Finite Group Databasebeta

← #10618653

Subgroups — #10618653

Order 640.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×D4×Q8C211yesmaximal, characteristic, complemented
2Q8×C22⋊D5C211yesmaximal, characteristic, complemented
3Q8×C22⋊D5C211yesmaximal, characteristic, complemented
4#66146C211yesmaximal, characteristic, complemented
5C2×Q8×Dic5C211yesmaximal, characteristic, complemented
6#66065C213yesmaximal, complemented
7#66055C213yesmaximal, complemented
8#65976C213yesmaximal, complemented
9#65975C213yesmaximal, complemented
10#65959C213yesmaximal, complemented
11#65956C213yesmaximal, complemented
12#65906C211yesmaximal, characteristic, complemented
13#65903C211yesmaximal, characteristic, complemented
14#65875C213yesmaximal, complemented
15#65872C213yesmaximal
16C22×C5×Q8C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic
18C2×Q8×D5C2211yescharacteristic
19C5×C4⋊Q8C2213yes—
20C5×C22⋊Q8C2213yes—
21C5×C22⋊Q8C2213yes—
22C4×C5×Q8C2211yescharacteristic
23C4×C5×D4C2213yes—
24#4092C2213yes—
25#4092C2213yes—
26Q8×Dic5C2213yes—
27Q8×Dic5C2213yes—
28Q8×Dic5C2213yes—
29Q8×Dic5C2213yes—
30Q8×Dic5C2211yescharacteristic
31Q8×Dic5C2211yescharacteristic
32Q8×Dic5C2211yescharacteristic
33#4090C2213yes—
34#4090C2213yes—
35#4085C2211yescharacteristic
36#4081C2213yes—
37C4×C22⋊D5C2213yes—
38C4×C22⋊D5C2213yes—
39#4072C2213yes—
40#4072C2213yes—
41C2×C4⋊Dic5C2213yes—
42#4070C2213yes—
43#4070C2213yes—
44C2×C4×Dic5C2213yes—
45#4044C2213yes—
46#4044C2216yes—
47#4038C2213yes—
48#4036C2216yes—
49#4036C2213yes—
50#4035C2213yes—
51#4035C2216yes—
52#4035C2216yes—
53#4034C2213yes—
54#4033C2216yes—
55#4033C2213yes—
56#4032C2213yes—
57#4031C2213yes—
58#4025C2213yes—
59#4025C2216yes—
60#4024C2213yes—
61#4023C2216yes—
62#4018C2213yes—
63#4016C2213yes—
64C4×Dic10C2213yes—
65C4×Dic10C2213yes—
66#3523—51nomaximal, Sylow
67C2×C5×Q8—24no—
68C2×C5×Q8—24no—
69C2×C5×Q8C2313yes—
70C2×C5×Q8C2313yes—
71C2×C5×Q8C2311yescharacteristic
72C2×C5×D4C2311yescharacteristic
73C22×C4×C5C2313yes—
74C22×C4×C5C2313yes—
75C2×C22⋊D5C2311yescharacteristic
76C2×C22⋊D5C2311yescharacteristic
77C22×Dic5C2313yes—
78C22×Dic5C2311yescharacteristic
79Q8×D5—24no—
80C2×C4×D5C2313yes—
81C2×Dic10C2313yes—
82C2×Dic10C2313yes—
83C5×C4⋊C4C2313yes—
84C5×C4⋊C4C2313yes—
85C5×C4⋊C4C2316yes—
86C5×C22⋊C4C2313yes—
87C5×C22⋊C4C2313yes—
88C42×C5C2313yes—
89C22⋊Dic5C2313yes—
90C22⋊Dic5C2313yes—
91C22⋊Dic5C2311yescharacteristic
92C22⋊Dic5C2313yes—
93#685C2311yescharacteristic
94#685C2313yes—
95#685C2313yes—
96#685C2313yes—
97C4⋊Dic5C2313yes—
98C4⋊Dic5C2313yes—
99C4⋊Dic5C2313yes—
100C4⋊Dic5C2313yes—
