Finite Group Databasebeta

← #10605584

Subgroups — #10605584

Order 640.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×C22⋊D5C211yesmaximal, characteristic
2C2×#4029C212yesmaximal, complemented
3C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
4C5×#379C211yesmaximal, characteristic, complemented
5#65536C211yesmaximal, characteristic, complemented
6#65250C212yesmaximal, complemented
7#65248C212yesmaximal, complemented
8#65245C212yesmaximal, complemented
9#65244C211yesmaximal, characteristic, complemented
10#65239C211yesmaximal, characteristic, complemented
11#65235C211yesmaximal, characteristic, complemented
12C25×C5C2211yescharacteristic
13C24×D5C2211yescharacteristic
14C22×C22⋊D5—24no—
15C22×C22⋊D5—28no—
16C22×C22⋊D5C2212yes—
17C22×C22⋊D5C2212yes—
18C23×Dic5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic, complemented
20C22×C4×D5C2211yescharacteristic, complemented
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2212yescomplemented
24C2×C5×C22⋊C4C2212yescomplemented
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2212yescomplemented
28C2×C22⋊Dic5C2212yescomplemented
29C2×#685C2211yescharacteristic, complemented
30C2×#685C2211yescharacteristic, complemented
31C2×#685C2212yescomplemented
32C2×#685C2212yescomplemented
33C2×C4⋊Dic5C2212yescomplemented
34C2×#683C2212yescomplemented
35#4029—28no—
36D5×C22⋊C4—24no—
37#3963C2212yescomplemented
38#3963C2212yescomplemented
39#2405—51nomaximal, Sylow
40C24×C5—24no—
41C24×C5—22no—
42C24×C5—22no—
43C24×C5—22no—
44C24×C5—42no—
45C24×C5C2312yes—
46C24×C5C2311yescharacteristic
47C23×D5—22no—
48C23×D5—22no—
49C23×D5—24no—
50C23×D5—42no—
51C23×D5D414yes—
52C23×D5C2311yescharacteristic
53C23×D5C2311yescharacteristic
54C22×C4×C5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2312yes—
57C2×C22⋊D5—232no—
58C2×C22⋊D5—24no—
59C2×C22⋊D5—24no—
60C2×C22⋊D5—24no—
61C2×C22⋊D5—24no—
62C2×C22⋊D5—24no—
63C2×C22⋊D5D418yes—
64C22×Dic5—22no—
65C22×Dic5—24no—
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2312yes—
69C22×Dic5C2312yes—
70C2×C4×D5—22no—
71C2×C4×D5—22no—
72C2×C4×D5—24no—
73C2×C4×D5—42no—
74C5×C22⋊C4—22no—
75C5×C22⋊C4—24no—
76C5×C22⋊C4—22no—
77C5×C22⋊C4—42no—
78C22⋊Dic5—22no—
79C22⋊Dic5—24no—
80C22⋊Dic5—22no—
81C22⋊Dic5—42no—
82#685—22no—
83#685—22no—
84#685—24no—
85#685—42no—
86C4⋊Dic5—24no—
87#683—24no—
88C23×D4—51no—
89C2×#123—52no—
90C22×C22⋊C4—51no—
91#394—52no—
92#394—52no—
93#386—52no—
94#385—51no—
95#385—51no—
96#379—51no—
97#379—51no—
98#379D511yescharacteristic, complemented
99C23×C5—22no—
100C23×C5—22no—
101C23×C5—28no—
102C23×C5—22no—
103C23×C5—22no—
104C23×C5—22no—
105C23×C5—22no—
106C23×C5—22no—
107C23×C5—22no—
108C23×C5—48no—
109C23×C5—48no—
110C23×C5—44no—
111C23×C5—44no—
112C23×C5C2×D414yes—
113C23×C5C2×D412yes—
