Finite Group Databasebeta

← #11997092

Subgroups — #11997092

Order 1280.

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

normal subgroups only

This group has 576 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
1#10614104C211yesmaximal, characteristic
2#10614033C211yesmaximal, characteristic
3C5×#1799C211yesmaximal, characteristic
4#10600294C212yesmaximal, complemented
5#10598116C212yesmaximal, complemented
6C2×C4×C5×D4C2211yescharacteristic
7C2×#4081—21no—
8C22×C4⋊Dic5C2211yescharacteristic
9C2×C5×#99—21no—
10C5×#380C2212yes—
11#65501—21no—
12#65501C2211yescharacteristic
13#65496—21no—
14C2×C22⋊#168C2212yes—
15#65300—24no—
16#65272C412yescomplemented
17#65242C412yescomplemented
18#65232—21no—
19#65217—21no—
20#65215—21no—
21C5×#342—21no—
22#64749—24no—
23#9474—51nomaximal, Sylow
24C22×C5×D4D411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5C2311yescharacteristic
27C23×Dic5—21no—
28C4×C5×D4—22no—
29C4×C5×D4—22no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4C2×C411yescharacteristic
32C2×C5×C22⋊C4—22no—
33C2×C5×C22⋊C4C2×C411yescharacteristic
34C2×C5×C22⋊C4D411yescharacteristic
35C2×C42×C5D411yescharacteristic
36C2×C22⋊Dic5—22no—
37C2×C22⋊Dic5—21no—
38#4081—44no—
39C2×C4⋊Dic5—22no—
40C2×C4⋊Dic5—22no—
41C2×C4⋊Dic5—21no—
42C2×C4⋊Dic5—21no—
43C2×C4⋊Dic5C2×C412yes—
44C2×#683—22no—
45C2×#681—22no—
46C22×#168D414yes—
47C5×#99—42no—
48C5×#95—22no—
49#3963—22no—
50#3963—22no—
51#3963C2×C412yes—
52C22⋊#168—22no—
53C22⋊#168—22no—
54C22⋊#168—22no—
55C4⋊#168—22no—
56#3935—22no—
57#2973—51no—
58#2486—51no—
59#1823—52no—
60#1799D511yescharacteristic
61#1289—52no—
62C24×C5C22⋊C411yescharacteristic
63C24×C5C2×D411yescharacteristic
64C2×C5×D4—22no—
65C2×C5×D4—21no—
66C2×C5×D4—22no—
67C2×C5×D4—21no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—42no—
71C22×C4×C5—42no—
72C22×C4×C5—42no—
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22×C411yescharacteristic
77C22×C4×C5C2×D411yescharacteristic
78C22×C4×C5C2×D412yes—
79C22×C4×C5#4112yes—
80C22×Dic5—22no—
81C22×Dic5—22no—
82C22×Dic5—42no—
83C22×Dic5—41no—
84C22×Dic5—42no—
85C22×Dic5—41no—
86C5×C4⋊C4—22no—
87C5×C4⋊C4—44no—
88C5×C22⋊C4—22no—
89C5×C22⋊C4—22no—
90C5×C22⋊C4—42no—
91C5×C22⋊C4—42no—
92C42×C5—21no—
93C42×C5—21no—
94C22⋊Dic5—44no—
95C22⋊Dic5—42no—
96C4⋊Dic5—24no—
97C4⋊Dic5—42no—
98C4⋊Dic5—41no—
99C4⋊Dic5—41no—
100#683—44no—
101#681—82no—
102C2×#168—24no—
103C2×#168—44no—
104C2×#123—101no—
105C2×C4×D4C2×D511yescharacteristic
106C22×C4⋊C4—51no—
107#435—104no—
108#432—101no—
109C2×#99—21no—
110#407—101no—
111C2×C22⋊C8—52no—
112#398—52no—
113#397—52no—
114#389—101no—
115#381—101no—
116#380—51no—
117#380—101no—
118#380C2×D512yes—
119#342—21no—
120#340—104no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5—41no—
127C23×C5—41no—
128C23×C5C2×C22⋊C411yescharacteristic
129C23×C5#12112yes—
130C23×C5AGL(1,ℤ/8)11yescharacteristic
131C23×C5AGL(1,ℤ/8)12yes—
132C23×C5#13711yescharacteristic
133C5×D4—42no—
134C5×D4—42no—
135C2×C4×C5—24no—
136C2×C4×C5—22no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—41no—
143C2×C4×C5—42no—
144C2×C4×C5—42no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—81no—
148C2×C4×C5—82no—
149C2×C4×C5—82no—
150C2×C4×C5—82no—
151C2×C4×C5C4×D412yes—
152C2×C4×C5#12011yescharacteristic
153C2×C4×C5#12311yescharacteristic
154C2×Dic5—42no—
155C2×Dic5—42no—
156C2×Dic5—42no—
157C2×Dic5—42no—
158C2×Dic5—42no—
159C2×Dic5—42no—
160C2×Dic5—82no—
161C2×Dic5—82no—
162#168—44no—
163C22×D4C22⋊D511yescharacteristic
164C23×C4—101no—
