Finite Group Databasebeta

← #10611039

Subgroups — #10611039

Order 640.

400 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 121 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#65813C211yesmaximal, characteristic, complemented
2#65811C211yesmaximal, characteristic, complemented
3C5×#431C211yesmaximal, characteristic, complemented
4#65138C211yesmaximal, characteristic, complemented
5#65137C211yesmaximal, characteristic, complemented
6#65132C211yesmaximal, characteristic, complemented
7#65130C211yesmaximal, characteristic, complemented
8#65023C212yesmaximal, complemented
9#65022C212yesmaximal, complemented
10#64986C212yesmaximal, complemented
11#64985C212yesmaximal, complemented
12C2×#709C2212yes—
13C2×C42×C5C2211yescharacteristic
14#4076—21no—
15#4076—21no—
16#4076C2211yescharacteristic
17#4075—21no—
18C4×C22⋊D5—21no—
19#4070C2211yescharacteristic
20C2×Dic20C2212yes—
21C2×D40C2212yes—
22C2×#677C2212yes—
23C2×#677C2212yes—
24#4022—21no—
25#4021C2212yes—
26#4021C2211yescharacteristic
27#4020C2211yescharacteristic
28C4×D20—21no—
29C4×D20—21no—
30C4×D20C2211yescharacteristic
31#4018—21no—
32#4016C2211yescharacteristic
33#4015C2211yescharacteristic
34C4×Dic10C2211yescharacteristic
35C5×C4⋊C8C2211yescharacteristic
36C5×C4⋊C8C2211yescharacteristic
37C5×C22⋊C8C2212yes—
38C4×C5×C8C2212yes—
39#3953C2212yes—
40#3953C2212yes—
41#3948C2212yes—
42#3948C2212yes—
43#3212—51nomaximal, Sylow
44C22×C4×C5D411yescharacteristic
45C22×C4×C5D411yescharacteristic
46C22×C4×C5C2311yescharacteristic
47C2×C22⋊D5—22no—
48C2×C22⋊D5—21no—
49#709—42no—
50#709—42no—
51C2×D20—21no—
52C2×D20—21no—
53C2×D20C2312yes—
54C2×D20C2311yescharacteristic
55C2×C4×D5—22no—
56C2×C4×D5—21no—
57C2×Dic10C2312yes—
58C2×Dic10C2311yescharacteristic
59C2×C5×C8C2312yes—
60C2×C5×C8C2312yes—
61C42×C5D411yescharacteristic
62C42×C5D411yescharacteristic
63C42×C5C2311yescharacteristic
64C42×C5C2311yescharacteristic
65C22⋊Dic5—21no—
66#685—21no—
67#685—21no—
68#685—22no—
69#685—22no—
70#685—21no—
71#685—21no—
72C4⋊Dic5—21no—
73C4⋊Dic5—21no—
74C4⋊Dic5C2311yescharacteristic
75C4⋊Dic5C2311yescharacteristic
76#683—21no—
77#683—21no—
78#683—21no—
79#683—21no—
80C4×Dic5—21no—
81Dic20—42no—
82D40—42no—
83#677—42no—
84#677—42no—
85#532—51no—
86#529—51no—
87#495—52no—
88#488—52no—
89#462—51no—
90#461—51no—
91#460—51no—
92#459—51no—
93#452—52no—
94#449—52no—
95#431D511yescharacteristic, complemented
96C23×C5C2×D411yescharacteristic
97C22×D5—21no—
98C22×D5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—21no—
101C2×C4×C5—21no—
102C2×C4×C5—41no—
103C2×C4×C5C2×D411yescharacteristic
104C2×C4×C5C2×D411yescharacteristic
105C2×C4×C5C2×D411yescharacteristic
106C2×C4×C5C2×D411yescharacteristic
107C2×C4×C5C2×D411yescharacteristic
108C2×C4×C5C2411yescharacteristic
109C22⋊D5—44no—
110C22⋊D5—42no—
111C2×Dic5—21no—
112C2×Dic5—21no—
113C2×Dic5—21no—
114C2×Dic5—22no—
115D20—21no—
116D20—22no—
117D20—21no—
118D20—22no—
119D20—42no—
120D20—42no—
121C4×D5—44no—
122C4×D5—42no—
123Dic10—21no—
124Dic10—22no—
125Dic10—21no—
126Dic10—22no—
127C5×C8—22no—
128C5×C8—22no—
129C2×#41—52no—
130C2×Q16—52no—
131C2×SD16—52no—
132C2×SD16—52no—
133C2×D8—52no—
134C4⋊Q8—51no—
135#127—51no—
136#126—101no—
137#125—51no—
138#124—51no—
139#124—52no—
140#123—101no—
141C22⋊Q8—51no—
142#121—51no—
143#121—101no—
144#121—101no—
145C4×Q8—51no—
146C4×D4—51no—
