Finite Group Databasebeta

← #12009921

Subgroups — #12009921

Order 1280.

516 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2542C211yesmaximal, characteristic
2C4×#3966C211yesmaximal, characteristic
3#10602395C211yesmaximal, characteristic
4#10598142C214yesmaximal, complemented
5C2×C5×C4⋊Q8C411yescharacteristic, complemented
6C2×C5×C22⋊Q8—21no—
7C2×C5×C22⋊Q8—21no—
8C2×C4×C5×D4C2211yescharacteristic
9C5×#399C412yescomplemented
10C5×#399C2212yes—
11C5×#390—21no—
12C5×#384—21no—
13C4×C5×C4⋊C4C411yescharacteristic, complemented
14C2×#3966—21no—
15C4×C22⋊Dic5—21no—
16#65496—21no—
17#64748—21no—
18#64746C2212yes—
19#64745C2212yes—
20#11788—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×C5×D4D411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C5×C4⋊Q8—41no—
26C5×C4⋊Q8—41no—
27C5×C22⋊Q8—44no—
28C5×C22⋊Q8—44no—
29C4×C5×D4—41no—
30C4×C5×D4—41no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—22no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4C2×C412yes—
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4—22no—
39C2×C5×C22⋊C4C2×C411yescharacteristic
40C2×C5×C22⋊C4C2311yescharacteristic
41C2×C42×C5—21no—
42C2×C42×C5C2×C411yescharacteristic
43C2×C22⋊Dic5—22no—
44C2×C4×Dic5—41no—
45C2×#681—41no—
46C5×#95C2×C412yes—
47#3966—24no—
48#3963—41no—
49C22⋊#168—22no—
50C22⋊#168—41no—
51#2542D511yescharacteristic
52C4×#99—51no—
53#1465—51no—
54#1352—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C2×D411yescharacteristic
57C2×C5×Q8—42no—
58C2×C5×Q8—42no—
59C2×C5×Q8—41no—
60C2×C5×Q8—41no—
61C2×C5×D4—22no—
62C2×C5×D4—22no—
63C2×C5×D4—41no—
64C22×C4×C5—22no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—22no—
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—24no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4—42no—
81C5×C4⋊C4—44no—
82C5×C4⋊C4—44no—
83C5×C4⋊C4—42no—
84C5×C22⋊C4—41no—
85C5×C22⋊C4—41no—
86C5×C22⋊C4—44no—
87C5×C22⋊C4—44no—
88C42×C5—21no—
89C42×C5—21no—
90C42×C5—42no—
91C22⋊Dic5—24no—
92C22⋊Dic5—42no—
93C4×Dic5—82no—
94#681—82no—
95C2×#168—42no—
96C2×C4⋊Q8Dic511yescharacteristic, complemented
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C4×D4C2×D511yescharacteristic
100C2×#99—101no—
101#407—101no—
102#399Dic512yescomplemented
103#399C2×D512yes—
104#390—21no—
105#384—21no—
106C4×C4⋊C4Dic511yescharacteristic, complemented
107C4×C22⋊C4—101no—
108#342—52no—
109#324—101no—
110#323—52no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—41no—
117C23×C5—41no—
118C23×C5—41no—
119C23×C5—41no—
120C23×C5C2×C22⋊C411yescharacteristic
121C23×C5#13712yes—
122C5×Q8—82no—
123C5×Q8—82no—
124C5×D4—44no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—22no—
128C2×C4×C5—24no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—41no—
137C2×C4×C5—41no—
138C2×C4×C5—41no—
139C2×C4×C5—42no—
140C2×C4×C5—42no—
141C2×C4×C5—42no—
142C2×C4×C5—81no—
143C2×C4×C5—81no—
144C2×C4×C5#9911yescharacteristic
145C2×C4×C5#9911yescharacteristic
146C2×C4×C5#10412yes—
147C2×Dic5—44no—
148C2×Dic5—82no—
149C2×Dic5—82no—
150#168—82no—
151C22×Q8—21no—
152C22×D4C22⋊D511yescharacteristic
153C23×C4C22⋊D511yescharacteristic
154C23×C4C22⋊D511yescharacteristic
155C2×M16—201no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C22⋊Q8—44no—
159C22⋊Q8—44no—
160C4×D4—41no—
161C4×D4—41no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C2×Dic512yes—
167C2×C4⋊C4C22⋊D511yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—102no—
171C2×C22⋊C4C2×Dic511yescharacteristic
172C2×C22⋊C4C22×D511yescharacteristic
173C2×C42—21no—
174C2×C42—201no—
175C2×C42C2×Dic511yescharacteristic
176#99—104no—
177C22⋊C8—102no—
178C22⋊C8—201no—
179#95—201no—
180#95C2×Dic512yes—
181C22×C5—21no—
182C22×C5—21no—
183C22×C5—22no—
184C22×C5—42no—
185C22×C5—42no—
186C22×C5—41no—
187C22×C5—41no—
188C22×C5—41no—
189C22×C5—41no—
190C22×C5—41no—
191C22×C5—81no—
192C22×C5—81no—
193C22×C5C2×#9911yescharacteristic
194C22×C5#41911yescharacteristic
195C22×C5C2×#10411yescharacteristic
196C4×C5—21no—
197C4×C5—22no—
198C4×C5—44no—
199C4×C5—41no—
200C4×C5—41no—
201C4×C5—41no—
202C4×C5—82no—
203C4×C5#35312yes—
204Dic5—84no—
205C24C22⋊Dic511yescharacteristic
206C24C2×C22⋊D511yescharacteristic
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—41no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—202no—
223C22×C4C22⋊Dic511yescharacteristic
224C22×C4C22⋊Dic511yescharacteristic
225C22×C4C22⋊Dic511yescharacteristic
226C22×C4C22×Dic511yescharacteristic
227C22×C4C2×C22⋊D511yescharacteristic
228M16—402no—
229C2×C8—202no—
230C4⋊C4—24no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—42no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—104no—
241C22⋊C4—202no—
242C42—21no—
243C42—21no—
244C42—42no—
245C42—402no—
246C2×C5—21no—
247C2×C5—21no—
248C2×C5—42no—
249C2×C5—81no—
250C2×C5#150811yescharacteristic
251C2×C5#212311yescharacteristic
252C2×C5C2×#35311yescharacteristic
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23#408812yes—
263C23C2×C22⋊Dic511yescharacteristic
264Q8—82no—
265Q8—82no—
266D4—44no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
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—81no—
285C2×C4—81no—
286C2×C4—204no—
287C2×C4—402no—
288C2×C4—402no—
289C2×C4#396611yescharacteristic
290C2×C4#396611yescharacteristic
291C2×C4#396912yes—
292C8—402no—
293C5#1178811yescharacteristic, complemented, Sylow
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22C2×#396611yescharacteristic
307C22#6551411yescharacteristic
308C22C2×#396911yescharacteristic
309C4—21no—
310C4—22no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—82no—
316C4—404no—
317C4#6476512yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2#1060248111yescharacteristic
323C2C2×#6476511yescharacteristic
324C2#1060266311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2542C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2542D5
hypocenterC5#11788
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
4. #10598142C2semidirect
5. C2×C5×C4⋊Q8C4semidirect
9. C5×#399C4semidirect
13. C4×C5×C4⋊C4C4semidirect
96. C2×C4⋊Q8Dic5semidirect
102. #399Dic5semidirect
106. C4×C4⋊C4Dic5semidirect
293. C5#11788semidirect

Maximal subgroups

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

subgroupindex
C5×#25422
C4×#39662
#106023952
#105981422
#117885