Finite Group Databasebeta

← C2×#10613696 #12979786

Subgroups — C2×#10613696 #12979786

Order 1280.

1854 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66056C211yesmaximal, characteristic, complemented
2C2×#66024C211yesmaximal, characteristic
3C2×C5×#501C211yesmaximal, characteristic, complemented
4C2×#65376C211yesmaximal, characteristic
5C2×#65352C211yesmaximal, characteristic, complemented
6#10613696C2116yesmaximal, direct product factor
7C2×#65164C211yesmaximal, characteristic, complemented
8C2×#65179C211yesmaximal, characteristic
9C2×#65163C211yesmaximal, characteristic, complemented
10C2×#65178C211yesmaximal, characteristic
11C2×#65146C211yesmaximal, characteristic, complemented
12C2×#65159C211yesmaximal, characteristic
13C2×#65145C211yesmaximal, characteristic, complemented
14C2×#65141C211yesmaximal, characteristic
15C2×#64999C211yesmaximal, characteristic
16C2×#64990C211yesmaximal, characteristic, complemented
17C2×C5×C4⋊Q8C2211yescharacteristic
18C2×C5×#125C2211yescharacteristic
19C2×#4090—21no—
20C2×#4090—21no—
21C22×#677C2211yescharacteristic
22#66056—24no—
23#66056C2218yescomplemented
24#66024—24no—
25#66024C2218yes—
26C2×#4043—21no—
27C2×#4042—21no—
28C2×#4042—21no—
29C2×#4039C2211yescharacteristic
30C2×#4039C2211yescharacteristic
31C2×D5×C4⋊C4—21no—
32C2×#4036C2211yescharacteristic
33C2×#4034—21no—
34C2×#4034C2211yescharacteristic
35C2×#4034C2211yescharacteristic
36C2×#4034C2211yescharacteristic
37C2×#4033—21no—
38C2×#4033C2211yescharacteristic
39C2×#4033C2211yescharacteristic
40C2×C4×D20C2211yescharacteristic
41C2×C4×Dic10C2211yescharacteristic
42C2×C4×Dic10C2211yescharacteristic
43C5×#501C2218yescomplemented
44C2×C5×#107C2211yescharacteristic
45C2×C5×#107C2211yescharacteristic
46C2×C5×#106C2211yescharacteristic
47C2×C5×#106C2211yescharacteristic
48C2×C5×#97C2211yescharacteristic
49C2×#3953C2211yescharacteristic
50C2×#3950C2211yescharacteristic
51C2×#3948C2211yescharacteristic
52C2×#3947C2211yescharacteristic
53C2×#3946C2211yescharacteristic
54C2×#3946C2211yescharacteristic
55C2×C8×Dic5C2211yescharacteristic
56#65376C2218yes—
57#65352C2218yescomplemented
58C2×#3942C2211yescharacteristic
59C2×#3942C2211yescharacteristic
60C2×#3941C2211yescharacteristic
61C2×#3941C2211yescharacteristic
62C2×#3940C2211yescharacteristic
63C2×#3939C2211yescharacteristic
64C2×#3939C2211yescharacteristic
65C2×#3939C2211yescharacteristic
66C2×C4⋊#168C2211yescharacteristic
67#65179C2218yes—
68#65178C2218yes—
69#65164C2218yescomplemented
70#65163C2218yescomplemented
