Finite Group Databasebeta

← #10609697

Subgroups — #10609697

Order 640.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#4043C211yesmaximal, characteristic
2C2×#4042C211yesmaximal, characteristic
3C5×#398C211yesmaximal, characteristic, complemented
4#65276C212yesmaximal
5#65273C212yesmaximal, complemented
6#65269C211yesmaximal, characteristic, complemented
7#65268C211yesmaximal, characteristic, complemented
8#64956C212yesmaximal
9D5×#95C211yesmaximal, characteristic
10#64945C212yesmaximal, complemented
11#64945C211yesmaximal, characteristic, complemented
12C22×C4×D5C2211yescharacteristic
13C22×C4×D5C2212yes—
14C22×Dic10C2211yescharacteristic
15C22×Dic10C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2212yes—
19C2×#685C2211yescharacteristic
20C2×#685C2211yescharacteristic
21C2×#685C2212yes—
22C2×C4⋊Dic5C2212yes—
23C2×C4⋊Dic5C2212yes—
24C2×#683C2211yescharacteristic
25C2×#683C2211yescharacteristic
26C2×#683C2212yes—
27C2×#683C2212yes—
28#4043—24no—
29#4042—24no—
30C5×#95C2211yescharacteristic
31C5×#95C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2211yescharacteristic
36#3963C2212yes—
37#2659—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×Dic5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2312yes—
49C2×C4×D5—42no—
50C2×C4×D5—42no—
51C2×C4×D5—42no—
52C2×C4×D5D414yes—
53C2×C4×D5Q818yescomplemented
54C2×Dic10—22no—
55C2×Dic10—22no—
56C2×Dic10—42no—
57C2×Dic10—42no—
58C5×C4⋊C4—22no—
59C5×C4⋊C4—22no—
60C5×C4⋊C4—42no—
61#685—22no—
62#685—22no—
63#685—24no—
64C4⋊Dic5—24no—
65C4⋊Dic5—42no—
66#683—22no—
67#683—22no—
68#683—42no—
69#683—42no—
70C2×C22⋊Q8—51no—
71C2×C22⋊Q8—51no—
72#398—51no—
73#398—52no—
74#398D511yescharacteristic, complemented
75#395—51no—
76#395—52no—
77#395—51no—
78#380—52no—
79#380—52no—
80C2×#95—51no—
81C23×C5C2411yescharacteristic
82C22×D5—21no—
83C22×D5—22no—
84C22×D5—21no—
85C22×D5C2×D412yes—
86C22×D5C2×Q814yes—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5—21no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—21no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5C2×D412yes—
99C2×C4×C5C2×Q814yes—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—21no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—21no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—21no—
111C2×Dic5C2×D412yes—
112C2×Dic5C2×Q814yes—
113C4×D5—44no—
114C4×D5—48no—
115Dic10—44no—
116Dic10—44no—
117C22×Q8—51no—
118C22×Q8—51no—
119C23×C4—51no—
120C23×C4—52no—
121C22⋊Q8—104no—
122C22⋊Q8—104no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—52no—
125C2×C4⋊C4—51no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4—52no—
128C2×C4⋊C4—52no—
129C2×C4⋊C4C2×D511yescharacteristic
130C2×C4⋊C4C2×D511yescharacteristic
131C2×C4⋊C4C2×D512yes—
132C2×C22⋊C4—51no—
133C2×C22⋊C4—51no—
134C2×C22⋊C4—52no—
135#95—51no—
136#95—52no—
137#95—52no—
138#95—52no—
139#95—52no—
140#95C2×D511yescharacteristic
141#95C2×D512yes—
142C22×C5C22×D411yescharacteristic
143C22×C5C22×Q812yes—
144C22×C5H(5,𝔽2)11yescharacteristic
145C22×C5H(5,𝔽2)12yes—
146C22×C52−1+411yescharacteristic
147C2×D5—22no—
148C2×D5—22no—
149C2×D5—24no—
150C2×D5—24no—
151C2×D5C4⋊Q814yes—
152C4×C5—22no—
153C4×C5—24no—
154C4×C5—41no—
155C4×C5—41no—
156C4×C5—42no—
157Dic5—22no—
158Dic5—24no—
159Dic5—41no—
160Dic5—42no—
161Dic5—41no—
162C24—51no—
163C2×Q8—102no—
164C2×Q8—102no—
165C2×Q8—202no—
166C2×Q8—202no—
167C22×C4—52no—
168C22×C4—51no—
169C22×C4—54no—
170C22×C4—58no—
171C22×C4—51no—
172C22×C4—51no—
173C22×C4—52no—
174C22×C4—202no—
175C22×C4—202no—
176C22×C4—202no—
177C22×C4C22×D511yescharacteristic
178C22×C4C22×D512yes—
179C22×C4C22×D511yescharacteristic
180C22×C4C22×D511yescharacteristic
181C22×C4C22×D512yes—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—102no—
186C4⋊C4—104no—
187C4⋊C4—102no—
188C4⋊C4—202no—
189C4⋊C4—202no—
190C4⋊C4—202no—
191C22⋊C4—102no—
192C22⋊C4—104no—
193C22⋊C4—102no—
194C2×C5C2×C4⋊Q811yescharacteristic
195C2×C5#53411yescharacteristic
196C2×C5#53711yescharacteristic
197C2×C5#54312yes—
198C2×C5#54412yes—
199D5—24no—
200C23—52no—
201C23—54no—
202C23—101no—
203C23—102no—
204C23—101no—
205C23C23×D511yescharacteristic
206Q8—204no—
207Q8—204no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—52no—
220C2×C4—54no—
221C2×C4—102no—
222C2×C4—102no—
223C2×C4—102no—
224C2×C4—101no—
225C2×C4—102no—
226C2×C4—101no—
227C2×C4—101no—
228C2×C4—102no—
229C2×C4—102no—
230C2×C4—102no—
231C2×C4—101no—
232C2×C4—208no—
233C2×C4—204no—
234C2×C4D4×D512yes—
235C2×C4Q8×D514yes—
236C5#265911yescharacteristic, complemented, Sylow
237C22—54no—
238C22—102no—
239C22—104no—
240C22—102no—
241C22—104no—
242C22C2×D4×D511yescharacteristic
243C22#414411yescharacteristic
244C22C2×Q8×D512yes—
245C22#414912yes—
246C22#415011yescharacteristic
247C4—24no—
248C4—22no—
249C4—41no—
250C4—42no—
251C4—41no—
252C4—104no—
253C4—102no—
254C4—201no—
255C4—202no—
256C4—201no—
257C2—104no—
258C2#6594211yescharacteristic
259C2#6596812yes—
260C2#6598711yescharacteristic
261C2#6602312yes—
262C2D5×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radicalC23×D5C23
hypercenter#398D5
hypocenterC5#2659
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C5×#398C2semidirect
5. #65273C2semidirect
6. #65269C2semidirect
7. #65268C2semidirect
10. #64945C2semidirect
11. #64945C2semidirect
53. C2×C4×D5Q8semidirect
74. #398D5semidirect
236. C5#2659semidirect

Maximal subgroups

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

subgroupindex
C2×#40432
C2×#40422
C5×#3982
#652762
#652732
#652692
#652682
#649562
D5×#952
#649452
#26595