Finite Group Databasebeta

← #10608067

Subgroups — #10608067

Order 640.

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

normal subgroups only

This group has 596 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×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C5×#390C211yesmaximal, characteristic, complemented
4#65510C211yesmaximal, characteristic, complemented
5#65266C211yesmaximal, characteristic, complemented
6#65245C214yesmaximal, complemented
7#65241C214yesmaximal, complemented
8C4×#685C211yesmaximal, characteristic, complemented
9#65225C211yesmaximal, characteristic, complemented
10C22×C5×D4C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×C4×D5C2211yescharacteristic
13C22×C4×D5C2212yes—
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C22⋊C4C2214yes—
16C2×C42×C5C2211yescharacteristic
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2214yes—
19#4084—24no—
20C2×#685C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2211yescharacteristic
23C2×#683C2212yes—
24C2×C4×Dic5C2211yescharacteristic
25D5×C4⋊C4—24no—
26#3963C2214yes—
27#3963C2212yes—
28#3963C2214yes—
29#2504—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—28no—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C22×Dic5C2311yescharacteristic
44C2×C4×D5—22no—
45C2×C4×D5—22no—
46C2×C4×D5—24no—
47C2×C4×D5—44no—
48C2×C4×D5D414yes—
49C5×C4⋊C4—22no—
50C5×C22⋊C4—44no—
51C42×C5—22no—
52C22⋊Dic5—24no—
53C22⋊Dic5—44no—
54#685—24no—
55C4⋊Dic5—22no—
56C4⋊Dic5—22no—
57#683—24no—
58C4×Dic5—22no—
59C2×#121—51no—
60C22×C4⋊C4—51no—
61#397—54no—
62#390—51no—
63#390D511yescharacteristic, complemented
64#386—54no—
65#382—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—21no—
75C22×D5—21no—
76C22×D5—22no—
77C22×D5C2×D412yes—
78C22×D5#4114yes—
79C5×D4—44no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—24no—
85C2×C4×C5C2×D412yes—
86C2×C4×C5#4112yes—
87C2×C4×C5#4112yes—
88C22⋊D5—48no—
89C2×Dic5—21no—
90C2×Dic5—21no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—24no—
96C2×Dic5—22no—
97C2×Dic5—24no—
98C2×Dic5C2×D412yes—
99C4×D5—28no—
100C4×D5—48no—
101C22×D4—52no—
102C22×D4C2×D511yescharacteristic
103C23×C4—51no—
104C23×C4—52no—
105#121—104no—
106C2×C4⋊C4—51no—
107C2×C4⋊C4—52no—
108C2×C4⋊C4—51no—
109C2×C4⋊C4—104no—
110C2×C4⋊C4C2×D511yescharacteristic
111C2×C22⋊C4—52no—
112C2×C22⋊C4—52no—
113C2×C22⋊C4—54no—
114C2×C22⋊C4C2×D514yes—
115C2×C42—51no—
116C2×C42C2×D511yescharacteristic
117#95—52no—
118#95—54no—
119#95—54no—
120C22×C5—42no—
121C22×C5—42no—
122C22×C5—42no—
123C22×C5—44no—
124C22×C5—44no—
125C22×C5C22×D411yescharacteristic
126C22×C5C2×#4111yescharacteristic
127C22×C5C2×#4111yescharacteristic
128C22×C5C2×#4112yes—
129C22×C5H(5,𝔽2)12yes—
130C2×D5—22no—
131C2×D5—22no—
132C2×D5—24no—
133C2×D5—24no—
134C2×D5#12414yes—
135C4×C5—42no—
136C4×C5—42no—
137C4×C5#12314yes—
138Dic5—22no—
139Dic5—42no—
140Dic5—42no—
141Dic5—42no—
142C24—51no—
143C24C22×D512yes—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—104no—
148C2×D4—108no—
149C22×C4—51no—
150C22×C4—54no—
151C22×C4—52no—
152C22×C4—52no—
153C22×C4—52no—
154C22×C4—104no—
155C22×C4—102no—
156C22×C4—102no—
157C22×C4—204no—
158C22×C4C22×D512yes—
159C22×C4C22×D512yes—
160C22×C4C22×D511yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—104no—
163C4⋊C4—102no—
164C4⋊C4—102no—
165C22⋊C4—44no—
166C22⋊C4—104no—
167C22⋊C4—104no—
168C22⋊C4—204no—
169C42—22no—
170C42—102no—
171C2×C5—82no—
172C2×C5C2×#12311yescharacteristic
173C2×C5C2×#12411yescharacteristic
174C2×C5#53511yescharacteristic
175C2×C5#55312yes—
176C2×C5#55312yes—
177D5—24no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—24no—
182C23—24no—
183C23—52no—
184C23—54no—
185C23—101no—
186C23—101no—
187C23—102no—
188C23C23×D511yescharacteristic
189D4—44no—
190D4—208no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—52no—
197C2×C4—101no—
198C2×C4—101no—
199C2×C4—108no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—104no—
205C2×C4—102no—
206C2×C4—104no—
207C2×C4—208no—
208C2×C4#70912yes—
209C2×C4D4×D512yes—
210C2×C4#71112yes—
211C5#250411yescharacteristic, complemented, Sylow
212C22—42no—
213C22—42no—
214C22—44no—
215C22—44no—
216C22—42no—
217C22—54no—
218C22—102no—
219C22—102no—
220C22—104no—
221C22—104no—
222C22C2×#70911yescharacteristic
223C22C2×D4×D511yescharacteristic
224C22C2×#71111yescharacteristic
225C22#414412yes—
226C22D5×#4112yes—
227C4—42no—
228C4—42no—
229C4—102no—
230C4—202no—
231C4—202no—
232C4—202no—
233C4#402914yes—
234C2—82no—
235C2—104no—
236C2C2×#402911yescharacteristic
237C2#6589212yes—
238C2#6594812yes—
239C2D5×#12411yescharacteristic
240C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radicalC2×#4084C2
hypercenter#390D5
hypocenterC5#2504
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×D5×C4⋊C4C2semidirect
3. C5×#390C2semidirect
4. #65510C2semidirect
5. #65266C2semidirect
6. #65245C2semidirect
7. #65241C2semidirect
8. C4×#685C2semidirect
9. #65225C2semidirect
63. #390D5semidirect
211. C5#2504semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
C2×D5×C4⋊C42
C5×#3902
#655102
#652662
#652452
#652412
C4×#6852
#652252
#25045