71#65159C2218yes—
72#65146C2218yescomplemented
73#65145C2218yescomplemented
74#65141C2218yes—
75#64999C2218yes—
76#64990C2218yescomplemented
77C2×#3385—51nomaximal, Sylow
78C22×C5×Q8—21no—
79C22×D20C2311yescharacteristic
80C22×C4×D5—21no—
81C22×C4×D5—21no—
82C22×C4×D5—21no—
83C22×Dic10—21no—
84C22×Dic10—21no—
85C22×Dic10—21no—
86C22×Dic10D411yescharacteristic
87C22×Dic10C2311yescharacteristic
88C22×C5×C8C2311yescharacteristic
89C22×C5×C8C2311yescharacteristic
90C5×C4⋊Q8—22no—
91C5×C4⋊Q8C2314yes—
92C5×#125—22no—
93C5×#125C2314yes—
94C2×C5×C4⋊C4—21no—
95C2×C5×C4⋊C4—21no—
96C2×C5×C4⋊C4—21no—
97C2×C5×C4⋊C4C2311yescharacteristic
98C2×C5×C4⋊C4C2311yescharacteristic
99C2×C5×C4⋊C4C2311yescharacteristic
100C2×C5×C4⋊C4C2311yescharacteristic
101C2×C42×C5C2311yescharacteristic
102#4090—28no—
103#4090—28no—
104C2×#685—21no—
105C2×#685—21no—
106C2×#685—21no—
107C2×C4⋊Dic5—21no—
108C2×C4⋊Dic5D411yescharacteristic
109C2×C4⋊Dic5C2311yescharacteristic
110C2×C4⋊Dic5C2311yescharacteristic
111C2×#683—21no—
112C2×#683—21no—
113C2×#683—21no—
114C2×#683—21no—
115C2×#683—21no—
116C2×#683—21no—
117C2×#683—21no—
118C2×#683D411yescharacteristic
119C2×#683D411yescharacteristic
120C2×C4×Dic5—21no—
121C2×C4×Dic5—21no—
122C2×C4×Dic5—21no—
123C2×C4×Dic5C2311yescharacteristic
124C2×C4×Dic5C2311yescharacteristic
125C22×#168C2311yescharacteristic
126C22×#168C2311yescharacteristic
127C2×#677Q818yescomplemented
128C2×#677C2314yes—
129#4043—28no—
130#4042—28no—
131#4042—28no—
132#4039—22no—
133#4039—22no—
134#4039C2314yes—
135#4039C2314yes—
136D5×C4⋊C4—28no—
137#4036—22no—
138#4036C2314yes—
139#4034—28no—
140#4034—22no—
141#4034—22no—
142#4034—22no—
143#4034C2314yes—
144#4034C2314yes—
145#4034C2314yes—
146#4033—28no—
147#4033—22no—
148#4033—22no—
149#4033C2314yes—
150#4033C2314yes—
151C4×D20—22no—
152C4×D20C2314yes—
153C4×Dic10—22no—
154C4×Dic10D414yes—
155C4×Dic10C2314yes—
156C4×Dic10C2314yes—
157C5×#107C2314yes—
158C5×#107C2314yes—
159C5×#106C2314yes—
160C5×#106C2314yes—
161C5×#97C2314yes—
162#3953C2314yes—
163#3950C2314yes—
164#3948C2314yes—
165#3947C2314yes—
166#3946C2314yes—
167#3946C2314yes—
168C8×Dic5C2314yes—
169#3942C2314yes—
170#3942C2314yes—
171#3941C2314yes—
172#3941C2314yes—
173#3940C2314yes—
174#3939C2314yes—
175#3939C2314yes—
176#3939C2314yes—
177C4⋊#168C2314yes—
178C2×Q82—51no—
179C2×#554—51no—
180#3385—516no—
181C2×#501—51no—
182C2×#501D511yescharacteristic, complemented
183C2×#500—51no—
184C2×#479—51no—
185C2×#479—51no—
186C2×#477—51no—
187C2×#476—51no—
188C2×#474—51no—
189C2×#474—51no—
