Finite Group Databasebeta

← #10618411

Subgroups — #10618411

Order 640.

992 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#541C211yesmaximal, characteristic, complemented
2#66065C211yesmaximal, characteristic, complemented
3D5×C4⋊Q8C211yesmaximal, characteristic, complemented
4#66042C214yesmaximal, complemented
5#66036C214yesmaximal, complemented
6#66031C211yesmaximal, characteristic, complemented
7#66027C214yesmaximal, complemented
8#66022C211yesmaximal, characteristic, complemented
9#66019C214yesmaximal, complemented
10#65962C212yesmaximal
11#65956C212yesmaximal, complemented
12#65912C212yesmaximal, complemented
13#65906C212yesmaximal, complemented
14#65856C211yesmaximal, characteristic
15#65852C211yesmaximal, characteristic, complemented
16C2×Q8×D5C2212yes—
17C2×#709C2211yescharacteristic
18C5×C4⋊Q8C2211yescharacteristic
19C5×#126C2214yes—
20C5×#125C2214yes—
21C5×#125C2211yescharacteristic
22C5×C22⋊Q8C2212yes—
23C4×C5×Q8C2212yes—
24C5×#117C2211yescharacteristic
25#4092C2212yes—
26Q8×Dic5C2212yes—
27#4090C2212yes—
28#4070C2212yes—
29C2×C4×Dic5C2211yescharacteristic
30#4044C2214yes—
31#4044C2214yes—
32#4043C2212yes—
33#4042C2214yes—
34#4042C2214yes—
35#4040C2214yes—
36#4039C2212yes—
37#4038C2212yes—
38#4038C2214yes—
39D5×C4⋊C4C2214yes—
40#4036C2214yes—
41#4035C2214yes—
42#4035C2214yes—
43#4035C2214yes—
44#4035C2214yes—
45#4034C2214yes—
46#4034C2212yes—
47#4034C2214yes—
48#4033C2212yes—
49#4033C2212yes—
50#4033C2214yes—
51#4033C2214yes—
52#4033C2214yes—
53#4031C2214yes—
54#4029C2214yes—
55#4027C2212yes—
56#4027C2212yes—
57#4025C2214yes—
58#4024C2214yes—
59#4023C2214yes—
60#4018C2211yescharacteristic
61#4018C2211yescharacteristic
62#4018C2214yes—
63#4018C2214yes—
64C42×D5C2211yescharacteristic
65C42×D5C2211yescharacteristic
66#4016C2211yescharacteristic
67#4015C2211yescharacteristic
68C4×Dic10C2212yes—
69C4×Dic10C2214yes—
70C4×Dic10C2214yes—
71#3509—51nomaximal, Sylow
72C2×C5×Q8C2312yes—
73C22×C4×C5C2311yescharacteristic
74C2×C22⋊D5C2311yescharacteristic
75C2×C22⋊D5C2311yescharacteristic
76C22×Dic5C2312yes—
77Q8×D5—28no—
78#709—24no—
79C2×D20C2311yescharacteristic
80C2×C4×D5C2311yescharacteristic
81C2×C4×D5C2314yes—
82C2×C4×D5C2312yes—
83C2×C4×D5C2312yes—
84C2×C4×D5C2311yescharacteristic
85C2×Dic10C2311yescharacteristic
86C2×Dic10C2312yes—
87C2×Dic10C2314yes—
88C2×Dic10C2314yes—
89C5×C4⋊C4C2312yes—
90C5×C4⋊C4C2312yes—
91C5×C4⋊C4C2314yes—
92C5×C4⋊C4C2314yes—
93C5×C4⋊C4C2314yes—
94C5×C4⋊C4C2314yes—
95C5×C22⋊C4C2312yes—
96C5×C22⋊C4C2314yes—
97C42×C5C2311yescharacteristic
98C42×C5C2311yescharacteristic
99C42×C5C2314yes—
100C22⋊Dic5C2314yes—
101#685C2312yes—
102#685C2312yes—
103#685C2314yes—
