Finite Group Databasebeta

← #13023929

Subgroups — #13023929

Order 1280.

426 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 86 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×#65161C211yesmaximal, characteristic
2#10604710C211yesmaximal, characteristic, complemented
3#10604696C211yesmaximal, characteristic, complemented
4#10598607C214yesmaximal, complemented
5C22×D40C411yescharacteristic
6C2×#4039—21no—
7#65758—21no—
8C2×C5×#107C411yescharacteristic
9C2×C8×Dic5C2211yescharacteristic
10C2×#3941—21no—
11#65161—21no—
12#65161—21no—
13#65161C412yes—
14#65161C2212yes—
15C22⋊#674—21no—
16#64888—21no—
17#64886C2212yescomplemented
18#64883—21no—
19#64881C2212yescomplemented
20#11663—51nomaximal, Sylow
21C22×D20—21no—
22C22×C4×D5—41no—
23C2×#704—41no—
24C22×C5×C8C2×C411yescharacteristic
25C2×C5×C4⋊C4—21no—
26C2×#685—41no—
27C2×C4×Dic5D411yescharacteristic
28C22×#168D411yescharacteristic
29C2×D40—21no—
30C2×D40—24no—
31C2×D40C2×C412yes—
32#4039—22no—
33#4039—22no—
34#4039—41no—
35#4039—41no—
36#4004—41no—
37C4⋊#170—22no—
38C2×#674—22no—
39C5×#107—21no—
40C5×#107C2×C412yes—
41C8×Dic5C2×C411yescharacteristic
42C8×Dic5D411yescharacteristic
43C8×Dic5D411yescharacteristic
44C8×Dic5C2311yescharacteristic
45#3941—22no—
46#3941—22no—
47C2×C4×D8—51no—
48#2163—51no—
49#1566—51no—
50#1333—54no—
51C23×D5—41no—
52C22×C4×C5—41no—
53C22×C4×C5C22⋊C411yescharacteristic
54C22×Dic5D811yescharacteristic
55C22×Dic5SD1611yescharacteristic
56C2×D20—24no—
57C2×D20—22no—
58C2×D20—42no—
59C2×D20—41no—
60C2×C4×D5—42no—
61C2×C4×D5—42no—
62C2×C4×D5—84no—
63#704—82no—
64C2×#170—42no—
65C2×C5×C8—22no—
66C2×C5×C8C22⋊C411yescharacteristic
67C2×C5×C8C22×C411yescharacteristic
68C5×C4⋊C4—22no—
69C5×C4⋊C4—41no—
70#685—42no—
71#685—42no—
72C4×Dic5C22⋊C411yescharacteristic
73C4×Dic5D811yescharacteristic
74C4×Dic5SD1611yescharacteristic
75C4×Dic5C2×D411yescharacteristic
76C2×#168—41no—
77C2×#168C22⋊C411yescharacteristic
78C2×#168C2×D411yescharacteristic
79D40—24no—
80D40—22no—
81D40M1614yescomplemented
82#674—42no—
83C22×D8—51no—
84C2×C4×D4—101no—
85C4×D8—52no—
86C4×D8—52no—
87C4×D8—101no—
88C4×D8—101no—
89C2×#107AGL(1,𝔽5)11yescharacteristic
90#423—101no—
91C2×#102—101no—
92C2×C4×C8—51no—
93C4⋊C16—101no—
94C22⋊C16—101no—
95#346—52no—
96#334—52no—
97#334—101no—
98C23×C5#10211yescharacteristic
99C22×D5—44no—
100C22×D5—42no—
101C22×D5—82no—
102C22×D5—82no—
103C2×C4×C5—21no—
104C2×C4×C5—21no—
105C2×C4×C5—42no—
106C2×C4×C5—82no—
107C2×C4×C5#10211yescharacteristic
108C2×C4×C5C2×C22⋊C411yescharacteristic
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5#10211yescharacteristic
112C2×Dic5#10211yescharacteristic
113C2×Dic5C2×D811yescharacteristic
114C2×Dic5C2×SD1611yescharacteristic
115D20—24no—
116D20—44no—
117D20—42no—
118C4×D5—84no—
119C4×D5—84no—
120#170—82no—
121C5×C8—21no—
122C5×C8C2×M1612yes—
123#168—21no—
124#168—21no—
125C22×D4—101no—
126C23×C4—201no—
127C2×D8—52no—
128C2×D8—104no—
129C2×D8—101no—
