Finite Group Databasebeta

← #11997105

Subgroups — #11997105

Order 1280.

818 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2406C211yesmaximal, characteristic
2#10613875C211yesmaximal, characteristic, complemented
3#10602931C212yesmaximal
4#10602345C211yesmaximal, characteristic, complemented
5#10598119C212yesmaximal, complemented
6C22×C5×#41C2211yescharacteristic
7C2×C5×#123—21no—
8C22×C5×C4⋊C4C2211yescharacteristic
9C2×#4072—21no—
10C5×#395C412yescomplemented
11C5×#394C412yescomplemented
12C5×#386—21no—
13C5×#385—22no—
14C5×#381C2211yescharacteristic
15C2×#3969—24no—
16C4×C22⋊Dic5—21no—
17C4×C22⋊Dic5C2212yes—
18#65496C2212yes—
19C4⋊C4×Dic5—24no—
20#64748—22no—
21#9476—51nomaximal, Sylow
22C2×C5×#41—22no—
23C2×C5×#41—22no—
24C2×C5×#41—44no—
25C22×C5×Q8C2×C411yescharacteristic
26C22×C5×D4—21no—
27C22×C5×D4C2×C411yescharacteristic
28C23×C4×C5—21no—
29C23×C4×C5D411yescharacteristic
30C23×C4×C5D411yescharacteristic
31C23×C4×C5C2311yescharacteristic
32C5×#123—44no—
33C2×C5×C4⋊C4—22no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4—22no—
36C2×C5×C4⋊C4C2×C412yes—
37C2×C5×C4⋊C4D412yes—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—22no—
40C2×C5×C22⋊C4—22no—
41C2×C22⋊Dic5—22no—
42#4072—28no—
43C2×C4⋊Dic5—22no—
44C2×C4×Dic5—22no—
45C2×C4×Dic5—22no—
46C2×C4×Dic5D414yes—
47C2×#681—22no—
48C5×#95—22no—
49C5×#95—21no—
50C5×#95C2×C412yes—
51#3969—48no—
52#3963—22no—
53#3963—22no—
54C22⋊#168—22no—
55C22⋊#168—22no—
56#2406D511yescharacteristic
57#2274—51no—
58#1789—52no—
59#1463—51no—
60#1284—52no—
61C24×C5—21no—
62C24×C5C2×D411yescharacteristic
63C5×#41—28no—
64C5×#41—44no—
65C5×#41—44no—
66C5×#41—44no—
67C2×C5×Q8—22no—
68C2×C5×Q8—42no—
69C2×C5×D4—24no—
70C2×C5×D4—22no—
71C2×C5×D4—42no—
72C2×C5×D4—42no—
73C2×C5×D4—41no—
74C2×C5×D4—41no—
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—24no—
78C22×C4×C5—21no—
79C22×C4×C5—21no—
80C22×C4×C5—21no—
81C22×C4×C5—22no—
82C22×C4×C5—42no—
83C22×C4×C5—42no—
84C22×C4×C5—42no—
85C22×C4×C5—42no—
86C22×C4×C5—42no—
87C22×C4×C5C22⋊C412yes—
88C22×C4×C5C22⋊C412yes—
89C22×C4×C5C22×C411yescharacteristic
90C22×C4×C5C2×D412yes—
91C22×C4×C5C2×D411yescharacteristic
92C22×C4×C5#4112yes—
93C22×Dic5—24no—
94C5×C4⋊C4—24no—
95C5×C4⋊C4—24no—
96C5×C4⋊C4—44no—
97C5×C22⋊C4—44no—
98C5×C22⋊C4—42no—
99C5×C22⋊C4—44no—
100C22⋊Dic5—28no—
101C4⋊Dic5—28no—
102C4×Dic5—28no—
103C4×Dic5—44no—
104C4×Dic5—82no—
105#681—44no—
106C2×#168—42no—
107C22×#41C2×D511yescharacteristic
108C2×#123—21no—
109C2×#117—101no—
110C22×C4⋊C4C2×D511yescharacteristic
111C2×#104—104no—
112#407—52no—
113#395Dic512yescomplemented
114#394Dic512yescomplemented
115#386—21no—
116#385—22no—
117#381C2×D511yescharacteristic
118C4×C4⋊C4—104no—
119C4×C22⋊C4—52no—
120C4×C22⋊C4—101no—
121#324—102no—
122C23×C5—22no—
123C23×C5—22no—
124C23×C5—22no—
125C23×C5—41no—
126C23×C5—41no—
127C23×C5—42no—
128C23×C5—41no—
129C23×C5—42no—
130C23×C5C2×C22⋊C411yescharacteristic
131C23×C5#12011yescharacteristic
132C23×C5#12311yescharacteristic
133C5×Q8—24no—
134C5×Q8—42no—
135C5×D4—24no—
136C5×D4—42no—
137C5×D4—44no—
138C5×D4—44no—
139C2×C4×C5—22no—
140C2×C4×C5—22no—
141C2×C4×C5—24no—
142C2×C4×C5—22no—
143C2×C4×C5—22no—
144C2×C4×C5—24no—
145C2×C4×C5—22no—
146C2×C4×C5—22no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5—42no—
150C2×C4×C5—48no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5—42no—
154C2×C4×C5—42no—
155C2×C4×C5—42no—
156C2×C4×C5—42no—
157C2×C4×C5—42no—
158C2×C4×C5—81no—
159C2×C4×C5#10414yescomplemented
