Finite Group Databasebeta

← #12026022

Subgroups — #12026022

Order 1280.

1566 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#380C211yesmaximal, characteristic
2C2×#65500C211yesmaximal, characteristic
3C2×C4×C22⋊Dic5C211yesmaximal, characteristic
4#10602361C212yesmaximal, complemented
5#10602360C212yesmaximal, complemented
6#10602340C214yesmaximal
7#10600199C214yesmaximal, complemented
8C24×C4×C5C2211yescharacteristic
9C22×C5×C4⋊C4C2212yes—
10C22×C5×C22⋊C4C2212yes—
11C22×C22⋊Dic5C2212yes—
12C22×C22⋊Dic5C2212yes—
13C22×#683C2214yes—
14C22×C4×Dic5C2212yes—
15C5×#380—24no—
16C2×C5×#95C2212yes—
17#65500C418yescomplemented
18C4×C22⋊Dic5—24no—
19C2×#3963C2214yes—
20C2×#3963C2211yescharacteristic
21C2×#3963C2211yescharacteristic
22C2×#3963C2214yes—
23C2×#3963C2214yes—
24C2×#3963C2212yes—
25C2×#3963C2212yes—
26#14295—51nomaximal, Sylow
27C25×C5C2311yescharacteristic
28C23×C4×C5—44no—
29C23×C4×C5C2×C414yes—
30C23×C4×C5D414yes—
31C23×C4×C5Q814yes—
32C23×C4×C5C2314yes—
33C23×C4×C5C2311yescharacteristic
34C23×C4×C5C2311yescharacteristic
35C23×Dic5C2314yes—
36C23×Dic5C2312yes—
37C23×Dic5C2312yes—
38C2×C5×C4⋊C4—24no—
39C2×C5×C4⋊C4—44no—
40C2×C5×C22⋊C4—24no—
41C2×C5×C22⋊C4—42no—
42C2×C5×C22⋊C4—42no—
43C2×C22⋊Dic5—24no—
44C2×C22⋊Dic5—24no—
45C2×C22⋊Dic5—24no—
46C2×C22⋊Dic5—44no—
47C2×C22⋊Dic5C2×C418yes—
48C2×#683—28no—
49C2×#683—44no—
50C2×#683—44no—
51C2×C4×Dic5—24no—
52C2×C4×Dic5—44no—
53C5×#95—24no—
54#3963—22no—
55#3963—22no—
56#3963—28no—
57#3963—44no—
58#3963—44no—
59#3963C2×C418yes—
60#3963C2×C418yes—
61C2×#381—51no—
62C2×#380D511yescharacteristic
63C2×C4×C22⋊C4—51no—
64#1444—52no—
65#1442—52no—
66#1439—54no—
67#1436—54no—
68C24×C5—22no—
69C24×C5—22no—
70C24×C5—24no—
71C24×C5—22no—
72C24×C5—22no—
73C24×C5C22×C412yes—
74C24×C5C2×D412yes—
75C24×C5C2×Q812yes—
76C24×C5C2411yescharacteristic
77C22×C4×C5—24no—
78C22×C4×C5—24no—
79C22×C4×C5—24no—
80C22×C4×C5—24no—
81C22×C4×C5—24no—
82C22×C4×C5—24no—
83C22×C4×C5—24no—
84C22×C4×C5—24no—
85C22×C4×C5—24no—
86C22×C4×C5—48no—
87C22×C4×C5—48no—
88C22×C4×C5—48no—
89C22×C4×C5C4⋊C4116yescomplemented
90C22×C4×C5C22×C412yes—
91C22×C4×C5C22×C412yes—
92C22×C4×C5C2×D412yes—
93C22×C4×C5C2×D412yes—
94C22×C4×C5C2×Q812yes—
95C22×C4×C5C2×Q812yes—
96C22×Dic5—22no—
97C22×Dic5—22no—
98C22×Dic5—22no—
99C22×Dic5—22no—
100C22×Dic5—24no—
101C22×Dic5—24no—
102C22×Dic5—24no—
103C22×Dic5—24no—
104C22×Dic5—24no—
105C22×Dic5—24no—
106C22×Dic5—24no—
107C22×Dic5—24no—
108C22×Dic5—24no—
109C22×Dic5—24no—
110C22×Dic5C22×C418yes—
111C22×Dic5#4114yes—
112C22×Dic5#4114yes—
113C5×C4⋊C4—48no—
114C5×C22⋊C4—48no—
115C22⋊Dic5—216no—
116C22⋊Dic5—48no—
117#683—416no—
118C4×Dic5—48no—
119C24×C4C2×D511yescharacteristic
120C22×C4⋊C4—54no—
121C22×C4⋊C4C2×D512yes—
122C22×C22⋊C4—52no—
123C22×C22⋊C4—52no—
124C22×C22⋊C4C2×D512yes—
125C22×C42—52no—
126#381—58no—
