Finite Group Databasebeta

← #12001641

Subgroups — #12001641

Order 1280.

712 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#65373C212yesmaximal, complemented
2#10606099C211yesmaximal, characteristic
3C5×#1956C211yesmaximal, characteristic, complemented
4C2×#64679C212yesmaximal, complemented
5#10598125C211yesmaximal, characteristic, complemented
6C2×C5×C4⋊Q8C2211yescharacteristic
7C2×#4041—22no—
8C2×C4×D20C2212yes—
9C2×C5×#103—22no—
10C2×C4×C5×C8C2211yescharacteristic
11C5×#384C2211yescharacteristic
12#65373—24no—
13#65373—24no—
14#65280—22no—
15#65276—22no—
16C2×#3941—22no—
17C2×#3940—22no—
18C2×C4⋊#168C2212yes—
19#64679—24no—
20#10235—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×D20—22no—
23C22×D20D412yes—
24C22×C4×D5—22no—
25C22×C5×C8—21no—
26C5×C4⋊Q8—22no—
27C5×C4⋊Q8—21no—
28C5×C4⋊Q8D412yescomplemented
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4D412yes—
32C2×C42×C5C2311yescharacteristic
33C2×#685—24no—
34C2×#685—22no—
35C2×C4⋊Dic5D412yes—
36C22×#168—22no—
37#4041—44no—
38#4041—44no—
39C4×D20—22no—
40C4×D20—22no—
41C4×D20—22no—
42C4×D20—22no—
43C5×#103—44no—
44C4×C5×C8—22no—
45#3963—22no—
46#3941—28no—
47#3940—28no—
48C4⋊#168—22no—
49C4⋊#168D414yescomplemented
50C2×#476—52no—
51#2537—51no—
52#1956D511yescharacteristic, complemented
53C2×D4⋊C8—52no—
54#1296—51no—
55C23×D5—22no—
56C2×C5×Q8—22no—
57C2×C5×Q8—42no—
58C2×C5×Q8—42no—
59C2×C5×Q8—41no—
60C22×C4×C5—22no—
61C22×C4×C5—41no—
62C22×C4×C5C2×D412yes—
63C22×C4×C5C2×D411yescharacteristic
64C22×Dic5—22no—
65C2×D20—22no—
66C2×D20—22no—
67C2×D20—22no—
68C2×D20—22no—
69C2×D20—22no—
70C2×D20—48no—
71C2×D20—82no—
72C2×D20SD1614yes—
73C2×C4×D5—42no—
74C2×C4×D5—42no—
75C2×C4×D5—42no—
76C2×C4×D5—42no—
77C2×C4×D5—42no—
78C2×C4×D5—42no—
79C2×C5×C8—22no—
80C2×C5×C8—41no—
81C2×C5×C8—41no—
82C5×C4⋊C4—24no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—42no—
85C5×C4⋊C4—42no—
86C5×C4⋊C4—42no—
87C42×C5C2×D411yescharacteristic
88C42×C5C2×D412yes—
89C42×C5#4111yescharacteristic
90#685—48no—
91#685—42no—
92#685—42no—
93C4⋊Dic5—22no—
94C4⋊Dic5—22no—
95C2×#168—24no—
96C2×#168—42no—
97C2×#168—42no—
98C2×C4⋊Q8C2×D511yescharacteristic
99C2×#121—102no—
100C2×C4×D4—52no—
101#476—104no—
102#476—104no—
103C2×#106—102no—
104C2×C4⋊C8—52no—
105C2×#103—22no—
106C2×#102—102no—
107C2×C4×C8C2×D511yescharacteristic
108#394—102no—
109#384C2×D511yescharacteristic
110#380—102no—
111D4⋊C8—104no—
112C23×C5#12011yescharacteristic
113C22×D5—24no—
114C22×D5—42no—
115C22×D5—42no—
116C22×D5—42no—
117C22×D5—42no—
118C22×D5—42no—
119C22×D5—42no—
120C5×Q8—44no—
121C5×Q8—82no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—42no—
125C2×C4×C5—44no—
126C2×C4×C5—42no—
127C2×C4×C5—81no—
128C2×C4×C5—81no—
129C2×C4×C5#12012yes—
130C2×C4×C5#12112yes—
131C2×C4×C5#12111yescharacteristic
132C2×C4×C5#12711yescharacteristic
133C2×C4×C5C2×SD1612yes—
134C2×C4×C5#13512yes—
135C2×C4×C5AGL(1,ℤ/8)12yes—
136C2×C4×C5#13712yes—
137C2×Dic5—42no—
138C2×Dic5—42no—
139C2×Dic5—42no—
140D20—24no—
141D20—24no—
142D20—42no—
143D20—42no—
144D20—88no—
145C4×D5—82no—
146C4×D5—82no—
147C4×D5—82no—
148C4×D5—82no—
149C5×C8—42no—
150#168—44no—
151C22×Q8—21no—
152C22×D4—52no—
153C22×D4—102no—
154C23×C4—102no—
155C22×C8—21no—
156C22×C8—102no—
157C4⋊Q8—21no—
158C4⋊Q8—22no—
159C4⋊Q8D2012yescomplemented
160#121—204no—
161#121—204no—
162C4×D4—102no—
163C4×D4—102no—
164C4×D4—102no—
