Finite Group Databasebeta

← #10611777

Subgroups — #10611777

Order 640.

414 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4092C211yesmaximal, characteristic, complemented
2#65940C211yesmaximal, characteristic
3C5×#448C211yesmaximal, characteristic, complemented
4#65517C211yesmaximal, characteristic, complemented
5#65463C212yesmaximal, complemented
6#65462C212yesmaximal, complemented
7#65323C211yesmaximal, characteristic, complemented
8#65105C212yesmaximal, complemented
9#65103C212yesmaximal, complemented
10#65015C211yesmaximal, characteristic, complemented
11#65011C211yesmaximal, characteristic, complemented
12C22×C5×Q8C2211yescharacteristic
13C2×#711C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C2×C5×SD16C2212yescomplemented
16C5×#121C2211yescharacteristic
17#4092—21no—
18#4092—21no—
19#4092C2212yes—
20#4092C2212yes—
21C2×#689C2212yescomplemented
22#4084C2211yescharacteristic
23#4083—21no—
24D4×Dic5—21no—
25C2×#685—21no—
26C2×C4⋊Dic5C2211yescharacteristic
27#4070C2211yescharacteristic
28C2×#683—21no—
29C2×#676C2212yes—
30#4043C2211yescharacteristic
31#4038C2211yescharacteristic
32#4029—21no—
33#4027—21no—
34C5×#103C2212yescomplemented
35C5×C22⋊C8C2211yescharacteristic
36#3967C2212yescomplemented
37#3964C2212yescomplemented
38C22⋊#168C2211yescharacteristic
39#3952C2212yes—
40#3950C2212yescomplemented
41#3948C2212yescomplemented
42#3940C2212yescomplemented
43#3196—51nomaximal, Sylow
44C23×D5D411yescharacteristic
45C2×C5×Q8—21no—
46C2×C5×Q8—42no—
47C2×C5×Q8C2312yes—
48C2×C5×D4—21no—
49C2×C5×D4C2311yescharacteristic
50C22×C4×C5—21no—
51C22×C4×C5C2311yescharacteristic
52C2×C22⋊D5—21no—
53C2×C22⋊D5—21no—
54C22×Dic5—21no—
55C22×Dic5—21no—
56C22×Dic5D411yescharacteristic
57#711—42no—
58C2×C4×D5—42no—
59C2×C4×D5D411yescharacteristic
60C2×C4×D5D411yescharacteristic
61C2×C4×D5C2311yescharacteristic
62C2×C4×D5C2311yescharacteristic
63C2×Dic10C2311yescharacteristic
64C5×SD16—42no—
65C2×C5×C8C2312yes—
66C5×C4⋊C4C2311yescharacteristic
67C5×C22⋊C4—21no—
68C22⋊Dic5—21no—
69C22⋊Dic5—21no—
70#689—42no—
71#685—21no—
72#685—21no—
73#685—22no—
74#685—22no—
75C4⋊Dic5—21no—
76C4⋊Dic5—21no—
77C4⋊Dic5C2312yes—
78C4⋊Dic5C2311yescharacteristic
79#683—21no—
80#683—22no—
81#683—22no—
82C4×Dic5—21no—
83C2×#168C2312yes—
84#676—42no—
85#525—51no—
86C2×C22⋊Q8—51no—
87#485—52no—
88#484—51no—
89#481—51no—
90#470—52no—
91#468—52no—
92#452—52no—
93#451—51no—
94#448D511yescharacteristic, complemented
95#407—51no—
96C23×C5—21no—
97C23×C5C2×D411yescharacteristic
98C22×D5—22no—
99C22×D5—41no—
100C22×D5—41no—
101C22×D5C2×D411yescharacteristic
102C22×D5C2×D411yescharacteristic
103C5×Q8—22no—
104C5×Q8—22no—
105C5×Q8—42no—
106C5×D4—22no—
107C5×D4—42no—
108C2×C4×C5—22no—
109C2×C4×C5—21no—
110C2×C4×C5—22no—
111C2×C4×C5—42no—
112C2×C4×C5C2×D411yescharacteristic
113C2×C4×C5C2411yescharacteristic
114C22⋊D5—42no—
115C22⋊D5—42no—
116C2×Dic5—21no—
117C2×Dic5—22no—
118C2×Dic5—21no—
119C2×Dic5—41no—
120C2×Dic5—42no—
121C2×Dic5—42no—
122C2×Dic5C2×D411yescharacteristic
123C2×Dic5C2×D411yescharacteristic
124C4×D5—22no—
125C4×D5—22no—
126C4×D5—41no—
127C4×D5—41no—
128Dic10—22no—
129C5×C8—22no—
130#168—22no—
131C2×#41—51no—
132C22×Q8C2×D511yescharacteristic
133C23×C4—51no—
134C2×Q16—52no—
135C2×SD16C2×D512yescomplemented
136C2×M16—52no—
137#123—101no—
138C22⋊Q8—52no—
139C22⋊Q8—52no—
140C22⋊Q8—51no—
141C22⋊Q8—51no—
142C22⋊Q8—101no—
143C22⋊Q8—101no—
144#121—51no—
145#121C2×D511yescharacteristic
