Finite Group Databasebeta

← C2×#10607721 #12258951

Subgroups — C2×#10607721 #12258951

Order 1280.

2510 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4035C211yesmaximal, characteristic
2C2×C5×#389C211yesmaximal, characteristic
3C2×#65261C211yesmaximal, characteristic
4C2×#65261C211yesmaximal, characteristic
5C2×#65261C211yesmaximal, characteristic
6C2×#65267C211yesmaximal, characteristic
7C2×#65270C211yesmaximal, characteristic
8C2×C4⋊C4×Dic5C211yesmaximal, characteristic
9C2×C4⋊C4×Dic5C211yesmaximal, characteristic
10C2×C4×#683C211yesmaximal, characteristic
11C2×C4×#683C211yesmaximal, characteristic
12#10607721C2116yesmaximal, direct product factor
13C2×#64944C211yesmaximal, characteristic
14C2×#64943C211yesmaximal, characteristic
15C2×#64938C211yesmaximal, characteristic
16C2×#64938C211yesmaximal, characteristic
17C22×C5×C4⋊C4C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C5×C4⋊C4C2211yescharacteristic
21C22×C42×C5C2211yescharacteristic
22C22×C4⋊Dic5C2211yescharacteristic
23C22×C4⋊Dic5C2211yescharacteristic
24C22×#683C2211yescharacteristic
25C22×#683C2211yescharacteristic
26C22×#683C2211yescharacteristic
27C22×#683C2211yescharacteristic
28C22×#683C2211yescharacteristic
29C22×#683C2211yescharacteristic
30C22×#683C2211yescharacteristic
31C22×#683C2211yescharacteristic
32C22×C4×Dic5C2211yescharacteristic
33C22×C4×Dic5C2211yescharacteristic
34C22×C4×Dic5C2211yescharacteristic
35C22×C4×Dic5C2211yescharacteristic
36C22×C4×Dic5C2211yescharacteristic
37C22×C4×Dic5C2211yescharacteristic
38C22×C4×Dic5C2211yescharacteristic
39C22×C4×Dic5C2211yescharacteristic
40C2×#4035C4116yescomplemented
41C2×#4035C2218yes—
42C5×#389C2218yes—
43C2×C5×#95C2211yescharacteristic
44C2×C5×#95C2211yescharacteristic
45C2×#3963C2211yescharacteristic
46C2×#3963C2211yescharacteristic
47C2×#3963C2211yescharacteristic
48C2×#3963C2211yescharacteristic
49C2×#3963C2211yescharacteristic
50C2×#3963C2211yescharacteristic
51C2×#3963C2211yescharacteristic
52C2×#3963C2211yescharacteristic
53C2×#3963C2211yescharacteristic
54C2×#3963C2211yescharacteristic
55#65270C2218yes—
56#65267C2218yes—
57C4⋊C4×Dic5C2218yes—
58C4⋊C4×Dic5C2218yes—
59#65261C2218yes—
60#65261C2218yes—
61#65261C2218yes—
62C4×#683C2218yes—
63C4×#683C2218yes—
64#64944C2218yes—
65#64943C2218yes—
66#64938C2218yes—
67#64938C2218yes—
68C2×#2370—51nomaximal, Sylow
69C23×C4×C5C2311yescharacteristic
70C23×C4×C5C2311yescharacteristic
71C23×C4×C5C2311yescharacteristic
72C23×C4×C5C2311yescharacteristic
73C23×C4×C5C2311yescharacteristic
74C23×C4×C5C2311yescharacteristic
75C23×C4×C5C2311yescharacteristic
76C23×Dic5C2311yescharacteristic
77C23×Dic5C2311yescharacteristic
78C23×Dic5C2311yescharacteristic
79C23×Dic5C2311yescharacteristic
80C23×Dic5C2311yescharacteristic
81C23×Dic5C2311yescharacteristic
82C23×Dic5C2311yescharacteristic
83C23×Dic5C2311yescharacteristic
