Finite Group Databasebeta

← #10615087

Subgroups — #10615087

Order 640.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C5×#533C211yesmaximal, characteristic, complemented
2#65811C211yesmaximal, characteristic
3#65382C211yesmaximal, characteristic, complemented
4#65381C211yesmaximal, characteristic, complemented
5#65377C211yesmaximal, characteristic, complemented
6#65374C211yesmaximal, characteristic, complemented
7#65336C214yesmaximal, complemented
8#65317C214yesmaximal, complemented
9#65214C211yesmaximal, characteristic, complemented
10C2×#709C2212yes—
11C5×C4⋊Q8C2211yescharacteristic
12C5×C4⋊Q8C2211yescharacteristic
13C5×#125—21no—
14C5×C22⋊Q8—21no—
15C5×C22⋊Q8C2212yes—
16C4×C5×Q8—21no—
17C4×C5×Q8C2212yes—
18C5×#117—21no—
19C2×C42×C5C2211yescharacteristic
20C2×#689C2214yes—
21C2×#688C2214yes—
22#4076—21no—
23#4076—21no—
24#4021C2212yes—
25#4020C2211yescharacteristic
26C4×D20—21no—
27C4×D20—21no—
28#4015C2211yescharacteristic
29C22⋊#168C2212yes—
30#3942C2214yes—
31#3941C2214yes—
32C4⋊#168C2212yes—
33C4×#168C2211yescharacteristic
34C4×#168C2211yescharacteristic
35#3210—51nomaximal, Sylow
36C2×C5×Q8—21no—
37C2×C5×Q8C2312yes—
38C22×C4×C5D411yescharacteristic
39C22×C4×C5D411yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C2×C22⋊D5—22no—
42#709—44no—
43C2×D20—21no—
44C2×D20—21no—
45C2×D20C2312yes—
46C2×C4×D5—22no—
47C2×Dic10C2312yes—
48C5×C4⋊C4—21no—
49C5×C4⋊C4—21no—
50C5×C4⋊C4—21no—
51C5×C4⋊C4—22no—
52C5×C4⋊C4—22no—
53C5×C4⋊C4C2312yes—
54C5×C22⋊C4—22no—
55C42×C5—22no—
56C42×C5D411yescharacteristic
57C42×C5D411yescharacteristic
58C42×C5C2311yescharacteristic
59C42×C5C2311yescharacteristic
60#689—44no—
61#688—44no—
62#685—22no—
63#685—22no—
64C4⋊Dic5—21no—
65C4⋊Dic5—21no—
66C2×#168C2312yes—
67C2×#168C2312yes—
68#533D511yescharacteristic, complemented
69#532—51no—
70#494—51no—
71#492—51no—
72#487—51no—
73#486—51no—
74#464—54no—
75#452—54no—
76#431—51no—
77C23×C5C2×D411yescharacteristic
78C22×D5—22no—
79C5×Q8—24no—
80C5×Q8—42no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5C2×D411yescharacteristic
87C2×C4×C5C2×D411yescharacteristic
88C2×C4×C5C2×D411yescharacteristic
89C2×C4×C5C2×D411yescharacteristic
90C2×C4×C5C2×D411yescharacteristic
91C2×C4×C5C2411yescharacteristic
92C22⋊D5—44no—
93C2×Dic5—22no—
94D20—24no—
95D20—42no—
96D20—42no—
97C4×D5—44no—
98Dic10—24no—
99#168—22no—
100#168—22no—
101C2×#41—52no—
102C2×Q16—54no—
103C2×SD16—54no—
104C4⋊Q8—51no—
105C4⋊Q8C2×D511yescharacteristic
106C4⋊Q8C2×D511yescharacteristic
107#127—51no—
108#125—21no—
109#124—52no—
110C22⋊Q8—21no—
111C22⋊Q8C2×D512yes—
112#121—101no—
113#121—101no—
114C4×Q8—21no—
115C4×Q8C2×D512yes—
116C4×D4—101no—
117C4×D4—101no—
118#117—21no—
119C2×C42C2×D511yescharacteristic
120C4⋊C8—52no—
121#103—54no—
122#102—54no—
123C22⋊C8—52no—
124C4×C8—51no—
125C4×C8—51no—
126C22×C5—21no—
127C22×C5—21no—
128C22×C5—21no—
129C22×C5C22×D411yescharacteristic
130C2×D5—42no—
131C2×D5—42no—
132C2×D5—42no—
133C4×C5—21no—
134C4×C5—21no—
135C4×C5—21no—
136C4×C5—42no—
137C4×C5—42no—
138C4×C5#13512yes—
139C4×C5#13712yes—
140C4×C5H(5,𝔽2)12yes—
141Dic5—42no—
142#41—204no—
143C2×Q8—21no—
144C2×Q8—52no—
145C2×Q8C22×D512yes—
146C2×D4—52no—
147C2×D4—102no—
148C2×D4—101no—
149C2×D4—101no—
150C22×C4—102no—
151C22×C4C22⋊D511yescharacteristic
152C22×C4C22⋊D511yescharacteristic
153C22×C4C22×D511yescharacteristic
154Q16—204no—
155SD16—204no—
156C2×C8—52no—
157C2×C8—52no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—101no—
164C4⋊C4—101no—
165C4⋊C4C22×D512yes—
166C22⋊C4—22no—
167C22⋊C4—102no—
168C22⋊C4—102no—
169C42—22no—
170C42C22⋊D511yescharacteristic
171C42C22⋊D511yescharacteristic
172C42C22×D511yescharacteristic
173C42C22×D511yescharacteristic
174C2×C5—41no—
175C2×C5#53511yescharacteristic
176C2×C5C2×#13511yescharacteristic
177C2×C5C2×#13711yescharacteristic
178D5—82no—
179C23—102no—
180C23C2×C22⋊D511yescharacteristic
181Q8—24no—
182Q8—42no—
183Q8—104no—
184D4—104no—
185D4—204no—
186D4—202no—
187D4—202no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—102no—
194C2×C4—204no—
195C2×C4C2×C22⋊D511yescharacteristic
196C2×C4C2×C22⋊D511yescharacteristic
197C2×C4C2×C22⋊D511yescharacteristic
198C2×C4C2×C22⋊D511yescharacteristic
199C2×C4C2×C22⋊D511yescharacteristic
200C2×C4C23×D511yescharacteristic
201C8—102no—
202C8—102no—
203C5#321011yescharacteristic, complemented, Sylow
204C22—21no—
205C22—21no—
206C22—21no—
207C22—202no—
208C22—202no—
209C22—202no—
210C22C22×C22⋊D511yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—202no—
217C4#408812yes—
218C4#409612yes—
219C4#414912yes—
220C2—41no—
221C2—402no—
222C2C2×#408811yescharacteristic
223C2C2×#409611yescharacteristic
224C2#6616411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#533C2
socleC22×C5C22×D4
prime radical#65811C2
hypercenter#533D5
hypocenterC5#3210
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C5×#533C2semidirect
3. #65382C2semidirect
4. #65381C2semidirect
5. #65377C2semidirect
6. #65374C2semidirect
7. #65336C2semidirect
8. #65317C2semidirect
9. #65214C2semidirect
68. #533D5semidirect
203. C5#3210semidirect

Maximal subgroups

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

subgroupindex
C5×#5332
#658112
#653822
#653812
#653772
#653742
#653362
#653172
#652142
#32105