Finite Group Databasebeta

← #10612862

Subgroups — #10612862

Order 640.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#66059C211yesmaximal, characteristic, complemented
2#65914C211yesmaximal, characteristic, complemented
3C5×#477C211yesmaximal, characteristic, complemented
4#65373C211yesmaximal, characteristic, complemented
5#65316C211yesmaximal, characteristic, complemented
6#65180C212yesmaximal, complemented
7#65172C212yesmaximal, complemented
8#65132C211yesmaximal, characteristic, complemented
9D5×C4⋊C8C211yesmaximal, characteristic, complemented
10#65108C212yesmaximal, complemented
11#65096C212yesmaximal, complemented
12C2×#713—21no—
13C5×C4⋊Q8C2211yescharacteristic
14C4×C5×Q8C2211yescharacteristic
15#4093—21no—
16#4093C2211yescharacteristic
17#4092C2211yescharacteristic
18C2×#688C2212yes—
19C2×D40C2212yes—
20C2×C8×D5C2212yes—
21#4044—21no—
22#4041—21no—
23#4041C2212yes—
24#4041C2211yescharacteristic
25#4040—21no—
26#4039—21no—
27#4039C2212yes—
28#4036C2211yescharacteristic
29#4020C2211yescharacteristic
30C4×D20—21no—
31C4×D20C2211yescharacteristic
32#4018—21no—
33C42×D5C2211yescharacteristic
34C5×#107C2212yes—
35C5×C4⋊C8C2211yescharacteristic
36C5×#103C2212yes—
37#3953C2212yes—
38#3946C2212yes—
39#3941C2212yes—
40#3941C2212yes—
41#3940C2212yes—
42C4⋊#168C2211yescharacteristic
43#3247—51nomaximal, Sylow
44C2×C5×Q8—21no—
45C2×C5×Q8C2311yescharacteristic
46#713—44no—
47C2×D20—22no—
48C2×D20—21no—
49C2×D20—21no—
50C2×D20—21no—
51C2×D20C2311yescharacteristic
52C2×D20C2312yes—
53C2×C4×D5—21no—
54C2×C4×D5—22no—
55C2×C4×D5D411yescharacteristic
56C2×C4×D5D411yescharacteristic
57C2×C4×D5C2311yescharacteristic
58C2×C5×C8C2312yes—
59C5×C4⋊C4—21no—
60C5×C4⋊C4—21no—
61C5×C4⋊C4C2311yescharacteristic
62C5×C4⋊C4C2312yes—
63C42×C5—21no—
64C42×C5C2311yescharacteristic
65#688—42no—
66#685—21no—
67#685—21no—
68#685—21no—
69#685—21no—
70#685—21no—
71#685—22no—
72#685—22no—
73C4⋊Dic5—21no—
74C4⋊Dic5C2311yescharacteristic
75#683—21no—
76#683—21no—
77C4×Dic5D411yescharacteristic
78C4×Dic5D411yescharacteristic
79C4×Dic5C2311yescharacteristic
80C2×#168C2312yes—
81D40—42no—
82C8×D5—42no—
83#532—51no—
84#529—51no—
85#478—52no—
86#477D511yescharacteristic, complemented
87#476—51no—
88#466—52no—
89#465—52no—
90#463—52no—
91#460—51no—
92#459—51no—
93C2×C4⋊C8—51no—
94C22×D5—21no—
95C22×D5—22no—
96C22×D5C2×D411yescharacteristic
97C5×Q8—21no—
98C5×Q8—21no—
99C5×Q8—42no—
100C2×C4×C5—21no—
101C2×C4×C5—21no—
102C2×C4×C5—22no—
103C2×C4×C5C2×D411yescharacteristic
104C2×C4×C5C2×D411yescharacteristic
105C2×C4×C5C2411yescharacteristic
106C2×Dic5—21no—
107C2×Dic5C2×D411yescharacteristic
108C2×Dic5C2×D411yescharacteristic
109C2×Dic5C2×D411yescharacteristic
110D20—21no—
111D20—22no—
112D20—21no—
113D20—22no—
114D20—42no—
115D20—42no—
116D20—44no—
117D20—42no—
118C4×D5—22no—
119C4×D5—22no—
120C4×D5—21no—
121C4×D5—21no—
122C4×D5—44no—
123C4×D5—42no—
124C5×C8—22no—
125#168—22no—
126C2×#41—101no—
127C2×SD16—52no—
128C2×D8—52no—
129C22×C8—52no—
130C4⋊Q8C2×D511yescharacteristic
131#127—51no—
132#126—101no—
133#125—51no—
134#124—51no—
135#124—101no—
136#123—101no—
137C22⋊Q8—51no—
138#121—51no—
139#121—52no—
140#121—101no—