84C2×C5×C4⋊C4—24no—
85C2×C5×C4⋊C4—24no—
86C2×C5×C4⋊C4—42no—
87C2×C5×C4⋊C4C2×C418yes—
88C2×C5×C4⋊C4C2314yes—
89C2×C5×C4⋊C4C2314yes—
90C2×C5×C4⋊C4C2314yes—
91C2×C5×C4⋊C4C2314yes—
92C2×C42×C5—42no—
93C2×C42×C5C2314yes—
94C2×C4⋊Dic5—24no—
95C2×C4⋊Dic5C2×C418yes—
96C2×C4⋊Dic5C2314yes—
97C2×C4⋊Dic5C2314yes—
98C2×#683—24no—
99C2×#683—24no—
100C2×#683—42no—
101C2×#683C2×C418yes—
102C2×#683C2×C418yes—
103C2×#683C2×C418yes—
104C2×#683C2×C418yes—
105C2×#683Q818yes—
106C2×#683C2314yes—
107C2×#683C2314yes—
108C2×#683C2314yes—
109C2×#683C2314yes—
110C2×#683C2314yes—
111C2×#683C2314yes—
112C2×#683C2314yes—
113C2×#683C2314yes—
114C2×C4×Dic5—24no—
115C2×C4×Dic5—24no—
116C2×C4×Dic5—24no—
117C2×C4×Dic5—24no—
118C2×C4×Dic5—42no—
119C2×C4×Dic5—42no—
120C2×C4×Dic5—42no—
121C2×C4×Dic5C2×C418yes—
122C2×C4×Dic5C2314yes—
123C2×C4×Dic5C2314yes—
124C2×C4×Dic5C2314yes—
125C2×C4×Dic5C2314yes—
126C2×C4×Dic5C2314yes—
127C2×C4×Dic5C2314yes—
128C2×C4×Dic5C2314yes—
129C2×C4×Dic5C2314yes—
130#4035C2×C4164yescomplemented
131C5×#95C2314yes—
132C5×#95C2314yes—
133#3963C2314yes—
134#3963C2314yes—
135#3963C2314yes—
136#3963C2314yes—
137#3963C2314yes—
138#3963C2314yes—
139#3963C2314yes—
140#3963C2314yes—
141#3963C2314yes—
142#3963C2314yes—
143C22×#125—51no—
144#2370—516no—
145C2×#389—51no—
146C2×#389—51no—
147C2×#389—51no—
148C2×#389D511yescharacteristic
149C2×#387—51no—
150C2×#387—51no—
151C2×#387—51no—
152C2×#387—51no—
153C2×C4×C4⋊C4—51no—
154C2×C4×C4⋊C4—51no—
155C2×C4×C4⋊C4—51no—
156C2×C4×C4⋊C4—51no—
157C2×C4×C4⋊C4—51no—
158C2×C4×C4⋊C4—51no—
159C24×C5C2411yescharacteristic
160C22×C4×C5—22no—
161C22×C4×C5—22no—
162C22×C4×C5—22no—
163C22×C4×C5—22no—
164C22×C4×C5—22no—
165C22×C4×C5—22no—
166C22×C4×C5—22no—
167C22×C4×C5—22no—
168C22×C4×C5—22no—
169C22×C4×C5—22no—
170C22×C4×C5—22no—
171C22×C4×C5—22no—
172C22×C4×C5—22no—
173C22×C4×C5—22no—
174C22×C4×C5—22no—
175C22×C4×C5—22no—
176C22×C4×C5C22×C414yes—
177C22×C4×C5C22×C414yes—
178C22×C4×C5C22×C414yes—
179C22×C4×C5C2×Q814yes—
180C22×C4×C5#4114yes—
181C22×C4×C5C2412yes—
182C22×C4×C5C2412yes—
183C22×C4×C5C2412yes—
184C22×C4×C5C2412yes—
185C22×C4×C5C2412yes—
186C22×C4×C5C2412yes—
187C22×C4×C5C2412yes—
188C22×Dic5—22no—
189C22×Dic5—22no—
190C22×Dic5—22no—
191C22×Dic5—22no—
192C22×Dic5—22no—
193C22×Dic5—22no—
194C22×Dic5—22no—
195C22×Dic5—22no—
196C22×Dic5—22no—
197C22×Dic5—22no—
198C22×Dic5—22no—
199C22×Dic5—22no—
200C22×Dic5—22no—
201C22×Dic5—22no—
202C22×Dic5—22no—
203C22×Dic5—22no—
204C22×Dic5C22×C414yes—
205C22×Dic5C22×C414yes—
206C22×Dic5C22×C414yes—
207C22×Dic5C22×C414yes—
208C22×Dic5C2×Q814yes—
209C22×Dic5C2×Q814yes—
210C22×Dic5#4114yes—
211C22×Dic5#4114yes—
212C22×Dic5C2412yes—
213C22×Dic5C2412yes—
214C22×Dic5C2412yes—
215C22×Dic5C2412yes—
216C22×Dic5C2412yes—
217C22×Dic5C2412yes—
218C22×Dic5C2412yes—
219C22×Dic5C2412yes—
220C5×C4⋊C4—28no—
221C5×C4⋊C4—28no—