147C4×D4—101no—
148C4×D4—101no—
149C4×D4—101no—
150#117—101no—
151C2×C42C2×D511yescharacteristic
152C4⋊C8C2×D511yescharacteristic
153C4⋊C8C2×D511yescharacteristic
154#103—52no—
155#103—52no—
156#102—52no—
157#102—52no—
158C22⋊C8C2×D512yes—
159C4×C8C2×D512yes—
160C22×C5—21no—
161C22×C5—21no—
162C22×C5—21no—
163C22×C5C22×D411yescharacteristic
164C2×D5—42no—
165C2×D5—41no—
166C2×D5—41no—
167C2×D5—42no—
168C2×D5—42no—
169C2×D5—41no—
170C4×C5—21no—
171C4×C5—21no—
172C4×C5—21no—
173C4×C5—21no—
174C4×C5#13512yes—
175C4×C5H(5,𝔽2)11yescharacteristic
176C4×C5H(5,𝔽2)11yescharacteristic
177Dic5—42no—
178Dic5—41no—
179Dic5—41no—
180Dic5—41no—
181#41—202no—
182#41—202no—
183C2×Q8—52no—
184C2×Q8—51no—
185C2×D4—52no—
186C2×D4—51no—
187C2×D4—101no—
188C2×D4—102no—
189C2×D4—101no—
190C2×D4—101no—
191C22×C4—101no—
192C22×C4—102no—
193C22×C4D2011yescharacteristic
194C22×C4D2011yescharacteristic
195C22×C4C22×D511yescharacteristic
196Q16—202no—
197SD16—202no—
198SD16—202no—
199D8—202no—
200C2×C8C22×D512yes—
201C2×C8C22×D512yes—
202C4⋊C4—51no—
203C4⋊C4—51no—
204C4⋊C4—101no—
205C4⋊C4—101no—
206C4⋊C4—101no—
207C4⋊C4—101no—
208C4⋊C4—101no—
209C4⋊C4—101no—
210C22⋊C4—101no—
211C22⋊C4—101no—
212C22⋊C4—101no—
213C22⋊C4—101no—
214C22⋊C4—102no—
215C22⋊C4—102no—
216C22⋊C4—101no—
217C42—101no—
218C42D2011yescharacteristic
219C42D2011yescharacteristic
220C42C22×D511yescharacteristic
221C42C22×D511yescharacteristic
222C2×C5—41no—
223C2×C5#53511yescharacteristic
224C2×C5C2×#13511yescharacteristic
225C2×C5#57511yescharacteristic
226D5—82no—
227D5—81no—
228C23—101no—
229C23—102no—
230C23C2×D2011yescharacteristic
231Q8—101no—
232Q8—102no—
233Q8—101no—
234Q8—102no—
235D4—101no—
236D4—102no—
237D4—101no—
238D4—102no—
239D4—202no—
240D4—204no—
241D4—202no—
242D4—202no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—101no—
248C2×C4—101no—
249C2×C4—101no—
250C2×C4—102no—
251C2×C4—204no—
252C2×C4—202no—
253C2×C4C2×D2011yescharacteristic
254C2×C4C2×D2011yescharacteristic
255C2×C4C2×D2011yescharacteristic
256C2×C4C2×D2011yescharacteristic
257C2×C4C2×D2011yescharacteristic
258C2×C4C23×D511yescharacteristic
259C8—22no—
260C8—22no—
261C5#321211yescharacteristic, complemented, Sylow
262C22—21no—
263C22—21no—
264C22—21no—
265C22—201no—
266C22—202no—
267C22—202no—
268C22—201no—
269C22—202no—
270C22—201no—
271C22C22×D2011yescharacteristic
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—201no—
277C4—201no—
278C4—202no—
279C4—201no—
280C4#405012yes—
281C4#414911yescharacteristic
282C4#414911yescharacteristic
283C2—41no—
284C2—401no—
285C2—402no—
286C2#6585911yescharacteristic
287C2C2×#405011yescharacteristic
288C2#6608211yescharacteristic
subgroupG/H
centerC22C22×D20
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#431C2
socleC22×C5C22×D4
hypercenter#431D5
hypocenterC5#3212
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #65813C2semidirect
2. #65811C2semidirect
3. C5×#431C2semidirect
4. #65138C2semidirect
5. #65137C2semidirect
6. #65132C2semidirect
7. #65130C2semidirect
8. #65023C2semidirect
9. #65022C2semidirect
10. #64986C2semidirect
11. #64985C2semidirect
95. #431D5semidirect
261. C5#3212semidirect

Maximal subgroups

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

subgroupindex
#658132
#658112
C5×#4312
#651382
#651372
#651322
#651302
#650232
#650222
#649862
#649852
#32125