Finite Group Databasebeta

← #10608713

Subgroups — #10608713

Order 640.

562 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 133 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, complemented
2C2×#4044C211yesmaximal, characteristic, complemented
3C5×#394C211yesmaximal, characteristic, complemented
4#65512C211yesmaximal, characteristic
5#65275C211yesmaximal, characteristic, complemented
6#65248C212yesmaximal, complemented
7#65247C212yesmaximal, complemented
8#65239C212yesmaximal, complemented
9#65237C212yesmaximal, complemented
10#64966C211yesmaximal, characteristic, complemented
11#64939C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C23×Dic5C2212yes—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20D4×Dic5—24no—
21C2×#685C2211yescharacteristic
22C2×#685C2211yescharacteristic
23C2×#685C2211yescharacteristic
24C2×#685C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×C4⋊Dic5C2212yes—
28C2×#683C2211yescharacteristic
29C2×#683C2212yes—
30C2×C4×Dic5C2211yescharacteristic
31#4044—24no—
32C5×#95C2211yescharacteristic
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2636—51nomaximal, Sylow
38C24×C5C2312yes—
39C23×D5C2311yescharacteristic
40C2×C5×D4—24no—
41C2×C5×D4—22no—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2312yes—
45C22×C4×C5C2311yescharacteristic
46C2×C22⋊D5—24no—
47C2×C22⋊D5—42no—
48C2×C22⋊D5—42no—
49C22×Dic5—24no—
50C22×Dic5—42no—
51C22×Dic5—42no—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2312yes—
54C22×Dic5C2312yes—
55C22×Dic5C2311yescharacteristic
56C22×Dic5C2311yescharacteristic
57C5×C4⋊C4—22no—
58C5×C22⋊C4—24no—
59C5×C22⋊C4—42no—
60C22⋊Dic5—24no—
61C22⋊Dic5—42no—
62#685—22no—
63#685—22no—
64#685—22no—
65#685—42no—
66C4⋊Dic5—22no—
67C4⋊Dic5—22no—
68C4⋊Dic5—42no—
69#683—22no—
70#683—42no—
71C4×Dic5D414yescomplemented
72C2×#126—51no—
73C2×C4×D4—51no—
74#400—51no—
75#399—51no—
76#396—52no—
77#394—52no—
78#394D511yescharacteristic, complemented
79#392—51no—
80#386—52no—
81#385—52no—
82#382—51no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5#4114yes—
90C23×C5C2411yescharacteristic
91C22×D5—21no—
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—21no—
95C22×D5—21no—
96C5×D4—28no—
97C2×C4×C5—22no—
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—21no—
105C2×C4×C5—21no—
106C2×C4×C5C2×D412yes—
107C22⋊D5—48no—
108C2×Dic5—21no—
109C2×Dic5—22no—
110C2×Dic5—21no—
111C2×Dic5—21no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—48no—
120C2×Dic5C2×D412yes—
121C2×Dic5C2×D412yes—
122C22×D4—52no—
123C22×D4C2×D511yescharacteristic
124C23×C4—52no—
125#126—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⋊C4C2×D511yescharacteristic
133C2×C22⋊C4—52no—
134C2×C22⋊C4—51no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—51no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4C2×D512yes—
140C2×C22⋊C4C2×D512yes—
141C2×C42—51no—
142#95—52no—
143#95—51no—
144#95—52no—
145#95—52no—
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×C5C2×#4112yes—
156C22×C5H(5,𝔽2)11yescharacteristic
157C22×C5H(5,𝔽2)12yes—
158C22×C52−1+411yescharacteristic
159C2×D5—41no—
160C2×D5—41no—
161C2×D5—41no—
162C2×D5—42no—
163C2×D5—42no—
164C4×C5—22no—
165C4×C5—41no—
166C4×C5—41no—
167C4×C5—42no—
168Dic5—22no—
169Dic5—22no—
170Dic5—41no—
171Dic5—42no—
172Dic5—42no—
173C24—51no—
174C24C22×D512yes—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—104no—
178C2×D4—202no—
179C2×D4—202no—
180C22×C4—51no—
181C22×C4—51no—
182C22×C4—52no—
183C22×C4—52no—
184C22×C4—51no—
185C22×C4—104no—
186C22×C4—202no—
187C22×C4—202no—
188C22×C4C22×D511yescharacteristic
189C22×C4C22×D511yescharacteristic
190C22×C4C22×D512yes—
191C22×C4C22×D511yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—102no—
194C4⋊C4—102no—
195C4⋊C4—102no—
196C4⋊C4—202no—
197C4⋊C4—202no—
198C22⋊C4—24no—
199C22⋊C4—42no—
200C22⋊C4—102no—
201C22⋊C4—102no—
202C22⋊C4—104no—
203C22⋊C4—102no—
204C22⋊C4—202no—
205C22⋊C4—202no—
206C42—54no—
207C2×C5—44no—
208C2×C5#52511yescharacteristic
209C2×C5#53511yescharacteristic
210C2×C5#53711yescharacteristic
211C2×C5#53812yes—
212C2×C5#53912yes—
213D5—81no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—101no—
221C23—102no—
222C23—101no—
223C23—102no—
224C23—101no—
225C23#71114yes—
226C23C23×D511yescharacteristic
227D4—28no—
228D4—208no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—52no—
239C2×C4—52no—
240C2×C4—101no—
241C2×C4—102no—
242C2×C4—101no—
243C2×C4—101no—
244C2×C4—102no—
245C2×C4—102no—
246C2×C4—102no—
247C2×C4—102no—
248C2×C4—102no—
249C2×C4—102no—
250C2×C4—102no—
251C2×C4—208no—
252C2×C4D4×D512yes—
253C5#263611yescharacteristic, complemented, Sylow
254C22—24no—
255C22—24no—
256C22—24no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—201no—
262C22—201no—
263C22—202no—
264C22—202no—
265C22—201no—
266C22C2×D4×D511yescharacteristic
267C22C2×#71112yes—
268C22#414411yescharacteristic
269C22#414912yes—
270C22#415011yescharacteristic
271C4—22no—
272C4—41no—
273C4—41no—
274C4—42no—
275C4—102no—
276C4—102no—
277C4—202no—
278C4—202no—
279C4—201no—
280C2—44no—
281C2—401no—
282C2#6592011yescharacteristic
283C2#6594211yescharacteristic
284C2#6595112yes—
285C2#6599912yes—
286C2#6600811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radical#65512C2
hypercenter#394D5
hypocenterC5#2636
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic5C2semidirect
2. C2×#4044C2semidirect
3. C5×#394C2semidirect
5. #65275C2semidirect
6. #65248C2semidirect
7. #65247C2semidirect
8. #65239C2semidirect
9. #65237C2semidirect
10. #64966C2semidirect
11. #64939C2semidirect
71. C4×Dic5D4semidirect
78. #394D5semidirect
253. C5#2636semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40442
C5×#3942
#655122
#652752
#652482
#652472
#652392
#652372
#649662
#649392
#26365