Finite Group Databasebeta

← #11995086

Subgroups — #11995086

Order 1280.

648 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2854C211yesmaximal, characteristic
2C2×#65226C211yesmaximal, characteristic, complemented
3C2×#64757C212yesmaximal
4#10602324C211yesmaximal, characteristic, complemented
5#10598115C212yesmaximal, complemented
6C2×C5×#123—21no—
7C2×C5×#121C2211yescharacteristic
8C22×C42×C5C2211yescharacteristic
9C5×#390C412yescomplemented
10C5×#389C412yescomplemented
11C5×#388—22no—
12C5×#386—21no—
13C5×#380C2211yescharacteristic
14C2×#3963—21no—
15C2×#3963—21no—
16C2×#3963C2212yes—
17C2×C22⋊#168C2212yes—
18#65226—24no—
19#64757—24no—
20#64744—22no—
21#9157—51nomaximal, Sylow
22C22×C5×D4—21no—
23C22×C5×D4C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5C2311yescharacteristic
27C23×Dic5—22no—
28C5×#123—44no—
29C5×#121—22no—
30C5×#121—22no—
31C2×C5×C4⋊C4—22no—
32C2×C5×C4⋊C4C2×C411yescharacteristic
33C2×C5×C4⋊C4C2×C412yes—
34C2×C5×C22⋊C4—21no—
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—21no—
38C2×C42×C5—22no—
39C2×C42×C5—22no—
40C2×C42×C5C2×C412yes—
41C2×C42×C5D412yes—
42C22×#168—22no—
43C5×#95—22no—
44C5×#95—21no—
45#3963—24no—
46#3963—24no—
47#3963—22no—
48#3963D414yes—
49C22⋊#168—22no—
50C22⋊#168—22no—
51#2854D511yescharacteristic
52C2×#383—51no—
53#1817—51no—
54C2×#327—52no—
55#1294—52no—
56C24×C5—21no—
57C24×C5C2×D411yescharacteristic
58C2×C5×D4—22no—
59C2×C5×D4—42no—
60C2×C5×D4—41no—
61C2×C5×D4—41no—
62C2×C5×D4—42no—
63C2×C5×D4—42no—
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—22no—
69C22×C4×C5—22no—
70C22×C4×C5—22no—
71C22×C4×C5—21no—
72C22×C4×C5—42no—
73C22×C4×C5—42no—
74C22×C4×C5C22⋊C412yes—
75C22×C4×C5C22⋊C412yes—
76C22×C4×C5C22×C411yescharacteristic
77C22×C4×C5C2×D412yes—
78C22×C4×C5C2×D411yescharacteristic
79C22×C4×C5#4112yes—
80C22×Dic5—24no—
81C22×Dic5—44no—
82C22×Dic5—44no—
83C22×Dic5—42no—
84C22×Dic5—42no—
85C5×C4⋊C4—22no—
86C5×C4⋊C4—42no—
87C5×C4⋊C4—44no—
88C5×C22⋊C4—44no—
89C5×C22⋊C4—44no—
90C5×C22⋊C4—42no—
91C5×C22⋊C4—42no—
92C42×C5—24no—
93C42×C5—24no—
94C2×#168—44no—
95C2×#168—42no—
96C2×#123—21no—
97C2×#121C2×D511yescharacteristic
98C22×C42C2×D511yescharacteristic
99C2×C22⋊C8—52no—
100#390Dic512yescomplemented
101#389Dic512yescomplemented
102#388—22no—
103#386—21no—
104#383—104no—
105#380C2×D511yescharacteristic
106C2×#95—52no—
107C2×#95—101no—
108C2×#95—101no—
109#336—102no—
110#327—104no—
111C23×C5—22no—
112C23×C5—22no—
113C23×C5—41no—
114C23×C5—42no—
115C23×C5—41no—
116C23×C5—41no—
117C23×C5—42no—
118C23×C5C2×C22⋊C411yescharacteristic
119C23×C5#12011yescharacteristic
120C23×C5#12311yescharacteristic
121C23×C5#13512yes—
122C23×C5AGL(1,ℤ/8)12yes—
123C5×D4—82no—
124C5×D4—82no—
125C5×D4—82no—
126C2×C4×C5—22no—
127C2×C4×C5—22no—
128C2×C4×C5—22no—
129C2×C4×C5—22no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—22no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—41no—
141C2×C4×C5—42no—
142C2×C4×C5—41no—
143C2×C4×C5—42no—
144C2×C4×C5—41no—
145C2×C4×C5—42no—
146C2×C4×C5C4×D412yes—
147C2×C4×C5#12112yes—
148C2×Dic5—44no—
149C2×Dic5—44no—
150C2×Dic5—44no—
151C2×Dic5—84no—
152C2×Dic5—84no—
153#168—82no—
154C22×D4—21no—
155C22×D4C2×Dic511yescharacteristic
156C23×C4—102no—
157C23×C4C22⋊D511yescharacteristic
