Finite Group Databasebeta

← #13028151

Subgroups — #13028151

Order 1280.

2166 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×C4⋊C4C211yesmaximal, characteristic
2C2×#65765C211yesmaximal, characteristic
3C2×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4#10604909C214yesmaximal
5#10604345C214yesmaximal, complemented
6#10604342C214yesmaximal, complemented
7C23×C4×D5C2211yescharacteristic
8C23×C4×D5C2212yes—
9C22×C22⋊AGL(1,𝔽5)C2212yes—
10C22×C22⋊AGL(1,𝔽5)C2212yes—
11C22×C4⋊AGL(1,𝔽5)C2212yes—
12C22×C4×AGL(1,𝔽5)C2212yes—
13C22×C5×C4⋊C4C411yescharacteristic, complemented
14C22×C4⋊Dic5C411yescharacteristic, complemented
15C22×#683C412yescomplemented
16C2×D5×C4⋊C4—24no—
17C2×D5×C4⋊C4—24no—
18C2×D5×C4⋊C4C412yescomplemented
19C2×D5×C4⋊C4C412yescomplemented
20C2×D5×C4⋊C4C2212yes—
21C2×D5×C4⋊C4C2212yes—
22#65765—24no—
23C4×C22⋊AGL(1,𝔽5)—24no—
24C2×(C2×AGL(1,𝔽5))⋊C4C2212yes—
25C2×(C2×AGL(1,𝔽5))⋊C4C2212yes—
26C2×(C2×AGL(1,𝔽5))⋊C4C22116yes—
27#13924—51nomaximal, Sylow
28C24×D5C2311yescharacteristic
29C23×AGL(1,𝔽5)C2314yes—
30C23×AGL(1,𝔽5)C2314yes—
31C23×C4×C5C2×C411yescharacteristic
32C23×C4×C5C2×C412yes—
33C23×Dic5C2×C411yescharacteristic
34C23×Dic5C2×C412yes—
35C22×C4×D5—28no—
36C22×C4×D5—24no—
37C22×C4×D5—24no—
38C22×C4×D5—48no—
39C22×C4×D5C2×C411yescharacteristic
40C22×C4×D5C2×C411yescharacteristic
41C22×C4×D5C2×C414yes—
42C22×C4×D5D414yes—
43C22×C4×D5D414yes—
44C22×C4×D5C2314yes—
45C22×C4×D5C2311yescharacteristic
46C22×C4×D5C2311yescharacteristic
47C2×C22⋊AGL(1,𝔽5)—24no—
48C2×C22⋊AGL(1,𝔽5)—24no—
49C2×C22⋊AGL(1,𝔽5)—24no—
50C2×C22⋊AGL(1,𝔽5)—28no—
51C2×C22⋊AGL(1,𝔽5)Q818yes—
52C2×C4⋊AGL(1,𝔽5)—24no—
53C2×C4⋊AGL(1,𝔽5)—28no—
54C2×C4×AGL(1,𝔽5)—24no—
55C2×C4×AGL(1,𝔽5)—28no—
56C2×C5×C4⋊C4—24no—
57C2×C5×C4⋊C4C2×C412yes—
58C2×C5×C4⋊C4C2×C412yes—
59C2×C4⋊Dic5—24no—
60C2×C4⋊Dic5C2×C412yes—
61C2×C4⋊Dic5C2×C412yes—
62C2×#683—28no—
63C2×#683C2×C414yes—
64C2×#683C2×C414yes—
65D5×C4⋊C4—28no—
66D5×C4⋊C4—28no—
67D5×C4⋊C4—24no—
68D5×C4⋊C4—24no—
69D5×C4⋊C4—24no—
70D5×C4⋊C4—24no—
71(C2×AGL(1,𝔽5))⋊C4—24no—
72(C2×AGL(1,𝔽5))⋊C4—24no—
73(C2×AGL(1,𝔽5))⋊C4—416no—
74C23×C4⋊C4—51no—
75C2×#380—51no—
76C2×C4×C22⋊C4—51no—
77#1445—54no—
78C4⋊#95—54no—
79C22⋊#95—54no—
80C24×C5C22×C411yescharacteristic
81C23×D5—24no—
82C23×D5—24no—
83C23×D5C22×C411yescharacteristic
84C23×D5C2×D412yes—
85C23×D5C2×D412yes—
86C23×D5C2×Q814yes—
87C23×D5#4114yes—
88C23×D5C2411yescharacteristic
89C22×AGL(1,𝔽5)—24no—
90C22×AGL(1,𝔽5)—28no—
91C22×AGL(1,𝔽5)—24no—
92C22×AGL(1,𝔽5)—28no—
93C22×AGL(1,𝔽5)—24no—
94C22×AGL(1,𝔽5)—28no—
95C22×AGL(1,𝔽5)—28no—
96C22×AGL(1,𝔽5)—28no—
97C22×AGL(1,𝔽5)C2×Q818yes—
