Finite Group Databasebeta

← #12030356

Subgroups — #12030356

Order 1280.

1530 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 411 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
1C22×C5×#126C211yesmaximal, characteristic
2C2×#65224C211yesmaximal, characteristic, complemented
3C2×#65226C211yesmaximal, characteristic, complemented
4#10602358C216yesmaximal, complemented
5#10602339C216yesmaximal
6C2×C5×#126—24no—
7C2×C5×#126—24no—
8C2×C5×#126C418yescomplemented
9C22×C5×C4⋊C4C2213yes—
10C22×C5×C22⋊C4C2213yes—
11C22×C42×C5C2211yescharacteristic
12C22×C22⋊Dic5C2214yes—
13C22×C4⋊Dic5C2216yes—
14C2×#3963C2216yes—
15C2×#3963C22112yes—
16#65226—24no—
17#65224—24no—
18#14665—51nomaximal, Sylow
19C25×C5C2311yescharacteristic
20C23×C4×C5C2313yes—
21C23×C4×C5C2313yes—
22C23×Dic5C2314yes—
23C23×Dic5C2314yes—
24C5×#126—232no—
25C2×C5×C4⋊C4—26no—
26C2×C5×C4⋊C4—26no—
27C2×C5×C4⋊C4C2×C4112yes—
28C2×C5×C22⋊C4—26no—
29C2×C5×C22⋊C4—26no—
30C2×C5×C22⋊C4C2×C4112yes—
31C2×C42×C5C2×C414yes—
32C2×C42×C5D414yes—
33C2×C42×C5D414yes—
34C2×C22⋊Dic5—412no—
35C2×C4⋊Dic5—212no—
36C2×C4⋊Dic5—412no—
37#3963—212no—
38#3963—412no—
39C22×#126D511yescharacteristic
40C2×#384—51no—
41C2×#383—51no—
42#1441—56no—
43#1439—56no—
44C24×C5—21no—
45C24×C5—21no—
46C24×C5—26no—
47C24×C5—26no—
48C24×C5C22×C412yes—
49C24×C5C2411yescharacteristic
50C22×C4×C5—26no—
51C22×C4×C5—26no—
52C22×C4×C5—23no—
53C22×C4×C5—23no—
54C22×C4×C5—26no—
55C22×C4×C5—26no—
56C22×C4×C5C22×C416yes—
57C22×C4×C5C22×C416yes—
58C22×C4×C5C2×D416yes—
59C22×C4×C5C2×D416yes—
60C22×Dic5—24no—
61C22×Dic5—24no—
62C22×Dic5—212no—
63C22×Dic5—212no—
64C22×Dic5—212no—
65C22×Dic5—212no—
66C5×C4⋊C4—224no—
67C5×C22⋊C4—224no—
68C42×C5C22⋊C4116yescomplemented
69C22⋊Dic5—416no—
70C4⋊Dic5—424no—
71C2×#126—24no—
72C2×#126—24no—
73C2×#126Dic518yescomplemented
74C22×C4⋊C4—56no—
75C22×C4⋊C4C2×D513yes—
76C22×C22⋊C4—54no—
77C22×C22⋊C4C2×D513yes—
78C22×C42C2×D511yescharacteristic
79#384—104no—
80#383—104no—
81C2×#95—56no—
82C2×#95—512no—
83C23×C5—22no—
84C23×C5—212no—
85C23×C5—43no—
86C23×C5—43no—
87C23×C5—46no—
88C23×C5—44no—
89C23×C5—412no—
90C23×C5C23×C411yescharacteristic
91C23×C5C22×D411yescharacteristic
92C23×C5C22×D411yescharacteristic
93C23×C5H(5,𝔽2)16yes—
94C23×C52−1+416yes—
95C2×C4×C5—212no—
96C2×C4×C5—26no—
97C2×C4×C5—26no—
98C2×C4×C5—212no—
99C2×C4×C5—26no—
100C2×C4×C5—212no—
101C2×C4×C5—43no—
102C2×C4×C5—43no—
103C2×C4×C5—46no—
104C2×C4×C5C2×C22⋊C4112yes—
105C2×Dic5—412no—
106C2×Dic5—412no—
107C2×Dic5—44no—
108C2×Dic5—44no—
109C2×Dic5—412no—
110C2×Dic5—412no—
111C25C22×D511yescharacteristic
112C23×C4—54no—
113C23×C4—54no—
114C23×C4C22×D513yes—
115C23×C4C22×D513yes—
116#126—232no—
117C2×C4⋊C4—26no—
