Finite Group Databasebeta

← #12100437

Subgroups — #12100437

Order 1280.

1898 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#525C211yesmaximal, characteristic
2C23×#681C211yesmaximal, characteristic, complemented
3C2×#65523C212yesmaximal, complemented
4C2×#65496C211yesmaximal, characteristic, complemented
5#10602404C218yesmaximal, complemented
6#10602372C212yesmaximal, complemented
7C22×C5×#41C2211yescharacteristic
8C24×C4×C5C2211yescharacteristic
9C5×#525—24no—
10C5×#525—24no—
11C2×C5×#123C412yes—
12C2×C5×C22⋊Q8C412yes—
13C2×C5×#121C412yes—
14C2×C5×#120C412yes—
15C2×C4×C5×D4C2214yes—
16C2×C5×#117C2211yescharacteristic
17C22×#681—28no—
18C22×#681—24no—
19C22×#681C2212yes—
20C22×#681C2214yes—
21C23×#168C2212yes—
22#65523—28no—
23#65496—24no—
24C2×C22⋊#168C2214yes—
25C2×C22⋊#168C2218yescomplemented
26#64744C2218yescomplemented
27#20896—51nomaximal, Sylow
28C25×C5C2×C411yescharacteristic
29C2×C5×#41—22no—
30C2×C5×#41—22no—
31C2×C5×#41—24no—
32C2×C5×#41—24no—
33C22×C5×Q8C2×C411yescharacteristic
34C22×C5×D4C2×C412yes—
35C22×C5×D4C2×C411yescharacteristic
36C22×C5×D4C2×C414yes—
37C23×C4×C5—24no—
38C23×C4×C5—24no—
39C23×C4×C5C2×C412yes—
40C23×C4×C5C2×C411yescharacteristic
41C23×C4×C5D414yes—
42C23×C4×C5D414yes—
43C23×C4×C5C2312yes—
44C23×C4×C5C2312yes—
45C23×C4×C5C2311yescharacteristic
46C5×#123—28no—
47C5×C22⋊Q8—28no—
48C5×#121—28no—
49C5×#120—28no—
50C4×C5×D4—28no—
51C4×C5×D4—28no—
52C5×#117—22no—
53C5×#117—22no—
54C2×C5×C4⋊C4C2×C412yes—
55C2×C5×C4⋊C4C2×C414yes—
56C2×C5×C22⋊C4C2×C412yes—
57C2×C5×C22⋊C4C2×C414yes—
58C2×C5×C22⋊C4C2×C414yes—
59C2×C42×C5C2312yes—
60C2×#681—232no—
61C2×#681—28no—
62C2×#681—24no—
63C2×#681—24no—
64C2×#681—24no—
65C2×#681—216no—
66C22×#168—24no—
67C22×#168—24no—
68C22×#168D418yescomplemented
69C22×#168C2314yes—
70C22×#168C2314yes—
71C22⋊#168—28no—
72C22⋊#168—216no—
73C23×M16—51no—
74C2×#525D511yescharacteristic
75C2×#408—52no—
76C2×#407—51no—
77#1870—58no—
78#1729—52no—
79C24×C5—24no—
80C24×C5—44no—
81C24×C5C22⋊C414yes—
82C24×C5C22×C412yes—
83C24×C5C22×C412yes—
84C24×C5C22×C411yescharacteristic
85C5×#41—44no—
86C5×#41—48no—
87C5×#41—44no—
88C2×C5×Q8—22no—
89C2×C5×Q8—24no—
90C2×C5×D4—28no—
91C2×C5×D4—24no—
92C2×C5×D4—28no—
93C2×C5×D4—22no—
94C2×C5×D4—24no—
95C2×C5×D4—48no—
96C22×C4×C5—24no—
97C22×C4×C5—28no—
98C22×C4×C5—28no—
99C22×C4×C5—216no—
100C22×C4×C5—28no—
101C22×C4×C5—24no—
102C22×C4×C5—24no—
103C22×C4×C5—22no—
104C22×C4×C5—22no—
105C22×C4×C5—24no—
106C22×C4×C5—24no—
107C22×C4×C5C22⋊C418yes—
108C22×C4×C5C22⋊C414yes—
109C22×C4×C5C22×C414yes—
110C22×C4×C5C22×C412yes—
111C22×C4×C5C22×C412yes—
112C22×C4×C5C22×C411yescharacteristic
113C22×C4×C5C2×D414yes—
114C22×C4×C5C2×D414yes—
115C22×C4×C5C2×D414yes—
116C22×C4×C5C2×D412yes—
117C22×C4×C5C2×D412yes—
118C22×C4×C5#4114yes—
119C22×C4×C5C2411yescharacteristic
120C5×C4⋊C4—24no—
121C5×C4⋊C4—28no—
122C5×C22⋊C4—24no—
123C5×C22⋊C4—28no—
124C5×C22⋊C4—28no—
