Finite Group Databasebeta

← #10609715

Subgroups — #10609715

Order 640.

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

normal subgroups only

This group has 504 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×#4040C212yesmaximal
2C5×#398C211yesmaximal, characteristic, complemented
3#65278C212yesmaximal, complemented
4#65277C211yesmaximal, characteristic
5#65272C211yesmaximal, characteristic, complemented
6#65261C212yesmaximal, complemented
7#64964C212yesmaximal, complemented
8#64964C212yesmaximal, complemented
9#64957C212yesmaximal, complemented
10C22×D20C2211yescharacteristic
11C22×C4×D5C2212yes—
12C2×C5×C4⋊C4C2212yes—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×#685C2212yes—
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×#683C2212yes—
24C2×C4×Dic5C2211yescharacteristic
25#4040—28no—
26C5×#95C2212yes—
27C5×#95C2211yescharacteristic
28#3963C2212yes—
29#3963C2211yescharacteristic
30#3963C2212yes—
31#3963C2212yes—
32#2724—51nomaximal, Sylow
33C23×D5C2312yes—
34C22×C4×C5C2312yes—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2311yescharacteristic
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C2×D20—41no—
43C2×D20—41no—
44C2×D20D414yes—
45C2×C4×D5—24no—
46C2×C4×D5—42no—
47C2×C4×D5—42no—
48C5×C4⋊C4—24no—
49C5×C4⋊C4—41no—
50C5×C4⋊C4—41no—
51#685—24no—
52#685—24no—
53#685—24no—
54#685—24no—
55#685—24no—
56C4⋊Dic5—41no—
57#683—24no—
58#683—24no—
59#683—42no—
60C4×Dic5—41no—
61C2×#123—52no—
62#398—51no—
63#398D511yescharacteristic, complemented
64#397—52no—
65#397—52no—
66#390—51no—
67#389—52no—
68#385—52no—
69#385—52no—
70C23×C5C2411yescharacteristic
71C22×D5—22no—
72C22×D5—22no—
73C22×D5—22no—
74C22×D5—22no—
75C22×D5—22no—
76C22×D5—22no—
77C22×D5C2×D414yes—
78C2×C4×C5—21no—
79C2×C4×C5—21no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5C2×D412yes—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5#4114yes—
100D20—44no—
101C4×D5—48no—
102C22×D4—51no—
103C23×C4—52no—
104#123—108no—
105C2×C4⋊C4—52no—
106C2×C4⋊C4—52no—
107C2×C4⋊C4—52no—
108C2×C4⋊C4—51no—
109C2×C4⋊C4C2×D512yes—
110C2×C4⋊C4C2×D511yescharacteristic
111C2×C4⋊C4C2×D511yescharacteristic
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—52no—
119#95—52no—
120#95—52no—
121#95—51no—
122#95C2×D512yes—
123#95C2×D511yescharacteristic
124C22×C5C22×D411yescharacteristic
125C22×C5C2×#4112yes—
126C22×C5H(5,𝔽2)11yescharacteristic
127C22×C5H(5,𝔽2)11yescharacteristic
128C22×C52−1+412yes—
129C2×D5—24no—
130C2×D5—24no—
131C2×D5—24no—
132C2×D5—42no—
133C2×D5—42no—
134C2×D5—42no—
135C2×D5—42no—
136C4×C5—22no—
137C4×C5—42no—
138C4×C5—42no—
139C4×C5—41no—
140C4×C5—41no—
141Dic5—24no—
142Dic5—42no—
143Dic5—42no—
144C24—52no—
145C2×D4—54no—
146C2×D4—201no—
147C2×D4—201no—
148C22×C4—52no—
149C22×C4—52no—
150C22×C4—52no—
151C22×C4—104no—
152C22×C4—202no—
153C22×C4—202no—
154C22×C4C22×D511yescharacteristic
155C22×C4C22×D512yes—
156C22×C4C22×D511yescharacteristic
157C22×C4C22×D512yes—
158C22×C4C22×D511yescharacteristic
159C4⋊C4—24no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—104no—
163C4⋊C4—104no—
164C4⋊C4—202no—
165C4⋊C4—201no—
166C22⋊C4—104no—
167C22⋊C4—104no—
168C22⋊C4—104no—
169C22⋊C4—104no—
170C22⋊C4—104no—
171C42—201no—
172C2×C5#53411yescharacteristic
173C2×C5#53912yes—
174C2×C5#54712yes—
175C2×C5#55911yescharacteristic
176C2×C5#56311yescharacteristic
177D5—44no—
178C23—54no—
179C23—102no—
180C23—102no—
181C23—102no—
182C23—102no—
183C23—102no—
184C23—102no—
185C23C23×D511yescharacteristic
186D4—204no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—54no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—208no—
209C2×C4D4×D512yes—
210C5#272411yescharacteristic, complemented, Sylow
211C22—104no—
212C22—104no—
213C22—104no—
214C22—202no—
215C22—202no—
216C22—202no—
217C22—202no—
218C22C2×D4×D511yescharacteristic
219C22#414411yescharacteristic
220C22#414712yes—
221C22D5×#4112yes—
222C22#414911yescharacteristic
223C4—22no—
224C4—42no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—104no—
229C4—202no—
230C4—202no—
231C2—204no—
232C2#6593911yescharacteristic
233C2#6596512yes—
234C2#6599811yescharacteristic
235C2#6602612yes—
236C2#6606911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radicalC22×D20C22
hypercenter#398D5
hypocenterC5#2724
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C5×#398C2semidirect
3. #65278C2semidirect
5. #65272C2semidirect
6. #65261C2semidirect
7. #64964C2semidirect
8. #64964C2semidirect
9. #64957C2semidirect
63. #398D5semidirect
210. C5#2724semidirect

Maximal subgroups

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

subgroupindex
C2×#40402
C5×#3982
#652782
#652772
#652722
#652612
#649642
#649572
#27245