Finite Group Databasebeta

← #10613413

Subgroups — #10613413

Order 640.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#66062C211yesmaximal, characteristic, complemented
2D5×#124C211yesmaximal, characteristic
3C5×#489C211yesmaximal, characteristic, complemented
4#65378C211yesmaximal, characteristic, complemented
5#65344C211yesmaximal, characteristic, complemented
6#65091C212yesmaximal, complemented
7#65089C212yesmaximal, complemented
8#65059C212yesmaximal
9#65057C212yesmaximal, complemented
10#65003C211yesmaximal, characteristic, complemented
11#64996C211yesmaximal, characteristic, complemented
12C2×Q8×D5C2211yescharacteristic
13C2×D4×D5C2211yescharacteristic
14C5×C4⋊Q8C2211yescharacteristic, complemented
15C5×#124C2211yescharacteristic
16#4093C2211yescharacteristic
17#4092—21no—
18Q8×Dic5—21no—
19#4082C2211yescharacteristic
20C2×#676C2212yes—
21#4043—21no—
22#4038C2212yes—
23#4036C2212yescomplemented
24#4033—21no—
25#4031—21no—
26#4031—21no—
27D5×C22⋊C4—21no—
28D5×C22⋊C4—21no—
29#4021C2211yescharacteristic
30C42×D5C2211yescharacteristic
31#4015C2211yescharacteristic, complemented
32C5×#103C2212yescomplemented
33C5×#102C2212yes—
34C5×#97C2211yescharacteristic, complemented
35#3967C2212yescomplemented
36#3964C2212yes—
37#3953C2212yes—
38#3948C2212yescomplemented
39C8×Dic5C2212yescomplemented
40#3942C2212yescomplemented
41#3941C2212yes—
42#3935C2211yescharacteristic, complemented
43#3140—51nomaximal, Sylow
44C23×D5—21no—
45C2×C5×Q8—21no—
46C2×C5×Q8C2311yescharacteristic
47C2×C5×D4C2311yescharacteristic
48C2×C22⋊D5—21no—
49Q8×D5—24no—
50D4×D5—24no—
51C2×D20C2311yescharacteristic
52C2×C4×D5—21no—
53C2×C4×D5D411yescharacteristic
54C2×C4×D5D411yescharacteristic
55C2×C4×D5C2311yescharacteristic
56C2×Dic10—21no—
57C2×Dic10—21no—
58C2×Dic10C2311yescharacteristic
59C2×C5×C8C2312yes—
60C5×C4⋊C4—21no—
61C5×C4⋊C4C2312yes—
62C5×C22⋊C4—21no—
63C5×C22⋊C4—21no—
64C42×C5C2311yescharacteristic
65C22⋊Dic5—21no—
66C22⋊Dic5—21no—
67#685—22no—
68#685—21no—
69#685—21no—
70#685—21no—
71#685—21no—
72C4⋊Dic5—21no—
73C4⋊Dic5—22no—
74C4⋊Dic5C2312yes—
75#683—22no—
76C4×Dic5—22no—
77C4×Dic5D411yescharacteristic
78C4×Dic5D411yescharacteristic
79C4×Dic5C2311yescharacteristic
80C2×#168C2312yes—
81#676—24no—
82#533—51no—
83C2×#124—51no—
84#489—51no—
85#489—51no—
86#489—51no—
87#489D511yescharacteristic, complemented
88#488—52no—
89#488—52no—
90#419—52no—
91#418—52no—
92C4×M16—51no—
93C23×C5—21no—
94C22×D5—21no—
95C22×D5—42no—
96C22×D5—42no—
97C22×D5—42no—
98C22×D5C2×D411yescharacteristic
99C5×Q8—22no—
100C5×Q8—42no—
101C5×D4—22no—
102C2×C4×C5—22no—
103C2×C4×C5—21no—
104C2×C4×C5C2×D411yescharacteristic
105C2×C4×C5C2×D411yescharacteristic
106C2×C4×C5C2411yescharacteristic
107C22⋊D5—42no—
108C2×Dic5—22no—
109C2×Dic5—21no—
110C2×Dic5C2×D411yescharacteristic
111C2×Dic5C2×D411yescharacteristic
112C2×Dic5C2×D411yescharacteristic
113D20—22no—
114C4×D5—41no—
115C4×D5—41no—
116C4×D5—42no—
117C4×D5#4114yes—
118Dic10—22no—
119Dic10—42no—
120Dic10—42no—
121C5×C8—22no—
122#168—22no—
123C22×Q8—51no—
124C22×D4—51no—
125C2×M16—52no—
126C4⋊Q8—51no—
127C4⋊Q8C2×D511yescharacteristic, complemented
128#125—52no—
129#124—51no—
130#124—51no—
131#124—51no—
132#124—101no—
133#124—101no—
134#124C2×D511yescharacteristic
135C22⋊Q8—101no—
136C22⋊Q8—101no—
137C4×Q8—101no—
138C4×Q8—101no—
