Finite Group Databasebeta

← #10614408

Subgroups — #10614408

Order 640.

612 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 157 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×#121C211yesmaximal, characteristic
2C2×#4082C211yesmaximal, characteristic, complemented
3C2×#4072C211yesmaximal, characteristic, complemented
4C2×#4070C211yesmaximal, characteristic, complemented
5#65511C214yesmaximal, complemented
6#65510C211yesmaximal, characteristic, complemented
7#65265C211yesmaximal, characteristic, complemented
8#65263C211yesmaximal, characteristic, complemented
9#65237C214yesmaximal, complemented
10C22×C5×D4C2211yescharacteristic
11C22×C5×D4C2212yes—
12C23×C4×C5C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C5×#121—24no—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2214yes—
20C2×C22⋊Dic5C2214yes—
21#4082—24no—
22#4072—24no—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×C4⋊Dic5C2212yes—
25#4070—24no—
26C2×#683C2212yes—
27C2×#683C2214yes—
28C2×C4×Dic5C2211yescharacteristic
29C2×C4×Dic5C2211yescharacteristic
30#3963C2214yes—
31#2620—51nomaximal, Sylow
32C24×C5C2312yes—
33C24×C5C2311yescharacteristic
34C2×C5×D4—24no—
35C2×C5×D4—22no—
36C2×C5×D4—22no—
37C2×C5×D4—22no—
38C2×C5×D4—44no—
39C22×C4×C5—22no—
40C22×C4×C5—22no—
41C22×C4×C5D414yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C2×Dic10—22no—
50C2×Dic10—22no—
51C2×Dic10—22no—
52C5×C4⋊C4—22no—
53C5×C22⋊C4—24no—
54C22⋊Dic5—24no—
55C22⋊Dic5—24no—
56C22⋊Dic5—28no—
57C22⋊Dic5—44no—
58C4⋊Dic5—22no—
59C4⋊Dic5—24no—
60#683—24no—
61#683—44no—
62C4×Dic5—22no—
63C4×Dic5D414yescomplemented
64C2×#124—51no—
65C2×C22⋊Q8—51no—
66C2×#121D511yescharacteristic
67C2×#117—51no—
68#396—54no—
69#394—54no—
70#391—51no—
71#390—51no—
72#384—51no—
73C23×C5—21no—
74C23×C5—21no—
75C23×C5—22no—
76C23×C5—22no—
77C23×C5—22no—
78C23×C5—22no—
79C23×C5—22no—
80C23×C5—24no—
81C23×C5—24no—
82C23×C5C2×D412yes—
83C23×C5C2411yescharacteristic
84C5×D4—48no—
85C5×D4—44no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—28no—
89C2×C4×C5—22no—
90C2×C4×C5—24no—
91C2×C4×C5C2×D412yes—
92C2×C4×C5C2×D412yes—
93C2×C4×C5C2×D412yes—
94C2×C4×C5#4112yes—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—24no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5C2×D414yes—
106Dic10—44no—
107C22×Q8—51no—
108C22×D4C2×D511yescharacteristic
109C22×D4C2×D512yes—
110C23×C4C2×D511yescharacteristic
111#124—104no—
112C22⋊Q8—104no—
113#121—24no—
114#117—104no—
115C2×C4⋊C4—52no—
116C2×C4⋊C4—51no—
117C2×C4⋊C4—52no—
118C2×C4⋊C4—54no—
119C2×C4⋊C4C2×D511yescharacteristic
120C2×C22⋊C4—52no—
121C2×C22⋊C4—52no—
122C2×C22⋊C4—54no—
123C2×C22⋊C4—54no—
124C2×C22⋊C4C2×D512yes—
125C2×C42—51no—
126C2×C42—51no—
127#95—54no—
128C22×C5—22no—
129C22×C5—22no—
130C22×C5—22no—
131C22×C5—42no—
132C22×C5—42no—
133C22×C5—42no—
134C22×C5—42no—
135C22×C5—42no—
136C22×C5—44no—
137C22×C5—44no—
138C22×C5C22×D411yescharacteristic
139C22×C5C22×D411yescharacteristic
140C22×C5C2×#4111yescharacteristic
141C22×C5H(5,𝔽2)12yes—
142C22×C52−1+412yes—
143C4×C5—22no—
144C4×C5—42no—
145C4×C5#12114yes—
146Dic5—24no—
147Dic5—42no—
148Dic5—42no—
149Dic5—42no—
150C24C22×D512yes—
151C24C22×D511yescharacteristic
152C2×Q8—102no—
153C2×Q8—102no—
154C2×Q8—102no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—44no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—52no—
163C22×C4—52no—
164C22×C4—52no—
165C22×C4—52no—
166C22×C4C22⋊D514yes—
167C22×C4C22×D512yes—
168C22×C4C22×D511yescharacteristic
169C22×C4C22×D511yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—104no—
172C4⋊C4—104no—
173C4⋊C4—102no—
174C4⋊C4—204no—
175C22⋊C4—24no—
176C22⋊C4—104no—
177C22⋊C4—108no—
178C22⋊C4—104no—
179C22⋊C4—204no—
180C42—54no—
181C42—102no—
182C2×C5—42no—
183C2×C5—82no—
184C2×C5C2×#12111yescharacteristic
185C2×C5#53511yescharacteristic
186C2×C5#53611yescharacteristic
187C2×C5#53712yes—
188C2×C5#54212yes—
189C23—21no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—22no—
198C23C2×C22⋊D512yes—
199C23C23×D511yescharacteristic
200Q8—204no—
201D4—48no—
202D4—44no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—28no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—54no—
209C2×C4—102no—
210C2×C4—102no—
211C2×C4—102no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—102no—
215C2×C4—104no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4D4×D512yes—
220C2×C4#71112yes—
221C2×C4C2×C22⋊D512yes—
222C2×C4C2×C22⋊D512yes—
223C5#262011yescharacteristic, complemented, Sylow
224C22—22no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—44no—
233C22—44no—
234C22C2×D4×D511yescharacteristic
235C22C2×#71111yescharacteristic
236C22#414412yes—
237C22#415012yes—
238C22C22×C22⋊D511yescharacteristic
239C4—22no—
240C4—42no—
241C4—104no—
242C4—202no—
243C4—202no—
244C4—202no—
245C4#408514yes—
246C2—42no—
247C2—82no—
248C2#6593712yes—
249C2#6594212yes—
250C2C2×#408511yescharacteristic
251C2#6613911yescharacteristic
252C2#6615911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC2×C5×#121C2
hypercenterC2×#121D5
hypocenterC5#2620
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#4082C2semidirect
3. C2×#4072C2semidirect
4. C2×#4070C2semidirect
5. #65511C2semidirect
6. #65510C2semidirect
7. #65265C2semidirect
8. #65263C2semidirect
9. #65237C2semidirect
63. C4×Dic5D4semidirect
223. C5#2620semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1212
C2×#40822
C2×#40722
C2×#40702
#655112
#655102
#652652
#652632
#652372
#26205