Finite Group Databasebeta

← #10608045

Subgroups — #10608045

Order 640.

1348 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×D4×D5C211yesmaximal, characteristic, complemented
2C2×#4086C211yesmaximal, characteristic, complemented
3C2×#4041C211yesmaximal, characteristic, complemented
4C2×#4028C214yesmaximal, complemented
5C2×#4020C211yesmaximal, characteristic, complemented
6C5×#390C211yesmaximal, characteristic, complemented
7#65277C211yesmaximal, characteristic, complemented
8#65249C214yesmaximal, complemented
9C4×#685C211yesmaximal, characteristic, complemented
10C24×D5C2212yes—
11C22×C5×D4C2211yescharacteristic
12C22×C22⋊D5C2212yes—
13C22×C22⋊D5C2214yescomplemented
14C2×D4×D5—24no—
15C2×D4×D5—24no—
16C2×D4×D5—24no—
17C22×D20C2212yes—
18C22×D20C2211yescharacteristic
19C22×D20C2211yescharacteristic, complemented
20C22×D20C2214yescomplemented
21C22×C4×D5C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic, complemented
23C2×C5×C22⋊C4C2214yescomplemented
24C2×C42×C5C2211yescharacteristic, complemented
25#4086—24no—
26C2×#685C2212yes—
27C2×#685C2214yescomplemented
28C2×#685C2212yescomplemented
29C2×C4×Dic5C2211yescharacteristic, complemented
30#4041—24no—
31#4028—216no—
32#4020—24no—
33#3963C2212yescomplemented
34#2433—51nomaximal, Sylow
35C24×C5C2312yes—
36C23×D5—24no—
37C23×D5—24no—
38C23×D5—24no—
39C23×D5—28no—
40C23×D5—44no—
41C23×D5C2312yes—
42C23×D5C2312yes—
43C23×D5C2311yescharacteristic
44C2×C5×D4—22no—
45C2×C5×D4—22no—
46C2×C5×D4D414yescomplemented
47C22×C4×C5C2312yes—
48C22×C4×C5C2312yes—
49C22×C4×C5C2311yescharacteristic
50C2×C22⋊D5—24no—
51C2×C22⋊D5—24no—
52C2×C22⋊D5—28no—
53C2×C22⋊D5—24no—
54C2×C22⋊D5—28no—
55C2×C22⋊D5—44no—
56C22×Dic5C2311yescharacteristic
57C22×Dic5C2312yes—
58D4×D5—232no—
59C2×D20—22no—
60C2×D20—22no—
61C2×D20—28no—
62C2×D20—24no—
63C2×D20—28no—
64C2×D20—22no—
65C2×D20—22no—
66C2×D20—22no—
67C2×D20—28no—
68C2×D20—44no—
69C2×D20D414yescomplemented
70C2×C4×D5—22no—
71C2×C4×D5—22no—
72C2×C4×D5D414yescomplemented
73C5×C4⋊C4—22no—
74C5×C22⋊C4—28no—
75C42×C5—22no—
76#685—24no—
77#685—28no—
78#685D418yescomplemented
79C4×Dic5—22no—
80C23×D4—51no—
81C2×#127—51no—
82C2×#127—51no—
83C2×#121—51no—
84C2×#120—54no—
85#392—54no—
86#390—51no—
87#390D511yescharacteristic, complemented
88C4×C22⋊C4—51no—
89C23×C5—24no—
90C23×C5—22no—
91C23×C5—22no—
92C23×C5—24no—
93C23×C5C2×D414yes—
94C23×C5C2411yescharacteristic
95C22×D5—21no—
96C22×D5—21no—
97C22×D5—22no—
98C22×D5—216no—
99C22×D5—24no—
100C22×D5—22no—
101C22×D5—22no—
102C22×D5—24no—
103C22×D5—22no—
104C22×D5—22no—
105C22×D5—24no—
106C22×D5—22no—
