Finite Group Databasebeta

← C2×#10613219 #12969844

Subgroups — C2×#10613219 #12969844

Order 1280.

2354 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×C22⋊Q8C211yesmaximal, characteristic
2C22×#4043C211yesmaximal, characteristic
3C2×C5×#484C211yesmaximal, characteristic, complemented
4C2×#65339C211yesmaximal, characteristic, complemented
5C2×#65259C211yesmaximal, characteristic, complemented
6#10613219C2116yesmaximal, direct product factor
7C2×#65173C212yesmaximal, complemented
8C2×#65176C212yesmaximal, complemented
9C2×#65102C212yesmaximal, complemented
10C2×#65097C212yesmaximal, complemented
11C2×#65028C211yesmaximal, characteristic, complemented
12C2×D5×C22⋊C8C211yesmaximal, characteristic
13C22×Q8×D5C2211yescharacteristic
14C23×C4×D5C2211yescharacteristic
15C23×Dic10C2211yescharacteristic
16C2×C5×C22⋊Q8C2211yescharacteristic
17C22×C5×C4⋊C4C2211yescharacteristic
18C2×#4092C2211yescharacteristic
19C22×#689C2212yescomplemented
20C22×#685—21no—
21C22×C4⋊Dic5—21no—
22C2×#4070C2211yescharacteristic
23C22×Dic20C2212yescomplemented
24C22×C8×D5C2212yes—
25D5×C22⋊Q8—24no—
26D5×C22⋊Q8C2218yes—
27C2×#4043—21no—
28C2×#4043—21no—
29C2×#4043—24no—
30C2×#4043C2212yes—
31C2×#4043C2212yes—
32C2×#4043C2218yes—
33C2×#4043C2211yescharacteristic
34C2×#4042—21no—
35C2×D5×C4⋊C4—21no—
36C2×D5×C4⋊C4C2211yescharacteristic
37C2×D5×C22⋊C4—21no—
38C2×#4024—21no—
39C5×#484C2218yescomplemented
40C2×C5×#107C2212yescomplemented
41C2×C5×#103C2212yescomplemented
42C2×C5×C22⋊C8C2211yescharacteristic
43C2×C22⋊#168C2211yescharacteristic
44C2×#3952C2212yes—
45C2×#3948C2212yescomplemented
46#65339C2218yescomplemented
47#65259C2218yescomplemented
48C2×#3942C2212yescomplemented
49C2×#3942C2212yescomplemented
50C2×#3939C2212yescomplemented
51#65176C22116yescomplemented
52#65173C22116yescomplemented
53#65102C22116yescomplemented
54#65097C22116yescomplemented
55#65028C2218yescomplemented
56D5×C22⋊C8C2218yes—
57C2×#3189—51nomaximal, Sylow
58C24×D5D411yescharacteristic
59C22×C5×Q8C2311yescharacteristic
60C23×C4×C5—21no—
61C23×C4×C5C2311yescharacteristic
62C23×Dic5—21no—
63C23×Dic5D411yescharacteristic
64C2×Q8×D5—22no—
65C2×Q8×D5—44no—
66C2×Q8×D5C2314yes—
67C22×C4×D5—21no—
68C22×C4×D5—21no—
69C22×C4×D5—44no—
70C22×C4×D5D411yescharacteristic
71C22×C4×D5D411yescharacteristic
72C22×C4×D5D414yes—
73C22×C4×D5C2311yescharacteristic
74C22×C4×D5C2311yescharacteristic
75C22×C4×D5C2314yes—