141C4×Q8C2×D511yescharacteristic
142C4×D4—52no—
143C4×D4—51no—
144C4×D4—101no—
145C4×D4—101no—
146#117—101no—
147C2×C42—51no—
148#107C2×D512yes—
149#106—52no—
150C4⋊C8—52no—
151C4⋊C8—51no—
152C4⋊C8C2×D511yescharacteristic
153#103C2×D512yes—
154#102—52no—
155#102—52no—
156#102—52no—
157C22×C5C22×D411yescharacteristic
158C2×D5—21no—
159C2×D5—21no—
160C2×D5—41no—
161C2×D5—42no—
162C2×D5—41no—
163C2×D5—42no—
164C2×D5—42no—
165C2×D5—41no—
166C2×D5AGL(1,ℤ/8)12yes—
167C4×C5—21no—
168C4×C5—41no—
169C4×C5—42no—
170C4×C5—41no—
171C4×C5#13512yes—
172C4×C5H(5,𝔽2)11yescharacteristic
173C4×C52−1+411yescharacteristic
174Dic5—21no—
175Dic5—21no—
176Dic5—21no—
177Dic5—41no—
178#41—204no—
179C2×Q8—21no—
180C2×Q8C22×D511yescharacteristic
181C2×D4—51no—
182C2×D4—52no—
183C2×D4—101no—
184C2×D4—102no—
185C2×D4—101no—
186C2×D4—101no—
187C22×C4—51no—
188C22×C4—51no—
189C22×C4—51no—
190C22×C4—102no—
191C22×C4—101no—
192SD16—202no—
193D8—202no—
194C2×C8—52no—
195C2×C8—202no—
196C2×C8C22×D512yes—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—51no—
200C4⋊C4—101no—
201C4⋊C4—101no—
202C4⋊C4—101no—
203C4⋊C4C22×D511yescharacteristic
204C4⋊C4C22×D512yes—
205C22⋊C4—101no—
206C22⋊C4—101no—
207C22⋊C4—102no—
208C22⋊C4—102no—
209C22⋊C4—101no—
210C22⋊C4—101no—
211C22⋊C4—101no—
212C42—21no—
213C42—51no—
214C42—51no—
215C42—51no—
216C42C22×D511yescharacteristic
217C2×C5#53711yescharacteristic
218C2×C5C2×#13511yescharacteristic
219C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
220D5—22no—
221D5—81no—
222D5—82no—
223C23—51no—
224C23—101no—
225C23—102no—
226Q8—21no—
227Q8—21no—
228Q8—42no—
229D4—101no—
230D4—102no—
231D4—101no—
232D4—102no—
233D4—202no—
234D4—204no—
235D4—202no—
236D4—202no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—51no—
241C2×C4—51no—
242C2×C4—51no—
243C2×C4—102no—
244C2×C4—101no—
245C2×C4—101no—
246C2×C4—102no—
247C2×C4—101no—
248C2×C4—204no—
249C2×C4—202no—
250C2×C4D4×D511yescharacteristic
251C2×C4D4×D511yescharacteristic
252C2×C4C23×D511yescharacteristic
253C8—22no—
254C8—102no—
255C5#324711yescharacteristic, complemented, Sylow
256C22—52no—
257C22—101no—
258C22—101no—
259C22—201no—
260C22—202no—
261C22—202no—
262C22—201no—
263C22—201no—
264C22—202no—
265C22C2×D4×D511yescharacteristic
266C4—21no—
267C4—42no—
268C4—41no—
269C4—41no—
270C4—101no—
271C4—101no—
272C4—101no—
273C4—201no—
274C4#406512yes—
275C4#414711yescharacteristic
276C4#414911yescharacteristic
277C2—102no—
278C2—401no—
279C2—402no—
280C2#6596311yescharacteristic
281C2C2×#406511yescharacteristic
282C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#477C2
socleC22×C5C22×D4
hypercenter#477D5
hypocenterC5#3247
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #66059C2semidirect
2. #65914C2semidirect
3. C5×#477C2semidirect
4. #65373C2semidirect
5. #65316C2semidirect
6. #65180C2semidirect
7. #65172C2semidirect
8. #65132C2semidirect
9. D5×C4⋊C8C2semidirect
10. #65108C2semidirect
11. #65096C2semidirect
86. #477D5semidirect
255. C5#3247semidirect

Maximal subgroups

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

subgroupindex
#660592
#659142
C5×#4772
#653732
#653162
#651802
#651722
#651322
D5×C4⋊C82
#651082
#650962
#32475