Finite Group Databasebeta

← #10614329

Subgroups — #10614329

Order 640.

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

normal subgroups only

This group has 688 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×#4082C211yesmaximal, characteristic, complemented
3#65536C212yesmaximal, complemented
4#65511C212yesmaximal, complemented
5#65509C212yesmaximal, complemented
6#65509C211yesmaximal, characteristic, complemented
7#65241C212yesmaximal, complemented
8#65236C212yesmaximal, complemented
9#65233C212yesmaximal, complemented
10C25×C5C2211yescharacteristic
11C22×C5×D4C2212yes—
12C22×C5×D4C2211yescharacteristic
13C23×Dic5C2211yescharacteristic, complemented
14C22×Dic10C2211yescharacteristic, complemented
15C5×#120—24no—
16C2×C5×C22⋊C4C2212yes—
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23C2×C22⋊Dic5C2212yes—
24C2×C22⋊Dic5C2212yescomplemented
25C2×C22⋊Dic5C2211yescharacteristic, complemented
26#4082—24no—
27C2×#683C2212yescomplemented
28C2×C4×Dic5C2211yescharacteristic, complemented
29#3963C2212yescomplemented
30#3963C2212yescomplemented
31#3963C2212yescomplemented
32#3963C2212yescomplemented
33#2684—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—41no—
51C22×Dic5—42no—
52C22×Dic5C2312yes—
53C22×Dic5C2312yes—
54C22×Dic5C2312yes—
55C22×Dic5C2312yes—
56C2×Dic10—22no—
57C2×Dic10—41no—
58C2×Dic10—41no—
59C5×C22⋊C4—22no—
60C5×C22⋊C4—24no—
61C22⋊Dic5—24no—
62C22⋊Dic5—24no—
63C22⋊Dic5—24no—
64C22⋊Dic5—42no—
65C22⋊Dic5—42no—
66C22⋊Dic5—42no—
67C22⋊Dic5—42no—
68C22⋊Dic5—41no—
69#683—42no—
70C4×Dic5—22no—
71C2×#124—51no—
72C2×#120D511yescharacteristic
73#397—52no—
74#394—52no—
75#393—52no—
76#388—52no—
77#386—51no—
78#386—52no—
79#379—52no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—21no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—21no—
90C23×C5—22no—
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—
124Dic10—44no—
125C25C2×D511yescharacteristic
126C22×Q8—51no—
127C22×D4C2×D512yes—
128C22×D4C2×D511yescharacteristic
129C23×C4—51no—
130#124—104no—
131#120—24no—
132C2×C4⋊C4—52no—
133C2×C22⋊C4—52no—
134C2×C22⋊C4—52no—
135C2×C22⋊C4—52no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—51no—
141C2×C22⋊C4C2×D512yes—
142C2×C22⋊C4C2×D511yescharacteristic
143C2×C42—51no—
144#95—52no—
145#95—52no—
146#95—52no—
147#95—52no—
148C22×C5—22no—
149C22×C5—22no—
150C22×C5—24no—
151C22×C5—24no—
152C22×C5—24no—
153C22×C5—24no—
154C22×C5—24no—
155C22×C5—416no—
156C22×C5—42no—
157C22×C5—42no—
158C22×C5—42no—
159C22×C5—42no—
160C22×C5—42no—
161C22×C5—41no—
162C22×C5—41no—
163C22×C5—41no—
164C22×C5—41no—
165C22×C5—42no—
166C22×C5C22×D411yescharacteristic
167C22×C5C2×#4112yes—
168C22×C5H(5,𝔽2)11yescharacteristic
169C22×C5H(5,𝔽2)11yescharacteristic
170C22×C5H(5,𝔽2)12yes—
171C4×C5—22no—
172C4×C5—42no—
173Dic5—42no—
174Dic5—42no—
175Dic5—42no—
176Dic5—42no—
177C24—24no—
178C24—22no—
179C24—42no—
180C24—41no—
181C24—41no—
182C24C22×D512yes—
183C24C22×D511yescharacteristic
184C24C22×D511yescharacteristic
185C2×Q8—102no—
186C2×Q8—201no—
187C2×Q8—201no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4C22⋊D514yes—
194C22×C4—52no—
195C22×C4—52no—
196C22×C4—52no—
197C22×C4—52no—
198C22×C4—201no—
199C22×C4—202no—
200C22×C4C22×D512yes—
201C22×C4C22×D511yescharacteristic
202C4⋊C4—202no—
203C22⋊C4—22no—
204C22⋊C4—24no—
205C22⋊C4—104no—
206C22⋊C4—104no—
207C22⋊C4—104no—
208C22⋊C4—201no—
209C22⋊C4—202no—
210C22⋊C4—202no—
211C22⋊C4—202no—
212C22⋊C4—202no—
213C42—102no—
214C2×C5—42no—
215C2×C5—42no—
216C2×C5—44no—
217C2×C5#53411yescharacteristic
218C2×C5#53812yes—
219C2×C5#54612yes—
220C2×C5#55111yescharacteristic
221C2×C5H(3,𝔽4)11yescharacteristic
222C23—22no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—21no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—48no—
237C23—44no—
238C23—44no—
239C23—48no—
240C23—44no—
241C23#71114yes—
242C23C2×C22⋊D512yes—
243C23C2×C22⋊D512yes—
244C23C23×D511yescharacteristic
245Q8—204no—
246D4—44no—
247D4—48no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
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—102no—
264C2×C4—102no—
265C2×C4—204no—
266C2×C4C2×C22⋊D512yes—
267C5#268411yescharacteristic, complemented, Sylow
268C22—22no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—416no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22C2×#71112yes—
287C22#414411yescharacteristic
288C22#414411yescharacteristic
289C22#414412yes—
290C22C22×C22⋊D511yescharacteristic
291C4—22no—
292C4—42no—
293C4—202no—
294C4—202no—
295C4—202no—
296C4—202no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2#6592111yescharacteristic
301C2#6592712yes—
302C2#6592811yescharacteristic
303C2#6613811yescharacteristic
304C2#6614012yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC2×C5×#120C2
hypercenterC2×#120D5
hypocenterC5#2684
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C2×#4082C2semidirect
3. #65536C2semidirect
4. #65511C2semidirect
5. #65509C2semidirect
6. #65509C2semidirect
7. #65241C2semidirect
8. #65236C2semidirect
9. #65233C2semidirect
13. C23×Dic5C22semidirect
14. C22×Dic10C22semidirect
24. C2×C22⋊Dic5C22semidirect
25. C2×C22⋊Dic5C22semidirect
27. C2×#683C22semidirect
28. C2×C4×Dic5C22semidirect
29. #3963C22semidirect
30. #3963C22semidirect
31. #3963C22semidirect
32. #3963C22semidirect
267. C5#2684semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1202
C2×#40822
#655362
#655112
#655092
#652412
#652362
#652332
#26845