Finite Group Databasebeta

← #12000624

Subgroups — #12000624

Order 1280.

600 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2819C211yesmaximal, characteristic
2C2×#65344C212yesmaximal, complemented
3C2×#64760C212yesmaximal, complemented
4#10600345C211yesmaximal, characteristic, complemented
5#10598121C211yesmaximal, characteristic, complemented
6C2×C5×#124C2212yes—
7C2×#4015C2211yescharacteristic
8C5×#394—22no—
9C5×#393—22no—
10C5×#392—22no—
11C5×#382C2211yescharacteristic
12C2×#3967—22no—
13C2×#3964—22no—
14#65344—24no—
15#65344—24no—
16C2×#3942—22no—
17C2×#3935C2211yescharacteristic
18C2×#3935C2212yes—
19#64760—24no—
20#10085—51nomaximal, Sylow
21C22×C5×Q8D412yes—
22C22×C5×D4—22no—
23C22×C5×D4D412yes—
24C22×Dic10—21no—
25C5×#124—22no—
26C5×#124—22no—
27C5×#124—22no—
28C5×#124—22no—
29C2×C5×C4⋊C4D412yes—
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4—22no—
32C2×C5×C22⋊C4—24no—
33C2×C42×C5C2311yescharacteristic
34C2×C4⋊Dic5—22no—
35C22×#168—22no—
36C22×#168—21no—
37#4015—22no—
38#4015—21no—
39#4015D412yescomplemented
40C5×#95—22no—
41C5×#95—22no—
42#3967—28no—
43#3964—28no—
44#3942—44no—
45#3935—22no—
46#3935—22no—
47#3935D414yescomplemented
48C2×#489—52no—
49#2819D511yescharacteristic
50#1960—51no—
51C2×#329—52no—
52#1295—51no—
53C24×C5—22no—
54C2×C5×Q8—22no—
55C2×C5×Q8—22no—
56C2×C5×Q8—44no—
57C2×C5×D4—22no—
58C2×C5×D4—22no—
59C2×C5×D4—44no—
60C2×C5×D4—82no—
61C2×C5×D4—84no—
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5C2×D412yes—
65C22×C4×C5C2×D411yescharacteristic
66C22×Dic5—41no—
67C2×Dic10—22no—
68C2×Dic10—42no—
69C2×Dic10—42no—
70C2×Dic10—41no—
71C5×C4⋊C4—42no—
72C5×C22⋊C4—42no—
73C5×C22⋊C4—42no—
74C5×C22⋊C4—42no—
75C5×C22⋊C4—42no—
76C5×C22⋊C4—84no—
77C42×C5C2×D411yescharacteristic
78C42×C5C2×D412yes—
79C42×C5#4111yescharacteristic
80C4⋊Dic5—24no—
81C4⋊Dic5—42no—
82C2×#168—22no—
83C2×#168—24no—
84C2×#168—41no—
85C2×#168—44no—
86C2×#168—41no—
87C2×C4⋊Q8—51no—
88C2×#124C2×D512yes—
89#489—104no—
90#489—104no—
91C2×#103—102no—
92C2×#103—102no—
93C2×#102—102no—
94C2×#97—52no—
95C2×#97—51no—
96#394—22no—
97#393—22no—
98#392—22no—
99#382C2×D511yescharacteristic
100#329—104no—
101C23×C5—42no—
102C23×C5—42no—
103C23×C5—42no—
104C23×C5—44no—
105C23×C5—44no—
106C23×C5#12011yescharacteristic
107C5×Q8—44no—
108C5×Q8—82no—
109C5×D4—44no—
110C5×D4—82no—
111C5×D4—88no—
112C2×C4×C5—21no—
113C2×C4×C5—21no—
114C2×C4×C5—24no—
115C2×C4×C5—42no—
116C2×C4×C5—42no—
117C2×C4×C5—44no—
118C2×C4×C5—44no—
119C2×C4×C5#12012yes—
120C2×C4×C5#12111yescharacteristic
121C2×C4×C5#12112yes—
122C2×C4×C5#12711yescharacteristic
123C2×Dic5—42no—
124C2×Dic5—81no—
125C2×Dic5—81no—
126Dic10—44no—
127Dic10—82no—
128#168—44no—
129#168—42no—
130C22×Q8—101no—
131C22×Q8C22⋊D512yes—
132C22×D4—22no—
133C22×D4C22⋊D512yes—
134C22×C8—102no—
135C22×C8—101no—
136C4⋊Q8—52no—
