Finite Group Databasebeta

← #10609377

Subgroups — #10609377

Order 640.

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

normal subgroups only

This group has 444 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
1C2×#4024C212yesmaximal
2C5×#397C211yesmaximal, characteristic
3#65268C212yesmaximal, complemented
4#65242C211yesmaximal, characteristic
5#65241C212yesmaximal
6#65232C211yesmaximal, characteristic
7#64949C212yesmaximal, complemented
8#64947C212yesmaximal, complemented
9#64945C212yesmaximal, complemented
10C23×Dic5C2211yescharacteristic
11C22×Dic10C2212yes—
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×C5×C22⋊C4C2212yes—
14C2×C5×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic5C2212yes—
16C2×C22⋊Dic5C2211yescharacteristic
17C2×C4⋊Dic5C2212yes—
18C2×C4⋊Dic5C2211yescharacteristic
19C2×#683C2212yes—
20C2×#683C2212yes—
21C2×#683C2212yes—
22#4024—28no—
23C5×#95C2212yes—
24C5×#95C2211yescharacteristic
25#3963C2212yes—
26#3963C2212yes—
27#3963C2211yescharacteristic
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#2817—51nomaximal, Sylow
33C24×C5C2311yescharacteristic
34C22×C4×C5C2312yes—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2312yes—
38C22×Dic5—41no—
39C22×Dic5—41no—
40C22×Dic5D414yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C2×Dic10—24no—
46C2×Dic10—42no—
47C2×Dic10—42no—
48C5×C4⋊C4—41no—
49C5×C22⋊C4—24no—
50C5×C22⋊C4—41no—
51C22⋊Dic5—24no—
52C22⋊Dic5—41no—
53C4⋊Dic5—24no—
54C4⋊Dic5—41no—
55#683—24no—
56#683—24no—
57#683—42no—
58C2×C22⋊Q8—52no—
59#400—52no—
60#400—52no—
61#397—52no—
62#397—51no—
63#397D511yescharacteristic
64#395—52no—
65#395—52no—
66#381—51no—
67C23×C5—21no—
68C23×C5—21no—
69C23×C5—22no—
70C23×C5—22no—
71C23×C5C2×D412yes—
72C23×C5C2411yescharacteristic
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—21no—
77C2×C4×C5—21no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×Dic5—22no—
84C2×Dic5—22no—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—44no—
95C2×Dic5C2×D414yes—
96Dic10—48no—
97C22×Q8—52no—
98C23×C4—51no—
99C22⋊Q8—108no—
100C2×C4⋊C4—52no—
101C2×C4⋊C4—52no—
102C2×C4⋊C4—52no—
103C2×C4⋊C4—52no—
104C2×C4⋊C4—51no—
105C2×C4⋊C4C2×D511yescharacteristic
106C2×C22⋊C4—52no—
107C2×C22⋊C4—51no—
108C2×C22⋊C4C2×D512yes—
109C2×C22⋊C4C2×D511yescharacteristic
110#95—52no—
111#95—52no—
112#95—52no—
113#95—52no—
114#95—52no—
115#95—52no—
116#95—51no—
117#95C2×D512yes—
118#95C2×D511yescharacteristic
119C22×C5—22no—
120C22×C5—24no—
121C22×C5—41no—
122C22×C5—41no—
123C22×C5—42no—
124C22×C5C22×D411yescharacteristic
125C22×C5H(5,𝔽2)11yescharacteristic
126C22×C5H(5,𝔽2)11yescharacteristic
127C22×C52−1+412yes—
128C22×C52−1+412yes—
129C4×C5—42no—
130C4×C5—42no—
131C4×C5—41no—
132C4×C5—41no—
133Dic5—24no—
134Dic5—42no—
135Dic5—42no—
136Dic5—42no—
137C24C22×D511yescharacteristic
138C2×Q8—104no—
139C2×Q8—202no—
140C2×Q8—202no—
141C22×C4—52no—
142C22×C4—54no—
143C22×C4—52no—
144C22×C4—52no—
145C22×C4—52no—
146C22×C4—201no—
147C22×C4—201no—
148C22×C4C22×D511yescharacteristic
149C22×C4C22×D511yescharacteristic
150C22×C4C22×D512yes—
151C22×C4C22×D512yes—
152C4⋊C4—41no—
153C4⋊C4—104no—
154C4⋊C4—104no—
155C4⋊C4—104no—
156C4⋊C4—202no—
157C4⋊C4—201no—
158C22⋊C4—24no—
159C22⋊C4—41no—
160C22⋊C4—104no—
161C22⋊C4—201no—
162C2×C5—42no—
163C2×C5#53411yescharacteristic
164C2×C5#53612yes—
165C2×C5#56311yescharacteristic
166C2×C5#56311yescharacteristic
167C2×C5#56312yes—
168C23—21no—
169C23—21no—
170C23—22no—
171C23—22no—
172C23D4×D512yes—
173C23C23×D511yescharacteristic
174Q8—208no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—54no—
186C2×C4—102no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—102no—
190C2×C4—102no—
191C2×C4—102no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—204no—
198C5#281711yescharacteristic, complemented, Sylow
199C22—22no—
200C22—24no—
201C22—41no—
202C22—41no—
203C22—42no—
204C22C2×D4×D511yescharacteristic
205C22#414411yescharacteristic
206C22#414911yescharacteristic
207C22#415012yes—
208C22#415012yes—
209C4—42no—
210C4—42no—
211C4—41no—
212C4—41no—
213C4—104no—
214C4—202no—
215C4—202no—
216C4—202no—
217C2—42no—
218C2#6593911yescharacteristic
219C2#6598212yes—
220C2#6598511yescharacteristic
221C2#6600611yescharacteristic
222C2#6604612yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC24×C5C23
hypercenter#397D5
hypocenterC5#2817
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
3. #65268C2semidirect
7. #64949C2semidirect
8. #64947C2semidirect
9. #64945C2semidirect
198. C5#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#40242
C5×#3972
#652682
#652422
#652412
#652322
#649492
#649472
#649452
#28175