Finite Group Databasebeta

← #12010319

Subgroups — #12010319

Order 1280.

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

normal subgroups only

This group has 472 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×#2486C211yesmaximal, characteristic
2#10602412C211yesmaximal, characteristic
3#10602394C211yesmaximal, characteristic
4#10598140C214yesmaximal, complemented
5C2×C5×#123—21no—
6C2×C4×C5×D4C2211yescharacteristic
7C22×C5×C4⋊C4C411yescharacteristic, complemented
8C5×#398C412yescomplemented
9C5×#397C2212yes—
10C5×#389—21no—
11C5×#381—21no—
12C5×#380—21no—
13C5×#380C411yescharacteristic, complemented
14C2×#3966—21no—
15#65501—21no—
16#65214—21no—
17#64746C2212yes—
18#64745—21no—
19#64745C2212yes—
20#11874—51nomaximal, Sylow
21C22×C5×D4D411yescharacteristic
22C23×C4×C5—21no—
23C23×C4×C5C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C5×#123—42no—
26C5×#123—42no—
27C4×C5×D4—41no—
28C4×C5×D4—41no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—22no—
32C2×C5×C4⋊C4—41no—
33C2×C5×C4⋊C4—41no—
34C2×C5×C4⋊C4C2×C412yes—
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4D411yescharacteristic
39C2×C5×C22⋊C4C2311yescharacteristic
40C2×C42×C5D411yescharacteristic
41C2×C22⋊Dic5—22no—
42C2×C4⋊Dic5—41no—
43C5×#95—22no—
44C5×#95—22no—
45C5×#95C2×C412yes—
46#3966—42no—
47#3963—41no—
48C22⋊#168—22no—
49C4⋊#168—41no—
50C4×#168—41no—
51#2486D511yescharacteristic
52#1799—51no—
53#1460—51no—
54#1344—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C2×D411yescharacteristic
57C2×C5×D4—41no—
58C2×C5×D4—41no—
59C2×C5×D4D812yes—
60C2×C5×D4SD1612yes—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—22no—
66C22×C4×C5—22no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5—42no—
70C22×C4×C5—42no—
71C22×C4×C5—42no—
72C22×C4×C5C22⋊C411yescharacteristic
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22×C411yescharacteristic
76C22×C4×C5C2×D411yescharacteristic
77C22×Dic5—42no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—42no—
80C5×C4⋊C4—42no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—44no—
84C5×C22⋊C4—41no—
85C5×C22⋊C4—41no—
86C5×C22⋊C4—42no—
87C5×C22⋊C4—42no—
88C5×C22⋊C4—42no—
89C42×C5—21no—
90C42×C5—21no—
91C22⋊Dic5—42no—
92C22⋊Dic5—42no—
93C4⋊Dic5—82no—
94C2×#168—42no—
95C2×#123—21no—
96C2×C4×D4C2×D511yescharacteristic
97C22×C4⋊C4Dic511yescharacteristic, complemented
98#431—101no—
99C2×#99—101no—
100#398Dic512yescomplemented
101#397C2×D512yes—
102#389—21no—
103#381—21no—
104#380—21no—
105#380—101no—
106#380Dic511yescharacteristic, complemented
107#342—52no—
108#323—52no—
109#323—101no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—41no—
115C23×C5—41no—
116C23×C5—41no—
117C23×C5—41no—
118C23×C5#10212yes—
119C23×C5C2×C22⋊C411yescharacteristic
120C23×C5C2×D811yescharacteristic
121C23×C5C2×SD1611yescharacteristic
122C5×D4—42no—
123C5×D4—42no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—22no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—42no—
131C2×C4×C5—42no—
132C2×C4×C5—42no—
133C2×C4×C5—41no—
134C2×C4×C5—41no—
135C2×C4×C5—41no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—42no—
141C2×C4×C5—42no—
142C2×C4×C5—81no—
143C2×C4×C5—84no—
144C2×C4×C5#9911yescharacteristic
