Finite Group Databasebeta

← #10609315

Subgroups — #10609315

Order 640.

556 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 129 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×#4031C212yesmaximal, complemented
2C5×#397C211yesmaximal, characteristic, complemented
3#65279C212yesmaximal, complemented
4#65249C211yesmaximal, characteristic
5#65247C212yesmaximal, complemented
6#65235C211yesmaximal, characteristic, complemented
7#64962C212yesmaximal, complemented
8#64961C212yesmaximal, complemented
9#64935C212yesmaximal, complemented
10C22×C22⋊D5C2212yes—
11C23×Dic5C2211yescharacteristic
12C22×D20C2211yescharacteristic
13C22×Dic10C2212yes—
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C22⋊C4C2212yes—
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C22⋊Dic5C2211yescharacteristic
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×#683C2212yes—
24C2×C4×Dic5C2212yes—
25#4031—28no—
26C5×#95C2212yes—
27C5×#95C2211yescharacteristic
28#3963C2211yescharacteristic
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#2712—51nomaximal, Sylow
33C24×C5C2311yescharacteristic
34C23×D5C2312yes—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2312yes—
37C22×C4×C5C2312yes—
38C22×C4×C5C2311yescharacteristic
39C2×C22⋊D5—24no—
40C2×C22⋊D5—24no—
41C2×C22⋊D5—42no—
42C22×Dic5—42no—
43C22×Dic5—41no—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C2×D20—42no—
48C2×D20—41no—
49C2×Dic10—24no—
50C2×Dic10—24no—
51C2×Dic10—42no—
52C5×C4⋊C4—41no—
53C5×C22⋊C4—24no—
54C5×C22⋊C4—41no—
55C22⋊Dic5D414yescomplemented
56#685—24no—
57#685—24no—
58#685—24no—
59#685—42no—
60#685—42no—
61#683—42no—
62C4×Dic5—24no—
63C2×#124—52no—
64#397D511yescharacteristic, complemented
65#394—52no—
66#393—52no—
67#392—51no—
68#391—52no—
69#388—52no—
70#386—52no—
71#385—51no—
72C23×C5—22no—
73C23×C5—21no—
74C23×C5—22no—
75C23×C5—21no—
76C23×C5C2×D412yes—
77C23×C5C2411yescharacteristic
78C22×D5—22no—
79C22×D5—22no—
80C22×D5—22no—
81C22×D5—22no—
82C22×D5—22no—
83C22×D5—22no—
84C22×D5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—21no—
93C2×C4×C5—21no—
94C2×C4×C5—22no—
95C22⋊D5—48no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—44no—
104C2×Dic5C2×D414yes—
105C2×Dic5#4114yes—
106D20—44no—
107Dic10—48no—
108C22×Q8—52no—
109C22×D4—52no—
110C22×D4—51no—
111C23×C4—51no—
112#124—108no—
113C2×C4⋊C4—52no—
114C2×C4⋊C4C2×D511yescharacteristic
115C2×C22⋊C4—51no—
116C2×C22⋊C4—52no—
117C2×C22⋊C4—52no—
118C2×C22⋊C4—52no—
119C2×C22⋊C4—52no—
120C2×C22⋊C4—52no—
121C2×C22⋊C4C2×D512yes—
122C2×C22⋊C4C2×D511yescharacteristic
123C2×C42—52no—
124#95—52no—
125#95—52no—
126#95—52no—
127#95—51no—
128#95C2×D512yes—
129#95C2×D511yescharacteristic
130C22×C5—22no—
131C22×C5—24no—
132C22×C5—42no—
133C22×C5—41no—
134C22×C5—41no—
135C22×C5C22×D411yescharacteristic
136C22×C5C2×#4112yes—
137C22×C5H(5,𝔽2)11yescharacteristic
138C22×C5H(5,𝔽2)12yes—
139C22×C52−1+411yescharacteristic
140C2×D5—42no—
141C2×D5—42no—
142C2×D5—42no—
143C2×D5—42no—
144C2×D5—42no—
145C2×D5—42no—
146C2×D5—42no—
147C4×C5—42no—
148C4×C5—42no—
149C4×C5—41no—
150C4×C5—41no—
151Dic5—24no—
152Dic5—24no—
153Dic5—42no—
154C24—52no—
155C24C22×D511yescharacteristic
156C2×Q8—104no—
157C2×Q8—104no—
158C2×Q8—202no—
159C2×D4—104no—
160C2×D4—104no—
161C2×D4—201no—
162C2×D4—202no—
163C2×D4—202no—
164C22×C4—52no—
165C22×C4—52no—
166C22×C4—52no—
167C22×C4—201no—
168C22×C4—202no—
169C22×C4C22×D511yescharacteristic
170C22×C4C22×D512yes—
171C22×C4C22×D511yescharacteristic
172C22×C4C22×D512yes—
173C4⋊C4—41no—
174C4⋊C4—202no—
175C22⋊C4—24no—
176C22⋊C4—41no—
177C22⋊C4—54no—
178C22⋊C4—104no—
179C22⋊C4—104no—
180C22⋊C4—104no—
181C22⋊C4—202no—
182C22⋊C4—202no—
183C42—104no—
184C2×C5—42no—
185C2×C5#53711yescharacteristic
186C2×C5#54212yes—
187C2×C5#54612yes—
188C2×C5#55911yescharacteristic
189C2×C5H(3,𝔽4)11yescharacteristic
190D5—82no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23—21no—
195C23—102no—
196C23—102no—
197C23—102no—
198C23—102no—
199C23—102no—
200C23—102no—
201C23—102no—
202C23D4×D512yes—
203C23C23×D511yescharacteristic
204Q8—208no—
205D4—204no—
206D4—208no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—54no—
218C2×C4—54no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—102no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—102no—
226C2×C4—204no—
227C5#271211yescharacteristic, complemented, Sylow
228C22—22no—
229C22—24no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—202no—
234C22—202no—
235C22—202no—
236C22—202no—
237C22—202no—
238C22—202no—
239C22—202no—
240C22C2×D4×D511yescharacteristic
241C22#414411yescharacteristic
242C22D5×#4112yes—
243C22#414912yes—
244C22#415011yescharacteristic
245C4—42no—
246C4—42no—
247C4—41no—
248C4—41no—
249C4—104no—
250C4—104no—
251C4—202no—
252C2—42no—
253C2—402no—
254C2#6593111yescharacteristic
255C2#6598812yes—
256C2#6599811yescharacteristic
257C2#6601711yescharacteristic
258C2#6604412yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radical#65249C2
hypercenter#397D5
hypocenterC5#2712
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#4031C2semidirect
2. C5×#397C2semidirect
3. #65279C2semidirect
5. #65247C2semidirect
6. #65235C2semidirect
7. #64962C2semidirect
8. #64961C2semidirect
9. #64935C2semidirect
55. C22⋊Dic5D4semidirect
64. #397D5semidirect
227. C5#2712semidirect

Maximal subgroups

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

subgroupindex
C2×#40312
C5×#3972
#652792
#652492
#652472
#652352
#649622
#649612
#649352
#27125