Finite Group Databasebeta

← #10610968

Subgroups — #10610968

Order 640.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65867C211yesmaximal, characteristic
2#65845C211yesmaximal, characteristic
3C5×#428C211yesmaximal, characteristic, complemented
4#65426C211yesmaximal, characteristic, complemented
5#65417C211yesmaximal, characteristic
6#65285C211yesmaximal, characteristic, complemented
7#65260C211yesmaximal, characteristic, complemented
8#65177C211yesmaximal, characteristic, complemented
9#65170C211yesmaximal, characteristic, complemented
10#65164C211yesmaximal, characteristic, complemented
11#65162C211yesmaximal, characteristic, complemented
12#65157C211yesmaximal, characteristic, complemented
13#65151C211yesmaximal, characteristic, complemented
14#65144C211yesmaximal, characteristic, complemented
15#65143C211yesmaximal, characteristic, complemented
16C2×C5×M16C2211yescharacteristic
17C5×#117C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C4×C22⋊D5—21no—
20C4×C22⋊D5C2211yescharacteristic
21#4070—21no—
22#4070C2211yescharacteristic
23#4070C2211yescharacteristic
24C2×Dic20C2211yescharacteristic
25C2×#676C2211yescharacteristic
26#4043C2211yescharacteristic
27#4043C2211yescharacteristic
28#4042—21no—
29#4038C2211yescharacteristic
30D5×C4⋊C4C2211yescharacteristic
31#4035—21no—
32#4034C2211yescharacteristic
33#4034C2211yescharacteristic
34#4033C2211yescharacteristic
35#4033C2211yescharacteristic
36#4031—21no—
37#4025—21no—
38#4022—21no—
39C4×Dic10—21no—
40C5×#107C2211yescharacteristic
41C5×#107C2211yescharacteristic
42C5×#106C2211yescharacteristic
43C5×#106C2211yescharacteristic
44C22⋊#168C2211yescharacteristic
45#3952C2211yescharacteristic
46#3949C2211yescharacteristic
47#3948C2211yescharacteristic
48#3948C2211yescharacteristic
49#3946C2211yescharacteristic
50C8×Dic5C2211yescharacteristic
51#3942C2211yescharacteristic
52#3942C2211yescharacteristic
53#3942C2211yescharacteristic
54#3942C2211yescharacteristic
55#3940C2211yescharacteristic
56#3940C2211yescharacteristic
57#3940C2211yescharacteristic
58#3939C2211yescharacteristic
59#3356—51nomaximal, Sylow
60C22×C4×C5—21no—
61C22×C4×C5C2311yescharacteristic
62C2×C22⋊D5D411yescharacteristic
63C2×C4×D5—21no—
64C2×C4×D5C2311yescharacteristic
65C2×Dic10—21no—
66C2×Dic10—21no—
67C2×Dic10C2311yescharacteristic
68C2×Dic10C2311yescharacteristic
69C5×M16—22no—
70C2×C5×C8C2311yescharacteristic
71C2×C5×C8C2311yescharacteristic
72C5×C4⋊C4—21no—
73C5×C4⋊C4C2311yescharacteristic
74C5×C4⋊C4C2311yescharacteristic
75C5×C4⋊C4C2311yescharacteristic
76C5×C4⋊C4C2311yescharacteristic
77C5×C22⋊C4—21no—
78C42×C5—21no—
79C22⋊Dic5—21no—
80C22⋊Dic5—21no—
81C22⋊Dic5D411yescharacteristic
82#685—21no—
83#685—21no—
84#685—21no—
85#685D411yescharacteristic
86C4⋊Dic5—21no—
87C4⋊Dic5—21no—
88C4⋊Dic5C2311yescharacteristic
89C4⋊Dic5C2311yescharacteristic
90#683—21no—
91#683—21no—
92#683—21no—
93#683—21no—
94#683—21no—
95#683—21no—
96#683—21no—
97#683D411yescharacteristic
98C4×Dic5—21no—
99C4×Dic5—21no—
100C4×Dic5—21no—
101C4×Dic5C2311yescharacteristic
102C2×#168C2311yescharacteristic
103C2×#168C2311yescharacteristic
104Dic20—22no—
105#676—22no—
106#556—51no—
107#554—51no—
108#498—51no—
109#485—51no—
110#485—51no—
111#483—51no—
112#483—51no—
113#477—51no—
114#475—51no—
115#470—51no—
116#470—51no—
117C4×Q16—51no—
118#436—51no—
119#428—51no—
120#428D511yescharacteristic, complemented
121C23×C5C2×D411yescharacteristic
122C22×D5C2×D411yescharacteristic
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—81no—
130C2×C4×C5C2×D411yescharacteristic
131C2×C4×C5C2411yescharacteristic
132C22⋊D5—22no—
133C2×Dic5—21no—
134C2×Dic5—21no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5C2×D411yescharacteristic
138C2×Dic5C2×D411yescharacteristic
139C2×Dic5C2×D411yescharacteristic
140C4×D5—21no—
141C4×D5—21no—
142C4×D5—81no—
143Dic10—21no—
144Dic10—21no—
145Dic10—21no—
