Finite Group Databasebeta

← #12019144

Subgroups — #12019144

Order 1280.

292 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 81 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×#64736C211yesmaximal, characteristic
2#10602233C211yesmaximal, characteristic
3#10600696C211yesmaximal, characteristic
4#10600689C211yesmaximal, characteristic, complemented
5C5×#1378C211yesmaximal, characteristic, complemented
6#10598313C211yesmaximal, characteristic, complemented
7#10598312C211yesmaximal, characteristic, complemented
8C22×D40C2211yescharacteristic
9C2×C5×M32C2211yescharacteristic
10C2×C5×M32C2211yescharacteristic
11C5×#405C2211yescharacteristic
12C2×#3955—21no—
13C2×#3951C2211yescharacteristic
14C2×#3951C2211yescharacteristic
15#65395C2211yescharacteristic
16#65290—21no—
17#64776—21no—
18#64751—21no—
19#64736—22no—
20#64729—22no—
21#13490—51nomaximal, Sylow
22C22×D20—21no—
23C2×C5×M16D411yescharacteristic
24C22×C5×C8C2311yescharacteristic
25C5×#117D411yescharacteristic
26#4072—21no—
27C2×#681—21no—
28C2×#681—21no—
29C2×D40—21no—
30C2×D40—21no—
31C2×D40—21no—
32C2×D40—21no—
33C2×D40—21no—
34C2×D40D412yes—
35C5×M32—21no—
36C5×M32—21no—
37C5×M32—21no—
38C5×M32—21no—
39C2×C5×C16—21no—
40C2×C5×C16—21no—
41C5×#97—21no—
42C4×C5×C8—21no—
43#3955—42no—
44#3951—21no—
45#3951—21no—
46#3951—21no—
47#3951—21no—
48#3950—21no—
49#3949—21no—
50#3941—42no—
51C2×#361—51no—
52#2140—51no—
53#1978—51no—
54#1833—51no—
55#1389—51no—
56#1388—51no—
57#1378D511yescharacteristic, complemented
58C23×D5—41no—
59C22×C4×C5C2×D411yescharacteristic
60C2×D20—22no—
61C2×D20—41no—
62C2×D20—41no—
63C2×D20—41no—
64C2×D20—41no—
65C2×D20—41no—
66C5×M16—21no—
67C5×M16—21no—
68C2×C5×C8—21no—
69C2×C5×C8C2×D411yescharacteristic
70C2×C5×C8#4111yescharacteristic
71C2×C5×C8#4111yescharacteristic
72C2×C5×C8#4111yescharacteristic
73C2×C5×C8#4111yescharacteristic
74C2×C5×C8#4111yescharacteristic
75C5×C4⋊C4—21no—
76C5×C22⋊C4—21no—
77C42×C5—21no—
78C22⋊Dic5—41no—
79C4⋊Dic5—41no—
80C4×Dic5—41no—
81#681—41no—
82#681—41no—
83#681—41no—
84#681—41no—
85C2×#168—41no—
86C2×#168—41no—
87D40—22no—
88D40—22no—
89D40—41no—
90D40—41no—
91C5×C16—41no—
92C5×C16—41no—
93C22×D8—51no—
94C2×M32C2×D511yescharacteristic
95C2×M32C2×D511yescharacteristic
96C2×#108—51no—
97C2×#108—51no—
98#427—51no—
99#418—101no—
100C2×#100—101no—
101#405C2×D511yescharacteristic
102#361—102no—
103#357—102no—
104#344—101no—
105H(3,ℤ/4)—101no—
106C23×C5#13511yescharacteristic
107C22×D5—42no—
108C22×D5—81no—
109C22×D5—81no—
110C22×D5—162no—
111C2×C4×C5—41no—
112C2×C4×C5#12111yescharacteristic
113C2×C4×C5#12311yescharacteristic
114C2×C4×C5#12411yescharacteristic
115C2×C4×C5#13511yescharacteristic
116C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
118C2×Dic5—81no—
119D20—42no—
120D20—42no—
121D20—81no—
122D20—81no—
123C5×C8—41no—
124C5×C8#12311yescharacteristic
125C5×C8#12311yescharacteristic
126C5×C8#12611yescharacteristic
127C5×C8#12611yescharacteristic
128#168—81no—
129#168—81no—
130C22×D4—101no—
131C2×D8—52no—
132C2×D8—101no—
133C2×D8—101no—
134C2×D8—101no—
135C2×D8—101no—
136C2×D8—101no—
