Finite Group Databasebeta

← #10608838

Subgroups — #10608838

Order 640.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#711C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#4029C212yesmaximal, complemented
4C5×#394C211yesmaximal, characteristic, complemented
5#65511C211yesmaximal, characteristic, complemented
6#65268C211yesmaximal, characteristic, complemented
7#65245C212yesmaximal, complemented
8#65239C212yesmaximal, complemented
9#65235C212yesmaximal, complemented
10#64953C211yesmaximal, characteristic, complemented
11#64945C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C23×Dic5C2212yes—
15C2×#711—28no—
16C2×#711—24no—
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C22×Dic10C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2212yes—
23C2×C5×C22⋊C4C2212yes—
24C2×C22⋊Dic5C2212yes—
25C2×C22⋊Dic5C2212yes—
26C2×#685C2212yes—
27C2×C4⋊Dic5C2212yes—
28C2×C4⋊Dic5C2211yescharacteristic
29C2×#683C2212yes—
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32D5×C4⋊C4—24no—
33#4029—28no—
34C5×#95C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#3963C2211yescharacteristic
38#3963C2212yes—
39#2406—51nomaximal, Sylow
40C24×C5C2312yes—
41C23×D5C2311yescharacteristic
42C2×C5×D4—24no—
43C2×C5×D4—22no—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2312yes—
46C22×C4×C5C2311yescharacteristic
47C22×C4×C5C2311yescharacteristic
48C2×C22⋊D5—24no—
49C2×C22⋊D5—24no—
50C2×C22⋊D5D418yes—
51C22×Dic5—24no—
52C22×Dic5—24no—
53C22×Dic5—24no—
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2312yes—
56C22×Dic5C2312yes—
57C22×Dic5C2311yescharacteristic
58C22×Dic5C2311yescharacteristic
59#711—232no—
60C2×C4×D5—24no—
61C2×C4×D5—22no—
62C2×C4×D5—22no—
63C2×C4×D5—22no—
64C2×C4×D5—24no—
65C2×C4×D5—22no—
66C2×C4×D5D414yes—
67C2×Dic10—22no—
68C2×Dic10—24no—
69C5×C4⋊C4—22no—
70C5×C22⋊C4—24no—
71C5×C22⋊C4—24no—
72C22⋊Dic5—24no—
73C22⋊Dic5—24no—
74#685—24no—
75C4⋊Dic5—24no—
76C4⋊Dic5—22no—
77#683—22no—
78#683—24no—
79#683—22no—
80C22×#41—51no—
81C2×#123—52no—
82C22×C4⋊C4—51no—
83#395—51no—
84#395—51no—
85#394—51no—
86#394D511yescharacteristic, complemented
87#386—52no—
88#385—52no—
89#385—52no—
90#381—51no—
91C23×C5—22no—
92C23×C5—22no—
93C23×C5—22no—
94C23×C5—22no—
95C23×C5—22no—
96C23×C5—22no—
97C23×C5C2×D414yes—
98C23×C5C2411yescharacteristic
99C22×D5—21no—
100C22×D5—21no—
101C22×D5—22no—
102C22×D5C2×D414yes—
103C22×D5C2×D412yes—
104C5×D4—28no—
105C2×C4×C5—21no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5—22no—
109C2×C4×C5—21no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5C2×D412yes—
114C2×C4×C5#4112yescomplemented
115C22⋊D5—216no—
116C2×Dic5—216no—
117C2×Dic5—21no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—21no—
122C2×Dic5—21no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—22no—
127C2×Dic5C2×D414yes—
128C2×Dic5C2×D412yes—
129C2×Dic5#4112yescomplemented
130C4×D5—28no—
131C4×D5—28no—
132C4×D5—44no—
133Dic10—28no—
134C2×#41—104no—
135C2×#41—108no—
136C22×Q8—51no—
137C22×D4—52no—
138C22×D4C2×D511yescharacteristic
139C23×C4—51no—
140C23×C4—52no—
141C23×C4—51no—
142C23×C4—51no—
143#123—108no—
144C2×C4⋊C4—51no—
145C2×C4⋊C4—51no—
146C2×C4⋊C4—51no—
147C2×C4⋊C4—52no—
148C2×C4⋊C4—52no—
149C2×C4⋊C4—104no—
150C2×C4⋊C4C2×D511yescharacteristic
151C2×C22⋊C4—52no—
152C2×C22⋊C4—52no—
