Finite Group Databasebeta

← #10613310

Subgroups — #10613310

Order 640.

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

normal subgroups only

This group has 408 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#66068C211yesmaximal, characteristic, complemented
2#66052C211yesmaximal, characteristic, complemented
3C5×#486C211yesmaximal, characteristic, complemented
4#65380C211yesmaximal, characteristic, complemented
5#65361C211yesmaximal, characteristic, complemented
6#65070C212yesmaximal, complemented
7#65068C212yesmaximal, complemented
8#65064C212yesmaximal, complemented
9#65061C212yesmaximal, complemented
10C4×#677C211yesmaximal, characteristic, complemented
11C8×D20C211yesmaximal, characteristic, complemented
12C2×#711C2212yes—
13C5×C4⋊Q8C2211yescharacteristic
14C5×#127C2211yescharacteristic
15#4093—22no—
16#4086—21no—
17#4085—22no—
18#4084C2212yes—
19D4×Dic5—21no—
20C2×#686C2212yes—
21C2×#677C2211yescharacteristic
22C2×C8×D5C2212yes—
23#4041—21no—
24#4041C2212yes—
25#4038—21no—
26#4038C2212yes—
27C4×D20C2211yescharacteristic
28C4×D20C2211yescharacteristic
29C4×Dic10C2211yescharacteristic
30C5×#102C2212yes—
31C5×#102C2212yes—
32C4×C5×C8C2211yescharacteristic
33#3964C2212yes—
34#3953C2211yescharacteristic
35#3952C2212yes—
36#3949C2211yescharacteristic
37#3948C2211yescharacteristic
38#3942C2212yes—
39#3939C2212yes—
40C4⋊#168C2211yescharacteristic
41#3305—51nomaximal, Sylow
42C2×C5×Q8—21no—
43C2×C5×D4—21no—
44C2×C5×D4—21no—
45C2×C5×D4C2312yes—
46C2×C22⋊D5—22no—
47C2×C22⋊D5—22no—
48C22×Dic5—22no—
49#711—24no—
50#711—42no—
51C2×D20—22no—
52C2×D20D411yescharacteristic
53C2×D20C2311yescharacteristic
54C2×C4×D5—21no—
55C2×C4×D5C2312yes—
56C2×Dic10C2311yescharacteristic
57C2×C5×C8C2311yescharacteristic
58C2×C5×C8C2311yescharacteristic
59C5×C4⋊C4—21no—
60C5×C4⋊C4C2312yes—
61C42×C5C2311yescharacteristic
62C22⋊Dic5—22no—
63#686—42no—
64#685—22no—
65#685—21no—
66#685—22no—
67#685D412yes—
68C4⋊Dic5D411yescharacteristic
69C4⋊Dic5C2311yescharacteristic
70C4⋊Dic5C2311yescharacteristic
71#683—21no—
72C4×Dic5—21no—
73C4×Dic5—22no—
74C2×#168C2312yes—
75#677—22no—
76C8×D5—42no—
77#555—51no—
78#550—51no—
79#486D511yescharacteristic, complemented
80#482—52no—
81#477—51no—
82#466—52no—
83#459—51no—
84#449—52no—
85C4×SD16—51no—
86C8×D4—51no—
87#416—52no—
88C23×C5—22no—
89C22×D5—21no—
90C22×D5C2×D412yes—
91C5×Q8—81no—
92C5×D4—22no—
93C5×D4—22no—
94C5×D4—42no—
95C5×D4—81no—
96C2×C4×C5—22no—
97C2×C4×C5C2×D411yescharacteristic
98C2×C4×C5C2×D411yescharacteristic
99C2×C4×C5C2411yescharacteristic
100C22⋊D5—44no—
101C22⋊D5—82no—
102C2×Dic5—21no—
103C2×Dic5—21no—
104C2×Dic5—21no—
105C2×Dic5—44no—
106C2×Dic5C2×D412yes—
107D20—22no—
108D20—21no—
109D20—21no—
110D20—82no—
111C4×D5—22no—
112C4×D5—22no—
113C4×D5—42no—
114Dic10—21no—
115Dic10#4112yes—
116C5×C8—21no—
117C5×C8—21no—
118#168—22no—
119C2×#41—52no—
120C2×SD16—51no—
121C2×D8—52no—
122C22×C8—52no—
123C4⋊Q8C2×D511yescharacteristic
124#127—101no—
125#127C2×D511yescharacteristic
126#124—102no—
127#121—52no—
128#121—52no—
