Finite Group Databasebeta

← #10612628

Subgroups — #10612628

Order 640.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65977C211yesmaximal, characteristic
2#65959C211yesmaximal, characteristic, complemented
3C5×#470C211yesmaximal, characteristic, complemented
4#65474C211yesmaximal, characteristic, complemented
5#65471C211yesmaximal, characteristic, complemented
6#65423C211yesmaximal, characteristic
7#65408C211yesmaximal, characteristic, complemented
8#65340C211yesmaximal, characteristic, complemented
9#65337C211yesmaximal, characteristic
10#65160C211yesmaximal, characteristic, complemented
11#65143C211yesmaximal, characteristic, complemented
12#65107C211yesmaximal, characteristic, complemented
13#65087C211yesmaximal, characteristic, complemented
14#65085C211yesmaximal, characteristic, complemented
15#65083C211yesmaximal, characteristic, complemented
16C2×C5×Q16C2211yescharacteristic
17C2×C5×M16C2211yescharacteristic
18C5×C22⋊Q8C2211yescharacteristic
19C5×C22⋊Q8C2211yescharacteristic
20#4093C2211yescharacteristic
21Q8×Dic5C2211yescharacteristic
22Q8×Dic5C2211yescharacteristic
23#4090C2211yescharacteristic
24C2×#689C2211yescharacteristic
25C2×#688C2211yescharacteristic
26#4076C2211yescharacteristic
27#4072C2211yescharacteristic
28#4070C2211yescharacteristic
29C22×#168C2211yescharacteristic
30C2×#677C2211yescharacteristic
31#4044—21no—
32#4043—21no—
33#4039C2211yescharacteristic
34#4036—21no—
35#4036C2211yescharacteristic
36#4035—21no—
37#4034C2211yescharacteristic
38#4033C2211yescharacteristic
39#4031—21no—
40#4029—21no—
41#4025—21no—
42#4025—21no—
43C5×#106C2211yescharacteristic
44C5×#103C2211yescharacteristic
45C5×#103C2211yescharacteristic
46#3967C2211yescharacteristic
47#3967C2211yescharacteristic
48#3953C2211yescharacteristic
49#3950C2211yescharacteristic
50#3948C2211yescharacteristic
51#3946C2211yescharacteristic
52#3946C2211yescharacteristic
53#3946C2211yescharacteristic
54#3946C2211yescharacteristic
55#3942C2211yescharacteristic
56#3941C2211yescharacteristic
57#3940C2211yescharacteristic
58#3939C2211yescharacteristic
59#3258—51nomaximal, Sylow
60C2×C5×Q8C2311yescharacteristic
61C2×C5×Q8C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C2×C22⋊D5—21no—
64C2×D20C2311yescharacteristic
65C2×C4×D5—21no—
66C2×Dic10—21no—
67C2×Dic10C2311yescharacteristic
68C5×Q16—41no—
69C5×M16—41no—
70C2×C5×C8C2311yescharacteristic
71C2×C5×C8C2311yescharacteristic
72C5×C4⋊C4—21no—
73C5×C4⋊C4—21no—
74C5×C4⋊C4C2311yescharacteristic
75C5×C4⋊C4C2311yescharacteristic
76C5×C22⋊C4—21no—
77C5×C22⋊C4—21no—
78C22⋊Dic5—21no—
79C22⋊Dic5D411yescharacteristic
80C22⋊Dic5D411yescharacteristic
81#689—41no—
82#688—41no—
83#685—21no—
84#685—21no—
85#685—21no—
86#685—21no—
87C4⋊Dic5—21no—
88C4⋊Dic5—21no—
89C4⋊Dic5—21no—
90C4⋊Dic5D411yescharacteristic
91C4⋊Dic5D411yescharacteristic
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×Dic5—21no—
103C4×Dic5C2311yescharacteristic
104C4×Dic5C2311yescharacteristic
105C2×#168—41no—
106C2×#168C2311yescharacteristic
107C2×#168C2311yescharacteristic
108#677—41no—
109#542—51no—
110#541—51no—
111#479—51no—
112#478—51no—
113#477—51no—
114#475—51no—
115#470D511yescharacteristic, complemented
116#468—51no—
117#467—51no—
118#465—51no—
119#464—51no—
120#464—51no—
121#462—51no—
122#461—51no—
123#424—51no—
124C23×C5C2×D411yescharacteristic
125C22×D5—21no—
126C5×Q8—21no—
127C5×Q8—21no—
128C5×Q8—21no—
129C5×Q8—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5C2×D411yescharacteristic
137C2×C4×C5C2411yescharacteristic
138C22⋊D5—81no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5C2×D411yescharacteristic
143C2×Dic5C2×D411yescharacteristic
144C2×Dic5C2×D411yescharacteristic
145C2×Dic5C2×D411yescharacteristic
146D20—21no—
147D20—21no—
148C4×D5—81no—
149Dic10—21no—
150Dic10—21no—
151Dic10—81no—
152C5×C8—21no—
153C5×C8—21no—
