Finite Group Databasebeta

← C2×#10604424 #13030200

Subgroups — C2×#10604424 #13030200

Order 1280.

1194 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65912C211yesmaximal, characteristic
2C2×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C2×#65678C211yesmaximal, characteristic
4C2×#64926C212yesmaximal
5#10604424C218yesmaximal, direct product factor
6C2×#64869C212yesmaximal
7C22×C4⋊AGL(1,𝔽5)—21no—
8C22×C4×AGL(1,𝔽5)—21no—
9C2×C4×C5×Q8C411yescharacteristic, complemented
10C2×#4092C2211yescharacteristic
11C2×#4090C411yescharacteristic, complemented
12#65912—22no—
13#65912—22no—
14#65912C414yescomplemented
15#65912C2214yes—
16C2×#4043C2211yescharacteristic
17C2×#4042—21no—
18C2×#4035—21no—
19C2×#4034C411yescharacteristic, complemented
20C2×#4018—21no—
21C2×C42×D5C2211yescharacteristic
22C2×C4×Dic10—21no—
23C2×C4×Dic10C411yescharacteristic, complemented
24C2×(C2×AGL(1,𝔽5))⋊C4C2212yes—
25C2×#4004—21no—
26C2×#4003C2212yes—
27C2×C4×#170—21no—
28C4×C4⋊AGL(1,𝔽5)—22no—
29C4×C4⋊AGL(1,𝔽5)C2214yes—
30#65678—22no—
31#65678C2214yes—
32#64926C2218yes—
33#64869C2218yes—
34C2×#1503—51nomaximal, Sylow
35C23×AGL(1,𝔽5)—22no—
36C22×C5×Q8C2×C411yescharacteristic
37C22×C4×D5D411yescharacteristic
38C22×C4×D5D411yescharacteristic
39C22×C4×D5C2311yescharacteristic
40C22×Dic10—21no—
41C22×Dic10C2×C411yescharacteristic
42C22×#170—22no—
43C2×C4⋊AGL(1,𝔽5)—28no—
44C2×C4⋊AGL(1,𝔽5)—24no—
45C2×C4×AGL(1,𝔽5)—24no—
46C2×C4×AGL(1,𝔽5)—44no—
47C4×C5×Q8—22no—
48C4×C5×Q8C2×C414yescomplemented
49C2×C5×C4⋊C4—21no—
50C2×C5×C4⋊C4C2×C411yescharacteristic
51C2×C42×C5—21no—
52C2×C42×C5C2×C411yescharacteristic
53#4092—21no—
54#4092—21no—
55#4092C2×C412yes—
56#4092C2312yes—
57#4090—22no—
58#4090C2×C414yescomplemented
59C2×#685—21no—
60C2×#685—21no—
61C2×C4⋊Dic5—21no—
62C2×C4⋊Dic5C2×C411yescharacteristic
63C2×#683—21no—
64C2×#683—21no—
65C2×#683—21no—
66C2×#683—21no—
67C2×#683—21no—
68C2×C4×Dic5—21no—
69C2×C4×Dic5C2×C411yescharacteristic
70C2×C4×Dic5D411yescharacteristic
71C2×C4×Dic5D411yescharacteristic
72#4043—21no—
73#4043—21no—
74#4043C2×C412yes—
75#4043C2312yes—
76#4042—24no—
77#4042—24no—
78#4035—24no—
79#4035—24no—
80#4034—22no—
81#4034C2×C414yescomplemented
82#4018—24no—
83#4018—24no—
84C42×D5C2×C412yes—
85C42×D5D412yes—
86C42×D5D412yes—
87C42×D5C2312yes—
88C4×Dic10—22no—
89C4×Dic10—24no—
90C4×Dic10—24no—
91C4×Dic10C2×C414yescomplemented
92(C2×AGL(1,𝔽5))⋊C4—22no—
93(C2×AGL(1,𝔽5))⋊C4C2314yes—
94#4004—24no—
95#4003—22no—
96#4003C2314yes—
97C4×#170—24no—
98C2×#533—51no—
99C2×#431—51no—
100C2×C4×C4⋊C4—51no—
101#1503—58no—
102C2×#328—52no—
103C2×#328—52no—
104C23×D5C2×D411yescharacteristic
105C22×AGL(1,𝔽5)—28no—
106C22×AGL(1,𝔽5)—24no—
107C22×AGL(1,𝔽5)—44no—
108C22×AGL(1,𝔽5)—44no—
109C2×C5×Q8—21no—
110C2×C5×Q8—42no—
