Finite Group Databasebeta

← #10609749

Subgroups — #10609749

Order 640.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×#4033C211yesmaximal, characteristic, complemented
2C2×#4023C211yesmaximal, characteristic
3C5×#399C211yesmaximal, characteristic
4#65271C211yesmaximal, characteristic, complemented
5#65269C212yesmaximal, complemented
6C4⋊C4×Dic5C212yesmaximal, complemented
7#65237C212yesmaximal
8#65236C211yesmaximal, characteristic
9C22⋊C4×Dic5C212yesmaximal
10#64945C211yesmaximal, characteristic, complemented
11#64938C211yesmaximal, characteristic, complemented
12C23×Dic5C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2212yes—
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19C2×C22⋊Dic5C2212yes—
20C2×C4⋊Dic5C2212yes—
21C2×#683C2211yescharacteristic
22C2×#683C2211yescharacteristic
23C2×#683C2212yes—
24C2×#683C2212yes—
25C2×C4×Dic5C2211yescharacteristic
26C2×C4×Dic5C2212yes—
27C2×C4×Dic5C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29#4033—24no—
30#4023—24no—
31C5×#95C2211yescharacteristic
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2396—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×Dic5—22no—
44C22×Dic5—22no—
45C22×Dic5Q814yescomplemented
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C2×Dic10—22no—
53C2×Dic10—24no—
54C5×C4⋊C4—22no—
55C5×C4⋊C4—24no—
56C5×C22⋊C4—22no—
57C5×C22⋊C4—24no—
58C22⋊Dic5—22no—
59C22⋊Dic5—24no—
60C4⋊Dic5—24no—
61#683—22no—
62#683—24no—
63#683—22no—
64#683—24no—
65C4×Dic5—22no—
66C4×Dic5—24no—
67C4×Dic5—24no—
68C4×Dic5D414yes—
69C2×C4×Q8—51no—
70C2×#117—51no—
71#400—51no—
72#399D511yescharacteristic
73#398—52no—
74#396—52no—
75#395—51no—
76#393—51no—
77C4×C4⋊C4—52no—
78C4×C4⋊C4—51no—
79C4×C22⋊C4—52no—
80C23×C5—21no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—21no—
84C23×C5C2×Q812yes—
85C23×C5C2411yescharacteristic
86C2×C4×C5—22no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—21no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5C2×D412yes—
95C2×C4×C5#4114yes—
96C2×Dic5—22no—
97C2×Dic5—21no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—21no—
101C2×Dic5—28no—
102C2×Dic5—21no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5C2×D412yes—
108C2×Dic5C2×D412yes—
109C2×Dic5C2×Q812yes—
110C2×Dic5C2×Q812yes—
111C2×Dic5#4114yes—
112C2×Dic5#4112yes—
113Dic10—28no—
114C22×Q8—51no—
115C23×C4—51no—
116C4×Q8—104no—
117#117—104no—
118C2×C4⋊C4—51no—
119C2×C4⋊C4—51no—
120C2×C4⋊C4—52no—
121C2×C4⋊C4—52no—
122C2×C4⋊C4—52no—
123C2×C4⋊C4C2×D511yescharacteristic
124C2×C4⋊C4C2×D512yes—
125C2×C22⋊C4—51no—
126C2×C22⋊C4—52no—
127C2×C22⋊C4C2×D511yescharacteristic
128C2×C22⋊C4C2×D512yes—
129C2×C42—51no—
130C2×C42—51no—
131C2×C42—52no—
132C2×C42—52no—
133#95—52no—
134#95—52no—
135#95—52no—
136#95—51no—
137#95—52no—
138#95C2×D511yescharacteristic
139C22×C5—22no—
140C22×C5—24no—
141C22×C5—41no—
142C22×C5—42no—
143C22×C5—41no—
144C22×C5C22×D411yescharacteristic
145C22×C5C22×Q811yescharacteristic
146C22×C5C2×#4112yes—
147C22×C5C2×#4111yescharacteristic
148C22×C5C2×#4112yes—
149C4×C5—22no—
150C4×C5—24no—
151C4×C5—41no—
152C4×C5—42no—
153Dic5—22no—
154Dic5—24no—
155Dic5—22no—
156Dic5—41no—
157Dic5—42no—
158Dic5C22⋊Q814yes—
159C24C22×D511yescharacteristic
160C2×Q8—104no—
161C2×Q8—102no—
162C22×C4—51no—
163C22×C4—54no—
164C22×C4—51no—
165C22×C4—52no—
166C22×C4—51no—
167C22×C4—52no—
168C22×C4—51no—
169C22×C4—102no—
170C22×C4—102no—
171C22×C4C22×D511yescharacteristic
172C22×C4C22×D512yes—
173C22×C4C22×D511yescharacteristic
174C22×C4C22×D512yes—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—102no—
178C4⋊C4—104no—
179C4⋊C4—104no—
180C4⋊C4—102no—
181C4⋊C4—104no—
182C22⋊C4—22no—
183C22⋊C4—24no—
184C22⋊C4—102no—
185C22⋊C4—104no—
186C42—54no—
187C42—102no—
188C42—104no—
189C42—104no—
190C2×C5—42no—
191C2×C5C2×C22⋊Q811yescharacteristic
192C2×C5#52511yescharacteristic
193C2×C5#52912yes—
194C2×C5#53211yescharacteristic
195C2×C5#53312yes—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—21no—
200C23Q8×D512yes—
201C23C23×D511yescharacteristic
202Q8—108no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—52no—
212C2×C4—52no—
213C2×C4—52no—
214C2×C4—52no—
215C2×C4—52no—
216C2×C4—54no—
217C2×C4—108no—
218C2×C4—102no—
219C2×C4—101no—
220C2×C4—102no—
221C2×C4—101no—
222C2×C4—102no—
223C2×C4—101no—
224C2×C4—102no—
225C2×C4—102no—
226C2×C4—102no—
227C2×C4—102no—
228C2×C4D4×D512yes—
229C2×C4#71114yes—
230C5#239611yescharacteristic, complemented, Sylow
231C22—24no—
232C22—22no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D511yescharacteristic
237C22C2×#71112yes—
238C22C2×Q8×D511yescharacteristic
239C22D5×#4112yes—
240C22D5×#4111yescharacteristic
241C4—24no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—54no—
246C4—102no—
247C4—102no—
248C4—104no—
249C4—202no—
250C4—201no—
251C2—42no—
252C2#6595612yes—
253C2D5×C22⋊Q811yescharacteristic
254C2#6598312yes—
255C2#6599011yescharacteristic
256C2#6605011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC24×C5C23
hypercenter#399D5
hypocenterC5#2396
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4033C2semidirect
4. #65271C2semidirect
5. #65269C2semidirect
6. C4⋊C4×Dic5C2semidirect
10. #64945C2semidirect
11. #64938C2semidirect
45. C22×Dic5Q8semidirect
230. C5#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#40332
C2×#40232
C5×#3992
#652712
#652692
C4⋊C4×Dic52
#652372
#652362
C22⋊C4×Dic52
#649452
#649382
#23965