127#380—24no—
128C4×C22⋊C4—104no—
129C2×#95—52no—
130C2×#95—52no—
131C2×#95—54no—
132C2×#95—51no—
133C2×#95—51no—
134C2×#95—54no—
135C2×#95—54no—
136C2×#95C2×D512yes—
137C23×C5—24no—
138C23×C5—24no—
139C23×C5—24no—
140C23×C5—24no—
141C23×C5—22no—
142C23×C5—22no—
143C23×C5—24no—
144C23×C5—22no—
145C23×C5—22no—
146C23×C5—24no—
147C23×C5—24no—
148C23×C5—44no—
149C23×C5—44no—
150C23×C5—44no—
151C23×C5—44no—
152C23×C5C2×C4⋊C414yes—
153C23×C5C23×C411yescharacteristic
154C23×C5C22×D411yescharacteristic
155C23×C5C22×Q811yescharacteristic
156C23×C5C2×#4112yes—
157C23×C5C2×#4112yes—
158C23×C5H(5,𝔽2)14yes—
159C23×C5H(5,𝔽2)12yes—
160C23×C52−1+412yes—
161C2×C4×C5—24no—
162C2×C4×C5—24no—
163C2×C4×C5—24no—
164C2×C4×C5—24no—
165C2×C4×C5—24no—
166C2×C4×C5—24no—
167C2×C4×C5—416no—
168C2×C4×C5—44no—
169C2×C4×C5—44no—
170C2×C4×C5—44no—
171C2×C4×C5—44no—
172C2×C4×C5—44no—
173C2×C4×C5—44no—
174C2×C4×C5—44no—
175C2×C4×C5C2×C4⋊C414yes—
176C2×C4×C5C2×C4⋊C414yes—
177C2×Dic5—28no—
178C2×Dic5—28no—
179C2×Dic5—28no—
180C2×Dic5—24no—
181C2×Dic5—24no—
182C2×Dic5—28no—
183C2×Dic5—28no—
184C2×Dic5—42no—
185C2×Dic5—42no—
186C2×Dic5—42no—
187C2×Dic5—42no—
188C2×Dic5—44no—
189C2×Dic5—44no—
190C2×Dic5—44no—
191C2×Dic5—44no—
192C2×Dic5—44no—
193C2×Dic5—44no—
194C25C22×D511yescharacteristic
195C23×C4—44no—
196C23×C4—52no—
197C23×C4—54no—
198C23×C4—52no—
199C23×C4Dic1014yes—
200C23×C4C4×D514yes—
201C23×C4C22⋊D514yes—
202C23×C4C22×D514yes—
203C23×C4C22×D511yescharacteristic
204C23×C4C22×D511yescharacteristic
205C2×C4⋊C4—24no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—108no—
208C2×C4⋊C4—204no—
209C2×C4⋊C4—204no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—58no—
214C2×C22⋊C4—104no—
215C2×C22⋊C4—104no—
216C2×C22⋊C4—104no—
217C2×C22⋊C4—204no—
218C2×C42—104no—
219C2×C42—204no—
220#95—24no—
221#95—58no—
222#95—58no—
223#95—108no—
224#95—102no—
225#95—102no—
226#95—204no—
227#95—204no—
228C22×C5—24no—
229C22×C5—24no—
230C22×C5—24no—
231C22×C5—48no—
232C22×C5—44no—
233C22×C5—44no—
234C22×C5—44no—
235C22×C5—44no—
236C22×C5C22×C4⋊C411yescharacteristic
237C22×C5C4×#4112yes—
238C22×C5#51812yes—
239C22×C5#52012yes—
240C22×C5#52512yes—
241C22×C5#53311yescharacteristic
242C22×C5#53311yescharacteristic
243C22×C5#53412yes—
244C22×C5#53511yescharacteristic
245C22×C5#53611yescharacteristic
246C22×C5#53814yes—
247C22×C5#53914yes—
248C22×C5#54014yes—
249C22×C5#54214yes—
250C22×C5#54312yes—
251C22×C5#54412yes—
252C4×C5—24no—
253C4×C5—24no—
254C4×C5—84no—
255Dic5—48no—
256Dic5—44no—
257Dic5—44no—
258C24—22no—
259C24—24no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24C2×Dic1012yes—
264C24C2×C4×D512yes—
265C24C2×C22⋊D512yes—
266C24C23×D511yescharacteristic
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—58no—
280C22×C4—54no—
281C22×C4—54no—
282C22×C4—104no—
283C22×C4—104no—
284C22×C4—104no—