137C4⋊Q8—102no—
138C4⋊Q8—101no—
139#124—22no—
140#124—22no—
141#124—22no—
142#124—22no—
143C2×C4⋊C4—102no—
144C2×C4⋊C4C22⋊D512yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—24no—
148C2×C42C22×D511yescharacteristic
149#103—108no—
150#103—204no—
151#102—108no—
152#97—54no—
153#97—102no—
154#97—102no—
155#95—22no—
156#95—22no—
157C22×C5—84no—
158C22×C5—84no—
159C22×C5—82no—
160C22×C5—82no—
161C22×C5—82no—
162C22×C5#39211yescharacteristic
163C22×C5#45311yescharacteristic
164C22×C5#45411yescharacteristic
165C22×C5#45514yes—
166C4×C5—22no—
167C4×C5—22no—
168C4×C5—21no—
169C4×C5—21no—
170C4×C5—82no—
171C4×C5—82no—
172Dic5—82no—
173C24—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—44no—
177C2×Q8—102no—
178C2×Q8—202no—
179C2×Q8—202no—
180C2×Q8—201no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—44no—
184C2×D4—82no—
185C2×D4—84no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—201no—
189C22×C4C2×C22⋊D512yes—
190C22×C4C2×C22⋊D511yescharacteristic
191C2×C8—102no—
192C2×C8—104no—
193C2×C8—204no—
194C2×C8—201no—
195C2×C8—201no—
196C4⋊C4—42no—
197C4⋊C4—104no—
198C4⋊C4—202no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—84no—
204C42D4×D511yescharacteristic
205C42D4×D512yes—
206C42#71111yescharacteristic
207C2×C5—162no—
208C2×C5#165912yes—
209C2×C5#166212yes—
210C2×C5#200412yes—
211C2×C5#200611yescharacteristic
212C23—42no—
213C23—42no—
214C23—44no—
215C23—44no—
216C23—42no—
217C23#409911yescharacteristic
218Q8—44no—
219Q8—82no—
220Q8—204no—
221Q8—402no—
222D4—44no—
223D4—82no—
224D4—88no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—202no—
233C2×C4—401no—
234C2×C4—401no—
235C2×C4#408312yes—
236C2×C4#408411yescharacteristic
237C2×C4#408512yes—
238C2×C4#408611yescharacteristic
239C8—204no—
240C8—202no—
241C5#1008511yescharacteristic, complemented, Sylow
242C22—82no—
243C22—82no—
244C22—82no—
245C22—84no—
246C22—84no—
247C22#6551211yescharacteristic
248C22#6553011yescharacteristic
249C22#6553114yes—
250C22#6553411yescharacteristic
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—82no—
256C4—82no—
257C4—402no—
258C2—162no—
259C2#1060265012yes—
260C2#1060267211yescharacteristic
261C2#1060274212yes—
262C2#1060274712yes—
subgroupG/H
centerC23#4099
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
Fitting, prime radicalC5×#2819C2
socleC23×C5#120
hypercenter#2819D5
hypocenterC5#10085
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2819

Complements

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

NH ≅ G/Ntype
2. C2×#65344C2semidirect
3. C2×#64760C2semidirect
4. #10600345C2semidirect
5. #10598121C2semidirect
39. #4015D4semidirect
47. #3935D4semidirect
241. C5#10085semidirect

Maximal subgroups

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

subgroupindex
C5×#28192
C2×#653442
C2×#647602
#106003452
#105981212
#100855