Finite Group Databasebeta

← C2×#10604454 #13033419

Subgroups — C2×#10604454 #13033419

Order 1280.

1686 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D5C211yesmaximal, characteristic
2C22×#4009C212yesmaximal
3C2×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4C2×#65676C211yesmaximal, characteristic
5#10604454C218yesmaximal, direct product factor
6C2×#64874C212yesmaximal
7C22×Q8×D5C2211yescharacteristic
8C22×C4⋊AGL(1,𝔽5)C2212yes—
9C22×C4×AGL(1,𝔽5)—21no—
10C22×D5⋊C8C2212yes—
11C2×C4×C5×Q8C411yescharacteristic, complemented
12C2×Q8×Dic5C411yescharacteristic, complemented
13C4×Q8×D5—22no—
14C4×Q8×D5—22no—
15C4×Q8×D5C414yescomplemented
16C4×Q8×D5C2214yes—
17C2×D5×C4⋊C4—21no—
18C2×D5×C4⋊C4C2211yescharacteristic
19C2×#4033—21no—
20C2×#4033C411yescharacteristic, complemented
21C2×C42×D5—21no—
22C2×C42×D5C2211yescharacteristic
23C2×C4×Dic10—21no—
24C2×C4×Dic10C411yescharacteristic, complemented
25C2×#4009C4116yescomplemented
26C2×#4009C2218yes—
27C2×(C2×AGL(1,𝔽5))⋊C4—21no—
28C2×#4002—21no—
29C2×#4002—21no—
30C4×C4⋊AGL(1,𝔽5)—22no—
31C4×C4⋊AGL(1,𝔽5)C2214yes—
32#65676—22no—
33#65676C2214yes—
34#64874C2218yes—
35C2×#1763—51nomaximal, Sylow
36C23×AGL(1,𝔽5)—22no—
37C22×C5×Q8C2×C411yescharacteristic
38C2×Q8×D5—21no—
39C2×Q8×D5—21no—
40C2×Q8×D5—24no—
41C2×Q8×D5C2×C418yes—
42C2×Q8×D5C2×C412yes—
43C2×Q8×D5C2312yes—
44C22×C4×D5—21no—
45C22×C4×D5—21no—
46C22×C4×D5D411yescharacteristic
47C22×C4×D5D411yescharacteristic
48C22×C4×D5C2311yescharacteristic
49C22×Dic10—21no—
50C22×Dic10C2×C411yescharacteristic
51C22×#170—22no—
52C2×C4⋊AGL(1,𝔽5)—22no—
53C2×C4⋊AGL(1,𝔽5)—24no—
54C2×C4⋊AGL(1,𝔽5)C2×C418yes—
55C2×C4⋊AGL(1,𝔽5)C2314yes—
56C2×C4×AGL(1,𝔽5)—24no—
57C2×C4×AGL(1,𝔽5)—44no—
58C2×D5⋊C8—22no—
59C2×D5⋊C8—24no—
60C2×D5⋊C8C2×C418yes—
61C2×D5⋊C8C2314yes—
62C4×C5×Q8—22no—
63C4×C5×Q8C2×C414yescomplemented
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4C2×C411yescharacteristic
66C2×C42×C5—21no—
67C2×C42×C5C2×C411yescharacteristic
68Q8×Dic5—22no—
69Q8×Dic5C2×C414yescomplemented
70C2×C4⋊Dic5—21no—
71C2×C4⋊Dic5C2×C411yescharacteristic
72C2×#683—21no—
73C2×#683—21no—
74C2×#683—21no—
75C2×C4×Dic5—21no—
76C2×C4×Dic5—21no—
77C2×C4×Dic5—21no—
78C2×C4×Dic5C2×C411yescharacteristic
79C2×C4×Dic5D411yescharacteristic
80C2×C4×Dic5D411yescharacteristic
81D5×C4⋊C4—24no—
82D5×C4⋊C4—24no—
83D5×C4⋊C4—21no—
84D5×C4⋊C4—21no—
85D5×C4⋊C4C2×C412yes—
86D5×C4⋊C4C2312yes—
87#4033—24no—
88#4033—22no—
89#4033—24no—
90#4033C2×C414yescomplemented
91C42×D5—24no—
92C42×D5—24no—
93C42×D5C2×C412yes—
94C42×D5D412yes—
95C42×D5D412yes—
96C42×D5C2312yes—
97C4×Dic10—22no—
