Finite Group Databasebeta

← #10609821

Subgroups — #10609821

Order 640.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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, complemented
2C2×#4030C211yesmaximal, characteristic
3C5×#399C211yesmaximal, characteristic, complemented
4#65281C211yesmaximal, characteristic, complemented
5#65280C212yesmaximal, complemented
6#65273C212yesmaximal, complemented
7#65250C212yesmaximal, complemented
8#65248C212yesmaximal, complemented
9#65236C211yesmaximal, characteristic, complemented
10#64962C211yesmaximal, characteristic, complemented
11#64953C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×C22⋊D5C2211yescharacteristic
14C22×D20C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C22×Dic10C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2212yes—
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×#685C2212yes—
23C2×#685C2211yescharacteristic
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×#685C2212yes—
27C2×C4⋊Dic5C2212yes—
28C2×#683C2211yescharacteristic
29C2×#683C2212yes—
30#4043—24no—
31#4030—24no—
32C5×#95C2211yescharacteristic
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2816—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C23×D5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C2×C22⋊D5—22no—
46C2×C22⋊D5—22no—
47C2×C22⋊D5—42no—
48C2×C22⋊D5—42no—
49C22×Dic5C2312yes—
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2311yescharacteristic
53C2×D20—22no—
54C2×D20—42no—
55C2×C4×D5—42no—
56C2×C4×D5D414yes—
57C2×Dic10—22no—
58C2×Dic10—42no—
59C5×C4⋊C4—22no—
60C5×C4⋊C4—42no—
61C5×C22⋊C4—22no—
62C5×C22⋊C4—42no—
63C22⋊Dic5—42no—
64#685—24no—
65#685—22no—
66#685—42no—
67#685—42no—
68#685—42no—
69C4⋊Dic5—24no—
70#683—22no—
71#683—42no—
72C2×C22⋊Q8—51no—
73C2×#121—51no—
74#399—51no—
75#399D511yescharacteristic, complemented
76#398—52no—
77#394—52no—
78#394—52no—
79#394—52no—
80#393—51no—
81#393—51no—
82#381—51no—
83C23×C5—21no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—21no—
87C23×C5—21no—
88C23×C5C2411yescharacteristic
89C22×D5—22no—
90C22×D5—22no—
91C22×D5—21no—
92C22×D5—21no—
93C22×D5—21no—
94C22×D5—22no—
95C22×D5—22no—
96C22×D5—21no—
97C22×D5—21no—
98C22×D5C2×D412yes—
99C2×C4×C5—22no—
100C2×C4×C5—21no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5C2×D412yes—
109C22⋊D5—44no—
110C22⋊D5—44no—
111C2×Dic5—22no—
112C2×Dic5—21no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5C2×D412yes—
121D20—44no—
122C4×D5—44no—
123Dic10—44no—
124C22×Q8—51no—
125C22×D4—51no—
126C22×D4—51no—
127C22×D4—51no—
128C23×C4—51no—
129C22⋊Q8—104no—
130#121—104no—
131C2×C4⋊C4—51no—
132C2×C4⋊C4—52no—
133C2×C4⋊C4—52no—
134C2×C4⋊C4C2×D511yescharacteristic
135C2×C4⋊C4C2×D512yes—
136C2×C22⋊C4—51no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4—52no—
142C2×C22⋊C4C2×D511yescharacteristic
143C2×C22⋊C4C2×D512yes—
144#95—51no—
145#95—52no—
146#95—52no—
147#95—52no—
148#95C2×D511yescharacteristic
149C22×C5—41no—
150C22×C5—41no—
151C22×C5—41no—
152C22×C5—42no—
153C22×C5—42no—
154C22×C5C22×D411yescharacteristic
155C22×C5H(5,𝔽2)12yes—
156C22×C5H(5,𝔽2)12yes—
157C22×C52−1+411yescharacteristic
158C22×C52−1+411yescharacteristic
159C2×D5—22no—
160C2×D5—22no—
161C2×D5—22no—
162C2×D5—42no—
163C2×D5—42no—
164C2×D5—41no—
165C2×D5—41no—
166C2×D5—41no—
167C2×D5—42no—
168C2×D5—42no—
169C4×C5—22no—
170C4×C5—41no—
171C4×C5—42no—
172C4×C5—42no—
173Dic5—22no—
174Dic5—42no—
175Dic5—41no—
176Dic5—42no—
177C24—51no—
178C24—51no—
179C24C22×D511yescharacteristic
180C2×Q8—102no—
181C2×Q8—202no—
182C2×D4—102no—
183C2×D4—102no—
184C2×D4—102no—
185C2×D4—202no—
186C2×D4—202no—
187C2×D4—202no—
188C22×C4—51no—
189C22×C4—54no—
190C22×C4—52no—
191C22×C4—52no—
192C22×C4—51no—
193C22×C4—202no—
194C22×C4C22×D512yes—
195C22×C4C22×D511yescharacteristic
196C22×C4C22×D512yes—
197C22×C4C22×D511yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—102no—
201C4⋊C4—104no—
202C4⋊C4—202no—
203C22⋊C4—22no—
204C22⋊C4—42no—
205C22⋊C4—104no—
206C22⋊C4—102no—
207C22⋊C4—202no—
208C22⋊C4—202no—
209C22⋊C4—202no—
210C22⋊C4—202no—
211C2×C5—81no—
212C2×C5#53411yescharacteristic
213C2×C5#53511yescharacteristic
214C2×C5#53611yescharacteristic
215C2×C5#56212yes—
216C2×C5#56212yes—
217D5—42no—
218D5—81no—
219C23—21no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—21no—
224C23—52no—
225C23—102no—
226C23—101no—
227C23—102no—
228C23—101no—
229C23—101no—
230C23—102no—
231C23—101no—
232C23—102no—
233C23—101no—
234C23C23×D511yescharacteristic
235Q8—204no—
236D4—204no—
237D4—204no—
238D4—204no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—52no—
249C2×C4—102no—
250C2×C4—101no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—102no—
256C2×C4—102no—
257C2×C4—102no—
258C2×C4—204no—
259C2×C4D4×D512yes—
260C5#281611yescharacteristic, complemented, Sylow
261C22—41no—
262C22—41no—
263C22—42no—
264C22—41no—
265C22—42no—
266C22—102no—
267C22—102no—
268C22—102no—
269C22—202no—
270C22—202no—
271C22—201no—
272C22—201no—
273C22—202no—
274C22—201no—
275C22—202no—
276C22C2×D4×D511yescharacteristic
277C22#414412yes—
278C22#414711yescharacteristic
279C22#414912yes—
280C22#415011yescharacteristic
281C4—22no—
282C4—41no—
283C4—42no—
284C4—42no—
285C4—102no—
286C4—201no—
287C4—202no—
288C4—202no—
289C2—81no—
290C2—202no—
291C2—401no—
292C2#6596211yescharacteristic
293C2#6598112yes—
294C2#6598711yescharacteristic
295C2#6600112yes—
296C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC2×#4030C2
hypercenter#399D5
hypocenterC5#2816
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4043C2semidirect
3. C5×#399C2semidirect
4. #65281C2semidirect
5. #65280C2semidirect
6. #65273C2semidirect
7. #65250C2semidirect
8. #65248C2semidirect
9. #65236C2semidirect
10. #64962C2semidirect
11. #64953C2semidirect
75. #399D5semidirect
260. C5#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#40432
C2×#40302
C5×#3992
#652812
#652802
#652732
#652502
#652482
#652362
#649622
#649532
#28165