Finite Group Databasebeta

← #12009934

Subgroups — #12009934

Order 1280.

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

normal subgroups only

This group has 572 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×#2409C211yesmaximal, characteristic
2C4×#3965C211yesmaximal, characteristic
3#10602395C211yesmaximal, characteristic
4#10598139C214yesmaximal, complemented
5C23×C5×Q8C411yescharacteristic
6C2×C5×C22⋊Q8—21no—
7C2×C4×C5×D4C2211yescharacteristic
8C5×#395C2212yes—
9C5×#393C412yes—
10C5×#391—21no—
11C5×#386—21no—
12C5×#385—21no—
13C5×#380C411yescharacteristic
14C2×#3965—21no—
15#65496—21no—
16C4×#681—21no—
17#64750C2212yes—
18#64748C2212yes—
19#64745—21no—
20#11791—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×C5×Q8—41no—
23C22×C5×Q8—41no—
24C22×C5×Q8—44no—
25C22×C5×Q8C2×C412yes—
26C22×C5×D4D411yescharacteristic
27C23×C4×C5—21no—
28C23×C4×C5C2×C411yescharacteristic
29C23×C4×C5D411yescharacteristic
30C5×C22⋊Q8—44no—
31C4×C5×D4—41no—
32C4×C5×D4—41no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4C2311yescharacteristic
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4C2×C411yescharacteristic
39C2×C5×C22⋊C4D411yescharacteristic
40C2×C42×C5D411yescharacteristic
41C2×#681—22no—
42C2×#681—41no—
43C5×#95—22no—
44C5×#95—22no—
45C5×#95C2×C412yes—
46#3965—24no—
47C22⋊#168—22no—
48C22⋊#168—41no—
49#3935—41no—
50C4×#168—41no—
51#2409D511yescharacteristic
52C4×#100—51no—
53#1465—51no—
54#1354—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C22⋊C411yescharacteristic
57C2×C5×Q8—24no—
58C2×C5×Q8—44no—
59C2×C5×Q8—42no—
60C2×C5×Q8—42no—
61C2×C5×Q8—44no—
62C2×C5×Q8—48no—
63C2×C5×Q8—44no—
64C2×C5×Q8—41no—
65C2×C5×Q8—41no—
66C2×C5×D4—22no—
67C2×C5×D4—22no—
68C2×C5×D4—41no—
69C22×C4×C5—22no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5—22no—
75C22×C4×C5—44no—
76C22×C4×C5—42no—
77C22×C4×C5—42no—
78C22×C4×C5—42no—
79C22×C4×C5C22⋊C411yescharacteristic
80C22×C4×C5C22⋊C411yescharacteristic
81C22×C4×C5C22×C411yescharacteristic
82C22×C4×C5C2×D411yescharacteristic
83C22×C4×C5C2×D411yescharacteristic
84C5×C4⋊C4—41no—
85C5×C4⋊C4—42no—
86C5×C4⋊C4—44no—
87C5×C22⋊C4—41no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—44no—
90C5×C22⋊C4—42no—
91C42×C5—21no—
92C42×C5—21no—
93#681—24no—
94#681—42no—
95#681—82no—
96C2×#168—42no—
97C2×#168—42no—
98C23×Q8Dic511yescharacteristic
99C2×C22⋊Q8—21no—
100C2×C4×D4C2×D511yescharacteristic
101C2×#100—101no—
102#407—101no—
103C4×M16—101no—
104#395C2×D512yes—
105#393Dic512yes—
106#391—21no—
107#386—21no—
108#385—21no—
109#380Dic511yescharacteristic
110#343—52no—
111#324—52no—
112#323—101no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—41no—
119C23×C5—41no—
120C23×C5—41no—
121C23×C5—41no—
122C23×C5#10412yes—
123C23×C5C2×C22⋊C411yescharacteristic
124C5×Q8—44no—
125C5×Q8—44no—
126C5×Q8—44no—
127C5×Q8—44no—
128C5×D4—44no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—22no—
132C2×C4×C5—24no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—48no—
139C2×C4×C5—42no—
140C2×C4×C5—42no—
141C2×C4×C5—42no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—42no—
146C2×C4×C5—42no—
147C2×C4×C5—42no—
148C2×C4×C5—81no—
149C2×C4×C5#9911yescharacteristic