129#121—102no—
130#121—101no—
131C4×Q8—51no—
132C4×D4—51no—
133C4×D4—51no—
134C4×D4—101no—
135#117—52no—
136#117—101no—
137#107—52no—
138#106—51no—
139C4⋊C8—51no—
140#103—51no—
141#103—52no—
142#102—52no—
143#102—51no—
144#102C2×D512yes—
145#102C2×D512yes—
146C22⋊C8—52no—
147C4×C8C2×D511yescharacteristic
148C22×C5—42no—
149C22×C5—42no—
150C22×C5—42no—
151C22×C5C22×D411yescharacteristic
152C2×D5—22no—
153C2×D5—22no—
154C2×D5—22no—
155C2×D5—41no—
156C2×D5—41no—
157C2×D5—41no—
158C4×C5—21no—
159C4×C5—42no—
160C4×C5#13512yes—
161C4×C5C2×#4111yescharacteristic
162C4×C5H(5,𝔽2)11yescharacteristic
163Dic5—22no—
164Dic5—22no—
165Dic5—41no—
166Dic5—41no—
167#41—104no—
168#41—202no—
169C2×Q8—21no—
170C2×Q8—51no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—51no—
174C2×D4—51no—
175C2×D4—102no—
176C2×D4—102no—
177C2×D4—102no—
178C2×D4C22×D512yes—
179C22×C4—52no—
180C22×C4—102no—
181C22×C4—101no—
182SD16—102no—
183D8—202no—
184C2×C8—52no—
185C2×C8—202no—
186C2×C8C22×D511yescharacteristic
187C2×C8C22×D511yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—51no—
190C4⋊C4—51no—
191C4⋊C4—51no—
192C4⋊C4—101no—
193C4⋊C4C22×D512yes—
194C22⋊C4—52no—
195C22⋊C4—102no—
196C22⋊C4—102no—
197C22⋊C4—102no—
198C22⋊C4—101no—
199C42—102no—
200C42—101no—
201C42C22×D511yescharacteristic
202C2×C5—82no—
203C2×C5#54611yescharacteristic
204C2×C5C2×#13511yescharacteristic
205C2×C5#57611yescharacteristic
206D5—42no—
207D5—81no—
208C23—22no—
209C23—52no—
210C23—101no—
211Q8—52no—
212Q8—81no—
213Q8—101no—
214D4—22no—
215D4—22no—
216D4—42no—
217D4—81no—
218D4—101no—
219D4—101no—
220D4—102no—
221D4—204no—
222D4—402no—
223D4—402no—
224C2×C4—22no—
225C2×C4—52no—
226C2×C4—101no—
227C2×C4—102no—
228C2×C4—102no—
229C2×C4—101no—
230C2×C4—101no—
231C2×C4—204no—
232C2×C4—202no—
233C2×C4D4×D511yescharacteristic
234C2×C4D4×D511yescharacteristic
235C2×C4C23×D511yescharacteristic
236C8—21no—
237C8—21no—
238C8—102no—
239C5#330511yescharacteristic, complemented, Sylow
240C22—42no—
241C22—42no—
242C22—42no—
243C22—102no—
244C22—102no—
245C22—102no—
246C22—201no—
247C22—201no—
248C22—201no—
249C22C2×D4×D511yescharacteristic
250C4—21no—
251C4—42no—
252C4—102no—
253C4—102no—
254C4—201no—
255C4—201no—
256C4#406212yes—
257C4D5×#4111yescharacteristic
258C4#414911yescharacteristic
259C2—82no—
260C2—202no—
261C2—401no—
262C2#6601011yescharacteristic
263C2#6609111yescharacteristic
264C2C2×#406211yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#486C2
socleC22×C5C22×D4
hypercenter#486D5
hypocenterC5#3305
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #66068C2semidirect
2. #66052C2semidirect
3. C5×#486C2semidirect
4. #65380C2semidirect
5. #65361C2semidirect
6. #65070C2semidirect
7. #65068C2semidirect
8. #65064C2semidirect
9. #65061C2semidirect
10. C4×#677C2semidirect
11. C8×D20C2semidirect
79. #486D5semidirect
239. C5#3305semidirect

Maximal subgroups

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

subgroupindex
#660682
#660522
C5×#4862
#653802
#653612
#650702
#650682
#650642
#650612
C4×#6772
C8×D202
#33055