76C22×Dic10—21no—
77C22×Dic10—22no—
78C22×Dic10—21no—
79C22×Dic10—44no—
80C22×Dic10C2311yescharacteristic
81C22×Dic10C2312yes—
82C22×Dic10C2314yes—
83C22×C5×C8C2312yes—
84C5×C22⋊Q8—22no—
85C5×C22⋊Q8C2314yes—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—22no—
89C2×C5×C4⋊C4C2312yes—
90C2×C5×C4⋊C4C2311yescharacteristic
91C2×C5×C4⋊C4C2314yes—
92C2×C5×C22⋊C4—21no—
93C2×C22⋊Dic5—21no—
94#4092—22no—
95#4092C2314yes—
96C2×#689—44no—
97C2×#689C2318yescomplemented
98C2×#685—21no—
99C2×#685—21no—
100C2×#685—22no—
101C2×#685—22no—
102C2×#685—28no—
103C2×C4⋊Dic5—21no—
104C2×C4⋊Dic5—22no—
105C2×C4⋊Dic5—22no—
106C2×C4⋊Dic5—28no—
107C2×C4⋊Dic5C2311yescharacteristic
108#4070—22no—
109#4070C2314yes—
110C2×#683—21no—
111C2×#683—21no—
112C2×#683—21no—
113C22×#168C2312yes—
114C2×Dic20—44no—
115C2×Dic20C2318yescomplemented
116C2×C8×D5—44no—
117C2×C8×D5C2318yes—
118#4043—24no—
119#4043—24no—
120#4043—28no—
121#4043—28no—
122#4043—22no—
123#4043C2318yes—
124#4043C2318yes—
125#4043C2314yes—
126#4042—28no—
127D5×C4⋊C4—28no—
128D5×C4⋊C4—22no—
129D5×C4⋊C4C2314yes—
130D5×C22⋊C4—28no—
131#4024—28no—
132C5×#107C2318yescomplemented
133C5×#103C2318yescomplemented
134C5×C22⋊C8C2314yes—
135C22⋊#168C2314yes—
136#3952C2318yes—
137#3948C2318yescomplemented
138#3942C2318yescomplemented
139#3942C2318yescomplemented
140#3939C2318yescomplemented
141C22×C22⋊Q8—51no—
142C22×C22⋊Q8—51no—
143#3189—516no—
144C2×#484—52no—
145C2×#484—51no—
146C2×#484D511yescharacteristic, complemented
147C2×#467—52no—
148C2×#467—52no—
149C2×#451—52no—
150C2×#451—51no—
151C2×#451—51no—
152C22×C22⋊C8—51no—
153C24×C5C2×D411yescharacteristic
154C23×D5—42no—
155C23×D5—42no—
156C23×D5Q1618yescomplemented
157C23×D5C2×D411yescharacteristic
158C23×D5C2×D411yescharacteristic
159C23×D5C2×D414yes—
160C2×C5×Q8—21no—
161C2×C5×Q8—24no—
162C2×C5×Q8C2412yes—
163C22×C4×C5—24no—
164C22×C4×C5—21no—
165C22×C4×C5—21no—
166C22×C4×C5—24no—
167C22×C4×C5—22no—
168C22×C4×C5—44no—
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D412yes—
171C22×C4×C5C2412yes—
172C22×C4×C5C2411yescharacteristic
173C22×Dic5—21no—
174C22×Dic5—21no—
175C22×Dic5—24no—
176C22×Dic5—22no—
177C22×Dic5—42no—
178C22×Dic5—44no—
179C22×Dic5C2×D411yescharacteristic
180C22×Dic5C2×D411yescharacteristic
181C22×Dic5C2×D414yes—
182Q8×D5—48no—
183Q8×D5—48no—
184C2×C4×D5—24no—
185C2×C4×D5—24no—
186C2×C4×D5—24no—
187C2×C4×D5—24no—
188C2×C4×D5—48no—
189C2×C4×D5—48no—
