Finite Group Databasebeta

← #10614720

Subgroups — #10614720

Order 640.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×C5×#124C211yesmaximal, characteristic
2C2×#4081C211yesmaximal, characteristic
3#65518C212yesmaximal, complemented
4#65509C211yesmaximal, characteristic
5#65241C212yesmaximal, complemented
6#65241C212yesmaximal, complemented
7#65235C212yesmaximal, complemented
8#65231C212yesmaximal, complemented
9#65222C212yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×C5×D4C2211yescharacteristic
12C23×Dic5C2211yescharacteristic
13C23×Dic5C2211yescharacteristic
14C5×#124—24no—
15C2×C5×C22⋊C4C2212yes—
16C2×C5×C22⋊C4C2212yes—
17C2×C42×C5C2211yescharacteristic
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2211yescharacteristic
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2211yescharacteristic
22#4081—24no—
23C2×#683C2212yes—
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×#683C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#2717—51nomaximal, Sylow
34C24×C5C2311yescharacteristic
35C24×C5C2311yescharacteristic
36C2×C5×Q8—22no—
37C2×C5×Q8—41no—
38C2×C5×Q8—41no—
39C2×C5×D4—41no—
40C2×C5×D4—41no—
41C2×C5×D4D414yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×Dic5—22no—
46C22×Dic5—24no—
47C22×Dic5—41no—
48C22×Dic5—41no—
49C22×Dic5—41no—
50C22×Dic5C2312yes—
51C22×Dic5C2312yes—
52C22×Dic5C2312yes—
53C22×Dic5C2312yes—
54C5×C22⋊C4—24no—
55C5×C22⋊C4—24no—
56C42×C5—22no—
57C22⋊Dic5—22no—
58C22⋊Dic5—24no—
59C22⋊Dic5—24no—
60C22⋊Dic5—41no—
61#683—24no—
62#683—24no—
63#683—42no—
64#683—42no—
65C2×#124D511yescharacteristic
66C2×#123—51no—
67#397—52no—
68#397—52no—
69#395—52no—
70#387—52no—
71#386—51no—
72#385—52no—
73#380—52no—
74C23×C5—22no—
75C23×C5—22no—
76C23×C5—21no—
77C23×C5—21no—
78C23×C5—21no—
79C23×C5—21no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5C2×D412yes—
83C23×C5C2×D412yes—
84C23×C5C2411yescharacteristic
85C5×Q8—44no—
86C5×D4—44no—
87C2×C4×C5—21no—
88C2×C4×C5—21no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5C2×D412yes—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—44no—
108C2×Dic5—44no—
109C2×Dic5#4114yes—
110C22×Q8C2×D511yescharacteristic
111C22×D4C2×D511yescharacteristic
112C23×C4—51no—
113C23×C4—51no—
114#124—24no—
115#123—104no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4—52no—
118C2×C4⋊C4—52no—
119C2×C4⋊C4—52no—
120C2×C22⋊C4—52no—
121C2×C22⋊C4—51no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4—51no—
124C2×C22⋊C4C2×D512yes—
125C2×C22⋊C4C2×D512yes—
126C2×C42C2×D511yescharacteristic
127#95—52no—
128#95—52no—
129#95—52no—
130#95—52no—
131#95—52no—
132#95—52no—
133C22×C5—22no—
134C22×C5—22no—
135C22×C5—24no—
136C22×C5—24no—
137C22×C5—42no—
138C22×C5—42no—
139C22×C5—41no—
140C22×C5—41no—
141C22×C5—41no—
142C22×C5—41no—
143C22×C5C22×D411yescharacteristic
144C22×C5C2×#4112yes—
145C22×C5H(5,𝔽2)11yescharacteristic
146C22×C5H(5,𝔽2)11yescharacteristic
147C22×C52−1+412yes—
148C4×C5—22no—
149C4×C5—42no—
150C4×C5—42no—
151Dic5—24no—
152Dic5—42no—
153Dic5—42no—
154Dic5—42no—
155C24C22×D511yescharacteristic
156C24C22×D511yescharacteristic
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4C22⋊D514yes—
163C22×C4—52no—
164C22×C4—52no—
165C22×C4—52no—
166C22×C4—52no—
167C22×C4—104no—
168C22×C4—102no—
169C22×C4—201no—
170C22×C4—201no—
171C22×C4—201no—
172C22×C4C22×D512yes—
173C22×C4C22×D512yes—
174C22×C4C22×D511yescharacteristic
175C4⋊C4—104no—
176C4⋊C4—104no—
177C4⋊C4—202no—
178C4⋊C4—202no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—104no—
182C22⋊C4—102no—
183C22⋊C4—104no—
184C22⋊C4—201no—
185C42—22no—
186C2×C5—42no—
187C2×C5—42no—
188C2×C5#53411yescharacteristic
189C2×C5#54212yes—
190C2×C5#54712yes—
191C2×C5#55111yescharacteristic
192C2×C5#56311yescharacteristic
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23C2×C22⋊D512yes—
202C23C2×C22⋊D512yes—
203C23C23×D511yescharacteristic
204Q8—44no—
205D4—44no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—54no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—102no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—102no—
226C2×C4—204no—
227C2×C4—204no—
228C2×C4C2×C22⋊D512yes—
229C5#271711yescharacteristic, complemented, Sylow
230C22—22no—
231C22—22no—
232C22—24no—
233C22—24no—
234C22—42no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22#414411yescharacteristic
241C22#414411yescharacteristic
242C22D5×#4112yes—
243C22#415012yes—
244C22C22×C22⋊D511yescharacteristic
245C4—22no—
246C4—42no—
247C4—42no—
248C4—104no—
249C4—202no—
250C4—202no—
251C4—202no—
252C2—42no—
253C2—42no—
254C2#6600312yes—
255C2#6600611yescharacteristic
256C2#6601311yescharacteristic
257C2#6613811yescharacteristic
258C2#6615812yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#124C2
prime radicalC22×C5×D4C22
hypercenterC2×#124D5
hypocenterC5#2717
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
3. #65518C2semidirect
5. #65241C2semidirect
6. #65241C2semidirect
7. #65235C2semidirect
8. #65231C2semidirect
9. #65222C2semidirect
229. C5#2717semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1242
C2×#40812
#655182
#655092
#652412
#652352
#652312
#652222
#27175