Finite Group Databasebeta

← #10609838

Subgroups — #10609838

Order 640.

900 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 159 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×#4041C211yesmaximal, characteristic, complemented
2C2×#4040C212yesmaximal, complemented
3C2×#4028C212yesmaximal, complemented
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5C5×#399C211yesmaximal, characteristic, complemented
6#65281C211yesmaximal, characteristic, complemented
7#65277C212yesmaximal, complemented
8#65249C211yesmaximal, characteristic, complemented
9#65246C212yesmaximal, complemented
10#64960C211yesmaximal, characteristic, complemented
11#64955C211yesmaximal, characteristic, complemented
12C24×D5C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C22×D20C2212yes—
15C22×D20C2212yes—
16C22×D20C2211yescharacteristic, complemented
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic, complemented
20C2×C5×C4⋊C4C2212yescomplemented
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2212yes—
23C2×C22⋊Dic5C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2212yes—
26C2×#685C2211yescharacteristic
27C2×#685C2212yes—
28C2×#685C2212yescomplemented
29C2×#685C2212yescomplemented
30C2×#685C2212yescomplemented
31C2×#683C2212yescomplemented
32C2×C4×Dic5C2212yescomplemented
33#4041—24no—
34#4040—28no—
35#4028—28no—
36D5×C22⋊C4—24no—
37C5×#95C2211yescharacteristic, complemented
38#3963C2211yescharacteristic, complemented
39#2539—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C23×D5—22no—
42C23×D5—22no—
43C23×D5—22no—
44C23×D5—24no—
45C23×D5—42no—
46C23×D5C2312yes—
47C23×D5C2311yescharacteristic
48C23×D5C2311yescharacteristic
49C22×C4×C5C2311yescharacteristic
50C22×C4×C5C2312yes—
51C22×C4×C5C2311yescharacteristic
52C22×C4×C5C2312yes—
53C2×C22⋊D5—24no—
54C2×C22⋊D5—24no—
55C2×C22⋊D5—42no—
56C22×Dic5C2311yescharacteristic
57C22×Dic5C2311yescharacteristic
58C22×Dic5C2312yes—
59C2×D20—24no—
60C2×D20—24no—
61C2×D20—24no—
62C2×D20—24no—
63C2×D20—24no—
64C2×D20—22no—
65C2×D20—42no—
66C2×D20—42no—
67C2×C4×D5—22no—
68C2×C4×D5—24no—
69C2×C4×D5—42no—
70C2×C4×D5D414yescomplemented
71C5×C4⋊C4—24no—
72C5×C4⋊C4—22no—
73C5×C22⋊C4—22no—
74C5×C22⋊C4—24no—
75C22⋊Dic5—22no—
76#685—22no—
77#685—24no—
78#685—22no—
79#685—24no—
80#685—24no—
81#685—24no—
82#685D418yescomplemented
83#683—24no—
84C4×Dic5—42no—
85C2×#123—52no—
86C2×#121—51no—
87C2×#120—52no—
88C22×C22⋊C4—51no—
89#399—51no—
90#399D511yescharacteristic, complemented
91#392—51no—
92#392—51no—
93#390—52no—
94#390—52no—
95C4×C22⋊C4—51no—
96C23×C5—21no—
97C23×C5—22no—
98C23×C5—21no—
99C23×C5—22no—
100C23×C5#4112yes—
101C23×C5C2411yescharacteristic
102C22×D5—21no—
103C22×D5—21no—
