Finite Group Databasebeta

← #11997080

Subgroups — #11997080

Order 1280.

716 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 123 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#10614033C211yesmaximal, characteristic
3#10602925C212yesmaximal, complemented
4#10602412C211yesmaximal, characteristic
5#10598116C212yesmaximal, complemented
6C23×C5×Q8C2211yescharacteristic
7C2×C5×C22⋊Q8—21no—
8C2×C4×C5×D4C2211yescharacteristic
9C5×#395C412yescomplemented
10C5×#393C412yescomplemented
11C5×#391—21no—
12C5×#386—21no—
13C5×#385—21no—
14C5×#380C2211yescharacteristic
15C2×#3967—24no—
16C2×#3966—21no—
17#65501C2212yes—
18#65496—21no—
19C2×C22⋊#168C2212yes—
20#65300—24no—
21#65215—21no—
22#64746—21no—
23#9471—51nomaximal, Sylow
24C22×C5×Q8—21no—
25C22×C5×Q8—22no—
26C22×C5×Q8—22no—
27C22×C5×Q8—44no—
28C22×C5×Q8C2×C412yes—
29C22×C5×D4D411yescharacteristic
30C23×C4×C5—21no—
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⋊C4C2×C411yescharacteristic
39C2×C5×C22⋊C4—22no—
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4C2×C411yescharacteristic
42C2×C5×C22⋊C4D411yescharacteristic
43C2×C42×C5D411yescharacteristic
44C2×C22⋊Dic5—22no—
45C2×C4⋊Dic5—22no—
46C2×#681—22no—
47C22×#168D414yes—
48C5×#95—22no—
49C5×#95—22no—
50C5×#95C2×C412yes—
51#3967—48no—
52#3966—42no—
53#3963—22no—
54C22⋊#168—22no—
55C22⋊#168—22no—
56C22⋊#168—22no—
57C4⋊#168—22no—
58#3935—22no—
59#2973—51no—
60#2409D511yescharacteristic
61#1799—51no—
62#1787—52no—
63#1289—52no—
64C24×C5C22⋊C411yescharacteristic
65C24×C5C2×D411yescharacteristic
66C2×C5×Q8—24no—
67C2×C5×Q8—28no—
68C2×C5×Q8—24no—
69C2×C5×Q8—44no—
70C2×C5×Q8—44no—
71C2×C5×Q8—42no—
72C2×C5×Q8—48no—
73C2×C5×Q8—41no—
74C2×C5×Q8—41no—
75C2×C5×D4—22no—
76C2×C5×D4—21no—
77C2×C5×D4—21no—
78C2×C5×D4—22no—
79C22×C4×C5—22no—
80C22×C4×C5—21no—
81C22×C4×C5—21no—
82C22×C4×C5—22no—
83C22×C4×C5—42no—
84C22×C4×C5—44no—
85C22×C4×C5—42no—
86C22×C4×C5—42no—
87C22×C4×C5C22⋊C411yescharacteristic
88C22×C4×C5C22⋊C411yescharacteristic
89C22×C4×C5C22⋊C411yescharacteristic
90C22×C4×C5C22×C411yescharacteristic
91C22×C4×C5C2×D412yes—
92C22×C4×C5C2×D411yescharacteristic
93C22×C4×C5#4112yes—
94C22×Dic5—42no—
95C5×C4⋊C4—22no—
96C5×C4⋊C4—42no—
97C5×C4⋊C4—44no—
98C5×C22⋊C4—22no—
99C5×C22⋊C4—22no—
100C5×C22⋊C4—44no—
101C5×C22⋊C4—41no—
102C5×C22⋊C4—41no—
103C42×C5—21no—
104C42×C5—21no—
105C22⋊Dic5—42no—
106C22⋊Dic5—42no—
107C4⋊Dic5—44no—
108#681—82no—
109C2×#168—24no—
110C2×#168—44no—
111C23×Q8C2×D511yescharacteristic
112C2×C22⋊Q8—21no—
113C2×C4×D4C2×D511yescharacteristic
114#435—104no—
115#432—101no—
116C2×#103—104no—
117C2×#99—101no—
118#407—101no—
119C2×C22⋊C8—52no—
120#395Dic512yescomplemented
121#393Dic512yescomplemented
122#391—21no—
123#386—21no—
124#385—21no—
125#380—52no—
126#380C2×D511yescharacteristic
127#342—101no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5—41no—
132C23×C5—41no—
133C23×C5—41no—
134C23×C5—41no—
135C23×C5C2×C22⋊C411yescharacteristic
136C23×C5#12112yes—
137C23×C5AGL(1,ℤ/8)11yescharacteristic
138C23×C5#13711yescharacteristic
139C23×C5#13712yes—
140C5×Q8—28no—
141C5×Q8—44no—
142C5×Q8—44no—
143C5×Q8—44no—
144C5×D4—42no—
145C5×D4—42no—
146C2×C4×C5—24no—
147C2×C4×C5—22no—
148C2×C4×C5—24no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—41no—
155C2×C4×C5—42no—
156C2×C4×C5—42no—
157C2×C4×C5—48no—
158C2×C4×C5—42no—
159C2×C4×C5—42no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—42no—
163C2×C4×C5—42no—
164C2×C4×C5—42no—
165C2×C4×C5—81no—
166C2×C4×C5C4×D412yes—
167C2×C4×C5#12011yescharacteristic
168C2×C4×C5#12311yescharacteristic
169C2×Dic5—82no—
170C2×Dic5—82no—
171C2×Dic5—82no—
172#168—44no—
