Finite Group Databasebeta

← #10609857

Subgroups — #10609857

Order 640.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C5×#399C211yesmaximal, characteristic, complemented
2#65283C212yesmaximal, complemented
3#65280C212yesmaximal, complemented
4#65275C212yesmaximal, complemented
5#65250C212yesmaximal, complemented
6#65249C211yesmaximal, characteristic
7#65247C212yesmaximal, complemented
8#65235C211yesmaximal, characteristic, complemented
9#64964C212yesmaximal, complemented
10C22×C22⋊D5C2212yes—
11C23×Dic5C2211yescharacteristic
12C22×D20C2211yescharacteristic
13C2×C5×C4⋊C4C2212yes—
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C22⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C22⋊Dic5C2211yescharacteristic
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2212yes—
24C2×#683C2212yes—
25C5×#95C2211yescharacteristic
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2211yescharacteristic
31#2826—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C23×D5C2312yes—
34C22×C4×C5C2311yescharacteristic
35C22×C4×C5C2312yes—
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C2×C22⋊D5—24no—
39C2×C22⋊D5—42no—
40C2×C22⋊D5—42no—
41C22×Dic5—42no—
42C22×Dic5—41no—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C2×D20—41no—
47C2×D20—42no—
48C5×C4⋊C4—42no—
49C5×C4⋊C4—41no—
50C5×C22⋊C4—42no—
51C5×C22⋊C4—41no—
52C22⋊Dic5—22no—
53#685—42no—
54#685—42no—
55#685—42no—
56#685—42no—
57#685—42no—
58C4⋊Dic5—42no—
59#683—42no—
60#400—52no—
61#399D511yescharacteristic, complemented
62#397—52no—
63#396—52no—
64#394—52no—
65#394—52no—
66#392—51no—
67#386—52no—
68#385—51no—
69C23×C5—21no—
70C23×C5—22no—
71C23×C5—22no—
72C23×C5—21no—
73C23×C5#4112yes—
74C23×C5C2411yescharacteristic
75C22×D5—22no—
76C22×D5—22no—
77C22×D5—22no—
78C22×D5—22no—
79C22×D5—22no—
80C22×D5—22no—
81C22×D5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5—21no—
88C2×C4×C5—22no—
89C2×C4×C5—21no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C22⋊D5—48no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—44no—
103D20—44no—
104C22×D4—52no—
105C22×D4—51no—
106C23×C4—51no—
107C2×C4⋊C4—52no—
108C2×C4⋊C4—52no—
109C2×C4⋊C4C2×D512yes—
110C2×C4⋊C4C2×D511yescharacteristic
111C2×C22⋊C4—51no—
112C2×C22⋊C4—52no—
113C2×C22⋊C4—52no—
114C2×C22⋊C4—52no—
115C2×C22⋊C4—52no—
116C2×C22⋊C4—52no—
117C2×C22⋊C4C2×D511yescharacteristic
118C2×C22⋊C4C2×D512yes—
119#95—52no—
120#95—52no—
121#95—52no—
122#95—51no—
123#95—52no—
124#95C2×D511yescharacteristic
125C22×C5—24no—
126C22×C5—22no—
127C22×C5—41no—
128C22×C5—41no—
129C22×C5—42no—
130C22×C5C2×#4111yescharacteristic
131C22×C5H(5,𝔽2)12yes—
132C22×C5H(5,𝔽2)11yescharacteristic
133C22×C5H(5,𝔽2)12yes—
134C22×C52−1+411yescharacteristic
135C2×D5—42no—
136C2×D5—42no—
137C2×D5—42no—
138C2×D5—42no—
139C2×D5—42no—
140C2×D5—42no—
141C2×D5—42no—
142C4×C5—41no—
143C4×C5—42no—
144C4×C5—41no—
145C4×C5—42no—
146Dic5—42no—
147Dic5—42no—
148Dic5—42no—
149C24—52no—
150C24C22×D511yescharacteristic
151C2×D4—104no—
152C2×D4—201no—
153C2×D4—202no—
154C2×D4—202no—
155C2×D4—202no—
156C22×C4—52no—
157C22×C4—52no—
158C22×C4—52no—
159C22×C4—201no—
160C22×C4—202no—
161C22×C4C22×D511yescharacteristic
162C22×C4C22×D512yes—
163C22×C4C22×D512yes—
164C22×C4C22×D511yescharacteristic
165C4⋊C4—41no—
166C4⋊C4—42no—
167C4⋊C4—202no—
168C4⋊C4—202no—
169C22⋊C4—42no—
170C22⋊C4—41no—
171C22⋊C4—102no—
172C22⋊C4—202no—
173C22⋊C4—202no—
174C22⋊C4—202no—
175C22⋊C4—202no—
176C22⋊C4—202no—
177C2×C5—42no—
178C2×C5#53812yes—
179C2×C5#53911yescharacteristic
180C2×C5#56011yescharacteristic
181C2×C5#56011yescharacteristic
182C2×C5#56212yes—
183D5—82no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23—21no—
188C23—102no—
189C23—102no—
190C23—102no—
191C23—102no—
192C23—102no—
193C23—102no—
194C23—102no—
195C23#71312yes—
196C23C23×D511yescharacteristic
197D4—204no—
198D4—208no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—102no—
210C2×C4—102no—
211C2×C4—102no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—204no—
219C5#282611yescharacteristic, complemented, Sylow
220C22—24no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—202no—
226C22—202no—
227C22—202no—
228C22—202no—
229C22—202no—
230C22—202no—
231C22—202no—
232C22#414412yes—
233C22#414411yescharacteristic
234C22C2×#71311yescharacteristic
235C22#414912yes—
236C22#415011yescharacteristic
237C4—41no—
238C4—42no—
239C4—41no—
240C4—42no—
241C4—202no—
242C4—202no—
243C4—202no—
244C2—42no—
245C2—402no—
246C2#6597812yes—
247C2#6597911yescharacteristic
248C2#6600011yescharacteristic
249C2#6600112yes—
250C2#6605411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radical#65249C2
hypercenter#399D5
hypocenterC5#2826
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C5×#399C2semidirect
2. #65283C2semidirect
3. #65280C2semidirect
4. #65275C2semidirect
5. #65250C2semidirect
7. #65247C2semidirect
8. #65235C2semidirect
9. #64964C2semidirect
61. #399D5semidirect
219. C5#2826semidirect

Maximal subgroups

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

subgroupindex
C5×#3992
#652832
#652802
#652752
#652502
#652492
#652472
#652352
#649642
#28265