104#685C2314yes—
105#685C2314yes—
106C4⋊Dic5C2312yes—
107C4⋊Dic5C2314yes—
108C4⋊Dic5C2314yes—
109C4⋊Dic5C2314yes—
110#683C2314yes—
111#683C2312yes—
112#683C2314yes—
113#683C2314yes—
114#683C2314yes—
115#683C2314yes—
116#683C2314yes—
117#683C2314yes—
118#683C2312yes—
119#683C2314yes—
120C4×Dic5C2312yes—
121C4×Dic5C2314yes—
122C4×Dic5C2314yes—
123C4×Dic5C2314yes—
124C4×Dic5C2311yescharacteristic
125C4×Dic5C2312yes—
126C4×Dic5C2311yescharacteristic
127C4×Dic5C2314yes—
128C4×Dic5C2312yes—
129#557—54no—
130#556—54no—
131#552—54no—
132#552—54no—
133#541—51no—
134#541—51no—
135#541—51no—
136#541D511yescharacteristic, complemented
137#536—52no—
138#533—51no—
139#533—52no—
140#533—52no—
141C2×C4⋊Q8—51no—
142#519—52no—
143C4×#41—51no—
144C23×C5C2411yescharacteristic
145C22×D5C2411yescharacteristic
146C22×D5C2411yescharacteristic
147C5×Q8—24no—
148C2×C4×C5—22no—
149C2×C4×C5C2412yes—
150C2×C4×C5C2411yescharacteristic
151C2×C4×C5C2414yes—
152C2×C4×C5C2412yes—
153C2×C4×C5C2414yes—
154C2×C4×C5C2411yescharacteristic
155C22⋊D5—22no—
156C22⋊D5—22no—
157C2×Dic5—24no—
158C2×Dic5C2412yes—
159C2×Dic5C2414yes—
160C2×Dic5C2411yescharacteristic
161C2×Dic5C2414yes—
162C2×Dic5C2411yescharacteristic
163C2×Dic5C2412yes—
164D20—22no—
165C4×D5—28no—
166C4×D5—22no—
167C4×D5—24no—
168C4×D5—24no—
169C4×D5#4114yes—
170Dic10—24no—
171Dic10—22no—
172Dic10—28no—
173Dic10—28no—
174C2×#41—51no—
175C22×Q8—52no—
176C4⋊Q8—52no—
177C4⋊Q8—54no—
178C4⋊Q8—52no—
179C4⋊Q8—54no—
180C4⋊Q8—51no—
181C4⋊Q8C2×D511yescharacteristic
182#126—54no—
183#126—54no—
184#126—54no—
185#126C2×D514yes—
186#125—54no—
187#125—54no—
188#125—54no—
189#125—54no—
190#125—54no—
191#125—51no—
192#125C2×D511yescharacteristic
193#125C2×D514yes—
194#124—54no—
195#123—54no—
196#123—54no—
197C22⋊Q8—52no—
198C22⋊Q8—52no—
199C22⋊Q8—52no—
200C22⋊Q8—54no—
201C22⋊Q8—54no—
202C22⋊Q8—54no—
203C22⋊Q8C2×D512yes—
204C4×Q8—52no—
205C4×Q8—52no—
206C4×Q8—52no—
207C4×Q8—54no—
208C4×Q8—54no—
209C4×Q8—54no—
210C4×Q8—52no—
211C4×Q8—54no—
212C4×Q8—54no—
213C4×Q8C2×D512yes—
214C4×D4—52no—
215C4×D4—52no—
216C4×D4—52no—
217#117—52no—
218#117—54no—
219#117—54no—
220#117—51no—
221#117—51no—
222#117—54no—
223#117—54no—
224#117C2×D511yescharacteristic
225C2×C4⋊C4—54no—
226C2×C42—51no—
227C2×C42—51no—
228C2×C42—51no—
229C22×C5—21no—
230C22×C5—22no—
231C22×C5C2511yescharacteristic
232C2×D5—22no—
233C2×D5—21no—
234C2×D5—22no—
235C2×D5C2×#4112yes—
236C4×C5—24no—
237C4×C5—21no—
238C4×C5—24no—
239C4×C5—22no—
240C4×C5—22no—
241C4×C5C2×#4112yes—
242Dic5—24no—
243Dic5—21no—
244Dic5—24no—
245Dic5—22no—
246Dic5C2×#4112yes—
247Dic52−1+414yes—
248#41—104no—
249C2×Q8—54no—
250C2×Q8—52no—
251C2×Q8—54no—
252C2×Q8—51no—