190C2×#446—51no—
191C2×#440—51no—
192C2×#440—51no—
193C2×C4×SD16—51no—
194C23×D5—21no—
195C2×C5×Q8—24no—
196C2×C5×Q8—82no—
197C22×C4×C5—21no—
198C22×C4×C5—21no—
199C22×C4×C5—21no—
200C22×C4×C5—21no—
201C22×C4×C5C2×D411yescharacteristic
202C22×C4×C5C2×D411yescharacteristic
203C22×C4×C5C2411yescharacteristic
204C22×Dic5—21no—
205C22×Dic5—21no—
206C22×Dic5—21no—
207C22×Dic5C2×D411yescharacteristic
208C22×Dic5C2×D411yescharacteristic
209C22×Dic5C2×D411yescharacteristic
210C22×Dic5C2×D411yescharacteristic
211C2×D20—21no—
212C2×D20—22no—
213C2×D20C2×Q814yes—
214C2×D20C2412yes—
215C2×C4×D5—24no—
216C2×C4×D5—24no—
217C2×C4×D5—24no—
218C2×C4×D5—44no—
219C2×C4×D5—82no—
220C2×C4×D5—82no—
221C2×Dic10—21no—
222C2×Dic10—22no—
223C2×Dic10—24no—
224C2×Dic10—24no—
225C2×Dic10—24no—
226C2×Dic10—24no—
227C2×Dic10—82no—
228C2×Dic10—82no—
229C2×Dic10—82no—
230C2×Dic10C2×D414yes—
231C2×Dic10C2×Q814yes—
232C2×Dic10C2412yes—
233C2×C5×C8—22no—
234C2×C5×C8C2×Q814yes—
235C2×C5×C8C2412yes—
236C2×C5×C8C2412yes—
237C5×C4⋊C4—24no—
238C5×C4⋊C4—24no—
239C5×C4⋊C4—24no—
240C5×C4⋊C4—21no—
241C5×C4⋊C4—21no—
242C5×C4⋊C4—21no—
243C5×C4⋊C4—21no—
244C5×C4⋊C4C2412yes—
245C5×C4⋊C4C2412yes—
246C5×C4⋊C4C2412yes—
247C5×C4⋊C4C2412yes—
248C42×C5C2×D412yes—
249C42×C5C2412yes—
250#685—24no—
251#685—24no—
252#685—24no—
253C4⋊Dic5—21no—
254C4⋊Dic5—21no—
255C4⋊Dic5—24no—
256C4⋊Dic5C2×D414yes—
257C4⋊Dic5C2412yes—
258C4⋊Dic5C2412yes—
259#683—24no—
260#683—24no—
261#683—24no—
262#683—24no—
263#683—24no—
264#683—24no—
265#683—24no—
266#683C2×D414yes—
267#683C2×D414yes—
268C4×Dic5—24no—
269C4×Dic5—24no—
270C4×Dic5—24no—
271C4×Dic5C2×D412yes—
272C4×Dic5C2×D412yes—
273C4×Dic5C2412yes—
274C4×Dic5C2412yes—
275C2×#168—22no—
276C2×#168—22no—
277C2×#168C2412yes—
278C2×#168C2412yes—
279#677C2×Q8116yescomplemented
280C22×SD16—51no—
281Q82—58no—
282Q82—104no—
283#554—58no—
284#554—104no—
285C2×C4⋊Q8—51no—
286C2×C4⋊Q8—51no—
287C2×C4⋊Q8—51no—
288C2×C4⋊Q8—101no—
289C2×C4⋊Q8—101no—
290C2×C4⋊Q8—101no—
291C2×C4⋊Q8C2×D511yescharacteristic
292C2×#125—51no—
293C2×#125C2×D511yescharacteristic
294C2×C22⋊Q8—101no—
295C2×C22⋊Q8—101no—
296C2×C22⋊Q8—101no—
297C2×C4×Q8—51no—
298C2×C4×Q8—51no—
299C2×C4×Q8—51no—
300C2×C4×Q8—51no—
301C2×C4×Q8—101no—
302C2×C4×D4—51no—
303C2×C4×D4—51no—
304C2×C4×D4—51no—
305C22×C4⋊C4—101no—
306#501—58no—
307#501C2×D518yescomplemented
308#500—58no—
309#479—58no—
310#479—58no—
311#477—58no—
312#476—58no—
313#474—58no—
314#474—58no—
315#446—58no—
316#440—58no—
317#440—58no—
318C4×SD16—58no—
319C2×#107—51no—
320C2×#107—51no—