153C2×C22⋊C4—52no—
154C2×C22⋊C4C2×D512yes—
155C2×C22⋊C4C2×D512yes—
156#95—52no—
157#95—51no—
158#95—52no—
159#95—52no—
160#95C2×D511yescharacteristic
161C22×C5—24no—
162C22×C5—24no—
163C22×C5—24no—
164C22×C5—42no—
165C22×C5—42no—
166C22×C5—42no—
167C22×C5—42no—
168C22×C5C22×D411yescharacteristic
169C22×C5C22×D412yes—
170C22×C5C2×#4111yescharacteristic
171C22×C5C2×#4111yescharacteristic
172C22×C52−1+412yes—
173C2×D5—22no—
174C2×D5—22no—
175C2×D5—24no—
176C2×D5—24no—
177C2×D5#12014yes—
178C4×C5—22no—
179C4×C5—22no—
180C4×C5—41no—
181C4×C5—42no—
182Dic5—24no—
183Dic5—22no—
184Dic5—22no—
185Dic5—41no—
186Dic5—42no—
187C24—51no—
188C24C22×D512yes—
189#41—1032no—
190C2×Q8—104no—
191C2×Q8—102no—
192C2×D4—24no—
193C2×D4—22no—
194C2×D4—58no—
195C2×D4—104no—
196C2×D4—104no—
197C22×C4—51no—
198C22×C4—51no—
199C22×C4—52no—
200C22×C4—54no—
201C22×C4—52no—
202C22×C4—51no—
203C22×C4—102no—
204C22×C4—102no—
205C22×C4—104no—
206C22×C4—102no—
207C22×C4—104no—
208C22×C4—104no—
209C22×C4—104no—
210C22×C4—104no—
211C22×C4—102no—
212C22×C4C22×D512yes—
213C22×C4C22×D511yescharacteristic
214C22×C4C22×D511yescharacteristic
215C22×C4C22×D511yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—102no—
218C4⋊C4—102no—
219C4⋊C4—104no—
220C4⋊C4—104no—
221C4⋊C4—102no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—104no—
225C22⋊C4—104no—
226C22⋊C4—104no—
227C2×C5—44no—
228C2×C5C2×#12011yescharacteristic
229C2×C5#52511yescharacteristic
230C2×C5#53611yescharacteristic
231C2×C5#54712yes—
232C2×C5#54712yes—
233D5—24no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—54no—
241C23—52no—
242C23—101no—
243C23—101no—
244C23—102no—
245C23D4×D514yes—
246C23C23×D511yescharacteristic
247Q8—108no—
248D4—28no—
249D4—1016no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—52no—
259C2×C4—54no—
260C2×C4—52no—
261C2×C4—108no—
262C2×C4—108no—
263C2×C4—1016no—
264C2×C4—101no—
265C2×C4—102no—
266C2×C4—102no—
267C2×C4—102no—
268C2×C4—101no—
269C2×C4—102no—
270C2×C4—102no—
271C2×C4—101no—
272C2×C4—102no—
273C2×C4—102no—
274C2×C4—204no—
275C2×C4D4×D512yes—
276C2×C4#71112yescomplemented
277C5#240611yescharacteristic, complemented, Sylow
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—54no—
286C22—102no—
287C22—104no—
288C22—104no—
289C22—102no—
290C22C2×D4×D511yescharacteristic
291C22C2×D4×D512yes—
292C22C2×#71111yescharacteristic
293C22D5×#4111yescharacteristic
294C22#415012yes—
295C4—22no—
296C4—22no—
297C4—41no—
298C4—42no—
299C4—102no—
300C4—104no—
301C4—102no—
302C4—201no—
303C4—202no—
304C2—44no—
305C2—104no—
306C2D5×#12011yescharacteristic
307C2#6594011yescharacteristic
308C2#6594512yes—
309C2#6598912yes—
310C2#6600911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC22×#711C2
hypercenter#394D5
hypocenterC5#2406
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×D5×C4⋊C4C2semidirect
3. C2×#4029C2semidirect
4. C5×#394C2semidirect
5. #65511C2semidirect
6. #65268C2semidirect
7. #65245C2semidirect
8. #65239C2semidirect
9. #65235C2semidirect
10. #64953C2semidirect
11. #64945C2semidirect
86. #394D5semidirect
114. C2×C4×C5#41semidirect
129. C2×Dic5#41semidirect
276. C2×C4#711semidirect
277. C5#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#7112
C2×D5×C4⋊C42
C2×#40292
C5×#3942
#655112
#652682
#652452
#652392
#652352
#649532
#649452
#24065