Finite Group Databasebeta

← #10608674

Subgroups — #10608674

Order 640.

604 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 151 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×D4×Dic5C211yesmaximal, characteristic
2C2×#4033C211yesmaximal, characteristic, complemented
3C2×#4024C212yesmaximal, complemented
4C5×#394C211yesmaximal, characteristic
5#65511C211yesmaximal, characteristic
6#65272C211yesmaximal, characteristic, complemented
7#65240C212yesmaximal, complemented
8#65234C212yesmaximal, complemented
9#65231C212yesmaximal, complemented
10#64950C211yesmaximal, characteristic, complemented
11#64939C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C23×Dic5C2212yes—
14C22×Dic10C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21D4×Dic5—24no—
22C2×C4⋊Dic5C2212yes—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×#683C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×C4×Dic5C2211yescharacteristic
29C2×C4×Dic5C2211yescharacteristic
30C2×C4×Dic5C2211yescharacteristic
31#4033—24no—
32#4024—28no—
33C5×#95C2211yescharacteristic
34#3963C2212yes—
35#3963C2211yescharacteristic
36#3963C2212yes—
37#3963C2212yes—
38#2536—51nomaximal, Sylow
39C24×C5C2312yes—
40C2×C5×D4—24no—
41C2×C5×D4—22no—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C22×Dic5—24no—
47C22×Dic5—24no—
48C22×Dic5—24no—
49C22×Dic5C2312yes—
50C22×Dic5C2311yescharacteristic
51C22×Dic5C2312yes—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C22×Dic5C2311yescharacteristic
55C2×Dic10—22no—
56C2×Dic10—22no—
57C2×Dic10D414yescomplemented
58C5×C4⋊C4—22no—
59C5×C22⋊C4—24no—
60C5×C22⋊C4—24no—
61C22⋊Dic5—24no—
62C22⋊Dic5—24no—
63C22⋊Dic5—24no—
64C4⋊Dic5—22no—
65C4⋊Dic5—24no—
66#683—24no—
67#683—22no—
68#683—24no—
69#683—22no—
70C4×Dic5—22no—
71C4×Dic5—22no—
72C4×Dic5D414yes—
73C2×C22⋊Q8—52no—
74C2×C4×Q8—51no—
75C2×C4×D4—51no—
76#398—51no—
77#398—51no—
78#394—51no—
79#394D511yescharacteristic
80#388—52no—
81#388—52no—
82#382—51no—
83#380—52no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5#4114yes—
91C23×C5C2411yescharacteristic
92C5×D4—28no—
93C2×C4×C5—21no—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5C2×D412yes—
102C2×C4×C5C2×D412yes—
103C2×Dic5—21no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—21no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—48no—
115C2×Dic5C2×D412yes—
116C2×Dic5C2×D412yes—
117C2×Dic5C2×D412yes—
118C2×Dic5C2×D412yes—
119C2×Dic5#4112yes—
120Dic10—28no—
121C22×Q8—51no—
122C22×D4C2×D511yescharacteristic
123C23×C4—52no—
124C22⋊Q8—108no—
125C4×Q8—104no—
126C4×D4—104no—
127C2×C4⋊C4—51no—
128C2×C4⋊C4—51no—
129C2×C4⋊C4—51no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—52no—
132C2×C4⋊C4—52no—
133C2×C4⋊C4C2×D511yescharacteristic
134C2×C22⋊C4—52no—
135C2×C22⋊C4—52no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4C2×D512yes—
138C2×C22⋊C4C2×D512yes—
139C2×C42—51no—
140C2×C42—51no—
141C2×C42—51no—
142#95—52no—
143#95—52no—
144#95—52no—
145#95—51no—
146#95C2×D511yescharacteristic
147C22×C5—24no—
148C22×C5—24no—
149C22×C5—24no—
150C22×C5—42no—
151C22×C5—42no—
152C22×C5—42no—
153C22×C5—42no—
154C22×C5C22×D411yescharacteristic
155C22×C5C22×D411yescharacteristic
156C22×C5C2×#4112yes—
157C22×C5C2×#4111yescharacteristic
158C22×C52−1+412yes—
159C4×C5—22no—
160C4×C5—22no—
161C4×C5—41no—
162C4×C5—42no—
163Dic5—22no—
164Dic5—22no—
165Dic5—42no—
166Dic5—41no—
167Dic5—42no—
168Dic5#12114yes—
169C24C22×D512yes—
170C2×Q8—54no—
171C2×Q8—102no—
172C2×Q8—102no—
173C2×D4—24no—
174C2×D4—22no—
175C22×C4—51no—
176C22×C4—51no—
177C22×C4—51no—
178C22×C4—52no—
179C22×C4—51no—
180C22×C4—52no—
181C22×C4—104no—
182C22×C4—104no—
183C22×C4—104no—
184C22×C4C22×D512yes—
185C22×C4C22×D511yescharacteristic
186C22×C4C22×D511yescharacteristic
187C22×C4C22×D511yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—102no—
190C4⋊C4—104no—
191C4⋊C4—104no—
192C4⋊C4—102no—
193C4⋊C4—102no—
194C4⋊C4—104no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—104no—
198C22⋊C4—104no—
199C22⋊C4—104no—
200C42—54no—
201C42—102no—
202C42—102no—
203C2×C5—44no—
204C2×C5C2×#12111yescharacteristic
205C2×C5#52511yescharacteristic
206C2×C5#53611yescharacteristic
207C2×C5#54812yes—
208C2×C5#55212yes—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23#71114yes—
216C23C23×D511yescharacteristic
217Q8—108no—
218D4—28no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—52no—
228C2×C4—52no—
229C2×C4—52no—
230C2×C4—52no—
231C2×C4—52no—
232C2×C4—101no—
233C2×C4—101no—
234C2×C4—102no—
235C2×C4—101no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—102no—
241C2×C4—102no—
242C2×C4—102no—
243C2×C4—208no—
244C2×C4D4×D512yes—
245C2×C4D4×D512yes—
246C5#253611yescharacteristic, complemented, Sylow
247C22—24no—
248C22—24no—
249C22—24no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22C2×D4×D511yescharacteristic
255C22C2×D4×D511yescharacteristic
256C22C2×#71112yes—
257C22D5×#4111yescharacteristic
258C22#415012yes—
259C4—22no—
260C4—22no—
261C4—41no—
262C4—42no—
263C4—54no—
264C4—102no—
265C4—102no—
266C4—201no—
267C4—202no—
268C4—202no—
269C2—44no—
270C2#6592011yescharacteristic
271C2#6593212yes—
272C2D5×#12111yescharacteristic
273C2#6598412yes—
274C2#6600911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC22×C5×D4C22
hypercenter#394D5
hypocenterC5#2536
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#4033C2semidirect
3. C2×#4024C2semidirect
6. #65272C2semidirect
7. #65240C2semidirect
8. #65234C2semidirect
9. #65231C2semidirect
10. #64950C2semidirect
11. #64939C2semidirect
57. C2×Dic10D4semidirect
246. C5#2536semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40332
C2×#40242
C5×#3942
#655112
#652722
#652402
#652342
#652312
#649502
#649392
#25365