321C2×#107—51no—
322C2×#107—51no—
323C2×#107C2×D511yescharacteristic
324C2×#107C2×D511yescharacteristic
325C2×#106—51no—
326C2×#106C2×D511yescharacteristic
327C2×#106C2×D511yescharacteristic
328C2×C4⋊C8—51no—
329C2×C4⋊C8—51no—
330C2×C4⋊C8—51no—
331C2×#103—51no—
332C2×#103—51no—
333C2×#103—51no—
334C2×#102—51no—
335C2×#102—51no—
336C2×#102—51no—
337C2×#97—51no—
338C2×#97C2×D511yescharacteristic
339C2×C4×C8—51no—
340C23×C5C22×D411yescharacteristic
341C22×D5—24no—
342C22×D5—22no—
343C22×D5—42no—
344C22×D5—42no—
345C5×Q8—84no—
346C2×C4×C5—22no—
347C2×C4×C5—22no—
348C2×C4×C5—22no—
349C2×C4×C5—22no—
350C2×C4×C5—22no—
351C2×C4×C5—22no—
352C2×C4×C5—42no—
353C2×C4×C5—42no—
354C2×C4×C5—42no—
355C2×C4×C5—42no—
356C2×C4×C5C22×D412yes—
357C2×C4×C5C22×D412yes—
358C2×C4×C5C22×D411yescharacteristic
359C2×C4×C5C22×Q812yes—
360C2×C4×C52−1+412yes—
361C2×C4×C5C2511yescharacteristic
362C2×Dic5—24no—
363C2×Dic5—22no—
364C2×Dic5—22no—
365C2×Dic5—22no—
366C2×Dic5—22no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—42no—
370C2×Dic5—42no—
371C2×Dic5#13714yes—
372C2×Dic5C22×D412yes—
373C2×Dic5C22×D412yes—
374C2×Dic5C22×D412yes—
375C2×Dic5C22×D412yes—
376D20—24no—
377D20—22no—
378D20C22×Q814yes—
379C4×D5—48no—
380C4×D5—84no—
381C4×D5—84no—
382Dic10—24no—
383Dic10—22no—
384Dic10—28no—
385Dic10—84no—
386Dic10—84no—
387Dic10—84no—
388Dic10C22×Q814yes—
389C5×C8—22no—
390C5×C8C22×Q814yes—
391#168—22no—
392#168—22no—
393C22×Q8—21no—
394C22×Q8—51no—
395C22×Q8—51no—
396C22×Q8—101no—
397C22×Q8—101no—
398C22×Q8—101no—
399C22×D4—51no—
400C23×C4—101no—
401C23×C4—101no—
402C23×C4—101no—
403C2×SD16—58no—
404C2×SD16—54no—
405C22×C8—51no—
406C22×C8—51no—
407C22×C8C22×D511yescharacteristic
408C22×C8C22×D511yescharacteristic
409C4⋊Q8—22no—
410C4⋊Q8—54no—
411C4⋊Q8—54no—
412C4⋊Q8—54no—
413C4⋊Q8—102no—
414C4⋊Q8—102no—
415C4⋊Q8—102no—
416C4⋊Q8—108no—
417C4⋊Q8—108no—
418C4⋊Q8—108no—
419C4⋊Q8C22×D514yes—
420#125—22no—
421#125—54no—
422#125—102no—
423#125C22×D514yes—
424C22⋊Q8—108no—
425C22⋊Q8—108no—
426C22⋊Q8—108no—
427C4×Q8—54no—
428C4×Q8—54no—
429C4×Q8—54no—
430C4×Q8—54no—
431C4×Q8—54no—
432C4×Q8—102no—
433C4×Q8—102no—
434C4×Q8—108no—
435C4×Q8—102no—
436C4×D4—54no—
437C4×D4—54no—
438C4×D4—54no—
439C4×D4—102no—
440C4×D4—102no—
441C4×D4—102no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—51no—
446C2×C4⋊C4—51no—
447C2×C4⋊C4—51no—
448C2×C4⋊C4—51no—
449C2×C4⋊C4—51no—
450C2×C4⋊C4—108no—
451C2×C4⋊C4—101no—
452C2×C4⋊C4—101no—
453C2×C4⋊C4—101no—
454C2×C4⋊C4—101no—
455C2×C4⋊C4—101no—
456C2×C4⋊C4—101no—
457C2×C4⋊C4—101no—