104C22×D5—22no—
105C22×D5—28no—
106C22×D5—21no—
107C22×D5—22no—
108C22×D5—21no—
109C22×D5—22no—
110C22×D5—22no—
111C22×D5—22no—
112C22×D5—22no—
113C22×D5—22no—
114C22×D5—22no—
115C22×D5—22no—
116C22×D5—22no—
117C22×D5—48no—
118C22×D5—48no—
119C22×D5—44no—
120C22×D5—44no—
121C22×D5C2×D414yes—
122C22×D5C2×D412yes—
123C22×D5#4112yes—
124C2×C4×C5—22no—
125C2×C4×C5—22no—
126C2×C4×C5—22no—
127C2×C4×C5—22no—
128C2×C4×C5—21no—
129C2×C4×C5—22no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5C2×D414yes—
133C2×C4×C5C2×D412yes—
134C22⋊D5—48no—
135C2×Dic5—22no—
136C2×Dic5—22no—
137C2×Dic5—22no—
138C2×Dic5—22no—
139C2×Dic5—21no—
140C2×Dic5C2×D414yes—
141C2×Dic5C2×D412yes—
142D20—48no—
143D20—44no—
144D20—48no—
145C4×D5—44no—
146C4×D5—44no—
147C25—51no—
148C22×D4—52no—
149C22×D4—52no—
150C22×D4—52no—
151C22×D4—51no—
152C23×C4—51no—
153C23×C4—51no—
154#123—108no—
155#121—104no—
156#120—108no—
157C2×C4⋊C4—52no—
158C2×C4⋊C4C2×D511yescharacteristic, complemented
159C2×C4⋊C4C2×D512yescomplemented
160C2×C22⋊C4—52no—
161C2×C22⋊C4—51no—
162C2×C22⋊C4—51no—
163C2×C22⋊C4—51no—
164C2×C22⋊C4—52no—
165C2×C22⋊C4—52no—
166C2×C22⋊C4—52no—
167C2×C22⋊C4—52no—
168C2×C22⋊C4—104no—
169C2×C22⋊C4C2×D511yescharacteristic
170C2×C22⋊C4C2×D512yes—
171C2×C42—52no—
172#95—51no—
173#95C2×D511yescharacteristic, complemented
174C22×C5—22no—
175C22×C5—24no—
176C22×C5—41no—
177C22×C5—41no—
178C22×C5—42no—
179C22×C5C22×D411yescharacteristic
180C22×C5C22×D412yes—
181C22×C5C2×#4111yescharacteristic
182C22×C5C2×#4111yescharacteristic
183C22×C5H(5,𝔽2)12yes—
184C2×D5—22no—
185C2×D5—22no—
186C2×D5—24no—
187C2×D5—24no—
188C2×D5—22no—
189C2×D5—24no—
190C2×D5—416no—
191C2×D5—41no—
192C2×D5—41no—
193C2×D5—42no—
194C2×D5—42no—
195C2×D5—42no—
196C2×D5—42no—
197C2×D5—42no—
198C2×D5—42no—
199C2×D5—42no—
200C2×D5—42no—
201C2×D5#12714yes—
202C4×C5—22no—
203C4×C5—24no—
204C4×C5—41no—
205C4×C5—42no—
206Dic5—22no—
207Dic5—24no—
208Dic5—41no—
209C24—51no—
210C24—51no—
211C24—52no—
212C24—102no—
213C24—102no—
214C24—104no—
215C24—102no—
216C24—202no—
217C24C22×D511yescharacteristic
218C2×D4—104no—
219C2×D4—104no—
220C2×D4—104no—
221C2×D4—104no—
222C2×D4—104no—
223C2×D4—104no—
224C2×D4—104no—
225C2×D4—102no—
226C2×D4—202no—
227C2×D4—202no—
228C2×D4—202no—
229C22×C4—52no—
230C22×C4—51no—
231C22×C4—54no—
232C22×C4—51no—
233C22×C4—102no—
234C22×C4—104no—
235C22×C4—202no—
236C22×C4C22×D511yescharacteristic
237C22×C4C22×D512yes—
238C22×C4C22×D511yescharacteristic
239C22×C4C22×D512yes—
240C4⋊C4—22no—
241C4⋊C4—24no—
242C4⋊C4—104no—
243C22⋊C4—22no—
244C22⋊C4—24no—
245C22⋊C4—58no—
246C22⋊C4—102no—
247C22⋊C4—104no—
248C22⋊C4—104no—
249C22⋊C4—104no—
250C22⋊C4—102no—
251C22⋊C4—102no—
252C22⋊C4—104no—