150C2×C4×C5#9911yescharacteristic
151C2×C4×C5#13712yes—
152#168—44no—
153#168—82no—
154C22×Q8—21no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—44no—
158C22×Q8C2×Dic512yes—
159C22×D4C22⋊D511yescharacteristic
160C23×C4—21no—
161C23×C4C2×Dic511yescharacteristic
162C23×C4C22⋊D511yescharacteristic
163C2×M16—102no—
164C2×M16—201no—
165C22⋊Q8—44no—
166C4×D4—41no—
167C4×D4—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22×D511yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4C2×Dic511yescharacteristic
174C2×C22⋊C4C22⋊D511yescharacteristic
175C2×C42C22⋊D511yescharacteristic
176#100—104no—
177C22⋊C8—102no—
178C22⋊C8—201no—
179#97—201no—
180C4×C8—201no—
181#95—22no—
182#95—22no—
183#95C2×Dic512yes—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—22no—
187C22×C5—42no—
188C22×C5—42no—
189C22×C5—42no—
190C22×C5—41no—
191C22×C5—41no—
192C22×C5—41no—
193C22×C5—41no—
194C22×C5—41no—
195C22×C5—81no—
196C22×C5C2×#9911yescharacteristic
197C22×C5#41911yescharacteristic
198C22×C5C2×#10411yescharacteristic
199C4×C5—21no—
200C4×C5—22no—
201C4×C5—41no—
202C4×C5—44no—
203C4×C5—41no—
204C4×C5—41no—
205C4×C5—82no—
206C4×C5AΣL(1,𝔽16)12yes—
207C24C22⋊Dic511yescharacteristic
208C24C22⋊Dic511yescharacteristic
209C2×Q8—24no—
210C2×Q8—44no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—44no—
214C2×Q8—48no—
215C2×Q8—44no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊Dic511yescharacteristic
232C22×C4C22⋊Dic511yescharacteristic
233C22×C4C22×Dic511yescharacteristic
234C22×C4C2×C22⋊D511yescharacteristic
235C22×C4C2×C22⋊D511yescharacteristic
236M16—104no—
237M16—202no—
238M16—402no—
239C2×C8—202no—
240C2×C8—202no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—44no—
247C22⋊C4—42no—
248C42—21no—
249C42—21no—
250C2×C5—21no—
251C2×C5—21no—
252C2×C5—42no—
253C2×C5—81no—
254C2×C5#150811yescharacteristic
255C2×C5C2×AΣL(1,𝔽16)11yescharacteristic
256C2×C5#213511yescharacteristic
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#396912yes—
267C23C2×C22⋊Dic511yescharacteristic
268Q8—44no—
269Q8—44no—
270Q8—44no—
271Q8—44no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—48no—
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—42no—
292C2×C4—81no—
293C2×C4#396611yescharacteristic
294C2×C4#396611yescharacteristic
295C2×C4#408812yes—
296C8—204no—
297C8—402no—
298C5#1179111yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—22no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#396611yescharacteristic
312C22#6551411yescharacteristic
313C22C2×#396911yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—44no—
319C4—41no—
320C4—82no—
321C4#6475612yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2C2×#6475611yescharacteristic
327C2#1060248111yescharacteristic
328C2#1060266611yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2409C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C4⋊C4C23
hypercenter#2409D5
hypocenterC5#11791
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
4. #10598139C2semidirect
298. C5#11791semidirect

Maximal subgroups

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

subgroupindex
C5×#24092
C4×#39652
#106023952
#105981392
#117915