458C2×C4⋊C4—101no—
459C2×C4⋊C4C22×D511yescharacteristic
460C2×C4⋊C4C22×D511yescharacteristic
461C2×C4⋊C4C22×D511yescharacteristic
462C2×C4⋊C4C22×D511yescharacteristic
463C2×C22⋊C4—101no—
464C2×C22⋊C4—101no—
465C2×C22⋊C4—101no—
466C2×C42—51no—
467C2×C42—51no—
468C2×C42—101no—
469C2×C42—101no—
470C2×C42—101no—
471C2×C42C22×D511yescharacteristic
472#107—54no—
473#107—54no—
474#107—54no—
475#107—54no—
476#107C22×D514yes—
477#107C22×D514yes—
478#106—54no—
479#106C22×D514yes—
480#106C22×D514yes—
481C4⋊C8—54no—
482C4⋊C8—54no—
483C4⋊C8—54no—
484#103—54no—
485#103—54no—
486#103—54no—
487#102—54no—
488#102—54no—
489#102—54no—
490#97—54no—
491#97C22×D514yes—
492C4×C8—54no—
493C22×C5D4×Q812yes—
494C22×C5C2×#13712yes—
495C22×C5#57612yes—
496C22×C5C23×D411yescharacteristic
497C2×D5—24no—
498C2×D5—48no—
499C2×D5—42no—
500C2×D5—42no—
501C4×C5—22no—
502C4×C5—22no—
503C4×C5—42no—
504C4×C5—42no—
505C4×C5—42no—
506C4×C5—42no—
507C4×C5D4×Q812yes—
508C4×C5C23×Q811yescharacteristic
509C4×C5C2×2−1+411yescharacteristic
510Dic5—24no—
511Dic5—22no—
512Dic5—22no—
513Dic5—22no—
514Dic5—42no—
515Dic5—42no—
516Dic5C2×#13714yes—
517C24—101no—
518C2×Q8—24no—
519C2×Q8—54no—
520C2×Q8—54no—
521C2×Q8—52no—
522C2×Q8—82no—
523C2×Q8—104no—
524C2×Q8—104no—
525C2×Q8—104no—
526C2×Q8—104no—
527C2×Q8—101no—
528C2×Q8—102no—
529C2×Q8—402no—
530C2×Q8—402no—
531C2×Q8—402no—
532C2×D4—54no—
533C2×D4—52no—
534C2×D4—101no—
535C2×D4—102no—
536C22×C4—21no—
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—51no—
541C22×C4—51no—
542C22×C4—51no—
543C22×C4—51no—
544C22×C4—104no—
545C22×C4—104no—
546C22×C4—101no—
547C22×C4—101no—
548C22×C4—101no—
549C22×C4—104no—
550C22×C4—204no—
551C22×C4—402no—
552C22×C4—402no—
553C22×C4D4×D511yescharacteristic
554C22×C4D4×D511yescharacteristic
555C22×C4C23×D511yescharacteristic
556SD16—516no—
557C2×C8—22no—
558C2×C8—52no—
559C2×C8—52no—
560C2×C8—102no—
561C2×C8—102no—
562C2×C8Q8×D514yes—
563C2×C8C23×D512yes—
564C2×C8C23×D512yes—
565C4⋊C4—24no—
566C4⋊C4—24no—
567C4⋊C4—24no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—54no—
573C4⋊C4—54no—
574C4⋊C4—54no—
575C4⋊C4—52no—
576C4⋊C4—52no—
577C4⋊C4—104no—
578C4⋊C4—104no—
579C4⋊C4—104no—
580C4⋊C4—104no—
581C4⋊C4—104no—
582C4⋊C4—104no—
583C4⋊C4—104no—
584C4⋊C4—101no—
585C4⋊C4—101no—
586C4⋊C4—104no—
587C4⋊C4C23×D512yes—
588C4⋊C4C23×D512yes—
589C4⋊C4C23×D512yes—
590C4⋊C4C23×D512yes—
591C22⋊C4—104no—
592C22⋊C4—104no—
593C22⋊C4—104no—
594C42—52no—
595C42—52no—
596C42—52no—
597C42—52no—
598C42—104no—
599C42—104no—
600C42—104no—
601C42D4×D512yes—