98C22×C4×C5—24no—
99C22×C4×C5—24no—
100C22×C4×C5—44no—
101C22×C4×C5C22⋊C414yes—
102C22×C4×C5C22×C414yes—
103C22×C4×C5C22×C411yescharacteristic
104C22×C4×C5C22×C411yescharacteristic
105C22×Dic5—24no—
106C22×Dic5—24no—
107C22×Dic5—44no—
108C22×Dic5C22⋊C414yes—
109C22×Dic5C22×C414yes—
110C22×Dic5C22×C411yescharacteristic
111C22×Dic5C22×C411yescharacteristic
112C2×C4×D5—24no—
113C2×C4×D5—24no—
114C2×C4×D5—24no—
115C2×C4×D5—216no—
116C2×C4×D5—216no—
117C2×C4×D5—28no—
118C2×C4×D5—24no—
119C2×C4×D5—24no—
120C2×C4×D5—216no—
121C2×C4×D5—48no—
122C2×C4×D5—416no—
123C2×C4×D5—416no—
124C2×C4×D5C22⋊C414yes—
125C2×C4×D5C22⋊C414yes—
126C2×C4×D5C2×D412yes—
127C2×C4×D5C2×D412yes—
128C2×C4×D5C2×D412yes—
129C2×C4×D5C2×D412yes—
130C2×C4×D5#4118yes—
131C22⋊AGL(1,𝔽5)—216no—
132C22⋊AGL(1,𝔽5)—48no—
133C4⋊AGL(1,𝔽5)—48no—
134C4×AGL(1,𝔽5)—48no—
135C5×C4⋊C4—24no—
136C5×C4⋊C4—24no—
137C4⋊Dic5—24no—
138C4⋊Dic5—24no—
139#683—28no—
140#683—28no—
141C24×C4—51no—
142C24×C4—52no—
143C22×C4⋊C4—52no—
144C22×C4⋊C4—51no—
145C22×C4⋊C4—52no—
146C22×C4⋊C4—52no—
147C22×C4⋊C4—52no—
148C22×C4⋊C4—52no—
149C22×C4⋊C4—52no—
150C22×C4⋊C4—104no—
151C22×C4⋊C4—104no—
152C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
153C22×C22⋊C4—52no—
154C22×C22⋊C4—52no—
155C22×C42—52no—
156#380—104no—
157C4×C22⋊C4—104no—
158C2×#95—52no—
159C2×#95—52no—
160C2×#95—516no—
161C23×C5—22no—
162C23×C5—22no—
163C23×C5C2×C22⋊C412yes—
164C23×C5C4×Q814yes—
165C23×C5C23×C411yescharacteristic
166C22×D5—22no—
167C22×D5—22no—
168C22×D5—24no—
169C22×D5—24no—
170C22×D5—24no—
171C22×D5—24no—
172C22×D5—216no—
173C22×D5—216no—
174C22×D5C2×C22⋊C412yes—
175C22×D5C4×Q814yes—
176C22×D5C22⋊Q818yes—
177C22×D5#12414yes—
178C22×D5C4⋊Q814yes—
179C22×D5C22×D411yescharacteristic
180C22×D5C22×D411yescharacteristic
181C22×D5C22×Q812yes—
182C22×D5C2×#4112yes—
183C22×D5C2×#4114yes—
184C22×D5H(5,𝔽2)14yes—
185C2×AGL(1,𝔽5)—28no—
186C2×AGL(1,𝔽5)—216no—
187C2×AGL(1,𝔽5)—44no—
188C2×AGL(1,𝔽5)—44no—
189C2×AGL(1,𝔽5)—48no—
190C2×AGL(1,𝔽5)—44no—
191C2×AGL(1,𝔽5)—44no—
192C2×AGL(1,𝔽5)—44no—
193C2×AGL(1,𝔽5)—416no—
194C2×C4×C5—22no—
195C2×C4×C5—28no—
196C2×C4×C5—24no—
197C2×C4×C5—28no—
198C2×C4×C5—48no—
199C2×C4×C5C2×C22⋊C412yes—
200C2×C4×C5C2×C22⋊C412yes—
201C2×C4×C5#11714yes—
202C2×Dic5—22no—
203C2×Dic5—28no—
204C2×Dic5—24no—
205C2×Dic5—28no—
206C2×Dic5—48no—
207C2×Dic5C2×C22⋊C412yes—
208C2×Dic5C2×C22⋊C412yes—
209C2×Dic5#11714yes—
210C4×D5—24no—
211C4×D5—24no—
212C4×D5—28no—
213C4×D5—28no—
214C4×D5—416no—
215C4×D5—48no—
216C4×D5—48no—
217C4×D5#12318yes—
218C4×D5#12318yes—
219C25—51no—
220C23×C4—51no—
221C23×C4—51no—
222C23×C4—54no—
223C23×C4—54no—
224C23×C4—51no—
