Finite Group Databasebeta

← #10609643

Subgroups — #10609643

Order 640.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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×#398C211yesmaximal, characteristic
2#65274C212yesmaximal
3#65272C212yesmaximal
4#65268C211yesmaximal, characteristic
5#65267C212yesmaximal
6#65265C211yesmaximal, characteristic
7#64945C212yesmaximal
8#64941C212yesmaximal
9#64936C212yesmaximal
10C22×Dic10C2211yescharacteristic
11C2×C5×C4⋊C4C2212yes—
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C4⋊Dic5C2212yes—
15C2×C4⋊Dic5C2212yes—
16C2×C4⋊Dic5C2211yescharacteristic
17C2×#683C2212yes—
18C2×#683C2212yes—
19C2×#683C2212yes—
20C2×#683C2212yes—
21C2×C4×Dic5C2211yescharacteristic
22C2×C4×Dic5C2212yes—
23C5×#95C2211yescharacteristic
24C5×#95C2212yes—
25#3963C2212yes—
26#3963C2211yescharacteristic
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#2792—51nomaximal, Sylow
32C22×C4×C5C2312yes—
33C22×C4×C5C2311yescharacteristic
34C22×C4×C5C2311yescharacteristic
35C22×C4×C5C2312yes—
36C22×C4×C5C2311yescharacteristic
37C22×Dic5C2312yes—
38C22×Dic5C2312yes—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C2×Dic10—42no—
42C2×Dic10—41no—
43C5×C4⋊C4—24no—
44C5×C4⋊C4—41no—
45C5×C4⋊C4—41no—
46C4⋊Dic5—42no—
47C4⋊Dic5—41no—
48C4⋊Dic5—42no—
49#683—42no—
50#683—42no—
51#683—42no—
52#683Q818yescomplemented
53C4×Dic5—22no—
54C4×Dic5—42no—
55#398—52no—
56#398D511yescharacteristic
57#395—51no—
58#395—52no—
59#389—52no—
60#389—52no—
61#387—52no—
62#387—52no—
63#384—51no—
64C23×C5C2411yescharacteristic
65C2×C4×C5—22no—
66C2×C4×C5—22no—
67C2×C4×C5—21no—
68C2×C4×C5—21no—
69C2×C4×C5—22no—
70C2×C4×C5—22no—
71C2×C4×C5—22no—
72C2×C4×C5—22no—
73C2×C4×C5—21no—
74C2×C4×C5—21no—
75C2×C4×C5—22no—
76C2×C4×C5C2×Q814yes—
77C2×C4×C5#4112yes—
78C2×Dic5—22no—
79C2×Dic5—22no—
80C2×Dic5—22no—
81C2×Dic5—22no—
82C2×Dic5—22no—
83C2×Dic5—22no—
84C2×Dic5—22no—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5C2×Q814yes—
89C2×Dic5C2×Q814yes—
90Dic10—44no—
91C22×Q8—51no—
92C2×C4⋊C4—52no—
93C2×C4⋊C4—52no—
94C2×C4⋊C4—51no—
95C2×C4⋊C4—52no—
96C2×C4⋊C4—52no—
97C2×C4⋊C4—52no—
98C2×C4⋊C4—52no—
99C2×C4⋊C4C2×D512yes—
100C2×C4⋊C4C2×D511yescharacteristic
101C2×C4⋊C4C2×D511yescharacteristic
102C2×C42—51no—
103C2×C42—52no—
104#95—51no—
105#95—52no—
106#95—52no—
107#95—52no—
108#95—52no—
109#95—52no—
110#95C2×D512yes—
111#95C2×D511yescharacteristic
112C22×C5C22×Q812yes—
113C22×C5C2×#4111yescharacteristic
114C22×C5H(5,𝔽2)11yescharacteristic
115C22×C5H(5,𝔽2)12yes—
116C22×C52−1+411yescharacteristic
117C4×C5—22no—
118C4×C5—24no—
119C4×C5—42no—
120C4×C5—41no—
121C4×C5—41no—
122Dic5—24no—
123Dic5—24no—
124Dic5—42no—
125Dic5—42no—
126C2×Q8—201no—
127C2×Q8—202no—
128C22×C4—52no—
129C22×C4—52no—
130C22×C4—52no—
131C22×C4—52no—
132C22×C4C22×D512yes—
133C22×C4C22×D511yescharacteristic
134C22×C4C22×D511yescharacteristic
135C22×C4C22×D511yescharacteristic
136C22×C4C22×D512yes—
137C4⋊C4—24no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—58no—
141C4⋊C4—202no—
142C4⋊C4—202no—
143C4⋊C4—202no—
144C4⋊C4—201no—
145C4⋊C4—202no—
146C4⋊C4—202no—
147C42—102no—
148C42—202no—
149C2×C5#53911yescharacteristic
150C2×C5#54412yes—
151C2×C5#55412yes—
152C2×C5Q8211yescharacteristic
153C2×C5#56011yescharacteristic
154C23C23×D511yescharacteristic
155Q8—204no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—54no—
168C2×C4—54no—
169C2×C4—102no—
170C2×C4—102no—
171C2×C4—102no—
172C2×C4—102no—
173C2×C4—102no—
174C2×C4—102no—
175C2×C4—102no—
176C2×C4—102no—
177C2×C4—102no—
178C2×C4—102no—
179C2×C4#71112yes—
180C2×C4Q8×D514yes—
181C5#279211yescharacteristic, complemented, Sylow
182C22C2×#71111yescharacteristic
183C22#414411yescharacteristic
184C22C2×Q8×D512yes—
185C22#414912yes—
186C22#415011yescharacteristic
187C4—22no—
188C4—24no—
189C4—42no—
190C4—41no—
191C4—41no—
192C4—104no—
193C4—104no—
194C4—202no—
195C4—202no—
196C2#6593611yescharacteristic
197C2#6596912yes—
198C2#6600011yescharacteristic
199C2#6602312yes—
200C2#6605611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, prime radical, cosocleC23×C5C24
FittingC5×#398C2
hypercenter#398D5
hypocenterC5#2792
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
52. #683Q8semidirect
181. C5#2792semidirect

Maximal subgroups

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

subgroupindex
C5×#3982
#652742
#652722
#652682
#652672
#652652
#649452
#649412
#649362
#27925