Finite Group Databasebeta

← #10614712

Subgroups — #10614712

Order 640.

534 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 137 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×#124C211yesmaximal, characteristic, complemented
2C2×#4022C211yesmaximal, characteristic, complemented
3#65520C211yesmaximal, characteristic, complemented
4#65518C211yesmaximal, characteristic, complemented
5#65512C211yesmaximal, characteristic
6#65511C211yesmaximal, characteristic, complemented
7#65250C212yesmaximal, complemented
8#65246C212yesmaximal, complemented
9#65237C212yesmaximal, complemented
10#65234C212yesmaximal, complemented
11C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×C5×D4C2211yescharacteristic
14C22×C22⋊D5C2212yes—
15C5×#124—24no—
16C2×C5×C22⋊C4C2212yes—
17C2×C5×C22⋊C4C2212yes—
18C2×C42×C5C2211yescharacteristic
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×#685C2211yescharacteristic
23C2×#685C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2212yes—
26C2×C4⋊Dic5C2212yes—
27C2×#683C2211yescharacteristic
28C2×#683C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2212yes—
31C2×C4×Dic5C2212yes—
32#4022—24no—
33#3963C2211yescharacteristic
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2662—51nomaximal, Sylow
38C24×C5C2312yes—
39C23×D5C2311yescharacteristic
40C2×C5×Q8—22no—
41C2×C5×Q8—42no—
42C2×C5×D4—22no—
43C2×C5×D4—42no—
44C22×C4×C5C2312yes—
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2311yescharacteristic
47C22×C4×C5C2311yescharacteristic
48C2×C22⋊D5—24no—
49C2×C22⋊D5—42no—
50C2×C22⋊D5—42no—
51C22×Dic5C2312yes—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2312yes—
56C5×C22⋊C4—24no—
57C5×C22⋊C4—24no—
58C42×C5D414yescomplemented
59C22⋊Dic5—24no—
60C22⋊Dic5—42no—
61C22⋊Dic5—42no—
62#685—22no—
63#685—22no—
64#685—22no—
65#685—42no—
66C4⋊Dic5—24no—
67#683—22no—
68#683—22no—
69#683—22no—
70#683—42no—
71C4×Dic5—42no—
72C2×#126—51no—
73C2×#124D511yescharacteristic, complemented
74#396—52no—
75#395—51no—
76#394—51no—
77#394—52no—
78#393—51no—
79#392—51no—
80#390—52no—
81#388—52no—
82C4×C4⋊C4—51no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5C2411yescharacteristic
91C22×D5—21no—
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—21no—
95C22×D5—21no—
96C5×Q8—44no—
97C5×D4—44no—
98C2×C4×C5—22no—
99C2×C4×C5—21no—
100C2×C4×C5—22no—
101C2×C4×C5—21no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5C2×D412yes—
106C2×C4×C5C2×D412yes—
107C2×C4×C5C2×D412yes—
108C22⋊D5—48no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—21no—
112C2×Dic5—21no—
113C2×Dic5—21no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—21no—
120C2×Dic5—21no—
121C2×Dic5#4114yes—
122C22×Q8C2×D511yescharacteristic
123C22×D4—52no—
124C22×D4C2×D511yescharacteristic
125#126—104no—
126#124—24no—
127C2×C4⋊C4—51no—
128C2×C4⋊C4—51no—
129C2×C4⋊C4—51no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—52no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4—51no—
134C2×C22⋊C4—51no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4C2×D512yes—
140C2×C22⋊C4C2×D512yes—
141C2×C42—52no—
142C2×C42C2×D511yescharacteristic
143#95—51no—
144#95—51no—
145#95—52no—
146#95—52no—
147C22×C5—42no—
148C22×C5—42no—
149C22×C5—42no—
150C22×C5—42no—
151C22×C5—42no—
152C22×C5—42no—
153C22×C5—42no—
154C22×C5C22×D411yescharacteristic
155C22×C5C2×#4112yes—
156C22×C5H(5,𝔽2)12yes—
157C22×C5H(5,𝔽2)11yescharacteristic
158C22×C52−1+411yescharacteristic
159C2×D5—41no—
160C2×D5—41no—
161C2×D5—41no—
162C2×D5—42no—
163C2×D5—42no—
164C4×C5—22no—
165C4×C5—22no—
166C4×C5—22no—
167C4×C5—42no—
168Dic5—24no—
169Dic5—41no—
170Dic5—41no—
171Dic5—41no—
172Dic5—42no—
173C24—51no—
174C24C22×D512yes—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×D4—22no—
178C2×D4—42no—
179C2×D4—104no—
180C2×D4—202no—
181C2×D4—202no—
182C22×C4—52no—
183C22×C4—52no—
184C22×C4—51no—
185C22×C4—51no—
186C22×C4—51no—
187C22×C4C22×D512yes—
188C22×C4C22×D511yescharacteristic
189C22×C4C22×D511yescharacteristic
190C22×C4C22×D511yescharacteristic
191C4⋊C4—104no—
192C4⋊C4—102no—
193C4⋊C4—102no—
194C4⋊C4—102no—
195C4⋊C4—202no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—104no—
199C22⋊C4—102no—
200C22⋊C4—102no—
201C22⋊C4—102no—
202C22⋊C4—202no—
203C22⋊C4—202no—
204C22⋊C4—202no—
205C42—202no—
206C42C22⋊D514yescomplemented
207C2×C5—82no—
208C2×C5#53211yescharacteristic
209C2×C5#53511yescharacteristic
210C2×C5#53711yescharacteristic
211C2×C5#53812yes—
212C2×C5#54212yes—
213D5—81no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—102no—
222C23—102no—
223C23—101no—
224C23—101no—
225C23—101no—
226C23C23×D511yescharacteristic
227Q8—44no—
228D4—44no—
229D4—208no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—54no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—102no—
241C2×C4—102no—
242C2×C4—102no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—101no—
246C2×C4—101no—
247C2×C4—101no—
248C2×C4—101no—
249C2×C4—101no—
250C2×C4C2×C22⋊D512yes—
251C2×C4C2×C22⋊D512yes—
252C2×C4C2×C22⋊D512yes—
253C5#266211yescharacteristic, complemented, Sylow
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—201no—
262C22—201no—
263C22—201no—
264C22—202no—
265C22—202no—
266C22#414412yes—
267C22D5×#4112yes—
268C22#414911yescharacteristic
269C22#415011yescharacteristic
270C22C22×C22⋊D511yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—104no—
276C4—201no—
277C4—201no—
278C4—201no—
279C4—202no—
280C2—82no—
281C2—401no—
282C2#6600312yes—
283C2#6601212yes—
284C2#6613911yescharacteristic
285C2#6616611yescharacteristic
286C2#6616711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#124C2
prime radical#65512C2
hypercenterC2×#124D5
hypocenterC5#2662
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C5×#124C2semidirect
2. C2×#4022C2semidirect
3. #65520C2semidirect
4. #65518C2semidirect
6. #65511C2semidirect
7. #65250C2semidirect
8. #65246C2semidirect
9. #65237C2semidirect
10. #65234C2semidirect
11. C4×C4⋊Dic5C2semidirect
58. C42×C5D4semidirect
73. C2×#124D5semidirect
206. C42C22⋊D5semidirect
253. C5#2662semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1242
C2×#40222
#655202
#655182
#655122
#655112
#652502
#652462
#652372
#652342
C4×C4⋊Dic52
#26625