190C2×C4×D5—42no—
191C2×C4×D5—42no—
192C2×C4×D5—44no—
193C2×C4×D5—44no—
194C2×C4×D5C2×D412yes—
195C2×C4×D5C2×D412yes—
196C2×C4×D5C2×D414yes—
197C2×C4×D5C2×D414yes—
198C2×C4×D5C2412yes—
199C2×C4×D5C2412yes—
200C2×Dic10—24no—
201C2×Dic10—24no—
202C2×Dic10—24no—
203C2×Dic10—24no—
204C2×Dic10—22no—
205C2×Dic10—21no—
206C2×Dic10—24no—
207C2×Dic10—48no—
208C2×Dic10—44no—
209C2×Dic10—44no—
210C2×Dic10—48no—
211C2×Dic10C2414yes—
212C2×Dic10C2412yes—
213C2×C5×C8—24no—
214C2×C5×C8C2414yes—
215C5×C4⋊C4—24no—
216C5×C4⋊C4—22no—
217C5×C4⋊C4—24no—
218C5×C4⋊C4—21no—
219C5×C4⋊C4C2414yes—
220C5×C4⋊C4C2412yes—
221C5×C22⋊C4—24no—
222C22⋊Dic5—24no—
223#689—48no—
224#685—24no—
225#685—24no—
226#685—28no—
227#685—28no—
228C4⋊Dic5—21no—
229C4⋊Dic5—24no—
230C4⋊Dic5—28no—
231C4⋊Dic5—28no—
232C4⋊Dic5C2412yes—
233#683—24no—
234#683—24no—
235#683—24no—
236C2×#168—24no—
237C2×#168C2414yes—
238Dic20—48no—
239C8×D5—48no—
240C23×Q8—51no—
241C23×Q8—51no—
242C24×C4—51no—
243C22×Q16—52no—
244C22×Q16—52no—
245C23×C8—52no—
246C2×C22⋊Q8—51no—
247C2×C22⋊Q8—51no—
248C2×C22⋊Q8—58no—
249C2×C22⋊Q8—52no—
250C2×C22⋊Q8—58no—
251C2×C22⋊Q8—51no—
252C2×C22⋊Q8—52no—
253C2×C22⋊Q8—101no—
254C2×C22⋊Q8—101no—
255C2×C22⋊Q8—101no—
256C2×C22⋊Q8—104no—
257C2×C22⋊Q8—101no—
258C2×C22⋊Q8—104no—
259C2×C22⋊Q8C2×D511yescharacteristic
260C22×C4⋊C4—51no—
261C22×C4⋊C4—101no—
262C22×C4⋊C4—101no—
263C22×C4⋊C4C2×D511yescharacteristic
264C22×C22⋊C4—101no—
265C22×C22⋊C4—101no—
266#484—58no—
267#484—516no—
268#484C2×D518yescomplemented
269#467—516no—
270#467—516no—
271#451—58no—
272#451—516no—
273#451—58no—
274C2×#107—52no—
275C2×#107C2×D512yescomplemented
276C2×#103—52no—
277C2×#103—52no—
278C2×#103—52no—
279C2×#103C2×D512yescomplemented
280C2×C22⋊C8—51no—
281C2×C22⋊C8—52no—
282C2×C22⋊C8—58no—
283C2×C22⋊C8C2×D511yescharacteristic
284C23×C5—22no—
285C23×C5—22no—
286C23×C5C2×Q1614yes—
287C23×C5C22×D412yes—
288C23×C5C22×D411yescharacteristic
289C22×D5—22no—
290C22×D5—22no—
291C22×D5—22no—
292C22×D5—22no—
293C22×D5—44no—
294C22×D5—416no—
295C22×D5—44no—
296C22×D5C2×Q1614yes—
297C22×D5C2×Q1614yes—
298C22×D5C2×Q16116yescomplemented
299C22×D5C22×D412yes—
300C22×D5C22×D412yes—
301C5×Q8—24no—
302C5×Q8—42no—
303C2×C4×C5—22no—
304C2×C4×C5—24no—
305C2×C4×C5—24no—
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—24no—
309C2×C4×C5—48no—
310C2×C4×C5—42no—
311C2×C4×C5—42no—
312C2×C4×C5—44no—
313C2×C4×C5C22×D412yes—
