Finite Group Databasebeta

← #10608878

Subgroups — #10608878

Order 640.

612 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 129 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×#4030C212yesmaximal, complemented
2C5×#394C211yesmaximal, characteristic, complemented
3#65512C212yesmaximal, complemented
4#65277C211yesmaximal, characteristic, complemented
5#65272C211yesmaximal, characteristic, complemented
6#65250C212yesmaximal, complemented
7#65245C212yesmaximal, complemented
8#65234C212yesmaximal, complemented
9#64953C212yesmaximal, complemented
10C22×C5×D4C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×C22⋊D5C2212yes—
13C22×D20C2211yescharacteristic
14C22×C4×D5C2212yes—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×#683C2212yes—
25C2×C4×Dic5C2211yescharacteristic
26#4030—28no—
27C5×#95C2211yescharacteristic
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2211yescharacteristic
32#2699—51nomaximal, Sylow
33C24×C5C2312yes—
34C23×D5C2312yes—
35C2×C5×D4—42no—
36C2×C5×D4—41no—
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2312yes—
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C2×C22⋊D5—24no—
42C2×C22⋊D5—24no—
43C2×C22⋊D5—24no—
44C2×C22⋊D5—42no—
45C2×C22⋊D5—42no—
46C2×C22⋊D5—42no—
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C22×Dic5C2312yes—
50C2×D20—41no—
51C2×D20—41no—
52C2×D20D414yescomplemented
53C2×C4×D5—24no—
54C2×C4×D5—24no—
55C2×C4×D5—42no—
56C5×C4⋊C4—41no—
57C5×C22⋊C4—24no—
58C5×C22⋊C4—42no—
59C22⋊Dic5—24no—
60C22⋊Dic5—42no—
61#685—24no—
62#685—42no—
63#685—42no—
64C4⋊Dic5—41no—
65#683—24no—
66C4×Dic5—41no—
67C2×#121—52no—
68#398—51no—
69#394—52no—
70#394D511yescharacteristic, complemented
71#392—52no—
72#390—51no—
73#388—52no—
74#386—52no—
75#381—52no—
76C23×C5—22no—
77C23×C5—22no—
78C23×C5—22no—
79C23×C5—22no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5C2411yescharacteristic
84C22×D5—22no—
85C22×D5—22no—
86C22×D5—22no—
87C22×D5—22no—
88C22×D5—22no—
89C22×D5—22no—
90C22×D5C2×D414yes—
91C5×D4—44no—
92C2×C4×C5—21no—
93C2×C4×C5—21no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—21no—
101C2×C4×C5C2×D412yes—
102C22⋊D5—48no—
103C22⋊D5—48no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5#4114yes—
113D20—44no—
114C4×D5—48no—
115C22×D4—52no—
116C22×D4—52no—
117C22×D4—51no—
118C22×D4C2×D511yescharacteristic
119C23×C4—52no—
120#121—108no—
121C2×C4⋊C4—52no—
122C2×C4⋊C4—51no—
123C2×C4⋊C4C2×D511yescharacteristic
124C2×C22⋊C4—52no—
125C2×C22⋊C4—52no—
126C2×C22⋊C4—52no—
127C2×C22⋊C4—52no—
128C2×C22⋊C4—52no—
129C2×C22⋊C4C2×D512yes—
130C2×C22⋊C4C2×D512yes—
131C2×C42—51no—
132#95—52no—
133#95—52no—
134#95—52no—
135#95—51no—
136#95C2×D511yescharacteristic
137C22×C5—42no—
138C22×C5—42no—
139C22×C5—42no—
140C22×C5—42no—
141C22×C5—42no—
142C22×C5—42no—
143C22×C5—42no—
144C22×C5C22×D411yescharacteristic
145C22×C5C2×#4112yes—
146C22×C5H(5,𝔽2)11yescharacteristic
147C22×C5H(5,𝔽2)12yes—
148C22×C5H(5,𝔽2)11yescharacteristic
149C2×D5—24no—
150C2×D5—24no—
151C2×D5—24no—
152C2×D5—42no—
153C2×D5—42no—
154C2×D5—42no—
155C2×D5—42no—
156C4×C5—22no—
157C4×C5—42no—
158C4×C5—41no—
159C4×C5—41no—
160Dic5—24no—
161Dic5—42no—
162Dic5—42no—
163C24—52no—
164C24C22×D512yes—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—54no—
168C2×D4—104no—
169C2×D4—104no—
170C2×D4—104no—
171C2×D4—201no—
172C2×D4—201no—
173C2×D4—202no—
174C2×D4—202no—
175C2×D4—202no—
176C22×C4—52no—
177C22×C4—52no—
178C22×C4—52no—
179C22×C4—104no—
180C22×C4—104no—
181C22×C4—202no—
182C22×C4C22×D511yescharacteristic
183C22×C4C22×D512yes—
184C22×C4C22×D511yescharacteristic
185C22×C4C22×D511yescharacteristic
186C4⋊C4—41no—
187C4⋊C4—104no—
188C4⋊C4—201no—
189C22⋊C4—24no—
190C22⋊C4—42no—
191C22⋊C4—104no—
192C22⋊C4—104no—
193C22⋊C4—202no—
194C22⋊C4—202no—
195C22⋊C4—202no—
196C42—201no—
197C2×C5—82no—
198C2×C5#53411yescharacteristic
199C2×C5#54012yes—
200C2×C5#54612yes—
201C2×C5#55511yescharacteristic
202C2×C5#56011yescharacteristic
203D5—44no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—54no—
212C23—102no—
213C23—102no—
214C23—102no—
215C23—102no—
216C23—102no—
217C23—102no—
218C23C23×D511yescharacteristic
219D4—44no—
220D4—204no—
221D4—208no—
222D4—208no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—54no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—102no—
241C2×C4—208no—
242C2×C4D4×D512yes—
243C5#269911yescharacteristic, complemented, Sylow
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—104no—
252C22—104no—
253C22—104no—
254C22—202no—
255C22—202no—
256C22—202no—
257C22—202no—
258C22C2×D4×D511yescharacteristic
259C22#414411yescharacteristic
260C22#414412yes—
261C22D5×#4112yes—
262C22#414911yescharacteristic
263C4—22no—
264C4—41no—
265C4—41no—
266C4—42no—
267C4—104no—
268C4—202no—
269C4—202no—
270C2—82no—
271C2—204no—
272C2#6592911yescharacteristic
273C2#6593911yescharacteristic
274C2#6594612yes—
275C2#6599712yes—
276C2#6600411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
hypercenter#394D5
hypocenterC5#2699
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#4030C2semidirect
2. C5×#394C2semidirect
3. #65512C2semidirect
4. #65277C2semidirect
5. #65272C2semidirect
6. #65250C2semidirect
7. #65245C2semidirect
8. #65234C2semidirect
9. #64953C2semidirect
52. C2×D20D4semidirect
70. #394D5semidirect
243. C5#2699semidirect

Maximal subgroups

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

subgroupindex
C2×#40302
C5×#3942
#655122
#652772
#652722
#652502
#652452
#652342
#649532
#26995