Finite Group Databasebeta

← #10614470

Subgroups — #10614470

Order 640.

414 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 121 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×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2#66125C211yesmaximal, characteristic
3#65517C211yesmaximal, characteristic, complemented
4#65516C211yesmaximal, characteristic, complemented
5#65496C211yesmaximal, characteristic, complemented
6#65341C212yesmaximal, complemented
7#65338C212yesmaximal, complemented
8#65336C212yesmaximal, complemented
9#65334C212yesmaximal, complemented
10#65259C211yesmaximal, characteristic, complemented
11#65257C211yesmaximal, characteristic, complemented
12C22×C5×Q8C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C2×#709C2211yescharacteristic
15C5×C22⋊Q8—21no—
16C5×C22⋊Q8—21no—
17C5×C22⋊Q8C2212yes—
18C5×C22⋊Q8C2212yes—
19C2×C5×C4⋊C4—21no—
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4—21no—
22#4099—21no—
23C2×#689C2212yescomplemented
24C2×#688C2212yescomplemented
25#4076C2211yescharacteristic
26#4076C2211yescharacteristic
27#4075—21no—
28C4×C22⋊D5—21no—
29C4×C22⋊D5—21no—
30#4072C2211yescharacteristic
31#4070C2211yescharacteristic
32#4070C2211yescharacteristic
33C2×#681C2212yes—
34#3967C2212yescomplemented
35#3967C2212yescomplemented
36C22⋊#168C2212yes—
37C22⋊#168C2211yescharacteristic
38C22⋊#168C2211yescharacteristic
39#3942C2212yescomplemented
40#3941C2212yescomplemented
41#3940C2212yescomplemented
42#3939C2212yescomplemented
43#3196—51nomaximal, Sylow
44C24×C5D411yescharacteristic
45C2×C5×Q8—21no—
46C2×C5×Q8—42no—
47C2×C5×Q8C2312yes—
48C22×C4×C5—21no—
49C22×C4×C5—21no—
50C22×C4×C5—42no—
51C22×C4×C5D411yescharacteristic
52C22×C4×C5D411yescharacteristic
53C22×C4×C5D411yescharacteristic
54C22×C4×C5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C2×C22⋊D5—21no—
58C2×C22⋊D5—21no—
59C2×C22⋊D5—21no—
60#709—42no—
61C2×D20C2311yescharacteristic
62C2×C4×D5—21no—
63C2×Dic10C2311yescharacteristic
64C5×C4⋊C4—22no—
65C5×C4⋊C4—22no—
66C5×C4⋊C4—21no—
67C5×C4⋊C4C2312yes—
68C5×C22⋊C4—22no—
69C5×C22⋊C4—22no—
70C22⋊Dic5—21no—
71C22⋊Dic5—21no—
72C22⋊Dic5—21no—
73#689—42no—
74#688—42no—
75#685—21no—
76#685—21no—
77C4⋊Dic5C2311yescharacteristic
78C4⋊Dic5C2311yescharacteristic
79#683—21no—
80#683—21no—
81C4×Dic5—21no—
82#681—42no—
83C2×#168C2312yes—
84C2×#168C2312yes—
85#525—51no—
86C2×C22⋊Q8D511yescharacteristic, complemented
87#485—52no—
88#484—51no—
89#481—51no—
90#470—52no—
91#468—52no—
92#452—52no—
93#451—51no—
94#448—51no—
95#407—51no—
96C23×C5—22no—
97C23×C5—41no—
98C23×C5—41no—
99C23×C5C2×D411yescharacteristic
100C23×C5C2×D411yescharacteristic
101C23×C5C2×D411yescharacteristic
102C22×D5—21no—
103C5×Q8—22no—
104C5×Q8—22no—
105C5×Q8—42no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—41no—
112C2×C4×C5—41no—
113C2×C4×C5—41no—
114C2×C4×C5—42no—
115C2×C4×C5—42no—
116C2×C4×C5C2×D411yescharacteristic
117C2×C4×C5C2×D411yescharacteristic
118C2×C4×C5C2×D411yescharacteristic
119C2×C4×C5C2411yescharacteristic
120C22⋊D5—42no—
121C22⋊D5—42no—
122C22⋊D5—42no—
123C2×Dic5—21no—
124C2×Dic5—21no—
125C2×Dic5—21no—
126D20—22no—
127C4×D5—42no—
128Dic10—22no—
129#168—22no—
130#168—22no—
131C2×#41—51no—
132C22×Q8C2×D511yescharacteristic
133C23×C4C2×D511yescharacteristic
134C2×Q16—52no—
135C2×SD16—52no—
136C2×M16—52no—
137#123—101no—
138C22⋊Q8—21no—
139C22⋊Q8—21no—
140C22⋊Q8—51no—
141C22⋊Q8—51no—
142C22⋊Q8C2×D512yes—
143C22⋊Q8C2×D512yes—
