Finite Group Databasebeta

← #13023084

Subgroups — #13023084

Order 1280.

666 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C22×#3952C211yesmaximal, characteristic
2#10604872C212yesmaximal, complemented
3#10598760C214yesmaximal, complemented
4C23×C4×D5C411yescharacteristic
5C23×C5×C8C411yescharacteristic
6C23×#168C2211yescharacteristic
7C2×#3998—22no—
8C2×#3952—22no—
9C2×#3952—22no—
10C2×#3952C412yes—
11C2×#3952C2212yes—
12C22⋊#674—22no—
13C22⋊#674C2214yescomplemented
14#11418—51nomaximal, Sylow
15C24×D5C811yescharacteristic
16C23×C4×C5C2×C411yescharacteristic
17C23×Dic5C811yescharacteristic
18C22×C4×D5—21no—
19C22×C4×D5—21no—
20C22×C4×D5—21no—
21C22×C4×D5—21no—
22C22×C4×D5—42no—
23C22×C4×D5—42no—
24C22×C4×D5C812yes—
25C22×C4×D5C2×C412yes—
26C22×C5×C8—21no—
27C22×C5×C8—21no—
28C22×C5×C8—42no—
29C22×C5×C8C2×C412yes—
30C22×#168—42no—
31C22×#168C2×C411yescharacteristic
32C22×#168D412yes—
33C22×#168D412yes—
34C22×#168C2311yescharacteristic
35#3998—44no—
36C2×#674—24no—
37#3952—24no—
38#3952—24no—
39C22×C22⋊C8—51no—
40#2112—52no—
41#1314—54no—
42C24×C5C2×C811yescharacteristic
43C23×D5—21no—
44C23×D5—21no—
45C23×D5—24no—
46C23×D5—42no—
47C23×D5—42no—
48C23×D5C2×C812yes—
49C22×C4×C5—22no—
50C22×C4×C5—22no—
51C22×C4×C5C22⋊C411yescharacteristic
52C22×C4×C5C22⋊C411yescharacteristic
53C22×C4×C5C2×C811yescharacteristic
54C22×C4×C5M1611yescharacteristic
55C22×C4×C5M1611yescharacteristic
56C22×C4×C5C22×C411yescharacteristic
57C22×Dic5—21no—
58C22×Dic5—21no—
59C22×Dic5—42no—
60C22×Dic5C2×C812yes—
61C2×C4×D5—22no—
62C2×C4×D5—22no—
63C2×C4×D5—22no—
64C2×C4×D5—22no—
65C2×C4×D5—44no—
66C2×C4×D5—44no—
67C2×C4×D5—42no—
68C2×C4×D5—42no—
69C2×C4×D5—42no—
70C2×C4×D5—42no—
71C2×C4×D5—44no—
72C2×C4×D5—42no—
73C2×C4×D5—42no—
74C2×C5×C8—22no—
75C2×C5×C8—42no—
76C2×C5×C8—42no—
77C2×C5×C8—42no—
78C2×#168—24no—
79C2×#168—41no—
80C2×#168—42no—
81C2×#168—41no—
82C2×#168C22⋊C412yes—
83C2×#168C2×D412yes—
84#674—44no—
85C24×C4—51no—
86C23×C8—51no—
87C23×C8AGL(1,𝔽5)11yescharacteristic
88C2×M32—102no—
89C2×C22⋊C8—52no—
90C2×C22⋊C8—52no—
91C2×C22⋊C8—102no—
92C2×C22⋊C8—102no—
93C22⋊C16—54no—
94C22⋊C16—102no—
95C23×C5—22no—
96C23×C5—22no—
97C23×C5—22no—
98C23×C5C22⋊C811yescharacteristic
99C23×C5C22⋊C811yescharacteristic
100C23×C5C22×C811yescharacteristic
101C22×D5—24no—
102C22×D5—24no—
103C22×D5—22no—
104C22×D5—22no—
105C22×D5—42no—
106C22×D5—416no—
107C22×D5—42no—
108C22×D5—42no—
109C22×D5—42no—
110C22×D5—42no—
111C22×D5—42no—
112C22×D5M3214yescomplemented
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—21no—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5C22⋊C811yescharacteristic
121C2×C4×C5C22⋊C811yescharacteristic
122C2×C4×C5#9912yes—
123C2×C4×C5#10012yes—
124C2×C4×C5C2×C22⋊C411yescharacteristic
125C2×C4×C5C2×M1611yescharacteristic
126C2×Dic5—22no—
127C2×Dic5—42no—
128C2×Dic5—42no—
129C2×Dic5—42no—
130C4×D5—44no—
131C4×D5—44no—
132C4×D5—44no—
133C4×D5—42no—
134C4×D5—42no—
135C5×C8—42no—
136#168—22no—
137#168—22no—
138C25—51no—
139C23×C4—51no—
140C23×C4—52no—
141C23×C4—52no—
142C23×C4—101no—
143C23×C4—101no—
144C23×C4—101no—
145C23×C4—101no—
146C23×C4—202no—
147C23×C4—202no—
148C23×C4C2×AGL(1,𝔽5)11yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—42no—
152C22×C8—52no—
153C22×C8—52no—
154C22×C8—51no—
155C22×C8—51no—
156C22×C8—202no—
157C22×C8C2×AGL(1,𝔽5)12yes—
158M32—204no—