98C4×Dic10—24no—
99C4×Dic10—24no—
100C4×Dic10C2×C414yescomplemented
101#4009C2×C4132yescomplemented
102(C2×AGL(1,𝔽5))⋊C4—24no—
103#4002—24no—
104#4002—24no—
105C22×C4×Q8—51no—
106C22×#103—52no—
107C22×#97—51no—
108C2×C4×C4⋊C4—51no—
109#1763—58no—
110C2×#340—52no—
111C23×D5C2×D411yescharacteristic
112C22×AGL(1,𝔽5)—28no—
113C22×AGL(1,𝔽5)—24no—
114C22×AGL(1,𝔽5)—44no—
115C22×AGL(1,𝔽5)—44no—
116C2×C5×Q8—21no—
117C2×C5×Q8—22no—
118C2×C5×Q8C4214yescomplemented
119C2×C5×Q8C22×C412yes—
120C22×C4×C5—21no—
121C22×C4×C5—21no—
122C22×C4×C5C22⋊C411yescharacteristic
123C22×C4×C5C22⋊C411yescharacteristic
124C22×C4×C5C22×C411yescharacteristic
125C22×Dic5—21no—
126C22×Dic5—21no—
127C22×Dic5C22⋊C411yescharacteristic
128C22×Dic5C22⋊C411yescharacteristic
129C22×Dic5C2×D411yescharacteristic
130Q8×D5—24no—
131Q8×D5—28no—
132Q8×D5—28no—
133Q8×D5—24no—
134Q8×D5C4218yescomplemented
135Q8×D5C22×C418yes—
136C2×C4×D5—22no—
137C2×C4×D5—28no—
138C2×C4×D5—22no—
139C2×C4×D5—22no—
140C2×C4×D5—24no—
141C2×C4×D5—22no—
142C2×C4×D5—22no—
143C2×C4×D5—22no—
144C2×C4×D5—42no—
145C2×C4×D5—42no—
146C2×C4×D5C22⋊C412yes—
147C2×C4×D5C22⋊C412yes—
148C2×C4×D5C22×C411yescharacteristic
149C2×C4×D5C22×C414yes—
150C2×C4×D5C2×D412yes—
151C2×C4×D5C2×D412yes—
152C2×C4×D5C2×D411yescharacteristic
153C2×C4×D5C2×D411yescharacteristic
154C2×C4×D5#4114yes—
155C2×C4×D5C2411yescharacteristic
156C2×Dic10—21no—
157C2×Dic10—22no—
158C2×Dic10—22no—
159C2×Dic10—28no—
160C2×Dic10—22no—
161C2×Dic10C4214yescomplemented
162C2×Dic10C22×C412yes—
163C2×#170—28no—
164C2×#170—24no—
165C4⋊AGL(1,𝔽5)—28no—
166C4⋊AGL(1,𝔽5)—24no—
167C4⋊AGL(1,𝔽5)C22×C418yes—
168C4×AGL(1,𝔽5)—48no—
169D5⋊C8—28no—
170D5⋊C8—24no—
171D5⋊C8C22×C418yes—
172C5×C4⋊C4—24no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4C22×C412yes—
175C42×C5—24no—
176C42×C5C22⋊C412yes—
177C42×C5C22×C412yes—
178C4⋊Dic5—21no—
179C4⋊Dic5—24no—
180C4⋊Dic5C22×C412yes—
181#683—24no—
182#683—22no—
183#683—24no—
184#683—22no—
185C4×Dic5—24no—
186C4×Dic5—24no—
187C4×Dic5—24no—
188C4×Dic5C22⋊C412yes—
189C4×Dic5C22⋊C412yes—
190C4×Dic5C22⋊C412yes—
191C4×Dic5C22×C412yes—
192C4×Dic5C2×D412yes—
193C4×Dic5C2×D412yes—
194C23×Q8—51no—
195C23×C8—52no—
196C2×C4×Q8—51no—
197C2×C4×Q8—54no—
198C2×C4×Q8—54no—
199C2×C4×Q8—51no—
200C2×C4×Q8—51no—
201C2×C4×Q8—101no—
202C2×C4×Q8—101no—
203C2×C4×Q8—102no—
204C2×C4×Q8—102no—
205C2×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
206C22×C4⋊C4—52no—
207C22×C4⋊C4—51no—
208C22×C4⋊C4—101no—
209C22×C42—51no—
210C22×C42—101no—
211C22×C42—101no—
212C2×#103—58no—
213C2×#103—516no—
214C2×#97—54no—
215C2×#97—102no—
216C2×#97—101no—
