Finite Group Databasebeta

← #12009986

Subgroups — #12009986

Order 1280.

478 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 95 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×C5×#398C211yesmaximal, characteristic
2#10602355C211yesmaximal, characteristic
3#10602345C211yesmaximal, characteristic
4#10598141C214yesmaximal, complemented
5C22×C5×C4⋊C4—21no—
6C22×C5×C4⋊C4C411yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C5×#398—21no—
9C5×#398—21no—
10C5×#398C412yes—
11C5×#398C2212yes—
12C2×C5×#95—21no—
13C2×C5×#95C411yescharacteristic
14#65496—21no—
15#65496—21no—
16#64750—21no—
17#64750C2212yes—
18#64748—21no—
19#64748C2212yes—
20#11803—51nomaximal, Sylow
21C23×C4×C5—21no—
22C23×C4×C5—21no—
23C23×C4×C5C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C2×C5×C4⋊C4—21no—
27C2×C5×C4⋊C4—22no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—41no—
30C2×C5×C4⋊C4—41no—
31C2×C5×C4⋊C4—41no—
32C2×C5×C4⋊C4—41no—
33C2×C5×C4⋊C4—44no—
34C2×C5×C4⋊C4C2×C412yes—
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C4⋊C4D411yescharacteristic
38C2×C5×C4⋊C4C2311yescharacteristic
39C2×#681—22no—
40C2×#681—41no—
41C5×#95—22no—
42C5×#95—21no—
43C5×#95—22no—
44C5×#95C2×C412yes—
45C22⋊#168—22no—
46C22⋊#168—41no—
47C22⋊#168—41no—
48C22⋊#168—41no—
49C2×#398D511yescharacteristic
50#1821—51no—
51#1463—51no—
52#1346—54no—
53C24×C5C22⋊C411yescharacteristic
54C22×C4×C5—24no—
55C22×C4×C5—21no—
56C22×C4×C5—21no—
57C22×C4×C5—21no—
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—22no—
61C22×C4×C5—21no—
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5—41no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5—42no—
70C22×C4×C5—42no—
71C22×C4×C5—41no—
72C22×C4×C5—41no—
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22×C411yescharacteristic
77C22×C4×C5C2×D411yescharacteristic
78C22×C4×C5C2×D411yescharacteristic
79C5×C4⋊C4—41no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—42no—
85C5×C4⋊C4—44no—
86C5×C4⋊C4—44no—
87#681—42no—
88#681—42no—
89#681—82no—
90C2×#168—42no—
91C2×#168—42no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4Dic511yescharacteristic
94C22×C4⋊C4C2×D511yescharacteristic
95#407—101no—
96#407—101no—
97#398—21no—
98#398—21no—
99#398Dic512yes—
100#398C2×D512yes—
101C2×#95—21no—
102C2×#95Dic511yescharacteristic
103#343—52no—
104#343—101no—
105#324—52no—
106#324—101no—
107C23×C5—22no—
108C23×C5—22no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5#9911yescharacteristic
112C23×C5#9911yescharacteristic
113C23×C5C2×C22⋊C411yescharacteristic
114C2×C4×C5—21no—
115C2×C4×C5—21no—
116C2×C4×C5—24no—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—42no—
125C2×C4×C5—42no—
126C2×C4×C5—48no—
127C2×C4×C5—42no—
128C2×C4×C5—42no—
129C2×C4×C5—42no—
130C2×C4×C5—42no—
131C2×C4×C5—42no—
132C2×C4×C5—41no—
133C2×C4×C5—41no—
134C2×C4×C5—41no—
135C2×C4×C5—41no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—81no—
140C2×C4×C5—82no—
141C2×C4×C5—82no—
142C2×C4×C5AGL(1,ℤ/8)12yes—
143#168—82no—
144#168—82no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×Dic511yescharacteristic
148C23×C4C22⋊D511yescharacteristic
149C23×C4C22⋊D511yescharacteristic
150C2×M16—102no—
151C2×M16—201no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C2×Dic512yes—
161C2×C4⋊C4C2×Dic511yescharacteristic
162C2×C4⋊C4C22⋊D511yescharacteristic
163C2×C4⋊C4C22⋊D511yescharacteristic
164C2×C4⋊C4C22×D511yescharacteristic
165C22⋊C8—102no—
166C22⋊C8—201no—
167C22⋊C8—201no—
168C22⋊C8—201no—
169#95—22no—
170#95—21no—
171#95—22no—
172#95C2×Dic512yes—
173C22×C5—24no—
174C22×C5—24no—
175C22×C5—21no—
176C22×C5—21no—
177C22×C5—21no—
178C22×C5—21no—
179C22×C5—21no—
180C22×C5—21no—
181C22×C5#35212yes—
182C22×C5#35612yes—
183C22×C5C2×#9911yescharacteristic
184C22×C5#41811yescharacteristic
185C22×C5#42111yescharacteristic
186C4×C5—22no—
187C4×C5—44no—
188C4×C5—41no—
189C4×C5—41no—
190C4×C5—41no—
191C4×C5—41no—
192C4×C5—41no—
193C4×C5—82no—
194C24C22⋊Dic511yescharacteristic
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4C22⋊Dic511yescharacteristic
215C22×C4C22⋊Dic511yescharacteristic
216C22×C4C22⋊Dic511yescharacteristic
217C22×C4C22×Dic511yescharacteristic
218C22×C4C2×C22⋊D511yescharacteristic
219C22×C4C2×C22⋊D511yescharacteristic
220M16—202no—
221M16—202no—
222M16—402no—
223C2×C8—202no—
224C2×C8—202no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C2×C5—21no—
234C2×C5—22no—
235C2×C5—22no—
236C2×C5—21no—
237C2×C5#151011yescharacteristic
238C2×C5C2×#35211yescharacteristic
239C2×C5C2×#35611yescharacteristic
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23#396611yescharacteristic
245C23#396611yescharacteristic
246C23C2×C22⋊Dic511yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—48no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4#407812yes—
276C8—402no—
277C8—402no—
278C5#1180311yescharacteristic, complemented, Sylow
279C22—24no—
280C22—24no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22#6475912yes—
288C22#6476812yes—
289C22#6550411yescharacteristic
290C22C2×#396611yescharacteristic
291C22#6552511yescharacteristic
292C4—22no—
293C4—44no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C2—21no—
301C2—22no—
302C2—21no—
303C2—22no—
304C2#1060246711yescharacteristic
305C2C2×#6475911yescharacteristic
306C2C2×#6476811yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC2×C5×#398C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenterC2×#398D5
hypocenterC5#11803
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
4. #10598141C2semidirect
278. C5#11803semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3982
#106023552
#106023452
#105981412
#118035