Finite Group Databasebeta

← #10609739

Subgroups — #10609739

Order 640.

456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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×#4044C212yesmaximal, complemented
2C5×#398C211yesmaximal, characteristic, complemented
3#65283C212yesmaximal, complemented
4#65283C212yesmaximal, complemented
5#65280C211yesmaximal, characteristic
6#65265C211yesmaximal, characteristic, complemented
7#64965C212yesmaximal, complemented
8#64961C212yesmaximal
9#64946C212yesmaximal, complemented
10C22×D20C2211yescharacteristic
11C2×C5×C4⋊C4C2212yes—
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×#685C2212yes—
15C2×#685C2212yes—
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×C4⋊Dic5C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×#683C2212yes—
23C2×#683C2212yes—
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×C4×Dic5C2211yescharacteristic
27#4044—28no—
28C5×#95C2212yes—
29C5×#95C2211yescharacteristic
30#3963C2211yescharacteristic
31#3963C2212yes—
32#2823—51nomaximal, Sylow
33C23×D5C2312yes—
34C22×C4×C5C2312yes—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2312yes—
38C22×C4×C5C2311yescharacteristic
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C2×D20—42no—
43C2×D20—41no—
44C5×C4⋊C4—24no—
45C5×C4⋊C4—41no—
46C5×C4⋊C4—41no—
47#685—24no—
48#685—24no—
49#685—24no—
50#685—42no—
51#685—42no—
52#685—42no—
53C4⋊Dic5—24no—
54C4⋊Dic5—41no—
55#683—24no—
56#683—42no—
57#683—42no—
58#683—42no—
59C4×Dic5D414yescomplemented
60C2×#126—52no—
61#398—52no—
62#398D511yescharacteristic, complemented
63#396—52no—
64#396—52no—
65#396—52no—
66#394—51no—
67#394—52no—
68#384—51no—
69C23×C5C2411yescharacteristic
70C22×D5—22no—
71C22×D5—22no—
72C22×D5—22no—
73C22×D5—22no—
74C22×D5—22no—
75C22×D5—22no—
76C22×D5—22no—
77C2×C4×C5—21no—
78C2×C4×C5—21no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—21no—
86C2×C4×C5—21no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5C2×D412yes—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5C2×D414yes—
99D20—44no—
100C22×D4—51no—
101#126—108no—
102C2×C4⋊C4—52no—
103C2×C4⋊C4—52no—
104C2×C4⋊C4—52no—
105C2×C4⋊C4—52no—
106C2×C4⋊C4—51no—
107C2×C4⋊C4—52no—
108C2×C4⋊C4C2×D512yes—
109C2×C4⋊C4C2×D511yescharacteristic
110C2×C4⋊C4C2×D511yescharacteristic
111C2×C22⋊C4—52no—
112C2×C22⋊C4—52no—
113C2×C22⋊C4—52no—
114C2×C22⋊C4—52no—
115C2×C22⋊C4—52no—
116C2×C22⋊C4—52no—
117C2×C42—51no—
118#95—51no—
119#95—52no—
120#95C2×D512yes—
121#95C2×D511yescharacteristic
122C22×C5C22×D411yescharacteristic
123C22×C5H(5,𝔽2)11yescharacteristic
124C22×C5H(5,𝔽2)12yes—
125C22×C52−1+412yes—
126C22×C52−1+411yescharacteristic
127C2×D5—42no—
128C2×D5—42no—
129C2×D5—42no—
130C2×D5—42no—
131C2×D5—42no—
132C2×D5—42no—
133C2×D5—42no—
134C4×C5—22no—
135C4×C5—42no—
136C4×C5—42no—
137C4×C5—41no—
138C4×C5—41no—
139Dic5—24no—
140Dic5—42no—
141Dic5—42no—
142C24—52no—
143C2×D4—201no—
144C2×D4—202no—
145C22×C4—52no—
146C22×C4—52no—
147C22×C4—52no—
148C22×C4C22×D511yescharacteristic
149C22×C4C22×D512yes—
150C22×C4C22×D512yes—
151C22×C4C22×D511yescharacteristic
152C22×C4C22×D511yescharacteristic
153C4⋊C4—24no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—104no—
157C4⋊C4—104no—
158C4⋊C4—202no—
159C4⋊C4—202no—
160C4⋊C4—202no—
161C4⋊C4—201no—
162C22⋊C4—104no—
163C22⋊C4—104no—
164C22⋊C4—104no—
165C22⋊C4—202no—
166C22⋊C4—202no—
167C22⋊C4—202no—
168C42—54no—
169C2×C5#53711yescharacteristic
170C2×C5#53712yes—
171C2×C5#56011yescharacteristic
172C2×C5#56311yescharacteristic
173C2×C5#56312yes—
174D5—82no—
175C23—102no—
176C23—102no—
177C23—102no—
178C23—102no—
179C23—102no—
180C23—102no—
181C23—102no—
182C23C23×D511yescharacteristic
183D4—204no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—54no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—102no—
205C2×C4D4×D512yes—
206C5#282311yescharacteristic, complemented, Sylow
207C22—202no—
208C22—202no—
209C22—202no—
210C22—202no—
211C22—202no—
212C22—202no—
213C22—202no—
214C22C2×D4×D511yescharacteristic
215C22#414411yescharacteristic
216C22#414712yes—
217C22#414912yes—
218C22#415011yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—104no—
225C4—202no—
226C4—202no—
227C2—402no—
228C2#6593111yescharacteristic
229C2#6596312yes—
230C2#6600011yescharacteristic
231C2#6603312yes—
232C2#6606911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radicalC22×D20C22
hypercenter#398D5
hypocenterC5#2823
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#4044C2semidirect
2. C5×#398C2semidirect
3. #65283C2semidirect
4. #65283C2semidirect
6. #65265C2semidirect
7. #64965C2semidirect
9. #64946C2semidirect
59. C4×Dic5D4semidirect
62. #398D5semidirect
206. C5#2823semidirect

Maximal subgroups

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

subgroupindex
C2×#40442
C5×#3982
#652832
#652802
#652652
#649652
#649612
#649462
#28235