Finite Group Databasebeta

← #10606316

Subgroups — #10606316

Order 640.

380 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 117 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×#385C211yesmaximal, characteristic
2#65499C212yesmaximal
3#65271C212yesmaximal, complemented
4#65268C212yesmaximal, complemented
5#65242C211yesmaximal, characteristic
6#65241C212yesmaximal
7#65236C211yesmaximal, characteristic
8#64949C212yesmaximal, complemented
9#64947C212yesmaximal, complemented
10C23×C4×C5C2211yescharacteristic
11C22×Dic10C2211yescharacteristic
12C2×C5×C4⋊C4C2212yes—
13C2×C5×C22⋊C4C2211yescharacteristic
14C2×C5×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic5C2212yes—
16C2×C22⋊Dic5C2212yes—
17C2×C4⋊Dic5C2211yescharacteristic
18C2×#683C2212yes—
19C2×#683C2212yes—
20C2×#683C2212yes—
21C5×#95C2212yes—
22#3963C2211yescharacteristic
23#3963C2212yes—
24#3963C2212yes—
25#3963C2212yes—
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2211yescharacteristic
30#3963C2212yes—
31#2835—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C22×C4×C5—41no—
34C22×C4×C5—42no—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2312yes—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C2×Dic10—41no—
44C2×Dic10—42no—
45C5×C4⋊C4—42no—
46C5×C22⋊C4—22no—
47C5×C22⋊C4—41no—
48C22⋊Dic5—24no—
49C22⋊Dic5—42no—
50C4⋊Dic5—41no—
51#683—42no—
52#683—42no—
53#683—42no—
54#400—52no—
55#400—52no—
56#400—52no—
57#397—52no—
58#397—51no—
59#395—52no—
60#393—51no—
61#385—52no—
62#385D511yescharacteristic
63C23×C5—22no—
64C23×C5—21no—
65C23×C5—21no—
66C23×C5—22no—
67C23×C5#4112yes—
68C23×C5C2411yescharacteristic
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×C5—22no—
77C2×C4×C5—22no—
78C2×C4×C5—22no—
79C2×C4×C5—44no—
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×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—22no—
92Dic10—44no—
93C22×Q8—51no—
94C23×C4C2×D511yescharacteristic
95C2×C4⋊C4—52no—
96C2×C4⋊C4—52no—
97C2×C4⋊C4—52no—
98C2×C4⋊C4—51no—
99C2×C4⋊C4C2×D512yes—
100C2×C22⋊C4—52no—
101C2×C22⋊C4—52no—
102C2×C22⋊C4C2×D511yescharacteristic
103C2×C22⋊C4C2×D511yescharacteristic
104#95—52no—
105#95—52no—
106#95—51no—
107#95—52no—
108#95—52no—
109#95—52no—
110#95—52no—
111#95—52no—
112#95—51no—
113#95C2×D512yes—
114C22×C5—22no—
115C22×C5—24no—
116C22×C5—41no—
117C22×C5—42no—
118C22×C5—41no—
119C22×C5C2×#4111yescharacteristic
120C22×C5H(5,𝔽2)11yescharacteristic
121C22×C5H(5,𝔽2)12yes—
122C22×C52−1+411yescharacteristic
123C22×C52−1+412yes—
124C4×C5—42no—
125C4×C5—41no—
126C4×C5—42no—
127C4×C5—41no—
128Dic5—42no—
129Dic5—42no—
130Dic5—42no—
131Dic5—42no—
132C24C22×D511yescharacteristic
133C2×Q8—201no—
134C2×Q8—202no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—52no—
138C22×C4—52no—
139C22×C4—52no—
140C22×C4—52no—
141C22×C4C22×D511yescharacteristic
142C22×C4C22×D512yes—
143C22×C4C22×D511yescharacteristic
144C22×C4C22×D512yes—
145C4⋊C4—42no—
146C4⋊C4—202no—
147C4⋊C4—202no—
148C4⋊C4—202no—
149C4⋊C4—201no—
150C22⋊C4—22no—
151C22⋊C4—41no—
152C22⋊C4—104no—
153C22⋊C4—202no—
154C2×C5—42no—
155C2×C5#53911yescharacteristic
156C2×C5#53912yes—
157C2×C5H(3,𝔽4)11yescharacteristic
158C2×C5#56312yes—
159C2×C5#56311yescharacteristic
160C23—21no—
161C23—22no—
162C23—21no—
163C23—22no—
164C23#70912yes—
165C23C23×D511yescharacteristic
166Q8—204no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—44no—
178C2×C4—102no—
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—
190C5#283511yescharacteristic, complemented, Sylow
191C22—22no—
192C22—24no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22C2×#70911yescharacteristic
197C22#414411yescharacteristic
198C22#414412yes—
199C22#414711yescharacteristic
200C22#415012yes—
201C4—42no—
202C4—42no—
203C4—41no—
204C4—41no—
205C4—202no—
206C4—202no—
207C4—202no—
208C4—202no—
209C2—42no—
210C2#6584911yescharacteristic
211C2#6589812yes—
212C2#6592811yescharacteristic
213C2#6598012yes—
214C2#6598511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#385C2
prime radicalC24×C5C23
hypercenter#385D5
hypocenterC5#2835
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #65271C2semidirect
4. #65268C2semidirect
8. #64949C2semidirect
9. #64947C2semidirect
190. C5#2835semidirect

Maximal subgroups

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

subgroupindex
C5×#3852
#654992
#652712
#652682
#652422
#652412
#652362
#649492
#649472
#28355