159C2×C16—104no—
160C22⋊C8—104no—
161C22⋊C8—104no—
162C22×C5—28no—
163C22×C5—21no—
164C22×C5—21no—
165C22×C5—21no—
166C22×C5—21no—
167C22×C5—21no—
168C22×C5—21no—
169C22×C5#32312yes—
170C22×C5#34912yes—
171C22×C5C2×C22⋊C811yescharacteristic
172C22×C5C2×M3212yes—
173C2×D5—24no—
174C2×D5—48no—
175C2×D5—48no—
176C2×D5—44no—
177C2×D5—42no—
178C2×D5—42no—
179C2×D5—44no—
180C4×C5—21no—
181C4×C5—21no—
182C4×C5#32312yes—
183C4×C5C2×#9911yescharacteristic
184C4×C5C2×#10011yescharacteristic
185Dic5—42no—
186C24—52no—
187C24—104no—
188C24—101no—
189C24—101no—
190C24—202no—
191C24—202no—
192C24D5⋊C811yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—52no—
196C22×C4—101no—
197C22×C4—101no—
198C22×C4—102no—
199C22×C4—102no—
200C22×C4—102no—
201C22×C4—102no—
202C22×C4—204no—
203C22×C4—202no—
204C22×C4—204no—
205C22×C4—202no—
206C22×C4—202no—
207C22×C4—204no—
208C22×C4—202no—
209C22×C4—202no—
210C22×C4—202no—
211C22×C4—202no—
212C22×C4D5⋊C811yescharacteristic
213C22×C4#70011yescharacteristic
214C22×C4#70011yescharacteristic
215C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
216C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
217C22×C4C22×AGL(1,𝔽5)11yescharacteristic
218C2×C8—22no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—52no—
223C2×C8—52no—
224C2×C8—104no—
225C2×C8—201no—
226C2×C8—202no—
227C2×C8—201no—
228C16—204no—
229C2×C5—22no—
230C2×C5—22no—
231C2×C5—22no—
232C2×C5C2×#32311yescharacteristic
233C2×C5#211211yescharacteristic
234C2×C5C2×#34911yescharacteristic
235D5—44no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—54no—
240C23—104no—
241C23—104no—
242C23—102no—
243C23—102no—
244C23—202no—
245C23—202no—
246C23—202no—
247C23—2016no—
248C23—202no—
249C23—202no—
250C23—202no—
251C23#400311yescharacteristic
252C23#400311yescharacteristic
253C23C2×D5⋊C811yescharacteristic
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—102no—
262C2×C4—204no—
263C2×C4—204no—
264C2×C4—202no—
265C2×C4—202no—
266C2×C4—202no—
267C2×C4—204no—
268C2×C4—202no—
269C2×C4—202no—
270C2×C4#399912yes—
271C2×C4#400311yescharacteristic
272C2×C4#400311yescharacteristic
273C2×C4#401312yes—
274C2×C4C2×#70011yescharacteristic
275C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
276C8—42no—
277C8—102no—
278C8—102no—
279C5#1141811yescharacteristic, complemented, Sylow
280C22—28no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—104no—
288C22—204no—
289C22—208no—
290C22—208no—
291C22—204no—
292C22—202no—
293C22—202no—
294C22#6489112yes—
295C22#6491512yes—
296C22D5⋊M3212yes—
297C22C2×#400311yescharacteristic
298C4—21no—
299C4—21no—
300C4—202no—
301C4#6491512yes—
302C4C2×#399911yescharacteristic
303C4C2×#401311yescharacteristic
304C2—22no—
305C2—22no—
306C2—22no—
307C2—204no—
308C2#1060470511yescharacteristic
309C2C2×#6489111yescharacteristic
310C2C2×#6491511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C8
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC23×C5×C8C4
socleC22×C5C2×C22⋊C8
prime radicalC24×D5C8
cosocleC22×#168C23
hypercenterC23×C8AGL(1,𝔽5)
hypocenterC5#11418
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
2. #10604872C2semidirect
3. #10598760C2semidirect
13. C22⋊#674C22semidirect
112. C22×D5M32semidirect
279. C5#11418semidirect

Maximal subgroups

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

subgroupindex
C22×#39522
#106048722
#105987602
#114185