114C23×C5#4112yes—
115C23×C5#4112yes—
116C23×C5C2411yescharacteristic
117C22×D5—21no—
118C22×D5—21no—
119C22×D5—22no—
120C22×D5—28no—
121C22×D5—22no—
122C22×D5—21no—
123C22×D5—21no—
124C22×D5—22no—
125C22×D5—28no—
126C22×D5—44no—
127C22×D5—48no—
128C22×D5—48no—
129C22×D5C2×D412yes—
130C22×D5C2×D414yes—
131C22×D5C2×D412yes—
132C2×C4×C5—22no—
133C2×C4×C5—22no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—22no—
137C2×C4×C5—22no—
138C2×C4×C5—22no—
139C22⋊D5—216no—
140C22⋊D5—216no—
141C2×Dic5—28no—
142C2×Dic5—22no—
143C2×Dic5—22no—
144C2×Dic5—22no—
145C2×Dic5—22no—
146C2×Dic5—21no—
147C2×Dic5—21no—
148C2×Dic5—22no—
149C2×Dic5—22no—
150C2×Dic5—22no—
151C2×Dic5C2×D414yes—
152C4×D5—44no—
153C4×D5—44no—
154C25—51no—
155C25C2×D511yescharacteristic
156C22×D4—52no—
157C22×D4—52no—
158C22×D4—108no—
159C22×D4—104no—
160C23×C4—51no—
161C23×C4—51no—
162C23×C4—51no—
163#123—108no—
164C2×C4⋊C4—52no—
165C2×C4⋊C4—52no—
166C2×C22⋊C4—51no—
167C2×C22⋊C4—51no—
168C2×C22⋊C4—52no—
169C2×C22⋊C4—51no—
170C2×C22⋊C4—51no—
171C2×C22⋊C4—52no—
172C2×C22⋊C4—52no—
173C2×C22⋊C4—52no—
174C2×C22⋊C4—104no—
175C2×C22⋊C4C2×D511yescharacteristic
176C2×C22⋊C4C2×D511yescharacteristic
177C2×C22⋊C4C2×D512yescomplemented
178C2×C22⋊C4C2×D512yescomplemented
179#95—52no—
180#95—52no—
181C22×C5—22no—
182C22×C5—22no—
183C22×C5—24no—
184C22×C5—24no—
185C22×C5—24no—
186C22×C5—24no—
187C22×C5—24no—
188C22×C5—416no—
189C22×C5—42no—
190C22×C5—42no—
191C22×C5—42no—
192C22×C5—42no—
193C22×C5#12314yescomplemented
194C22×C5C22×D411yescharacteristic
195C22×C5C22×D412yes—
196C22×C5C2×#4111yescharacteristic
197C22×C5C2×#4111yescharacteristic
198C22×C5H(5,𝔽2)12yes—
199C2×D5—22no—
200C2×D5—22no—
201C2×D5—24no—
202C2×D5—24no—
203C2×D5—22no—
204C2×D5—22no—
205C2×D5—22no—
206C2×D5—416no—
207C2×D5—42no—
208C2×D5—42no—
209C2×D5#12014yes—
210C4×C5—41no—
211C4×C5—41no—
212C4×C5—42no—
213Dic5—24no—
214Dic5—41no—
215Dic5—41no—
216Dic5—42no—
217C24—24no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—42no—
222C24—51no—
223C24—51no—
224C24—54no—
225C24—104no—
226C24—102no—
227C24—102no—
228C24—202no—
229C24C22×D512yes—
230C24C22×D511yescharacteristic
231C2×D4—58no—
232C2×D4—104no—
233C2×D4—104no—
234C2×D4—104no—
235C2×D4—104no—
236C2×D4—1032no—
237C2×D4—104no—
238C22×C4—52no—
239C22×C4—51no—
240C22×C4—52no—
241C22×C4—51no—
242C22×C4—102no—
243C22×C4—102no—
244C22×C4—104no—
245C22×C4—104no—
246C22×C4—102no—
247C22×C4—202no—
248C22×C4C22×D511yescharacteristic
249C22×C4C22×D512yes—
250C22×C4C22×D511yescharacteristic
251C4⋊C4—104no—
252C4⋊C4—104no—
253C22⋊C4—22no—
254C22⋊C4—24no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—102no—
258C22⋊C4—102no—
259C22⋊C4—104no—
260C22⋊C4—102no—
261C22⋊C4—102no—
262C22⋊C4—104no—
263C22⋊C4—202no—
264C22⋊C4—202no—
265C2×C5—24no—
