Finite Group Databasebeta

← #12012625

Subgroups — #12012625

Order 1280.

376 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 59 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×#3192C211yesmaximal, characteristic
2#10601881C212yesmaximal, complemented
3#10600780C211yesmaximal, characteristic, complemented
4#10598270C212yesmaximal, complemented
5#10598172C211yesmaximal, characteristic, complemented
6C2×C5×#121—21no—
7C5×#481—22no—
8C5×#469—21no—
9C5×#469C2212yes—
10C5×#447—22no—
11C5×#407C2211yescharacteristic
12#65496—21no—
13#65414C2211yescharacteristic, complemented
14#65413—21no—
15#64783—22no—
16#64780C2211yescharacteristic, complemented
17#64773C2212yescomplemented
18#64707—22no—
19#12298—51nomaximal, Sylow
20C22×C5×D4—21no—
21C22×C5×D4—41no—
22C23×C4×C5D411yescharacteristic
23C2×C5×D8—21no—
24C2×C5×D8D412yes—
25C2×C5×M16D411yescharacteristic
26C2×C5×M16C2311yescharacteristic
27C5×#121—22no—
28C5×#121—22no—
29C5×#121—41no—
30C5×#121—41no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C22⋊C4—41no—
33C2×#681—21no—
34C2×#681—21no—
35C5×#106—21no—
36C5×#106D412yes—
37C5×#102—22no—
38C5×#102—22no—
39C5×C22⋊C8—22no—
40C22⋊#168—42no—
41#3954—42no—
42#3951—22no—
43#3944—22no—
44#3192D511yescharacteristic
45#2238—52no—
46#2091—51no—
47#1341—52no—
48#1320—51no—
49C24×C5—41no—
50C24×C5SD1611yescharacteristic
51C2×C5×D4—22no—
52C2×C5×D4—41no—
53C2×C5×D4—42no—
54C2×C5×D4—42no—
55C2×C5×D4—81no—
56C2×C5×D4—81no—
57C2×C5×D4—84no—
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—41no—
61C22×C4×C5—41no—
62C22×C4×C5—41no—
63C22×C4×C5SD1611yescharacteristic
64C22×C4×C5C2×D411yescharacteristic
65C5×D8—42no—
66C5×D8—81no—
67C5×M16—41no—
68C5×M16—41no—
69C2×C5×C8—21no—
70C2×C5×C8C2×D412yes—
71C5×C4⋊C4—22no—
72C5×C4⋊C4—41no—
73C5×C22⋊C4—42no—
74C5×C22⋊C4—42no—
75#681—41no—
76#681—41no—
77#681—41no—
78#681—41no—
79C2×#168—22no—
80C2×#168—41no—
81#672—42no—
82C2×#121—21no—
83#481—22no—
84#469—21no—
85#469C2×D512yes—
86#447—22no—
87#430—51no—
88#430—101no—
89#407—101no—
90#407C2×D511yescharacteristic
91#365—102no—
92#361—102no—
93#349—52no—
94#338—51no—
95C23×C5—21no—
96C23×C5—41no—
97C23×C5—42no—
98C23×C5—42no—
99C23×C5—82no—
100C23×C5—82no—
101C23×C5—82no—
102C23×C5C2×SD1611yescharacteristic
103C5×D4—42no—
104C5×D4—42no—
105C5×D4—82no—
106C5×D4—84no—
107C5×D4—84no—
108C2×C4×C5—21no—
109C2×C4×C5—21no—
110C2×C4×C5—21no—
111C2×C4×C5—41no—
112C2×C4×C5—41no—
113C2×C4×C5—41no—
114C2×C4×C5—42no—
115C2×C4×C5—81no—
116C2×C4×C5—82no—
117C2×C4×C5#12011yescharacteristic
118C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
119C5×C8—22no—
120C5×C8—41no—
121#168—42no—
122#168—81no—
123C22×D4—21no—
124C22×D4—41no—
125C23×C4C22⋊D511yescharacteristic
126C2×D8—21no—
127C2×D8C22⋊D512yes—
128C2×M16—101no—
129C2×M16—101no—
130C2×M16C22⋊D511yescharacteristic
131C2×M16C22×D511yescharacteristic
132#121—22no—
133#121—22no—
134#121—41no—
135#121—41no—
136C2×C4⋊C4—21no—
137C2×C22⋊C4—41no—
138M32—102no—
139#108—102no—
140#108—202no—
141#106—21no—
142#106C22⋊D512yes—
143#102—22no—
144#102—22no—
145C22⋊C8—22no—
146C22⋊C8—202no—
147C22×C5—22no—
148C22×C5—21no—
149C22×C5—41no—
150C22×C5—41no—
151C22×C5—42no—
152C22×C5—41no—
153C22×C5—82no—
154C22×C5—82no—
155C22×C5—81no—
156C22×C5—82no—
157C22×C5—164no—
158C22×C5#45011yescharacteristic
159C4×C5—21no—
160C4×C5—41no—
161C4×C5—82no—
162C4×C5#45311yescharacteristic
163C4×C5#45811yescharacteristic
164C24—41no—
165C24#68711yescharacteristic
166C2×D4—22no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—84no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4#68711yescharacteristic
179C22×C4C2×C22⋊D511yescharacteristic
180D8—42no—
181D8—81no—
182M16—41no—
183M16—41no—
184M16—201no—
185M16—201no—
186M16—201no—
187M16—201no—
188C2×C8—21no—
189C2×C8—102no—
190C2×C8—201no—
191C2×C8C2×C22⋊D512yes—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C16—202no—
197C2×C5—21no—
198C2×C5—41no—
199C2×C5—42no—
200C2×C5—162no—
201C2×C5#161311yescharacteristic
202C23—21no—
203C23—41no—
204C23—42no—
205C23—42no—
206C23—82no—
207C23—82no—
208C23—82no—
209C23C2×#68711yescharacteristic
210D4—42no—
211D4—42no—
212D4—82no—
213D4—84no—
214D4—84no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—81no—
223C2×C4—82no—
224C2×C4#407811yescharacteristic
225C2×C4#409911yescharacteristic
226C8—22no—
227C8—41no—
228C8—202no—
229C8—401no—
230C5#1229811yescharacteristic, complemented, Sylow
231C22—22no—
232C22—21no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—82no—
238C22—82no—
239C22—81no—
240C22—82no—
241C22—164no—
242C22#6550711yescharacteristic
243C4—21no—
244C4—41no—
245C4—82no—
246C4#6553011yescharacteristic
247C4#6553211yescharacteristic
248C2—21no—
249C2—42no—
250C2—41no—
251C2—162no—
252C2#1060263711yescharacteristic
subgroupG/H
centerC2#10602637
commutator, cosocleC2×C5×M16C23
FrattiniC2×M16C22×D5
Fitting, prime radicalC5×#3192C2
socleC2×C5#1613
hypercenter#3192D5
hypocenterC5#12298
LCSG ▷ C2×C5×M16 ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3192

Complements

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

NH ≅ G/Ntype
2. #10601881C2semidirect
3. #10600780C2semidirect
4. #10598270C2semidirect
5. #10598172C2semidirect
13. #65414C22semidirect
16. #64780C22semidirect
17. #64773C22semidirect
230. C5#12298semidirect

Maximal subgroups

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

subgroupindex
C5×#31922
#106018812
#106007802
#105982702
#105981722
#122985