165C23×C4D2011yescharacteristic
166C23×C4C22×D511yescharacteristic
167C2×M16—102no—
168C22×C8—54no—
169#123—204no—
170C4×D4—22no—
171C4×D4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—52no—
174C2×C4⋊C4—102no—
175C2×C4⋊C4—102no—
176C2×C4⋊C4—102no—
177C2×C4⋊C4—101no—
178C2×C4⋊C4—101no—
179C2×C4⋊C4C4×D511yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—102no—
182C2×C22⋊C4—101no—
183C2×C22⋊C4C4×D511yescharacteristic
184C2×C22⋊C4C22⋊D511yescharacteristic
185C2×C42D2011yescharacteristic
186C4⋊C8—102no—
187#99—42no—
188C22⋊C8—102no—
189C22⋊C8—102no—
190C22⋊C8—102no—
191#97—102no—
192#95—22no—
193#95—52no—
194#95—102no—
195#95—102no—
196C22×C5—21no—
197C22×C5—22no—
198C22×C5—21no—
199C22×C5—22no—
200C22×C5—41no—
201C22×C5—42no—
202C22×C5—41no—
203C22×C5—42no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—81no—
208C22×C5#38611yescharacteristic
209C22×C5#41711yescharacteristic
210C22×C5#41811yescharacteristic
211C22×C5#46812yes—
212C22×C5#46912yes—
213C4×C5—21no—
214C4×C5—22no—
215C4×C5—41no—
216C4×C5—41no—
217C4×C5—41no—
218C4×C5—162no—
219C4×C5#45812yes—
220Dic5—82no—
221Dic5—82no—
222C24#68511yescharacteristic
223C24C2×C22⋊D511yescharacteristic
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—21no—
227C2×D4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—102no—
234C22×C4—102no—
235C22×C4—201no—
236C22×C4—202no—
237C22×C4—201no—
238C22×C4—202no—
239C22×C4#68511yescharacteristic
240C22×C4#68511yescharacteristic
241C22×C4#68511yescharacteristic
242C22×C4C2×C4×D511yescharacteristic
243C22×C4C2×D2011yescharacteristic
244C22×C4D4×D512yes—
245C22×C4#71112yes—
246M16—402no—
247C2×C8—104no—
248C2×C8—204no—
249C4⋊C4—22no—
250C4⋊C4—44no—
251C4⋊C4—104no—
252C4⋊C4—204no—
253C4⋊C4—202no—
254C4⋊C4—201no—
255C4⋊C4—201no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—204no—
261C22⋊C4—202no—
262C42—21no—
263C42—21no—
264C2×C5—21no—
265C2×C5—22no—
266C2×C5—21no—
267C2×C5—22no—
268C2×C5—81no—
269C2×C5#189011yescharacteristic
270C2×C5#189511yescharacteristic
271C2×C5#190911yescharacteristic
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×#68511yescharacteristic
280C23#407811yescharacteristic
281C23#408512yes—
282C23#408811yescharacteristic
283C23#409512yes—
284D4—42no—
285D4—42no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—202no—
303C2×C4—202no—
304C2×C4—202no—
305C2×C4—202no—
306C2×C4—202no—
307C2×C4—202no—
308C2×C4—402no—
309C2×C4—402no—
310C2×C4#402712yes—
311C2×C4#402811yescharacteristic
312C2×C4#403211yescharacteristic
313C8—204no—
314C5#947411yescharacteristic, complemented, Sylow
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#6524711yescharacteristic
328C22#6525511yescharacteristic
329C22#6529011yescharacteristic
330C22#6532812yes—
331C22#6533812yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—162no—
338C4—402no—
339C4—402no—
340C4#6503112yes—
341C2—22no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#1060031011yescharacteristic
347C2#1060081611yescharacteristic
348C2#1060083911yescharacteristic
subgroupG/H
centerC22#65247
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#1799C2
socleC22×C5#386
prime radicalC22×C5×D4D4
cosocleC23×C4×C5C23
hypercenter#1799D5
hypocenterC5#9474
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
4. #10600294C2semidirect
5. #10598116C2semidirect
16. #65272C4semidirect
17. #65242C4semidirect
314. C5#9474semidirect

Maximal subgroups

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

subgroupindex
#106141042
#106140332
C5×#17992
#106002942
#105981162
#94745