Finite Group Databasebeta

← C2×#10612211 #12970490

Subgroups — C2×#10612211 #12970490

Order 1280.

2134 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×C4⋊Q8C211yesmaximal, characteristic
2C2×#65912C211yesmaximal, characteristic
3C2×C5×#462C211yesmaximal, characteristic, complemented
4C2×#65475C212yesmaximal
5C2×#65470C212yesmaximal, complemented
6C2×#65379C211yesmaximal, characteristic, complemented
7C2×#65320C211yesmaximal, characteristic, complemented
8#10612211C2116yesmaximal, direct product factor
9C2×#65138C211yesmaximal, characteristic, complemented
10C2×#65122C211yesmaximal, characteristic
11C2×#65097C212yesmaximal
12C2×#65081C212yesmaximal, complemented
13C22×Q8×D5C2212yes—
14C22×C5×Q16C2212yes—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C4×C5×Q8C2211yescharacteristic
17C2×#4092C2211yescharacteristic
18C2×#4090C2211yescharacteristic
19C2×#4090C2212yes—
20C22×#689C2212yes—
21C22×#689C2212yes—
22C22×Dic20C2212yes—
23D5×C4⋊Q8—24no—
24D5×C4⋊Q8C2218yes—
25#65912—24no—
26#65912C2218yes—
27C2×#4043C2211yescharacteristic
28C2×#4042—21no—
29C2×D5×C4⋊C4—21no—
30C2×D5×C4⋊C4—21no—
31C2×#4035—21no—
32C2×#4034—21no—
33C2×#4034—21no—
34C2×#4034C2211yescharacteristic
35C2×#4018—21no—
36C2×C42×D5C2211yescharacteristic
37C2×#4015C2211yescharacteristic
38C2×C4×Dic10—21no—
39C2×C4×Dic10C2211yescharacteristic
40C5×#462C2218yescomplemented
41C2×C5×C4⋊C8C2211yescharacteristic
42C2×C5×#103C2212yes—
43C2×#3967C2212yes—
44#65475C22116yes—
45#65470C22116yescomplemented
46C2×#3952C2212yes—
47C2×#3948C2212yes—
48C2×C8×Dic5C2212yes—
49#65379C2218yescomplemented
50#65320C2218yescomplemented
51C2×#3942C2212yes—
52C2×C4⋊#168C2211yescharacteristic
53#65138C2218yescomplemented
54#65122C2218yes—
55#65097C22116yes—
56#65081C22116yescomplemented
57C2×#3209—51nomaximal, Sylow
58C22×C5×Q8C2312yes—
59C22×C5×Q8C2311yescharacteristic
60C2×Q8×D5—24no—
61C2×Q8×D5—44no—
62C2×Q8×D5—44no—
63C2×Q8×D5C2318yes—
64C22×C4×D5—22no—
65C22×C4×D5D411yescharacteristic
66C22×C4×D5D411yescharacteristic
67C22×C4×D5C2311yescharacteristic
68C22×Dic10—21no—
69C22×Dic10—22no—
70C22×Dic10C2311yescharacteristic
71C22×Dic10C2312yes—
72C2×C5×Q16—44no—
73C2×C5×Q16C2318yes—
74C22×C5×C8C2312yes—
75C5×C4⋊Q8—22no—
76C5×C4⋊Q8C2314yes—
77C4×C5×Q8—22no—
78C4×C5×Q8C2314yes—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C42×C5—21no—
84C2×C42×C5C2311yescharacteristic
85#4092—22no—
86#4092C2314yes—
87#4090—24no—
88#4090—22no—
89#4090C2318yes—
90#4090C2314yes—
91C2×#689—44no—
92C2×#689—44no—
93C2×#689C2318yes—
94C2×#689C2318yes—
