Finite Group Databasebeta

← #10608574

Subgroups — #10608574

Order 640.

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

normal subgroups only

This group has 424 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×#393C211yesmaximal, characteristic
2#65519C212yesmaximal, complemented
3#65239C212yesmaximal
4#65236C211yesmaximal, characteristic
5#65235C211yesmaximal, characteristic
6#65233C212yesmaximal
7#64948C212yesmaximal, complemented
8#64945C212yesmaximal, complemented
9#64939C212yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C23×Dic5C2211yescharacteristic
12C22×Dic10C2211yescharacteristic
13C2×C5×C22⋊C4C2212yes—
14C2×C5×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic5C2212yes—
16C2×C22⋊Dic5C2211yescharacteristic
17C2×C4⋊Dic5C2212yes—
18C2×C4⋊Dic5C2212yes—
19C2×#683C2212yes—
20C2×#683C2212yes—
21C2×C4×Dic5C2212yes—
22C5×#95C2212yes—
23C5×#95C2211yescharacteristic
24#3963C2211yescharacteristic
25#3963C2212yes—
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#2763—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C2×C5×Q8—24no—
34C2×C5×Q8—41no—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2312yes—
38C22×C4×C5C2312yes—
39C22×Dic5—42no—
40C22×Dic5—41no—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C2×Dic10—41no—
46C2×Dic10—42no—
47C5×C22⋊C4—24no—
48C5×C22⋊C4—41no—
49C22⋊Dic5—42no—
50C22⋊Dic5Q814yescomplemented
51C4⋊Dic5—42no—
52C4⋊Dic5—42no—
53#683—42no—
54#683—42no—
55C4×Dic5—24no—
56#400—52no—
57#395—52no—
58#393—51no—
59#393D511yescharacteristic
60#391—52no—
61#388—52no—
62#385—51no—
63#385—52no—
64#382—52no—
65C23×C5—22no—
66C23×C5—22no—
67C23×C5—21no—
68C23×C5—21no—
69C23×C5C2×Q812yes—
70C23×C5C2411yescharacteristic
71C5×Q8—44no—
72C2×C4×C5—22no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—22no—
77C2×C4×C5—22no—
78C2×C4×C5—21no—
79C2×C4×C5—21no—
80C2×C4×C5—22no—
81C2×C4×C5#4114yes—
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—22no—
93C2×Dic5—44no—
94C2×Dic5C2×Q814yes—
95Dic10—44no—
96C22×Q8—51no—
97C22×Q8C2×D511yescharacteristic
98C23×C4—51no—
99C2×C4⋊C4—52no—
100C2×C4⋊C4—52no—
101C2×C4⋊C4—52no—
102C2×C4⋊C4—52no—
103C2×C22⋊C4—51no—
104C2×C22⋊C4—52no—
105C2×C22⋊C4C2×D512yes—
106C2×C22⋊C4C2×D511yescharacteristic
107C2×C42—52no—
108#95—51no—
109#95—52no—
110#95—52no—
111#95—52no—
112#95—52no—
113#95—52no—
114#95—52no—
115#95C2×D512yes—
116#95C2×D511yescharacteristic
117C22×C5—22no—
118C22×C5—24no—
119C22×C5—42no—
120C22×C5—41no—
121C22×C5—41no—
122C22×C5C22×Q811yescharacteristic
123C22×C5C2×#4112yes—
124C22×C5H(5,𝔽2)11yescharacteristic
125C22×C5H(5,𝔽2)12yes—
126C22×C52−1+411yescharacteristic
127C4×C5—24no—
128C4×C5—42no—
129C4×C5—41no—
130C4×C5—41no—
131Dic5—24no—
132Dic5—42no—
133Dic5—42no—
134Dic5—42no—
135C24C22×D511yescharacteristic
136C2×Q8—24no—
137C2×Q8—41no—
138C2×Q8—201no—
139C2×Q8—202no—
140C22×C4—52no—
141C22×C4—52no—
142C22×C4—52no—
143C22×C4—52no—
144C22×C4—201no—
145C22×C4—202no—
146C22×C4C22×D511yescharacteristic
147C22×C4C22×D512yes—
148C22×C4C22×D511yescharacteristic
149C22×C4C22×D512yes—
150C4⋊C4—202no—
151C4⋊C4—202no—
152C4⋊C4—202no—
153C4⋊C4—202no—
154C22⋊C4—24no—
155C22⋊C4—41no—
156C22⋊C4—54no—
157C22⋊C4—202no—
158C42—104no—
159C2×C5—42no—
160C2×C5#54212yes—
161C2×C5#54411yescharacteristic
162C2×C5#55111yescharacteristic
163C2×C5#55412yes—
164C2×C5H(3,𝔽4)11yescharacteristic
165C23—22no—
166C23—21no—
167C23—22no—
168C23—21no—
169C23Q8×D512yes—
170C23C23×D511yescharacteristic
171Q8—44no—
172Q8—204no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
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—102no—
194C2×C4—204no—
195C2×C4#71114yes—
196C5#276311yescharacteristic, complemented, Sylow
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C2×#71112yes—
203C22#414411yescharacteristic
204C22C2×Q8×D511yescharacteristic
205C22#414912yes—
206C22#415011yescharacteristic
207C4—24no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—104no—
212C4—202no—
213C4—202no—
214C4—202no—
215C2—42no—
216C2#6592111yescharacteristic
217C2#6595711yescharacteristic
218C2#6596912yes—
219C2#6601612yes—
220C2#6601711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC24×C5C23
hypercenter#393D5
hypocenterC5#2763
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #65519C2semidirect
7. #64948C2semidirect
8. #64945C2semidirect
9. #64939C2semidirect
50. C22⋊Dic5Q8semidirect
196. C5#2763semidirect

Maximal subgroups

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

subgroupindex
C5×#3932
#655192
#652392
#652362
#652352
#652332
#649482
#649452
#649392
#27635