217C2×#97—101no—
218C4×C4⋊C4—54no—
219C4×C4⋊C4—102no—
220C2×#95—101no—
221#340—58no—
222C23×C5C2×C22⋊C411yescharacteristic
223C22×D5C2×C22⋊C411yescharacteristic
224C22×D5C4×D414yes—
225C22×D5AGL(1,ℤ/8)12yes—
226C22×D5#13712yes—
227C22×D5#13714yes—
228C22×D5C22×D411yescharacteristic
229C2×AGL(1,𝔽5)—28no—
230C2×AGL(1,𝔽5)—44no—
231C2×AGL(1,𝔽5)—44no—
232C2×AGL(1,𝔽5)—416no—
233C5×Q8—22no—
234C5×Q8—24no—
235C5×Q8C2×C4214yescomplemented
236C2×C4×C5—22no—
237C2×C4×C5—24no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—42no—
242C2×C4×C5C2×C4212yes—
243C2×C4×C5C2×C22⋊C412yes—
244C2×C4×C5C2×C22⋊C412yes—
245C2×C4×C5C2×C22⋊C411yescharacteristic
246C2×C4×C5#11712yes—
247C2×C4×C5C23×C411yescharacteristic
248C2×Dic5—22no—
249C2×Dic5—24no—
250C2×Dic5—22no—
251C2×Dic5—22no—
252C2×Dic5—22no—
253C2×Dic5—42no—
254C2×Dic5C2×C22⋊C412yes—
255C2×Dic5C2×C22⋊C412yes—
256C2×Dic5C2×C22⋊C411yescharacteristic
257C2×Dic5C4×D414yes—
258C2×Dic5C22×D411yescharacteristic
259C4×D5—28no—
260C4×D5—22no—
261C4×D5—24no—
262C4×D5—22no—
263C4×D5—28no—
264C4×D5—24no—
265C4×D5—24no—
266C4×D5—42no—
267C4×D5—42no—
268C4×D5—44no—
269C4×D5C2×C4212yes—
270C4×D5#11712yes—
271C4×D5C4×D414yes—
272C4×D5C4×D414yes—
273C4×D5C23×C412yes—
274C4×D5C2×#4112yes—
275Dic10—22no—
276Dic10—24no—
277Dic10—28no—
278Dic10—28no—
279Dic10C2×C4214yescomplemented
280#170—28no—
281C22×Q8—58no—
282C22×Q8—52no—
283C22×Q8—52no—
284C22×Q8—51no—
285C22×Q8—101no—
286C22×Q8—101no—
287C22×Q8—101no—
288C22×Q8—104no—
289C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
290C23×C4—51no—
291C23×C4—51no—
292C23×C4—51no—
293C23×C4—101no—
294C23×C4—101no—
295C23×C4—102no—
296C22×C8—58no—
297C22×C8—54no—
298C22×C8—104no—
299C22×C8—102no—
300C22×C8—102no—
301C4×Q8—22no—
302C4×Q8—54no—
303C4×Q8—54no—
304C4×Q8—54no—
305C4×Q8—104no—
306C4×Q8—104no—
307C4×Q8—104no—
308C4×Q8—104no—
309C4×Q8—102no—
310C4×Q8—102no—
311C4×Q8—102no—
312C4×Q8C2×AGL(1,𝔽5)14yescomplemented
313C2×C4⋊C4—21no—
314C2×C4⋊C4—58no—
315C2×C4⋊C4—54no—
316C2×C4⋊C4—51no—
317C2×C4⋊C4—52no—
318C2×C4⋊C4—52no—
319C2×C4⋊C4—101no—
320C2×C4⋊C4—101no—
321C2×C4⋊C4—104no—
322C2×C4⋊C4—104no—
323C2×C4⋊C4—101no—
324C2×C4⋊C4—101no—
325C2×C4⋊C4—101no—
326C2×C4⋊C4—104no—
327C2×C4⋊C4—102no—
328C2×C4⋊C4—101no—
329C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
330C2×C42—21no—
331C2×C42—51no—
332C2×C42—51no—
333C2×C42—51no—
334C2×C42—52no—
335C2×C42—52no—
336C2×C42—52no—
337C2×C42—52no—
338C2×C42—104no—
339C2×C42—104no—
340C2×C42—101no—
341C2×C42—101no—
342C2×C42—101no—
343C2×C42—104no—
344C2×C42—204no—
345C2×C42C2×AGL(1,𝔽5)11yescharacteristic