225C23×C4—51no—
226C23×C4—51no—
227C23×C4—54no—
228C23×C4—52no—
229C23×C4—54no—
230C23×C4—54no—
231C23×C4—54no—
232C23×C4—108no—
233C23×C4—104no—
234C23×C4—104no—
235C23×C4—208no—
236C23×C4C2×AGL(1,𝔽5)11yescharacteristic
237C23×C4C2×AGL(1,𝔽5)12yes—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—54no—
240C2×C4⋊C4—54no—
241C2×C4⋊C4—52no—
242C2×C4⋊C4—52no—
243C2×C4⋊C4—104no—
244C2×C4⋊C4—104no—
245C2×C4⋊C4—104no—
246C2×C4⋊C4—104no—
247C2×C4⋊C4—108no—
248C2×C4⋊C4—108no—
249C2×C4⋊C4—108no—
250C2×C4⋊C4—104no—
251C2×C4⋊C4—108no—
252C2×C4⋊C4—104no—
253C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
254C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
255C2×C22⋊C4—58no—
256C2×C22⋊C4—104no—
257C2×C22⋊C4—104no—
258C2×C22⋊C4—108no—
259C2×C22⋊C4—104no—
260C2×C42—104no—
261C2×C42—108no—
262#95—104no—
263#95—104no—
264#95—2016no—
265C22×C5—22no—
266C22×C5—28no—
267C22×C5—22no—
268C22×C5#39114yes—
269C22×C5C22×C22⋊C411yescharacteristic
270C22×C5C2×#11712yes—
271C22×C5C2×C4×Q812yes—
272C22×C5#51812yes—
273C2×D5—22no—
274C2×D5—28no—
275C2×D5—22no—
276C2×D5—216no—
277C2×D5—28no—
278C2×D5—28no—
279C2×D5#39114yes—
280C2×D5C2×C22⋊Q812yes—
281C2×D5C2×#12312yes—
282C2×D5C2×#12312yes—
283C2×D5C2×#12411yescharacteristic
284C2×D5C2×C4⋊Q811yescharacteristic
285C2×D5#53512yes—
286C2×D5#53512yes—
287C2×D5#55318yes—
288C2×D5#55418yes—
289AGL(1,𝔽5)—48no—
290AGL(1,𝔽5)—84no—
291C4×C5—22no—
292C4×C5—44no—
293C4×C5#38114yes—
294Dic5—22no—
295Dic5—44no—
296Dic5#38114yes—
297C24—51no—
298C24—54no—
299C24—54no—
300C24—52no—
301C24—52no—
302C24—51no—
303C24—104no—
304C24—104no—
305C24C22×AGL(1,𝔽5)11yescharacteristic
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—44no—
309C22×C4—51no—
310C22×C4—58no—
311C22×C4—52no—
312C22×C4—52no—
313C22×C4—54no—
314C22×C4—51no—
315C22×C4—52no—
316C22×C4—52no—
317C22×C4—54no—
318C22×C4—54no—
319C22×C4—58no—
320C22×C4—54no—
321C22×C4—104no—
322C22×C4—1016no—
323C22×C4—1016no—
324C22×C4—104no—
325C22×C4—1016no—
326C22×C4—104no—
327C22×C4—104no—
328C22×C4—108no—
329C22×C4—104no—
330C22×C4—104no—
331C22×C4—104no—
332C22×C4—108no—
333C22×C4—108no—
334C22×C4—104no—
335C22×C4—104no—
336C22×C4—104no—
337C22×C4—108no—
338C22×C4—108no—
339C22×C4—108no—
340C22×C4—208no—
341C22×C4—2016no—
342C22×C4—2016no—
343C22×C4—204no—
344C22×C4C22⋊AGL(1,𝔽5)14yes—
345C22×C4C22×AGL(1,𝔽5)14yes—
346C22×C4C22×AGL(1,𝔽5)11yescharacteristic
347C22×C4C22×AGL(1,𝔽5)11yescharacteristic
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—108no—
351C4⋊C4—108no—
352C4⋊C4—104no—
353C4⋊C4—104no—
354C4⋊C4—208no—
355C22⋊C4—1016no—
356C22⋊C4—208no—
357C42—208no—
