#395512yes—
261C23C2×#68511yescharacteristic
262Q8—404no—
263D4—44no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—202no—
277C2×C4—202no—
278C2×C4—202no—
279C2×C4—202no—
280C2×C4—202no—
281C2×C4—202no—
282C2×C4C4×C22⋊D512yes—
283C2×C4#407511yescharacteristic
284C2×C4#407612yes—
285C2×C4#409911yescharacteristic
286C8—44no—
287C8—402no—
288C5#945111yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22C2×#395511yescharacteristic
302C22#6542111yescharacteristic
303C22#6550211yescharacteristic
304C4—22no—
305C4—22no—
306C4—21no—
307C4—41no—
308C4—41no—
309C4—402no—
310C4—402no—
311C4#6553112yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#1060244211yescharacteristic
317C2#1060279511yescharacteristic
318C2#1060281711yescharacteristic
subgroupG/H
centerC22#65502
commutatorC22×C4×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#2973C2
socleC22×C5#386
prime radicalC22×C5×D4C2×C4
cosocleC2×C42×C5C23
hypercenter#2973D5
hypocenterC5#9451
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #10601617C2semidirect
5. #10598241C2semidirect
288. C5#9451semidirect

Maximal subgroups

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

subgroupindex
C5×#29732
#106141132
#106023962
#106016172
#105982412
#94515