Finite Group Databasebeta

← #12008717

Subgroups — #12008717

Order 1280.

750 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 107 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×#385C211yesmaximal, characteristic
2#10602332C211yesmaximal, characteristic
3#10602332C211yesmaximal, characteristic
4#10598127C214yesmaximal, complemented
5C24×C4×C5C411yescharacteristic
6C22×C5×C4⋊C4—21no—
7C22×C5×C22⋊C4C411yescharacteristic
8C22×C5×C22⋊C4C2211yescharacteristic
9C5×#385—21no—
10C5×#385—21no—
11C5×#385C412yes—
12C5×#385C2212yes—
13C2×C5×#95—21no—
14#65496—21no—
15#65496—21no—
16#64745—21no—
17#64745—21no—
18#64745C2212yes—
19#64745C2212yes—
20#11517—51nomaximal, Sylow
21C25×C5C2×C411yescharacteristic
22C23×C4×C5—21no—
23C23×C4×C5—21no—
24C23×C4×C5—21no—
25C23×C4×C5—44no—
26C23×C4×C5—41no—
27C23×C4×C5—41no—
28C23×C4×C5C2×C412yes—
29C23×C4×C5D411yescharacteristic
30C23×C4×C5D411yescharacteristic
31C2×C5×C4⋊C4—22no—
32C2×C5×C4⋊C4—22no—
33C2×C5×C4⋊C4—44no—
34C2×C5×C22⋊C4—21no—
35C2×C5×C22⋊C4—24no—
36C2×C5×C22⋊C4—41no—
37C2×C5×C22⋊C4—41no—
38C2×C5×C22⋊C4C2×C412yes—
39C2×C5×C22⋊C4C2×C411yescharacteristic
40C2×C5×C22⋊C4D411yescharacteristic
41C2×C5×C22⋊C4D411yescharacteristic
42C2×C5×C22⋊C4C2311yescharacteristic
43C2×#681—41no—
44C2×#681—41no—
45C5×#95—22no—
46C5×#95—22no—
47C22⋊#168—22no—
48C22⋊#168—22no—
49C22⋊#168—41no—
50C22⋊#168—41no—
51C2×#385D511yescharacteristic
52#1462—51no—
53#1462—51no—
54#1326—54no—
55C24×C5—21no—
56C24×C5—21no—
57C24×C5—22no—
58C24×C5—22no—
59C24×C5—42no—
60C24×C5—42no—
61C24×C5C22⋊C411yescharacteristic
62C24×C5C22⋊C411yescharacteristic
63C24×C5C22×C411yescharacteristic
64C22×C4×C5—24no—
65C22×C4×C5—24no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—22no—
69C22×C4×C5—22no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—44no—
73C22×C4×C5—44no—
74C22×C4×C5—42no—
75C22×C4×C5—42no—
76C22×C4×C5—42no—
77C22×C4×C5—42no—
78C22×C4×C5—48no—
79C22×C4×C5—44no—
80C22×C4×C5—41no—
81C22×C4×C5—41no—
82C22×C4×C5—42no—
83C22×C4×C5—44no—
84C22×C4×C5—42no—
85C22×C4×C5—42no—
86C22×C4×C5C22⋊C411yescharacteristic
87C22×C4×C5C22⋊C411yescharacteristic
88C22×C4×C5C2×D411yescharacteristic
89C22×C4×C5C2×D411yescharacteristic
90C5×C4⋊C4—44no—
91C5×C4⋊C4—44no—
92C5×C22⋊C4—24no—
93C5×C22⋊C4—22no—
94C5×C22⋊C4—41no—
95C5×C22⋊C4—41no—
96C5×C22⋊C4—41no—
97C5×C22⋊C4—41no—
98C5×C22⋊C4M1614yescomplemented
99#681—82no—
100#681—82no—
101C2×#168—42no—
102C2×#168—42no—
103C24×C4Dic511yescharacteristic
104C22×C4⋊C4—21no—
105C22×C22⋊C4Dic511yescharacteristic
106C22×C22⋊C4C2×D511yescharacteristic
107#407—101no—
108#407—101no—
109#385—21no—
110#385—21no—
111#385Dic512yes—
112#385C2×D512yes—
113C2×#95—21no—
114#323—52no—
115#323—52no—
116#323—101no—
117#323—101no—
118C23×C5—22no—
119C23×C5—22no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—24no—
123C23×C5—24no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—41no—
129C23×C5—41no—
130C23×C5—41no—
131C23×C5—41no—
132C23×C5—42no—
133C23×C5—48no—
134C23×C5—48no—
135C23×C5—42no—
136C23×C5—42no—
137C23×C5—42no—
138C23×C5#9911yescharacteristic
139C23×C5#9911yescharacteristic
140C23×C5#10112yes—
141C23×C5C2×C22⋊C411yescharacteristic
142C23×C5C2×M1612yes—
143C2×C4×C5—24no—
144C2×C4×C5—24no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—42no—
150C2×C4×C5—48no—
151C2×C4×C5—42no—
152C2×C4×C5—42no—
153C2×C4×C5—44no—
154C2×C4×C5—48no—
155C2×C4×C5—44no—
156C2×C4×C5—42no—
157C2×C4×C5—44no—
158C2×C4×C5—42no—
159C2×C4×C5—42no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5—41no—
