Finite Group Databasebeta

← #10611870

Subgroups — #10611870

Order 640.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×#4082C211yesmaximal, characteristic, complemented
2#65958C211yesmaximal, characteristic, complemented
3C5×#450C211yesmaximal, characteristic, complemented
4#65506C211yesmaximal, characteristic
5#65457C212yesmaximal, complemented
6#65455C212yesmaximal, complemented
7#65339C211yesmaximal, characteristic, complemented
8#65076C212yesmaximal, complemented
9#65057C212yesmaximal, complemented
10#65027C211yesmaximal, characteristic, complemented
11#65007C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C2×C5×SD16C2212yes—
15C5×C22⋊Q8C2211yescharacteristic
16C2×C22⋊Dic5—21no—
17C2×C22⋊Dic5—21no—
18#4093C2211yescharacteristic, complemented
19Q8×Dic5C2211yescharacteristic, complemented
20C2×#689C2212yescomplemented
21#4082—21no—
22#4082—21no—
23#4082C2212yes—
24#4082C2212yes—
25C2×#686C2212yes—
26#4076C2211yescharacteristic
27C2×C4×Dic5C2211yescharacteristic
28C2×#677C2212yescomplemented
29#4044—21no—
30#4039C2211yescharacteristic, complemented
31#4038—21no—
32#4036C2211yescharacteristic, complemented
33#4032—21no—
34#4027—21no—
35C5×#102C2212yes—
36C5×C22⋊C8C2211yescharacteristic
37#3964C2212yes—
38C22⋊#168C2211yescharacteristic
39#3948C2212yescomplemented
40#3946C2212yescomplemented
41C8×Dic5C2212yescomplemented
42#3942C2212yescomplemented
43#3211—51nomaximal, Sylow
44C24×C5—21no—
45C2×C5×Q8C2311yescharacteristic
46C2×C5×D4—21no—
47C2×C5×D4—42no—
48C2×C5×D4C2312yes—
49C22×C4×C5C2311yescharacteristic
50C2×C22⋊D5—21no—
51C22×Dic5—21no—
52C22×Dic5D411yescharacteristic
53C22×Dic5D411yescharacteristic
54C2×D20C2311yescharacteristic
55C2×C4×D5—21no—
56C2×Dic10—21no—
57C2×Dic10—42no—
58C2×Dic10C2312yes—
59C5×SD16—42no—
60C2×C5×C8C2312yes—
61C5×C4⋊C4—21no—
62C5×C4⋊C4C2311yescharacteristic
63C5×C22⋊C4—21no—
64C22⋊Dic5—22no—
65C22⋊Dic5—22no—
66C22⋊Dic5—22no—
67C22⋊Dic5—22no—
68#689—42no—
69#686—42no—
70#685—21no—
71#685—21no—
72#685—21no—
73#685—21no—
74C4⋊Dic5—21no—
75C4⋊Dic5—21no—
76C4⋊Dic5C2311yescharacteristic
77#683—21no—
78C4×Dic5—21no—
79C4×Dic5D411yescharacteristic
80C4×Dic5D411yescharacteristic
81C4×Dic5C2311yescharacteristic
82C4×Dic5C2311yescharacteristic
83C2×#168C2312yes—
84#677—42no—
85#529—51no—
86C2×#124—51no—
87#495—52no—
88#488—52no—
89#464—52no—
90#463—52no—
91#451—51no—
92#450D511yescharacteristic, complemented
93#448—51no—
94#447—51no—
95#431—51no—
96C23×C5—22no—
97C23×C5—42no—
98C23×C5—42no—
99C23×C5—42no—
100C23×C5C2×D411yescharacteristic
101C22×D5—21no—
102C5×Q8—22no—
103C5×D4—22no—
104C5×D4—22no—
105C5×D4—42no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5C2×D411yescharacteristic
110C2×C4×C5C2411yescharacteristic
111C22⋊D5—42no—
112C2×Dic5—22no—
113C2×Dic5—21no—
114C2×Dic5—22no—
115C2×Dic5—41no—
116C2×Dic5—42no—
117C2×Dic5C2×D411yescharacteristic
118C2×Dic5C2×D411yescharacteristic
119C2×Dic5C2×D411yescharacteristic
120C2×Dic5C2×D411yescharacteristic
121D20—22no—
122C4×D5—42no—
123Dic10—22no—
124Dic10—22no—
125Dic10—42no—
126C5×C8—22no—
127#168—22no—
128C22×Q8—51no—
129C22×D4C2×D511yescharacteristic
130C2×Q16—52no—
131C2×SD16—52no—
132C2×SD16C2×D512yes—
133C2×D8—52no—
134#126—101no—
135#125—51no—
136#124—51no—
137#124—52no—
138#124—52no—
139#124—101no—
140#124—101no—
141#123—101no—
142C22⋊Q8C2×D511yescharacteristic
