Finite Group Databasebeta

← #12012046

Subgroups — #12012046

Order 1280.

388 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2817C211yesmaximal, characteristic
2#10600503C212yesmaximal
3#10600329C211yesmaximal, characteristic
4#10598140C212yesmaximal
5#10598118C211yesmaximal, characteristic
6C2×C5×C22⋊Q8—21no—
7#65817C2211yescharacteristic
8C5×#400—22no—
9C5×#397—21no—
10C5×#397C2212yes—
11C5×#395—22no—
12C5×#381C2211yescharacteristic
13#65500—21no—
14#65260—22no—
15#64747—21no—
16#64746C2211yescharacteristic
17#64745C2212yes—
18#64672—22no—
19#12207—51nomaximal, Sylow
20C22×C5×Q8—21no—
21C23×C4×C5D411yescharacteristic
22C5×C22⋊Q8—42no—
23C5×C22⋊Q8—42no—
24C2×C5×C4⋊C4—22no—
25C2×C5×C4⋊C4—21no—
26C2×C5×C4⋊C4—21no—
27C2×C5×C4⋊C4D412yes—
28C2×C5×C22⋊C4—22no—
29C2×C5×C22⋊C4D411yescharacteristic
30C2×C5×C22⋊C4C2311yescharacteristic
31C2×C22⋊Dic5—21no—
32C2×C22⋊Dic5—21no—
33#4070—22no—
34#4024—42no—
35C5×#95—22no—
36C5×#95—21no—
37C5×#95—22no—
38C5×#95—22no—
39C5×#95D412yes—
40#3963—42no—
41C22⋊#168—22no—
42#3942—42no—
43#3940—42no—
44#2817D511yescharacteristic
45#2030—51no—
46#1608—52no—
47#1344—52no—
48#1317—51no—
49C24×C5C2×D411yescharacteristic
50C2×C5×Q8—42no—
51C2×C5×Q8—81no—
52C2×C5×Q8—81no—
53C22×C4×C5—21no—
54C22×C4×C5—21no—
55C22×C4×C5—22no—
56C22×C4×C5—22no—
57C22×C4×C5—21no—
58C22×C4×C5—41no—
59C22×C4×C5—41no—
60C22×C4×C5SD1612yes—
61C22×C4×C5C2×D412yes—
62C22×Dic5—22no—
63C22×Dic5—41no—
64C2×Dic10—22no—
65C5×C4⋊C4—42no—
66C5×C4⋊C4—42no—
67C5×C4⋊C4—44no—
68C5×C4⋊C4—81no—
69C5×C22⋊C4—41no—
70C5×C22⋊C4—42no—
71C5×C22⋊C4—42no—
72C5×C22⋊C4—41no—
73C22⋊Dic5—41no—
74C22⋊Dic5—41no—
75C22⋊Dic5—41no—
76C22⋊Dic5—41no—
77C4⋊Dic5—22no—
78#683—42no—
79C2×#168—42no—
80#543—51no—
81C2×C22⋊Q8—21no—
82#483—102no—
83#400—22no—
84#397—21no—
85#397C2×D512yes—
86#395—22no—
87#381—101no—
88#381C2×D511yescharacteristic
89#342—51no—
90#342—101no—
91#328—102no—
92#323—52no—
93C23×C5—21no—
94C23×C5—21no—
95C23×C5—42no—
96C23×C5#12011yescharacteristic
97C23×C5C2×SD1611yescharacteristic
98C23×C5AGL(1,ℤ/8)11yescharacteristic
99C5×Q8—84no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—41no—
103C2×C4×C5—41no—
104C2×C4×C5—42no—
105C2×C4×C5—42no—
106C2×C4×C5—44no—
107C2×C4×C5—42no—
108C2×C4×C5—42no—
109C2×C4×C5—42no—
110C2×C4×C5—41no—
111C2×C4×C5—42no—
112C2×C4×C5—42no—
113C2×C4×C5—81no—
114C2×Dic5—42no—
115C2×Dic5—82no—
116C2×Dic5—81no—
117C2×Dic5—82no—
118Dic10—82no—
119#168—82no—
120C22×Q8—21no—
121C23×C4C22⋊D511yescharacteristic
122C22⋊Q8—42no—
123C22⋊Q8—42no—
124C22⋊Q8—102no—
125C22⋊Q8—202no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4C22⋊D512yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—101no—
132C2×C22⋊C4—101no—
133C2×C22⋊C4C22⋊D511yescharacteristic
134C2×C22⋊C4C22×D511yescharacteristic
135#106—202no—
136#103—202no—
137C22⋊C8—102no—
138#95—22no—
139#95—21no—
140#95—22no—
141#95—22no—
142#95—202no—
143#95C22⋊D512yes—
144C22×C5—22no—
145C22×C5—21no—
146C22×C5—21no—
147C22×C5—41no—
148C22×C5—41no—
149C22×C5—42no—
150C22×C5—81no—
151C22×C5#45011yescharacteristic
152C22×C5#45511yescharacteristic
153C22×C5B(4,𝔽2)11yescharacteristic
154C4×C5—42no—
155C4×C5—42no—
156C4×C5—82no—
157C4×C5—81no—
158C4×C5—82no—
159Dic5—82no—
160Dic5—161no—
161C24C2×C22⋊D511yescharacteristic
162C2×Q8—42no—
163C2×Q8—81no—
164C2×Q8—81no—
165C2×Q8—102no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—102no—
174C22×C4—201no—
175C22×C4#68712yes—
176C22×C4C2×C22⋊D512yes—
177C2×C8—202no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—81no—
182C4⋊C4—102no—
183C4⋊C4—202no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—41no—
188C22⋊C4—201no—
189C22⋊C4—201no—
190C22⋊C4—201no—
191C22⋊C4—201no—
192C2×C5—21no—
193C2×C5—21no—
194C2×C5—41no—
195C2×C5—41no—
196C2×C5#160111yescharacteristic
197C2×C5#220112yes—
198C23—21no—
199C23—21no—
200C23—42no—
201C23#407811yescharacteristic
202C23C2×#68711yescharacteristic
203C23#409911yescharacteristic
204Q8—84no—
205Q8—402no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—202no—
221C2×C4—402no—
222C2×C4—401no—
223C2×C4—402no—
224C8—402no—
225C5#1220711yescharacteristic, complemented, Sylow
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#6550711yescharacteristic
234C22#6553111yescharacteristic
235C22#6553311yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—402no—
242C4—801no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#1060263911yescharacteristic
248C2#1060275112yes—
subgroupG/H
centerC22#65533
commutator, cosocleC2×C5×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2817C2
socleC22×C5B(4,𝔽2)
prime radicalC24×C5C2×D4
hypercenter#2817D5
hypocenterC5#12207
LCSG ▷ C2×C5×C22⋊C4 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
225. C5#12207semidirect

Maximal subgroups

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

subgroupindex
C5×#28172
#106005032
#106003292
#105981402
#105981182
#122075