Finite Group Databasebeta

← #10608082

Subgroups — #10608082

Order 640.

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

normal subgroups only

This group has 576 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×#4082C211yesmaximal, characteristic
2C2×#4082C211yesmaximal, characteristic
3C2×#4033C211yesmaximal, characteristic, complemented
4C2×#4015C211yesmaximal, characteristic, complemented
5C5×#390C211yesmaximal, characteristic
6#65263C211yesmaximal, characteristic, complemented
7#65240C214yesmaximal, complemented
8#65237C214yesmaximal, complemented
9C4×#683C211yesmaximal, characteristic, complemented
10C22×C5×D4C2211yescharacteristic
11C22×Dic10C2211yescharacteristic
12C22×Dic10C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C22⋊C4C2214yes—
15C2×C42×C5C2211yescharacteristic
16C2×C22⋊Dic5C2212yes—
17C2×C22⋊Dic5C2214yes—
18C2×C22⋊Dic5C2212yes—
19#4082—24no—
20#4082—24no—
21C2×C4⋊Dic5C2214yes—
22C2×#683C2212yes—
23C2×#683C2212yes—
24C2×C4×Dic5C2212yes—
25C2×C4×Dic5C2211yescharacteristic
26C2×C4×Dic5C2211yescharacteristic
27#4033—24no—
28#4015—24no—
29#3963C2212yes—
30#3963C2214yes—
31#2545—51nomaximal, Sylow
32C24×C5C2312yes—
33C2×C5×D4—22no—
34C2×C5×D4—22no—
35C2×C5×D4—22no—
36C22×C4×C5C2312yes—
37C22×C4×C5C2312yes—
38C22×C4×C5C2311yescharacteristic
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2311yescharacteristic
43C22×Dic5C2311yescharacteristic
44C2×Dic10—22no—
45C2×Dic10—22no—
46C2×Dic10—22no—
47C2×Dic10—22no—
48C2×Dic10—22no—
49C2×Dic10D414yes—
50C5×C4⋊C4—22no—
51C5×C22⋊C4—28no—
52C42×C5—22no—
53C22⋊Dic5—28no—
54C22⋊Dic5—24no—
55C22⋊Dic5—24no—
56C4⋊Dic5—28no—
57#683—24no—
58#683—24no—
59C4×Dic5—22no—
60C4×Dic5—24no—
61C4×Dic5D414yes—
62C2×C4⋊Q8—51no—
63C2×#124—51no—
64C2×#124—51no—
65C2×C4×Q8—51no—
66#396—54no—
67#391—51no—
68#390D511yescharacteristic
69#388—54no—
70C4×C4⋊C4—51no—
71C23×C5—22no—
72C23×C5—22no—
73C23×C5—24no—
74C23×C5—22no—
75C23×C5—24no—
76C23×C5C2411yescharacteristic
77C5×D4—44no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—24no—
82C2×C4×C5C2×D412yes—
83C2×C4×C5C2×D412yes—
84C2×C4×C5#4112yes—
85C2×C4×C5#4114yes—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5—22no—
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×Dic5C2×D414yes—
99Dic10—28no—
100Dic10—44no—
101C22×Q8—51no—
102C22×Q8—51no—
103C22×D4C2×D511yescharacteristic
104C4⋊Q8—104no—
105#124—104no—
106#124—104no—
107C4×Q8—104no—
108C2×C4⋊C4—52no—
109C2×C4⋊C4—54no—
110C2×C4⋊C4—52no—
111C2×C4⋊C4C2×D511yescharacteristic
112C2×C22⋊C4—52no—
113C2×C22⋊C4—54no—
114C2×C22⋊C4—52no—
115C2×C22⋊C4C2×D514yes—
116C2×C42—51no—
117C2×C42—52no—
118C2×C42—51no—
119C2×C42C2×D511yescharacteristic
120#95—52no—
121#95—54no—
122C22×C5—42no—
123C22×C5—42no—
124C22×C5—42no—
125C22×C5—44no—
126C22×C5—44no—
127C22×C5C22×D411yescharacteristic
128C22×C5C22×D411yescharacteristic
129C22×C5C2×#4111yescharacteristic
130C22×C5C2×#4112yes—
131C22×C52−1+412yes—
132C4×C5—24no—
133C4×C5—42no—
134C4×C5#12114yes—
135Dic5—24no—
136Dic5—22no—
137Dic5—22no—
138Dic5—42no—
139Dic5—42no—
140C24C22×D512yes—
141C2×Q8—54no—
142C2×Q8—102no—
143C2×Q8—102no—
144C2×Q8—102no—
145C2×Q8—102no—
146C2×Q8—102no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C22×C4—51no—
151C22×C4—51no—
152C22×C4—52no—
153C22×C4—52no—
154C22×C4—52no—
155C22×C4C22×D512yes—
156C22×C4C22×D512yes—
157C22×C4C22×D511yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—104no—
160C4⋊C4—108no—
161C4⋊C4—104no—
162C22⋊C4—28no—
163C22⋊C4—104no—
164C22⋊C4—104no—
165C22⋊C4—108no—
166C42—22no—
167C42—54no—
168C42—104no—
169C42—102no—
170C2×C5—82no—
171C2×C5C2×#12111yescharacteristic
172C2×C5#53211yescharacteristic
173C2×C5#53611yescharacteristic
174C2×C5#54812yes—
175C2×C5#55612yes—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23C23×D511yescharacteristic
182Q8—108no—
183Q8—204no—
184D4—44no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—52no—
190C2×C4—52no—
191C2×C4—54no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4#70912yes—
203C2×C4D4×D512yes—
204C2×C4D4×D512yes—
205C2×C4#71114yes—
206C5#254511yescharacteristic, complemented, Sylow
207C22—42no—
208C22—42no—
209C22—44no—
210C22—44no—
211C22—42no—
212C22C2×#70911yescharacteristic
213C22C2×D4×D511yescharacteristic
214C22C2×D4×D511yescharacteristic
215C22C2×#71112yes—
216C22#415012yes—
217C4—24no—
218C4—42no—
219C4—102no—
220C4—104no—
221C4—102no—
222C4—202no—
223C4—202no—
224C4#403014yes—
225C2—82no—
226C2C2×#403011yescharacteristic
227C2#6587612yes—
228C2#6593212yes—
229C2#6600911yescharacteristic
230C2#6605011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radicalC22×C5×D4C22
hypercenter#390D5
hypocenterC5#2545
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
3. C2×#4033C2semidirect
4. C2×#4015C2semidirect
6. #65263C2semidirect
7. #65240C2semidirect
8. #65237C2semidirect
9. C4×#683C2semidirect
206. C5#2545semidirect

Maximal subgroups

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

subgroupindex
C2×#40822
C2×#40332
C2×#40152
C5×#3902
#652632
#652402
#652372
C4×#6832
#25455