95C2×#685—21no—
96C2×#685—21no—
97C2×C4⋊Dic5—21no—
98C2×C4⋊Dic5—21no—
99C2×C4⋊Dic5—21no—
100C2×C4⋊Dic5C2311yescharacteristic
101C2×#683—21no—
102C2×#683—21no—
103C2×#683—21no—
104C2×#683—21no—
105C2×#683—21no—
106C2×#683—22no—
107C2×#683—22no—
108C2×C4×Dic5—21no—
109C2×C4×Dic5D411yescharacteristic
110C2×C4×Dic5D411yescharacteristic
111C2×C4×Dic5C2311yescharacteristic
112C22×#168C2312yes—
113C2×Dic20—44no—
114C2×Dic20C2318yes—
115#4043—22no—
116#4043C2314yes—
117#4042—28no—
118D5×C4⋊C4—28no—
119D5×C4⋊C4—28no—
120#4035—28no—
121#4034—28no—
122#4034—28no—
123#4034—22no—
124#4034C2314yes—
125#4018—28no—
126C42×D5D414yes—
127C42×D5C2314yes—
128#4015—22no—
129#4015C2314yes—
130C4×Dic10—22no—
131C4×Dic10—28no—
132C4×Dic10C2314yes—
133C5×C4⋊C8C2314yes—
134C5×#103C2318yes—
135#3967C2318yes—
136#3952C2318yes—
137#3948C2318yes—
138C8×Dic5C2318yes—
139#3942C2318yes—
140C4⋊#168C2314yes—
141C2×#533—51no—
142C22×C4⋊Q8—51no—
143#3209—516no—
144C2×#494—52no—
145C2×#487—52no—
146C2×#462—51no—
147C2×#462—51no—
148C2×#462—51no—
149C2×#462D511yescharacteristic, complemented
150C2×#451—52no—
151C2×#451—52no—
152C2×#431—51no—
153C23×D5C2×D411yescharacteristic
154C2×C5×Q8—21no—
155C2×C5×Q8—22no—
156C2×C5×Q8—22no—
157C2×C5×Q8—24no—
158C2×C5×Q8—24no—
159C2×C5×Q8—22no—
160C2×C5×Q8C2412yes—
161C2×C5×Q8C2414yes—
162C22×C4×C5—21no—
163C22×C4×C5—21no—
164C22×C4×C5—22no—
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2411yescharacteristic
168C22×Dic5—22no—
169C22×Dic5—21no—
170C22×Dic5—21no—
171C22×Dic5C2×D411yescharacteristic
172C22×Dic5C2×D411yescharacteristic
173C22×Dic5C2×D411yescharacteristic
174Q8×D5—48no—
175Q8×D5—48no—
176Q8×D5—416no—
177C2×C4×D5—22no—
178C2×C4×D5—22no—
179C2×C4×D5—24no—
180C2×C4×D5—28no—
181C2×C4×D5—48no—
182C2×C4×D5Q1618yescomplemented
183C2×C4×D5C2×D414yes—
184C2×C4×D5C2×D412yes—
185C2×C4×D5C2×D414yes—
186C2×C4×D5C2412yes—
187C2×Dic10—24no—
188C2×Dic10—24no—
189C2×Dic10—22no—
190C2×Dic10—22no—
191C2×Dic10—22no—
192C2×Dic10—21no—
193C2×Dic10—28no—
194C2×Dic10—24no—
195C2×Dic10—48no—
196C2×Dic10—44no—
197C2×Dic10C2414yes—
198C2×Dic10C2412yes—
199C5×Q16—48no—
200C2×C5×C8—24no—
201C2×C5×C8C2414yes—
202C5×C4⋊C4—24no—
203C5×C4⋊C4—24no—
204C5×C4⋊C4—24no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4C2412yes—
207C42×C5—24no—
208C42×C5C2×D412yes—
209C42×C5C2412yes—
210#689—48no—
211#689—48no—
212#685—24no—
213#685—24no—
214C4⋊Dic5—21no—
215C4⋊Dic5—24no—
216C4⋊Dic5—24no—
217C4⋊Dic5—24no—
218C4⋊Dic5C2412yes—
219#683—24no—
220#683—24no—
221#683—24no—
222#683—24no—
223#683—24no—
224#683—28no—
