Finite Group Databasebeta

← #12001609

Subgroups — #12001609

Order 1280.

676 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 137 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×#2582C211yesmaximal, characteristic, complemented
2C2×#65316C212yesmaximal
3C2×Q8⋊#168C212yesmaximal, complemented
4#10600373C211yesmaximal, characteristic
5#10598125C211yesmaximal, characteristic, complemented
6C2×C4×C5×Q8C2212yes—
7C22×#688—22no—
8C2×#4020C2211yescharacteristic
9C5×#398—22no—
10C5×#391—22no—
11C5×#389—22no—
12C5×#384C2211yescharacteristic
13#65316—24no—
14#65316—24no—
15C2×#3941—22no—
16C2×#3941—22no—
17C2×C4⋊#168C2212yes—
18C2×C4×#168C2211yescharacteristic
19Q8⋊#168—24no—
20#10230—51nomaximal, Sylow
21C22×C5×Q8D412yes—
22C22×D20—22no—
23C22×D20—21no—
24C4×C5×Q8—22no—
25C4×C5×Q8—22no—
26C4×C5×Q8—22no—
27C4×C5×Q8—22no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4D412yes—
32C2×C5×C4⋊C4D412yes—
33C2×C42×C5—22no—
34C2×C42×C5C2311yescharacteristic
35C2×#688—28no—
36C2×#688—44no—
37C2×#688—44no—
38C22×#168—22no—
39C22×#168—21no—
40#4020—22no—
41#4020—21no—
42#4020D412yes—
43C5×#95—24no—
44C5×#95—22no—
45#3941—28no—
46#3941—44no—
47C4⋊#168—22no—
48C4⋊#168D414yes—
49C4×#168—22no—
50C2×#460—52no—
51#2582D511yescharacteristic, complemented
52#1957—51no—
53C2×Q8⋊C8—52no—
54#1296—51no—
55C23×D5—41no—
56C2×C5×Q8—22no—
57C2×C5×Q8—22no—
58C2×C5×Q8—22no—
59C2×C5×Q8—22no—
60C2×C5×Q8—22no—
61C2×C5×Q8SD1614yescomplemented
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5—22no—
65C22×C4×C5C2×D411yescharacteristic
66C22×C4×C5C2×D412yes—
67C2×D20—24no—
68C2×D20—22no—
69C2×D20—42no—
70C2×D20—42no—
71C2×D20—42no—
72C2×D20—42no—
73C2×D20—42no—
74C2×D20—42no—
75C2×D20—42no—
76C2×D20—41no—
77C5×C4⋊C4—22no—
78C5×C4⋊C4—22no—
79C5×C4⋊C4—42no—
80C5×C4⋊C4—42no—
81C5×C4⋊C4—42no—
82C5×C4⋊C4—42no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—82no—
85C42×C5—42no—
86C42×C5—42no—
87C42×C5C2×D411yescharacteristic
88C42×C5C2×D412yes—
89C42×C5#4111yescharacteristic
90#688—416no—
91C2×#168—22no—
92C2×#168—24no—
93C2×#168—41no—
94C2×#168—41no—
95C2×#168—42no—
96C2×#168—42no—
97C22×SD16—102no—
98C2×#127—51no—
99C2×C4×Q8C2×D512yes—
100#460—104no—
101#460—104no—
102C2×C4⋊C8—52no—
103C2×#102—102no—
104C2×#102—102no—
105C2×C4×C8—51no—
106#398—22no—
107#391—22no—
108#389—22no—
109#384C2×D511yescharacteristic
110Q8⋊C8—104no—
111C23×C5#12011yescharacteristic
112C22×D5—42no—
113C22×D5—82no—
114C22×D5—82no—
115C22×D5—81no—
116C22×D5—81no—
117C5×Q8—24no—
118C5×Q8—24no—
119C5×Q8—42no—
120C5×Q8—42no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—24no—
124C2×C4×C5—42no—
125C2×C4×C5—42no—
126C2×C4×C5—42no—
127C2×C4×C5—44no—
128C2×C4×C5—42no—
129C2×C4×C5—42no—
130C2×C4×C5—42no—
131C2×C4×C5#12012yes—
132C2×C4×C5#12111yescharacteristic
133C2×C4×C5#12112yes—
134C2×C4×C5#12711yescharacteristic
135C2×C4×C5C2×SD1612yes—
136C2×C4×C5#13512yes—
137C2×C4×C5AGL(1,ℤ/8)12yes—
138C2×C4×C5#13712yes—
139D20—44no—
140D20—44no—
141D20—44no—
142D20—82no—
143D20—82no—
144D20—82no—
145#168—42no—
146#168—44no—
147C22×Q8C22⋊D512yes—
148C22×D4—102no—
149C22×D4—101no—
150C2×SD16—108no—
151C2×SD16—204no—
152C2×SD16—204no—