154#168—21no—
155#168—21no—
156C2×Q16—51no—
157C2×Q16C2×D511yescharacteristic
158C2×SD16—51no—
159C2×SD16—51no—
160C2×M16C2×D511yescharacteristic
161C22×C8—51no—
162C4⋊Q8—51no—
163C4⋊Q8—51no—
164#126—101no—
165#126—101no—
166#126—101no—
167#125—51no—
168#125—101no—
169#125—101no—
170#124—51no—
171#124—101no—
172#123—101no—
173C22⋊Q8—51no—
174C22⋊Q8—101no—
175C22⋊Q8C2×D511yescharacteristic
176C22⋊Q8C2×D511yescharacteristic
177#121—51no—
178C4×Q8—51no—
179C4×Q8—51no—
180C4×Q8—51no—
181C4×D4—51no—
182#117—51no—
183#107—51no—
184#107—51no—
185#106—51no—
186#106C2×D511yescharacteristic
187C4⋊C8—51no—
188C4⋊C8—51no—
189C4⋊C8—51no—
190C4⋊C8—51no—
191#103—51no—
192#103—51no—
193#103—51no—
194#103—51no—
195#103C2×D511yescharacteristic
196#103C2×D511yescharacteristic
197#102—51no—
198#102—51no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—21no—
202C22×C5C22×D411yescharacteristic
203C2×D5—41no—
204C2×D5—41no—
205C2×D5—41no—
206C4×C5—21no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5H(5,𝔽2)11yescharacteristic
212C4×C52−1+411yescharacteristic
213Dic5—21no—
214Dic5—21no—
215Dic5—21no—
216Dic5—21no—
217Dic5—41no—
218Dic5—41no—
219Dic5—41no—
220C2×Q8—51no—
221C2×Q8—101no—
222C2×Q8C22×D511yescharacteristic
223C2×Q8C22×D511yescharacteristic
224C2×D4—51no—
225C2×D4—101no—
226C22×C4—101no—
227C22×C4C22×D511yescharacteristic
228Q16—41no—
229Q16—201no—
230SD16—201no—
231SD16—201no—
232M16—41no—
233C2×C8—51no—
234C2×C8—51no—
235C2×C8—201no—
236C2×C8C22×D511yescharacteristic
237C2×C8C22×D511yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—51no—
241C4⋊C4—51no—
242C4⋊C4—51no—
243C4⋊C4—51no—
244C4⋊C4—101no—
245C4⋊C4—101no—
246C4⋊C4—101no—
247C4⋊C4—101no—
248C4⋊C4—101no—
249C4⋊C4—101no—
250C4⋊C4—101no—
251C4⋊C4—101no—
252C4⋊C4—101no—
253C4⋊C4C22×D511yescharacteristic
254C4⋊C4C22×D511yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—51no—
258C22⋊C4—51no—
259C22⋊C4—101no—
260C22⋊C4—101no—
261C22⋊C4—101no—
262C22⋊C4—101no—
263C22⋊C4—101no—
264C42—51no—
265C42—51no—
266C42—101no—
267C42—101no—
268C42—101no—
269C2×C5—41no—
270C2×C5#53711yescharacteristic
271C2×C5#57711yescharacteristic
272C2×C5#57811yescharacteristic
273D5—81no—
274C23—101no—
275C23D4×D511yescharacteristic
276Q8—21no—
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—101no—
281Q8—101no—
282Q8—401no—
283D4—101no—
284D4—101no—
285D4—401no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—51no—
293C2×C4—51no—
294C2×C4—51no—
295C2×C4—51no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—401no—
300C2×C4D4×D511yescharacteristic
301C2×C4C23×D511yescharacteristic
302C8—21no—
303C8—21no—
304C8—101no—
305C8—101no—
306C5#325811yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22—21no—
310C22—201no—
311C22—201no—
312C22—201no—
313C22C2×D4×D511yescharacteristic
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—101no—
320C4—101no—
321C4—101no—
322C4—101no—
323C4—201no—
324C4—201no—
325C4—201no—
326C4#414411yescharacteristic
327C4#415011yescharacteristic
328C2—41no—
329C2—401no—
330C2#6593111yescharacteristic
331C2#6611211yescharacteristic
332C2#6611311yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#470C2
socleC22×C5C22×D4
prime radical#4076C22
hypercenter#470D5
hypocenterC5#3258
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #65959C2semidirect
3. C5×#470C2semidirect
4. #65474C2semidirect
5. #65471C2semidirect
7. #65408C2semidirect
8. #65340C2semidirect
10. #65160C2semidirect
11. #65143C2semidirect
12. #65107C2semidirect
13. #65087C2semidirect
14. #65085C2semidirect
15. #65083C2semidirect
115. #470D5semidirect
306. C5#3258semidirect

Maximal subgroups

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

subgroupindex
#659772
#659592
C5×#4702
#654742
#654712
#654232
#654082
#653402
#653372
#651602
#651432
#651072
#650872
#650852
#650832
#32585