253C42—202no—
254C2×C5—42no—
255C2×C5#52511yescharacteristic
256C2×C5C2×#12711yescharacteristic
257C2×C5#53411yescharacteristic
258C2×C5#54612yes—
259C2×C5#54612yes—
260D5—24no—
261D5—42no—
262D5—82no—
263C23—21no—
264C23—22no—
265C23—21no—
266C23—22no—
267C23—54no—
268C23—52no—
269C23—52no—
270C23—108no—
271C23—101no—
272C23—101no—
273C23—102no—
274C23—101no—
275C23—102no—
276C23—101no—
277C23—102no—
278C23—102no—
279C23—102no—
280C23—102no—
281C23—102no—
282C23—102no—
283C23—102no—
284C23—102no—
285C23—208no—
286C23—204no—
287C23—208no—
288C23—204no—
289C23#71312yes—
290C23C23×D511yescharacteristic
291D4—208no—
292D4—204no—
293D4—208no—
294D4—208no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—54no—
304C2×C4—52no—
305C2×C4—102no—
306C2×C4—102no—
307C2×C4—102no—
308C2×C4—101no—
309C2×C4—102no—
310C2×C4—204no—
311C2×C4—204no—
312C2×C4D4×D514yes—
313C2×C4D4×D512yes—
314C5#253911yescharacteristic, complemented, Sylow
315C22—22no—
316C22—24no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—54no—
321C22—102no—
322C22—104no—
323C22—102no—
324C22—104no—
325C22—102no—
326C22—104no—
327C22—2016no—
328C22—201no—
329C22—202no—
330C22—201no—
331C22—202no—
332C22—202no—
333C22—202no—
334C22—202no—
335C22—202no—
336C22—202no—
337C22—202no—
338C22C2×D4×D511yescharacteristic
339C22C2×D4×D512yes—
340C22C2×#71311yescharacteristic
341C22D5×#4111yescharacteristic
342C22#414912yes—
343C4—22no—
344C4—24no—
345C4—41no—
346C4—42no—
347C4—104no—
348C4—102no—
349C4—201no—
350C2—42no—
351C2—104no—
352C2—202no—
353C2—402no—
354C2#6596111yescharacteristic
355C2#6596412yes—
356C2#6598711yescharacteristic
357C2#6598812yes—
358C2D5×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
hypercenter#399D5
hypocenterC5#2539
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4041C2semidirect
2. C2×#4040C2semidirect
3. C2×#4028C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. C5×#399C2semidirect
6. #65281C2semidirect
7. #65277C2semidirect
8. #65249C2semidirect
9. #65246C2semidirect
10. #64960C2semidirect
11. #64955C2semidirect
16. C22×D20C22semidirect
19. C2×C5×C4⋊C4C22semidirect
20. C2×C5×C4⋊C4C22semidirect
28. C2×#685C22semidirect
29. C2×#685C22semidirect
30. C2×#685C22semidirect
31. C2×#683C22semidirect
32. C2×C4×Dic5C22semidirect
37. C5×#95C22semidirect
38. #3963C22semidirect
70. C2×C4×D5D4semidirect
82. #685D4semidirect
90. #399D5semidirect
158. C2×C4⋊C4C2×D5semidirect
159. C2×C4⋊C4C2×D5semidirect
173. #95C2×D5semidirect
314. C5#2539semidirect

Maximal subgroups

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

subgroupindex
C2×#40412
C2×#40402
C2×#40282
C2×D5×C22⋊C42
C5×#3992
#652812
#652772
#652492
#652462
#649602
#649552
#25395