Finite Group Databasebeta

← #11997255

Subgroups — #11997255

Order 1280.

682 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 111 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
2#10614035C211yesmaximal, characteristic
3#10602926C212yesmaximal, complemented
4#10602395C211yesmaximal, characteristic
5#10598117C212yesmaximal, complemented
6C23×C5×Q8C2211yescharacteristic
7C2×C5×C22⋊Q8—21no—
8C2×C4×C5×D4C2211yescharacteristic
9C5×#395—21no—
10C5×#393—21no—
11C5×#391—21no—
12C5×#386C412yes—
13C5×#385C412yes—
14C5×#380C2211yescharacteristic
15C2×#3968—24no—
16#65496—21no—
17#65496C2212yes—
18#65496C2212yes—
19#65301—24no—
20#65214—21no—
21#64748—21no—
22#64745—21no—
23#9506—51nomaximal, Sylow
24C22×C5×Q8—21no—
25C22×C5×Q8—22no—
26C22×C5×Q8—21no—
27C22×C5×Q8—22no—
28C22×C5×Q8—44no—
29C22×C5×D4C2×C411yescharacteristic
30C23×C4×C5C2×C412yes—
31C23×C4×C5D411yescharacteristic
32C23×C4×C5C2311yescharacteristic
33C5×C22⋊Q8—44no—
34C4×C5×D4—22no—
35C4×C5×D4—22no—
36C2×C5×C4⋊C4—22no—
37C2×C5×C4⋊C4—21no—
38C2×C5×C4⋊C4D411yescharacteristic
39C2×C5×C22⋊C4—22no—
40C2×C5×C22⋊C4C2×C411yescharacteristic
41C2×C5×C22⋊C4C2×C412yes—
42C2×C5×C22⋊C4D411yescharacteristic
43C2×C42×C5D411yescharacteristic
44C2×#681—22no—
45C2×#681—22no—
46C2×#681D414yes—
47C5×#95—22no—
48C5×#95—22no—
49C5×#95—21no—
50#3968—48no—
51C22⋊#168—22no—
52C22⋊#168—22no—
53C22⋊#168—22no—
54C22⋊#168—22no—
55C22⋊#168—22no—
56C4⋊#168—22no—
57C4×#168—22no—
58#2975—51no—
59#2409D511yescharacteristic
60#1784—52no—
61#1465—51no—
62#1305—52no—
63C24×C5C22⋊C411yescharacteristic
64C24×C5C22×C411yescharacteristic
65C2×C5×Q8—24no—
66C2×C5×Q8—28no—
67C2×C5×Q8—44no—
68C2×C5×Q8—42no—
69C2×C5×Q8—44no—
70C2×C5×Q8—48no—
71C2×C5×Q8—41no—
72C2×C5×Q8—42no—
73C2×C5×Q8—41no—
74C2×C5×D4—22no—
75C2×C5×D4—22no—
76C2×C5×D4—41no—
77C22×C4×C5—22no—
78C22×C4×C5—24no—
79C22×C4×C5—21no—
80C22×C4×C5—21no—
81C22×C4×C5—22no—
82C22×C4×C5—44no—
83C22×C4×C5—42no—
84C22×C4×C5—42no—
85C22×C4×C5C22⋊C411yescharacteristic
86C22×C4×C5C22⋊C411yescharacteristic
87C22×C4×C5C22⋊C411yescharacteristic
88C22×C4×C5C2×D412yes—
89C22×C4×C5C2×D411yescharacteristic
90C22×C4×C5C2×D411yescharacteristic
91C22×C4×C5#4112yes—
92C5×C4⋊C4—22no—
93C5×C4⋊C4—42no—
94C5×C4⋊C4—44no—
95C5×C22⋊C4—22no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4—44no—
98C5×C22⋊C4—42no—
99C42×C5—21no—
100C42×C5—21no—
101#681—44no—
102#681—44no—
103#681—82no—
104C2×#168—24no—
105C2×#168—42no—
106C23×Q8C2×D511yescharacteristic
107C2×C22⋊Q8—21no—
108C2×C4×D4C2×D511yescharacteristic
109#436—104no—
110#431—101no—
111C2×#101—104no—
112#407—52no—
113#407—52no—
114#407—101no—
115#395—21no—
116#393—21no—
117#391—21no—
118#386Dic512yes—
119#385Dic512yes—
120#380C2×D511yescharacteristic
121#324—101no—
122#323—101no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—41no—
129C23×C5—41no—
130C23×C5—41no—
131C23×C5—41no—
132C23×C5C2×C22⋊C411yescharacteristic
133C23×C5C4×D412yes—
134C5×Q8—48no—
135C5×Q8—44no—
136C5×Q8—44no—
137C5×D4—44no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—24no—
141C2×C4×C5—22no—
142C2×C4×C5—24no—
143C2×C4×C5—28no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—41no—
147C2×C4×C5—42no—
148C2×C4×C5—42no—
149C2×C4×C5—44no—
150C2×C4×C5—42no—
151C2×C4×C5—42no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—42no—
155C2×C4×C5—42no—
156C2×C4×C5—42no—
157C2×C4×C5—81no—
158C2×C4×C5#10112yescomplemented