160C2×C4×C5C4×D412yes—
161C2×C4×C5#12112yes—
162C2×Dic5—28no—
163C2×Dic5—44no—
164C2×Dic5—44no—
165#168—82no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—44no—
169C22×Q8C2×Dic511yescharacteristic
170C22×D4—21no—
171C22×D4C2×Dic511yescharacteristic
172C23×C4—21no—
173C23×C4C22⋊D511yescharacteristic
174C23×C4C22⋊D511yescharacteristic
175C23×C4C22×D511yescharacteristic
176C2×M16—102no—
177#123—44no—
178#117—108no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—102no—
183C2×C4⋊C4C2×Dic512yes—
184C2×C4⋊C4C22⋊D512yes—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—102no—
189C2×C42—54no—
190C2×C42—102no—
191C2×C42—102no—
192#104—208no—
193C22⋊C8—102no—
194C22⋊C8—102no—
195#95—22no—
196#95—21no—
197#95—102no—
198#95—102no—
199#95C2×Dic512yes—
200C22×C5—21no—
201C22×C5—21no—
202C22×C5—21no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5—44no—
208C22×C5—44no—
209C22×C5—42no—
210C22×C5—82no—
211C22×C5—81no—
212C22×C5—81no—
213C22×C5#38611yescharacteristic
214C22×C5C2×#10412yes—
215C22×C5B(4,𝔽2)12yes—
216C4×C5—22no—
217C4×C5—22no—
218C4×C5—24no—
219C4×C5—42no—
220C4×C5—42no—
221C4×C5—82no—
222Dic5—48no—
223C24—21no—
224C24C2×C22⋊D511yescharacteristic
225#41—28no—
226#41—44no—
227#41—44no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—41no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—104no—
250C22×C4C22⋊Dic512yes—
251C22×C4C22⋊Dic512yes—
252C22×C4D4×D512yes—
253C22×C4#71112yes—
254C22×C4C22×Dic511yescharacteristic
255C22×C4C2×C22⋊D511yescharacteristic
256M16—204no—
257C2×C8—202no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—44no—
261C4⋊C4—108no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—44no—
265C22⋊C4—108no—
266C42—108no—
267C42—204no—
268C42—402no—
269C2×C5—22no—
270C2×C5—22no—
271C2×C5—41no—
272C2×C5—82no—
273C2×C5#188911yescharacteristic
274C2×C5#189611yescharacteristic
275C2×C5#191211yescharacteristic
276C23—22no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—41no—
283C23—42no—
284C23#408111yescharacteristic
285C23#408311yescharacteristic
286C23C2×C22⋊Dic511yescharacteristic
287Q8—24no—
288Q8—42no—
289D4—24no—
290D4—42no—
291D4—44no—
292D4—44no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—48no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—108no—
314C2×C4—204no—
315C2×C4—204no—
316C2×C4#396914yescomplemented
317C2×C4D4×Dic512yes—
318C2×C4#408512yes—
319C8—402no—
320C5#947611yescharacteristic, complemented, Sylow
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22—44no—
330C22—42no—
331C22—82no—
332C22—81no—
333C22—81no—
334C22#6532112yes—
335C22#6550911yescharacteristic
336C22C2×#396912yes—
337C4—24no—
338C4—22no—
339C4—22no—
340C4—42no—
341C4—42no—
342C4—82no—
343C4—208no—
344C2—22no—
345C2—22no—
346C2—41no—
347C2—82no—
348C2#1060246211yescharacteristic
349C2#1060250411yescharacteristic
350C2#1060265711yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2406C2
socleC22×C5#386
prime radicalC22×C5×#41C22
cosocleC23×C4×C5C23
hypercenter#2406D5
hypocenterC5#9476
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. #10613875C2semidirect
4. #10602345C2semidirect
5. #10598119C2semidirect
10. C5×#395C4semidirect
11. C5×#394C4semidirect
113. #395Dic5semidirect
114. #394Dic5semidirect
159. C2×C4×C5#104semidirect
316. C2×C4#3969semidirect
320. C5#9476semidirect

Maximal subgroups

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

subgroupindex
C5×#24062
#106138752
#106029312
#106023452
#105981192
#94765