Finite Group Databasebeta

← #10614331

Subgroups — #10614331

Order 640.

680 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 141 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×C5×#120C211yesmaximal, characteristic
2C2×#4081C211yesmaximal, characteristic, complemented
3#65536C212yesmaximal, complemented
4#65511C212yesmaximal, complemented
5#65510C211yesmaximal, characteristic, complemented
6#65509C212yesmaximal, complemented
7#65241C212yesmaximal, complemented
8#65237C212yesmaximal, complemented
9#65234C212yesmaximal, complemented
10C25×C5C2211yescharacteristic
11C22×C5×D4C2212yes—
12C22×C5×D4C2211yescharacteristic
13C23×Dic5C2211yescharacteristic, complemented
14C5×#120—24no—
15C2×C5×C22⋊C4C2212yes—
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23C2×C22⋊Dic5C2211yescharacteristic, complemented
24C2×C22⋊Dic5C2212yescomplemented
25#4081—24no—
26C2×C4⋊Dic5C2211yescharacteristic, complemented
27C2×#683C2212yescomplemented
28C2×#683C2212yescomplemented
29C2×C4×Dic5C2211yescharacteristic, complemented
30#3963C2212yescomplemented
31#3963C2212yescomplemented
32#3963C2212yescomplemented
33#2697—51nomaximal, Sylow
34C24×C5—22no—
35C24×C5—24no—
36C24×C5—42no—
37C24×C5—41no—
38C24×C5—41no—
39C24×C5C2311yescharacteristic
40C24×C5C2311yescharacteristic
41C24×C5C2312yes—
42C2×C5×D4—24no—
43C2×C5×D4—24no—
44C2×C5×D4—41no—
45C2×C5×D4—41no—
46C2×C5×D4—42no—
47C2×C5×D4D414yes—
48C22×C4×C5C2312yes—
49C22×C4×C5C2311yescharacteristic
50C22×Dic5—22no—
51C22×Dic5—41no—
52C22×Dic5—41no—
53C22×Dic5C2312yes—
54C22×Dic5C2312yes—
55C22×Dic5C2312yes—
56C22×Dic5C2312yes—
57C5×C22⋊C4—22no—
58C5×C22⋊C4—24no—
59C22⋊Dic5—24no—
60C22⋊Dic5—22no—
61C22⋊Dic5—24no—
62C22⋊Dic5—42no—
63C22⋊Dic5—42no—
64C22⋊Dic5—42no—
65C22⋊Dic5—42no—
66C22⋊Dic5—42no—
67C4⋊Dic5—41no—
68#683—24no—
69#683—42no—
70C4×Dic5—41no—
71C2×#123—51no—
72C2×#120D511yescharacteristic
73#397—52no—
74#396—52no—
75#394—52no—
76#390—51no—
77#388—52no—
78#386—52no—
79#379—52no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—21no—
90C23×C5—21no—
91C23×C5—21no—
92C23×C5—21no—
93C23×C5—22no—
94C23×C5—48no—
95C23×C5—44no—
96C23×C5—44no—
97C23×C5—48no—
98C23×C5—44no—
99C23×C5C2×D412yes—
100C23×C5C2×D412yes—
101C23×C5#4114yes—
102C23×C5C2411yescharacteristic
103C5×D4—44no—
104C5×D4—48no—
105C2×C4×C5—21no—
106C2×C4×C5—21no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5C2×D412yes—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—44no—
124C25C2×D511yescharacteristic
125C22×D4C2×D512yes—
126C22×D4C2×D511yescharacteristic
127C23×C4—51no—
128#123—104no—
129#120—24no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—51no—
132C2×C4⋊C4—52no—
133C2×C22⋊C4—52no—
134C2×C22⋊C4—52no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4C2×D512yes—
142C2×C22⋊C4C2×D511yescharacteristic
143C2×C42—51no—
144#95—52no—
145#95—52no—
146#95—52no—
147C22×C5—22no—
148C22×C5—22no—
149C22×C5—24no—
150C22×C5—24no—
151C22×C5—24no—
152C22×C5—24no—
153C22×C5—24no—