222C5×C4⋊C4—44no—
223C5×C4⋊C4C22×C4116yes—
224C42×C5—44no—
225C4⋊Dic5—28no—
226C4⋊Dic5C22×C4116yes—
227#683—28no—
228#683—28no—
229#683—44no—
230#683C22×C4116yes—
231#683C22×C4116yes—
232#683C22×C4116yes—
233#683C22×C4116yes—
234#683C2×Q8116yes—
235C4×Dic5—28no—
236C4×Dic5—28no—
237C4×Dic5—28no—
238C4×Dic5—28no—
239C4×Dic5—44no—
240C4×Dic5—44no—
241C4×Dic5—44no—
242C4×Dic5C22×C4116yes—
243C2×#125—516no—
244C2×#125—58no—
245C22×C4⋊C4—51no—
246C22×C4⋊C4—51no—
247C22×C4⋊C4—51no—
248C22×C4⋊C4—51no—
249C22×C4⋊C4—51no—
250C22×C4⋊C4—51no—
251C22×C4⋊C4—51no—
252C22×C4⋊C4—51no—
253C22×C4⋊C4—51no—
254C22×C4⋊C4—51no—
255C22×C4⋊C4C2×D511yescharacteristic
256C22×C4⋊C4C2×D511yescharacteristic
257C22×C4⋊C4C2×D511yescharacteristic
258C22×C4⋊C4C2×D511yescharacteristic
259C22×C42—51no—
260C22×C42—51no—
261C22×C42—51no—
262C22×C42—51no—
263C22×C42—51no—
264C22×C42—51no—
265C22×C42—51no—
266C22×C42—51no—
267C22×C42C2×D511yescharacteristic
268#389—58no—
269#389—58no—
270#389—58no—
271#389C2×D518yes—
272#387—58no—
273#387—58no—
274#387—58no—
275#387—58no—
276C4×C4⋊C4—58no—
277C4×C4⋊C4—58no—
278C4×C4⋊C4—58no—
279C4×C4⋊C4—58no—
280C4×C4⋊C4—58no—
281C4×C4⋊C4—58no—
282C2×#95—51no—
283C2×#95—51no—
284C2×#95—51no—
285C2×#95—51no—
286C2×#95—51no—
287C2×#95—51no—
288C2×#95—51no—
289C2×#95—51no—
290C2×#95—51no—
291C2×#95—51no—
292C2×#95C2×D511yescharacteristic
293C2×#95C2×D511yescharacteristic
294C23×C5C23×C412yes—
295C23×C5C22×Q812yes—
296C23×C5C2×#4112yes—
297C23×C5C2×#4112yes—
298C23×C5C2×#4112yes—
299C23×C5H(5,𝔽2)12yes—
300C23×C52−1+412yes—
301C23×C5C2511yescharacteristic
302C2×C4×C5—28no—
303C2×C4×C5—28no—
304C2×C4×C5—28no—
305C2×C4×C5—28no—
306C2×C4×C5—28no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—22no—
312C2×C4×C5—22no—
313C2×C4×C5—22no—
314C2×C4×C5—22no—
315C2×C4×C5—22no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5—22no—
320C2×C4×C5—22no—
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—44no—
324C2×C4×C5—44no—
325C2×C4×C5C23×C414yes—
326C2×C4×C5C23×C414yes—
327C2×C4×C5C23×C414yes—
328C2×C4×C5C22×Q814yes—
329C2×C4×C5C2×#4114yes—
330C2×Dic5—28no—
331C2×Dic5—28no—
332C2×Dic5—28no—
333C2×Dic5—28no—
334C2×Dic5—28no—
335C2×Dic5—22no—
336C2×Dic5—22no—
337C2×Dic5—22no—
338C2×Dic5—22no—
339C2×Dic5—22no—
340C2×Dic5—22no—
341C2×Dic5—22no—
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—22no—
346C2×Dic5—22no—
347C2×Dic5—22no—
348C2×Dic5—22no—
349C2×Dic5—22no—
350C2×Dic5—22no—
351C2×Dic5—44no—
352C2×Dic5—44no—
353C2×Dic5C4×Q8116yes—
354C2×Dic5C23×C414yes—
355C2×Dic5C23×C414yes—
356C2×Dic5C23×C414yes—
357C2×Dic5C23×C414yes—
358C2×Dic5C22×Q814yes—
359C2×Dic5C22×Q814yes—
360C2×Dic5C2×#4114yes—
361C2×Dic5C2×#4114yes—
362C23×C4—51no—
363C23×C4—51no—
364C23×C4—51no—
365C23×C4—51no—
366C23×C4—51no—
367C23×C4—51no—
368C23×C4—51no—
369C23×C4—51no—
370C23×C4C22×D511yescharacteristic