130C2×M16—201no—
131C22×C8—51no—
132C22×C8C2×AGL(1,𝔽5)11yescharacteristic
133C4×D4—102no—
134C4×D4—102no—
135C4×D4—201no—
136C4×D4—201no—
137C2×C4⋊C4—21no—
138C2×C22⋊C4—201no—
139C2×C42—51no—
140C2×C16—102no—
141#107—21no—
142#107C2×AGL(1,𝔽5)12yes—
143C4⋊C8—102no—
144C4⋊C8—201no—
145#102—102no—
146#102—102no—
147C4×C8—51no—
148C4×C8—51no—
149C4×C8—51no—
150C4×C8—51no—
151C22×C5—21no—
152C22×C5—21no—
153C22×C5#36012yes—
154C22×C5C2×#10211yescharacteristic
155C2×D5—44no—
156C2×D5—88no—
157C2×D5—82no—
158C2×D5—82no—
159C4×C5—21no—
160C4×C5—82no—
161C4×C5#40711yescharacteristic
162C4×C5#42111yescharacteristic
163Dic5—21no—
164Dic5—21no—
165Dic5—21no—
166Dic5#36112yes—
167C24—201no—
168C2×D4—104no—
169C2×D4—102no—
170C2×D4—202no—
171C2×D4—201no—
172C22×C4—41no—
173C22×C4—51no—
174C22×C4—51no—
175C22×C4—202no—
176C22×C4—202no—
177C22×C4—404no—
178C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
179D8—54no—
180D8—104no—
181D8—102no—
182M16—402no—
183C2×C8—22no—
184C2×C8—51no—
185C2×C8—51no—
186C2×C8—201no—
187C2×C8—202no—
188C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
189C2×C8C22×AGL(1,𝔽5)11yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—41no—
192C22⋊C4—202no—
193C22⋊C4—202no—
194C42—51no—
195C42—51no—
196C42—51no—
197C42—51no—
198C16—202no—
199C2×C5—22no—
200C2×C5#148611yescharacteristic
201C2×C5C2×#36011yescharacteristic
202C2×C5C2×#36111yescharacteristic
203D5—84no—
204C23—204no—
205C23—202no—
206C23—402no—
207C23—402no—
208C23#400711yescharacteristic
209D4—104no—
210D4—204no—
211D4—202no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—51no—
216C2×C4—51no—
217C2×C4—51no—
218C2×C4—51no—
219C2×C4—82no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—404no—
223C2×C4—404no—
224C2×C4#400711yescharacteristic
225C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
226C8—21no—
227C8—101no—
228C8—101no—
229C8—402no—
230C8D5⋊M1612yes—
231C5#1166311yescharacteristic, complemented, Sylow
232C22—21no—
233C22—21no—
234C22—204no—
235C22—408no—
236C22—402no—
237C22—402no—
238C22#6490712yes—
239C22C2×#400711yescharacteristic
240C4—21no—
241C4—52no—
242C4—82no—
243C4—101no—
244C4—101no—
245C4—101no—
246C4#6575511yescharacteristic
247C4#6578411yescharacteristic
248C2—22no—
249C2—404no—
250C2#1060427211yescharacteristic
251C2D5⋊#36111yescharacteristic
252C2C2×#6490711yescharacteristic
subgroupG/H
centerC22C2×#4007
commutatorC2×C5×C8C22×C4
FrattiniC2×C8C22×AGL(1,𝔽5)
FittingC2×C5×#107C4
socleC22×C5C2×#102
prime radicalC22×D40C4
cosocleC8×Dic5C23
hypercenterC2×#107AGL(1,𝔽5)
hypocenterC5#11663
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. #10604710C2semidirect
3. #10604696C2semidirect
4. #10598607C2semidirect
17. #64886C22semidirect
19. #64881C22semidirect
81. D40M16semidirect
231. C5#11663semidirect

Maximal subgroups

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

subgroupindex
C2×#651612
#106047102
#106046962
#105986072
#116635