Finite Group Databasebeta

← #13023923

Subgroups — #13023923

Order 1280.

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

normal subgroups only

This group has 422 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#10604709C211yesmaximal, characteristic, complemented
3#10604699C211yesmaximal, characteristic, complemented
4#10598609C214yesmaximal, complemented
5C22×D40C411yescharacteristic
6C2×#4039—21no—
7#65759—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#64885C2212yescomplemented
18#64883—21no—
19#64882C2212yescomplemented
20#11661—51nomaximal, Sylow
21C22×D20—21no—
22C22×C4×D5—41no—
23C22×C5×C8C2×C411yescharacteristic
24C2×C5×C4⋊C4—21no—
25C2×#685—41no—
26C2×C4×Dic5D411yescharacteristic
27C22×#168D411yescharacteristic
28C2×D40—21no—
29C2×D40—24no—
30C2×D40C2×C412yes—
31#4039—22no—
32#4039—22no—
33#4039—41no—
34#4039—41no—
35C22⋊#170—41no—
36#4002—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#2162—51no—
49#1568—51no—
50#1331—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—
63C2×#170—42no—
64C2×C5×C8—22no—
65C2×C5×C8C22⋊C411yescharacteristic
66C2×C5×C8C22×C411yescharacteristic
67C5×C4⋊C4—22no—
68C5×C4⋊C4—41no—
69#685—42no—
70#685—42no—
71C4×Dic5C22⋊C411yescharacteristic
72C4×Dic5D811yescharacteristic
73C4×Dic5SD1611yescharacteristic
74C4×Dic5C2×D411yescharacteristic
75C2×#168—41no—
76C2×#168C22⋊C411yescharacteristic
77C2×#168C2×D411yescharacteristic
78D40—24no—
79D40—22no—
80D40M1614yes—
81#674—42no—
82C22×D8—51no—
83C2×C4×D4—101no—
84C4×D8—52no—
85C4×D8—52no—
86C4×D8—101no—
87C4×D8—101no—
88#432—101no—
89C2×#107AGL(1,𝔽5)11yescharacteristic
90C2×#102—101no—
91C2×C4×C8—51no—
92C4⋊C16—101no—
93C22⋊C16—101no—
94C4×C16—52no—
95#335—52no—
96#334—101no—
97C23×C5#10211yescharacteristic
98C22×D5—44no—
99C22×D5—42no—
100C22×D5—82no—
101C22×D5—82no—
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5—42no—
105C2×C4×C5—82no—
106C2×C4×C5#10211yescharacteristic
107C2×C4×C5C2×C22⋊C411yescharacteristic
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5#10211yescharacteristic
111C2×Dic5#10211yescharacteristic
112C2×Dic5C2×D811yescharacteristic
113C2×Dic5C2×SD1611yescharacteristic
114D20—24no—
115D20—44no—
116D20—42no—
117C4×D5—84no—
118C4×D5—84no—
119#170—82no—
120C5×C8—21no—
121C5×C8C2×M1612yes—
122#168—21no—
123#168—21no—
124C22×D4—101no—
125C23×C4—201no—
126C2×D8—52no—
127C2×D8—104no—
128C2×D8—101no—
129C22×C8—51no—
130C22×C8C2×AGL(1,𝔽5)11yescharacteristic
131C4×D4—102no—
132C4×D4—102no—
133C4×D4—201no—
134C4×D4—201no—
135C2×C4⋊C4—21no—
136C2×C22⋊C4—201no—
137C2×C42—51no—
138C2×C16—102no—
139#107—21no—
140#107C2×AGL(1,𝔽5)12yes—
141C4⋊C8—102no—
142#102—102no—
143#102—102no—
144C22⋊C8—201no—
145#97—201no—
146C4×C8—51no—
147C4×C8—51no—
148C4×C8—51no—
149C4×C8—51no—
150C22×C5—21no—
151C22×C5—21no—
152C22×C5#35912yes—
153C22×C5C2×#10211yescharacteristic
154C2×D5—44no—
155C2×D5—88no—
156C2×D5—82no—
157C2×D5—82no—
158C4×C5—21no—
159C4×C5—82no—
160C4×C5#40711yescharacteristic
161C4×C5#42111yescharacteristic
162Dic5—21no—
163Dic5—21no—
164Dic5—21no—
165Dic5#50912yes—
166C24—201no—
167C2×D4—104no—
168C2×D4—102no—
169C2×D4—202no—
170C2×D4—201no—
171C22×C4—41no—
172C22×C4—51no—
173C22×C4—51no—
174C22×C4—202no—
175C22×C4—202no—
176C22×C4—404no—
177C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
178D8—54no—
179D8—104no—
180D8—102no—
181C2×C8—22no—
182C2×C8—51no—
183C2×C8—51no—
184C2×C8—201no—
185C2×C8—202no—
186C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
187C2×C8C22×AGL(1,𝔽5)11yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—41no—
190C22⋊C4—202no—
191C22⋊C4—202no—
192C42—51no—
193C42—51no—
194C42—51no—
195C42—51no—
196C16—202no—
197C2×C5—22no—
198C2×C5#148611yescharacteristic
199C2×C5#214011yescharacteristic
200C2×C5C2×#35911yescharacteristic
201D5—84no—
202C23—204no—
203C23—202no—
204C23—402no—
205C23—402no—
206C23#400711yescharacteristic
207D4—104no—
208D4—204no—
209D4—202no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—42no—
213C2×C4—51no—
214C2×C4—51no—
215C2×C4—51no—
216C2×C4—51no—
217C2×C4—82no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—404no—
221C2×C4—404no—
222C2×C4#400711yescharacteristic
223C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
224C8—21no—
225C8—101no—
226C8—101no—
227C8—402no—
228C8D5⋊M1612yes—
229C5#1166111yescharacteristic, complemented, Sylow
230C22—21no—
231C22—21no—
232C22—204no—
233C22—408no—
234C22—402no—
235C22—402no—
236C22#6490912yes—
237C22C2×#400711yescharacteristic
238C4—21no—
239C4—52no—
240C4—82no—
241C4—101no—
242C4—101no—
243C4—101no—
244C4#6575511yescharacteristic
245C4#6578411yescharacteristic
246C2—22no—
247C2—404no—
248C2#1060427211yescharacteristic
249C2#1060478611yescharacteristic
250C2C2×#6490911yescharacteristic
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#11661
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. #10604709C2semidirect
3. #10604699C2semidirect
4. #10598609C2semidirect
17. #64885C22semidirect
19. #64882C22semidirect
229. C5#11661semidirect

Maximal subgroups

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

subgroupindex
C2×#651612
#106047092
#106046992
#105986092
#116615