Finite Group Databasebeta

← #13020903

Subgroups — #13020903

Order 1280.

582 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 134 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#10608211C211yesmaximal, characteristic
2C2×Q8⋊#170C212yesmaximal, complemented
3C2×#64871C212yesmaximal
4#10604191C211yesmaximal, characteristic, complemented
5#10604173C211yesmaximal, characteristic, complemented
6C2×#4093C411yescharacteristic
7C2×Q8×Dic5C2211yescharacteristic
8C2×#4039C2211yescharacteristic
9C2×#4004—22no—
10C2×C4⋊#170C2212yes—
11C2×C4×#170C2212yes—
12C5×#391C411yescharacteristic
13#65266C411yescharacteristic
14#65228C411yescharacteristic
15C42×Dic5C2211yescharacteristic
16#64959—22no—
17#64955—22no—
18Q8⋊#170—24no—
19#64918—22no—
20#64871—24no—
21#9640—51nomaximal, Sylow
22C22×C5×Q8C2×C411yescharacteristic
23C22×D20C2×C411yescharacteristic
24C22×C4×D5—21no—
25C22×#170—22no—
26C22×#170—22no—
27C2×C5×C4⋊C4C2×C411yescharacteristic
28C2×C42×C5C2×C411yescharacteristic
29#4093—22no—
30#4093C814yescomplemented
31Q8×Dic5—22no—
32Q8×Dic5—22no—
33C2×#685—22no—
34C2×#685—22no—
35C2×#685—21no—
36C2×C4⋊Dic5—21no—
37C2×C4⋊Dic5C2×C411yescharacteristic
38C2×C4×Dic5—21no—
39C2×C4×Dic5C2×C411yescharacteristic
40C2×C4×Dic5D412yes—
41C2×C4×Dic5D412yes—
42C2×C4×Dic5C2311yescharacteristic
43#4039—22no—
44#4039—22no—
45#4004—44no—
46C4⋊#170—22no—
47C4⋊#170—22no—
48C4×#170—22no—
49C4×#170D414yes—
50C5×#95—22no—
51#3963—22no—
52C4×#124—51no—
53#1842—51no—
54#1744—51no—
55C2×Q8⋊C8—52no—
56C2×D4⋊C8—52no—
57C23×D5—21no—
58C2×C5×Q8—21no—
59C2×C5×Q8—22no—
60C2×C5×Q8—22no—
61C2×C5×Q8C2×C812yes—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5C22⋊C412yes—
65C22×C4×C5C22×C411yescharacteristic
66C22×Dic5—21no—
67C22×Dic5C22⋊C412yes—
68C22×Dic5C2×D411yescharacteristic
69C22×Dic5C2×D411yescharacteristic
70C2×D20—22no—
71C2×D20—22no—
72C2×D20—21no—
73C2×D20C2×C812yes—
74C2×C4×D5—41no—
75C2×C4×D5—41no—
76C2×C4×D5—42no—
77C2×C4×D5—42no—
78C2×#170—24no—
79C2×#170—44no—
80C2×#170—42no—
81C2×#170—44no—
82C5×C4⋊C4—22no—
83C42×C5—22no—
84#685—28no—
85#685—44no—
86#685—42no—
87C4⋊Dic5—22no—
88C4⋊Dic5—42no—
89C4×Dic5—22no—
90C4×Dic5—22no—
91C4×Dic5—42no—
92C4×Dic5—41no—
93C4×Dic5—41no—
94C4×Dic5C2×C812yes—
95C4×Dic5M1612yes—
96C4×Dic5C2×D412yes—
97C4×Dic5#4112yes—
98C2×#124—51no—
99C2×C4×Q8—51no—
100C2×C4×D4—51no—
101C2×C4⋊C8—52no—
102C2×C4⋊C8—102no—
103C2×C4×C8—52no—
104#391AGL(1,𝔽5)11yescharacteristic
105#390—51no—
106#388—102no—
107#382—51no—
108C4×C22⋊C4—102no—
109C43—51no—
110#336—102no—
111Q8⋊C8—104no—
112D4⋊C8—104no—
113C23×C5C2×C22⋊C411yescharacteristic
114C22×D5—22no—
115C22×D5—41no—
116C22×D5—41no—
117C22×D5—42no—
118C22×D5—42no—
119C5×Q8—24no—
120C5×Q8—42no—
121C2×C4×C5—22no—
122C2×C4×C5—22no—
123C2×C4×C5—22no—
124C2×C4×C5—22no—
125C2×C4×C5—41no—
126C2×C4×C5—41no—
127C2×C4×C5—41no—
128C2×C4×C5—42no—
129C2×C4×C5C4×D412yes—
130C2×C4×C5C22×C811yescharacteristic
131C2×C4×C5C2×M1611yescharacteristic
132C2×C4×C5#13112yes—
133C2×Dic5—22no—
134C2×Dic5—22no—
135C2×Dic5—22no—
136C2×Dic5—41no—
137C2×Dic5—42no—
138C2×Dic5C22⋊C814yes—
139C2×Dic5#12112yes—
140C2×Dic5#12411yescharacteristic
141C2×Dic5#12711yescharacteristic
142C2×Dic5AGL(1,ℤ/8)12yes—
143C2×Dic5#13712yes—
144D20—24no—
145D20—42no—
146C4×D5—82no—
147C4×D5—81no—
148C4×D5—81no—
149#170—44no—
150#170—82no—
151C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
152C22×D4—51no—
153C23×C4—101no—
154C22×C8—102no—
155C22×C8—102no—