225#683—28no—
226C4×Dic5—24no—
227C4×Dic5C2×D414yes—
228C4×Dic5C2×D412yes—
229C4×Dic5C2×D414yes—
230C4×Dic5C2412yes—
231C2×#168—24no—
232C2×#168C2414yes—
233Dic20—48no—
234C23×Q8—52no—
235C22×Q16—52no—
236C22×Q16—52no—
237C22×Q16—52no—
238C22×Q16C2×D512yes—
239#533—58no—
240#533—104no—
241C2×C4⋊Q8—51no—
242C2×C4⋊Q8—52no—
243C2×C4⋊Q8—51no—
244C2×C4⋊Q8—58no—
245C2×C4⋊Q8—51no—
246C2×C4⋊Q8—101no—
247C2×C4⋊Q8—101no—
248C2×C4⋊Q8—104no—
249C2×C4⋊Q8C2×D511yescharacteristic
250C2×#125—101no—
251C2×C22⋊Q8—51no—
252C2×C22⋊Q8—51no—
253C2×C22⋊Q8—101no—
254C2×C4×Q8—51no—
255C2×C4×Q8—101no—
256C2×C4×Q8C2×D511yescharacteristic
257C2×#117—101no—
258C22×C4⋊C4—101no—
259C22×C4⋊C4—101no—
260C22×C42—51no—
261#494—516no—
262#487—516no—
263#462—58no—
264#462—58no—
265#462—58no—
266#462C2×D518yescomplemented
267#451—516no—
268#451—516no—
269#431—58no—
270C2×C4⋊C8—51no—
271C2×C4⋊C8C2×D511yescharacteristic
272C2×#103—52no—
273C2×#103—52no—
274C2×#103—52no—
275C2×#103C2×D512yes—
276C2×C22⋊C8—52no—
277C2×C4×C8—52no—
278C23×C5C22×D411yescharacteristic
279C22×D5—22no—
280C22×D5—22no—
281C22×D5C2×Q1614yes—
282C22×D5C22×D412yes—
283C5×Q8—22no—
284C5×Q8—24no—
285C5×Q8—22no—
286C5×Q8—24no—
287C5×Q8—44no—
288C5×Q8—42no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—24no—
292C2×C4×C5—22no—
293C2×C4×C5—44no—
294C2×C4×C5—42no—
295C2×C4×C5—42no—
296C2×C4×C5C2×Q1614yes—
297C2×C4×C5C22×D412yes—
298C2×C4×C5C22×D411yescharacteristic
299C2×C4×C5C22×D412yes—
300C2×C4×C5H(5,𝔽2)12yes—
301C2×C4×C5H(5,𝔽2)12yes—
302C2×C4×C5C2511yescharacteristic
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—24no—
307C2×Dic5—44no—
308C2×Dic5—42no—
309C2×Dic5—42no—
310C2×Dic5C2×Q1614yes—
311C2×Dic5#13714yes—
312C2×Dic5C22×D412yes—
313C2×Dic5C22×D412yes—
314C2×Dic5C22×D412yes—
315C4×D5—24no—
316C4×D5—22no—
317C4×D5—22no—
318C4×D5—28no—
319C4×D5—416no—
320C4×D5C2×Q16116yescomplemented
321Dic10—22no—
322Dic10—24no—
323Dic10—22no—
324Dic10—24no—
325Dic10—416no—
326Dic10—42no—
327Dic10—44no—
328Dic10—48no—
329C5×C8—24no—
330#168—24no—
331C22×Q8—58no—
332C22×Q8—51no—
333C22×Q8—52no—
334C22×Q8—101no—
335C22×Q8—102no—
336C22×Q8—104no—
337C22×Q8—204no—
338C22×Q8—204no—
339C22×Q8C22×D511yescharacteristic
340C22×Q8C22×D512yes—
341C23×C4—51no—
342C23×C4—51no—
343C23×C4—51no—
344C23×C4—102no—
345C2×Q16—44no—
346C2×Q16—58no—
347C2×Q16—58no—
348C2×Q16—58no—
349C2×Q16—204no—
350C2×Q16—204no—
351C2×Q16—204no—
352C2×Q16C22×D518yes—
353C22×C8—52no—
354C22×C8C22×D512yes—
355C4⋊Q8—22no—
356C4⋊Q8—54no—
357C4⋊Q8—58no—
358C4⋊Q8—54no—
359C4⋊Q8—54no—
360C4⋊Q8—108no—
