Finite Group Databasebeta

← #10608188

Subgroups — #10608188

Order 640.

612 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 149 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×#4082C211yesmaximal, characteristic, complemented
3C2×#4043C211yesmaximal, characteristic, complemented
4C2×#4018C211yesmaximal, characteristic, complemented
5C5×#390C211yesmaximal, characteristic, complemented
6#65265C211yesmaximal, characteristic, complemented
7#65248C214yesmaximal, complemented
8#65237C214yesmaximal, complemented
9#65218C211yesmaximal, characteristic, complemented
10C22×C5×D4C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×C4×D5C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C22⋊C4C2214yes—
16C2×C42×C5C2211yescharacteristic
17C2×C22⋊Dic5C2214yes—
18C2×C22⋊Dic5C2212yes—
19#4084—24no—
20#4082—24no—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2212yes—
24C2×C4⋊Dic5C2211yescharacteristic
25C2×#683C2212yes—
26C2×#683C2214yes—
27C2×C4×Dic5C2211yescharacteristic
28#4043—24no—
29#4018—24no—
30#3963C2214yes—
31#2620—51nomaximal, Sylow
32C24×C5C2312yes—
33C23×D5C2311yescharacteristic
34C2×C5×D4—22no—
35C2×C5×D4—22no—
36C2×C5×D4—22no—
37C22×C4×C5C2312yes—
38C22×C4×C5C2312yes—
39C22×C4×C5C2311yescharacteristic
40C2×C22⋊D5—24no—
41C2×C22⋊D5—44no—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C22×Dic5C2311yescharacteristic
45C22×Dic5C2312yes—
46C2×C4×D5—22no—
47C2×C4×D5—22no—
48C2×C4×D5D414yes—
49C2×Dic10—22no—
50C2×Dic10—22no—
51C2×Dic10—22no—
52C5×C4⋊C4—22no—
53C5×C22⋊C4—44no—
54C42×C5—22no—
55C22⋊Dic5—28no—
56C22⋊Dic5—24no—
57#685—24no—
58#685—24no—
59C4⋊Dic5—22no—
60C4⋊Dic5—24no—
61#683—24no—
62#683—44no—
63C4×Dic5D414yescomplemented
64C2×#124—51no—
65C2×C22⋊Q8—51no—
66C2×#121—51no—
67C2×#117—51no—
68#396—54no—
69#394—54no—
70#391—51no—
71#390D511yescharacteristic, complemented
72#384—51no—
73C23×C5—22no—
74C23×C5—22no—
75C23×C5—22no—
76C23×C5—24no—
77C23×C5—24no—
78C23×C5C2411yescharacteristic
79C22×D5—21no—
80C22×D5—21no—
81C22×D5—22no—
82C22×D5—22no—
83C22×D5C2×D412yes—
84C5×D4—44no—
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5—24no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5C2×D412yes—
91C2×C4×C5C2×D412yes—
92C2×C4×C5#4112yes—
93C22⋊D5—48no—
94C2×Dic5—21no—
95C2×Dic5—21no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—24no—
103C2×Dic5C2×D414yes—
104C2×Dic5C2×D412yes—
105C4×D5—28no—
106Dic10—44no—
107C22×Q8—51no—
108C22×D4—52no—
109C22×D4C2×D511yescharacteristic
110C23×C4—51no—
111#124—104no—
112C22⋊Q8—104no—
113#121—104no—
114#117—104no—
115C2×C4⋊C4—52no—
116C2×C4⋊C4—51no—
117C2×C4⋊C4—52no—
118C2×C4⋊C4—54no—
119C2×C4⋊C4C2×D511yescharacteristic
120C2×C22⋊C4—52no—
121C2×C22⋊C4—54no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4—52no—
124C2×C22⋊C4C2×D514yes—
125C2×C42—51no—
126C2×C42C2×D511yescharacteristic
127#95—54no—
128C22×C5—42no—
129C22×C5—42no—
130C22×C5—42no—
131C22×C5—44no—
132C22×C5—44no—
133C22×C5C22×D411yescharacteristic
134C22×C5C22×D411yescharacteristic
135C22×C5C2×#4111yescharacteristic
136C22×C5H(5,𝔽2)12yes—
137C22×C52−1+412yes—
138C2×D5—22no—
139C2×D5—22no—
140C2×D5—22no—
141C2×D5—42no—
142C2×D5—42no—
143C4×C5—42no—
144C4×C5—42no—
145C4×C5#12114yes—
146Dic5—22no—
147Dic5—24no—
148Dic5—42no—
149Dic5—42no—
150C24—51no—
151C24C22×D512yes—
152C2×Q8—102no—
153C2×Q8—102no—
154C2×Q8—102no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—104no—
159C2×D4—204no—
160C22×C4—52no—
161C22×C4—51no—
162C22×C4—54no—
163C22×C4—52no—
164C22×C4—52no—
165C22×C4—102no—
166C22×C4—102no—
167C22×C4C22×D512yes—
168C22×C4C22×D512yes—
169C22×C4C22×D511yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—104no—
172C4⋊C4—104no—
173C4⋊C4—102no—
174C4⋊C4—204no—
175C22⋊C4—44no—
176C22⋊C4—104no—
177C22⋊C4—104no—
178C22⋊C4—104no—
179C22⋊C4—108no—
180C42—22no—
181C42—54no—
182C2×C5—82no—
183C2×C5C2×#12111yescharacteristic
184C2×C5#53511yescharacteristic
185C2×C5#53611yescharacteristic
186C2×C5#53712yes—
187C2×C5#54212yes—
188D5—42no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—24no—
193C23—24no—
194C23—52no—
195C23—101no—
196C23—101no—
197C23—102no—
198C23—102no—
199C23C23×D511yescharacteristic
200Q8—204no—
201D4—44no—
202D4—208no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—54no—
209C2×C4—52no—
210C2×C4—101no—
211C2×C4—101no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—108no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—104no—
220C2×C4#70912yes—
221C2×C4D4×D512yes—
222C2×C4D4×D512yes—
223C5#262011yescharacteristic, complemented, Sylow
224C22—42no—
225C22—42no—
226C22—44no—
227C22—44no—
228C22—42no—
229C22—102no—
230C22—102no—
231C22—102no—
232C22—202no—
233C22—202no—
234C22C2×#70911yescharacteristic
235C22C2×D4×D511yescharacteristic
236C22C2×D4×D511yescharacteristic
237C22#414412yes—
238C22#415012yes—
239C4—42no—
240C4—42no—
241C4—102no—
242C4—104no—
243C4—202no—
244C4—202no—
245C4#403014yes—
246C2—82no—
247C2—202no—
248C2C2×#403011yescharacteristic
249C2#6589512yes—
250C2#6594212yes—
251C2#6600911yescharacteristic
252C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radicalC2×#4084C2
hypercenter#390D5
hypocenterC5#2620
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#4082C2semidirect
3. C2×#4043C2semidirect
4. C2×#4018C2semidirect
5. C5×#390C2semidirect
6. #65265C2semidirect
7. #65248C2semidirect
8. #65237C2semidirect
9. #65218C2semidirect
63. C4×Dic5D4semidirect
71. #390D5semidirect
223. C5#2620semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
C2×#40822
C2×#40432
C2×#40182
C5×#3902
#652652
#652482
#652372
#652182
#26205