Finite Group Databasebeta

← #11997109

Subgroups — #11997109

Order 1280.

734 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2273C211yesmaximal, characteristic
2#10613844C211yesmaximal, characteristic
3#10602332C211yesmaximal, characteristic
4#10600235C212yesmaximal, complemented
5#10598107C212yesmaximal, complemented
6C22×C5×C22⋊C4—21no—
7C22×C5×C22⋊C4C2211yescharacteristic
8C2×#4072C2211yescharacteristic
9C4×C5×C22⋊C4—21no—
10C4×C5×C22⋊C4—24no—
11C4×C5×C22⋊C4C2212yes—
12#65536—21no—
13#65500C2211yescharacteristic
14#65499—21no—
15#65496C2212yes—
16#65242C412yescomplemented
17#65238C412yescomplemented
18#65233—22no—
19#64745—22no—
20#64743—24no—
21#9477—51nomaximal, Sylow
22C25×C5D411yescharacteristic
23C23×C4×C5—22no—
24C23×C4×C5D411yescharacteristic
25C23×C4×C5C2311yescharacteristic
26C2×C5×C22⋊C4—22no—
27C2×C5×C22⋊C4—28no—
28C2×C5×C22⋊C4—22no—
29C2×C5×C22⋊C4—22no—
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4C2×C412yes—
32C2×C5×C22⋊C4D412yes—
33C2×C42×C5—22no—
34C2×C42×C5D414yes—
35C2×C22⋊Dic5—22no—
36C2×C22⋊Dic5—21no—
37C2×C22⋊Dic5—21no—
38C2×C22⋊Dic5—22no—
39#4072—22no—
40#4072—22no—
41C2×C4⋊Dic5C2×C411yescharacteristic, complemented
42C2×C4⋊Dic5C2×C411yescharacteristic, complemented
43C2×#683—22no—
44C2×C4×Dic5—21no—
45C2×#681—22no—
46C5×#95—22no—
47C5×#95—22no—
48#3963—22no—
49#3963—21no—
50#3963C2×C412yescomplemented
51C22⋊#168—22no—
52C22⋊#168—22no—
53#2482—51no—
54#2273D511yescharacteristic
55#1826—52no—
56#1462—51no—
57#1284—52no—
58C24×C5—21no—
59C24×C5—24no—
60C24×C5—21no—
61C24×C5—42no—
62C24×C5—81no—
63C24×C5C22⋊C412yes—
64C24×C5C2×D411yescharacteristic
65C22×C4×C5—21no—
66C22×C4×C5—24no—
67C22×C4×C5—28no—
68C22×C4×C5—21no—
69C22×C4×C5—42no—
70C22×C4×C5—81no—
71C22×C4×C5—82no—
72C22×C4×C5—82no—
73C22×C4×C5C22⋊C412yes—
74C22×C4×C5C22×C411yescharacteristic
75C22×C4×C5C2×D411yescharacteristic
76C22×C4×C5C2×D412yes—
77C22×C4×C5#4112yes—
78C22×Dic5—22no—
79C22×Dic5—21no—
80C22×Dic5—21no—
81C5×C22⋊C4—28no—
82C5×C22⋊C4—28no—
83C5×C22⋊C4—24no—
84C5×C22⋊C4—24no—
85C42×C5—44no—
86C42×C5—82no—
87C22⋊Dic5—42no—
88C22⋊Dic5—44no—
89C22⋊Dic5—41no—
90C22⋊Dic5—41no—
91C22⋊Dic5—82no—
92C4⋊Dic5—22no—
93C4⋊Dic5—22no—
94#683—82no—
95C4×Dic5—42no—
96#681—44no—
97C2×#168—42no—
98C2×#117—51no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4C2×D511yescharacteristic
101#407—52no—
102#399—52no—
103#397—52no—
104#388—102no—
105#385—101no—
106#381—51no—
107#379—101no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4C2×D512yes—
111#339—104no—
112#323—102no—
113C23×C5—22no—
114C23×C5—22no—
115C23×C5—24no—
116C23×C5—22no—
117C23×C5—22no—
118C23×C5—22no—
119C23×C5—22no—
120C23×C5—41no—
121C23×C5—42no—
122C23×C5—42no—
123C23×C5—42no—
124C23×C5—42no—
125C23×C5—81no—
126C23×C5—81no—
127C23×C5—88no—
128C23×C5#10414yes—
129C23×C5C2×C22⋊C411yescharacteristic
130C23×C5#12011yescharacteristic
131C23×C5#12311yescharacteristic
132C2×C4×C5—24no—
133C2×C4×C5—28no—
134C2×C4×C5—22no—
135C2×C4×C5—41no—
136C2×C4×C5—41no—
137C2×C4×C5—42no—
138C2×C4×C5—44no—
139C2×C4×C5—42no—
140C2×C4×C5—44no—
141C2×C4×C5—42no—
142C2×C4×C5—81no—
143C2×C4×C5—88no—
144C2×C4×C5C4×D412yes—
145C2×C4×C5#12112yes—
146C2×Dic5—42no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149C2×Dic5—41no—
150C2×Dic5—41no—
151C2×Dic5—42no—
152C2×Dic5—42no—
153#168—82no—
154C25C22⋊D511yescharacteristic
155C23×C4—22no—
156C23×C4C22⋊D511yescharacteristic