154C22×C5—416no—
155C22×C5—42no—
156C22×C5—42no—
157C22×C5—42no—
158C22×C5—42no—
159C22×C5—42no—
160C22×C5—41no—
161C22×C5—41no—
162C22×C5—41no—
163C22×C5—41no—
164C22×C5—42no—
165C22×C5C22×D411yescharacteristic
166C22×C5C2×#4112yes—
167C22×C5H(5,𝔽2)11yescharacteristic
168C22×C5H(5,𝔽2)11yescharacteristic
169C22×C5H(5,𝔽2)12yes—
170C4×C5—22no—
171C4×C5—42no—
172Dic5—42no—
173Dic5—42no—
174Dic5—42no—
175Dic5—42no—
176C24—24no—
177C24—22no—
178C24—42no—
179C24—41no—
180C24—41no—
181C24C22×D512yes—
182C24C22×D511yescharacteristic
183C24C22×D511yescharacteristic
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4C22⋊D514yes—
190C22×C4—52no—
191C22×C4—52no—
192C22×C4—52no—
193C22×C4—52no—
194C22×C4—102no—
195C22×C4—201no—
196C22×C4—201no—
197C22×C4C22×D512yes—
198C22×C4C22×D511yescharacteristic
199C4⋊C4—104no—
200C4⋊C4—202no—
201C4⋊C4—201no—
202C22⋊C4—22no—
203C22⋊C4—24no—
204C22⋊C4—104no—
205C22⋊C4—102no—
206C22⋊C4—104no—
207C22⋊C4—202no—
208C22⋊C4—202no—
209C22⋊C4—202no—
210C22⋊C4—202no—
211C22⋊C4—202no—
212C42—201no—
213C2×C5—42no—
214C2×C5—42no—
215C2×C5—44no—
216C2×C5#53411yescharacteristic
217C2×C5#53812yes—
218C2×C5#54612yes—
219C2×C5#55111yescharacteristic
220C2×C5#56011yescharacteristic
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—48no—
236C23—44no—
237C23—44no—
238C23—48no—
239C23—44no—
240C23#71114yes—
241C23C2×C22⋊D512yes—
242C23C2×C22⋊D512yes—
243C23C23×D511yescharacteristic
244D4—44no—
245D4—48no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—102no—
256C2×C4—102no—
257C2×C4—102no—
258C2×C4—102no—
259C2×C4—102no—
260C2×C4—102no—
261C2×C4—102no—
262C2×C4—102no—
263C2×C4—204no—
264C2×C4C2×C22⋊D512yes—
265C5#269711yescharacteristic, complemented, Sylow
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—416no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—42no—
283C22—41no—
284C22C2×#71112yes—
285C22#414411yescharacteristic
286C22#414411yescharacteristic
287C22#414412yes—
288C22C22×C22⋊D511yescharacteristic
289C4—22no—
290C4—42no—
291C4—202no—
292C4—202no—
293C4—202no—
294C4—202no—
295C2—42no—
296C2—44no—
297C2—42no—
298C2#6592111yescharacteristic
299C2#6592712yes—
300C2#6592911yescharacteristic
301C2#6613811yescharacteristic
302C2#6614012yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC2×C5×#120C2
hypercenterC2×#120D5
hypocenterC5#2697
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C2×#4081C2semidirect
3. #65536C2semidirect
4. #65511C2semidirect
5. #65510C2semidirect
6. #65509C2semidirect
7. #65241C2semidirect
8. #65237C2semidirect
9. #65234C2semidirect
13. C23×Dic5C22semidirect
23. C2×C22⋊Dic5C22semidirect
24. C2×C22⋊Dic5C22semidirect
26. C2×C4⋊Dic5C22semidirect
27. C2×#683C22semidirect
28. C2×#683C22semidirect
29. C2×C4×Dic5C22semidirect
30. #3963C22semidirect
31. #3963C22semidirect
32. #3963C22semidirect
265. C5#2697semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1202
C2×#40812
#655362
#655112
#655102
#655092
#652412
#652372
#652342
#26975