143#121—51no—
144C4×Q8—51no—
145C4×D4—51no—
146C4×D4—101no—
147#117—101no—
148C2×C22⋊C4—101no—
149C2×C22⋊C4—101no—
150C2×C42—51no—
151C4⋊C8—52no—
152#103—52no—
153#103—52no—
154#102—52no—
155#102C2×D512yes—
156C22⋊C8—51no—
157C22⋊C8C2×D511yescharacteristic
158C4×C8—52no—
159C22×C5—21no—
160C22×C5—21no—
161C22×C5—42no—
162C22×C5—42no—
163C22×C5—42no—
164C22×C5—84no—
165C22×C5#13512yes—
166C22×C5C22×D411yescharacteristic
167C2×D5—41no—
168C2×D5—41no—
169C2×D5—41no—
170C4×C5—21no—
171C4×C5—41no—
172C4×C5—41no—
173C4×C5H(5,𝔽2)12yes—
174Dic5—21no—
175Dic5—21no—
176Dic5—21no—
177Dic5—21no—
178Dic5—42no—
179Dic5—41no—
180C24—21no—
181C2×Q8—52no—
182C2×Q8—101no—
183C2×Q8—202no—
184C2×Q8C22×D511yescharacteristic
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—51no—
188C2×D4—101no—
189C2×D4C22×D512yes—
190C22×C4—51no—
191C22×C4—51no—
192C22×C4—101no—
193C22×C4—101no—
194C22×C4C22×D511yescharacteristic
195Q16—202no—
196SD16—42no—
197SD16—202no—
198D8—202no—
199C2×C8—52no—
200C2×C8C22×D512yes—
201C4⋊C4—21no—
202C4⋊C4—51no—
203C4⋊C4—101no—
204C4⋊C4—101no—
205C4⋊C4—101no—
206C4⋊C4C22×D511yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—101no—
209C22⋊C4—101no—
210C22⋊C4—101no—
211C22⋊C4—102no—
212C22⋊C4—102no—
213C22⋊C4—102no—
214C22⋊C4—102no—
215C22⋊C4—101no—
216C42—51no—
217C42—51no—
218C42—51no—
219C42—51no—
220C42—101no—
221C2×C5—22no—
222C2×C5—82no—
223C2×C5#53511yescharacteristic
224C2×C5C2×#13511yescharacteristic
225C2×C5#57511yescharacteristic
226D5—81no—
227C23—22no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—101no—
232C23D4×D511yescharacteristic
233Q8—22no—
234Q8—102no—
235Q8—102no—
236Q8—202no—
237D4—22no—
238D4—22no—
239D4—42no—
240D4—102no—
241D4—202no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—51no—
246C2×C4—51no—
247C2×C4—51no—
248C2×C4—51no—
249C2×C4—102no—
250C2×C4—101no—
251C2×C4—102no—
252C2×C4—202no—
253C2×C4—201no—
254C2×C4—202no—
255C2×C4D4×D511yescharacteristic
256C2×C4C23×D511yescharacteristic
257C8—22no—
258C8—102no—
259C5#321111yescharacteristic, complemented, Sylow
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—201no—
267C22—201no—
268C22—201no—
269C22#406212yes—
270C22C2×D4×D511yescharacteristic
271C4—21no—
272C4—41no—
273C4—41no—
274C4—101no—
275C4—101no—
276C4—101no—
277C4—101no—
278C4—201no—
279C4—202no—
280C4#414412yes—
281C2—22no—
282C2—82no—
283C2—401no—
284C2#6592611yescharacteristic
285C2C2×#406211yescharacteristic
286C2#6610711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#450C2
socleC22×C5C22×D4
prime radical#65506C2
hypercenter#450D5
hypocenterC5#3211
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#4082C2semidirect
2. #65958C2semidirect
3. C5×#450C2semidirect
5. #65457C2semidirect
6. #65455C2semidirect
7. #65339C2semidirect
8. #65076C2semidirect
9. #65057C2semidirect
10. #65027C2semidirect
11. #65007C2semidirect
18. #4093C22semidirect
19. Q8×Dic5C22semidirect
20. C2×#689C22semidirect
28. C2×#677C22semidirect
30. #4039C22semidirect
32. #4036C22semidirect
39. #3948C22semidirect
40. #3946C22semidirect
41. C8×Dic5C22semidirect
42. #3942C22semidirect
92. #450D5semidirect
259. C5#3211semidirect

Maximal subgroups

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

subgroupindex
C2×#40822
#659582
C5×#4502
#655062
#654572
#654552
#653392
#650762
#650572
#650272
#650072
#32115