Finite Group Databasebeta

← #10611725

Subgroups — #10611725

Order 640.

532 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 123 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×#4086C211yesmaximal, characteristic, complemented
2#65930C211yesmaximal, characteristic, complemented
3C5×#447C211yesmaximal, characteristic, complemented
4#65506C211yesmaximal, characteristic, complemented
5#65440C212yesmaximal, complemented
6#65439C212yesmaximal, complemented
7#65325C211yesmaximal, characteristic, complemented
8#65075C212yesmaximal, complemented
9#65049C212yesmaximal, complemented
10#65021C211yesmaximal, characteristic, complemented
11#65007C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5—21no—
14C22×C22⋊D5—21no—
15C2×#711—21no—
16C22×D20C2211yescharacteristic
17C2×C5×D8C2212yescomplemented
18C5×#121C2211yescharacteristic
19#4086—21no—
20#4086—21no—
21#4086C2212yes—
22#4086C2212yes—
23#4086C2211yescharacteristic, complemented
24#4085—21no—
25D4×Dic5C2211yescharacteristic, complemented
26C2×#686C2212yescomplemented
27C2×#686C2212yescomplemented
28#4076C2211yescharacteristic
29C2×C4×Dic5C2211yescharacteristic
30C2×D40C2212yescomplemented
31#4039C2211yescharacteristic, complemented
32#4034C2211yescharacteristic, complemented
33#4031—21no—
34#4027—21no—
35C5×#102C2212yescomplemented
36C5×C22⋊C8C2211yescharacteristic
37#3964C2212yescomplemented
38C22⋊#168C2211yescharacteristic
39#3953C2212yescomplemented
40#3946C2212yescomplemented
41C8×Dic5C2212yescomplemented
42#3941C2212yescomplemented
43#3205—51nomaximal, Sylow
44C24×C5—21no—
45C23×D5—21no—
46C2×C5×D4—21no—
47C2×C5×D4—21no—
48C2×C5×D4—42no—
49C2×C5×D4C2311yescharacteristic
50C2×C5×D4C2312yes—
51C22×C4×C5C2311yescharacteristic
52C2×C22⋊D5—21no—
53C2×C22⋊D5—21no—
54C2×C22⋊D5—22no—
55C2×C22⋊D5—22no—
56C2×C22⋊D5—22no—
57C2×C22⋊D5—22no—
58C2×C22⋊D5—21no—
59C2×C22⋊D5—44no—
60C2×C22⋊D5—44no—
61C22×Dic5—21no—
62C22×Dic5D411yescharacteristic
63C22×Dic5D411yescharacteristic
64#711—44no—
65C2×D20—21no—
66C2×D20—42no—
67C2×D20C2312yes—
68C2×D20C2311yescharacteristic
69C2×C4×D5—21no—
70C2×Dic10—21no—
71C5×D8—42no—
72C2×C5×C8C2312yes—
73C5×C4⋊C4C2311yescharacteristic
74C5×C22⋊C4—21no—
75C22⋊Dic5—21no—
76#686—42no—
77#686—42no—
78#685—21no—
79#685—21no—
80C4⋊Dic5C2311yescharacteristic
81#683—21no—
82C4×Dic5D411yescharacteristic
83C4×Dic5D411yescharacteristic
84C4×Dic5C2311yescharacteristic
85C4×Dic5C2311yescharacteristic
86C2×#168C2312yes—
87D40—42no—
88#532—51no—
89C2×#127—51no—
90#493—52no—
91#486—52no—
92#459—52no—
93#459—52no—
94#447—51no—
95#447—51no—
96#447—51no—
97#447D511yescharacteristic, complemented
98#431—51no—
99C23×C5—21no—
100C23×C5—22no—
101C23×C5—42no—
102C23×C5—42no—
103C23×C5—42no—
104C23×C5C2×D411yescharacteristic
105C22×D5—21no—
106C22×D5—22no—
107C22×D5—42no—
108C22×D5—42no—
109C22×D5—42no—
110C5×D4—22no—
111C5×D4—22no—
112C5×D4—22no—
113C5×D4—42no—
114C5×D4—42no—
115C2×C4×C5—22no—
116C2×C4×C5—21no—
117C2×C4×C5C2×D411yescharacteristic
118C2×C4×C5C2411yescharacteristic
119C22⋊D5—44no—
120C22⋊D5—42no—
121C22⋊D5—42no—
122C22⋊D5—44no—
123C22⋊D5—44no—
124C22⋊D5—42no—
125C22⋊D5—44no—
126C2×Dic5—22no—
127C2×Dic5—21no—
128C2×Dic5—42no—
129C2×Dic5D814yescomplemented
130C2×Dic5C2×D411yescharacteristic
131C2×Dic5C2×D411yescharacteristic
132C2×Dic5C2×D411yescharacteristic
133C2×Dic5C2×D411yescharacteristic
134D20—22no—
135D20—22no—
136D20—22no—
137D20—42no—
138C4×D5—42no—
139Dic10—42no—
140C5×C8—22no—
141#168—22no—
142C2×#41—101no—
143C22×D4—51no—
144C22×D4—101no—
145C22×D4—101no—
146C22×D4C2×D511yescharacteristic
147C2×D8—52no—
148C2×D8—52no—
149C2×D8—52no—
150C2×D8C2×D512yescomplemented
151C4⋊Q8—51no—
152#127—52no—
153#127—52no—
154#127—51no—
155#127—101no—
156#127—101no—
157#124—101no—
158#121—51no—
159#121—101no—
160#121C2×D511yescharacteristic
161C4×D4—51no—
162C4×D4—51no—
