Finite Group Databasebeta

← #10608267

Subgroups — #10608267

Order 640.

604 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 157 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×#4093C211yesmaximal, characteristic
2C2×#4090C211yesmaximal, characteristic, complemented
3C2×#4039C211yesmaximal, characteristic
4C2×#4021C211yesmaximal, characteristic
5C5×#391C211yesmaximal, characteristic, complemented
6#65263C211yesmaximal, characteristic, complemented
7C4×#683C211yesmaximal, characteristic, complemented
8#64964C214yesmaximal, complemented
9#64956C214yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×D20C2211yescharacteristic
12C22×C4×D5C2212yes—
13C22×Dic10C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C42×C5C2211yescharacteristic
16#4093—24no—
17#4090—24no—
18C2×#685C2214yes—
19C2×#685C2212yes—
20C2×#685C2214yes—
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×#683C2212yes—
24C2×C4×Dic5C2211yescharacteristic
25C2×C4×Dic5C2211yescharacteristic
26#4039—24no—
27#4021—24no—
28C5×#95C2214yes—
29#3963C2214yes—
30#3963C2212yes—
31#2547—51nomaximal, Sylow
32C23×D5C2312yes—
33C2×C5×Q8—22no—
34C2×C5×Q8—22no—
35C2×C5×Q8—22no—
36C22×C4×C5C2312yes—
37C22×C4×C5C2312yes—
38C22×C4×C5C2312yes—
39C22×C4×C5C2311yescharacteristic
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2311yescharacteristic
43C22×Dic5C2311yescharacteristic
44C2×D20—22no—
45C2×D20—22no—
46C2×D20D414yes—
47C2×C4×D5—24no—
48C2×C4×D5—28no—
49C2×Dic10—22no—
50C2×Dic10—22no—
51C2×Dic10—22no—
52C5×C4⋊C4—22no—
53C42×C5—22no—
54#685—28no—
55#685—28no—
56#685—24no—
57#683—24no—
58#683—24no—
59#683—24no—
60C4×Dic5—22no—
61C4×Dic5D414yes—
62C2×C4⋊Q8—51no—
63C2×#124—51no—
64C2×#124—51no—
65C2×C4×D4—51no—
66#397—54no—
67#391—51no—
68#391D511yescharacteristic, complemented
69#380—54no—
70C4×C4⋊C4—51no—
71C23×C5C2411yescharacteristic
72C22×D5—22no—
73C22×D5—24no—
74C22×D5—22no—
75C22×D5—24no—
76C22×D5C2×D414yes—
77C5×Q8—44no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—24no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5C2×D412yes—
87C2×C4×C5C2×D412yes—
88C2×C4×C5#4112yes—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5C2×D412yes—
97C2×Dic5C2×D412yes—
98C2×Dic5#4114yes—
99D20—28no—
100C4×D5—48no—
101Dic10—44no—
102C22×Q8—51no—
103C22×Q8C2×D511yescharacteristic
104C22×D4—51no—
105C23×C4—52no—
106C4⋊Q8—104no—
107#124—104no—
108#124—104no—
109C4×D4—104no—
110C2×C4⋊C4—52no—
111C2×C4⋊C4—52no—
112C2×C4⋊C4—52no—
113C2×C4⋊C4C2×D511yescharacteristic
114C2×C22⋊C4—52no—
115C2×C22⋊C4—54no—
116C2×C22⋊C4—54no—
117C2×C42—51no—
118C2×C42—51no—
119C2×C42C2×D511yescharacteristic
120#95—52no—
121#95—54no—
122#95C2×D514yes—
123C22×C5C22×D411yescharacteristic
124C22×C5C22×D411yescharacteristic
125C22×C5C2×#4111yescharacteristic
126C22×C5C2×#4112yes—
127C22×C52−1+412yes—
128C2×D5—24no—
129C2×D5—28no—
130C2×D5—42no—
131C2×D5—42no—
132C2×D5—44no—
133C4×C5—42no—
134C4×C5—42no—
135C4×C5—42no—
136C4×C5#12114yes—
137Dic5—22no—
138Dic5—42no—
139Dic5—42no—
140Dic5#12414yescomplemented
141C24—52no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—102no—
146C2×Q8—102no—
147C2×Q8—102no—
148C2×D4—54no—
149C2×D4—102no—
150C2×D4—102no—
151C22×C4—51no—
152C22×C4—51no—
153C22×C4—52no—
154C22×C4—52no—
155C22×C4—104no—
156C22×C4—108no—
157C22×C4C22×D512yes—
158C22×C4C22×D512yes—
159C22×C4C22×D512yes—
160C22×C4C22×D511yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—104no—
163C4⋊C4—104no—
164C4⋊C4—104no—
165C22⋊C4—104no—
166C22⋊C4—108no—
167C22⋊C4—108no—
168C42—22no—
169C42—54no—
170C42—102no—
171C2×C5C2×#12111yescharacteristic
172C2×C5C2×#12411yescharacteristic
173C2×C5#53611yescharacteristic
174C2×C5#54712yes—
175C2×C5#55612yes—
176D5—44no—
177C23—54no—
178C23—102no—
179C23—102no—
180C23—104no—
181C23—104no—
182C23C23×D511yescharacteristic
183Q8—44no—
184Q8—204no—
185D4—108no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—52no—
195C2×C4—54no—
196C2×C4—52no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—208no—
205C2×C4#70912yes—
206C2×C4D4×D512yes—
207C2×C4D4×D512yes—
208C5#254711yescharacteristic, complemented, Sylow
209C22—104no—
210C22—108no—
211C22—202no—
212C22—204no—
213C22—202no—
214C22C2×#70911yescharacteristic
215C22C2×D4×D511yescharacteristic
216C22C2×D4×D511yescharacteristic
217C22#414712yes—
218C22D5×#4112yes—
219C4—42no—
220C4—42no—
221C4—42no—
222C4—54no—
223C4—102no—
224C4—202no—
225C4—202no—
226C4#403014yes—
227C2—204no—
228C2C2×#403011yescharacteristic
229C2#6590212yes—
230C2#6596512yes—
231C2D5×#12411yescharacteristic
232C2#6605811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#391C2
prime radicalC22×D20C22
hypercenter#391D5
hypocenterC5#2547
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#4090C2semidirect
5. C5×#391C2semidirect
6. #65263C2semidirect
7. C4×#683C2semidirect
8. #64964C2semidirect
9. #64956C2semidirect
68. #391D5semidirect
140. Dic5#124semidirect
208. C5#2547semidirect

Maximal subgroups

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

subgroupindex
C2×#40932
C2×#40902
C2×#40392
C2×#40212
C5×#3912
#652632
C4×#6832
#649642
#649562
#25475