146#120—101no—
147C4×D4—101no—
148C4×D4—101no—
149#117—51no—
150C2×C4⋊C4—51no—
151C2×C4⋊C4—101no—
152C2×C22⋊C4—101no—
153#107—52no—
154#106—52no—
155#103—52no—
156#103—52no—
157#103C2×D512yescomplemented
158#102—52no—
159C22⋊C8—51no—
160C22⋊C8—52no—
161C22⋊C8C2×D511yescharacteristic
162C22×C5—21no—
163C22×C5—21no—
164C22×C5—41no—
165C22×C5—41no—
166C22×C5—41no—
167C22×C5#13712yes—
168C22×C5C22×D411yescharacteristic
169C2×D5—21no—
170C2×D5—21no—
171C2×D5—21no—
172C2×D5—21no—
173C2×D5—21no—
174C2×D5—21no—
175C2×D5—44no—
176C4×C5—21no—
177C4×C5—42no—
178C4×C5—41no—
179C4×C5H(5,𝔽2)12yes—
180Dic5—21no—
181Dic5—21no—
182Dic5—41no—
183Dic5—42no—
184Dic5—41no—
185C24—51no—
186#41—202no—
187C2×Q8—21no—
188C2×Q8—42no—
189C2×Q8—51no—
190C2×Q8C22×D512yes—
191C2×D4—21no—
192C2×D4—101no—
193C2×D4—101no—
194C2×D4C22×D511yescharacteristic
195C22×C4—21no—
196C22×C4—51no—
197C22×C4—51no—
198C22×C4—51no—
199C22×C4—51no—
200C22×C4—51no—
201C22×C4—101no—
202C22×C4—101no—
203C22×C4—202no—
204C22×C4C22×D511yescharacteristic
205Q16—202no—
206SD16—42no—
207M16—202no—
208C2×C8—52no—
209C2×C8C22×D512yes—
210C4⋊C4—52no—
211C4⋊C4—51no—
212C4⋊C4—102no—
213C4⋊C4—102no—
214C4⋊C4—101no—
215C4⋊C4—101no—
216C4⋊C4—101no—
217C4⋊C4C22×D511yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—102no—
220C22⋊C4—102no—
221C22⋊C4—101no—
222C22⋊C4—101no—
223C22⋊C4—101no—
224C22⋊C4—101no—
225C42—101no—
226C2×C5—22no—
227C2×C5—81no—
228C2×C5#53411yescharacteristic
229C2×C5C2×#13711yescharacteristic
230C2×C5#57511yescharacteristic
231D5—41no—
232D5—41no—
233C23—21no—
234C23—51no—
235C23—51no—
236C23—102no—
237C23—201no—
238C23—201no—
239C23D4×D511yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—42no—
243Q8—102no—
244D4—22no—
245D4—42no—
246D4—202no—
247D4—202no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—51no—
253C2×C4—51no—
254C2×C4—102no—
255C2×C4—102no—
256C2×C4—101no—
257C2×C4—102no—
258C2×C4—101no—
259C2×C4—201no—
260C2×C4—201no—
261C2×C4—201no—
262C2×C4—202no—
263C2×C4—202no—
264C2×C4D4×D511yescharacteristic
265C2×C4C23×D511yescharacteristic
266C8—22no—
267C8—102no—
268C5#319611yescharacteristic, complemented, Sylow
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—101no—
275C22—101no—
276C22—101no—
277C22—101no—
278C22—101no—
279C22—101no—
280C22—204no—
281C22#406112yes—
282C22C2×D4×D511yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—101no—
287C4—101no—
288C4—202no—
289C4—201no—
290C4—201no—
291C4#414412yes—
292C2—22no—
293C2—81no—
294C2—201no—
295C2—201no—
296C2#6592311yescharacteristic
297C2C2×#406111yescharacteristic
298C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#448C2
socleC22×C5C22×D4
prime radical#65940C2
hypercenter#448D5
hypocenterC5#3196
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#4092C2semidirect
3. C5×#448C2semidirect
4. #65517C2semidirect
5. #65463C2semidirect
6. #65462C2semidirect
7. #65323C2semidirect
8. #65105C2semidirect
9. #65103C2semidirect
10. #65015C2semidirect
11. #65011C2semidirect
15. C2×C5×SD16C22semidirect
21. C2×#689C22semidirect
34. C5×#103C22semidirect
36. #3967C22semidirect
37. #3964C22semidirect
40. #3950C22semidirect
41. #3948C22semidirect
42. #3940C22semidirect
94. #448D5semidirect
135. C2×SD16C2×D5semidirect
157. #103C2×D5semidirect
268. C5#3196semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
#659402
C5×#4482
#655172
#654632
#654622
#653232
#651052
#651032
#650152
#650112
#31965