Finite Group Databasebeta

← C2×#10604282 #13034385

Subgroups — C2×#10604282 #13034385

Order 1280.

1878 conjugacy classes of subgroups.

The subgroup lattice has 526 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×D5⋊(C22⋊C8)C211yesmaximal, characteristic
4C2×#64874C214yesmaximal, complemented
5#10604282C218yesmaximal, direct product factor
6C23×C4×D5—21no—
7C23×C4×D5C2211yescharacteristic
8C22×C22⋊AGL(1,𝔽5)—21no—
9C22×C4⋊AGL(1,𝔽5)C2212yes—
10C22×D5⋊C8C2212yes—
11C22×C5×C4⋊C4C411yescharacteristic, complemented
12C22×C4⋊Dic5C411yescharacteristic, complemented
13C22×#683—21no—
14C2×D5×C4⋊C4—21no—
15C2×D5×C4⋊C4—22no—
16C2×D5×C4⋊C4—21no—
17C2×D5×C4⋊C4—22no—
18C2×D5×C4⋊C4C412yescomplemented
19C2×D5×C4⋊C4C414yescomplemented
20C2×D5×C4⋊C4C2212yes—
21C2×D5×C4⋊C4C2214yes—
22C2×C22⋊#170—21no—
23#65765—22no—
24#65765C2214yes—
25C2×(C2×AGL(1,𝔽5))⋊C4—21no—
26D5⋊(C22⋊C8)—22no—
27D5⋊(C22⋊C8)C2214yes—
28C2×#4003—21no—
29#64874C22116yescomplemented
30C2×#1823—51nomaximal, Sylow
31C24×D5D411yescharacteristic
32C23×AGL(1,𝔽5)—22no—
33C23×C4×C5—21no—
34C23×C4×C5C2×C411yescharacteristic
35C23×Dic5—21no—
36C23×Dic5D411yescharacteristic
37C22×C4×D5—28no—
38C22×C4×D5—24no—
39C22×C4×D5—22no—
40C22×C4×D5—22no—
41C22×C4×D5—24no—
42C22×C4×D5—44no—
43C22×C4×D5—44no—
44C22×C4×D5C2×C411yescharacteristic
45C22×C4×D5C2×C412yes—
46C22×C4×D5D411yescharacteristic
47C22×C4×D5D412yes—
48C22×C4×D5D411yescharacteristic
49C22×C4×D5D412yes—
50C22×C4×D5C2312yes—
51C22×C4×D5C2311yescharacteristic
52C2×C22⋊AGL(1,𝔽5)—24no—
53C2×C22⋊AGL(1,𝔽5)—44no—
54C22×#170—22no—
55C2×C4⋊AGL(1,𝔽5)—28no—
56C2×C4⋊AGL(1,𝔽5)—22no—
57C2×C4⋊AGL(1,𝔽5)C2314yes—
58C2×D5⋊C8—28no—
59C2×D5⋊C8—22no—
60C2×D5⋊C8C2314yes—
61C2×C5×C4⋊C4—22no—
62C2×C5×C4⋊C4—21no—
63C2×C5×C4⋊C4C2×C412yes—
64C2×C5×C4⋊C4C2×C414yescomplemented
65C2×C4⋊Dic5—22no—
66C2×C4⋊Dic5—21no—
67C2×C4⋊Dic5C2×C412yes—
68C2×C4⋊Dic5C2×C414yescomplemented
69C2×#683—24no—
70C2×#683—22no—
71C2×#683—22no—
72C2×#683—24no—
73D5×C4⋊C4—24no—
74D5×C4⋊C4—24no—
75D5×C4⋊C4—24no—
76D5×C4⋊C4—22no—
77D5×C4⋊C4—24no—
78D5×C4⋊C4—22no—
79D5×C4⋊C4—24no—
80D5×C4⋊C4C2×C414yes—
81D5×C4⋊C4C2×C418yescomplemented
82D5×C4⋊C4C2314yes—
83C22⋊#170—24no—
84(C2×AGL(1,𝔽5))⋊C4—24no—
85#4003—24no—
86C23×C4⋊C4—51no—
87C22×C22⋊C8—51no—
88C2×#380—51no—
89#1823—58no—
90C2×#340—54no—
91C24×C5C22⋊C411yescharacteristic
92C23×D5—24no—
93C23×D5—24no—
94C23×D5C22⋊C412yes—
95C23×D5C22⋊C411yescharacteristic
96C23×D5D814yes—
97C23×D5SD1614yes—
98C23×D5C2×D412yes—
99C23×D5C2×D411yescharacteristic
100C22×AGL(1,𝔽5)—24no—
101C22×AGL(1,𝔽5)—44no—
102C22×AGL(1,𝔽5)—44no—
103C22×AGL(1,𝔽5)—44no—
104C22×C4×C5—24no—
105C22×C4×C5—22no—
106C22×C4×C5—24no—
107C22×C4×C5—44no—
108C22×C4×C5C22⋊C412yes—
109C22×C4×C5C22⋊C411yescharacteristic
110C22×C4×C5C22×C412yes—
111C22×C4×C5C22×C411yescharacteristic
112C22×Dic5—24no—
113C22×Dic5—22no—
114C22×Dic5—24no—
115C22×Dic5—44no—
