Finite Group Databasebeta

← #10609807

Subgroups — #10609807

Order 640.

416 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 129 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×#399C211yesmaximal, characteristic
2#65268C212yesmaximal, complemented
3#65267C212yesmaximal, complemented
4#65261C212yesmaximal, complemented
5#65236C211yesmaximal, characteristic
6#65235C211yesmaximal, characteristic
7#65234C212yesmaximal
8#65231C212yesmaximal
9#64947C212yesmaximal, complemented
10C23×Dic5C2211yescharacteristic
11C22×Dic10C2211yescharacteristic
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×C5×C4⋊C4C2212yes—
14C2×C5×C22⋊C4C2212yes—
15C2×C5×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic5C2212yes—
17C2×C22⋊Dic5C2211yescharacteristic
18C2×#683C2212yes—
19C2×#683C2212yes—
20C2×#683C2212yes—
21C2×#683C2212yes—
22C2×C4×Dic5C2212yes—
23C5×#95C2211yescharacteristic
24#3963C2212yes—
25#3963C2212yes—
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2211yescharacteristic
31#2742—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C22×C4×C5C2311yescharacteristic
34C22×C4×C5C2312yes—
35C22×C4×C5C2312yes—
36C22×C4×C5C2311yescharacteristic
37C22×Dic5—24no—
38C22×Dic5—41no—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C2×Dic10—41no—
44C2×Dic10—42no—
45C5×C4⋊C4—42no—
46C5×C4⋊C4—41no—
47C5×C22⋊C4—42no—
48C5×C22⋊C4—41no—
49C22⋊Dic5—42no—
50C22⋊Dic5Q814yescomplemented
51#683—24no—
52#683—24no—
53#683—42no—
54#683—42no—
55C4×Dic5—42no—
56#400—52no—
57#399D511yescharacteristic
58#395—52no—
59#393—51no—
60#389—52no—
61#388—52no—
62#387—52no—
63#385—51no—
64#380—52no—
65C23×C5—22no—
66C23×C5—22no—
67C23×C5—21no—
68C23×C5—21no—
69C23×C5C2×Q812yes—
70C23×C5C2411yescharacteristic
71C2×C4×C5—22no—
72C2×C4×C5—22no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—22no—
77C2×C4×C5—21no—
78C2×C4×C5—21no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×Dic5—22no—
82C2×Dic5—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—44no—
92C2×Dic5C2×Q814yes—
93C2×Dic5#4114yes—
94Dic10—44no—
95C22×Q8—51no—
96C23×C4—51no—
97C2×C4⋊C4—52no—
98C2×C4⋊C4—52no—
99C2×C4⋊C4—52no—
100C2×C4⋊C4—52no—
101C2×C4⋊C4C2×D512yes—
102C2×C4⋊C4C2×D511yescharacteristic
103C2×C22⋊C4—51no—
104C2×C22⋊C4—52no—
105C2×C22⋊C4C2×D512yes—
106C2×C22⋊C4C2×D511yescharacteristic
107C2×C42—52no—
108#95—52no—
109#95—52no—
110#95—52no—
111#95—51no—
112#95—52no—
113#95—52no—
114#95—52no—
115#95C2×D511yescharacteristic
116C22×C5—22no—
117C22×C5—24no—
118C22×C5—42no—
119C22×C5—41no—
120C22×C5—41no—
121C22×C5C22×Q811yescharacteristic
122C22×C5C2×#4112yes—
123C22×C5H(5,𝔽2)12yes—
124C22×C5H(5,𝔽2)11yescharacteristic
125C22×C52−1+411yescharacteristic
126C4×C5—42no—
127C4×C5—41no—
128C4×C5—41no—
129C4×C5—42no—
130Dic5—24no—
131Dic5—24no—
132Dic5—42no—
133Dic5—42no—
134C24C22×D511yescharacteristic
135C2×Q8—201no—
136C2×Q8—202no—
137C22×C4—52no—
138C22×C4—52no—
139C22×C4—52no—
140C22×C4—52no—
141C22×C4—104no—
142C22×C4—201no—
143C22×C4C22×D512yes—
144C22×C4C22×D511yescharacteristic
145C22×C4C22×D511yescharacteristic
146C22×C4C22×D512yes—
147C4⋊C4—42no—
148C4⋊C4—41no—
149C4⋊C4—104no—
150C4⋊C4—104no—
151C4⋊C4—202no—
152C4⋊C4—202no—
153C22⋊C4—42no—
154C22⋊C4—41no—
155C22⋊C4—54no—
156C22⋊C4—202no—
157C42—202no—
158C2×C5—42no—
159C2×C5#54212yes—
160C2×C5#54411yescharacteristic
161C2×C5#55111yescharacteristic
162C2×C5#55412yes—
163C2×C5#56011yescharacteristic
164C23—22no—
165C23—21no—
166C23—22no—
167C23—21no—
168C23Q8×D512yes—
169C23C23×D511yescharacteristic
170Q8—204no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—54no—
182C2×C4—54no—
183C2×C4—102no—
184C2×C4—102no—
185C2×C4—102no—
186C2×C4—102no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—102no—
190C2×C4—102no—
191C2×C4—102no—
192C2×C4—102no—
193C2×C4—204no—
194C5#274211yescharacteristic, complemented, Sylow
195C22—22no—
196C22—24no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22#414411yescharacteristic
201C22#414412yes—
202C22C2×Q8×D511yescharacteristic
203C22D5×#4112yes—
204C22#415011yescharacteristic
205C4—41no—
206C4—42no—
207C4—41no—
208C4—42no—
209C4—104no—
210C4—104no—
211C4—202no—
212C4—202no—
213C2—42no—
214C2#6595711yescharacteristic
215C2#6597012yes—
216C2#6599311yescharacteristic
217C2#6599612yes—
218C2#6605411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC24×C5C23
hypercenter#399D5
hypocenterC5#2742
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #65268C2semidirect
3. #65267C2semidirect
4. #65261C2semidirect
9. #64947C2semidirect
50. C22⋊Dic5Q8semidirect
194. C5#2742semidirect

Maximal subgroups

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

subgroupindex
C5×#3992
#652682
#652672
#652612
#652362
#652352
#652342
#652312
#649472
#27425