159C2×C4×C5#12011yescharacteristic
160C2×C4×C5#12112yes—
161C2×C4×C5#12311yescharacteristic
162#168—44no—
163#168—82no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—44no—
169C22×D4C2×Dic511yescharacteristic
170C23×C4C2×Dic512yes—
171C23×C4C22⋊D511yescharacteristic
172C23×C4C22×D511yescharacteristic
173C2×M16—54no—
174C2×M16—102no—
175C2×M16—102no—
176C22⋊Q8—44no—
177C4×D4—22no—
178C4×D4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C22⋊D511yescharacteristic
182C2×C22⋊C4—22no—
183C2×C22⋊C4C2×Dic511yescharacteristic
184C2×C22⋊C4C2×Dic512yes—
185C2×C22⋊C4C22⋊D511yescharacteristic
186C2×C42C22⋊D511yescharacteristic
187C4⋊C8—102no—
188#101—208no—
189C22⋊C8—102no—
190C22⋊C8—102no—
191C22⋊C8—102no—
192C22⋊C8—102no—
193C22⋊C8—102no—
194C4×C8—102no—
195#95—22no—
196#95—22no—
197#95—21no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—22no—
201C22×C5—22no—
202C22×C5—22no—
203C22×C5—42no—
204C22×C5—42no—
205C22×C5—42no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5—81no—
210C22×C5#38611yescharacteristic
211C22×C5C2×#10111yescharacteristic
212C22×C5#41311yescharacteristic
213C4×C5—22no—
214C4×C5—21no—
215C4×C5—22no—
216C4×C5—44no—
217C4×C5—41no—
218C4×C5—41no—
219C4×C5—82no—
220C4×C5B(4,𝔽2)12yes—
221C24C22⋊Dic511yescharacteristic
222C24C22×Dic511yescharacteristic
223C2×Q8—28no—
224C2×Q8—24no—
225C2×Q8—42no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—48no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C22⋊Dic511yescharacteristic
244C22×C4C22⋊Dic511yescharacteristic
245C22×C4C22⋊Dic511yescharacteristic
246C22×C4D4×D512yes—
247C22×C4#71112yes—
248C22×C4C2×C22⋊D511yescharacteristic
249C22×C4C2×C22⋊D511yescharacteristic
250M16—204no—
251M16—204no—
252M16—402no—
253C2×C8—104no—
254C2×C8—202no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C42—21no—
263C42—21no—
264C2×C5—21no—
265C2×C5—21no—
266C2×C5—42no—
267C2×C5—81no—
268C2×C5#188911yescharacteristic
269C2×C5#190111yescharacteristic
270C2×C5#191511yescharacteristic
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23D4×Dic512yes—
281C23C2×C22⋊Dic511yescharacteristic
282Q8—48no—
283Q8—44no—
284Q8—44no—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—28no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4#396812yescomplemented
307C2×C4#408111yescharacteristic
308C2×C4#408311yescharacteristic
309C2×C4#408512yes—
310C8—204no—
311C8—402no—
312C5#950611yescharacteristic, complemented, Sylow
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#6550911yescharacteristic
326C22C2×#396811yescharacteristic
327C22#6552611yescharacteristic
328C4—22no—
329C4—21no—
330C4—22no—
331C4—41no—
332C4—44no—
333C4—41no—
334C4—82no—
335C4#6532112yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2#1060246211yescharacteristic
341C2#1060249111yescharacteristic
342C2#1060265911yescharacteristic
subgroupG/H
centerC22#65509
commutatorC24×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2409C2
socleC22×C5#386
prime radicalC22×C5×D4C2×C4
cosocleC23×C4×C5C23
hypercenter#2409D5
hypocenterC5#9506
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #10602926C2semidirect
5. #10598117C2semidirect
158. C2×C4×C5#101semidirect
306. C2×C4#3968semidirect
312. C5#9506semidirect

Maximal subgroups

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

subgroupindex
C5×#24092
#106140352
#106029262
#106023952
#105981172
#95065