Finite Group Databasebeta

← #10608251

Subgroups — #10608251

Order 640.

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

normal subgroups only

This group has 512 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×#4092C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic
3C5×#391C211yesmaximal, characteristic, complemented
4#65519C211yesmaximal, characteristic, complemented
5#65266C211yesmaximal, characteristic, complemented
6C4×#685C211yesmaximal, characteristic
7#65225C211yesmaximal, characteristic, complemented
8#64953C214yesmaximal
9#64947C214yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×C4×D5C2212yes—
12C22×C4×D5C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C42×C5C2211yescharacteristic
15#4092—24no—
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×C4⋊Dic5C2211yescharacteristic
19C2×C4⋊Dic5C2211yescharacteristic
20C2×#683C2212yes—
21C2×#683C2212yes—
22C2×C4×Dic5C2211yescharacteristic
23D5×C4⋊C4—24no—
24C5×#95C2214yes—
25#3963C2212yes—
26#3963C2214yes—
27#3963C2214yes—
28#3963C2214yes—
29#2503—51nomaximal, Sylow
30C23×D5C2311yescharacteristic
31C2×C5×Q8—22no—
32C2×C5×Q8—22no—
33C2×C5×Q8—22no—
34C22×C4×C5C2312yes—
35C22×C4×C5C2312yes—
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×Dic5C2312yes—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2311yescharacteristic
42C2×C4×D5—22no—
43C2×C4×D5—22no—
44C2×C4×D5—24no—
45C2×C4×D5—28no—
46C2×C4×D5D414yes—
47C5×C4⋊C4—22no—
48C42×C5—22no—
49#685—24no—
50#685—24no—
51C4⋊Dic5—22no—
52C4⋊Dic5—22no—
53#683—24no—
54#683—24no—
55C4×Dic5—22no—
56C2×C22⋊Q8—51no—
57C22×C4⋊C4—51no—
58#400—54no—
59#391—51no—
60#391D511yescharacteristic, complemented
61#382—51no—
62#382—51no—
63#381—54no—
64C4×C22⋊C4—51no—
65C23×C5C2411yescharacteristic
66C22×D5—21no—
67C22×D5—21no—
68C22×D5—22no—
69C22×D5C2×D412yes—
70C22×D5#4114yes—
71C5×Q8—44no—
72C2×C4×C5—22no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—24no—
77C2×C4×C5—22no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5C2×D412yes—
81C2×C4×C5#4112yes—
82C2×C4×C5#4112yes—
83C2×Dic5—21no—
84C2×Dic5—21no—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5—22no—
89C2×Dic5—24no—
90C2×Dic5—24no—
91C2×Dic5—22no—
92C2×Dic5C2×D412yes—
93C4×D5—28no—
94C4×D5—48no—
95C22×Q8C2×D511yescharacteristic
96C23×C4—51no—
97C23×C4—52no—
98C22⋊Q8—104no—
99C2×C4⋊C4—51no—
100C2×C4⋊C4—52no—
101C2×C4⋊C4—51no—
102C2×C4⋊C4—52no—
103C2×C4⋊C4—104no—
104C2×C4⋊C4C2×D511yescharacteristic
105C2×C22⋊C4—52no—
106C2×C22⋊C4—52no—
107C2×C42—51no—
108C2×C42C2×D511yescharacteristic
109#95—52no—
110#95—54no—
111#95—54no—
112#95—54no—
113#95C2×D514yes—
114C22×C5C22×D411yescharacteristic
115C22×C5C2×#4111yescharacteristic
116C22×C5C2×#4111yescharacteristic
117C22×C5C2×#4112yes—
118C22×C52−1+412yes—
119C2×D5—22no—
120C2×D5—22no—
121C2×D5—24no—
122C2×D5—24no—
123C2×D5#12414yes—
124C4×C5—42no—
125C4×C5—42no—
126C4×C5—42no—
127C4×C5#12314yes—
128Dic5—22no—
129Dic5—42no—
130Dic5—42no—
131Dic5—42no—
132C24—51no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—22no—
136C22×C4—51no—
137C22×C4—54no—
138C22×C4—52no—
139C22×C4—52no—
140C22×C4—52no—
141C22×C4—104no—
142C22×C4—108no—
143C22×C4—102no—
144C22×C4—102no—
145C22×C4C22×D512yes—
146C22×C4C22×D512yes—
147C22×C4C22×D512yes—
148C22×C4C22×D511yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—104no—
151C4⋊C4—104no—
152C4⋊C4—102no—
153C4⋊C4—102no—
154C22⋊C4—104no—
155C22⋊C4—104no—
156C42—22no—
157C42—102no—
158C2×C5C2×#12311yescharacteristic
159C2×C5C2×#12411yescharacteristic
160C2×C5#53611yescharacteristic
161C2×C5#55212yes—
162C2×C5#55212yes—
163D5—24no—
164C23—52no—
165C23—54no—
166C23—101no—
167C23—101no—
168C23—102no—
169C23C23×D511yescharacteristic
170Q8—44no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—52no—
180C2×C4—101no—
181C2×C4—101no—
182C2×C4—108no—
183C2×C4—102no—
184C2×C4—102no—
185C2×C4—102no—
186C2×C4—104no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—104no—
190C2×C4—208no—
191C2×C4#70912yes—
192C2×C4D4×D512yes—
193C2×C4#71112yes—
194C5#250311yescharacteristic, complemented, Sylow
195C22—54no—
196C22—102no—
197C22—102no—
198C22—104no—
199C22—104no—
200C22C2×#70911yescharacteristic
201C22C2×D4×D511yescharacteristic
202C22C2×#71111yescharacteristic
203C22#414712yes—
204C22D5×#4112yes—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—102no—
209C4—202no—
210C4—202no—
211C4—202no—
212C4#402914yes—
213C2—104no—
214C2C2×#402911yescharacteristic
215C2#6591512yes—
216C2#6597312yes—
217C2D5×#12411yescharacteristic
218C2#6605811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#391C2
prime radicalC23×D5C23
hypercenter#391D5
hypocenterC5#2503
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
3. C5×#391C2semidirect
4. #65519C2semidirect
5. #65266C2semidirect
7. #65225C2semidirect
9. #64947C2semidirect
60. #391D5semidirect
194. C5#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
C2×D5×C4⋊C42
C5×#3912
#655192
#652662
C4×#6852
#652252
#649532
#649472
#25035