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—44no—
177C22×Q8C2×Dic512yes—
178C22×D4C22⋊D511yescharacteristic
179C23×C4—21no—
180C23×C4C22⋊D511yescharacteristic
181C23×C4C22×D511yescharacteristic
182C2×M16—102no—
183C22×C8—54no—
184C22⋊Q8—44no—
185C4×D4—22no—
186C4×D4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—102no—
190C2×C4⋊C4C2×Dic511yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—102no—
194C2×C22⋊C4C2×Dic511yescharacteristic
195C2×C22⋊C4C22⋊D511yescharacteristic
196C2×C42C22⋊D511yescharacteristic
197C4⋊C8—102no—
198#103—208no—
199#99—202no—
200C22⋊C8—102no—
201C22⋊C8—102no—
202C22⋊C8—102no—
203#97—102no—
204#95—22no—
205#95—22no—
206#95—102no—
207#95C2×Dic512yes—
208C22×C5—22no—
209C22×C5—21no—
210C22×C5—22no—
211C22×C5—21no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5—42no—
215C22×C5—42no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—81no—
220C22×C5#38611yescharacteristic
221C22×C5#41711yescharacteristic
222C22×C5#41911yescharacteristic
223C22×C5#46812yes—
224C22×C5#47012yes—
225C4×C5—21no—
226C4×C5—22no—
227C4×C5—44no—
228C4×C5—41no—
229C4×C5—41no—
230C4×C5—41no—
231C4×C5—82no—
232C4×C5B(4,𝔽2)12yes—
233Dic5—162no—
234C24C22⋊Dic511yescharacteristic
235C24C2×C22⋊D511yescharacteristic
236C2×Q8—24no—
237C2×Q8—28no—
238C2×Q8—24no—
239C2×Q8—44no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—48no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—202no—
258C22×C4C22⋊Dic511yescharacteristic
259C22×C4C22⋊Dic511yescharacteristic
260C22×C4C22⋊Dic511yescharacteristic
261C22×C4D4×D512yes—
262C22×C4#71112yes—
263C22×C4C22×Dic511yescharacteristic
264C22×C4C2×C22⋊D511yescharacteristic
265M16—402no—
266C2×C8—104no—
267C2×C8—204no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—204no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—44no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—202no—
278C22⋊C4—202no—
279C42—21no—
280C42—21no—
281C2×C5—22no—
282C2×C5—21no—
283C2×C5—21no—
284C2×C5—22no—
285C2×C5—81no—
286C2×C5#188911yescharacteristic
287C2×C5#189411yescharacteristic
288C2×C5#190911yescharacteristic
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#408512yes—
297C23#408812yes—
298C23#409511yescharacteristic
299C23#409711yescharacteristic
300C23C2×C22⋊Dic511yescharacteristic
301Q8—28no—
302Q8—44no—
303Q8—44no—
304Q8—44no—
305D4—42no—
306D4—42no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—48no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—402no—
328C2×C4—402no—
329C2×C4—402no—
330C2×C4D4×Dic512yes—
331C2×C4#408111yescharacteristic
332C2×C4#408311yescharacteristic
333C8—204no—
334C5#947111yescharacteristic, complemented, Sylow
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#6533812yes—
348C22#6534112yes—
349C22#6550911yescharacteristic
350C22#6551411yescharacteristic
351C22#6552111yescharacteristic
352C4—21no—
353C4—22no—
354C4—41no—
355C4—44no—
356C4—41no—
357C4—41no—
358C4—82no—
359C4—802no—
360C4#6532112yes—
361C2—21no—
362C2—22no—
363C2—21no—
364C2—22no—
365C2—81no—
366C2#1060246211yescharacteristic
367C2#1060249011yescharacteristic
368C2#1060265811yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2409C2
socleC22×C5#386
prime radicalC22×C5×D4D4
cosocleC23×C4×C5C23
hypercenter#2409D5
hypocenterC5#9471
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #10602925C2semidirect
5. #10598116C2semidirect
9. C5×#395C4semidirect
10. C5×#393C4semidirect
120. #395Dic5semidirect
121. #393Dic5semidirect
334. C5#9471semidirect

Maximal subgroups

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

subgroupindex
C5×#24092
#106140332
#106029252
#106024122
#105981162
#94715