116C22×Dic5C22⋊C412yes—
117C22×Dic5C22⋊C411yescharacteristic
118C22×Dic5C2×D412yes—
119C22×Dic5C2×D411yescharacteristic
120C2×C4×D5—24no—
121C2×C4×D5—24no—
122C2×C4×D5—24no—
123C2×C4×D5—28no—
124C2×C4×D5—28no—
125C2×C4×D5—28no—
126C2×C4×D5—28no—
127C2×C4×D5—24no—
128C2×C4×D5—28no—
129C2×C4×D5—24no—
130C2×C4×D5—48no—
131C2×C4×D5—416no—
132C2×C4×D5—48no—
133C2×C4×D5—44no—
134C2×C4×D5—48no—
135C2×C4×D5—44no—
136C2×C4×D5C22⋊C412yes—
137C2×C4×D5C22⋊C412yes—
138C2×C4×D5C22⋊C412yes—
139C2×C4×D5C22×C412yes—
140C2×C4×D5C22×C411yescharacteristic
141C2×C4×D5C2×D412yes—
142C2×C4×D5C2×D412yes—
143C2×C4×D5C2×D411yescharacteristic
144C2×C4×D5C2×D411yescharacteristic
145C2×C4×D5#4118yes—
146C2×C4×D5C2411yescharacteristic
147C22⋊AGL(1,𝔽5)—48no—
148C2×#170—24no—
149C2×#170—44no—
150C4⋊AGL(1,𝔽5)—28no—
151C4⋊AGL(1,𝔽5)—44no—
152D5⋊C8—28no—
153D5⋊C8—44no—
154C5×C4⋊C4—22no—
155C5×C4⋊C4—24no—
156C5×C4⋊C4C22×C414yes—
157C4⋊Dic5—22no—
158C4⋊Dic5—24no—
159C4⋊Dic5C22×C414yes—
160#683—24no—
161#683—28no—
162#683—24no—
163C24×C4—51no—
164C24×C4—101no—
165C23×C8—52no—
166C22×C4⋊C4—51no—
167C22×C4⋊C4—52no—
168C22×C4⋊C4—52no—
169C22×C4⋊C4—54no—
170C22×C4⋊C4—54no—
171C22×C4⋊C4—52no—
172C22×C4⋊C4—101no—
173C22×C4⋊C4—102no—
174C22×C4⋊C4—102no—
175C22×C4⋊C4—101no—
176C22×C4⋊C4—101no—
177C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
178C22×C22⋊C4—101no—
179C2×C22⋊C8—54no—
180C2×C22⋊C8—101no—
181C2×C22⋊C8—101no—
182C2×C22⋊C8—102no—
183#380—54no—
184#380—102no—
185C2×#95—101no—
186#340—516no—
187C23×C5—22no—
188C23×C5—22no—
189C23×C5#10214yes—
190C23×C5C2×C22⋊C412yes—
191C23×C5C2×C22⋊C411yescharacteristic
192C22×D5—22no—
193C22×D5—22no—
194C22×D5—24no—
195C22×D5—24no—
196C22×D5—24no—
197C22×D5—24no—
198C22×D5—216no—
199C22×D5—216no—
200C22×D5#10214yes—
201C22×D5#10218yes—
202C22×D5C2×C22⋊C412yes—
203C22×D5C2×C22⋊C411yescharacteristic
204C22×D5#12314yes—
205C22×D5C2×D814yes—
206C22×D5C2×D812yes—
207C22×D5C2×SD1614yes—
208C22×D5C2×SD1612yes—
209C22×D5#13714yes—
210C22×D5C22×D411yescharacteristic
211C2×AGL(1,𝔽5)—44no—
212C2×AGL(1,𝔽5)—44no—
213C2×AGL(1,𝔽5)—44no—
214C2×AGL(1,𝔽5)—88no—
215C2×C4×C5—22no—
216C2×C4×C5—24no—
217C2×C4×C5—24no—
218C2×C4×C5—24no—
219C2×C4×C5—48no—
220C2×C4×C5—44no—
221C2×C4×C5C2×C22⋊C412yes—
222C2×C4×C5C2×C22⋊C411yescharacteristic
223C2×C4×C5#11714yes—
224C2×C4×C5C23×C411yescharacteristic
225C2×Dic5—22no—
226C2×Dic5—24no—
227C2×Dic5—24no—
228C2×Dic5—24no—
229C2×Dic5—48no—
230C2×Dic5—44no—
231C2×Dic5C2×C22⋊C412yes—
232C2×Dic5C2×C22⋊C411yescharacteristic
233C2×Dic5#12314yes—
234C2×Dic5C22×D411yescharacteristic
235C4×D5—24no—
236C4×D5—24no—
237C4×D5—24no—
238C4×D5—24no—
239C4×D5—24no—
240C4×D5—24no—
241C4×D5—44no—
242C4×D5—44no—
243C4×D5—416no—
244C4×D5—48no—
245C4×D5#11714yes—
246C4×D5#12314yes—
247C4×D5#12314yes—
248C4×D5C2×#4114yes—
249#170—44no—
250C25—51no—
251C23×C4—21no—
252C23×C4—51no—
253C23×C4—51no—