111C2×C5×Q8C22×C412yes—
112C22×C4×C5—21no—
113C22×C4×C5—21no—
114C22×C4×C5C22⋊C411yescharacteristic
115C22×C4×C5C22⋊C411yescharacteristic
116C22×C4×C5C22×C411yescharacteristic
117C22×Dic5—21no—
118C22×Dic5—21no—
119C22×Dic5C22⋊C411yescharacteristic
120C22×Dic5C22⋊C411yescharacteristic
121C22×Dic5C2×D411yescharacteristic
122C2×C4×D5—24no—
123C2×C4×D5—42no—
124C2×C4×D5C22⋊C412yes—
125C2×C4×D5C22⋊C412yes—
126C2×C4×D5SD1614yes—
127C2×C4×D5Q1614yes—
128C2×C4×D5C22×C411yescharacteristic
129C2×C4×D5C2×D412yes—
130C2×C4×D5C2×D411yescharacteristic
131C2×C4×D5C2×D411yescharacteristic
132C2×C4×D5C2×D412yes—
133C2×C4×D5C2411yescharacteristic
134C2×Dic10—21no—
135C2×Dic10—24no—
136C2×Dic10—44no—
137C2×Dic10—42no—
138C2×Dic10C22×C412yes—
139C2×#170—24no—
140C2×#170—44no—
141C4⋊AGL(1,𝔽5)—216no—
142C4×AGL(1,𝔽5)—48no—
143C5×C4⋊C4—24no—
144C5×C4⋊C4—21no—
145C5×C4⋊C4C22×C412yes—
146C42×C5—24no—
147C42×C5C22⋊C412yes—
148C42×C5C22×C412yes—
149#685—22no—
150#685—22no—
151#685—22no—
152#685—22no—
153C4⋊Dic5—21no—
154C4⋊Dic5—22no—
155C4⋊Dic5—22no—
156C4⋊Dic5C22×C412yes—
157#683—24no—
158#683—22no—
159#683—24no—
160#683—24no—
161#683—24no—
162#683—22no—
163C4×Dic5—24no—
164C4×Dic5C22⋊C412yes—
165C4×Dic5C22⋊C412yes—
166C4×Dic5C22⋊C412yes—
167C4×Dic5C22×C412yes—
168C4×Dic5C2×D412yes—
169C4×Dic5C2×D412yes—
170#533—54no—
171#533—54no—
172#533—102no—
173#533—102no—
174C2×C4⋊Q8—51no—
175C2×C4⋊Q8—51no—
176C2×#125—101no—
177C2×C22⋊Q8—51no—
178C2×C22⋊Q8—51no—
179C2×C22⋊Q8—101no—
180C2×C4×Q8—51no—
181C2×C4×Q8—101no—
182C2×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
183C2×#117—101no—
184C22×C4⋊C4—101no—
185C22×C42—51no—
186C22×C42—101no—
187#431—54no—
188#431—102no—
189C2×C4⋊C8—101no—
190C2×C22⋊C8—52no—
191C2×C4×C8—101no—
192C4×C4⋊C4—54no—
193C4×C4⋊C4—102no—
194C2×#95—52no—
195#328—58no—
196#328—58no—
197C23×C5C2×C22⋊C411yescharacteristic
198C22×D5—22no—
199C22×D5—22no—
200C22×D5C2×C22⋊C411yescharacteristic
201C22×D5C2×SD1612yes—
202C22×D5C2×Q1612yes—
203C22×D5C22×D411yescharacteristic
204C2×AGL(1,𝔽5)—28no—
205C2×AGL(1,𝔽5)—44no—
206C2×AGL(1,𝔽5)—416no—
207C2×AGL(1,𝔽5)—44no—
208C5×Q8—42no—
209C5×Q8—42no—
210C2×C4×C5—22no—
211C2×C4×C5—22no—
212C2×C4×C5—22no—
213C2×C4×C5—22no—
214C2×C4×C5—42no—
215C2×C4×C5—42no—
216C2×C4×C5#10314yes—
217C2×C4×C5C2×C22⋊C411yescharacteristic
218C2×C4×C5C2×C22⋊C412yes—
219C2×C4×C5C2×C22⋊C412yes—
220C2×C4×C5C23×C411yescharacteristic
221C2×Dic5—22no—
222C2×Dic5—22no—
223C2×Dic5—22no—
224C2×Dic5—42no—
225C2×Dic5—42no—
226C2×Dic5#10314yes—
227C2×Dic5#10414yes—
228C2×Dic5C2×C22⋊C411yescharacteristic
229C2×Dic5C2×C22⋊C412yes—
230C2×Dic5C2×C22⋊C412yes—
231C2×Dic5C22×D411yescharacteristic
232C4×D5—24no—
233C4×D5—24no—
234C4×D5—41no—
235C4×D5—41no—
236C4×D5—41no—
237C4×D5—41no—
238C4×D5#10318yes—
