Finite Group Databasebeta

← #10608985

Subgroups — #10608985

Order 640.

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

normal subgroups only

This group has 584 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×#4092C211yesmaximal, characteristic, complemented
2C2×#4039C211yesmaximal, characteristic
3C5×#395C211yesmaximal, characteristic, complemented
4#65520C211yesmaximal, characteristic, complemented
5#65278C212yesmaximal, complemented
6#65276C212yesmaximal, complemented
7#65275C211yesmaximal, characteristic, complemented
8#64966C211yesmaximal, characteristic, complemented
9#64954C212yesmaximal
10D5×#95C211yesmaximal, characteristic, complemented
11#64943C212yesmaximal, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×D20C2211yescharacteristic
14C22×C4×D5C2212yes—
15C22×C4×D5C2211yescharacteristic
16C22×C4×D5C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2212yes—
19#4092—24no—
20C2×#685C2211yescharacteristic
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2212yes—
27C2×#683C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29#4039—24no—
30C5×#95C2212yes—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2211yescharacteristic
36#3963C2212yes—
37#2564—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C2×C5×Q8—22no—
41C2×C5×Q8—42no—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2312yes—
50C22×Dic5C2311yescharacteristic
51C2×D20—22no—
52C2×D20—22no—
53C2×D20Q814yescomplemented
54C2×C4×D5—24no—
55C2×C4×D5—24no—
56C2×C4×D5—22no—
57C2×C4×D5—42no—
58C2×C4×D5—42no—
59C2×C4×D5—42no—
60C2×C4×D5D414yes—
61C5×C4⋊C4—22no—
62C5×C4⋊C4—42no—
63#685—22no—
64#685—24no—
65#685—22no—
66#685—24no—
67#685—42no—
68C4⋊Dic5—22no—
69C4⋊Dic5—24no—
70#683—24no—
71C4×Dic5—22no—
72C2×C22⋊Q8—51no—
73C2×C4×D4—51no—
74#400—51no—
75#399—51no—
76#395D511yescharacteristic, complemented
77#393—51no—
78#387—52no—
79#386—52no—
80#385—52no—
81#380—52no—
82C2×#95—51no—
83C23×C5C2411yescharacteristic
84C22×D5—22no—
85C22×D5—22no—
86C22×D5—21no—
87C22×D5—22no—
88C22×D5—21no—
89C22×D5—22no—
90C22×D5C2×D412yes—
91C22×D5C2×Q812yes—
92C22×D5C2×Q812yes—
93C22×D5#4112yes—
94C5×Q8—44no—
95C2×C4×C5—21no—
96C2×C4×C5—22no—
97C2×C4×C5—21no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—21no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5C2×D412yes—
106C2×C4×C5C2×Q812yes—
107C2×C4×C5#4114yes—
108C2×Dic5—22no—
109C2×Dic5—21no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5C2×D412yes—
118D20—28no—
119C4×D5—44no—
120C4×D5—44no—
121C4×D5—48no—
122C22×Q8C2×D511yescharacteristic
123C22×D4—51no—
124C23×C4—51no—
125C23×C4—52no—
126C23×C4—51no—
127C22⋊Q8—104no—
128C4×D4—104no—
129C2×C4⋊C4—52no—
130C2×C4⋊C4—51no—
131C2×C4⋊C4—52no—
132C2×C4⋊C4C2×D511yescharacteristic
133C2×C4⋊C4C2×D512yes—
134C2×C22⋊C4—51no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C42—51no—
140#95—52no—
141#95—51no—
142#95—52no—
143#95—52no—
144#95—52no—
145#95C2×D511yescharacteristic
146#95C2×D512yes—
147C22×C5C22×D411yescharacteristic
148C22×C5C22×Q811yescharacteristic
149C22×C5C2×#4112yes—
150C22×C5C2×#4111yescharacteristic
151C22×C5H(5,𝔽2)12yes—
152C2×D5—22no—
153C2×D5—22no—
154C2×D5—24no—
155C2×D5—24no—
156C2×D5—22no—
157C2×D5—24no—
158C2×D5—41no—
159C2×D5—41no—
160C2×D5—42no—
161C2×D5C22⋊Q814yes—
162C4×C5—22no—
163C4×C5—22no—
164C4×C5—24no—
165C4×C5—41no—
166C4×C5—42no—
167Dic5—22no—
168Dic5—41no—
169Dic5—42no—
170Dic5—42no—
171C24—51no—
172C24—51no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×D4—54no—
176C2×D4—102no—
177C2×D4—102no—
178C22×C4—51no—
179C22×C4—54no—
180C22×C4—52no—
181C22×C4—51no—
182C22×C4—52no—
183C22×C4—102no—
184C22×C4—104no—
185C22×C4—104no—
186C22×C4—202no—
187C22×C4—202no—
188C22×C4—202no—
189C22×C4C22×D511yescharacteristic
190C22×C4C22×D512yes—
191C22×C4C22×D511yescharacteristic
192C22×C4C22×D511yescharacteristic
193C22×C4C22×D512yes—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—104no—
197C4⋊C4—102no—
198C4⋊C4—104no—
199C22⋊C4—102no—
200C22⋊C4—104no—
201C22⋊C4—102no—
202C22⋊C4—104no—
203C22⋊C4—202no—
204C42—102no—
205C2×C5C2×C22⋊Q811yescharacteristic
206C2×C5#53211yescharacteristic
207C2×C5#53411yescharacteristic
208C2×C5#55112yes—
209C2×C5#55412yes—
210D5—24no—
211D5—42no—
212C23—52no—
213C23—52no—
214C23—52no—
215C23—52no—
216C23—102no—
217C23—102no—
218C23—101no—
219C23—102no—
220C23—102no—
221C23—101no—
222C23C23×D511yescharacteristic
223Q8—44no—
224D4—108no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—52no—
236C2×C4—102no—
237C2×C4—101no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—102no—
241C2×C4—102no—
242C2×C4—102no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—204no—
246C2×C4—208no—
247C2×C4—204no—
248C2×C4D4×D512yes—
249C2×C4Q8×D512yes—
250C2×C4#71314yes—
251C5#256411yescharacteristic, complemented, Sylow
252C22—54no—
253C22—102no—
254C22—104no—
255C22—102no—
256C22—104no—
257C22—104no—
258C22—102no—
259C22—201no—
260C22—201no—
261C22—202no—
262C22C2×D4×D511yescharacteristic
263C22#414412yes—
264C22C2×Q8×D511yescharacteristic
265C22C2×#71312yes—
266C22D5×#4111yescharacteristic
267C4—22no—
268C4—22no—
269C4—24no—
270C4—41no—
271C4—42no—
272C4—102no—
273C4—201no—
274C4—202no—
275C4—202no—
276C2—104no—
277C2—202no—
278C2#6592311yescharacteristic
279C2D5×C22⋊Q811yescharacteristic
280C2#6597112yes—
281C2#6605911yescharacteristic
282C2#6606412yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#395C2
prime radicalC22×D20C22
hypercenter#395D5
hypocenterC5#2564
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#4092C2semidirect
3. C5×#395C2semidirect
4. #65520C2semidirect
5. #65278C2semidirect
6. #65276C2semidirect
7. #65275C2semidirect
8. #64966C2semidirect
10. D5×#95C2semidirect
11. #64943C2semidirect
53. C2×D20Q8semidirect
76. #395D5semidirect
251. C5#2564semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
C2×#40392
C5×#3952
#655202
#652782
#652762
#652752
#649662
#649542
D5×#952
#649432
#25645