Finite Group Databasebeta

← #12010428

Subgroups — #12010428

Order 1280.

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

normal subgroups only

This group has 452 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×#2487C211yesmaximal, characteristic
2#10602412C211yesmaximal, characteristic
3#10602394C211yesmaximal, characteristic
4#10598142C214yesmaximal, complemented
5C2×C4×C5×D4C2211yescharacteristic
6C22×C5×C4⋊C4C411yescharacteristic, complemented
7C5×#400C412yescomplemented
8C5×#399C2212yes—
9C5×#389—21no—
10C5×#386—21no—
11C5×#385—21no—
12C5×#380—21no—
13C5×#380C411yescharacteristic, complemented
14C2×#3966—21no—
15#65501—21no—
16#65214—21no—
17#64746C2212yes—
18#64745—21no—
19#64745C2212yes—
20#11899—51nomaximal, Sylow
21C22×C5×D4D411yescharacteristic
22C23×C4×C5—21no—
23C23×C4×C5C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C4×C5×D4—41no—
26C4×C5×D4—41no—
27C2×C5×C4⋊C4—21no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—41no—
31C2×C5×C4⋊C4—41no—
32C2×C5×C4⋊C4C2×C412yes—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C22⋊C4—21no—
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4D411yescharacteristic
37C2×C5×C22⋊C4C2311yescharacteristic
38C2×C42×C5D411yescharacteristic
39C2×C22⋊Dic5—22no—
40C2×C4⋊Dic5—41no—
41C5×#95—22no—
42C5×#95—22no—
43C5×#95C2×C412yes—
44#3966—42no—
45#3963—41no—
46C22⋊#168—22no—
47C4⋊#168—41no—
48C4×#168—41no—
49#2487D511yescharacteristic
50#1799—51no—
51#1460—51no—
52#1352—54no—
53C24×C5C22⋊C411yescharacteristic
54C24×C5C2×D411yescharacteristic
55C2×C5×D4—41no—
56C2×C5×D4—41no—
57C2×C5×D4SD1612yes—
58C2×C5×D4Q1612yes—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—22no—
64C22×C4×C5—22no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—42no—
68C22×C4×C5—82no—
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5C22×C411yescharacteristic
73C22×C4×C5C2×D411yescharacteristic
74C22×Dic5—42no—
75C5×C4⋊C4—41no—
76C5×C4⋊C4—42no—
77C5×C4⋊C4—42no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4—82no—
81C5×C22⋊C4—41no—
82C5×C22⋊C4—41no—
83C5×C22⋊C4—42no—
84C5×C22⋊C4—82no—
85C42×C5—21no—
86C42×C5—21no—
87C22⋊Dic5—42no—
88C22⋊Dic5—42no—
89C4⋊Dic5—82no—
90C2×#168—42no—
91C2×C4×D4C2×D511yescharacteristic
92C22×C4⋊C4Dic511yescharacteristic, complemented
93#431—101no—
94C2×#99—101no—
95#400Dic512yescomplemented
96#399C2×D512yes—
97#389—21no—
98#386—21no—
99#385—21no—
100#380—21no—
101#380—101no—
102#380Dic511yescharacteristic, complemented
103#342—52no—
104#323—52no—
105#323—101no—
106C23×C5—21no—
107C23×C5—21no—
108C23×C5—21no—
109C23×C5—21no—
110C23×C5—41no—
111C23×C5—41no—
112C23×C5—41no—
113C23×C5—41no—
114C23×C5#10312yes—
115C23×C5C2×C22⋊C411yescharacteristic
116C23×C5C2×SD1611yescharacteristic
117C23×C5C2×Q1611yescharacteristic
118C5×D4—42no—
119C5×D4—42no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—22no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—42no—
127C2×C4×C5—42no—
128C2×C4×C5—42no—
129C2×C4×C5—41no—
130C2×C4×C5—41no—
131C2×C4×C5—41no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—81no—
139C2×C4×C5—84no—
140C2×C4×C5#9911yescharacteristic
141C2×C4×C5#9911yescharacteristic
