Finite Group Databasebeta

← #13020923

Subgroups — #13020923

Order 1280.

610 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 128 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
1Dic5×#124C211yesmaximal, characteristic
2C2×Q8⋊#170C212yesmaximal, complemented
3C2×D4⋊#170C212yesmaximal
4#10604191C211yesmaximal, characteristic, complemented
5C4×#4002C211yesmaximal, characteristic, complemented
6C2×C5×#124C411yescharacteristic
7C2×Q8×Dic5C2211yescharacteristic
8C2×D4×Dic5C2211yescharacteristic
9C2×#4004—22no—
10C2×#4002—22no—
11C2×C4⋊#170C2212yes—
12C2×C4×#170C2212yes—
13#65519C411yescharacteristic
14#65510C411yescharacteristic
15#65240—22no—
16C22⋊C4×Dic5—22no—
17#65225C411yescharacteristic
18C42×Dic5C2211yescharacteristic
19Q8⋊#170—24no—
20D4⋊#170—24no—
21#9646—51nomaximal, Sylow
22C22×C5×Q8C2×C411yescharacteristic
23C22×C5×D4C2×C411yescharacteristic
24C23×Dic5—21no—
25C22×#170—22no—
26C22×#170—22no—
27C5×#124—22no—
28C5×#124—22no—
29C2×C5×C22⋊C4—22no—
30C2×C42×C5C2×C411yescharacteristic
31C2×C22⋊Dic5—22no—
32C2×C22⋊Dic5—21no—
33Q8×Dic5—22no—
34Q8×Dic5—22no—
35D4×Dic5—22no—
36D4×Dic5—22no—
37C2×C4⋊Dic5—21no—
38C2×C4⋊Dic5C2×C411yescharacteristic
39C2×C4⋊Dic5C2×C411yescharacteristic
40C2×C4×Dic5—21no—
41C2×C4×Dic5C2×C411yescharacteristic
42C2×C4×Dic5D412yes—
43C2×C4×Dic5D412yes—
44C2×C4×Dic5C2311yescharacteristic
45#4004—44no—
46#4002—28no—
47C4⋊#170—22no—
48C4⋊#170—22no—
49C4×#170—22no—
50C4×#170D414yes—
51#3963—22no—
52#3963—22no—
53C4×#124—51no—
54#1842—51no—
55C4×#97—51no—
56C2×Q8⋊C8—52no—
57C2×D4⋊C8—52no—
58C24×C5—21no—
59C2×C5×Q8—21no—
60C2×C5×Q8—22no—
61C2×C5×Q8—22no—
62C2×C5×Q8M1612yes—
63C2×C5×D4—22no—
64C2×C5×D4—21no—
65C2×C5×D4—22no—
66C2×C5×D4M1612yes—
67C22×C4×C5—21no—
68C22×C4×C5C22⋊C412yes—
69C22×C4×C5C22×C411yescharacteristic
70C22×Dic5—21no—
71C22×Dic5—21no—
72C22×Dic5—41no—
73C22×Dic5—41no—
74C22×Dic5—42no—
75C22×Dic5—42no—
76C22×Dic5C22⋊C412yes—
77C22×Dic5C2×D411yescharacteristic
78C22×Dic5C2×D411yescharacteristic
79C2×#170—28no—
80C2×#170—24no—
81C2×#170—42no—
82C2×#170—44no—
83C5×C22⋊C4—28no—
84C42×C5—21no—
85C42×C5—21no—
86C22⋊Dic5—44no—
87C22⋊Dic5—42no—
88C4⋊Dic5—22no—
89C4⋊Dic5—22no—
90C4⋊Dic5—42no—
91C4×Dic5—22no—
92C4×Dic5—24no—
93C4×Dic5—22no—
94C4×Dic5—22no—
95C4×Dic5—41no—
96C4×Dic5—41no—
97C4×Dic5C2×D412yes—
98C4×Dic5#4112yes—
99C2×#124AGL(1,𝔽5)11yescharacteristic
100C2×C4×Q8—51no—
101C2×C4×D4—51no—
102C2×C4⋊C8—52no—
103C2×C4⋊C8—102no—
104C2×#97—102no—
105C2×C4×C8—52no—
106#391—51no—
107#390—51no—
108#388—102no—
109#382—51no—
110C4×C22⋊C4—102no—
111C43—51no—
112Q8⋊C8—104no—
113D4⋊C8—104no—
114C23×C5—22no—
115C23×C5—41no—
116C23×C5—41no—
117C23×C5—42no—
118C23×C5—42no—
119C23×C5C2×C22⋊C411yescharacteristic
120C5×Q8—24no—
121C5×Q8—42no—
122C5×D4—24no—
123C5×D4—42no—
124C2×C4×C5—22no—
125C2×C4×C5—22no—
126C2×C4×C5—22no—
127C2×C4×C5—22no—
128C2×C4×C5—41no—
129C2×C4×C5—41no—
130C2×C4×C5C4×D412yes—
131C2×C4×C5C2×M1611yescharacteristic
132C2×C4×C5C2×M1611yescharacteristic
133C2×C4×C5#13112yes—
134C2×Dic5—22no—
135C2×Dic5—22no—
136C2×Dic5—22no—
137C2×Dic5—41no—
138C2×Dic5—41no—
139C2×Dic5—42no—
140C2×Dic5—42no—
141C2×Dic5—82no—
142C2×Dic5—81no—
143C2×Dic5—81no—
144C2×Dic5#10414yes—
145C2×Dic5#12112yes—
146C2×Dic5#12411yescharacteristic
147C2×Dic5#12711yescharacteristic
148C2×Dic5AGL(1,ℤ/8)12yes—
149C2×Dic5#13712yes—
150#170—28no—
151#170—82no—
152C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
