Finite Group Databasebeta

← #10609819

Subgroups — #10609819

Order 640.

408 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 133 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#65274C212yesmaximal, complemented
3#65272C212yesmaximal, complemented
4#65271C212yesmaximal, complemented
5#65239C212yesmaximal
6#65238C211yesmaximal, characteristic
7#65235C211yesmaximal, characteristic
8#65234C212yesmaximal
9#64936C212yesmaximal, complemented
10C23×Dic5C2211yescharacteristic
11C2×C5×C4⋊C4C2212yes—
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×C4⋊Dic5C2212yes—
20C2×#683C2212yes—
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×C4×Dic5C2212yes—
24C5×#95C2211yescharacteristic
25#3963C2212yes—
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2211yescharacteristic
30#3963C2212yes—
31#2807—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C22×C4×C5C2312yes—
34C22×C4×C5C2311yescharacteristic
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2312yes—
37C22×Dic5—42no—
38C22×Dic5—41no—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C5×C4⋊C4—24no—
44C5×C4⋊C4—41no—
45C5×C22⋊C4—24no—
46C5×C22⋊C4—41no—
47C22⋊Dic5—42no—
48C22⋊Dic5Q814yes—
49C4⋊Dic5—42no—
50C4⋊Dic5—41no—
51C4⋊Dic5—42no—
52#683—24no—
53#683—42no—
54#683—42no—
55C4×Dic5—42no—
56#400—52no—
57#399—51no—
58#399D511yescharacteristic
59#398—52no—
60#389—52no—
61#389—52no—
62#388—52no—
63#385—51no—
64#385—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—22no—
79C2×C4×C5—21no—
80C2×C4×C5#4114yes—
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—22no—
92C2×Dic5—44no—
93C2×Dic5C2×Q814yes—
94C23×C4—51no—
95C2×C4⋊C4—52no—
96C2×C4⋊C4—52no—
97C2×C4⋊C4—52no—
98C2×C4⋊C4—52no—
99C2×C4⋊C4—51no—
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—52no—
112#95—52no—
113#95—51no—
114#95C2×D511yescharacteristic
115C22×C5—22no—
116C22×C5—24no—
117C22×C5—42no—
118C22×C5—41no—
119C22×C5—41no—
120C22×C5C22×Q811yescharacteristic
121C22×C5C2×#4112yes—
122C22×C5H(5,𝔽2)11yescharacteristic
123C22×C5H(5,𝔽2)12yes—
124C22×C52−1+411yescharacteristic
125C4×C5—24no—
126C4×C5—42no—
127C4×C5—41no—
128C4×C5—41no—
129Dic5—24no—
130Dic5—42no—
131Dic5—42no—
132Dic5—42no—
133C24C22×D511yescharacteristic
134C22×C4—52no—
135C22×C4—52no—
136C22×C4—52no—
137C22×C4—52no—
138C22×C4—201no—
139C22×C4—202no—
140C22×C4C22×D512yes—
141C22×C4C22×D511yescharacteristic
142C22×C4C22×D511yescharacteristic
143C22×C4C22×D512yes—
144C4⋊C4—24no—
145C4⋊C4—41no—
146C4⋊C4—104no—
147C4⋊C4—202no—
148C4⋊C4—202no—
149C4⋊C4—202no—
150C4⋊C4—201no—
151C4⋊C4—202no—
152C22⋊C4—24no—
153C22⋊C4—41no—
154C22⋊C4—54no—
155C22⋊C4—202no—
156C42—202no—
157C2×C5—42no—
158C2×C5#53912yes—
159C2×C5#54411yescharacteristic
160C2×C5#55412yes—
161C2×C5#55911yescharacteristic
162C2×C5#56011yescharacteristic
163C23—22no—
164C23—21no—
165C23—22no—
166C23—21no—
167C23Q8×D512yes—
168C23C23×D511yescharacteristic
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—54no—
179C2×C4—102no—
180C2×C4—102no—
181C2×C4—102no—
182C2×C4—102no—
183C2×C4—102no—
184C2×C4—102no—
185C2×C4—102no—
186C2×C4—102no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—102no—
190C2×C4—204no—
191C2×C4#71114yes—
192C5#280711yescharacteristic, complemented, Sylow
193C22—22no—
194C22—24no—
195C22—42no—
196C22—41no—
197C22—41no—
198C22C2×#71112yes—
199C22#414411yescharacteristic
200C22C2×Q8×D511yescharacteristic
201C22#414912yes—
202C22#415011yescharacteristic
203C4—24no—
204C4—41no—
205C4—42no—
206C4—41no—
207C4—104no—
208C4—202no—
209C4—202no—
210C4—202no—
211C2—42no—
212C2#6595711yescharacteristic
213C2#6596912yes—
214C2#6599912yes—
215C2#6600011yescharacteristic
216C2#6604711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC24×C5C23
hypercenter#399D5
hypocenterC5#2807
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #65274C2semidirect
3. #65272C2semidirect
4. #65271C2semidirect
9. #64936C2semidirect
192. C5#2807semidirect

Maximal subgroups

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

subgroupindex
C5×#3992
#652742
#652722
#652712
#652392
#652382
#652352
#652342
#649362
#28075