107C22×D5—24no—
108C22×D5—416no—
109C22×D5—416no—
110C22×D5—48no—
111C22×D5—48no—
112C22×D5C2×D414yes—
113C22×D5C2×D414yes—
114C22×D5C2×D412yes—
115C5×D4—28no—
116C2×C4×C5—22no—
117C2×C4×C5—22no—
118C2×C4×C5—24no—
119C2×C4×C5—22no—
120C2×C4×C5C2×D412yes—
121C2×C4×C5C2×D412yes—
122C2×C4×C5C2×D412yes—
123C2×C4×C5C2×D414yescomplemented
124C22⋊D5—216no—
125C22⋊D5—416no—
126C2×Dic5—22no—
127C2×Dic5—21no—
128C2×Dic5—21no—
129C2×Dic5—22no—
130C2×Dic5C2×D412yes—
131C2×Dic5C2×D414yescomplemented
132D20—28no—
133D20—416no—
134D20—48no—
135D20—44no—
136C4×D5—28no—
137C25—52no—
138C22×D4—52no—
139C22×D4—54no—
140C22×D4—51no—
141C22×D4—52no—
142C22×D4—51no—
143C22×D4—54no—
144C22×D4—104no—
145C22×D4—104no—
146C22×D4—104no—
147C22×D4C2×D511yescharacteristic
148C23×C4—51no—
149#127—104no—
150#127—104no—
151#121—104no—
152#120—1016no—
153C2×C4⋊C4C2×D511yescharacteristic, complemented
154C2×C22⋊C4—52no—
155C2×C22⋊C4—54no—
156C2×C22⋊C4—52no—
157C2×C22⋊C4C2×D514yescomplemented
158C2×C42—51no—
159C2×C42C2×D511yescharacteristic, complemented
160#95—52no—
161C22×C5—24no—
162C22×C5—28no—
163C22×C5—42no—
164C22×C5—42no—
165C22×C5—44no—
166C22×C5C22×D411yescharacteristic
167C22×C5C22×D411yescharacteristic
168C22×C5C22×D411yescharacteristic
169C22×C5C22×D412yes—
170C22×C5H(5,𝔽2)12yes—
171C2×D5—22no—
172C2×D5—22no—
173C2×D5—24no—
174C2×D5—24no—
175C2×D5—24no—
176C2×D5—28no—
177C2×D5—432no—
178C2×D5—42no—
179C2×D5—42no—
180C2×D5—44no—
181C2×D5—42no—
182C2×D5—42no—
183C2×D5—42no—
184C2×D5—44no—
185C2×D5—44no—
186C2×D5#12714yescomplemented
187C4×C5—24no—
188C4×C5—42no—
189C4×C5#12014yescomplemented
190Dic5—22no—
191Dic5—24no—
192C24—51no—
193C24—52no—
194C24—52no—
195C24—104no—
196C24—104no—
197C24—108no—
198C24—104no—
199C24—204no—
200C24C22×D512yes—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—54no—
204C2×D4—108no—
205C2×D4—104no—
206C2×D4—108no—
207C2×D4—104no—
208C2×D4—108no—
209C2×D4—1032no—
210C2×D4—104no—
211C2×D4—104no—
212C2×D4—108no—
213C2×D4—102no—
214C2×D4—102no—
215C2×D4—102no—
216C2×D4—102no—
217C2×D4—102no—
218C2×D4—108no—
219C2×D4—204no—
220C2×D4—204no—
221C2×D4D2014yescomplemented
222C22×C4—51no—
223C22×C4—52no—
224C22×C4—54no—
225C22×C4—102no—
226C22×C4—102no—
227C22×C4C22×D512yes—
228C22×C4C22×D512yes—
229C22×C4C22×D511yescharacteristic
230C4⋊C4—22no—
231C22⋊C4—28no—
232C22⋊C4—58no—
233C22⋊C4—108no—
234C22⋊C4—104no—
235C42—22no—
236C42—102no—
237C2×C5—44no—
238C2×C5C2×#12011yescharacteristic
239C2×C5C2×#12711yescharacteristic
240C2×C5#53511yescharacteristic
241C2×C5D4212yes—
242C2×C5D4212yes—
243D5—24no—
244D5—44no—
245D5—82no—
246C23—24no—