314C2×C4×C5C22×D411yescharacteristic
315C2×C4×C5H(5,𝔽2)14yes—
316C2×C4×C5C2511yescharacteristic
317C2×Dic5—22no—
318C2×Dic5—22no—
319C2×Dic5—22no—
320C2×Dic5—22no—
321C2×Dic5—24no—
322C2×Dic5—44no—
323C2×Dic5—48no—
324C2×Dic5—42no—
325C2×Dic5—44no—
326C2×Dic5—42no—
327C2×Dic5C22×D412yes—
328C2×Dic5C22×D412yes—
329C4×D5—24no—
330C4×D5—24no—
331C4×D5—24no—
332C4×D5—24no—
333C4×D5—44no—
334C4×D5—44no—
335C4×D5—48no—
336C4×D5—48no—
337Dic10—24no—
338Dic10—24no—
339Dic10—24no—
340Dic10—48no—
341Dic10—48no—
342Dic10—44no—
343Dic10—42no—
344C5×C8—24no—
345#168—24no—
346C25—51no—
347C22×Q8—54no—
348C22×Q8—51no—
349C22×Q8—52no—
350C22×Q8—54no—
351C22×Q8—101no—
352C22×Q8—102no—
353C22×Q8—102no—
354C22×Q8—101no—
355C22×Q8—204no—
356C22×Q8—204no—
357C22×Q8C22×D511yescharacteristic
358C23×C4—21no—
359C23×C4—51no—
360C23×C4—51no—
361C23×C4—51no—
362C23×C4—51no—
363C23×C4—51no—
364C23×C4—54no—
365C23×C4—54no—
366C23×C4—101no—
367C23×C4—101no—
368C23×C4—101no—
369C23×C4—204no—
370C23×C4C22×D511yescharacteristic
371C2×Q16—58no—
372C2×Q16—58no—
373C2×Q16—204no—
374C2×Q16—204no—
375C22×C8—52no—
376C22×C8—58no—
377C22×C8—204no—
378C22×C8C22×D512yes—
379C22⋊Q8—22no—
380C22⋊Q8—54no—
381C22⋊Q8—54no—
382C22⋊Q8—58no—
383C22⋊Q8—54no—
384C22⋊Q8—58no—
385C22⋊Q8—108no—
386C22⋊Q8—108no—
387C22⋊Q8—104no—
388C22⋊Q8—104no—
389C22⋊Q8—108no—
390C22⋊Q8—108no—
391C22⋊Q8—102no—
392C22⋊Q8—102no—
393C22⋊Q8—102no—
394C22⋊Q8C22×D514yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—51no—
399C2×C4⋊C4—54no—
400C2×C4⋊C4—102no—
401C2×C4⋊C4—108no—
402C2×C4⋊C4—101no—
403C2×C4⋊C4—101no—
404C2×C4⋊C4—108no—
405C2×C4⋊C4—101no—
406C2×C4⋊C4—102no—
407C2×C4⋊C4—102no—
408C2×C4⋊C4—101no—
409C2×C4⋊C4C22×D511yescharacteristic
410C2×C4⋊C4C22×D512yes—
411C2×C4⋊C4C22×D514yes—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—108no—
414C2×C22⋊C4—108no—
415C2×C22⋊C4—101no—
416C2×C22⋊C4—101no—
417C2×C22⋊C4—102no—
418C2×C22⋊C4—102no—
419C2×C22⋊C4—101no—
420#107—58no—
421#107C22×D518yescomplemented
422#103—58no—
423#103—58no—
424#103—58no—
425#103C22×D518yescomplemented
426C22⋊C8—54no—
427C22⋊C8—58no—
428C22⋊C8C22×D514yes—
429C22×C5—28no—
430C22×C5—22no—
431C22×C5—22no—
432C22×C5#53412yes—
433C22×C5C22×Q1614yes—
434C22×C5C22×Q1612yes—
435C22×C5#57512yes—
436C22×C5C23×D411yescharacteristic
437C2×D5—28no—
438C2×D5—28no—
439C2×D5—22no—
440C2×D5—22no—
441C2×D5—22no—
442C2×D5—22no—
443C2×D5—416no—
444C2×D5C22×Q1614yes—
