Finite Group Databasebeta

← #10611925

Subgroups — #10611925

Order 640.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66151C211yesmaximal, characteristic
2#65976C211yesmaximal, characteristic, complemented
3C5×#451C211yesmaximal, characteristic, complemented
4C2×#3967C211yesmaximal, characteristic, complemented
5#65477C211yesmaximal, characteristic, complemented
6#65476C211yesmaximal, characteristic, complemented
7#65474C211yesmaximal, characteristic, complemented
8Q16×Dic5C211yesmaximal, characteristic, complemented
9#65337C211yesmaximal, characteristic, complemented
10#65103C211yesmaximal, characteristic, complemented
11#65093C211yesmaximal, characteristic, complemented
12#65089C211yesmaximal, characteristic, complemented
13#65085C211yesmaximal, characteristic, complemented
14#65025C211yesmaximal, characteristic, complemented
15#65014C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C2×#713C2211yescharacteristic
18C2×C5×Q16C2211yescharacteristic
19C2×C5×Q16C2211yescharacteristic
20C5×C22⋊Q8C2211yescharacteristic
21#4093—21no—
22#4093C2211yescharacteristic
23#4092—21no—
24#4092C2211yescharacteristic
25#4092C2211yescharacteristic
26Q8×Dic5C2211yescharacteristic
27Q8×Dic5C2211yescharacteristic
28C2×#688C2211yescharacteristic
29#4076—21no—
30#4076C2211yescharacteristic
31C4×C22⋊D5—21no—
32C4×C22⋊D5C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C22×#168C2211yescharacteristic
35C2×C8×D5C2211yescharacteristic
36#4044—21no—
37#4038C2211yescharacteristic
38#4036—21no—
39#4036C2211yescharacteristic
40#4032—21no—
41#4023—21no—
42C5×#103C2211yescharacteristic
43C5×#103C2211yescharacteristic
44C5×C22⋊C8C2211yescharacteristic
45#3967C2211yescharacteristic
46#3967C2211yescharacteristic
47#3967C2211yescharacteristic
48#3967C2211yescharacteristic
49#3967C2211yescharacteristic
50#3953C2211yescharacteristic
51#3953C2211yescharacteristic
52#3952C2211yescharacteristic
53#3950C2211yescharacteristic
54#3950C2211yescharacteristic
55#3946C2211yescharacteristic
56C8×Dic5C2211yescharacteristic
57#3941C2211yescharacteristic
58#3940C2211yescharacteristic
59#3300—51nomaximal, Sylow
60C2×C5×Q8—21no—
61C2×C5×Q8—22no—
62C2×C5×Q8—41no—
63C2×C5×Q8C2311yescharacteristic
64C2×C5×Q8C2311yescharacteristic
65C2×C5×Q8C2311yescharacteristic
66C22×C4×C5—21no—
67C22×C4×C5C2311yescharacteristic
68C2×C22⋊D5—21no—
69C2×C22⋊D5D411yescharacteristic
70C22×Dic5—21no—
71#713—22no—
72#713—41no—
73C2×D20—21no—
74C2×D20C2311yescharacteristic
75C2×C4×D5—21no—
76C2×C4×D5C2311yescharacteristic
77C5×Q16—22no—
78C5×Q16—41no—
79C2×C5×C8C2311yescharacteristic
80C2×C5×C8C2311yescharacteristic
81C5×C4⋊C4—21no—
82C5×C4⋊C4C2311yescharacteristic
83C5×C22⋊C4—21no—
84C22⋊Dic5D411yescharacteristic
85#688—41no—
86#685—21no—
87#685—21no—
88#685—21no—
89#685—21no—
90#685—21no—
91#685—21no—
92#685D411yescharacteristic
93C4⋊Dic5—21no—
94C4⋊Dic5—21no—
95C4⋊Dic5—21no—
96C4⋊Dic5—21no—
97C4⋊Dic5C2311yescharacteristic
98C4⋊Dic5C2311yescharacteristic
99C4⋊Dic5C2311yescharacteristic
100#683—21no—
101#683—21no—
102#683—21no—
103#683D411yescharacteristic
104C4×Dic5—21no—
105C4×Dic5—21no—
106C4×Dic5—21no—
107C4×Dic5C2311yescharacteristic
108C2×#168—41no—
109C2×#168C2311yescharacteristic
110C2×#168C2311yescharacteristic
111C8×D5—41no—
112#552—51no—
113#548—51no—
114#488—51no—
115#485—51no—
116#480—51no—
117#479—51no—
118#467—51no—
119#465—51no—
120#464—51no—
121#452—51no—
122#451D511yescharacteristic, complemented
123C4×Q16—51no—
124C8×D4—51no—
125#416—51no—
126C2×#103—51no—
127C23×C5C2×D411yescharacteristic
128C22×D5—21no—
129C22×D5C2×D411yescharacteristic
130C5×Q8—21no—
131C5×Q8—21no—
132C5×Q8—21no—
133C5×Q8—21no—
134C5×Q8—21no—
135C5×Q8—42no—
136C5×Q8#4112yes—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—42no—
144C2×C4×C5C2×D411yescharacteristic
145C2×C4×C5C2411yescharacteristic
146C22⋊D5—22no—
147C22⋊D5—42no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—42no—
152C2×Dic5C2×D411yescharacteristic
153C2×Dic5C2×D411yescharacteristic
154C2×Dic5C2×D411yescharacteristic
155D20—21no—
156D20—21no—
157D20—42no—
158C4×D5—21no—
159C4×D5—21no—
160C4×D5—42no—