361C4⋊Q8—108no—
362C4⋊Q8—102no—
363C4⋊Q8—104no—
364C4⋊Q8—102no—
365C4⋊Q8—102no—
366C4⋊Q8C22×D514yes—
367#125—108no—
368C22⋊Q8—54no—
369C22⋊Q8—54no—
370C22⋊Q8—102no—
371C22⋊Q8—108no—
372C22⋊Q8—102no—
373C4×Q8—22no—
374C4×Q8—54no—
375C4×Q8—108no—
376C4×Q8—102no—
377C4×Q8C22×D514yes—
378#117—108no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—51no—
383C2×C4⋊C4—108no—
384C2×C4⋊C4—108no—
385C2×C4⋊C4—102no—
386C2×C4⋊C4—102no—
387C2×C4⋊C4—101no—
388C2×C4⋊C4—101no—
389C2×C4⋊C4—101no—
390C2×C4⋊C4—101no—
391C2×C4⋊C4—101no—
392C2×C4⋊C4—101no—
393C2×C4⋊C4—101no—
394C2×C4⋊C4—101no—
395C2×C4⋊C4C22×D511yescharacteristic
396C2×C22⋊C4—101no—
397C2×C22⋊C4—101no—
398C2×C42—21no—
399C2×C42—51no—
400C2×C42—51no—
401C2×C42—51no—
402C2×C42—54no—
403C2×C42—54no—
404C2×C42—101no—
405C2×C42C22×D511yescharacteristic
406C4⋊C8—54no—
407C4⋊C8C22×D514yes—
408#103—58no—
409#103—58no—
410#103—58no—
411#103C22×D518yes—
412C22⋊C8—58no—
413C4×C8—58no—
414C22×C5#53512yes—
415C22×C5C22×Q1612yes—
416C22×C5C2×#13712yes—
417C22×C5C23×D411yescharacteristic
418C2×D5—22no—
419C2×D5—28no—
420C2×D5—22no—
421C2×D5C22×Q1614yes—
422C4×C5—22no—
423C4×C5—44no—
424C4×C5—42no—
425C4×C5—42no—
426C4×C5#53512yes—
427C4×C5C22×Q1614yes—
428C4×C5C2×H(5,𝔽2)11yescharacteristic
429C4×C5C2×H(5,𝔽2)11yescharacteristic
430Dic5—22no—
431Dic5—44no—
432Dic5—42no—
433Dic5—42no—
434Dic5C22×Q1614yes—
435Dic5C2×#13714yes—
436C24—51no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—54no—
444C2×Q8—52no—
445C2×Q8—104no—
446C2×Q8—108no—
447C2×Q8—101no—
448C2×Q8—102no—
449C2×Q8—102no—
450C2×Q8—104no—
451C2×Q8—102no—
452C2×Q8—104no—
453C2×Q8—204no—
454C2×Q8—208no—
455C2×Q8—208no—
456C2×Q8—208no—
457C2×Q8—2016no—
458C2×Q8C23×D514yes—
459C2×Q8C23×D512yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—51no—
464C22×C4—51no—
465C22×C4—54no—
466C22×C4—58no—
467C22×C4—51no—
468C22×C4—54no—
469C22×C4—52no—
470C22×C4—52no—
471C22×C4—104no—
472C22×C4—108no—
473C22×C4—102no—
474C22×C4—102no—
475C22×C4—101no—
476C22×C4—101no—
477C22×C4—102no—
478C22×C4—208no—
479C22×C4D4×D511yescharacteristic
480C22×C4D4×D511yescharacteristic
481C22×C4C23×D511yescharacteristic
482Q16—48no—
483Q16—208no—
484Q16—208no—
485Q16—208no—
486C2×C8—24no—
487C2×C8—54no—
488C2×C8—104no—
489C2×C8C23×D514yes—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—52no—
495C4⋊C4—108no—
496C4⋊C4—108no—
497C4⋊C4—104no—
498C4⋊C4—104no—
499C4⋊C4—104no—
500C4⋊C4—104no—
501C4⋊C4—104no—
502C4⋊C4—101no—
503C4⋊C4—104no—
504C4⋊C4—104no—
505C4⋊C4—104no—
506C4⋊C4C23×D512yes—
507C22⋊C4—104no—
508C22⋊C4—104no—