144#121—51no—
145#121—51no—
146#120—101no—
147C4×D4—101no—
148C4×D4—101no—
149#117—51no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4C2×D511yescharacteristic
152C2×C22⋊C4—21no—
153#107—52no—
154#106—52no—
155#103—52no—
156#103—52no—
157#103—52no—
158#102—52no—
159C22⋊C8—51no—
160C22⋊C8—51no—
161C22⋊C8—52no—
162C22×C5—21no—
163C22×C5—21no—
164C22×C5—21no—
165C22×C5—21no—
166C22×C5—21no—
167C22×C5—21no—
168C22×C5—21no—
169C22×C5—21no—
170C22×C5—44no—
171C22×C5#13712yes—
172C22×C5C22×D411yescharacteristic
173C2×D5—41no—
174C2×D5—41no—
175C2×D5—41no—
176C4×C5—21no—
177C4×C5—21no—
178C4×C5—21no—
179C4×C5—42no—
180C4×C5—42no—
181C4×C5H(5,𝔽2)12yes—
182Dic5—41no—
183Dic5—41no—
184Dic5—41no—
185C24C22⋊D511yescharacteristic
186#41—202no—
187C2×Q8—21no—
188C2×Q8—42no—
189C2×Q8—51no—
190C2×Q8C22×D512yes—
191C2×D4—51no—
192C2×D4—101no—
193C2×D4—101no—
194C2×D4—101no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—101no—
199C22×C4C22⋊D511yescharacteristic
200C22×C4C22⋊D511yescharacteristic
201C22×C4C22⋊D511yescharacteristic
202C22×C4C22×D511yescharacteristic
203C22×C4C22×D511yescharacteristic
204C22×C4C22×D511yescharacteristic
205Q16—202no—
206SD16—202no—
207M16—202no—
208C2×C8—52no—
209C2×C8—52no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—51no—
214C4⋊C4—51no—
215C4⋊C4—101no—
216C4⋊C4—101no—
217C4⋊C4C22×D512yes—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—101no—
221C22⋊C4—101no—
222C22⋊C4—101no—
223C22⋊C4—101no—
224C22⋊C4—101no—
225C42—101no—
226C2×C5—22no—
227C2×C5—41no—
228C2×C5—41no—
229C2×C5#53411yescharacteristic
230C2×C5C2×#13711yescharacteristic
231C2×C5#57511yescharacteristic
232D5—81no—
233C23—22no—
234C23—41no—
235C23—41no—
236C23—101no—
237C23C2×C22⋊D511yescharacteristic
238C23C2×C22⋊D511yescharacteristic
239C23C2×C22⋊D511yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—42no—
243Q8—102no—
244D4—102no—
245D4—202no—
246D4—202no—
247D4—202no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—101no—
259C2×C4—101no—
260C2×C4—101no—
261C2×C4—202no—
262C2×C4C2×C22⋊D511yescharacteristic
263C2×C4C2×C22⋊D511yescharacteristic
264C2×C4C2×C22⋊D511yescharacteristic
265C2×C4C23×D511yescharacteristic
266C8—102no—
267C8—102no—
268C5#319611yescharacteristic, complemented, Sylow
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—201no—
279C22—201no—
280C22—201no—
281C22#408812yes—
282C22C22×C22⋊D511yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—201no—
289C4—201no—
290C4—201no—
291C4#414412yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—401no—
296C2#6613811yescharacteristic
297C2C2×#408811yescharacteristic
298C2#6615611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC2×C5×C22⋊Q8C2
socleC22×C5C22×D4
prime radical#66125C2
hypercenterC2×C22⋊Q8D5
hypocenterC5#3196
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×C22⋊Q8C2semidirect
3. #65517C2semidirect
4. #65516C2semidirect
5. #65496C2semidirect
6. #65341C2semidirect
7. #65338C2semidirect
8. #65336C2semidirect
9. #65334C2semidirect
10. #65259C2semidirect
11. #65257C2semidirect
23. C2×#689C22semidirect
24. C2×#688C22semidirect
34. #3967C22semidirect
35. #3967C22semidirect
39. #3942C22semidirect
40. #3941C22semidirect
41. #3940C22semidirect
42. #3939C22semidirect
86. C2×C22⋊Q8D5semidirect
268. C5#3196semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊Q82
#661252
#655172
#655162
#654962
#653412
#653382
#653362
#653342
#652592
#652572
#31965