Finite Group Databasebeta

← #10611946

Subgroups — #10611946

Order 640.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×Dic5C211yesmaximal, characteristic
2#65956C211yesmaximal, characteristic, complemented
3C5×#452C211yesmaximal, characteristic
4#65528C211yesmaximal, characteristic
5#65473C211yesmaximal, characteristic, complemented
6#65471C211yesmaximal, characteristic, complemented
7#65453C211yesmaximal, characteristic, complemented
8#65451C211yesmaximal, characteristic, complemented
9#65334C211yesmaximal, characteristic, complemented
10#65083C211yesmaximal, characteristic, complemented
11#65078C211yesmaximal, characteristic, complemented
12#65050C211yesmaximal, characteristic, complemented
13#65046C211yesmaximal, characteristic, complemented
14#65012C211yesmaximal, characteristic, complemented
15#65008C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C2×C5×Q16C2211yescharacteristic
18C2×C5×SD16C2211yescharacteristic
19C5×C22⋊Q8C2211yescharacteristic
20Q8×Dic5C2211yescharacteristic
21Q8×Dic5C2211yescharacteristic
22Q8×Dic5C2211yescharacteristic
23#4090C2211yescharacteristic
24C2×#689C2211yescharacteristic
25D4×Dic5—21no—
26D4×Dic5—21no—
27D4×Dic5C2211yescharacteristic
28D4×Dic5C2211yescharacteristic
29C2×#687C2211yescharacteristic
30#4072—21no—
31#4072C2211yescharacteristic
32#4070C2211yescharacteristic
33C2×C4×Dic5—21no—
34C2×C4×Dic5C2211yescharacteristic
35#4035—21no—
36#4034C2211yescharacteristic
37#4033—21no—
38#4033C2211yescharacteristic
39#4024—21no—
40#4023—21no—
41C5×#103C2211yescharacteristic
42C5×#102C2211yescharacteristic
43C5×C22⋊C8C2211yescharacteristic
44#3967C2211yescharacteristic
45#3967C2211yescharacteristic
46#3967C2211yescharacteristic
47#3964C2211yescharacteristic
48C22⋊#168C2211yescharacteristic
49#3950C2211yescharacteristic
50#3949C2211yescharacteristic
51#3948C2211yescharacteristic
52#3948C2211yescharacteristic
53#3947C2211yescharacteristic
54#3947C2211yescharacteristic
55#3946C2211yescharacteristic
56#3946C2211yescharacteristic
57#3940C2211yescharacteristic
58#3939C2211yescharacteristic
59#3116—51nomaximal, Sylow
60C5×#41—22no—
61C5×#41—22no—
62C2×C5×Q8C2311yescharacteristic
63C2×C5×Q8C2311yescharacteristic
64C2×C5×D4—21no—
65C2×C5×D4C2311yescharacteristic
66C22×C4×C5—21no—
67C22×C4×C5C2311yescharacteristic
68C22×Dic5—21no—
69C22×Dic5—21no—
70C22×Dic5D411yescharacteristic
71C22×Dic5D411yescharacteristic
72C2×Dic10—21no—
73C2×Dic10C2311yescharacteristic
74C5×Q16—22no—
75C5×SD16—22no—
76C2×C5×C8C2311yescharacteristic
77C2×C5×C8C2311yescharacteristic
78C5×C4⋊C4—21no—
79C5×C4⋊C4C2311yescharacteristic
80C5×C22⋊C4—21no—
81C22⋊Dic5—21no—
82C22⋊Dic5—21no—
83C22⋊Dic5—21no—
84C22⋊Dic5—21no—
85#689—22no—
86#687—22no—
87C4⋊Dic5—21no—
88C4⋊Dic5—21no—
89C4⋊Dic5—21no—
90C4⋊Dic5C2311yescharacteristic
91C4⋊Dic5C2311yescharacteristic
92C4⋊Dic5C2311yescharacteristic
93#683—21no—
94#683—21no—
95#683—21no—
96#683—21no—
97#683—21no—
98C4×Dic5—21no—
99C4×Dic5—21no—
100C4×Dic5—21no—
101C4×Dic5—21no—
102C4×Dic5—21no—
103C4×Dic5D411yescharacteristic
104C4×Dic5D411yescharacteristic
105C4×Dic5C2311yescharacteristic
106C4×Dic5C2311yescharacteristic
107C2×#168C2311yescharacteristic
108C2×#168C2311yescharacteristic
109#533—51no—
110C4×#41—51no—
111#485—51no—
112#485—51no—
113#477—51no—
114#476—51no—
115#462—51no—
116#461—51no—
117#452—51no—
118#452D511yescharacteristic
119#441—51no—
120#441—51no—
121#440—51no—
122#440—51no—
123#432—51no—
124C23×C5—21no—
125C23×C5C2×D411yescharacteristic
126C5×Q8—21no—
127C5×Q8—21no—
128C5×Q8—21no—
129C5×Q8#4112yes—
130C5×D4—24no—
131C5×D4—21no—
132C5×D4#4112yes—
133C2×C4×C5—24no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5C2×D411yescharacteristic
140C2×C4×C5C2411yescharacteristic
141C2×Dic5—22no—
142C2×Dic5—21no—
143C2×Dic5—21no—
144C2×Dic5—21no—
145C2×Dic5—21no—
146C2×Dic5—41no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149C2×Dic5C2×D411yescharacteristic
150C2×Dic5C2×D411yescharacteristic
151C2×Dic5C2×D411yescharacteristic