142C2×C4×C5#10312yes—
143C2×Dic5—82no—
144C2×Dic5—82no—
145C2×Dic5—82no—
146#168—82no—
147C22×D4C22⋊D511yescharacteristic
148C23×C4—21no—
149C23×C4C2×Dic511yescharacteristic
150C23×C4C22⋊D511yescharacteristic
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—201no—
159C2×C4⋊C4C2×Dic512yes—
160C2×C4⋊C4C2×Dic511yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—102no—
164C2×C22⋊C4C22⋊D511yescharacteristic
165C2×C22⋊C4C22×D511yescharacteristic
166C2×C42C22⋊D511yescharacteristic
167C4⋊C8—201no—
168#99—202no—
169C22⋊C8—102no—
170C4×C8—201no—
171#95—22no—
172#95—22no—
173#95—201no—
174#95C2×Dic512yes—
175C22×C5—21no—
176C22×C5—21no—
177C22×C5—21no—
178C22×C5—22no—
179C22×C5—21no—
180C22×C5—42no—
181C22×C5—41no—
182C22×C5—41no—
183C22×C5—42no—
184C22×C5—41no—
185C22×C5—41no—
186C22×C5—41no—
187C22×C5—41no—
188C22×C5—81no—
189C22×C5C2×#9911yescharacteristic
190C22×C5C2×#10311yescharacteristic
191C22×C5#42111yescharacteristic
192C4×C5—21no—
193C4×C5—22no—
194C4×C5—41no—
195C4×C5—41no—
196C4×C5—41no—
197C4×C5—82no—
198C4×C5—82no—
199C4×C5#35212yes—
200Dic5—162no—
201C24C22⋊Dic511yescharacteristic
202C24C2×C22⋊D511yescharacteristic
203C2×D4—41no—
204C2×D4—41no—
205C2×D4#68812yes—
206C2×D4#68912yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—82no—
217C22×C4—202no—
218C22×C4C22⋊Dic511yescharacteristic
219C22×C4C22⋊Dic511yescharacteristic
220C22×C4C22⋊Dic511yescharacteristic
221C22×C4C22×Dic511yescharacteristic
222C22×C4C2×C22⋊D511yescharacteristic
223C2×C8—202no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—82no—
230C4⋊C4—402no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—82no—
235C22⋊C4—202no—
236C22⋊C4—202no—
237C42—21no—
238C42—21no—
239C2×C5—21no—
240C2×C5—21no—
241C2×C5—41no—
242C2×C5—42no—
243C2×C5—41no—
244C2×C5#152011yescharacteristic
245C2×C5C2×#35211yescharacteristic
246C2×C5#212811yescharacteristic
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23#396712yes—
256C23C2×#68811yescharacteristic
257C23C2×#68911yescharacteristic
258C23C2×C22⋊Dic511yescharacteristic
259D4—42no—
260D4—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4—402no—
282C2×C4—402no—
283C2×C4—402no—
284C2×C4#396611yescharacteristic
285C2×C4#396611yescharacteristic
286C2×C4#396712yes—
287C8—402no—
288C5#1189911yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#396611yescharacteristic
304C22C2×#396711yescharacteristic
305C22#6552511yescharacteristic
306C4—21no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4—802no—
314C4#6475912yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2#1060248311yescharacteristic
321C2C2×#6475911yescharacteristic
322C2#1060265411yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2487C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2487D5
hypocenterC5#11899
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
4. #10598142C2semidirect
6. C22×C5×C4⋊C4C4semidirect
7. C5×#400C4semidirect
13. C5×#380C4semidirect
92. C22×C4⋊C4Dic5semidirect
95. #400Dic5semidirect
102. #380Dic5semidirect
288. C5#11899semidirect

Maximal subgroups

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

subgroupindex
C5×#24872
#106024122
#106023942
#105981422
#118995