Finite Group Databasebeta

← #10611910

Subgroups — #10611910

Order 640.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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×#4090C211yesmaximal, characteristic, complemented
2#65958C211yesmaximal, characteristic
3C5×#451C211yesmaximal, characteristic, complemented
4#65516C211yesmaximal, characteristic
5#65479C212yesmaximal, complemented
6#65471C212yesmaximal, complemented
7#65339C211yesmaximal, characteristic, complemented
8#65107C212yesmaximal, complemented
9#65086C212yesmaximal, complemented
10#65027C211yesmaximal, characteristic
11#65008C211yesmaximal, characteristic, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C2×C5×Q16C2212yes—
15C5×C22⋊Q8C2211yescharacteristic
16#4093C2211yescharacteristic
17Q8×Dic5C2211yescharacteristic
18#4090—21no—
19#4090—21no—
20#4090C2212yes—
21#4090C2212yes—
22C2×#689C2212yes—
23C2×#688C2212yes—
24#4076C2211yescharacteristic
25C2×#683—21no—
26C2×#683—21no—
27C2×C4×Dic5C2211yescharacteristic
28C2×#677C2212yes—
29#4044—21no—
30#4039C2211yescharacteristic
31#4038—21no—
32#4036C2211yescharacteristic
33#4032—21no—
34#4027—21no—
35C5×#103C2212yes—
36C5×C22⋊C8C2211yescharacteristic
37#3967C2212yes—
38C22⋊#168C2211yescharacteristic
39#3948C2212yes—
40#3947C2212yes—
41#3946C2212yes—
42#3942C2212yes—
43#3218—51nomaximal, Sylow
44C2×C5×Q8—21no—
45C2×C5×Q8—42no—
46C2×C5×Q8C2311yescharacteristic
47C2×C5×Q8C2312yes—
48C22×C4×C5—21no—
49C22×C4×C5C2311yescharacteristic
50C2×C22⋊D5—21no—
51C22×Dic5—21no—
52C22×Dic5D411yescharacteristic
53C22×Dic5D411yescharacteristic
54C2×D20C2311yescharacteristic
55C2×C4×D5—21no—
56C2×Dic10—21no—
57C2×Dic10—42no—
58C2×Dic10C2312yes—
59C5×Q16—42no—
60C2×C5×C8C2312yes—
61C5×C4⋊C4—21no—
62C5×C4⋊C4C2311yescharacteristic
63C5×C22⋊C4—21no—
64#689—42no—
65#688—42no—
66#685—21no—
67#685—21no—
68#685—21no—
69#685—21no—
70C4⋊Dic5—21no—
71C4⋊Dic5—21no—
72C4⋊Dic5C2311yescharacteristic
73#683—21no—
74#683—22no—
75#683—22no—
76#683—22no—
77#683—22no—
78C4×Dic5—21no—
79C4×Dic5D411yescharacteristic
80C4×Dic5D411yescharacteristic
81C4×Dic5C2311yescharacteristic
82C4×Dic5C2311yescharacteristic
83C2×#168C2312yes—
84#677—42no—
85C2×C4⋊Q8—51no—
86#529—51no—
87#497—52no—
88#491—52no—
89#462—52no—
90#461—52no—
91#451—51no—
92#451D511yescharacteristic, complemented
93#448—51no—
94#448—51no—
95#432—51no—
96C23×C5C2×D411yescharacteristic
97C22×D5—21no—
98C5×Q8—22no—
99C5×Q8—22no—
100C5×Q8—22no—
101C5×Q8—42no—
102C2×C4×C5—22no—
103C2×C4×C5—21no—
104C2×C4×C5—22no—
105C2×C4×C5—21no—
106C2×C4×C5—42no—
107C2×C4×C5C2×D411yescharacteristic
108C2×C4×C5C2411yescharacteristic
109C22⋊D5—42no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—21no—
113C2×Dic5—22no—
114C2×Dic5—42no—
115C2×Dic5C2×D411yescharacteristic
116C2×Dic5C2×D411yescharacteristic
117C2×Dic5C2×D411yescharacteristic
118C2×Dic5C2×D411yescharacteristic
119D20—22no—
120C4×D5—42no—
121Dic10—22no—
122Dic10—22no—
123Dic10—42no—
124C5×C8—22no—
125#168—22no—
126C22×Q8—51no—
127C22×Q8C2×D511yescharacteristic
128C2×Q16—52no—
129C2×Q16C2×D512yes—