165C4×D4—102no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—52no—
169C2×C4⋊C4C22⋊D512yes—
170C2×C22⋊C4—104no—
171C2×C22⋊C4—102no—
172C2×C42C22×D511yescharacteristic
173#106—108no—
174C4⋊C8—54no—
175C4⋊C8—102no—
176#103—44no—
177#102—108no—
178C4×C8—22no—
179#95—102no—
180C22×C5#39211yescharacteristic
181C22×C5#45012yes—
182C22×C5#45212yes—
183C22×C5#45311yescharacteristic
184C22×C5#45411yescharacteristic
185C2×D5—44no—
186C2×D5—44no—
187C2×D5—44no—
188C2×D5—82no—
189C2×D5—82no—
190C2×D5—82no—
191C2×D5—82no—
192C4×C5—21no—
193C4×C5—21no—
194C4×C5—82no—
195C4×C5—82no—
196C4×C5#44812yes—
197C4×C5#44912yes—
198C4×C5#46512yes—
199C4×C5#46812yes—
200Dic5—82no—
201C24—102no—
202C2×Q8—22no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—41no—
206C2×D4—54no—
207C2×D4—102no—
208C2×D4—102no—
209C2×D4—102no—
210C2×D4—102no—
211C2×D4—102no—
212C2×D4—208no—
213C2×D4—402no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—102no—
217C22×C4—202no—
218C22×C4—202no—
219C22×C4—202no—
220C22×C4—202no—
221C22×C4—202no—
222C22×C4—202no—
223C22×C4D4×D512yes—
224C22×C4C2×C22⋊D511yescharacteristic
225C2×C8—22no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—104no—
229C2×C8—202no—
230C2×C8—202no—
231C4⋊C4—24no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—102no—
237C4⋊C4—102no—
238C22⋊C4—208no—
239C22⋊C4—202no—
240C22⋊C4—202no—
241C42C2×D2011yescharacteristic
242C42#70911yescharacteristic
243C42D4×D512yes—
244C2×C5#162911yescharacteristic
245C2×C5#163411yescharacteristic
246C2×C5#163512yes—
247C2×C5#200012yes—
248C2×C5#200611yescharacteristic
249D5—84no—
250C23—104no—
251C23—202no—
252C23—202no—
253C23—202no—
254C23—202no—
255C23—202no—
256C23—202no—
257C23#408311yescharacteristic
258Q8—44no—
259Q8—82no—
260D4—104no—
261D4—104no—
262D4—202no—
263D4—202no—
264D4—408no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—202no—
273C2×C4—202no—
274C2×C4—202no—
275C2×C4—402no—
276C2×C4—402no—
277C2×C4—402no—
278C2×C4—402no—
279C2×C4#402812yes—
280C2×C4#403012yes—
281C2×C4D5×SD1612yes—
282C2×C4#406012yes—
283C2×C4#406412yes—
284C2×C4#406512yes—
285C2×C4#407611yescharacteristic
286C2×C4#408611yescharacteristic
287C8—42no—
288C8—204no—
289C5#1023511yescharacteristic, complemented, Sylow
290C22—204no—
291C22—204no—
292C22—204no—
293C22—402no—
294C22—402no—
295C22—402no—
296C22—402no—
297C22#6524911yescharacteristic
298C22#6535111yescharacteristic
299C22#6536611yescharacteristic
300C22#6545812yes—
301C22#6547612yes—
302C4—21no—
303C4—21no—
304C4—82no—
305C4—82no—
306C4—402no—
307C4#6509512yes—
308C4#6509612yes—
309C4#6510012yes—
310C4#6510112yes—
311C2—404no—
312C2#1060045711yescharacteristic
313C2#1060106512yes—
314C2#1060202112yes—
315C2#1060205911yescharacteristic
316C2#1060209711yescharacteristic
subgroupG/H
centerC23#4083
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#1956C2
socleC23×C5#120
prime radical#10606099C2
hypercenter#1956D5
hypocenterC5#10235
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. C2×#65373C2semidirect
3. C5×#1956C2semidirect
4. C2×#64679C2semidirect
5. #10598125C2semidirect
28. C5×C4⋊Q8D4semidirect
49. C4⋊#168D4semidirect
52. #1956D5semidirect
159. C4⋊Q8D20semidirect
289. C5#10235semidirect

Maximal subgroups

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

subgroupindex
C2×#653732
#106060992
C5×#19562
C2×#646792
#105981252
#102355