125C42×C5—22no—
126C42×C5—22no—
127#681—232no—
128#681—48no—
129C2×#168—216no—
130C2×#168—24no—
131C2×#168—24no—
132C2×#168—24no—
133C2×#168—24no—
134C2×#168—24no—
135C2×#168C2×D418yes—
136C22×#41C2×D511yescharacteristic
137C24×C4C2×D511yescharacteristic
138C22×M16—52no—
139C22×M16—54no—
140C22×M16—108no—
141C22×M16—104no—
142C23×C8—52no—
143#525—24no—
144#525—24no—
145C2×#123Dic512yes—
146C2×C22⋊Q8Dic512yes—
147C2×#121Dic512yes—
148C2×#120Dic512yes—
149C2×C4×D4C2×D514yes—
150C2×#117C2×D511yescharacteristic
151#408—108no—
152#407—104no—
153C2×C22⋊C8—54no—
154C2×C22⋊C8—58no—
155#336—58no—
156C23×C5—24no—
157C23×C5—24no—
158C23×C5—24no—
159C23×C5—24no—
160C23×C5—28no—
161C23×C5—22no—
162C23×C5—24no—
163C23×C5—28no—
164C23×C5—48no—
165C23×C5—416no—
166C23×C5C2×C22⋊C414yes—
167C23×C5C2×C22⋊C412yes—
168C23×C5C4×D414yes—
169C23×C5C23×C411yescharacteristic
170C5×Q8—44no—
171C5×D4—416no—
172C5×D4—48no—
173C5×D4—44no—
174C2×C4×C5—24no—
175C2×C4×C5—28no—
176C2×C4×C5—24no—
177C2×C4×C5—28no—
178C2×C4×C5—24no—
179C2×C4×C5—216no—
180C2×C4×C5—24no—
181C2×C4×C5—22no—
182C2×C4×C5—24no—
183C2×C4×C5—28no—
184C2×C4×C5—42no—
185C2×C4×C5—44no—
186C2×C4×C5—42no—
187C2×C4×C5C2×C22⋊C414yes—
188C2×C4×C5C2×C22⋊C412yes—
189C2×C4×C5C4×D414yes—
190C2×C4×C5#12014yes—
191C2×C4×C5#12118yes—
192C2×C4×C5#12314yes—
193C2×C4×C5C22×D412yes—
194C2×C4×C5C22×D411yescharacteristic
195C2×C4×C5C22×D411yescharacteristic
196C2×C4×C5C2×#4112yes—
197#168—28no—
198#168—44no—
199C25C2×Dic511yescharacteristic
200C2×#41—24no—
201C2×#41—24no—
202C2×#41—22no—
203C2×#41—22no—
204C22×Q8C2×Dic511yescharacteristic
205C22×D4C2×Dic512yes—
206C22×D4C2×Dic511yescharacteristic
207C22×D4C2×Dic514yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4C2×Dic512yes—
211C23×C4C2×Dic511yescharacteristic
212C23×C4C22⋊D514yes—
213C23×C4C22⋊D514yes—
214C23×C4C22×D512yes—
215C23×C4C22×D512yes—
216C23×C4C22×D511yescharacteristic
217C2×M16—1016no—
218C2×M16—1032no—
219C2×M16—108no—
220C2×M16—104no—
221C2×M16—104no—
222C2×M16—104no—
223C22×C8—58no—
224C22×C8—54no—
225C22×C8—54no—
226C22×C8—104no—
227C22×C8—104no—
228#123—28no—
229C22⋊Q8—28no—
230#121—28no—
231#120—28no—
232C4×D4—28no—
233C4×D4—28no—
234#117—22no—
235#117—22no—
236C2×C4⋊C4C2×Dic512yes—
237C2×C4⋊C4C2×Dic514yes—
238C2×C22⋊C4C2×Dic512yes—
239C2×C22⋊C4C2×Dic514yes—
240C2×C22⋊C4C2×Dic514yes—
241C2×C42C22×D512yes—
242C22⋊C8—1016no—
243C22⋊C8—108no—
244C22×C5—24no—
245C22×C5—28no—
246C22×C5—24no—
247C22×C5—24no—
248C22×C5—24no—
249C22×C5—416no—
250C22×C5—48no—
251C22×C5—44no—
252C22×C5—48no—
253C22×C5—42no—
254C22×C5#38614yes—
255C22×C5C22×C22⋊C411yescharacteristic
256C22×C5C2×C4×D412yes—
257C22×C5#56814yes—
258C4×C5—24no—
259C4×C5—44no—
260C4×C5—42no—
261C4×C5#38614yes—
262C4×C5C2×#12011yescharacteristic
263C4×C5C2×#12112yes—
264C4×C5C2×#12311yescharacteristic
265C24—24no—
266C24—44no—
267C24C22⋊Dic514yes—
268C24C22×Dic512yes—