346#103—532no—
347#97—104no—
348#97—104no—
349#95—104no—
350C22×C5C4×C22⋊C412yes—
351C22×C5#41712yes—
352C22×C5#41912yes—
353C22×C5C22×C22⋊C411yescharacteristic
354C2×D5C4×C22⋊C412yes—
355C2×D5#41712yes—
356C2×D5#41912yes—
357C2×D5#44018yes—
358C2×D5#44118yes—
359C2×D5C2×C4×D412yes—
360C2×D5C2×AGL(1,ℤ/8)11yescharacteristic
361C2×D5C2×#13712yes—
362C2×D5C2×#13711yescharacteristic
363AGL(1,𝔽5)—48no—
364C4×C5—22no—
365C4×C5—22no—
366C4×C5—24no—
367C4×C5—42no—
368C4×C5C4×C22⋊C412yes—
369C4×C5C22×C4211yescharacteristic
370C4×C5C2×#11711yescharacteristic
371Dic5—22no—
372Dic5—22no—
373Dic5—24no—
374Dic5—42no—
375Dic5C4×C22⋊C412yes—
376Dic5C2×C4×D412yes—
377C24—51no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—58no—
381C2×Q8—58no—
382C2×Q8—54no—
383C2×Q8—52no—
384C2×Q8—102no—
385C2×Q8—102no—
386C2×Q8—108no—
387C2×Q8—104no—
388C2×Q8—104no—
389C2×Q8—108no—
390C2×Q8—108no—
391C2×Q8—102no—
392C2×Q8—101no—
393C2×Q8C4×AGL(1,𝔽5)14yescomplemented
394C2×Q8C22×AGL(1,𝔽5)12yes—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—51no—
398C22×C4—54no—
399C22×C4—54no—
400C22×C4—51no—
401C22×C4—51no—
402C22×C4—51no—
403C22×C4—51no—
404C22×C4—51no—
405C22×C4—51no—
406C22×C4—52no—
407C22×C4—52no—
408C22×C4—52no—
409C22×C4—52no—
410C22×C4—102no—
411C22×C4—102no—
412C22×C4—102no—
413C22×C4—102no—
414C22×C4—104no—
415C22×C4—108no—
416C22×C4—102no—
417C22×C4—102no—
418C22×C4—101no—
419C22×C4—104no—
420C22×C4—108no—
421C22×C4—101no—
422C22×C4—202no—
423C22×C4—202no—
424C22×C4—204no—
425C22×C4—204no—
426C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
427C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
428C22×C4C22×AGL(1,𝔽5)11yescharacteristic
429C2×C8—58no—
430C2×C8—104no—
431C2×C8—104no—
432C2×C8—108no—
433C2×C8—108no—
434C4⋊C4—24no—
435C4⋊C4—21no—
436C4⋊C4—58no—
437C4⋊C4—52no—
438C4⋊C4—102no—
439C4⋊C4—102no—
440C4⋊C4—104no—
441C4⋊C4—104no—
442C4⋊C4—104no—
443C4⋊C4—108no—
444C4⋊C4—101no—
445C4⋊C4—104no—
446C4⋊C4C22×AGL(1,𝔽5)12yes—
447C42—24no—
448C42—52no—
449C42—52no—
450C42—52no—
451C42—52no—
452C42—52no—
453C42—52no—
454C42—104no—
455C42—104no—
456C42—104no—
457C42—208no—
458C42C22⋊AGL(1,𝔽5)12yes—
459C42C22×AGL(1,𝔽5)12yes—
460C2×C5#176314yescomplemented
461C2×C5C2×C4×C22⋊C411yescharacteristic
462C2×C5C2×#41911yescharacteristic
463C2×C5C2×#41711yescharacteristic
464D5#176314yescomplemented
465D5C2×#44012yes—
466D5C2×#44112yes—
467C23—52no—
468C23—52no—
469C23—51no—
470C23—51no—
471C23—54no—
472C23—54no—
473C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
474Q8—22no—
475Q8—24no—
476Q8—54no—
477Q8—108no—
478Q8—108no—