358C2×C5—24no—
359C2×C5C2×#38111yescharacteristic
360C2×C5C2×#39111yescharacteristic
361C2×C5#230911yescharacteristic
362C2×C5#234514yes—
363D5—24no—
364D5#248714yes—
365D5#250212yes—
366D5#250412yes—
367C23—22no—
368C23—22no—
369C23—54no—
370C23—54no—
371C23—52no—
372C23—52no—
373C23—51no—
374C23—51no—
375C23—54no—
376C23—52no—
377C23—58no—
378C23—54no—
379C23—54no—
380C23—104no—
381C23—104no—
382C23—104no—
383C23—104no—
384C23—1016no—
385C23—1016no—
386C23—102no—
387C23—102no—
388C23Q8×AGL(1,𝔽5)14yes—
389C23C2×C22⋊AGL(1,𝔽5)12yes—
390C23C23×AGL(1,𝔽5)11yescharacteristic
391C2×C4—22no—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—48no—
396C2×C4—54no—
397C2×C4—52no—
398C2×C4—58no—
399C2×C4—58no—
400C2×C4—52no—
401C2×C4—108no—
402C2×C4—104no—
403C2×C4—104no—
404C2×C4—108no—
405C2×C4—102no—
406C2×C4—108no—
407C2×C4—108no—
408C2×C4—1016no—
409C2×C4—108no—
410C2×C4—104no—
411C2×C4—208no—
412C2×C4—208no—
413C2×C4—2016no—
414C2×C4—208no—
415C2×C4—204no—
416C2×C4—208no—
417C2×C4—204no—
418C2×C4—204no—
419C2×C4—204no—
420C2×C4—2016no—
421C2×C4—204no—
422C2×C4#413014yes—
423C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
424C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
425C5#1392411yescharacteristic, complemented, Sylow
426C22—28no—
427C22—22no—
428C22—22no—
429C22—58no—
430C22—58no—
431C22—52no—
432C22—52no—
433C22—52no—
434C22—52no—
435C22—52no—
436C22—51no—
437C22—51no—
438C22—54no—
439C22—1016no—
440C22—108no—
441C22—108no—
442C22—108no—
443C22—102no—
444C22—102no—
445C22(C2×AGL(1,𝔽5))⋊Q814yes—
446C22C2×#413012yes—
447C22#6625812yes—
448C22C2×Q8×AGL(1,𝔽5)12yes—
449C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
450C4—22no—
451C4—44no—
452C4—54no—
453C4—102no—
454C4—208no—
455C4—204no—
456C4—404no—
457C4#6576414yes—
458C2—24no—
459C2—54no—
460C2—52no—
461C2—52no—
462C2—104no—
463C2#1061710414yes—
464C2C2×#6576411yescharacteristic
465C2#1061750711yescharacteristic
466C2C2×(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutator, socleC23×C5C23×C4
FittingC22×C5×C4⋊C4C4
prime radicalC24×D5C23
cosocleC23×D5C24
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5#13924
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #10604345C2semidirect
6. #10604342C2semidirect
13. C22×C5×C4⋊C4C4semidirect
14. C22×C4⋊Dic5C4semidirect
15. C22×#683C4semidirect
18. C2×D5×C4⋊C4C4semidirect
19. C2×D5×C4⋊C4C4semidirect
152. C22×C4⋊C4AGL(1,𝔽5)semidirect
425. C5#13924semidirect

Maximal subgroups

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

subgroupindex
C22×D5×C4⋊C42
C2×#657652
C2×C4×C22⋊AGL(1,𝔽5)2
#106049092
#106043452
#106043422
#139245