Finite Group Databasebeta

← #10612803

Subgroups — #10612803

Order 640.

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

normal subgroups only

This group has 348 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
1#66062C211yesmaximal, characteristic, complemented
2#65891C211yesmaximal, characteristic
3C5×#476C211yesmaximal, characteristic, complemented
4#65380C211yesmaximal, characteristic, complemented
5#65308C211yesmaximal, characteristic, complemented
6#65157C212yesmaximal, complemented
7#65146C212yesmaximal, complemented
8#65138C211yesmaximal, characteristic, complemented
9#65126C211yesmaximal, characteristic, complemented
10#65069C212yesmaximal
11#65056C212yesmaximal, complemented
12C5×C4⋊Q8C2211yescharacteristic
13C4×C5×D4C2211yescharacteristic
14#4092—21no—
15Q8×Dic5—21no—
16#4084C2211yescharacteristic
17#4082C2211yescharacteristic
18C2×#687C2212yes—
19C4×C22⋊D5—21no—
20#4072—21no—
21C2×Dic20C2212yes—
22C2×#676C2212yes—
23#4043C2212yes—
24#4043C2211yescharacteristic
25#4038—21no—
26#4036—21no—
27#4036C2211yescharacteristic
28#4033C2212yes—
29#4029—21no—
30#4025—21no—
31C42×D5C2211yescharacteristic
32#4015C2211yescharacteristic
33C4×Dic10C2211yescharacteristic
34C5×#106C2212yes—
35C5×C4⋊C8C2211yescharacteristic
36C5×#102C2212yes—
37#3948C2212yes—
38#3946C2212yes—
39#3942C2212yes—
40#3942C2212yes—
41#3939C2212yes—
42C4⋊#168C2211yescharacteristic
43#3252—51nomaximal, Sylow
44C2×C5×Q8—21no—
45C2×C5×D4C2311yescharacteristic
46C22×C4×C5—21no—
47C2×C22⋊D5—21no—
48C2×C4×D5D411yescharacteristic
49C2×C4×D5D411yescharacteristic
50C2×C4×D5C2311yescharacteristic
51C2×Dic10C2311yescharacteristic
52C2×Dic10C2312yes—
53C2×C5×C8C2312yes—
54C5×C4⋊C4—21no—
55C5×C4⋊C4C2311yescharacteristic
56C5×C4⋊C4C2312yes—
57C5×C22⋊C4—21no—
58C42×C5C2311yescharacteristic
59C22⋊Dic5—21no—
60C22⋊Dic5—21no—
61C22⋊Dic5—21no—
62#687—42no—
63#685—21no—
64#685—22no—
65C4⋊Dic5—21no—
66C4⋊Dic5—21no—
67C4⋊Dic5—21no—
68C4⋊Dic5—21no—
69C4⋊Dic5—22no—
70C4⋊Dic5C2311yescharacteristic
71#683—21no—
72#683—21no—
73#683—22no—
74C4×Dic5—21no—
75C4×Dic5—22no—
76C4×Dic5D411yescharacteristic
77C4×Dic5D411yescharacteristic
78C4×Dic5C2311yescharacteristic
79C2×#168C2312yes—
80Dic20—42no—
81#676—42no—
82#533—51no—
83#529—51no—
84#479—52no—
85#477—51no—
86#476D511yescharacteristic, complemented
87#470—52no—
88#468—52no—
89#464—52no—
90#462—51no—
91#461—51no—
92#423—51no—
93C23×C5—21no—
94C22×D5C2×D411yescharacteristic
95C5×Q8—42no—
96C5×D4—21no—
97C5×D4—21no—
98C2×C4×C5—22no—
99C2×C4×C5—21no—
100C2×C4×C5—42no—
101C2×C4×C5C2×D411yescharacteristic
102C2×C4×C5C2×D411yescharacteristic
103C2×C4×C5C2411yescharacteristic
104C22⋊D5—42no—
105C2×Dic5—21no—
106C2×Dic5—22no—
107C2×Dic5—21no—
108C2×Dic5C2×D411yescharacteristic
109C2×Dic5C2×D411yescharacteristic
110C2×Dic5C2×D411yescharacteristic
111C4×D5—22no—
112C4×D5—21no—
113C4×D5—21no—
114C4×D5—41no—
115Dic10—21no—
116Dic10—22no—
117Dic10—21no—
118Dic10—22no—
119C5×C8—22no—
120#168—22no—
121C2×Q16—52no—
122C2×SD16—52no—
123C2×M16—52no—
124C4⋊Q8—51no—
125C4⋊Q8C2×D511yescharacteristic
126#126—101no—
127#125—51no—