285C22×C4—104no—
286C22×C4—104no—
287C22×C4—104no—
288C22×C4—104no—
289C22×C4—104no—
290C22×C4—104no—
291C22×C4—102no—
292C22×C4—102no—
293C22×C4—104no—
294C22×C4—102no—
295C22×C4—102no—
296C22×C4#683116yescomplemented
297C22×C4C2×Dic1012yes—
298C22×C4C2×Dic1012yes—
299C22×C4C2×C4×D512yes—
300C22×C4C2×C4×D512yes—
301C22×C4C2×C22⋊D512yes—
302C22×C4C2×C22⋊D512yes—
303C4⋊C4—48no—
304C4⋊C4—2016no—
305C22⋊C4—48no—
306C22⋊C4—1016no—
307C22⋊C4—208no—
308C42—208no—
309C2×C5—44no—
310C2×C5#229611yescharacteristic
311C2×C5#231511yescharacteristic
312C2×C5#231711yescharacteristic
313C2×C5#233514yes—
314C2×C5#263512yes—
315C2×C5#263812yes—
316C2×C5#266512yes—
317C2×C5#266712yes—
318C23—24no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—24no—
326C23—22no—
327C23—22no—
328C23—24no—
329C23—44no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23C2×#68314yes—
334C23C22×Dic1011yescharacteristic
335C23C22×C4×D511yescharacteristic
336C23#414414yes—
337C23#414412yes—
338C23#414712yes—
339C23D5×#4112yes—
340C23D5×#4112yes—
341C23C22×C22⋊D511yescharacteristic
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—416no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—108no—
357C2×C4—108no—
358C2×C4—108no—
359C2×C4—104no—
360C2×C4—104no—
361C2×C4—108no—
362C2×C4—108no—
363C2×C4—202no—
364C2×C4—202no—
365C2×C4—202no—
366C2×C4—202no—
367C2×C4—204no—
368C2×C4—204no—
369C2×C4—204no—
370C2×C4—204no—
371C2×C4—204no—
372C2×C4—204no—
373C2×C4C2×#68314yes—
374C2×C4C2×#68314yes—
375C5#1429511yescharacteristic, complemented, Sylow
376C22—24no—
377C22—24no—
378C22—24no—
379C22—44no—
380C22—48no—
381C22—44no—
382C22—44no—
383C22—44no—
384C22#6581712yes—
385C22#6582012yes—
386C22#6583412yes—
387C22#6583812yes—
388C22#6585111yescharacteristic
389C22#6585111yescharacteristic
390C22#6585612yes—
391C22#6594714yes—
392C22#6594914yes—
393C22#6597214yes—
394C22#6597414yes—
395C22C22×#68311yescharacteristic
396C22#6613812yes—
397C22#6613911yescharacteristic
398C22#6615011yescharacteristic
399C22#6616212yes—
400C4—24no—
401C4—24no—
402C4—84no—
403C4—204no—
404C4—208no—
405C4—204no—
406C2—44no—
407C2#1060560614yes—
408C2#1060568912yes—
409C2#1060569012yes—
410C2#1061382611yescharacteristic
411C2#1061388811yescharacteristic
412C2#1061395711yescharacteristic
413C2#1061444212yes—
414C2#1061455412yes—
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#380C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×#380D5
hypocenterC5#14295
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. #10602361C2semidirect
5. #10602360C2semidirect
7. #10600199C2semidirect
17. #65500C4semidirect
89. C22×C4×C5C4⋊C4semidirect
296. C22×C4#683semidirect
375. C5#14295semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3802
C2×#655002
C2×C4×C22⋊Dic52
#106023612
#106023602
#106023402
#106001992
#142955