Finite Group Databasebeta

← #10609772

Subgroups — #10609772

Order 640.

564 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 147 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×#4027C211yesmaximal, characteristic
3C5×#399C211yesmaximal, characteristic, complemented
4#65281C211yesmaximal, characteristic, complemented
5#65278C212yesmaximal, complemented
6#65261C212yesmaximal, complemented
7#65246C212yesmaximal
8#65236C211yesmaximal, characteristic, complemented
9#65235C212yesmaximal, complemented
10#64962C211yesmaximal, characteristic, complemented
11#64939C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2211yescharacteristic
13C23×Dic5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C22×Dic10C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2212yes—
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#683C2211yescharacteristic
25C2×#683C2212yes—
26C2×#683C2212yes—
27C2×#683C2212yes—
28C2×C4×Dic5C2212yes—
29C2×C4×Dic5C2211yescharacteristic
30#4033—24no—
31#4027—24no—
32C5×#95C2211yescharacteristic
33#3963C2211yescharacteristic
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2580—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C2×C22⋊D5—22no—
45C2×C22⋊D5—22no—
46C2×C22⋊D5Q814yescomplemented
47C22×Dic5—22no—
48C22×Dic5—42no—
49C22×Dic5C2312yes—
50C22×Dic5C2311yescharacteristic
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2312yes—
53C22×Dic5C2311yescharacteristic
54C2×C4×D5—22no—
55C2×C4×D5—42no—
56C2×Dic10—22no—
57C2×Dic10—24no—
58C5×C4⋊C4—22no—
59C5×C4⋊C4—42no—
60C5×C22⋊C4—22no—
61C5×C22⋊C4—42no—
62C22⋊Dic5—24no—
63#685—22no—
64#685—24no—
65#685—42no—
66#683—22no—
67#683—24no—
68#683—24no—
69#683—42no—
70C4×Dic5—24no—
71C4×Dic5D414yes—
72C2×C4×Q8—51no—
73C2×C4×D4—51no—
74#399—51no—
75#399D511yescharacteristic, complemented
76#393—51no—
77#393—51no—
78#390—52no—
79#389—52no—
80#385—52no—
81#385—52no—
82#382—51no—
83C23×C5—21no—
84C23×C5—22no—
85C23×C5—21no—
86C23×C5—22no—
87C23×C5C2×Q812yes—
88C23×C5C2411yescharacteristic
89C22×D5—21no—
90C22×D5—22no—
91C22×D5—21no—
92C22×D5—22no—
93C22×D5C2×Q812yes—
94C2×C4×C5—22no—
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×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5C2×D412yes—
104C22⋊D5—28no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—21no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—44no—
114C2×Dic5C2×D412yes—
115C2×Dic5C2×D412yes—
116C2×Dic5C2×Q812yes—
117C2×Dic5#4114yes—
118C2×Dic5#4112yes—
119C4×D5—44no—
120Dic10—28no—
121C22×Q8—51no—
122C22×D4—51no—
123C23×C4—51no—
124C23×C4—51no—
125C4×Q8—104no—
126C4×D4—104no—
127C2×C4⋊C4—51no—
128C2×C4⋊C4—52no—
129C2×C4⋊C4—52no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4C2×D511yescharacteristic
132C2×C4⋊C4C2×D512yes—
133C2×C22⋊C4—52no—
134C2×C22⋊C4—51no—
135C2×C22⋊C4—52no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4C2×D511yescharacteristic
138C2×C22⋊C4C2×D512yes—
139C2×C42—51no—
140C2×C42—52no—
141#95—52no—
142#95—51no—
143#95—52no—
144#95—52no—
145#95C2×D511yescharacteristic
146C22×C5—22no—
147C22×C5—24no—
148C22×C5—41no—
149C22×C5—41no—
150C22×C5—42no—
151C22×C5C22×D411yescharacteristic
152C22×C5C22×Q811yescharacteristic
153C22×C5C2×#4111yescharacteristic
154C22×C5C2×#4112yes—
155C22×C5H(5,𝔽2)12yes—
156C2×D5—22no—
157C2×D5—24no—
158C2×D5—41no—
159C2×D5—41no—
160C2×D5—42no—
161C4×C5—22no—
162C4×C5—41no—
163C4×C5—42no—
164C4×C5—42no—
165Dic5—24no—
166Dic5—22no—
167Dic5—41no—
168Dic5—42no—
169Dic5C22⋊Q814yes—
170C24—51no—
171C24C22×D511yescharacteristic
172C2×Q8—104no—
173C2×Q8—102no—
174C2×D4—54no—
175C2×D4—102no—
176C2×D4—102no—
177C22×C4—51no—
178C22×C4—52no—
179C22×C4—51no—
180C22×C4—52no—
181C22×C4—51no—
182C22×C4—102no—
183C22×C4—102no—
184C22×C4—202no—
185C22×C4—202no—
186C22×C4C22×D512yes—
187C22×C4C22×D511yescharacteristic
188C22×C4C22×D512yes—
189C22×C4C22×D511yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—102no—
193C4⋊C4—104no—
194C4⋊C4—104no—
195C4⋊C4—202no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—104no—
199C22⋊C4—104no—
200C22⋊C4—102no—
201C22⋊C4—202no—
202C42—54no—
203C42—104no—
204C2×C5—42no—
205C2×C5C2×C22⋊Q811yescharacteristic
206C2×C5#52511yescharacteristic
207C2×C5#53511yescharacteristic
208C2×C5#55412yes—
209C2×C5#55912yes—
210D5—42no—
211C23—21no—
212C23—22no—
213C23—22no—
214C23—21no—
215C23—52no—
216C23—101no—
217C23—102no—
218C23—102no—
219C23—101no—
220C23Q8×D512yes—
221C23C23×D511yescharacteristic
222Q8—108no—
223D4—108no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—52no—
234C2×C4—52no—
235C2×C4—52no—
236C2×C4—52no—
237C2×C4—54no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—102no—
241C2×C4—101no—
242C2×C4—102no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—102no—
246C2×C4—204no—
247C2×C4—204no—
248C2×C4D4×D512yes—
249C5#258011yescharacteristic, complemented, Sylow
250C22—22no—
251C22—24no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22—102no—
256C22—104no—
257C22—201no—
258C22—201no—
259C22—202no—
260C22C2×D4×D511yescharacteristic
261C22#414412yes—
262C22C2×Q8×D511yescharacteristic
263C22D5×#4111yescharacteristic
264C22D5×#4112yes—
265C4—22no—
266C4—41no—
267C4—42no—
268C4—42no—
269C4—54no—
270C4—102no—
271C4—104no—
272C4—201no—
273C4—202no—
274C2—42no—
275C2—202no—
276C2D5×C22⋊Q811yescharacteristic
277C2#6597012yes—
278C2#6599011yescharacteristic
279C2#6599812yes—
280C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC22×C22⋊D5C22
hypercenter#399D5
hypocenterC5#2580
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4033C2semidirect
3. C5×#399C2semidirect
4. #65281C2semidirect
5. #65278C2semidirect
6. #65261C2semidirect
8. #65236C2semidirect
9. #65235C2semidirect
10. #64962C2semidirect
11. #64939C2semidirect
46. C2×C22⋊D5Q8semidirect
75. #399D5semidirect
249. C5#2580semidirect

Maximal subgroups

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

subgroupindex
C2×#40332
C2×#40272
C5×#3992
#652812
#652782
#652612
#652462
#652362
#652352
#649622
#649392
#25805