Finite Group Databasebeta

← #10608065

Subgroups — #10608065

Order 640.

568 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#4084C211yesmaximal, characteristic
2C2×#4038C211yesmaximal, characteristic, complemented
3C5×#390C211yesmaximal, characteristic, complemented
4#65510C211yesmaximal, characteristic, complemented
5#65265C211yesmaximal, characteristic, complemented
6#65248C214yesmaximal, complemented
7#65234C214yesmaximal, complemented
8C4×#685C211yesmaximal, characteristic, complemented
9#65225C211yesmaximal, characteristic, complemented
10C22×C5×D4C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×C4×D5C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C22⋊C4C2214yes—
15C2×C42×C5C2211yescharacteristic
16C2×C22⋊Dic5C2212yes—
17C2×C22⋊Dic5C2214yes—
18#4084—24no—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2211yescharacteristic
23C2×#683C2214yes—
24C2×C4×Dic5C2212yes—
25C2×C4×Dic5C2211yescharacteristic
26#4038—24no—
27#3963C2214yes—
28#3963C2212yes—
29#2500—51nomaximal, Sylow
30C24×C5C2312yes—
31C23×D5C2311yescharacteristic
32C2×C5×D4—22no—
33C2×C5×D4—22no—
34C2×C5×D4—22no—
35C22×C4×C5C2312yes—
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C2×C22⋊D5—24no—
39C2×C22⋊D5—44no—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2311yescharacteristic
43C22×Dic5C2312yes—
44C2×C4×D5—22no—
45C2×C4×D5—22no—
46C2×C4×D5D414yes—
47C5×C4⋊C4—22no—
48C5×C22⋊C4—44no—
49C42×C5—22no—
50C22⋊Dic5—24no—
51C22⋊Dic5—28no—
52#685—24no—
53#685—24no—
54C4⋊Dic5—22no—
55C4⋊Dic5—22no—
56#683—44no—
57C4×Dic5—22no—
58C4×Dic5—24no—
59C2×#121—51no—
60C2×#117—51no—
61#394—54no—
62#390—51no—
63#390D511yescharacteristic, complemented
64#388—54no—
65#384—51no—
66#382—51no—
67C4×C22⋊C4—51no—
68C23×C5—22no—
69C23×C5—24no—
70C23×C5—22no—
71C23×C5—22no—
72C23×C5—24no—
73C23×C5C2411yescharacteristic
74C22×D5—22no—
75C22×D5—21no—
76C22×D5—21no—
77C22×D5—22no—
78C22×D5C2×D412yes—
79C5×D4—44no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—24no—
84C2×C4×C5—22no—
85C2×C4×C5C2×D412yes—
86C2×C4×C5#4112yes—
87C2×C4×C5#4112yes—
88C22⋊D5—48no—
89C2×Dic5—22no—
90C2×Dic5—21no—
91C2×Dic5—21no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—24no—
96C2×Dic5—24no—
97C2×Dic5C2×D412yes—
98C2×Dic5#4114yes—
99C4×D5—28no—
100C22×D4—52no—
101C22×D4C2×D511yescharacteristic
102C23×C4—51no—
103#121—104no—
104#117—104no—
105C2×C4⋊C4—51no—
106C2×C4⋊C4—51no—
107C2×C4⋊C4—54no—
108C2×C4⋊C4C2×D511yescharacteristic
109C2×C22⋊C4—52no—
110C2×C22⋊C4—54no—
111C2×C22⋊C4—52no—
112C2×C22⋊C4—52no—
113C2×C22⋊C4C2×D514yes—
114C2×C42—52no—
115C2×C42—51no—
116C2×C42C2×D511yescharacteristic
117#95—52no—
118#95—54no—
119C22×C5—42no—
120C22×C5—42no—
121C22×C5—42no—
122C22×C5—44no—
123C22×C5—44no—
124C22×C5C22×D411yescharacteristic
125C22×C5C2×#4111yescharacteristic
126C22×C5C2×#4111yescharacteristic
127C22×C5C2×#4112yes—
128C22×C5H(5,𝔽2)12yes—
129C2×D5—22no—
130C2×D5—22no—
131C2×D5—22no—
132C2×D5—42no—
133C2×D5—42no—
134C4×C5—42no—
135C4×C5—42no—
136C4×C5#12314yes—
137Dic5—22no—
138Dic5—24no—
139Dic5—42no—
140Dic5—42no—
141C24—51no—
142C24C22×D512yes—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—104no—
147C2×D4—204no—
148C22×C4—51no—
149C22×C4—52no—
150C22×C4—54no—
151C22×C4—52no—
152C22×C4—52no—
153C22×C4—102no—
154C22×C4—102no—
155C22×C4C22×D512yes—
156C22×C4C22×D512yes—
157C22×C4C22×D511yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—102no—
160C4⋊C4—102no—
161C4⋊C4—204no—
162C22⋊C4—44no—
163C22⋊C4—104no—
164C22⋊C4—104no—
165C22⋊C4—104no—
166C22⋊C4—108no—
167C42—22no—
168C42—104no—
169C42—102no—
170C2×C5—82no—
171C2×C5C2×#12311yescharacteristic
172C2×C5#53211yescharacteristic
173C2×C5#53511yescharacteristic
174C2×C5#55312yes—
175C2×C5#55512yes—
176D5—42no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—24no—
181C23—24no—
182C23—52no—
183C23—101no—
184C23—101no—
185C23—102no—
186C23—102no—
187C23C23×D511yescharacteristic
188D4—44no—
189D4—208no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—52no—
196C2×C4—54no—
197C2×C4—102no—
198C2×C4—101no—
199C2×C4—101no—
200C2×C4—102no—
201C2×C4—108no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—104no—
205C2×C4—104no—
206C2×C4#70912yes—
207C2×C4D4×D512yes—
208C2×C4#71112yes—
209C5#250011yescharacteristic, complemented, Sylow
210C22—42no—
211C22—42no—
212C22—44no—
213C22—44no—
214C22—42no—
215C22—102no—
216C22—102no—
217C22—102no—
218C22—202no—
219C22—202no—
220C22C2×#70911yescharacteristic
221C22C2×D4×D511yescharacteristic
222C22C2×#71111yescharacteristic
223C22#414412yes—
224C22D5×#4112yes—
225C4—42no—
226C4—42no—
227C4—102no—
228C4—104no—
229C4—202no—
230C4—202no—
231C4#402914yes—
232C2—82no—
233C2—202no—
234C2C2×#402911yescharacteristic
235C2#6589212yes—
236C2#6595012yes—
237C2#6600211yescharacteristic
238C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radicalC2×#4084C2
hypercenter#390D5
hypocenterC5#2500
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#4038C2semidirect
3. C5×#390C2semidirect
4. #65510C2semidirect
5. #65265C2semidirect
6. #65248C2semidirect
7. #65234C2semidirect
8. C4×#685C2semidirect
9. #65225C2semidirect
63. #390D5semidirect
209. C5#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
C2×#40382
C5×#3902
#655102
#652652
#652482
#652342
C4×#6852
#652252
#25005