254C23×C4—51no—
255C23×C4—52no—
256C23×C4—51no—
257C23×C4—52no—
258C23×C4—52no—
259C23×C4—51no—
260C23×C4—52no—
261C23×C4—108no—
262C23×C4—104no—
263C23×C4—104no—
264C23×C4—102no—
265C23×C4—102no—
266C23×C4—101no—
267C23×C4—102no—
268C23×C4—204no—
269C23×C4—204no—
270C23×C4C2×AGL(1,𝔽5)11yescharacteristic
271C22×C8—54no—
272C22×C8—108no—
273C22×C8—102no—
274C22×C8—102no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—52no—
278C2×C4⋊C4—54no—
279C2×C4⋊C4—54no—
280C2×C4⋊C4—54no—
281C2×C4⋊C4—58no—
282C2×C4⋊C4—54no—
283C2×C4⋊C4—102no—
284C2×C4⋊C4—102no—
285C2×C4⋊C4—104no—
286C2×C4⋊C4—104no—
287C2×C4⋊C4—104no—
288C2×C4⋊C4—104no—
289C2×C4⋊C4—104no—
290C2×C4⋊C4—104no—
291C2×C4⋊C4—104no—
292C2×C4⋊C4—102no—
293C2×C4⋊C4—102no—
294C2×C4⋊C4—102no—
295C2×C4⋊C4—101no—
296C2×C4⋊C4—108no—
297C2×C4⋊C4—102no—
298C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
299C2×C4⋊C4C2×AGL(1,𝔽5)14yescomplemented
300C2×C22⋊C4—104no—
301C2×C22⋊C4—204no—
302C22⋊C8—104no—
303C22⋊C8—104no—
304#95—104no—
305C22×C5—28no—
306C22×C5—22no—
307C22×C5—22no—
308C22×C5#38112yes—
309C22×C5C2×#10214yes—
310C22×C5C2×#10212yes—
311C22×C5#41912yes—
312C22×C5C22×C22⋊C411yescharacteristic
313C2×D5—28no—
314C2×D5—22no—
315C2×D5—22no—
316C2×D5—28no—
317C2×D5—28no—
318C2×D5—216no—
319C2×D5#38112yes—
320C2×D5C2×#10214yes—
321C2×D5C2×#10212yes—
322C2×D5#41912yes—
323C2×D5#48018yes—
324C2×D5#48318yes—
325C2×D5C2×#12312yes—
326C2×D5C22×D811yescharacteristic
327C2×D5C22×SD1611yescharacteristic
328C2×D5C2×#13712yes—
329AGL(1,𝔽5)—84no—
330C4×C5—22no—
331C4×C5—44no—
332C4×C5#38112yes—
333C4×C5C2×#11712yes—
334Dic5—22no—
335Dic5—44no—
336Dic5#38112yes—
337Dic5C2×#12312yes—
338C24—51no—
339C24—52no—
340C24—52no—
341C24—51no—
342C24—54no—
343C24—54no—
344C24—104no—
345C24—104no—
346C24C22⋊AGL(1,𝔽5)11yescharacteristic
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—44no—
351C22×C4—51no—
352C22×C4—58no—
353C22×C4—51no—
354C22×C4—51no—
355C22×C4—51no—
356C22×C4—51no—
357C22×C4—52no—
358C22×C4—52no—
359C22×C4—51no—
360C22×C4—52no—
361C22×C4—52no—
362C22×C4—52no—
363C22×C4—52no—
364C22×C4—52no—
365C22×C4—52no—
366C22×C4—104no—
367C22×C4—108no—
368C22×C4—108no—
369C22×C4—108no—
370C22×C4—108no—
371C22×C4—104no—
372C22×C4—104no—
373C22×C4—104no—
374C22×C4—108no—
375C22×C4—104no—
376C22×C4—104no—
377C22×C4—104no—
378C22×C4—102no—
379C22×C4—104no—
380C22×C4—204no—
381C22×C4—204no—
382C22×C4—208no—
383C22×C4—2016no—
384C22×C4—208no—
385C22×C4—208no—
386C22×C4—204no—
387C22×C4—204no—
388C22×C4—204no—
389C22×C4—204no—
390C22×C4C22⋊AGL(1,𝔽5)12yes—
391C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
392C22×C4C22×AGL(1,𝔽5)12yes—
393C22×C4C22×AGL(1,𝔽5)11yescharacteristic
394C2×C8—104no—
395C2×C8—108no—
396C2×C8—204no—
397C2×C8—204no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—54no—
401C4⋊C4—104no—
402C4⋊C4—104no—
403C4⋊C4—108no—
404C4⋊C4—108no—
405C4⋊C4—104no—