152C2×Dic5C2×D411yescharacteristic
153Dic10—21no—
154Dic10—21no—
155Dic10—42no—
156C5×C8—21no—
157C5×C8—21no—
158#168—21no—
159#168—21no—
160C2×#41C2×D511yescharacteristic
161C2×Q16—51no—
162C2×Q16C2×D511yescharacteristic
163C2×SD16—51no—
164C2×SD16C2×D511yescharacteristic
165C4⋊Q8—51no—
166C4⋊Q8—51no—
167#125—101no—
168C22⋊Q8—51no—
169C22⋊Q8—101no—
170C22⋊Q8C2×D511yescharacteristic
171C4×Q8—51no—
172C4×Q8—51no—
173C4×Q8—51no—
174C4×Q8—51no—
175C4×Q8—101no—
176C4×D4—51no—
177C4×D4—51no—
178C4×D4—101no—
179C4×D4—101no—
180#117—51no—
181#117—101no—
182#117—101no—
183C2×C42—51no—
184C2×C42—101no—
185#107—51no—
186#107—51no—
187#106—51no—
188#106—51no—
189C4⋊C8—51no—
190C4⋊C8—51no—
191#103—51no—
192#103—51no—
193#103—51no—
194#103—51no—
195#103—51no—
196#103C2×D511yescharacteristic
197#102—51no—
198#102C2×D511yescharacteristic
199C22⋊C8—51no—
200C22⋊C8C2×D511yescharacteristic
201#97—51no—
202#97—51no—
203C22×C5—21no—
204C22×C5—22no—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5C22×D411yescharacteristic
210C4×C5—21no—
211C4×C5—22no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5C2×#4111yescharacteristic
215C4×C5C2×#4111yescharacteristic
216Dic5—21no—
217Dic5—21no—
218Dic5—21no—
219Dic5—41no—
220Dic5—41no—
221Dic5—41no—
222Dic5—41no—
223Dic5#13712yes—
224#41—22no—
225#41—22no—
226C2×Q8—51no—
227C2×Q8—101no—
228C2×Q8C22×D511yescharacteristic
229C2×Q8C22×D511yescharacteristic
230C2×D4—21no—
231C2×D4C22×D511yescharacteristic
232C22×C4—21no—
233C22×C4—51no—
234C22×C4—51no—
235C22×C4—101no—
236C22×C4—101no—
237C22×C4C22×D511yescharacteristic
238Q16—22no—
239Q16—102no—
240SD16—22no—
241SD16—102no—
242C2×C8—51no—
243C2×C8—51no—
244C2×C8C22×D511yescharacteristic
245C2×C8C22×D511yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—51no—
248C4⋊C4—51no—
249C4⋊C4—51no—
250C4⋊C4—101no—
251C4⋊C4—101no—
252C4⋊C4—101no—
253C4⋊C4—101no—
254C4⋊C4—101no—
255C4⋊C4—101no—
256C4⋊C4—101no—
257C4⋊C4—101no—
258C4⋊C4C22×D511yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—101no—
261C22⋊C4—101no—
262C22⋊C4—101no—
263C22⋊C4—101no—
264C42—51no—
265C42—51no—
266C42—51no—
267C42—51no—
268C42—101no—
269C42—101no—
270C42—101no—
271C42—101no—
272C42—101no—
273C2×C5—41no—
274C2×C5—42no—
275C2×C5#52511yescharacteristic
276C2×C5C2×#13711yescharacteristic
277C2×C5#57511yescharacteristic
278C23—21no—
279C23D4×D511yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—101no—
284Q8—101no—
285Q8—202no—
286Q8#71112yes—
287D4—24no—
288D4—21no—
289D4#71112yes—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—51no—
297C2×C4—51no—
298C2×C4—51no—
299C2×C4—51no—
300C2×C4—102no—
301C2×C4—101no—
302C2×C4—101no—
303C2×C4—101no—
304C2×C4—101no—
305C2×C4—202no—
306C2×C4—201no—
307C2×C4—202no—
308C2×C4D4×D511yescharacteristic
309C2×C4C23×D511yescharacteristic
310C8—21no—
311C8—21no—
312C8—101no—
313C8—101no—
314C5#311611yescharacteristic, complemented, Sylow
315C22—21no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22C2×D4×D511yescharacteristic
322C4—21no—
323C4—22no—
324C4—41no—
325C4—41no—
326C4—52no—
327C4—101no—
328C4—101no—
329C4—101no—
330C4—201no—
331C4—201no—
332C4—201no—
333C4—201no—
334C4C2×#71111yescharacteristic
335C4C2×#71111yescharacteristic
336C2—41no—
337C2—42no—
338C2#6592011yescharacteristic
339C2#6610211yescharacteristic
340C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#452C2
socleC22×C5C22×D4
prime radicalC2×C5×#41C22
hypercenter#452D5
hypocenterC5#3116
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #65956C2semidirect
5. #65473C2semidirect
6. #65471C2semidirect
7. #65453C2semidirect
8. #65451C2semidirect
9. #65334C2semidirect
10. #65083C2semidirect
11. #65078C2semidirect
12. #65050C2semidirect
13. #65046C2semidirect
14. #65012C2semidirect
15. #65008C2semidirect
314. C5#3116semidirect

Maximal subgroups

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

subgroupindex
#41×Dic52
#659562
C5×#4522
#655282
#654732
#654712
#654532
#654512
#653342
#650832
#650782
#650502
#650462
#650122
#650082
#31165