128#125—101no—
129#124—51no—
130#123—101no—
131C22⋊Q8—51no—
132C22⋊Q8—52no—
133C22⋊Q8—101no—
134#121—51no—
135C4×Q8—52no—
136C4×Q8—51no—
137C4×Q8—101no—
138C4×D4—101no—
139C4×D4C2×D511yescharacteristic
140#117—101no—
141#117—101no—
142C2×C42—51no—
143#107—52no—
144#106C2×D512yes—
145C4⋊C8—52no—
146C4⋊C8—51no—
147C4⋊C8C2×D511yescharacteristic
148#103—52no—
149#103—52no—
150#103—52no—
151#102C2×D512yes—
152C22×C5—41no—
153C22×C5—41no—
154C22×C5—41no—
155C22×C5C22×D411yescharacteristic
156C2×D5—21no—
157C2×D5—21no—
158C2×D5—21no—
159C4×C5—21no—
160C4×C5—41no—
161C4×C5—42no—
162C4×C5#13712yes—
163C4×C5H(5,𝔽2)11yescharacteristic
164C4×C52−1+411yescharacteristic
165Dic5—21no—
166Dic5—21no—
167Dic5—21no—
168Dic5—41no—
169Dic5—42no—
170Dic5—41no—
171C2×Q8—21no—
172C2×Q8—51no—
173C2×Q8—52no—
174C2×D4—101no—
175C2×D4C22×D511yescharacteristic
176C22×C4—21no—
177C22×C4—51no—
178C22×C4—51no—
179C22×C4—51no—
180Q16—202no—
181SD16—202no—
182M16—202no—
183C2×C8—52no—
184C2×C8C22×D512yes—
185C4⋊C4—21no—
186C4⋊C4—51no—
187C4⋊C4—101no—
188C4⋊C4—101no—
189C4⋊C4—101no—
190C4⋊C4—102no—
191C4⋊C4—102no—
192C4⋊C4—101no—
193C4⋊C4—101no—
194C4⋊C4—101no—
195C4⋊C4C22×D512yes—
196C4⋊C4C22×D511yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—102no—
199C22⋊C4—101no—
200C22⋊C4—101no—
201C22⋊C4—101no—
202C22⋊C4—101no—
203C42—51no—
204C42—51no—
205C42—51no—
206C42—102no—
207C42—101no—
208C42C22×D511yescharacteristic
209C2×C5—81no—
210C2×C5#53711yescharacteristic
211C2×C5C2×#13711yescharacteristic
212C2×C5#57511yescharacteristic
213D5—41no—
214C23—21no—
215C23—51no—
216Q8—42no—
217Q8—101no—
218Q8—102no—
219Q8—102no—
220Q8—101no—
221D4—21no—
222D4—21no—
223D4—202no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—51no—
228C2×C4—51no—
229C2×C4—51no—
230C2×C4—101no—
231C2×C4—101no—
232C2×C4—102no—
233C2×C4—101no—
234C2×C4—101no—
235C2×C4—102no—
236C2×C4—201no—
237C2×C4D4×D511yescharacteristic
238C2×C4D4×D511yescharacteristic
239C2×C4C23×D511yescharacteristic
240C8—22no—
241C8—102no—
242C5#325211yescharacteristic, complemented, Sylow
243C22—41no—
244C22—41no—
245C22—41no—
246C22—101no—
247C22—101no—
248C22—101no—
249C22C2×D4×D511yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—101no—
254C4—101no—
255C4—101no—
256C4—201no—
257C4—201no—
258C4—202no—
259C4#406112yes—
260C4#414711yescharacteristic
261C4#414911yescharacteristic
262C2—81no—
263C2—201no—
264C2#6596311yescharacteristic
265C2C2×#406111yescharacteristic
266C2#6610711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#476C2
socleC22×C5C22×D4
prime radical#4084C22
hypercenter#476D5
hypocenterC5#3252
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #66062C2semidirect
3. C5×#476C2semidirect
4. #65380C2semidirect
5. #65308C2semidirect
6. #65157C2semidirect
7. #65146C2semidirect
8. #65138C2semidirect
9. #65126C2semidirect
11. #65056C2semidirect
86. #476D5semidirect
242. C5#3252semidirect

Maximal subgroups

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

subgroupindex
#660622
#658912
C5×#4762
#653802
#653082
#651572
#651462
#651382
#651262
#650692
#650562
#32525