509C42—24no—
510C42—54no—
511C42—54no—
512C42—52no—
513C42—52no—
514C42—104no—
515C42D4×D512yes—
516C42C23×D512yes—
517C2×C5#320914yescomplemented
518C2×C5C2×#53511yescharacteristic
519C2×C5C23×Q1611yescharacteristic
520C2×C5C22×#13711yescharacteristic
521D5—24no—
522C23—54no—
523C23—52no—
524C23—102no—
525C23—102no—
526C23C2×D4×D511yescharacteristic
527Q8—22no—
528Q8—24no—
529Q8—22no—
530Q8—24no—
531Q8—44no—
532Q8—42no—
533Q8—102no—
534Q8—104no—
535Q8—102no—
536Q8—104no—
537Q8—208no—
538Q8—2016no—
539Q8—204no—
540Q8—202no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—54no—
549C2×C4—54no—
550C2×C4—52no—
551C2×C4—52no—
552C2×C4—516no—
553C2×C4—52no—
554C2×C4—108no—
555C2×C4—102no—
556C2×C4—104no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—102no—
560C2×C4—102no—
561C2×C4—104no—
562C2×C4—2016no—
563C2×C4—202no—
564C2×C4—204no—
565C2×C4—202no—
566C2×C4D5×Q1614yes—
567C2×C4C2×D4×D512yes—
568C2×C4C2×D4×D512yes—
569C2×C4C2×D4×D511yescharacteristic
570C2×C4#414412yes—
571C2×C4#414912yes—
572C2×C4C24×D511yescharacteristic
573C8—24no—
574C8—104no—
575C5C2×#320911yescharacteristic, complemented, Sylow
576C22—54no—
577C22—102no—
578C22—102no—
579C22—108no—
580C22#6594112yes—
581C22C2×D5×Q1612yes—
582C22D5×#13712yes—
583C22C22×D4×D511yescharacteristic
584C4—22no—
585C4—42no—
586C4—44no—
587C4—42no—
588C4—54no—
589C4—54no—
590C4—102no—
591C4—202no—
592C4—204no—
593C4—202no—
594C4#6594112yes—
595C4C2×D5×Q1614yes—
596C4C2×#414411yescharacteristic
597C4C2×#414911yescharacteristic
598C2—104no—
599C2#1061221114yesdirect product factor
600C2C2×#6594111yescharacteristic
601C2C22×D5×Q1611yescharacteristic
602C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#462C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
hypercenterC2×#462D5
hypocenterC5C2×#3209
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. C2×C5×#462C2semidirect
5. C2×#65470C2semidirect
6. C2×#65379C2semidirect
7. C2×#65320C2semidirect
8. #10612211C2direct (N × H)
9. C2×#65138C2semidirect
12. C2×#65081C2semidirect
40. C5×#462C22semidirect
45. #65470C22semidirect
49. #65379C22semidirect
50. #65320C22semidirect
53. #65138C22semidirect
56. #65081C22semidirect
149. C2×#462D5semidirect
182. C2×C4×D5Q16semidirect
266. #462C2×D5semidirect
320. C4×D5C2×Q16semidirect
517. C2×C5#3209semidirect
575. C5C2×#3209semidirect
599. C2#10612211direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊Q82
C2×#659122
C2×C5×#4622
C2×#654752
C2×#654702
C2×#653792
C2×#653202
#106122112
C2×#651382
C2×#651222
C2×#650972
C2×#650812
C2×#32095