101C4⋊Dic5C2313yes—
102C4⋊Dic5C2316yes—
103C4⋊Dic5C2313yes—
104#683C2311yescharacteristic
105#683C2313yes—
106#683C2313yes—
107#683C2316yes—
108#683C2313yes—
109#683C2316yes—
110#683C2313yes—
111#683C2316yes—
112#683C2313yes—
113C4×Dic5C2316yes—
114C4×Dic5C2313yes—
115C4×Dic5C2313yes—
116C4×Dic5C2313yes—
117C4×Dic5C2316yes—
118C4×Dic5C2313yes—
119C4×Dic5C2313yes—
120#557—53no—
121#557—51no—
122#556—53no—
123#552—53no—
124#552—53no—
125D4×Q8—51no—
126D4×Q8—51no—
127D4×Q8D511yescharacteristic, complemented
128#541—53no—
129#541—53no—
130#533—53no—
131#529—53no—
132#519—51no—
133#519—51no—
134C2×C4×Q8—51no—
135C23×C5C2411yescharacteristic
136C23×C5C2411yescharacteristic
137C22×D5C2411yescharacteristic
138C5×Q8—26no—
139C5×Q8—26no—
140C5×Q8#4114yes—
141C5×D4—22no—
142C2×C4×C5—26no—
143C2×C4×C5—26no—
144C2×C4×C5C2413yes—
145C2×C4×C5C2413yes—
146C2×C4×C5C2413yes—
147C2×C4×C5C2411yescharacteristic
148C2×C4×C5C2413yes—
149C22⋊D5—22no—
150C22⋊D5—22no—
151C2×Dic5—26no—
152C2×Dic5—22no—
153C2×Dic5C2411yescharacteristic
154C2×Dic5C2413yes—
155C2×Dic5C2413yes—
156C2×Dic5C2411yescharacteristic
157C2×Dic5C2413yes—
158C2×Dic5C2413yes—
159C2×Dic5C2411yescharacteristic
160C4×D5—26no—
161Dic10—26no—
162Dic10—26no—
163C22×Q8—51no—
164C22×Q8C2×D511yescharacteristic
165C22×Q8C2×D511yescharacteristic
166C4⋊Q8—53no—
167C4⋊Q8—53no—
168C4⋊Q8—53no—
169C4⋊Q8C2×D513yes—
170#126—53no—
171#126—56no—
172#126—53no—
173#126—56no—
174#125—53no—
175#125—56no—
176#125—56no—
177#125—56no—
178#125—53no—
179#125—53no—
180#124—53no—
181#123—53no—
182#123—53no—
183C22⋊Q8—53no—
184C22⋊Q8—53no—
185C22⋊Q8—53no—
186C22⋊Q8—53no—
187C22⋊Q8—53no—
188C22⋊Q8C2×D513yes—
189C22⋊Q8C2×D513yes—
190#121—51no—
191C4×Q8—51no—
192C4×Q8—53no—
193C4×Q8—51no—
194C4×Q8—53no—
195C4×Q8—53no—
196C4×Q8—51no—
197C4×Q8—53no—
198C4×Q8—53no—
199C4×Q8—56no—
200C4×Q8—53no—
201C4×Q8—53no—
202C4×Q8C2×D511yescharacteristic
203C4×D4—53no—
204C4×D4—53no—
205C4×D4C2×D513yes—
206#117—53no—
207#117—53no—
208#117—53no—
209#117—56no—
210#117—53no—
211C2×C4⋊C4—53no—
212C2×C42—53no—
213C22×C5—21no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—21no—
218C22×C52−1+412yes—
219C22×C5C2511yescharacteristic
220C2×D5—21no—
221C2×D5—21no—
222C2×D5—21no—
223C4×C5—23no—
224C4×C5—23no—
225C4×C5—23no—
226C4×C5—21no—
227C4×C5C2×#4116yes—
228Dic5—23no—
229Dic5—23no—
230Dic5—23no—
231Dic5—23no—
232Dic5—21no—
233Dic5—21no—
234Dic52−1+412yes—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—53no—
238C2×Q8—53no—
239C2×Q8—104no—
240C2×Q8C22×D513yes—
241C2×Q8C22×D513yes—
242C2×Q8C22×D511yescharacteristic
243C2×D4—51no—
