Finite Group Databasebeta

← #12000623

Subgroups — #12000623

Order 1280.

600 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 95 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×#65378C212yesmaximal, complemented
2#10605985C211yesmaximal, characteristic
3C5×#1960C211yesmaximal, characteristic, complemented
4C2×#64684C212yesmaximal, complemented
5#10598121C211yesmaximal, characteristic, complemented
6C2×C5×C4⋊Q8C2211yescharacteristic
7C2×#4021C2212yes—
8C2×C5×#103—22no—
9C2×C5×#97C2211yescharacteristic
10C5×#382C2211yescharacteristic
11#65378—24no—
12#65378—24no—
13#65280—22no—
14C2×#3942—22no—
15C2×#3941—22no—
16C2×#3935C2212yes—
17#64962—22no—
18#64960—22no—
19#64684—24no—
20#10085—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×D20—22no—
23C22×D20D412yes—
24C22×Dic10D412yes—
25C22×C5×C8—21no—
26C5×C4⋊Q8—22no—
27C5×C4⋊Q8—21no—
28C5×C4⋊Q8D412yescomplemented
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4D412yes—
31C2×C42×C5C2311yescharacteristic
32C2×#685—22no—
33C2×#685—22no—
34C2×#685—24no—
35C22×#168—22no—
36#4021—22no—
37#4021—22no—
38#4021—22no—
39#4021—22no—
40C5×#103—44no—
41C5×#97—22no—
42C5×#95—22no—
43#3963—22no—
44#3942—28no—
45#3941—28no—
46#3935—22no—
47#3935D414yescomplemented
48C2×#489—52no—
49#2819—51no—
50#1960D511yescharacteristic, complemented
51C2×#329—52no—
52#1295—51no—
53C23×D5—22no—
54C2×C5×Q8—22no—
55C2×C5×Q8—42no—
56C2×C5×Q8—42no—
57C2×C5×Q8—41no—
58C22×C4×C5—22no—
59C22×C4×C5—41no—
60C22×C4×C5C2×D412yes—
61C22×C4×C5C2×D411yescharacteristic
62C22×Dic5—22no—
63C2×D20—22no—
64C2×D20—22no—
65C2×D20—44no—
66C2×D20—82no—
67C2×D20—84no—
68C2×Dic10—22no—
69C2×Dic10—22no—
70C2×Dic10—44no—
71C2×C5×C8—22no—
72C2×C5×C8—41no—
73C2×C5×C8—41no—
74C5×C4⋊C4—24no—
75C5×C4⋊C4—42no—
76C5×C4⋊C4—42no—
77C42×C5C2×D411yescharacteristic
78C42×C5C2×D412yes—
79C42×C5#4111yescharacteristic
80#685—42no—
81#685—42no—
82#685—42no—
83#685—42no—
84#685—84no—
85C2×#168—24no—
86C2×#168—44no—
87C2×C4⋊Q8C2×D511yescharacteristic
88C2×#124—52no—
89#489—104no—
90#489—104no—
91C2×#103—22no—
92C2×#103—102no—
93C2×#102—102no—
94C2×#97—52no—
95C2×#97C2×D511yescharacteristic
96#394—102no—
97#393—102no—
98#392—102no—
99#382C2×D511yescharacteristic
100#329—104no—
101C23×C5#12011yescharacteristic
102C22×D5—42no—
103C22×D5—42no—
104C22×D5—42no—
105C22×D5—44no—
106C22×D5—44no—
107C5×Q8—44no—
108C5×Q8—82no—
109C2×C4×C5—21no—
110C2×C4×C5—21no—
111C2×C4×C5—24no—
112C2×C4×C5—42no—
113C2×C4×C5—42no—
114C2×C4×C5—44no—
115C2×C4×C5—81no—
116C2×C4×C5—81no—
117C2×C4×C5#12012yes—
118C2×C4×C5#12112yes—
119C2×C4×C5#12111yescharacteristic
120C2×C4×C5#12711yescharacteristic
121C2×Dic5—42no—
122C2×Dic5—44no—
123D20—44no—
124D20—82no—
125D20—88no—
126Dic10—44no—
127Dic10—82no—
128C5×C8—42no—
129#168—44no—
130C22×Q8—21no—
131C22×Q8—52no—
132C22×D4—52no—
133C22×D4—102no—
134C22×C8—21no—
135C22×C8—102no—