602C42C23×D512yes—
603C2×C5#338514yescomplemented
604C2×C5C2×D4×Q811yescharacteristic
605C2×C5C22×#13711yescharacteristic
606C2×C5C2×#57611yescharacteristic
607D5—44no—
608C23—104no—
609C23—102no—
610C23—202no—
611C23—202no—
612C23C2×D4×D511yescharacteristic
613Q8—54no—
614Q8—84no—
615Q8—108no—
616Q8—104no—
617Q8—102no—
618Q8—404no—
619Q8—404no—
620Q8—404no—
621D4—54no—
622D4—104no—
623D4—102no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—54no—
635C2×C4—52no—
636C2×C4—52no—
637C2×C4—52no—
638C2×C4—52no—
639C2×C4—104no—
640C2×C4—102no—
641C2×C4—102no—
642C2×C4—102no—
643C2×C4—102no—
644C2×C4—102no—
645C2×C4—102no—
646C2×C4—208no—
647C2×C4—202no—
648C2×C4—202no—
649C2×C4—404no—
650C2×C4—404no—
651C2×C4C2×D4×D512yes—
652C2×C4C2×D4×D512yes—
653C2×C4C2×D4×D511yescharacteristic
654C2×C4C2×Q8×D512yes—
655C2×C4#414712yes—
656C2×C4C24×D511yescharacteristic
657C8—22no—
658C8—102no—
659C8—102no—
660C8C2×Q8×D514yes—
661C5C2×#338511yescharacteristic, complemented, Sylow
662C22—104no—
663C22—208no—
664C22—202no—
665C22—202no—
666C22#6606112yes—
667C22#6610812yes—
668C22D5×#13712yes—
669C22C22×D4×D511yescharacteristic
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—54no—
677C4—104no—
678C4—102no—
679C4—102no—
680C4—102no—
681C4—202no—
682C4—202no—
683C4#6606112yes—
684C4C22×Q8×D511yescharacteristic
685C4C2×#414711yescharacteristic
686C2—204no—
687C2#1061369614yesdirect product factor
688C2C2×#6606111yescharacteristic
689C2C2×#6610811yescharacteristic
690C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#501C2
socleC23×C5C22×D4
prime radicalC22×D20C23
hypercenterC2×#501D5
hypocenterC5C2×#3385
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×#66056C2semidirect
3. C2×C5×#501C2semidirect
5. C2×#65352C2semidirect
6. #10613696C2direct (N × H)
7. C2×#65164C2semidirect
9. C2×#65163C2semidirect
11. C2×#65146C2semidirect
13. C2×#65145C2semidirect
16. C2×#64990C2semidirect
23. #66056C22semidirect
43. C5×#501C22semidirect
57. #65352C22semidirect
69. #65164C22semidirect
70. #65163C22semidirect
72. #65146C22semidirect
73. #65145C22semidirect
76. #64990C22semidirect
127. C2×#677Q8semidirect
182. C2×#501D5semidirect
279. #677C2×Q8semidirect
307. #501C2×D5semidirect
603. C2×C5#3385semidirect
661. C5C2×#3385semidirect
687. C2#10613696direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660562
C2×#660242
C2×C5×#5012
C2×#653762
C2×#653522
#106136962
C2×#651642
C2×#651792
C2×#651632
C2×#651782
C2×#651462
C2×#651592
C2×#651452
C2×#651412
C2×#649992
C2×#649902
C2×#33855