146Dic10—21no—
147Dic10—42no—
148Dic10—81no—
149C5×C8—21no—
150C5×C8#4112yes—
151#168—21no—
152#168—21no—
153C2×Q16—51no—
154C2×M16—51no—
155C2×M16C2×D511yescharacteristic
156C4⋊Q8—51no—
157C4⋊Q8—51no—
158#126—101no—
159#126—101no—
160#125—101no—
161#124—101no—
162C22⋊Q8—51no—
163C22⋊Q8—51no—
164C22⋊Q8—51no—
165C22⋊Q8—51no—
166C22⋊Q8—101no—
167C22⋊Q8—101no—
168C4×Q8—51no—
169C4×Q8—51no—
170C4×Q8—101no—
171C4×D4—51no—
172C4×D4—101no—
173#117—51no—
174#117C2×D511yescharacteristic
175C2×C4⋊C4—51no—
176C2×C4⋊C4C2×D511yescharacteristic
177#107—51no—
178#107C2×D511yescharacteristic
179#107C2×D511yescharacteristic
180#106—51no—
181#106—51no—
182#106—51no—
183#106—51no—
184#106C2×D511yescharacteristic
185#106C2×D511yescharacteristic
186C4⋊C8—51no—
187#103—51no—
188#103—51no—
189#103—51no—
190#103—51no—
191#103—51no—
192#103—51no—
193C22⋊C8—51no—
194C22⋊C8—51no—
195C4×C8—51no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—21no—
199C22×C5C22×D411yescharacteristic
200C2×D5—21no—
201C2×D5—21no—
202C2×D5—21no—
203C4×C5—21no—
204C4×C5—41no—
205C4×C5—41no—
206C4×C5—41no—
207C4×C5—41no—
208C4×C5C2×#4111yescharacteristic
209C4×C52−1+411yescharacteristic
210Dic5—21no—
211Dic5—21no—
212Dic5—41no—
213Dic5—41no—
214Dic5—41no—
215Dic5—41no—
216Dic5#13712yes—
217C2×Q8—51no—
218C2×Q8—51no—
219C2×Q8—101no—
220C2×Q8—101no—
221C2×D4—51no—
222C22×C4—21no—
223C22×C4—51no—
224C22×C4—101no—
225C22×C4C22×D511yescharacteristic
226Q16—102no—
227M16—22no—
228M16—102no—
229C2×C8—51no—
230C2×C8—51no—
231C2×C8C22×D511yescharacteristic
232C2×C8C22×D511yescharacteristic
233C4⋊C4—21no—
234C4⋊C4—51no—
235C4⋊C4—51no—
236C4⋊C4—51no—
237C4⋊C4—101no—
238C4⋊C4—101no—
239C4⋊C4—101no—
240C4⋊C4—101no—
241C4⋊C4—101no—
242C4⋊C4—101no—
243C4⋊C4—101no—
244C4⋊C4—101no—
245C4⋊C4—101no—
246C4⋊C4C22×D511yescharacteristic
247C4⋊C4C22×D511yescharacteristic
248C4⋊C4C22×D511yescharacteristic
249C4⋊C4C22×D511yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—51no—
252C22⋊C4—51no—
253C22⋊C4—101no—
254C22⋊C4—101no—
255C22⋊C4—101no—
256C22⋊C4—101no—
257C22⋊C4—101no—
258C42—21no—
259C42—51no—
260C42—101no—
261C42—101no—
262C42—101no—
263C2×C5—41no—
264C2×C5#54711yescharacteristic
265C2×C5C2×#13711yescharacteristic
266C2×C5#57711yescharacteristic
267D5—41no—
268C23—51no—
269C23D4×D511yescharacteristic
270Q8—101no—
271Q8—101no—
272Q8—101no—
273Q8—101no—
274Q8—202no—
275Q8—401no—
276D4—102no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—51no—
286C2×C4—81no—
287C2×C4—101no—
288C2×C4—101no—
289C2×C4—101no—
290C2×C4—101no—
291C2×C4—101no—
292C2×C4—101no—
293C2×C4—401no—
294C2×C4D4×D511yescharacteristic
295C2×C4C23×D511yescharacteristic
296C8—21no—
297C8—101no—
298C8—101no—
299C8#70912yes—
300C5#335611yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22—21no—
304C22—101no—
305C22—101no—
306C22—101no—
307C22C2×D4×D511yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—52no—
314C4—101no—
315C4—101no—
316C4—201no—
317C4—201no—
318C4—201no—
319C4—201no—
320C4C2×#70911yescharacteristic
321C4#414711yescharacteristic
322C2—41no—
323C2—201no—
324C2#6584111yescharacteristic
325C2#6610911yescharacteristic
326C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#428C2
socleC22×C5C22×D4
prime radicalC2×C22⋊D5D4
hypercenter#428D5
hypocenterC5#3356
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. C5×#428C2semidirect
4. #65426C2semidirect
6. #65285C2semidirect
7. #65260C2semidirect
8. #65177C2semidirect
9. #65170C2semidirect
10. #65164C2semidirect
11. #65162C2semidirect
12. #65157C2semidirect
13. #65151C2semidirect
14. #65144C2semidirect
15. #65143C2semidirect
120. #428D5semidirect
300. C5#3356semidirect

Maximal subgroups

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

subgroupindex
#658672
#658452
C5×#4282
#654262
#654172
#652852
#652602
#651772
#651702
#651642
#651622
#651572
#651512
#651442
#651432
#33565