Finite Group Databasebeta

← #10604615

Subgroups — #10604615

Order 640.

362 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 66 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#66258C211yesmaximal, characteristic, complemented
2#66049C211yesmaximal, characteristic
3D5⋊#100C211yesmaximal, characteristic, complemented
4#65769C212yesmaximal, complemented
5#64863C212yesmaximal, complemented
6C2×#711C2211yescharacteristic
7C2×D4×D5C2211yescharacteristic
8C2×C22⋊AGL(1,𝔽5)—22no—
9D4×AGL(1,𝔽5)—24no—
10#4130C2212yes—
11D5⋊M16C2212yes—
12C5×#127C411yescharacteristic, complemented
13#4086C411yescharacteristic, complemented
14#4085—22no—
15#4083—22no—
16#4082C411yescharacteristic, complemented
17#4021C411yescharacteristic, complemented
18#4018C2211yescharacteristic
19#4013—22no—
20#4008—24no—
21#1911—51nomaximal, Sylow
22C23×D5D412yes—
23C22×AGL(1,𝔽5)—24no—
24C2×C5×D4—22no—
25C2×C5×D4C2×C411yescharacteristic
26C2×C5×D4C2×C411yescharacteristic, complemented
27C2×C22⋊D5—21no—
28C2×C22⋊D5—22no—
29C2×C22⋊D5D412yes—
30C22×Dic5—21no—
31#711—22no—
32#711—22no—
33D4×D5—22no—
34D4×D5—22no—
35C2×D20C2×C411yescharacteristic
36C2×C4×D5C2311yescharacteristic
37C2×Dic10C2×C411yescharacteristic, complemented
38C22⋊AGL(1,𝔽5)—24no—
39C22⋊AGL(1,𝔽5)—22no—
40#704—22no—
41C4⋊AGL(1,𝔽5)—22no—
42C4×AGL(1,𝔽5)D414yescomplemented
43#700—22no—
44D5⋊C8—22no—
45C42×C5C2×C411yescharacteristic, complemented
46C22⋊Dic5—22no—
47#685—22no—
48#685—21no—
49#683—21no—
50C4×Dic5C2×C411yescharacteristic, complemented
51#535—51no—
52#518—51no—
53#421—52no—
54C2×#100—51no—
55H(3,ℤ/4)—52no—
56C23×C5—21no—
57C23×C5C22⋊C412yes—
58C22×D5—22no—
59C22×D5—22no—
60C22×D5—82no—
61C22×D5C22⋊C412yes—
62C22×D5C2×D411yescharacteristic
63C2×AGL(1,𝔽5)—24no—
64C2×AGL(1,𝔽5)—84no—
65C5×D4—22no—
66C5×D4—22no—
67C5×D4—82no—
68C2×C4×C5—21no—
69C2×C4×C5C22×C411yescharacteristic
70C22⋊D5—24no—
71C22⋊D5—41no—
72C22⋊D5—41no—
73C22⋊D5—82no—
74C2×Dic5—21no—
75C2×Dic5—21no—
76C2×Dic5—42no—
77C2×Dic5C2×D411yescharacteristic
78D20—22no—
79C4×D5C2×D412yes—
80C4×D5#4112yes—
81Dic10—22no—
82#170—42no—
83C2×#41—51no—
84C22×D4—51no—
85C2×M16—52no—
86#127—51no—
87#127AGL(1,𝔽5)11yescharacteristic, complemented
88#124—51no—
89#124—51no—
90#121—102no—
91#120—102no—
92C4×D4—104no—
93#117—52no—
94#117—51no—
95C2×C22⋊C4—102no—
96#104—104no—
97#100—102no—
98C22×C5—22no—
99C22×C5—42no—
100C22×C5—42no—
101C22×C5—41no—
102C22×C5C2×C22⋊C411yescharacteristic
103C2×D5—22no—
104C2×D5—41no—
105C2×D5—44no—
106C2×D5—42no—
107C2×D5—82no—
108C2×D5#12011yescharacteristic
109C2×D5#12311yescharacteristic
110AGL(1,𝔽5)—44no—
111C4×C5—41no—
112C4×C5C4×D412yes—
113Dic5—41no—
114Dic5—41no—
115Dic5#12112yes—
116C24—52no—
117#41—102no—
118#41—102no—
119C2×Q8—51no—
120C2×D4—22no—
121C2×D4—52no—
122C2×D4—51no—
123C2×D4—102no—
124C2×D4—102no—
125C2×D4—101no—
126C2×D4—102no—
127C2×D4C2×AGL(1,𝔽5)11yescharacteristic
128C2×D4C2×AGL(1,𝔽5)11yescharacteristic, complemented
129C22×C4—51no—
130C22×C4—101no—
131C22×C4—104no—
132M16—102no—
133M16—102no—
134C2×C8—102no—
135C4⋊C4—101no—
136C4⋊C4—102no—
137C22⋊C4—101no—
138C22⋊C4—102no—
139C22⋊C4—102no—
140C22⋊C4—104no—
141C22⋊C4—102no—
142C42—54no—
143C42—51no—
144C42C2×AGL(1,𝔽5)11yescharacteristic, complemented
145C2×C5—21no—
146C2×C5—42no—
147C2×C5—81no—
148C2×C5#38611yescharacteristic
149D5—21no—
150D5—21no—
151D5—42no—
152C23—21no—
153C23—51no—
154C23—52no—
155C23—102no—
156C23—102no—
157C23—402no—
158C23C22⋊AGL(1,𝔽5)12yes—
159Q8—102no—
160D4—22no—
161D4—22no—
162D4—82no—
163D4—104no—
164D4—102no—
165D4—201no—
166D4—201no—
167D4—402no—
168C2×C4—21no—
169C2×C4—51no—
170C2×C4—52no—
171C2×C4—52no—
172C2×C4—104no—
173C2×C4—101no—
174C2×C4—101no—
175C2×C4—202no—
176C2×C4—404no—
177C2×C4C22×AGL(1,𝔽5)11yescharacteristic
178C8—202no—
179C5#191111yescharacteristic, complemented, Sylow
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—51no—
185C22—51no—
186C22—102no—
187C22—201no—
188C22—202no—
189C22—204no—
190C22—402no—
191C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
192C4—41no—
193C4—52no—
194C4—201no—
195C4—204no—
196C4—201no—
197C4D4×AGL(1,𝔽5)12yes—
198C2—21no—
199C2—42no—
200C2—81no—
201C2—101no—
202C2—101no—
203C2—202no—
204C2#6577911yescharacteristic
subgroupG/H
centerC2#65779
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#127C4
socleC2×C5#386
prime radical#66049C2
cosocleC2×C4×D5C23
hypercenter#127AGL(1,𝔽5)
hypocenterC5#1911
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #66258C2semidirect
3. D5⋊#100C2semidirect
4. #65769C2semidirect
5. #64863C2semidirect
12. C5×#127C4semidirect
13. #4086C4semidirect
16. #4082C4semidirect
17. #4021C4semidirect
26. C2×C5×D4C2×C4semidirect
37. C2×Dic10C2×C4semidirect
42. C4×AGL(1,𝔽5)D4semidirect
45. C42×C5C2×C4semidirect
50. C4×Dic5C2×C4semidirect
87. #127AGL(1,𝔽5)semidirect
128. C2×D4C2×AGL(1,𝔽5)semidirect
144. C42C2×AGL(1,𝔽5)semidirect
179. C5#1911semidirect

Maximal subgroups

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

subgroupindex
#662582
#660492
D5⋊#1002
#657692
#648632
#19115