Finite Group Databasebeta

← #10612521

Subgroups — #10612521

Order 640.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65956C211yesmaximal, characteristic, complemented
2#65930C211yesmaximal, characteristic
3C5×#468C211yesmaximal, characteristic, complemented
4#65464C211yesmaximal, characteristic, complemented
5#65453C211yesmaximal, characteristic, complemented
6#65423C211yesmaximal, characteristic, complemented
7M16×Dic5C211yesmaximal, characteristic, complemented
8#65338C211yesmaximal, characteristic, complemented
9#65331C211yesmaximal, characteristic, complemented
10#65178C211yesmaximal, characteristic, complemented
11#65163C211yesmaximal, characteristic, complemented
12#65106C211yesmaximal, characteristic, complemented
13#65081C211yesmaximal, characteristic, complemented
14#65049C211yesmaximal, characteristic, complemented
15#65046C211yesmaximal, characteristic, complemented
16C2×#711—21no—
17C2×C5×SD16C2211yescharacteristic
18C2×C5×M16C2211yescharacteristic
19C5×C22⋊Q8C2211yescharacteristic
20C5×#121C2211yescharacteristic
21Q8×Dic5C2211yescharacteristic
22#4090C2211yescharacteristic
23C2×#688C2211yescharacteristic
24#4086C2211yescharacteristic
25#4085—21no—
26D4×Dic5C2211yescharacteristic
27C2×#687C2211yescharacteristic
28#4076C2211yescharacteristic
29#4070C2211yescharacteristic
30C2×C4×Dic5C2211yescharacteristic
31C2×#681C2211yescharacteristic
32C2×#677C2211yescharacteristic
33#4039C2211yescharacteristic
34#4035—21no—
35#4034C2211yescharacteristic
36#4034C2211yescharacteristic
37#4033—21no—
38#4033C2211yescharacteristic
39#4031—21no—
40#4027—21no—
41#4024—21no—
42#4023—21no—
43C5×#107C2211yescharacteristic
44C5×#103C2211yescharacteristic
45C5×#102C2211yescharacteristic
46#3967C2211yescharacteristic
47#3964C2211yescharacteristic
48#3953C2211yescharacteristic
49#3950C2211yescharacteristic
50#3948C2211yescharacteristic
51#3947C2211yescharacteristic
52#3947C2211yescharacteristic
53C8×Dic5C2211yescharacteristic
54C8×Dic5C2211yescharacteristic
55#3942C2211yescharacteristic
56#3941C2211yescharacteristic
57#3939C2211yescharacteristic
58#3939C2211yescharacteristic
59#3182—51nomaximal, Sylow
60C2×C5×Q8C2311yescharacteristic
61C2×C5×D4—21no—
62C2×C5×D4C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C2×C22⋊D5—21no—
65C2×C22⋊D5—21no—
66C2×C22⋊D5—21no—
67C22×Dic5—21no—
68C22×Dic5D411yescharacteristic
69C22×Dic5D411yescharacteristic
70#711—44no—
71C2×D20C2311yescharacteristic
72C2×C4×D5—21no—
73C2×Dic10—21no—
74C2×Dic10—21no—
75C2×Dic10C2311yescharacteristic
76C5×SD16—22no—
77C5×M16—22no—
78C2×C5×C8C2311yescharacteristic
79C2×C5×C8C2311yescharacteristic
80C5×C4⋊C4—21no—
81C5×C4⋊C4C2311yescharacteristic
82C5×C4⋊C4C2311yescharacteristic
83C5×C22⋊C4—21no—
84C5×C22⋊C4—21no—
85C22⋊Dic5—21no—
86C22⋊Dic5—21no—
87#688—22no—
88#687—22no—
89#685—21no—
90#685—21no—
91C4⋊Dic5—21no—
92C4⋊Dic5C2311yescharacteristic
93C4⋊Dic5C2311yescharacteristic
94#683—21no—
95#683—21no—
96#683—21no—
97#683—21no—
98#683—21no—
99#683—21no—
100C4×Dic5—21no—
101C4×Dic5—21no—
102C4×Dic5D411yescharacteristic
103C4×Dic5D411yescharacteristic
104C4×Dic5C2311yescharacteristic
105C4×Dic5C2311yescharacteristic
106#681—22no—
107C2×#168C2311yescharacteristic
108C2×#168C2311yescharacteristic
109#677—22no—
110#533—51no—
111#532—51no—
112#500—51no—
113#492—51no—
114#487—51no—
115#486—51no—
116#468—51no—
117#468—51no—
118#468—51no—
119#468D511yescharacteristic, complemented
120#440—51no—
121#440—51no—
122#440—51no—
123#440—51no—
124C4×M16—51no—
125C23×C5—21no—
126C23×C5C2×D411yescharacteristic
127C22×D5—21no—
128C5×Q8—21no—
129C5×Q8—21no—
130C5×D4—21no—
131C5×D4—21no—
132C5×D4—42no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5C2×D411yescharacteristic
139C2×C4×C5C2411yescharacteristic
140C22⋊D5—42no—
141C22⋊D5—42no—
142C22⋊D5—42no—
143C2×Dic5—22no—
144C2×Dic5—22no—
145C2×Dic5—21no—
146C2×Dic5—21no—
147C2×Dic5—21no—
148C2×Dic5—42no—
149C2×Dic5C2×D411yescharacteristic
150C2×Dic5C2×D411yescharacteristic
151C2×Dic5C2×D411yescharacteristic
152C2×Dic5C2×D411yescharacteristic
153D20—21no—
154D20—21no—
155C4×D5—42no—
156Dic10—21no—
157Dic10—21no—
158Dic10—42no—
159Dic10—42no—
160C5×C8—21no—
161C5×C8#4112yescomplemented
162#168—21no—