153C22×D4C2×AGL(1,𝔽5)11yescharacteristic
154C23×C4—101no—
155C22×C8—102no—
156C22×C8—102no—
157#124—22no—
158#124—22no—
159C4×Q8—102no—
160C4×Q8—102no—
161C4×D4—102no—
162C4×D4—102no—
163C2×C4⋊C4—51no—
164C2×C4⋊C4—51no—
165C2×C4⋊C4—101no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—101no—
168C2×C22⋊C4—102no—
169C2×C42—51no—
170C2×C42—52no—
171C2×C42—52no—
172C2×C42—51no—
173C2×C42—101no—
174C2×C42C2×AGL(1,𝔽5)11yescharacteristic
175C4⋊C8—102no—
176C4⋊C8—102no—
177C4⋊C8—204no—
178#97—108no—
179C4×C8—54no—
180C4×C8—102no—
181#95—102no—
182#95—102no—
183C22×C5—42no—
184C22×C5—44no—
185C22×C5—81no—
186C22×C5—81no—
187C22×C5—82no—
188C22×C5#39011yescharacteristic
189C22×C5#40711yescharacteristic
190C22×C5#40811yescharacteristic
191C22×C5#41712yes—
192C22×C5C2×#10412yes—
193C4×C5—42no—
194C4×C5—42no—
195C4×C5#43512yes—
196C4×C5#43512yes—
197Dic5—22no—
198Dic5—24no—
199Dic5—81no—
200Dic5—81no—
201Dic5#48912yes—
202Dic5#49611yescharacteristic
203Dic5#49711yescharacteristic
204C24—21no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8#70412yes—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4#70412yes—
213C22×C4—21no—
214C22×C4—51no—
215C22×C4—52no—
216C22×C4—51no—
217C22×C4—101no—
218C22×C4—101no—
219C22×C4—201no—
220C22×C4—201no—
221C22×C4—202no—
222C22×C4—202no—
223C22×C4C22⋊AGL(1,𝔽5)12yes—
224C22×C4C22×AGL(1,𝔽5)11yescharacteristic
225C2×C8—104no—
226C2×C8—108no—
227C2×C8—202no—
228C2×C8—204no—
229C4⋊C4—102no—
230C4⋊C4—102no—
231C4⋊C4—202no—
232C22⋊C4—28no—
233C22⋊C4—202no—
234C22⋊C4—204no—
235C42—21no—
236C42—21no—
237C42—52no—
238C42—52no—
239C42—104no—
240C42—102no—
241C42—102no—
242C42—102no—
243C42—201no—
244C42—201no—
245C2×C5—82no—
246C2×C5#148312yes—
247C2×C5#148912yes—
248C2×C5#195511yescharacteristic
249C2×C5#195911yescharacteristic
250C2×C5#196211yescharacteristic
251C23—22no—
252C23—42no—
253C23—42no—
254C23—41no—
255C23—41no—
256C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
257Q8—24no—
258Q8—42no—
259D4—24no—
260D4—42no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
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—202no—
277C2×C4—202no—
278C2×C4—201no—
279C2×C4—201no—
280C2×C4—401no—
281C2×C4—402no—
282C2×C4—401no—
283C2×C4#413112yes—
284C2×C4D4×AGL(1,𝔽5)12yes—
285C2×C4C2×#70411yescharacteristic
286C2×C4C2×#70411yescharacteristic
287C8—108no—
288C8—402no—
289C5#964611yescharacteristic, complemented, Sylow
290C22—42no—
291C22—44no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22#6577611yescharacteristic
296C22#6578011yescharacteristic
297C22D5⋊#10412yes—
298C22#6579312yes—
299C22#6579811yescharacteristic
300C4—42no—
301C4—42no—
302C4—51no—
303C4—52no—
304C4—51no—
305C4—104no—
306C4—102no—
307C4—401no—
308C4—401no—
309C4#6577312yes—
310C4#6577312yes—
311C2—82no—
312C2#1060455312yes—
313C2#1060455911yescharacteristic
314C2#1060456711yescharacteristic
315C2#1060494911yescharacteristic
316C2#1060500312yes—
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC2×C5×#124C4
socleC23×C5C2×C22⋊C4
prime radicalC22×C5×D4C2×C4
cosocleC2×C4×Dic5C23
hypercenterC2×#124AGL(1,𝔽5)
hypocenterC5#9646
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×Q8⋊#170C2semidirect
4. #10604191C2semidirect
5. C4×#4002C2semidirect
289. C5#9646semidirect

Maximal subgroups

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

subgroupindex
Dic5×#1242
C2×Q8⋊#1702
C2×D4⋊#1702
#106041912
C4×#40022
#96465