Finite Group Databasebeta

← #10608898

Subgroups — #10608898

Order 640.

508 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C5×#394C211yesmaximal, characteristic, complemented
2#65512C212yesmaximal, complemented
3#65277C211yesmaximal, characteristic, complemented
4#65275C211yesmaximal, characteristic, complemented
5#65250C212yesmaximal, complemented
6#65250C212yesmaximal, complemented
7#65249C212yesmaximal, complemented
8#65240C212yesmaximal, complemented
9#64963C212yesmaximal, complemented
10C22×C5×D4C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×C22⋊D5C2212yes—
13C22×D20C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C22⋊C4C2212yes—
16C2×C5×C22⋊C4C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2212yes—
24C2×C4⋊Dic5C2211yescharacteristic
25C2×C4×Dic5C2211yescharacteristic
26C5×#95C2211yescharacteristic
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2211yescharacteristic
31#2828—51nomaximal, Sylow
32C24×C5C2312yes—
33C23×D5C2312yes—
34C2×C5×D4—42no—
35C2×C5×D4—41no—
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2312yes—
38C22×C4×C5C2311yescharacteristic
39C22×C4×C5C2311yescharacteristic
40C2×C22⋊D5—42no—
41C2×C22⋊D5—42no—
42C2×C22⋊D5—42no—
43C2×C22⋊D5—42no—
44C2×C22⋊D5—42no—
45C2×C22⋊D5—42no—
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C2×D20—22no—
50C2×D20—41no—
51C2×D20—41no—
52C5×C4⋊C4—41no—
53C5×C22⋊C4—24no—
54C5×C22⋊C4—42no—
55C22⋊Dic5—42no—
56C22⋊Dic5—42no—
57#685—42no—
58#685—42no—
59#685—42no—
60#685—42no—
61C4⋊Dic5—42no—
62C4⋊Dic5—41no—
63C4×Dic5—41no—
64#400—51no—
65#397—52no—
66#394—52no—
67#394—52no—
68#394D511yescharacteristic, complemented
69#392—52no—
70#392—52no—
71#390—51no—
72#388—52no—
73C23×C5—22no—
74C23×C5—22no—
75C23×C5—22no—
76C23×C5—22no—
77C23×C5—22no—
78C23×C5—22no—
79C23×C5—22no—
80C23×C5C2411yescharacteristic
81C22×D5—22no—
82C22×D5—22no—
83C22×D5—22no—
84C22×D5—22no—
85C22×D5—22no—
86C22×D5—22no—
87C22×D5—22no—
88C5×D4—44no—
89C2×C4×C5—21no—
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—21no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—21no—
98C2×C4×C5#4112yes—
99C22⋊D5—48no—
100C22⋊D5—48no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110D20—44no—
111C22×D4—52no—
112C22×D4—52no—
113C22×D4—51no—
114C22×D4C2×D511yescharacteristic
115C2×C4⋊C4—52no—
116C2×C4⋊C4—51no—
117C2×C4⋊C4C2×D511yescharacteristic
118C2×C22⋊C4—52no—
119C2×C22⋊C4—52no—
120C2×C22⋊C4—52no—
121C2×C22⋊C4—52no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4—52no—
124C2×C22⋊C4C2×D512yes—
125C2×C22⋊C4C2×D512yes—
126C2×C42—51no—
127#95—52no—
128#95—52no—
129#95—52no—
130#95—51no—
131#95C2×D511yescharacteristic
132C22×C5—42no—
133C22×C5—42no—
134C22×C5—42no—
135C22×C5—42no—
136C22×C5—42no—
137C22×C5—42no—
138C22×C5—42no—
139C22×C5C2×#4111yescharacteristic
140C22×C5H(5,𝔽2)11yescharacteristic
141C22×C5H(5,𝔽2)12yes—
142C22×C5H(5,𝔽2)11yescharacteristic
143C22×C5H(5,𝔽2)12yes—
144C2×D5—42no—
145C2×D5—42no—
146C2×D5—42no—
147C2×D5—42no—
148C2×D5—42no—
149C2×D5—42no—
150C2×D5—42no—
151C4×C5—22no—
152C4×C5—42no—
153C4×C5—41no—
154C4×C5—41no—
155Dic5—42no—
156Dic5—42no—
157Dic5—42no—
158C24—52no—
159C24C22×D512yes—
160C2×D4—42no—
161C2×D4—41no—
162C2×D4—102no—
163C2×D4—201no—
164C2×D4—201no—
165C2×D4—202no—
166C2×D4—202no—
167C2×D4—202no—
168C2×D4—202no—
169C2×D4—202no—
170C2×D4—202no—
171C22×C4—52no—
172C22×C4—52no—
173C22×C4—52no—
174C22×C4C22×D511yescharacteristic
175C22×C4C22×D512yes—
176C22×C4C22×D511yescharacteristic
177C22×C4C22×D511yescharacteristic
178C4⋊C4—41no—
179C4⋊C4—201no—
180C4⋊C4—202no—
181C22⋊C4—24no—
182C22⋊C4—42no—
183C22⋊C4—202no—
184C22⋊C4—202no—
185C22⋊C4—202no—
186C22⋊C4—202no—
187C22⋊C4—202no—
188C22⋊C4—202no—
189C42—201no—
190C2×C5—82no—
191C2×C5#53811yescharacteristic
192C2×C5#54012yes—
193C2×C5#56011yescharacteristic
194C2×C5#56012yes—
195C2×C5H(3,𝔽4)11yescharacteristic
196D5—82no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—102no—
205C23—102no—
206C23—102no—
207C23—102no—
208C23—102no—
209C23—102no—
210C23—102no—
211C23C23×D511yescharacteristic
212D4—44no—
213D4—204no—
214D4—208no—
215D4—208no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—102no—
226C2×C4—102no—
227C2×C4—102no—
228C2×C4—102no—
229C2×C4—102no—
230C2×C4—102no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4#71112yes—
235C5#282811yescharacteristic, complemented, Sylow
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—202no—
244C22—202no—
245C22—202no—
246C22—202no—
247C22—202no—
248C22—202no—
249C22—202no—
250C22C2×#71111yescharacteristic
251C22#414411yescharacteristic
252C22#414412yes—
253C22#414911yescharacteristic
254C22#414912yes—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—202no—
260C4—202no—
261C4—202no—
262C2—82no—
263C2—402no—
264C2#6592911yescharacteristic
265C2#6595111yescharacteristic
266C2#6595312yes—
267C2#6600012yes—
268C2#6601711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
hypercenter#394D5
hypocenterC5#2828
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C5×#394C2semidirect
2. #65512C2semidirect
3. #65277C2semidirect
4. #65275C2semidirect
5. #65250C2semidirect
6. #65250C2semidirect
7. #65249C2semidirect
8. #65240C2semidirect
9. #64963C2semidirect
68. #394D5semidirect
235. C5#2828semidirect

Maximal subgroups

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

subgroupindex
C5×#3942
#655122
#652772
#652752
#652502
#652492
#652402
#649632
#28285