371C23×C4C22×D511yescharacteristic
372C23×C4C22×D511yescharacteristic
373C23×C4C22×D511yescharacteristic
374C23×C4C22×D511yescharacteristic
375C23×C4C22×D511yescharacteristic
376C23×C4C22×D511yescharacteristic
377#125—564no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—58no—
382C2×C4⋊C4—58no—
383C2×C4⋊C4—58no—
384C2×C4⋊C4—58no—
385C2×C4⋊C4—58no—
386C2×C4⋊C4—58no—
387C2×C4⋊C4—54no—
388C2×C4⋊C4—54no—
389C2×C4⋊C4—54no—
390C2×C4⋊C4—54no—
391C2×C4⋊C4—54no—
392C2×C4⋊C4—54no—
393C2×C4⋊C4—54no—
394C2×C4⋊C4—54no—
395C2×C4⋊C4—54no—
396C2×C4⋊C4—54no—
397C2×C4⋊C4—104no—
398C2×C4⋊C4—104no—
399C2×C4⋊C4—104no—
400C2×C4⋊C4—202no—
401C2×C4⋊C4C4×D518yes—
402C2×C4⋊C4C22×D514yes—
403C2×C4⋊C4C22×D514yes—
404C2×C4⋊C4C22×D514yes—
405C2×C4⋊C4C22×D514yes—
406C2×C42—42no—
407C2×C42—58no—
408C2×C42—54no—
409C2×C42—54no—
410C2×C42—54no—
411C2×C42—54no—
412C2×C42—54no—
413C2×C42—54no—
414C2×C42—54no—
415C2×C42—54no—
416C2×C42—104no—
417C2×C42—104no—
418C2×C42—104no—
419C2×C42—104no—
420C2×C42—202no—
421C2×C42—202no—
422C2×C42—202no—
423C2×C42C22×D514yes—
424#95—54no—
425#95—54no—
426#95—54no—
427#95—54no—
428#95—54no—
429#95—54no—
430#95—54no—
431#95—54no—
432#95—54no—
433#95—54no—
434#95C22×D514yes—
435#95C22×D514yes—
436C22×C5C2×C4×Q814yes—
437C22×C5C4×#4114yes—
438C22×C5#52014yes—
439C22×C5#55214yes—
440C22×C5#55314yes—
441C22×C5#55414yes—
442C22×C5#55714yes—
443C22×C5C24×C411yescharacteristic
444C22×C5C23×Q811yescharacteristic
445C22×C5C22×#4111yescharacteristic
446C22×C5C22×#4111yescharacteristic
447C22×C5C22×#4111yescharacteristic
448C22×C5C2×H(5,𝔽2)11yescharacteristic
449C22×C5C2×2−1+411yescharacteristic
450C4×C5—24no—
451C4×C5—24no—
452C4×C5—24no—
453C4×C5—24no—
454C4×C5—24no—
455C4×C5—42no—
456C4×C5—42no—
457Dic5—24no—
458Dic5—24no—
459Dic5—24no—
460Dic5—24no—
461Dic5—24no—
462Dic5—42no—
463Dic5—42no—
464Dic5C2×C4×Q818yes—
465C24C23×D511yescharacteristic
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—54no—
483C22×C4—54no—
484C22×C4—54no—
485C22×C4—52no—
486C22×C4—54no—
487C22×C4—54no—
488C22×C4—54no—
489C22×C4—54no—
490C22×C4—54no—
491C22×C4—52no—
492C22×C4—52no—
493C22×C4—52no—
494C22×C4—52no—
495C22×C4—52no—
496C22×C4—52no—
497C22×C4—52no—
498C22×C4—102no—
499C22×C4—102no—
500C22×C4—102no—
501C22×C4—102no—
502C22×C4—102no—
503C22×C4—102no—
504C22×C4—102no—
505C22×C4—102no—
506C22×C4—102no—
507C22×C4—102no—
508C22×C4—102no—
509C22×C4—102no—
510C22×C4—102no—
511C22×C4—102no—
512C22×C4—102no—
513C22×C4—102no—
514C22×C4C2×C4×D514yes—
515C22×C4C2×C4×D514yes—
516C22×C4C2×C4×D514yes—
517C22×C4#71114yes—
518C22×C4Q8×D514yes—
519C22×C4C23×D512yes—
520C22×C4C23×D512yes—
521C22×C4C23×D512yes—
522C22×C4C23×D512yes—
523C22×C4C23×D512yes—
524C22×C4C23×D512yes—
525C22×C4C23×D512yes—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—44no—
529C4⋊C4—516no—
530C4⋊C4—516no—
531C4⋊C4—516no—
532C4⋊C4—516no—
533C4⋊C4—516no—