153C22×C8—102no—
154C22×C8—101no—
155#127—52no—
156#127—102no—
157#127—101no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊D512yes—
166C2×C4⋊C4C22⋊D512yes—
167C2×C42—22no—
168C2×C42C22×D511yescharacteristic
169C4⋊C8—54no—
170C4⋊C8—102no—
171#102—108no—
172#102—204no—
173C4×C8—102no—
174#95—24no—
175#95—22no—
176C22×C5#39211yescharacteristic
177C22×C5#44812yes—
178C22×C5#44912yes—
179C22×C5#45311yescharacteristic
180C22×C5#45411yescharacteristic
181C2×D5—84no—
182C2×D5—82no—
183C2×D5—161no—
184C2×D5—161no—
185C2×D5—162no—
186C4×C5—21no—
187C4×C5—21no—
188C4×C5—44no—
189C4×C5—82no—
190C4×C5—82no—
191C4×C5#45012yes—
192C4×C5#45212yes—
193C4×C5#46512yes—
194C4×C5#46812yes—
195C24—201no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8#68814yescomplemented
202C2×D4—104no—
203C2×D4—102no—
204C2×D4—202no—
205C2×D4—202no—
206C2×D4—202no—
207C2×D4—202no—
208C2×D4—201no—
209C2×D4—202no—
210C2×D4—202no—
211C2×D4—202no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4C2×C22⋊D511yescharacteristic
216C22×C4C2×C22⋊D512yes—
217SD16—2016no—
218C2×C8—102no—
219C2×C8—104no—
220C2×C8—201no—
221C2×C8—201no—
222C2×C8—202no—
223C2×C8—202no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—82no—
232C42—42no—
233C42—42no—
234C42D4×D511yescharacteristic
235C42D4×D512yes—
236C42#71111yescharacteristic
237C2×C5#162011yescharacteristic
238C2×C5#162311yescharacteristic
239C2×C5#162612yes—
240C2×C5#200012yes—
241C2×C5#200611yescharacteristic
242D5—162no—
243C23—202no—
244C23—402no—
245C23—402no—
246C23—401no—
247C23—401no—
248C23#409911yescharacteristic
249Q8—24no—
250Q8—24no—
251Q8—42no—
252Q8—42no—
253D4—204no—
254D4—204no—
255D4—204no—
256D4—402no—
257D4—402no—
258D4—402no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4#408312yes—
270C2×C4#408411yescharacteristic
271C2×C4#408512yes—
272C2×C4#408611yescharacteristic
273C2×C4C2×#68812yes—
274C2×C4#408812yes—
275C2×C4#409512yes—
276C2×C4#409612yes—
277C8—202no—
278C8—204no—
279C5#1023011yescharacteristic, complemented, Sylow
280C22—402no—
281C22—404no—
282C22—802no—
283C22—801no—
284C22—801no—
285C22#6551211yescharacteristic
286C22#6551612yes—
287C22#6552712yes—
288C22#6553011yescharacteristic
289C22#6553411yescharacteristic
290C4—21no—
291C4—21no—
292C4—44no—
293C4—82no—
294C4—82no—
295C4#6533512yes—
296C4#6533612yes—
297C4#6533712yes—
298C4#6533812yes—
299C2—802no—
300C2#1060249212yes—
301C2#1060267211yescharacteristic
302C2#1060272812yes—
303C2#1060273211yescharacteristic
304C2#1060273611yescharacteristic
subgroupG/H
centerC23#4099
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#2582C2
socleC23×C5#120
prime radicalC2×#4020C22
hypercenter#2582D5
hypocenterC5#10230
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2582

Complements

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

NH ≅ G/Ntype
1. C5×#2582C2semidirect
3. C2×Q8⋊#168C2semidirect
5. #10598125C2semidirect
51. #2582D5semidirect
61. C2×C5×Q8SD16semidirect
201. C2×Q8#688semidirect
279. C5#10230semidirect

Maximal subgroups

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

subgroupindex
C5×#25822
C2×#653162
C2×Q8⋊#1682
#106003732
#105981252
#102305