139#117—52no—
140C2×C22⋊C4—101no—
141C2×C22⋊C4—101no—
142C2×C42—51no—
143#103—52no—
144#103—52no—
145#103—52no—
146#103C2×D512yescomplemented
147#102—52no—
148#102—52no—
149#102—52no—
150#102C2×D512yes—
151#97—51no—
152#97C2×D511yescharacteristic, complemented
153C4×C8—52no—
154C22×C5—41no—
155C22×C5—41no—
156C22×C5—41no—
157C22×C5C22×D411yescharacteristic
158C2×D5—21no—
159C2×D5—21no—
160C2×D5—41no—
161C2×D5—41no—
162C2×D5—41no—
163C2×D5—84no—
164C2×D5#12412yes—
165C4×C5—21no—
166C4×C5—21no—
167C4×C5—41no—
168C4×C5—42no—
169C4×C5C2×#4112yes—
170Dic5—21no—
171Dic5—21no—
172Dic5—41no—
173Dic5—42no—
174Dic5#12412yes—
175C24—101no—
176C2×Q8—21no—
177C2×Q8—51no—
178C2×Q8—101no—
179C2×Q8—104no—
180C2×Q8—101no—
181C2×Q8C22×D511yescharacteristic
182C2×D4—51no—
183C2×D4—104no—
184C2×D4—101no—
185C2×D4C22×D511yescharacteristic
186C22×C4—51no—
187C22×C4—51no—
188C22×C4—51no—
189C22×C4—101no—
190M16—104no—
191C2×C8—52no—
192C2×C8C22×D512yes—
193C4⋊C4—21no—
194C4⋊C4—52no—
195C4⋊C4—102no—
196C4⋊C4—102no—
197C4⋊C4—101no—
198C4⋊C4C22×D512yes—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—101no—
202C22⋊C4—101no—
203C22⋊C4—102no—
204C22⋊C4—101no—
205C22⋊C4—101no—
206C22⋊C4—101no—
207C22⋊C4—101no—
208C42—51no—
209C42—51no—
210C42—51no—
211C42—102no—
212C42C22×D511yescharacteristic
213C2×C5—81no—
214C2×C5C2×#12411yescharacteristic
215C2×C5#57511yescharacteristic
216C2×C5#57511yescharacteristic
217D5—22no—
218D5—81no—
219C23—21no—
220C23—51no—
221C23—101no—
222C23—202no—
223C23—202no—
224C23—202no—
225Q8—22no—
226Q8—42no—
227Q8—102no—
228Q8—202no—
229Q8—202no—
230D4—22no—
231D4—102no—
232D4—202no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—51no—
236C2×C4—54no—
237C2×C4—51no—
238C2×C4—51no—
239C2×C4—102no—
240C2×C4—101no—
241C2×C4—201no—
242C2×C4—201no—
243C2×C4—202no—
244C2×C4D4×D511yescharacteristic
245C2×C4D4×D511yescharacteristic
246C2×C4C23×D511yescharacteristic
247C8—22no—
248C8—102no—
249C5#314011yescharacteristic, complemented, Sylow
250C22—41no—
251C22—41no—
252C22—41no—
253C22—52no—
254C22—101no—
255C22—101no—
256C22—201no—
257C22—201no—
258C22—201no—
259C22—404no—
260C22C2×D4×D511yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—41no—
265C4—52no—
266C4—101no—
267C4—101no—
268C4—202no—
269C4—201no—
270C4D5×#4112yes—
271C2—81no—
272C2—102no—
273C2—401no—
274C2D5×#12411yescharacteristic
275C2#6610711yescharacteristic
276C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#489C2
socleC22×C5C22×D4
prime radicalC2×D4×D5C22
hypercenter#489D5
hypocenterC5#3140
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #66062C2semidirect
3. C5×#489C2semidirect
4. #65378C2semidirect
5. #65344C2semidirect
6. #65091C2semidirect
7. #65089C2semidirect
9. #65057C2semidirect
10. #65003C2semidirect
11. #64996C2semidirect
14. C5×C4⋊Q8C22semidirect
23. #4036C22semidirect
31. #4015C22semidirect
32. C5×#103C22semidirect
34. C5×#97C22semidirect
35. #3967C22semidirect
38. #3948C22semidirect
39. C8×Dic5C22semidirect
40. #3942C22semidirect
42. #3935C22semidirect
87. #489D5semidirect
127. C4⋊Q8C2×D5semidirect
146. #103C2×D5semidirect
152. #97C2×D5semidirect
249. C5#3140semidirect

Maximal subgroups

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

subgroupindex
#660622
D5×#1242
C5×#4892
#653782
#653442
#650912
#650892
#650592
#650572
#650032
#649962
#31405