239C4×D5C2×SD1614yes—
240C4×D5C2×Q1614yes—
241Dic10—42no—
242Dic10—48no—
243Dic10—42no—
244#170—44no—
245C22×Q8—51no—
246C22×Q8—101no—
247C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
248C23×C4—51no—
249C23×C4—51no—
250C23×C4—51no—
251C23×C4—102no—
252C22×C8—102no—
253C4⋊Q8—54no—
254C4⋊Q8—54no—
255C4⋊Q8—102no—
256C4⋊Q8—102no—
257#125—104no—
258#125—104no—
259C22⋊Q8—52no—
260C22⋊Q8—52no—
261C22⋊Q8—52no—
262C22⋊Q8—52no—
263C22⋊Q8—101no—
264C22⋊Q8—101no—
265C22⋊Q8—104no—
266C22⋊Q8—104no—
267C22⋊Q8—101no—
268C22⋊Q8—101no—
269C4×Q8—22no—
270C4×Q8—54no—
271C4×Q8—104no—
272C4×Q8—104no—
273C4×Q8—102no—
274C4×Q8C2×AGL(1,𝔽5)14yescomplemented
275#117—104no—
276#117—104no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—51no—
279C2×C4⋊C4—101no—
280C2×C4⋊C4—101no—
281C2×C4⋊C4—101no—
282C2×C4⋊C4—101no—
283C2×C4⋊C4—101no—
284C2×C4⋊C4—104no—
285C2×C4⋊C4—108no—
286C2×C4⋊C4—101no—
287C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
288C2×C22⋊C4—101no—
289C2×C22⋊C4—101no—
290C2×C42—21no—
291C2×C42—51no—
292C2×C42—51no—
293C2×C42—51no—
294C2×C42—52no—
295C2×C42—52no—
296C2×C42—52no—
297C2×C42—52no—
298C2×C42—101no—
299C2×C42—104no—
300C2×C42—204no—
301C2×C42C2×AGL(1,𝔽5)11yescharacteristic
302C4⋊C8—104no—
303C22⋊C8—54no—
304C22⋊C8—102no—
305C4×C8—104no—
306#95—54no—
307#95—102no—
308C22×C5#41012yes—
309C22×C5C2×#10312yes—
310C22×C5C2×#10412yes—
311C22×C5C22×C22⋊C411yescharacteristic
312C2×D5—21no—
313C2×D5—21no—
314C2×D5—21no—
315C2×D5—21no—
316C2×D5—24no—
317C2×D5—24no—
318C2×D5C2×#10312yes—
319C2×D5C22×SD1611yescharacteristic
320C2×D5C22×Q1611yescharacteristic
321AGL(1,𝔽5)—48no—
322C4×C5—21no—
323C4×C5—21no—
324C4×C5—22no—
325C4×C5—42no—
326C4×C5—42no—
327C4×C5C2×#10314yes—
328Dic5—21no—
329Dic5—21no—
330Dic5—42no—
331Dic5—42no—
332Dic5C2×#10314yes—
333Dic5C2×#10414yes—
334C24—51no—
335C2×Q8—21no—
336C2×Q8—42no—
337C2×Q8—52no—
338C2×Q8—104no—
339C2×Q8—101no—
340C2×Q8—202no—
341C2×Q8—204no—
342C2×Q8C22×AGL(1,𝔽5)12yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—51no—
346C22×C4—51no—
347C22×C4—52no—
348C22×C4—52no—
349C22×C4—54no—
350C22×C4—54no—
351C22×C4—51no—
352C22×C4—52no—
353C22×C4—52no—
354C22×C4—51no—
355C22×C4—51no—
356C22×C4—51no—
357C22×C4—51no—
358C22×C4—104no—
359C22×C4—104no—
360C22×C4—108no—
361C22×C4—101no—
362C22×C4—101no—
363C22×C4—202no—
364C22×C4—204no—
365C22×C4—204no—
366C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
367C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
368C22×C4C22×AGL(1,𝔽5)11yescharacteristic
369C2×C8—104no—
370C2×C8—204no—
371C4⋊C4—24no—
372C4⋊C4—21no—
373C4⋊C4—52no—
374C4⋊C4—104no—
375C4⋊C4—104no—
376C4⋊C4—102no—
377C4⋊C4—102no—
378C4⋊C4—102no—
379C4⋊C4—102no—
380C4⋊C4—104no—
381C4⋊C4—1016no—
382C4⋊C4—101no—
383C4⋊C4—104no—
384C4⋊C4C22×AGL(1,𝔽5)12yes—
