Finite Group Databasebeta

← #10613533

Subgroups — #10613533

Order 640.

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

normal subgroups only

This group has 384 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
1#66062C211yesmaximal, characteristic
2D5×C4⋊Q8C211yesmaximal, characteristic
3C5×#494C211yesmaximal, characteristic, complemented
4#65475C214yesmaximal
5#65471C214yesmaximal, complemented
6#65381C211yesmaximal, characteristic, complemented
7#65370C211yesmaximal, characteristic, complemented
8#64974C211yesmaximal, characteristic
9#64970C211yesmaximal, characteristic, complemented
10C2×Q8×D5C2212yes—
11C2×C5×Q16C2214yes—
12C5×C4⋊Q8C2211yescharacteristic
13C5×C4⋊Q8C2211yescharacteristic
14#4092C2212yes—
15Q8×Dic5C2212yes—
16#4090C2212yes—
17C2×#689C2214yes—
18#4043—21no—
19#4038—21no—
20D5×C4⋊C4—21no—
21D5×C4⋊C4—21no—
22#4036—21no—
23#4034—21no—
24#4034—21no—
25#4033—21no—
26C42×D5C2211yescharacteristic
27#4015C2211yescharacteristic
28#4015C2211yescharacteristic
29C4×C5×C8C2211yescharacteristic
30#3967C2214yes—
31#3952C2212yes—
32#3948C2214yes—
33#3946C2212yes—
34C4×#168C2211yescharacteristic
35#3209—51nomaximal, Sylow
36C2×C5×Q8C2312yes—
37C2×C5×Q8C2312yes—
38Q8×D5—44no—
39C2×C4×D5—22no—
40C2×C4×D5D411yescharacteristic
41C2×C4×D5D411yescharacteristic
42C2×C4×D5C2311yescharacteristic
43C2×Dic10—21no—
44C2×Dic10—22no—
45C2×Dic10C2312yes—
46C5×Q16—44no—
47C2×C5×C8C2312yes—
48C5×C4⋊C4—21no—
49C5×C4⋊C4—21no—
50C5×C4⋊C4—21no—
51C5×C4⋊C4—21no—
52C42×C5C2311yescharacteristic
53#689—44no—
54#685—22no—
55C4⋊Dic5—21no—
56C4⋊Dic5—21no—
57C4⋊Dic5—21no—
58C4⋊Dic5—22no—
59C4⋊Dic5C2312yes—
60#683—22no—
61#683—22no—
62#683—22no—
63C4×Dic5—22no—
64C4×Dic5D411yescharacteristic
65C4×Dic5D411yescharacteristic
66C4×Dic5C2311yescharacteristic
67C2×#168C2312yes—
68#533—51no—
69C2×C4⋊Q8—51no—
70#494—51no—
71#494D511yescharacteristic, complemented
72#487—51no—
73#487—51no—
74#462—54no—
75#451—54no—
76#431—51no—
77C22×D5C2×D411yescharacteristic
78C5×Q8—24no—
79C5×Q8—24no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5C2×D411yescharacteristic
83C2×C4×C5C2×D411yescharacteristic
84C2×C4×C5C2411yescharacteristic
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5C2×D411yescharacteristic
88C2×Dic5C2×D411yescharacteristic
89C2×Dic5C2×D411yescharacteristic
90C4×D5—22no—
91C4×D5—22no—
92C4×D5—44no—
93C4×D5Q1614yescomplemented
94Dic10—24no—
95Dic10—44no—
96Dic10—42no—
97C5×C8—22no—
98#168—22no—
99C22×Q8—52no—
100C2×Q16—54no—
101C2×Q16C2×D514yes—
102C4⋊Q8—52no—
103C4⋊Q8—51no—
104C4⋊Q8—51no—
105C4⋊Q8—101no—
106C4⋊Q8—101no—
107C4⋊Q8C2×D511yescharacteristic
108C4⋊Q8C2×D511yescharacteristic
109#125—101no—
110C22⋊Q8—52no—
111C22⋊Q8—101no—
112C4×Q8—52no—
113C4×Q8—101no—
114#117—101no—
115C2×C4⋊C4—101no—
116C2×C4⋊C4—101no—
117C2×C42—51no—
118C4⋊C8—52no—
119#103—54no—
120#103—54no—
121C22⋊C8—52no—
122C4×C8—51no—
123C4×C8C2×D511yescharacteristic
124C22×C5C22×D411yescharacteristic
125C2×D5—21no—
126C2×D5—21no—
127C2×D5C2×Q1612yes—
128C4×C5—42no—
129C4×C5—42no—
130C4×C5C2×Q1612yes—
131C4×C5#13712yes—
132C4×C5H(5,𝔽2)12yes—
133Dic5—21no—
134Dic5—21no—
135Dic5—42no—
136Dic5—42no—
137Dic5C2×Q1612yes—
138C2×Q8—52no—
139C2×Q8—102no—
140C2×Q8—101no—
141C2×Q8—204no—
142C2×Q8C22×D512yes—
143C2×Q8C22×D512yes—
144C22×C4—51no—
145C22×C4—51no—
146C22×C4—51no—
147C22×C4—102no—
148Q16—44no—
149Q16—204no—
150C2×C8—52no—
151C2×C8C22×D512yes—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—52no—
157C4⋊C4—102no—
158C4⋊C4—102no—
159C4⋊C4—102no—
160C4⋊C4—102no—
161C4⋊C4—101no—
162C4⋊C4—101no—
163C4⋊C4—101no—
164C22⋊C4—102no—
165C42—51no—
166C42—51no—
167C42—51no—
168C42—102no—
169C42C22×D511yescharacteristic
170C2×C5#53511yescharacteristic
171C2×C5C22×Q1611yescharacteristic
172C2×C5C2×#13711yescharacteristic
173D5—22no—
174C23—51no—
175Q8—24no—
176Q8—24no—
177Q8—104no—
178Q8—204no—
179Q8—202no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—51no—
183C2×C4—54no—
184C2×C4—51no—
185C2×C4—51no—
186C2×C4—102no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—102no—
190C2×C4—204no—
191C2×C4D4×D511yescharacteristic
192C2×C4D4×D511yescharacteristic
193C2×C4C23×D511yescharacteristic
194C8—22no—
195C8—102no—
196C5#320911yescharacteristic, complemented, Sylow
197C22—52no—
198C22—101no—
199C22—101no—
200C22C2×D4×D511yescharacteristic
201C4—42no—
202C4—42no—
203C4—52no—
204C4—101no—
205C4—101no—
206C4—202no—
207C4—202no—
208C4D5×Q1612yes—
209C4#406412yes—
210C4#414412yes—
211C2—102no—
212C2#6604911yescharacteristic
213C2C2×D5×Q1611yescharacteristic
214C2C2×#406411yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#494C2
socleC22×C5C22×D4
prime radicalC22×D5C2×D4
hypercenter#494D5
hypocenterC5#3209
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
3. C5×#494C2semidirect
5. #65471C2semidirect
6. #65381C2semidirect
7. #65370C2semidirect
9. #64970C2semidirect
71. #494D5semidirect
93. C4×D5Q16semidirect
196. C5#3209semidirect

Maximal subgroups

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

subgroupindex
#660622
D5×C4⋊Q82
C5×#4942
#654752
#654712
#653812
#653702
#649742
#649702
#32095