118C2×C4⋊C4—26no—
119C2×C4⋊C4—1012no—
120C2×C4⋊C4—2012no—
121C2×C4⋊C4C2×Dic5112yes—
122C2×C22⋊C4—26no—
123C2×C22⋊C4—26no—
124C2×C22⋊C4—2012no—
125C2×C22⋊C4C2×Dic5112yes—
126C2×C42C2×Dic514yes—
127C2×C42C22⋊D514yes—
128C2×C42C22⋊D514yes—
129#95—1012no—
130#95—2012no—
131C22×C5—46no—
132C22×C5—48no—
133C22×C5—81no—
134C22×C5—83no—
135C22×C5—84no—
136C22×C5C22×C22⋊C411yescharacteristic
137C22×C5#52016yes—
138C22×C5#53513yes—
139C22×C5#53613yes—
140C22×C5#53716yes—
141C22×C5#56014yes—
142C22×C5#563112yes—
143C4×C5—212no—
144C4×C5—46no—
145Dic5—84no—
146Dic5—84no—
147C24—26no—
148C24—26no—
149C24—21no—
150C24—21no—
151C24C22×Dic512yes—
152C24C23×D511yescharacteristic
153C22×C4—26no—
154C22×C4—26no—
155C22×C4—26no—
156C22×C4—26no—
157C22×C4—23no—
158C22×C4—23no—
159C22×C4—1012no—
160C22×C4—1012no—
161C22×C4—1012no—
162C22×C4—1012no—
163C22×C4—104no—
164C22×C4—104no—
165C22×C4C22×Dic516yes—
166C22×C4C22×Dic516yes—
167C22×C4C2×C22⋊D516yes—
168C22×C4C2×C22⋊D516yes—
169C4⋊C4—224no—
170C4⋊C4—2024no—
171C22⋊C4—224no—
172C22⋊C4—2016no—
173C42C22⋊Dic5116yescomplemented
174C2×C5—82no—
175C2×C5#231113yes—
176C2×C5#238314yes—
177C2×C5#281811yescharacteristic
178C2×C5#282113yes—
179C2×C5#282314yes—
180C23—212no—
181C23—22no—
182C23—43no—
183C23—43no—
184C23—46no—
185C23—44no—
186C23—412no—
187C23#414916yes—
188C23#415016yes—
189C23C23×Dic511yescharacteristic
190C23C22×C22⋊D511yescharacteristic
191C23C22×C22⋊D511yescharacteristic
192C2×C4—26no—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—212no—
196C2×C4—212no—
197C2×C4—26no—
198C2×C4—43no—
199C2×C4—43no—
200C2×C4—46no—
201C2×C4—2012no—
202C2×C4—2012no—
203C2×C4—204no—
204C2×C4—204no—
205C2×C4—2012no—
206C2×C4—2012no—
207C2×C4C2×C22⋊Dic5112yes—
208C5#1466511yescharacteristic, complemented, Sylow
209C22—46no—
210C22—48no—
211C22—83no—
212C22—81no—
213C22—84no—
214C22#6604514yes—
215C22#66046112yes—
216C22#6615913yes—
217C22#6616116yes—
218C22#6616413yes—
219C22#6616716yes—
220C22C22×C22⋊Dic511yescharacteristic
221C4—212no—
222C4—46no—
223C4—404no—
224C4—404no—
225C2—82no—
226C2#1061480714yes—
227C2#1061483413yes—
228C2#1061484911yescharacteristic
229C2#1061485014yes—
230C2#1061581813yes—
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC22×C5×#126C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC22×#126D5
hypocenterC5#14665
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
2. C2×#65224C2semidirect
3. C2×#65226C2semidirect
4. #10602358C2semidirect
8. C2×C5×#126C4semidirect
68. C42×C5C22⋊C4semidirect
73. C2×#126Dic5semidirect
173. C42C22⋊Dic5semidirect
208. C5#14665semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#1262
C2×#652242
C2×#652262
#106023582
#106023392
#146655