163#168#4112yescomplemented
164C2×#41—101no—
165C2×SD16—51no—
166C2×SD16—51no—
167C2×SD16—51no—
168C2×SD16C2×D511yescharacteristic
169C2×M16—51no—
170C2×M16C2×D511yescharacteristic
171C4⋊Q8—51no—
172C4⋊Q8—51no—
173C4⋊Q8—51no—
174#127—51no—
175#125—101no—
176#124—101no—
177C22⋊Q8—51no—
178C22⋊Q8—101no—
179C22⋊Q8C2×D511yescharacteristic
180#121—51no—
181#121—101no—
182#121C2×D511yescharacteristic
183C4×Q8—51no—
184C4×Q8—51no—
185C4×Q8—101no—
186C4×D4—51no—
187C4×D4—51no—
188C4×D4—101no—
189#117—101no—
190C2×C42—51no—
191#107—51no—
192#107—51no—
193#107—51no—
194#107C2×D511yescharacteristic
195#103—51no—
196#103—51no—
197#103—51no—
198#103C2×D511yescharacteristic
199#102—51no—
200#102—51no—
201#102—51no—
202#102C2×D511yescharacteristic
203#97—51no—
204#97—51no—
205C4×C8—51no—
206C4×C8—51no—
207C22×C5—21no—
208C22×C5—21no—
209C22×C5—21no—
210C22×C5—41no—
211C22×C5—41no—
212C22×C5—41no—
213C22×C5C22×D411yescharacteristic
214C2×D5—41no—
215C2×D5—41no—
216C2×D5—41no—
217C4×C5—21no—
218C4×C5—41no—
219C4×C5—41no—
220C4×C5—41no—
221C4×C5C2×#4111yescharacteristic
222C4×C5C2×#4111yescharacteristic
223Dic5—21no—
224Dic5—21no—
225Dic5—41no—
226Dic5—41no—
227Dic5—41no—
228Dic5AGL(1,ℤ/8)12yes—
229Dic5#13712yes—
230#41—204no—
231C2×Q8—51no—
232C2×Q8—101no—
233C2×Q8—101no—
234C2×Q8C22×D511yescharacteristic
235C2×D4—21no—
236C2×D4—51no—
237C2×D4—101no—
238C2×D4—101no—
239C2×D4—101no—
240C2×D4C22×D511yescharacteristic
241C22×C4—51no—
242C22×C4—51no—
243C22×C4—101no—
244C22×C4—101no—
245C22×C4C22×D511yescharacteristic
246SD16—22no—
247SD16—102no—
248SD16—102no—
249SD16—102no—
250M16—22no—
251M16—102no—
252C2×C8—51no—
253C2×C8—51no—
254C2×C8C22×D511yescharacteristic
255C2×C8C22×D511yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—51no—
258C4⋊C4—51no—
259C4⋊C4—101no—
260C4⋊C4—101no—
261C4⋊C4—101no—
262C4⋊C4—101no—
263C4⋊C4—101no—
264C4⋊C4—101no—
265C4⋊C4—101no—
266C4⋊C4C22×D511yescharacteristic
267C4⋊C4C22×D511yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—101no—
271C22⋊C4—101no—
272C22⋊C4—101no—
273C22⋊C4—101no—
274C42—51no—
275C42—51no—
276C42—51no—
277C42—51no—
278C42—101no—
279C42—101no—
280C2×C5—41no—
281C2×C5—81no—
282C2×C5#53211yescharacteristic
283C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
284C2×C5C2×#13711yescharacteristic
285D5—81no—
286C23—21no—
287C23—101no—
288C23D4×D511yescharacteristic
289Q8—21no—
290Q8—21no—
291Q8—101no—
292Q8—101no—
293Q8—202no—
294Q8—202no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—101no—
299D4—101no—
300D4—202no—
301D4—202no—
302D4—202no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—51no—
309C2×C4—51no—
310C2×C4—51no—
311C2×C4—51no—
312C2×C4—102no—
313C2×C4—102no—
314C2×C4—101no—
315C2×C4—101no—
316C2×C4—101no—
317C2×C4—202no—
318C2×C4—202no—
319C2×C4D4×D511yescharacteristic
320C2×C4C23×D511yescharacteristic
321C8—21no—
322C8—52no—
323C8—101no—
324C8#71112yescomplemented
325C5#318211yescharacteristic, complemented, Sylow
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—201no—
333C22—201no—
334C22—201no—
335C22C2×D4×D511yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—52no—
341C4—52no—
342C4—101no—
343C4—101no—
344C4—201no—
345C4—201no—
346C4—201no—
347C4C2×#71111yescharacteristic
348C4D5×#4111yescharacteristic
349C2—41no—
350C2—81no—
351C2—401no—
352C2#6593011yescharacteristic
353C2D5×AGL(1,ℤ/8)11yescharacteristic
354C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#468C2
socleC22×C5C22×D4
prime radical#65930C2
hypercenter#468D5
hypocenterC5#3182
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #65956C2semidirect
3. C5×#468C2semidirect
4. #65464C2semidirect
5. #65453C2semidirect
6. #65423C2semidirect
7. M16×Dic5C2semidirect
8. #65338C2semidirect
9. #65331C2semidirect
10. #65178C2semidirect
11. #65163C2semidirect
12. #65106C2semidirect
13. #65081C2semidirect
14. #65049C2semidirect
15. #65046C2semidirect
119. #468D5semidirect
161. C5×C8#41semidirect
163. #168#41semidirect
324. C8#711semidirect
325. C5#3182semidirect

Maximal subgroups

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

subgroupindex
#659562
#659302
C5×#4682
#654642
#654532
#654232
M16×Dic52
#653382
#653312
#651782
#651632
#651062
#650812
#650492
#650462
#31825