137C2×M16—101no—
138C2×M16—101no—
139C2×M16D2011yescharacteristic
140C22×C8C22×D511yescharacteristic
141#117—101no—
142#117D2011yescharacteristic
143M32—21no—
144M32—21no—
145M32—21no—
146M32—21no—
147C2×C16—21no—
148C2×C16—21no—
149#108—101no—
150#108—101no—
151#108—101no—
152#108—101no—
153#107—101no—
154#106—101no—
155#102—202no—
156#100—202no—
157#97—21no—
158C4×C8—21no—
159C22×C5—41no—
160C22×C5#46311yescharacteristic
161C22×C5#48211yescharacteristic
162C22×C5#48811yescharacteristic
163C2×D5—82no—
164C2×D5—162no—
165C2×D5—162no—
166C2×D5—162no—
167C4×C5—81no—
168C4×C5#39711yescharacteristic
169C4×C5#46311yescharacteristic
170C4×C5#48211yescharacteristic
171C4×C5#49011yescharacteristic
172Dic5—161no—
173C24—201no—
174C2×D4—102no—
175C2×D4—201no—
176C2×D4—201no—
177C2×D4—201no—
178C2×D4—201no—
179C2×D4—201no—
180C22×C4C2×D2011yescharacteristic
181D8—102no—
182D8—102no—
183D8—201no—
184D8—201no—
185M16—21no—
186M16—21no—
187M16—201no—
188M16—201no—
189M16—201no—
190M16—201no—
191C2×C8—21no—
192C2×C8—201no—
193C2×C8—201no—
194C2×C8#70911yescharacteristic
195C2×C8#70911yescharacteristic
196C2×C8D4×D511yescharacteristic
197C2×C8#71111yescharacteristic
198C2×C8#71111yescharacteristic
199C2×C8#71311yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—201no—
202C22⋊C4—21no—
203C22⋊C4—201no—
204C42—21no—
205C42—201no—
206C16—41no—
207C16—41no—
208C2×C5—21no—
209C2×C5—21no—
210C2×C5—21no—
211C2×C5#207311yescharacteristic
212D5—162no—
213C23—202no—
214C23—401no—
215C23—401no—
216C23—802no—
217C23#405011yescharacteristic
218D4—202no—
219D4—202no—
220D4—401no—
221D4—401no—
222C2×C4—41no—
223C2×C4—401no—
224C2×C4#402111yescharacteristic
225C2×C4#403211yescharacteristic
226C2×C4#404111yescharacteristic
227C2×C4#405011yescharacteristic
228C2×C4#405411yescharacteristic
229C2×C4#405411yescharacteristic
230C8—41no—
231C8—401no—
232C8—401no—
233C8#402911yescharacteristic
234C8#402911yescharacteristic
235C8#404411yescharacteristic
236C8#404411yescharacteristic
237C5#1349011yescharacteristic, complemented, Sylow
238C22—41no—
239C22—402no—
240C22—802no—
241C22—802no—
242C22—802no—
243C22#6498611yescharacteristic
244C22#6502611yescharacteristic
245C22#6513311yescharacteristic
246C4—81no—
247C4—801no—
248C4#6496311yescharacteristic
249C4#6500211yescharacteristic
250C4#6502611yescharacteristic
251C4#6513311yescharacteristic
252C2—21no—
253C2—21no—
254C2—21no—
255C2—802no—
256C2#1059943011yescharacteristic
subgroupG/H
centerC2#10599430
commutator, cosocleC22×C5×C8C23
FrattiniC22×C8C22×D5
FittingC5×#1378C2
socleC2×C5#2073
prime radicalC22×D40C22
hypercenter#1378D5
hypocenterC5#13490
LCSG ▷ C22×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #1378

Complements

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

NH ≅ G/Ntype
4. #10600689C2semidirect
5. C5×#1378C2semidirect
6. #10598313C2semidirect
7. #10598312C2semidirect
57. #1378D5semidirect
237. C5#13490semidirect

Maximal subgroups

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

subgroupindex
C2×#647362
#106022332
#106006962
#106006892
C5×#13782
#105983132
#105983122
#134905