156#124—54no—
157#124—102no—
158C4×Q8—102no—
159C4×Q8—102no—
160C4×D4—102no—
161C4×D4—102no—
162C2×C4⋊C4—51no—
163C2×C4⋊C4—101no—
164C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
165C2×C22⋊C4—102no—
166C2×C22⋊C4—101no—
167C2×C22⋊C4—102no—
168C2×C42—51no—
169C2×C42—51no—
170C2×C42—52no—
171C2×C42—52no—
172C2×C42—101no—
173C2×C42C2×AGL(1,𝔽5)11yescharacteristic
174C4⋊C8—102no—
175C4⋊C8—102no—
176C4⋊C8—204no—
177C4×C8—54no—
178C4×C8—102no—
179#95—22no—
180#95—102no—
181C22×C5#39011yescharacteristic
182C22×C5C2×C22⋊C811yescharacteristic
183C22×C5#40811yescharacteristic
184C22×C5#41712yes—
185C22×C5#42112yes—
186C2×D5—42no—
187C2×D5—44no—
188C2×D5—81no—
189C2×D5—81no—
190C2×D5—82no—
191C4×C5—42no—
192C4×C5—42no—
193C4×C5—81no—
194C4×C5C8×D412yes—
195C4×C5#43612yes—
196Dic5—22no—
197Dic5—24no—
198Dic5—81no—
199Dic5#48912yes—
200Dic5#49611yescharacteristic
201Dic5#49711yescharacteristic
202C24—101no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8D5⋊C812yes—
207C2×D4—52no—
208C2×D4—101no—
209C2×D4—102no—
210C2×D4—102no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—51no—
214C22×C4—52no—
215C22×C4—51no—
216C22×C4—101no—
217C22×C4—201no—
218C22×C4—201no—
219C22×C4—202no—
220C22×C4—202no—
221C22×C4C22⋊AGL(1,𝔽5)12yes—
222C22×C4C22×AGL(1,𝔽5)11yescharacteristic
223C2×C8—104no—
224C2×C8—204no—
225C2×C8—202no—
226C2×C8—204no—
227C4⋊C4—22no—
228C4⋊C4—102no—
229C4⋊C4—202no—
230C22⋊C4—108no—
231C22⋊C4—204no—
232C22⋊C4—202no—
233C42—22no—
234C42—52no—
235C42—52no—
236C42—52no—
237C42—52no—
238C42—102no—
239C42—102no—
240C42—202no—
241C42—201no—
242C42—201no—
243C2×C5#147712yes—
244C2×C5#149512yes—
245C2×C5#195411yescharacteristic
246C2×C5#195911yescharacteristic
247C2×C5#196311yescharacteristic
248D5—82no—
249C23—102no—
250C23—202no—
251C23—201no—
252C23—201no—
253C23—202no—
254C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
255Q8—24no—
256Q8—42no—
257D4—104no—
258D4—202no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—52no—
268C2×C4—52no—
269C2×C4—51no—
270C2×C4—51no—
271C2×C4—54no—
272C2×C4—52no—
273C2×C4—102no—
274C2×C4—102no—
275C2×C4—102no—
276C2×C4—201no—
277C2×C4—202no—
278C2×C4—401no—
279C2×C4—402no—
280C2×C4—401no—
281C2×C4C2×D5⋊C811yescharacteristic
282C2×C4D5⋊M1611yescharacteristic
283C2×C4#413112yes—
284C2×C4D4×AGL(1,𝔽5)12yes—
285C8—204no—
286C8—402no—
287C5#964011yescharacteristic, complemented, Sylow
288C22—204no—
289C22—202no—
290C22—401no—
291C22—401no—
292C22—402no—
293C22D5⋊(C22⋊C8)11yescharacteristic
294C22#6577511yescharacteristic
295C22#6578011yescharacteristic
296C22#6578412yes—
297C22#6579312yes—
298C4—42no—
299C4—42no—
300C4—51no—
301C4—52no—
302C4—51no—
303C4—81no—
304C4—102no—
305C4—104no—
306C4—401no—
307C4#6569212yes—
308C4#6569512yes—
309C2—402no—
310C2#1060424111yescharacteristic
311C2#1060443712yes—
312C2#1060451512yes—
313C2#1060456711yescharacteristic
314C2#1060466411yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C8
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#391C4
socleC23×C5C2×C22⋊C4
prime radicalC22×D20C2×C4
cosocleC2×C4×Dic5C23
hypercenter#391AGL(1,𝔽5)
hypocenterC5#9640
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×Q8⋊#170C2semidirect
4. #10604191C2semidirect
5. #10604173C2semidirect
30. #4093C8semidirect
287. C5#9640semidirect

Maximal subgroups

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

subgroupindex
#106082112
C2×Q8⋊#1702
C2×#648712
#106041912
#106041732
#96405