158C23×C4C22⋊D511yescharacteristic
159C23×C4C22×D511yescharacteristic
160C22×C8—102no—
161#123—44no—
162#121—22no—
163#121—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C2×Dic511yescharacteristic
166C2×C4⋊C4C2×Dic512yes—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42C2×Dic512yes—
174C2×C42C22⋊D512yes—
175C22⋊C8—102no—
176C22⋊C8—102no—
177#95—22no—
178#95—21no—
179#95—54no—
180#95—104no—
181#95—104no—
182#95—102no—
183C22×C5—21no—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—21no—
187C22×C5—24no—
188C22×C5—24no—
189C22×C5—81no—
190C22×C5—81no—
191C22×C5—81no—
192C22×C5—82no—
193C22×C5—82no—
194C22×C5#38611yescharacteristic
195C22×C5#41012yes—
196C22×C5#41611yescharacteristic
197C22×C5#41811yescharacteristic
198C22×C5#42112yes—
199C22×C5#44912yes—
200C22×C5#48212yes—
201C4×C5—22no—
202C4×C5—22no—
203C4×C5—42no—
204C4×C5—42no—
205C4×C5—81no—
206C4×C5—82no—
207Dic5—84no—
208C24—21no—
209C24C2×C22⋊D511yescharacteristic
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—41no—
215C2×D4—41no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—104no—
227C22×C4—204no—
228C22×C4—204no—
229C22×C4—202no—
230C22×C4—202no—
231C22×C4C22⋊Dic512yes—
232C22×C4C22⋊Dic512yes—
233C22×C4D4×D512yes—
234C22×C4#71112yes—
235C22×C4C22×Dic511yescharacteristic
236C22×C4C2×C22⋊D511yescharacteristic
237C2×C8—204no—
238C2×C8—202no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—24no—
247C42—24no—
248C2×C5—22no—
249C2×C5—22no—
250C2×C5—161no—
251C2×C5#151412yes—
252C2×C5#152112yes—
253C2×C5#187511yescharacteristic
254C2×C5#189711yescharacteristic
255C2×C5#189911yescharacteristic
256C23—22no—
257C23—22no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—42no—
262C23—42no—
263C23#407812yes—
264C23#408111yescharacteristic
265C23#408311yescharacteristic
266C23#409612yes—
267C23C2×C22⋊Dic511yescharacteristic
268D4—82no—
269D4—82no—
270D4—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—204no—
292C2×C4—204no—
293C2×C4—204no—
294C2×C4—404no—
295C2×C4—404no—
296C2×C4D4×Dic512yes—
297C2×C4#408512yes—
298C8—402no—
299C5#915711yescharacteristic, complemented, Sylow
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—24no—
306C22—81no—
307C22—81no—
308C22—82no—
309C22—82no—
310C22—81no—
311C22#6532412yes—
312C22#6532612yes—
313C22#6550411yescharacteristic
314C22#6550911yescharacteristic
315C22#6552211yescharacteristic
316C22#6552512yes—
317C22#6552912yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—82no—
323C4—81no—
324C4—404no—
325C2—22no—
326C2—22no—
327C2—161no—
328C2#1060247111yescharacteristic
329C2#1060249811yescharacteristic
330C2#1060250511yescharacteristic
331C2#1060255012yes—
332C2#1060260612yes—
subgroupG/H
centerC23C2×C22⋊Dic5
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2854C2
socleC23×C5C2×C22⋊C4
prime radicalC2×C5×#121C22
cosocleC23×C4×C5C23
hypercenter#2854D5
hypocenterC5#9157
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
2. C2×#65226C2semidirect
4. #10602324C2semidirect
5. #10598115C2semidirect
9. C5×#390C4semidirect
10. C5×#389C4semidirect
100. #390Dic5semidirect
101. #389Dic5semidirect
299. C5#9157semidirect

Maximal subgroups

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

subgroupindex
C5×#28542
C2×#652262
C2×#647572
#106023242
#105981152
#91575