244C2×D4—51no—
245C2×D4C22×D511yescharacteristic
246C22×C4—51no—
247C22×C4—53no—
248C22×C4—53no—
249C22×C4C22×D513yes—
250C22×C4C22×D513yes—
251C4⋊C4—53no—
252C4⋊C4—53no—
253C4⋊C4—56no—
254C4⋊C4—53no—
255C4⋊C4—53no—
256C4⋊C4—53no—
257C4⋊C4—53no—
258C4⋊C4—53no—
259C4⋊C4—56no—
260C4⋊C4—56no—
261C4⋊C4—53no—
262C4⋊C4—53no—
263C4⋊C4—51no—
264C4⋊C4—56no—
265C4⋊C4—53no—
266C4⋊C4—53no—
267C4⋊C4C22×D513yes—
268C4⋊C4C22×D513yes—
269C4⋊C4C22×D516yes—
270C22⋊C4—51no—
271C22⋊C4—53no—
272C22⋊C4—53no—
273C22⋊C4—53no—
274C22⋊C4—51no—
275C22⋊C4—53no—
276C22⋊C4—53no—
277C22⋊C4—53no—
278C22⋊C4C22×D513yes—
279C22⋊C4C22×D513yes—
280C42—53no—
281C42—53no—
282C42—53no—
283C42—53no—
284C42—56no—
285C42—56no—
286C42—53no—
287C42C22×D513yes—
288C2×C5—22no—
289C2×C5—41no—
290C2×C5C22×#4111yescharacteristic
291C2×C5C2×2−1+411yescharacteristic
292C2×C5C2×2−1+411yescharacteristic
293D5—41no—
294C23—51no—
295C23C23×D511yescharacteristic
296C23C23×D511yescharacteristic
297Q8—26no—
298Q8—26no—
299Q8—106no—
300Q8—106no—
301Q8#71114yes—
302D4—22no—
303D4—102no—
304D4—102no—
305C2×C4—26no—
306C2×C4—26no—
307C2×C4—51no—
308C2×C4—53no—
309C2×C4—51no—
310C2×C4—53no—
311C2×C4—51no—
312C2×C4—53no—
313C2×C4—53no—
314C2×C4—102no—
315C2×C4—106no—
316C2×C4—106no—
317C2×C4C23×D513yes—
318C2×C4C23×D513yes—
319C2×C4C23×D513yes—
320C2×C4C23×D511yescharacteristic
321C2×C4C23×D513yes—
322C5#352311yescharacteristic, complemented, Sylow
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—101no—
329C22—101no—
330C22—101no—
331C22#414712yes—
332C22C24×D511yescharacteristic
333C4—23no—
334C4—23no—
335C4—21no—
336C4—23no—
337C4—52no—
338C4—103no—
339C4—103no—
340C4—103no—
341C4—101no—
342C4—101no—
343C4—103no—
344C4C2×#71116yes—
345C2—22no—
346C2—41no—
347C2—201no—
348C2C22×#71111yescharacteristic
349C2C2×#414711yescharacteristic
350C2D5×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×D4×Q8C2
prime radical#4085C22
hypercenterD4×Q8D5
hypocenterC5#3523
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C5×D4×Q8C2semidirect
2. Q8×C22⋊D5C2semidirect
3. Q8×C22⋊D5C2semidirect
4. #66146C2semidirect
5. C2×Q8×Dic5C2semidirect
6. #66065C2semidirect
7. #66055C2semidirect
8. #65976C2semidirect
9. #65975C2semidirect
10. #65959C2semidirect
11. #65956C2semidirect
12. #65906C2semidirect
13. #65903C2semidirect
14. #65875C2semidirect
127. D4×Q8D5semidirect
322. C5#3523semidirect

Maximal subgroups

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

subgroupindex
C5×D4×Q82
Q8×C22⋊D52
#661462
C2×Q8×Dic52
#660652
#660552
#659762
#659752
#659592
#659562
#659062
#659032
#658752
#658722
#35235