161C5×C8—21no—
162C5×C8—21no—
163#168—21no—
164#168—21no—
165C2×#41—51no—
166C22×Q8C2×D511yescharacteristic
167C2×Q16C2×D511yescharacteristic
168C2×Q16C2×D511yescharacteristic
169C2×SD16—51no—
170C22×C8—51no—
171C22×C8—51no—
172#126—101no—
173#125—51no—
174#125—101no—
175#124—51no—
176#124—101no—
177#123—101no—
178C22⋊Q8—51no—
179C22⋊Q8—51no—
180C22⋊Q8—101no—
181C22⋊Q8C2×D511yescharacteristic
182#121—51no—
183#121—101no—
184C4×Q8—51no—
185C4×Q8—51no—
186C4×D4—51no—
187C4×D4—101no—
188#117—51no—
189#117—101no—
190C2×C4⋊C4—51no—
191#107—51no—
192#107—51no—
193#106—51no—
194C4⋊C8—51no—
195#103—51no—
196#103—51no—
197#103—51no—
198#103—51no—
199#103—51no—
200#103C2×D511yescharacteristic
201#103C2×D511yescharacteristic
202#102—51no—
203#102—51no—
204#102—51no—
205C22⋊C8—51no—
206C22⋊C8C2×D511yescharacteristic
207C4×C8—51no—
208C22×C5—21no—
209C22×C5—21no—
210C22×C5#13512yes—
211C22×C5C22×D411yescharacteristic
212C2×D5—21no—
213C2×D5—21no—
214C2×D5—21no—
215C2×D5—41no—
216C2×D5—41no—
217C2×D5—41no—
218C4×C5—22no—
219C4×C5—21no—
220C4×C5—41no—
221C4×C5—41no—
222C4×C5—41no—
223C4×C5C2×#4111yescharacteristic
224C4×C5H(5,𝔽2)11yescharacteristic
225Dic5—21no—
226Dic5—21no—
227Dic5—21no—
228Dic5—41no—
229Dic5—41no—
230Dic5—41no—
231#41—102no—
232#41—201no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8C22×D511yescharacteristic
237C2×Q8C22×D511yescharacteristic
238C2×Q8C22×D511yescharacteristic
239C2×D4—51no—
240C2×D4—51no—
241C2×D4—101no—
242C2×D4—101no—
243C22×C4—21no—
244C22×C4—51no—
245C22×C4—101no—
246C22×C4—101no—
247C22×C4C22×D511yescharacteristic
248Q16—22no—
249Q16—41no—
250SD16—201no—
251C2×C8—51no—
252C2×C8—51no—
253C2×C8—201no—
254C2×C8—201no—
255C2×C8C22×D511yescharacteristic
256C2×C8C22×D511yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—51no—
259C4⋊C4—51no—
260C4⋊C4—51no—
261C4⋊C4—51no—
262C4⋊C4—101no—
263C4⋊C4—101no—
264C4⋊C4—101no—
265C4⋊C4—101no—
266C4⋊C4—101no—
267C4⋊C4—101no—
268C4⋊C4—101no—
269C4⋊C4C22×D511yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—51no—
272C22⋊C4—51no—
273C22⋊C4—101no—
274C22⋊C4—101no—
275C22⋊C4—101no—
276C22⋊C4—101no—
277C22⋊C4—101no—
278C22⋊C4—101no—
279C42—51no—
280C42—101no—
281C42—101no—
282C42—101no—
283C2×C5—22no—
284C2×C5#54611yescharacteristic
285C2×C5C2×#13511yescharacteristic
286C2×C5#57811yescharacteristic
287D5—41no—
288D5—81no—
289C23—51no—
290C23—101no—
291C23D4×D511yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—42no—
298Q8#71112yes—
299D4—102no—
300D4—101no—
301D4—101no—
302D4—202no—
303D4—202no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—51no—
312C2×C4—51no—
313C2×C4—51no—
314C2×C4—101no—
315C2×C4—101no—
316C2×C4—101no—
317C2×C4—101no—
318C2×C4—101no—
319C2×C4—202no—
320C2×C4—202no—
321C2×C4D4×D511yescharacteristic
322C2×C4C23×D511yescharacteristic
323C8—21no—
324C8—21no—
325C8—101no—
326C8—101no—
327C5#330011yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—101no—
331C22—101no—
332C22—101no—
333C22—201no—
334C22—201no—
335C22—201no—
336C22#406512yes—
337C22C2×D4×D511yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—101no—
344C4—101no—
345C4—101no—
346C4—201no—
347C4—201no—
348C4—201no—
349C4C2×#71111yescharacteristic
350C4#414411yescharacteristic
351C2—22no—
352C2—201no—
353C2—401no—
354C2#6592511yescharacteristic
355C2C2×#406511yescharacteristic
356C2#6611311yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#451C2
socleC22×C5C22×D4
prime radical#66151C2
hypercenter#451D5
hypocenterC5#3300
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #65976C2semidirect
3. C5×#451C2semidirect
4. C2×#3967C2semidirect
5. #65477C2semidirect
6. #65476C2semidirect
7. #65474C2semidirect
8. Q16×Dic5C2semidirect
9. #65337C2semidirect
10. #65103C2semidirect
11. #65093C2semidirect
12. #65089C2semidirect
13. #65085C2semidirect
14. #65025C2semidirect
15. #65014C2semidirect
122. #451D5semidirect
327. C5#3300semidirect

Maximal subgroups

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

subgroupindex
#661512
#659762
C5×#4512
C2×#39672
#654772
#654762
#654742
Q16×Dic52
#653372
#651032
#650932
#650892
#650852
#650252
#650142
#33005