157C23×C4C22×D511yescharacteristic
158C2×M16—102no—
159#117—102no—
160#117—102no—
161C2×C4⋊C4—51no—
162C2×C4⋊C4—51no—
163C2×C4⋊C4—102no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—102no—
170C2×C22⋊C4—101no—
171C2×C22⋊C4—101no—
172C2×C22⋊C4—102no—
173C2×C22⋊C4C4×D512yes—
174C2×C22⋊C4D2012yes—
175C2×C42—22no—
176C2×C42—101no—
177C2×C42C22⋊D514yes—
178C22⋊C8—102no—
179C22⋊C8—102no—
180#95—22no—
181#95—22no—
182#95—52no—
183#95—102no—
184#95—101no—
185C22×C5—21no—
186C22×C5—21no—
187C22×C5—21no—
188C22×C5—21no—
189C22×C5—21no—
190C22×C5—24no—
191C22×C5—21no—
192C22×C5—21no—
193C22×C5—44no—
194C22×C5—44no—
195C22×C5—44no—
196C22×C5—44no—
197C22×C5—42no—
198C22×C5—42no—
199C22×C5—82no—
200C22×C5—84no—
201C22×C5#38611yescharacteristic
202C22×C5C2×#10412yes—
203C22×C5#45812yes—
204C4×C5—22no—
205C4×C5—22no—
206C4×C5—48no—
207C4×C5—42no—
208Dic5—82no—
209Dic5—81no—
210Dic5—81no—
211C24—21no—
212C24—24no—
213C24—21no—
214C24—42no—
215C24—81no—
216C24#68512yes—
217C24C2×C22⋊D511yescharacteristic
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—21no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—102no—
227C22×C4—101no—
228C22×C4—101no—
229C22×C4#68512yes—
230C22×C4C2×C4×D511yescharacteristic
231C22×C4C2×D2011yescharacteristic
232C22×C4#70912yes—
233C22×C4C2×C22⋊D512yes—
234M16—204no—
235C2×C8—202no—
236C4⋊C4—102no—
237C4⋊C4—102no—
238C4⋊C4—402no—
239C22⋊C4—28no—
240C22⋊C4—28no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—202no—
244C22⋊C4—201no—
245C22⋊C4—204no—
246C22⋊C4—201no—
247C22⋊C4—402no—
248C42—44no—
249C42—82no—
250C42—202no—
251C2×C5—22no—
252C2×C5—22no—
253C2×C5—41no—
254C2×C5—44no—
255C2×C5#189011yescharacteristic
256C2×C5#189611yescharacteristic
257C2×C5#191111yescharacteristic
258C23—22no—
259C23—22no—
260C23—24no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—41no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—88no—
273C23#393714yes—
274C23C2×#68511yescharacteristic
275C23#407511yescharacteristic
276C23#409911yescharacteristic
277C2×C4—24no—
278C2×C4—28no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—88no—
289C2×C4—202no—
290C2×C4—202no—
291C2×C4—202no—
292C2×C4—202no—
293C2×C4—202no—
294C2×C4—201no—
295C2×C4—201no—
296C2×C4C4×C22⋊D512yes—
297C2×C4#407612yes—
298C8—402no—
299C5#947711yescharacteristic, complemented, Sylow
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22C2×#393712yes—
317C22#6550211yescharacteristic
318C22#6553212yes—
319C4—22no—
320C4—22no—
321C4—48no—
322C4—42no—
323C4—402no—
324C4—401no—
325C4—401no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2#1060232511yescharacteristic
331C2#1060244511yescharacteristic
332C2#1060244811yescharacteristic
subgroupG/H
centerC22#65502
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2273C2
socleC22×C5#386
prime radicalC25×C5D4
cosocleC23×C4×C5C23
hypercenter#2273D5
hypocenterC5#9477
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2273

Complements

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

NH ≅ G/Ntype
4. #10600235C2semidirect
5. #10598107C2semidirect
16. #65242C4semidirect
17. #65238C4semidirect
41. C2×C4⋊Dic5C2×C4semidirect
42. C2×C4⋊Dic5C2×C4semidirect
50. #3963C2×C4semidirect
299. C5#9477semidirect

Maximal subgroups

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

subgroupindex
C5×#22732
#106138442
#106023322
#106002352
#105981072
#94775