479Q8—102no—
480Q8—104no—
481Q8C2×C4×AGL(1,𝔽5)14yescomplemented
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—51no—
489C2×C4—51no—
490C2×C4—54no—
491C2×C4—52no—
492C2×C4—52no—
493C2×C4—52no—
494C2×C4—52no—
495C2×C4—54no—
496C2×C4—54no—
497C2×C4—52no—
498C2×C4—52no—
499C2×C4—102no—
500C2×C4—104no—
501C2×C4—104no—
502C2×C4—102no—
503C2×C4—104no—
504C2×C4—108no—
505C2×C4—108no—
506C2×C4—102no—
507C2×C4—102no—
508C2×C4—104no—
509C2×C4—102no—
510C2×C4—108no—
511C2×C4—102no—
512C2×C4—202no—
513C2×C4—202no—
514C2×C4—204no—
515C2×C4—204no—
516C2×C4—204no—
517C2×C4—2016no—
518C2×C4—202no—
519C2×C4C2×C4×AGL(1,𝔽5)12yes—
520C2×C4#413012yes—
521C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
522C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
523C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
524C2×C4C23×AGL(1,𝔽5)11yescharacteristic
525C8—108no—
526C5C2×#176311yescharacteristic, complemented, Sylow
527C22—52no—
528C22—52no—
529C22—52no—
530C22—52no—
531C22—58no—
532C22—58no—
533C22—51no—
534C22—51no—
535C22—52no—
536C22C4×C22⋊AGL(1,𝔽5)12yes—
537C22#6578212yes—
538C22#6579312yes—
539C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
540C4—24no—
541C4—22no—
542C4—22no—
543C4—42no—
544C4—52no—
545C4—52no—
546C4—104no—
547C4—102no—
548C4—102no—
549C4—208no—
550C4—202no—
551C4C4×C22⋊AGL(1,𝔽5)12yes—
552C4C22×C4×AGL(1,𝔽5)11yescharacteristic
553C4C2×#413011yescharacteristic
554C2—52no—
555C2—52no—
556C2—54no—
557C2#1060445414yesdirect product factor
558C2C2×C4×C22⋊AGL(1,𝔽5)11yescharacteristic
559C2C2×#6578211yescharacteristic
560C2C2×#6579311yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC4×C5C22×C42
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×#1763
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
5. #10604454C2direct (N × H)
11. C2×C4×C5×Q8C4semidirect
12. C2×Q8×Dic5C4semidirect
15. C4×Q8×D5C4semidirect
20. C2×#4033C4semidirect
24. C2×C4×Dic10C4semidirect
25. C2×#4009C4semidirect
63. C4×C5×Q8C2×C4semidirect
69. Q8×Dic5C2×C4semidirect
90. #4033C2×C4semidirect
100. C4×Dic10C2×C4semidirect
101. #4009C2×C4semidirect
118. C2×C5×Q8C42semidirect
134. Q8×D5C42semidirect
161. C2×Dic10C42semidirect
205. C2×C4×Q8AGL(1,𝔽5)semidirect
235. C5×Q8C2×C42semidirect
279. Dic10C2×C42semidirect
312. C4×Q8C2×AGL(1,𝔽5)semidirect
393. C2×Q8C4×AGL(1,𝔽5)semidirect
460. C2×C5#1763semidirect
464. D5#1763semidirect
481. Q8C2×C4×AGL(1,𝔽5)semidirect
526. C5C2×#1763semidirect
557. C2#10604454direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D52
C22×#40092
C2×C4×C4⋊AGL(1,𝔽5)2
C2×#656762
#106044542
C2×#648742
C2×#17635