534C4⋊C4—516no—
535C4⋊C4—108no—
536C4⋊C4—108no—
537C4⋊C4—108no—
538C4⋊C4—204no—
539C4⋊C4C2×C4×D5116yes—
540C42—44no—
541C42—516no—
542C42—108no—
543C42—108no—
544C42—108no—
545C42—108no—
546C42—204no—
547C42—204no—
548C42—204no—
549C2×C5#237018yescomplemented
550C2×C5C22×C4×Q811yescharacteristic
551C2×C5C2×C4×#4111yescharacteristic
552C2×C5C2×#52011yescharacteristic
553C2×C5C2×#55211yescharacteristic
554C2×C5C2×#55311yescharacteristic
555C2×C5C2×#55411yescharacteristic
556C2×C5C2×#55711yescharacteristic
557C23C22×C4×D512yes—
558C23C2×#71112yes—
559C23#414412yes—
560C23C2×Q8×D512yes—
561C23#414712yes—
562C23D5×#4112yes—
563C23D5×#4112yes—
564C23C24×D511yescharacteristic
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—516no—
589C2×C4—54no—
590C2×C4—54no—
591C2×C4—54no—
592C2×C4—54no—
593C2×C4—54no—
594C2×C4—54no—
595C2×C4—54no—
596C2×C4—54no—
597C2×C4—108no—
598C2×C4—108no—
599C2×C4—108no—
600C2×C4—108no—
601C2×C4—102no—
602C2×C4—102no—
603C2×C4—102no—
604C2×C4—102no—
605C2×C4—102no—
606C2×C4—102no—
607C2×C4—102no—
608C2×C4—102no—
609C2×C4—102no—
610C2×C4—102no—
611C2×C4—108no—
612C2×C4—102no—
613C2×C4—102no—
614C2×C4—102no—
615C2×C4—102no—
616C2×C4—102no—
617C2×C4—102no—
618C2×C4—204no—
619C2×C4—204no—
620C2×C4C22×C4×D514yes—
621C2×C4C22×C4×D514yes—
622C2×C4C22×C4×D514yes—
623C2×C4C2×#71114yes—
624C2×C4C2×Q8×D514yes—
625C5C2×#237011yescharacteristic, complemented, Sylow
626C22#6583814yes—
627C22C4×#71114yes—
628C22C4×Q8×D514yes—
629C22#6593514yes—
630C22#6597014yes—
631C22#6602714yes—
632C22#6605514yes—
633C22C23×C4×D511yescharacteristic
634C22C22×#71111yescharacteristic
635C22C2×#414411yescharacteristic
636C22C22×Q8×D511yescharacteristic
637C22C2×#414711yescharacteristic
638C22C2×D5×#4111yescharacteristic
639C22C2×D5×#4111yescharacteristic
640C4—24no—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—24no—
645C4—42no—
646C4—42no—
647C4—58no—
648C4—104no—
649C4—104no—
650C4—104no—
651C4—104no—
652C4—104no—
653C4—202no—
654C4—202no—
655C2#1060772118yesdirect product factor
656C2C2×#6583811yescharacteristic
657C2C2×C4×#71111yescharacteristic
658C2C2×C4×Q8×D511yescharacteristic
659C2C2×#6593511yescharacteristic
660C2C2×#6597011yescharacteristic
661C2C2×#6602711yescharacteristic
662C2C2×#6605511yescharacteristic
subgroupG/H
centerC24C23×D5
commutatorC22×C5C24×C4
FrattiniC23C24×D5
FittingC2×C5×#389C2
socle, prime radicalC24×C5C24
cosocleC23×C5C25
hypercenterC2×#389D5
hypocenterC5C2×#2370
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
12. #10607721C2direct (N × H)
40. C2×#4035C4semidirect
130. #4035C2×C4semidirect
549. C2×C5#2370semidirect
625. C5C2×#2370semidirect
655. C2#10607721direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40352
C2×C5×#3892
C2×#652612
C2×#652672
C2×#652702
C2×C4⋊C4×Dic52
C2×C4×#6832
#106077212
C2×#649442
C2×#649432
C2×#649382
C2×#23705