406C4⋊C4—102no—
407C4⋊C4—204no—
408C4⋊C4C22×AGL(1,𝔽5)14yes—
409C22⋊C4—208no—
410C2×C5—24no—
411C2×C5#182314yescomplemented
412C2×C5C2×#38111yescharacteristic
413C2×C5C22×#10211yescharacteristic
414C2×C5C2×#41911yescharacteristic
415D5—24no—
416D5#182314yescomplemented
417D5C2×#48012yes—
418D5C2×#48312yes—
419C23—22no—
420C23—22no—
421C23—51no—
422C23—51no—
423C23—54no—
424C23—52no—
425C23—52no—
426C23—54no—
427C23—52no—
428C23—54no—
429C23—58no—
430C23—54no—
431C23—54no—
432C23—104no—
433C23—104no—
434C23—104no—
435C23—104no—
436C23—1016no—
437C23—1016no—
438C23—102no—
439C23—102no—
440C23#400714yes—
441C23C2×C22⋊AGL(1,𝔽5)12yes—
442C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—51no—
450C2×C4—51no—
451C2×C4—54no—
452C2×C4—54no—
453C2×C4—54no—
454C2×C4—54no—
455C2×C4—54no—
456C2×C4—52no—
457C2×C4—104no—
458C2×C4—104no—
459C2×C4—104no—
460C2×C4—104no—
461C2×C4—104no—
462C2×C4—104no—
463C2×C4—102no—
464C2×C4—104no—
465C2×C4—104no—
466C2×C4—104no—
467C2×C4—204no—
468C2×C4—204no—
469C2×C4—208no—
470C2×C4—2016no—
471C2×C4—208no—
472C2×C4—204no—
473C2×C4—204no—
474C2×C4—204no—
475C2×C4—204no—
476C2×C4—408no—
477C2×C4#413014yes—
478C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
479C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
480C2×C4C23×AGL(1,𝔽5)11yescharacteristic
481C8—204no—
482C5C2×#182311yescharacteristic, complemented, Sylow
483C22—22no—
484C22—22no—
485C22—28no—
486C22—52no—
487C22—52no—
488C22—51no—
489C22—51no—
490C22—52no—
491C22—52no—
492C22—52no—
493C22—58no—
494C22—58no—
495C22—54no—
496C22—108no—
497C22—108no—
498C22—1016no—
499C22—102no—
500C22—102no—
501C22—108no—
502C22#6576412yes—
503C22C2×#400714yes—
504C22C2×#400712yes—
505C22#6578212yes—
506C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
507C4—22no—
508C4—44no—
509C4—52no—
510C4—52no—
511C4—102no—
512C4—204no—
513C4—404no—
514C4#6576412yes—
515C4C2×#413012yes—
516C2—24no—
517C2—52no—
518C2—52no—
519C2—54no—
520C2—104no—
521C2#1060428214yesdirect product factor
522C2C2×#6576411yescharacteristic
523C2C22×#400711yescharacteristic
524C2C2×#6578211yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC23×C5C2×C22⋊C4
prime radicalC24×D5D4
cosocleC2×C4×D5C24
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5C2×#1823
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#64874C2semidirect
5. #10604282C2direct (N × H)
11. C22×C5×C4⋊C4C4semidirect
12. C22×C4⋊Dic5C4semidirect
18. C2×D5×C4⋊C4C4semidirect
19. C2×D5×C4⋊C4C4semidirect
29. #64874C22semidirect
64. C2×C5×C4⋊C4C2×C4semidirect
68. C2×C4⋊Dic5C2×C4semidirect
81. D5×C4⋊C4C2×C4semidirect
177. C22×C4⋊C4AGL(1,𝔽5)semidirect
299. C2×C4⋊C4C2×AGL(1,𝔽5)semidirect
411. C2×C5#1823semidirect
416. D5#1823semidirect
482. C5C2×#1823semidirect
521. C2#10604282direct (N × H)

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×D5⋊(C22⋊C8)2
C2×#648742
#106042822
C2×#18235