247C23—22no—
248C23—22no—
249C23—24no—
250C23—54no—
251C23—52no—
252C23—54no—
253C23—1016no—
254C23—101no—
255C23—101no—
256C23—102no—
257C23—102no—
258C23—104no—
259C23—102no—
260C23—104no—
261C23—102no—
262C23—102no—
263C23—104no—
264C23—102no—
265C23—104no—
266C23—2016no—
267C23—208no—
268C23—2016no—
269C23—208no—
270C23C2×D2014yes—
271C23C23×D511yescharacteristic
272D4—28no—
273D4—1016no—
274D4—108no—
275D4—2016no—
276D4—208no—
277D4—2016no—
278D4—204no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—54no—
284C2×C4—52no—
285C2×C4—102no—
286C2×C4—101no—
287C2×C4—101no—
288C2×C4—102no—
289C2×C4—108no—
290C2×C4C2×D2012yes—
291C2×C4D4×D512yes—
292C2×C4D4×D512yes—
293C2×C4D4×D514yescomplemented
294C5#243311yescharacteristic, complemented, Sylow
295C22—24no—
296C22—28no—
297C22—42no—
298C22—44no—
299C22—42no—
300C22—54no—
301C22—102no—
302C22—104no—
303C22—104no—
304C22—102no—
305C22—104no—
306C22—108no—
307C22—2032no—
308C22—202no—
309C22—204no—
310C22—202no—
311C22—202no—
312C22—202no—
313C22—204no—
314C22—204no—
315C22—202no—
316C22C22×D2011yescharacteristic
317C22C2×D4×D511yescharacteristic
318C22C2×D4×D512yes—
319C22C2×D4×D511yescharacteristic
320C22#414912yes—
321C4—24no—
322C4—42no—
323C4—102no—
324C4—104no—
325C4#402814yescomplemented
326C2—44no—
327C2—104no—
328C2—204no—
329C2—402no—
330C2C2×#402811yescharacteristic
331C2D4×D2012yes—
332C2#6594312yes—
333C2#6600811yescharacteristic
334C2D5×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
hypercenter#390D5
hypocenterC5#2433
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×D4×D5C2semidirect
2. C2×#4086C2semidirect
3. C2×#4041C2semidirect
4. C2×#4028C2semidirect
5. C2×#4020C2semidirect
6. C5×#390C2semidirect
7. #65277C2semidirect
8. #65249C2semidirect
9. C4×#685C2semidirect
13. C22×C22⋊D5C22semidirect
19. C22×D20C22semidirect
20. C22×D20C22semidirect
22. C2×C5×C4⋊C4C22semidirect
23. C2×C5×C22⋊C4C22semidirect
24. C2×C42×C5C22semidirect
27. C2×#685C22semidirect
28. C2×#685C22semidirect
29. C2×C4×Dic5C22semidirect
33. #3963C22semidirect
46. C2×C5×D4D4semidirect
69. C2×D20D4semidirect
72. C2×C4×D5D4semidirect
78. #685D4semidirect
87. #390D5semidirect
123. C2×C4×C5C2×D4semidirect
131. C2×Dic5C2×D4semidirect
153. C2×C4⋊C4C2×D5semidirect
157. C2×C22⋊C4C2×D5semidirect
159. C2×C42C2×D5semidirect
186. C2×D5#127semidirect
189. C4×C5#120semidirect
221. C2×D4D20semidirect
293. C2×C4D4×D5semidirect
294. C5#2433semidirect
325. C4#4028semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D52
C2×#40862
C2×#40412
C2×#40282
C2×#40202
C5×#3902
#652772
#652492
C4×#6852
#24335