266C2×C5—44no—
267C2×C5C2×#12011yescharacteristic
268C2×C5C2×#12311yescharacteristic
269C2×C5#53411yescharacteristic
270C2×C5#54612yes—
271C2×C5#54612yes—
272D5—24no—
273D5—42no—
274C23—22no—
275C23—22no—
276C23—28no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—48no—
284C23—48no—
285C23—44no—
286C23—44no—
287C23—54no—
288C23—52no—
289C23—52no—
290C23—108no—
291C23—101no—
292C23—101no—
293C23—102no—
294C23—101no—
295C23—102no—
296C23—101no—
297C23—102no—
298C23—108no—
299C23—204no—
300C23—208no—
301C23—208no—
302C23#70912yes—
303C23D4×D514yes—
304C23D4×D512yes—
305C23#71112yes—
306C23C23×D511yescharacteristic
307D4—1016no—
308D4—1016no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—54no—
317C2×C4—108no—
318C2×C4—102no—
319C2×C4—102no—
320C2×C4—102no—
321C2×C4—102no—
322C2×C4—101no—
323C2×C4—101no—
324C2×C4—102no—
325C2×C4—102no—
326C2×C4—102no—
327C2×C4—204no—
328C2×C4—204no—
329C5#240511yescharacteristic, complemented, Sylow
330C22—24no—
331C22—22no—
332C22—22no—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—416no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—54no—
343C22—102no—
344C22—104no—
345C22—102no—
346C22—104no—
347C22—102no—
348C22—102no—
349C22—102no—
350C22—2016no—
351C22—202no—
352C22—202no—
353C22#402914yescomplemented
354C22C2×#70911yescharacteristic
355C22C2×D4×D511yescharacteristic
356C22C2×D4×D512yes—
357C22C2×#71111yescharacteristic
358C22#414412yes—
359C4—41no—
360C4—41no—
361C4—42no—
362C4—104no—
363C4—202no—
364C4—201no—
365C4—201no—
366C2—24no—
367C2—44no—
368C2—104no—
369C2—202no—
370C2C2×#402911yescharacteristic
371C2#6582412yes—
372C2D5×#12011yescharacteristic
373C2#6592311yescharacteristic
374C2#6592512yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#379C2
prime radicalC23×C22⋊D5C2
hypercenter#379D5
hypocenterC5#2405
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#4029C2semidirect
3. C2×D5×C22⋊C4C2semidirect
4. C5×#379C2semidirect
5. #65536C2semidirect
6. #65250C2semidirect
7. #65248C2semidirect
8. #65245C2semidirect
9. #65244C2semidirect
10. #65239C2semidirect
11. #65235C2semidirect
19. C22×C4×D5C22semidirect
20. C22×C4×D5C22semidirect
23. C2×C5×C22⋊C4C22semidirect
24. C2×C5×C22⋊C4C22semidirect
27. C2×C22⋊Dic5C22semidirect
28. C2×C22⋊Dic5C22semidirect
29. C2×#685C22semidirect
30. C2×#685C22semidirect
31. C2×#685C22semidirect
32. C2×#685C22semidirect
33. C2×C4⋊Dic5C22semidirect
34. C2×#683C22semidirect
37. #3963C22semidirect
38. #3963C22semidirect
98. #379D5semidirect
177. C2×C22⋊C4C2×D5semidirect
178. C2×C22⋊C4C2×D5semidirect
193. C22×C5#123semidirect
329. C5#2405semidirect
353. C22#4029semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D52
C2×#40292
C2×D5×C22⋊C42
C5×#3792
#655362
#652502
#652482
#652452
#652442
#652392
#652352
#24055