130C2×SD16—52no—
131C2×SD16—52no—
132C4⋊Q8—52no—
133C4⋊Q8—52no—
134C4⋊Q8—101no—
135C4⋊Q8—101no—
136#126—101no—
137#125—51no—
138#124—51no—
139#123—101no—
140C22⋊Q8C2×D511yescharacteristic
141#121—51no—
142C4×Q8—51no—
143C4×D4—51no—
144C4×D4—101no—
145#117—101no—
146C2×C4⋊C4—101no—
147C2×C4⋊C4—101no—
148C2×C42—51no—
149C4⋊C8—52no—
150#103—52no—
151#103—52no—
152#103—52no—
153#103C2×D512yes—
154C22⋊C8—51no—
155C22⋊C8C2×D511yescharacteristic
156#97—52no—
157C22×C5—21no—
158C22×C5—21no—
159C22×C5#13712yes—
160C22×C5C22×D411yescharacteristic
161C2×D5—41no—
162C2×D5—41no—
163C2×D5—41no—
164C4×C5—21no—
165C4×C5—42no—
166C4×C5—41no—
167C4×C5—41no—
168C4×C5H(5,𝔽2)12yes—
169Dic5—21no—
170Dic5—21no—
171Dic5—21no—
172Dic5—42no—
173Dic5—41no—
174Dic5#13712yes—
175C2×Q8—21no—
176C2×Q8—42no—
177C2×Q8—52no—
178C2×Q8—101no—
179C2×Q8—202no—
180C2×Q8C22×D512yes—
181C2×Q8C22×D511yescharacteristic
182C2×D4—51no—
183C2×D4—101no—
184C22×C4—21no—
185C22×C4—51no—
186C22×C4—51no—
187C22×C4—101no—
188C22×C4—101no—
189C22×C4C22×D511yescharacteristic
190Q16—42no—
191Q16—202no—
192SD16—202no—
193SD16—202no—
194C2×C8—52no—
195C2×C8C22×D512yes—
196C4⋊C4—21no—
197C4⋊C4—51no—
198C4⋊C4—102no—
199C4⋊C4—102no—
200C4⋊C4—102no—
201C4⋊C4—102no—
202C4⋊C4—101no—
203C4⋊C4—101no—
204C4⋊C4—101no—
205C4⋊C4C22×D511yescharacteristic
206C22⋊C4—21no—
207C22⋊C4—101no—
208C22⋊C4—101no—
209C22⋊C4—101no—
210C22⋊C4—101no—
211C42—51no—
212C42—51no—
213C42—51no—
214C42—51no—
215C42—101no—
216C2×C5—22no—
217C2×C5#53511yescharacteristic
218C2×C5C2×#13711yescharacteristic
219C2×C5C2×#13711yescharacteristic
220D5—81no—
221C23—101no—
222C23D4×D511yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—22no—
226Q8—42no—
227Q8—102no—
228Q8—102no—
229Q8—202no—
230D4—102no—
231D4—202no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—51no—
238C2×C4—51no—
239C2×C4—51no—
240C2×C4—51no—
241C2×C4—102no—
242C2×C4—102no—
243C2×C4—101no—
244C2×C4—102no—
245C2×C4—202no—
246C2×C4—202no—
247C2×C4D4×D511yescharacteristic
248C2×C4C23×D511yescharacteristic
249C8—22no—
250C8—102no—
251C5#321811yescharacteristic, complemented, Sylow
252C22—21no—
253C22—21no—
254C22—201no—
255C22—201no—
256C22—201no—
257C22#406412yes—
258C22C2×D4×D511yescharacteristic
259C4—21no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—52no—
264C4—101no—
265C4—101no—
266C4—101no—
267C4—201no—
268C4—202no—
269C4#414412yes—
270C2—22no—
271C2—401no—
272C2#6592611yescharacteristic
273C2C2×#406411yescharacteristic
274C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#451C2
socleC22×C5C22×D4
prime radical#4076C22
hypercenter#451D5
hypocenterC5#3218
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#4090C2semidirect
3. C5×#451C2semidirect
5. #65479C2semidirect
6. #65471C2semidirect
7. #65339C2semidirect
8. #65107C2semidirect
9. #65086C2semidirect
11. #65008C2semidirect
92. #451D5semidirect
251. C5#3218semidirect

Maximal subgroups

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

subgroupindex
C2×#40902
#659582
C5×#4512
#655162
#654792
#654712
#653392
#651072
#650862
#650272
#650082
#32185