163C4×D4—101no—
164C2×C42—51no—
165C4⋊C8—52no—
166#102—52no—
167#102—52no—
168#102—52no—
169#102C2×D512yescomplemented
170C22⋊C8—51no—
171C22⋊C8C2×D511yescharacteristic
172C4×C8—52no—
173C22×C5—21no—
174C22×C5—21no—
175C22×C5—42no—
176C22×C5—41no—
177C22×C5—41no—
178C22×C5—42no—
179C22×C5—42no—
180C22×C5—41no—
181C22×C5—84no—
182C22×C5C2×D812yes—
183C22×C5C22×D411yescharacteristic
184C2×D5—42no—
185C2×D5—41no—
186C2×D5—41no—
187C2×D5—42no—
188C2×D5—42no—
189C2×D5—41no—
190C2×D5—84no—
191C4×C5—21no—
192C4×C5—41no—
193C4×C5H(5,𝔽2)12yes—
194Dic5—21no—
195Dic5—41no—
196Dic5C2×D812yes—
197Dic5C2×D812yes—
198Dic5AGL(1,ℤ/8)12yes—
199C24—21no—
200C24—101no—
201#41—204no—
202C2×Q8—101no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—52no—
207C2×D4—51no—
208C2×D4—101no—
209C2×D4—101no—
210C2×D4—102no—
211C2×D4—102no—
212C2×D4—102no—
213C2×D4—102no—
214C2×D4—101no—
215C2×D4—101no—
216C2×D4—202no—
217C2×D4—204no—
218C2×D4—204no—
219C2×D4C22×D512yes—
220C2×D4C22×D511yescharacteristic
221C22×C4—51no—
222C22×C4—51no—
223C22×C4—101no—
224C22×C4—101no—
225C22×C4C22×D511yescharacteristic
226D8—42no—
227D8—202no—
228D8—202no—
229D8—202no—
230C2×C8—52no—
231C2×C8C22×D512yes—
232C4⋊C4—51no—
233C4⋊C4—101no—
234C4⋊C4C22×D511yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—101no—
237C22⋊C4—101no—
238C22⋊C4—101no—
239C42—51no—
240C42—51no—
241C42—51no—
242C42—51no—
243C2×C5—22no—
244C2×C5—81no—
245C2×C5—82no—
246C2×C5#53511yescharacteristic
247C2×C5C22×D811yescharacteristic
248C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
249D5—81no—
250D5—82no—
251C23—21no—
252C23—22no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—101no—
257C23—102no—
258C23—202no—
259C23—202no—
260C23—202no—
261C23D4×D511yescharacteristic
262Q8—202no—
263D4—22no—
264D4—22no—
265D4—22no—
266D4—42no—
267D4—42no—
268D4—102no—
269D4—102no—
270D4—102no—
271D4—202no—
272D4—204no—
273D4—204no—
274D4—204no—
275D4—202no—
276D4—202no—
277D4—204no—
278D4—202no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—51no—
282C2×C4—51no—
283C2×C4—51no—
284C2×C4—54no—
285C2×C4—51no—
286C2×C4—102no—
287C2×C4—101no—
288C2×C4—202no—
289C2×C4—202no—
290C2×C4D4×D511yescharacteristic
291C2×C4C23×D511yescharacteristic
292C8—22no—
293C8—102no—
294C5#320511yescharacteristic, complemented, Sylow
295C22—21no—
296C22—21no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—41no—
303C22—84no—
304C22—201no—
305C22—202no—
306C22—202no—
307C22—201no—
308C22—202no—
309C22—201no—
310C22—404no—
311C22D5×D812yes—
312C22C2×D4×D511yescharacteristic
313C4—21no—
314C4—41no—
315C4—52no—
316C4—52no—
317C4—52no—
318C4—101no—
319C4—201no—
320C4#414412yes—
321C2—22no—
322C2—82no—
323C2—81no—
324C2—402no—
325C2—401no—
326C2#6592611yescharacteristic
327C2C2×D5×D811yescharacteristic
328C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#447C2
socleC22×C5C22×D4
hypercenter#447D5
hypocenterC5#3205
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#4086C2semidirect
2. #65930C2semidirect
3. C5×#447C2semidirect
4. #65506C2semidirect
5. #65440C2semidirect
6. #65439C2semidirect
7. #65325C2semidirect
8. #65075C2semidirect
9. #65049C2semidirect
10. #65021C2semidirect
11. #65007C2semidirect
17. C2×C5×D8C22semidirect
23. #4086C22semidirect
25. D4×Dic5C22semidirect
26. C2×#686C22semidirect
27. C2×#686C22semidirect
30. C2×D40C22semidirect
31. #4039C22semidirect
32. #4034C22semidirect
35. C5×#102C22semidirect
37. #3964C22semidirect
39. #3953C22semidirect
40. #3946C22semidirect
41. C8×Dic5C22semidirect
42. #3941C22semidirect
97. #447D5semidirect
129. C2×Dic5D8semidirect
150. C2×D8C2×D5semidirect
169. #102C2×D5semidirect
294. C5#3205semidirect

Maximal subgroups

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

subgroupindex
C2×#40862
#659302
C5×#4472
#655062
#654402
#654392
#653252
#650752
#650492
#650212
#650072
#32055