385C22⋊C4—102no—
386C22⋊C4—102no—
387C22⋊C4—102no—
388C22⋊C4—102no—
389C42—24no—
390C42—52no—
391C42—52no—
392C42—52no—
393C42—52no—
394C42—52no—
395C42—52no—
396C42—104no—
397C42—208no—
398C42C22⋊AGL(1,𝔽5)12yes—
399C42C22×AGL(1,𝔽5)12yes—
400C2×C5#150314yescomplemented
401C2×C5C2×#41011yescharacteristic
402C2×C5C22×#10311yescharacteristic
403C2×C5C22×#10411yescharacteristic
404D5—21no—
405D5—21no—
406D5—22no—
407C23—51no—
408C23—51no—
409C23—52no—
410C23—52no—
411C23—102no—
412C23—102no—
413C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
414Q8—42no—
415Q8—42no—
416Q8—202no—
417Q8—202no—
418Q8—208no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—54no—
426C2×C4—52no—
427C2×C4—52no—
428C2×C4—54no—
429C2×C4—54no—
430C2×C4—58no—
431C2×C4—54no—
432C2×C4—51no—
433C2×C4—51no—
434C2×C4—104no—
435C2×C4—104no—
436C2×C4—102no—
437C2×C4—108no—
438C2×C4—102no—
439C2×C4—102no—
440C2×C4—201no—
441C2×C4—201no—
442C2×C4—201no—
443C2×C4—201no—
444C2×C4—204no—
445C2×C4—204no—
446C2×C4—2016no—
447C2×C4—202no—
448C2×C4—202no—
449C2×C4#400914yes—
450C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
451C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
452C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
453C2×C4C23×AGL(1,𝔽5)11yescharacteristic
454C8—204no—
455C5C2×#150311yescharacteristic, complemented, Sylow
456C22—51no—
457C22—51no—
458C22—52no—
459C22—101no—
460C22—101no—
461C22—104no—
462C22—104no—
463C22—101no—
464C22—101no—
465C22#6574512yes—
466C22C2×#400912yes—
467C22D5⋊#10412yes—
468C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
469C4—21no—
470C4—21no—
471C4—22no—
472C4—42no—
473C4—42no—
474C4—54no—
475C4—54no—
476C4—101no—
477C4—101no—
478C4—208no—
479C4—202no—
480C4—202no—
481C4C2×#400914yes—
482C2—101no—
483C2—101no—
484C2—102no—
485C2#1060442414yesdirect product factor
486C2C2×#6574511yescharacteristic
487C2C22×#400911yescharacteristic
488C2C2×D5⋊#10411yescharacteristic
subgroupG/H
centerC22×C4C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC2×C4×C5×Q8C4
socleC23×C5C2×C22⋊C4
prime radicalC23×D5C2×D4
cosocleC2×C4×D5C24
hypercenterC2×C4×Q8AGL(1,𝔽5)
hypocenterC5C2×#1503
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
5. #10604424C2direct (N × H)
9. C2×C4×C5×Q8C4semidirect
11. C2×#4090C4semidirect
14. #65912C4semidirect
19. C2×#4034C4semidirect
23. C2×C4×Dic10C4semidirect
48. C4×C5×Q8C2×C4semidirect
58. #4090C2×C4semidirect
81. #4034C2×C4semidirect
91. C4×Dic10C2×C4semidirect
182. C2×C4×Q8AGL(1,𝔽5)semidirect
274. C4×Q8C2×AGL(1,𝔽5)semidirect
400. C2×C5#1503semidirect
455. C5C2×#1503semidirect
485. C2#10604424direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#659122
C2×C4×C4⋊AGL(1,𝔽5)2
C2×#656782
C2×#649262
#106044242
C2×#648692
C2×#15035