253C2×Q8—108no—
254C2×Q8C22×D512yes—
255C2×D4—51no—
256C2×D4—51no—
257C2×D4—51no—
258C22×C4—51no—
259C22×C4—52no—
260C22×C4—52no—
261C22×C4—52no—
262C22×C4—51no—
263C22×C4—54no—
264C22×C4C22×D511yescharacteristic
265C4⋊C4—54no—
266C4⋊C4—54no—
267C4⋊C4—54no—
268C4⋊C4—54no—
269C4⋊C4—52no—
270C4⋊C4—52no—
271C4⋊C4—52no—
272C4⋊C4—54no—
273C4⋊C4—54no—
274C4⋊C4—54no—
275C4⋊C4—54no—
276C4⋊C4—54no—
277C4⋊C4—54no—
278C4⋊C4—54no—
279C4⋊C4C22×D514yes—
280C4⋊C4C22×D512yes—
281C4⋊C4C22×D514yes—
282C4⋊C4C22×D512yes—
283C4⋊C4C22×D514yes—
284C4⋊C4C22×D514yes—
285C22⋊C4—54no—
286C22⋊C4—52no—
287C22⋊C4—54no—
288C22⋊C4—52no—
289C22⋊C4—54no—
290C22⋊C4—54no—
291C22⋊C4C22×D514yes—
292C22⋊C4C22×D512yes—
293C42—51no—
294C42—51no—
295C42—52no—
296C42—52no—
297C42—52no—
298C42—54no—
299C42—54no—
300C42—54no—
301C42—54no—
302C42C22×D511yescharacteristic
303C42C22×D511yescharacteristic
304C42C22×D514yes—
305C2×C5—41no—
306C2×C5C22×#4111yescharacteristic
307C2×C5C2×2−1+412yes—
308D5—22no—
309D5—41no—
310C23—51no—
311C23—51no—
312C23C23×D511yescharacteristic
313Q8—24no—
314Q8—108no—
315Q8—102no—
316Q8—108no—
317Q8—104no—
318D4—102no—
319D4—102no—
320D4—102no—
321C2×C4—22no—
322C2×C4—51no—
323C2×C4—52no—
324C2×C4—54no—
325C2×C4—52no—
326C2×C4—54no—
327C2×C4—54no—
328C2×C4—51no—
329C2×C4—104no—
330C2×C4—104no—
331C2×C4—108no—
332C2×C4—104no—
333C2×C4—102no—
334C2×C4C23×D512yes—
335C2×C4C23×D514yes—
336C2×C4C23×D511yescharacteristic
337C2×C4C23×D512yes—
338C2×C4C23×D514yes—
339C2×C4C23×D511yescharacteristic
340C5#350911yescharacteristic, complemented, Sylow
341C22—21no—
342C22—22no—
343C22—52no—
344C22—102no—
345C22—101no—
346C22—102no—
347C22C24×D511yescharacteristic
348C4—21no—
349C4—24no—
350C4—22no—
351C4—22no—
352C4—24no—
353C4—52no—
354C4—54no—
355C4—101no—
356C4—104no—
357C4—102no—
358C4—104no—
359C4D5×#4112yes—
360C2—41no—
361C2—102no—
362C2—201no—
363C2C2×D5×#4111yescharacteristic
364C2D5×2−1+412yes—
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#541C2
prime radicalC2×#709C22
hypercenter#541D5
hypocenterC5#3509
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. C5×#541C2semidirect
2. #66065C2semidirect
3. D5×C4⋊Q8C2semidirect
4. #66042C2semidirect
5. #66036C2semidirect
6. #66031C2semidirect
7. #66027C2semidirect
8. #66022C2semidirect
9. #66019C2semidirect
11. #65956C2semidirect
12. #65912C2semidirect
13. #65906C2semidirect
15. #65852C2semidirect
136. #541D5semidirect
340. C5#3509semidirect

Maximal subgroups

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

subgroupindex
C5×#5412
#660652
D5×C4⋊Q82
#660422
#660362
#660312
#660272
#660222
#660192
#659622
#659562
#659122
#659062
#658562
#658522
#35095