269C24C22×Dic512yes—
270C24C22×Dic511yescharacteristic
271#41—44no—
272#41—44no—
273#41—48no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×D4—28no—
277C2×D4—24no—
278C2×D4—28no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—48no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—216no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4C22⋊Dic518yes—
294C22×C4C22⋊Dic514yes—
295C22×C4D4×D514yes—
296C22×C4#71114yes—
297C22×C4C22×Dic514yes—
298C22×C4C22×Dic512yes—
299C22×C4C22×Dic512yes—
300C22×C4C22×Dic511yescharacteristic
301C22×C4C2×C22⋊D514yes—
302C22×C4C2×C22⋊D514yes—
303C22×C4C2×C22⋊D512yes—
304C22×C4C2×C22⋊D512yes—
305C22×C4C23×D511yescharacteristic
306M16—1032no—
307M16—208no—
308C2×C8—58no—
309C2×C8—104no—
310C2×C8—104no—
311C2×C8—1016no—
312C2×C8—104no—
313C2×C8—104no—
314C2×C8—104no—
315C4⋊C4—24no—
316C4⋊C4—28no—
317C22⋊C4—24no—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C42—22no—
321C42—22no—
322C2×C5—24no—
323C2×C5—44no—
324C2×C5—82no—
325C2×C5C2×#38611yescharacteristic
326C2×C5#287912yes—
327C2×C5#293014yes—
328C23—24no—
329C23—24no—
330C23—24no—
331C23—24no—
332C23—28no—
333C23—24no—
334C23—28no—
335C23—22no—
336C23—48no—
337C23—416no—
338C23D4×Dic514yes—
339C23C2×C22⋊Dic514yes—
340C23C2×C22⋊Dic512yes—
341C23C23×Dic511yescharacteristic
342Q8—44no—
343D4—416no—
344D4—48no—
345D4—44no—
346C2×C4—24no—
347C2×C4—28no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—28no—
351C2×C4—216no—
352C2×C4—24no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4D4×Dic514yes—
360C2×C4#408114yes—
361C2×C4#408314yes—
362C2×C4#408518yes—
363C2×C4C2×C22⋊Dic514yes—
364C2×C4C2×C22⋊Dic512yes—
365C2×C4C2×D4×D512yes—
366C2×C4C2×#71112yes—
367C2×C4C22×C22⋊D511yescharacteristic
368C2×C4C22×C22⋊D511yescharacteristic
369C8—108no—
370C8—204no—
371C5#2089611yescharacteristic, complemented, Sylow
372C22—28no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—416no—
378C22—48no—
379C22—48no—
380C22—42no—
381C22—44no—
382C22#6550914yes—
383C22C2×D4×Dic512yes—
384C22#6615314yes—
385C22C22×C22⋊Dic511yescharacteristic
386C4—24no—
387C4—42no—
388C4—44no—
389C4#6550914yes—
390C4C2×#408111yescharacteristic
391C4C2×#408311yescharacteristic
392C4C2×#408512yes—
393C2—24no—
394C2—44no—
395C2—82no—
396C2#1061464814yes—
397C2C2×#6550911yescharacteristic
398C2#1061580512yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutator, socleC23×C5C23×C4
Fitting, prime radicalC2×C5×#525C2
cosocleC22×C4×C5C24
hypercenterC2×#525D5
hypocenterC5#20896
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
2. C23×#681C2semidirect
3. C2×#65523C2semidirect
4. C2×#65496C2semidirect
5. #10602404C2semidirect
6. #10602372C2semidirect
25. C2×C22⋊#168C22semidirect
26. #64744C22semidirect
68. C22×#168D4semidirect
371. C5#20896semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5252
C23×#6812
C2×#655232
C2×#654962
#106024042
#106023722
#208965