136C4⋊Q8—21no—
137C4⋊Q8—22no—
138C4⋊Q8D2012yescomplemented
139#124—102no—
140#124—102no—
141#124—102no—
142#124—102no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C22⋊D512yes—
145C2×C22⋊C4—102no—
146C2×C22⋊C4—102no—
147C2×C22⋊C4—104no—
148C2×C42C22×D511yescharacteristic
149#103—44no—
150#103—108no—
151#102—108no—
152#97—22no—
153#97—54no—
154#97—102no—
155#95—22no—
156#95—102no—
157C22×C5#39211yescharacteristic
158C22×C5#45311yescharacteristic
159C22×C5#45411yescharacteristic
160C22×C5#45514yes—
161C2×D5—84no—
162C2×D5—84no—
163C2×D5—82no—
164C2×D5—82no—
165C2×D5—82no—
166C4×C5—22no—
167C4×C5—22no—
168C4×C5—21no—
169C4×C5—21no—
170C4×C5—82no—
171C4×C5—82no—
172Dic5—82no—
173C24—102no—
174C2×Q8—22no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—41no—
178C2×Q8—102no—
179C2×Q8—102no—
180C2×Q8—204no—
181C2×D4—102no—
182C2×D4—102no—
183C2×D4—204no—
184C2×D4—402no—
185C2×D4—404no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4—102no—
189C22×C4D4×D512yes—
190C22×C4C2×C22⋊D511yescharacteristic
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—104no—
195C2×C8—204no—
196C4⋊C4—24no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C22⋊C4—202no—
200C22⋊C4—202no—
201C22⋊C4—202no—
202C22⋊C4—202no—
203C22⋊C4—404no—
204C42C2×D2011yescharacteristic
205C42#70911yescharacteristic
206C42D4×D512yes—
207C2×C5#165912yes—
208C2×C5#166212yes—
209C2×C5#200412yes—
210C2×C5#200611yescharacteristic
211D5—162no—
212C23—202no—
213C23—202no—
214C23—204no—
215C23—204no—
216C23—202no—
217C23#408311yescharacteristic
218Q8—44no—
219Q8—82no—
220Q8—204no—
221Q8—402no—
222D4—204no—
223D4—402no—
224D4—408no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—204no—
234C2×C4—202no—
235C2×C4#402812yes—
236C2×C4#403012yes—
237C2×C4#407611yescharacteristic
238C2×C4#408611yescharacteristic
239C8—42no—
240C8—204no—
241C5#1008511yescharacteristic, complemented, Sylow
242C22—402no—
243C22—404no—
244C22—404no—
245C22—402no—
246C22—402no—
247C22#6524911yescharacteristic
248C22#6535111yescharacteristic
249C22#6536611yescharacteristic
250C22#6549114yes—
251C4—21no—
252C4—21no—
253C4—22no—
254C4—22no—
255C4—82no—
256C4—82no—
257C4—402no—
258C2—802no—
259C2#1060045711yescharacteristic
260C2#1060113712yes—
261C2#1060204412yes—
262C2#1060212212yes—
subgroupG/H
centerC23#4083
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#1960C2
socleC23×C5#120
prime radical#10605985C2
hypercenter#1960D5
hypocenterC5#10085
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. C2×#65378C2semidirect
3. C5×#1960C2semidirect
4. C2×#64684C2semidirect
5. #10598121C2semidirect
28. C5×C4⋊Q8D4semidirect
47. #3935D4semidirect
50. #1960D5semidirect
138. C4⋊Q8D20semidirect
241. C5#10085semidirect

Maximal subgroups

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

subgroupindex
C2×#653782
#106059852
C5×#19602
C2×#646842
#105981212
#100855