145C2×C4×C5#9911yescharacteristic
146C2×C4×C5#10212yes—
147C2×Dic5—82no—
148C2×Dic5—82no—
149C2×Dic5—82no—
150#168—82no—
151C22×D4C22⋊D511yescharacteristic
152C23×C4—21no—
153C23×C4C2×Dic511yescharacteristic
154C23×C4C22⋊D511yescharacteristic
155#123—42no—
156#123—42no—
157C4×D4—41no—
158C4×D4—41no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—201no—
165C2×C4⋊C4C2×Dic512yes—
166C2×C4⋊C4C2×Dic511yescharacteristic
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—102no—
170C2×C22⋊C4C22⋊D511yescharacteristic
171C2×C22⋊C4C22×D511yescharacteristic
172C2×C42C22⋊D511yescharacteristic
173C4⋊C8—201no—
174#99—202no—
175C22⋊C8—102no—
176C4×C8—201no—
177#95—22no—
178#95—22no—
179#95—201no—
180#95C2×Dic512yes—
181C22×C5—21no—
182C22×C5—21no—
183C22×C5—21no—
184C22×C5—22no—
185C22×C5—21no—
186C22×C5—42no—
187C22×C5—41no—
188C22×C5—41no—
189C22×C5—42no—
190C22×C5—41no—
191C22×C5—41no—
192C22×C5—41no—
193C22×C5—41no—
194C22×C5—81no—
195C22×C5C2×#9911yescharacteristic
196C22×C5C2×#10211yescharacteristic
197C22×C5#42111yescharacteristic
198C4×C5—21no—
199C4×C5—22no—
200C4×C5—41no—
201C4×C5—41no—
202C4×C5—41no—
203C4×C5—82no—
204C4×C5—82no—
205C4×C5#35212yes—
206Dic5—162no—
207C24C22⋊Dic511yescharacteristic
208C24C2×C22⋊D511yescharacteristic
209C2×D4—41no—
210C2×D4—41no—
211C2×D4#68612yes—
212C2×D4#68712yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—202no—
225C22×C4C22⋊Dic511yescharacteristic
226C22×C4C22⋊Dic511yescharacteristic
227C22×C4C22⋊Dic511yescharacteristic
228C22×C4C22×Dic511yescharacteristic
229C22×C4C2×C22⋊D511yescharacteristic
230C2×C8—202no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—402no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—202no—
244C22⋊C4—202no—
245C42—21no—
246C42—21no—
247C2×C5—21no—
248C2×C5—21no—
249C2×C5—41no—
250C2×C5—42no—
251C2×C5—41no—
252C2×C5#151911yescharacteristic
253C2×C5C2×#35211yescharacteristic
254C2×C5#212911yescharacteristic
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23#396412yes—
264C23C2×#68611yescharacteristic
265C23C2×#68711yescharacteristic
266C23C2×C22⋊Dic511yescharacteristic
267D4—42no—
268D4—42no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—84no—
289C2×C4—402no—
290C2×C4—402no—
291C2×C4—402no—
292C2×C4#396412yes—
293C2×C4#396611yescharacteristic
294C2×C4#396611yescharacteristic
295C8—402no—
296C5#1187411yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#396411yescharacteristic
312C22C2×#396611yescharacteristic
313C22#6552511yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4—82no—
321C4—802no—
322C4#6475912yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#1060246811yescharacteristic
329C2C2×#6475911yescharacteristic
330C2#1060266411yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2486C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2486D5
hypocenterC5#11874
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
4. #10598140C2semidirect
7. C22×C5×C4⋊C4C4semidirect
8. C5×#398C4semidirect
13. C5×#380C4semidirect
97. C22×C4⋊C4Dic5semidirect
100. #398Dic5semidirect
106. #380Dic5semidirect
296. C5#11874semidirect

Maximal subgroups

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

subgroupindex
C5×#24862
#106024122
#106023942
#105981402
#118745