164C2×C4×C5—44no—
165C2×C4×C5—81no—
166C2×C4×C5—81no—
167#168—82no—
168#168—82no—
169C25C2×Dic511yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—44no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4C2×Dic512yes—
177C23×C4C22⋊D511yescharacteristic
178C23×C4C22⋊D511yescharacteristic
179C2×M16—201no—
180C2×M16—201no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4C2×Dic512yes—
189C2×C22⋊C4C2×Dic511yescharacteristic
190C2×C22⋊C4C22⋊D511yescharacteristic
191C2×C22⋊C4C22⋊D511yescharacteristic
192C2×C22⋊C4C22×D511yescharacteristic
193C22⋊C8—102no—
194C22⋊C8—102no—
195C22⋊C8—201no—
196C22⋊C8—201no—
197#95—22no—
198#95—22no—
199C22×C5—24no—
200C22×C5—24no—
201C22×C5—21no—
202C22×C5—21no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5—22no—
208C22×C5—22no—
209C22×C5—42no—
210C22×C5—42no—
211C22×C5—42no—
212C22×C5—48no—
213C22×C5—48no—
214C22×C5—42no—
215C22×C5—41no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5AΣL(1,𝔽16)12yes—
220C22×C5#35212yes—
221C22×C5#40711yescharacteristic
222C22×C5C2×#9911yescharacteristic
223C22×C5C2×#10111yescharacteristic
224C4×C5—44no—
225C4×C5—41no—
226C4×C5—44no—
227C4×C5—41no—
228C4×C5—41no—
229C4×C5—41no—
230C24—21no—
231C24—21no—
232C24—22no—
233C24—22no—
234C24—42no—
235C24—42no—
236C24C22⋊Dic511yescharacteristic
237C24C22⋊Dic511yescharacteristic
238C24C22×Dic511yescharacteristic
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4C22⋊Dic511yescharacteristic
262C22×C4C22⋊Dic511yescharacteristic
263C22×C4C2×C22⋊D511yescharacteristic
264C22×C4C2×C22⋊D511yescharacteristic
265M16—402no—
266M16—402no—
267C2×C8—202no—
268C2×C8—202no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C22⋊C4—24no—
272C22⋊C4—22no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4#68114yescomplemented
278C2×C5—21no—
279C2×C5—22no—
280C2×C5—22no—
281C2×C5—21no—
282C2×C5—42no—
283C2×C5—42no—
284C2×C5#146311yescharacteristic
285C2×C5C2×AΣL(1,𝔽16)11yescharacteristic
286C2×C5C2×#35211yescharacteristic
287C23—21no—
288C23—21no—
289C23—22no—
290C23—22no—
291C23—24no—
292C23—24no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—48no—
303C23—48no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23#396611yescharacteristic
308C23#396611yescharacteristic
309C23#396812yes—
310C23C2×#68112yes—
311C23C2×C22⋊Dic511yescharacteristic
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—48no—
322C2×C4—42no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—81no—
335C2×C4—81no—
336C8—402no—
337C8—402no—
338C5#1151711yescharacteristic, complemented, Sylow
339C22—24no—
340C22—24no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—48no—
354C22—48no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#6475612yes—
360C22#6475912yes—
361C22#6549611yescharacteristic
362C22C2×#396611yescharacteristic
363C22C2×#396811yescharacteristic
364C4—44no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—44no—
369C4—41no—
370C2—21no—
371C2—21no—
372C2—22no—
373C2—22no—
374C2—42no—
375C2—42no—
376C2#1060234511yescharacteristic
377C2C2×#6475611yescharacteristic
378C2C2×#6475911yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC24×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC2×C5×#385C2
socleC22×C5C2×#99
prime radicalC25×C5C2×C4
cosocleC2×C5×C22⋊C4C23
hypercenterC2×#385D5
hypocenterC5#11517
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
4. #10598127C2semidirect
98. C5×C22⋊C4M16semidirect
277. C22⋊C4#681semidirect
338. C5#11517semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3852
#106023322
#105981272
#115175