445C2×D5C22×Q1614yes—
446C4×C5—22no—
447C4×C5—42no—
448C4×C5—42no—
449C4×C5—44no—
450C4×C5#53412yes—
451C4×C5C2×H(5,𝔽2)12yes—
452Dic5—22no—
453Dic5—22no—
454Dic5—42no—
455Dic5—44no—
456Dic5—42no—
457C24—51no—
458C24—51no—
459C24—54no—
460C24—58no—
461C24—202no—
462C24—202no—
463C24D4×D511yescharacteristic
464C2×Q8—21no—
465C2×Q8—24no—
466C2×Q8—54no—
467C2×Q8—52no—
468C2×Q8—104no—
469C2×Q8—104no—
470C2×Q8—102no—
471C2×Q8—101no—
472C2×Q8—104no—
473C2×Q8—104no—
474C2×Q8—104no—
475C2×Q8—208no—
476C2×Q8—204no—
477C2×Q8—208no—
478C2×Q8—204no—
479C2×Q8—208no—
480C2×Q8—208no—
481C2×Q8C23×D512yes—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—51no—
489C22×C4—51no—
490C22×C4—54no—
491C22×C4—52no—
492C22×C4—52no—
493C22×C4—52no—
494C22×C4—52no—
495C22×C4—54no—
496C22×C4—54no—
497C22×C4—104no—
498C22×C4—104no—
499C22×C4—104no—
500C22×C4—104no—
501C22×C4—104no—
502C22×C4—101no—
503C22×C4—102no—
504C22×C4—101no—
505C22×C4—208no—
506C22×C4—208no—
507C22×C4—202no—
508C22×C4—202no—
509C22×C4—204no—
510C22×C4—204no—
511C22×C4—202no—
512C22×C4—204no—
513C22×C4D4×D512yes—
514C22×C4D4×D511yescharacteristic
515C22×C4C23×D512yes—
516C22×C4C23×D511yescharacteristic
517Q16—208no—
518Q16—208no—
519C2×C8—24no—
520C2×C8—54no—
521C2×C8—104no—
522C2×C8—208no—
523C2×C8C23×D514yes—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—21no—
528C4⋊C4—52no—
529C4⋊C4—104no—
530C4⋊C4—104no—
531C4⋊C4—104no—
532C4⋊C4—108no—
533C4⋊C4—108no—
534C4⋊C4—101no—
535C4⋊C4—104no—
536C4⋊C4C23×D512yes—
537C4⋊C4C23×D514yes—
538C22⋊C4—24no—
539C22⋊C4—104no—
540C22⋊C4—104no—
541C22⋊C4—108no—
542C22⋊C4—108no—
543C22⋊C4—104no—
544C2×C5—24no—
545C2×C5#318914yescomplemented
546C2×C5C2×#53411yescharacteristic
547C2×C5C23×Q1611yescharacteristic
548C2×C5C2×#57511yescharacteristic
549D5—24no—
550D5—24no—
551C23—22no—
552C23—22no—
553C23—54no—
554C23—54no—
555C23—52no—
556C23—52no—
557C23—516no—
558C23—102no—
559C23—102no—
560C23—102no—
561C23—102no—
562C23—204no—
563C23—204no—
564C23—2016no—
565C23D5×Q1614yes—
566C23C2×D4×D512yes—
567C23C2×D4×D511yescharacteristic
568Q8—24no—
569Q8—42no—
570Q8—104no—
571Q8—104no—
572Q8—104no—
573Q8—208no—
574Q8—208no—
575Q8—204no—
576Q8—202no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—52no—
588C2×C4—52no—
589C2×C4—102no—
590C2×C4—102no—
591C2×C4—104no—
592C2×C4—104no—
593C2×C4—104no—
594C2×C4—104no—
595C2×C4—102no—
596C2×C4—104no—
597C2×C4—102no—
598C2×C4—204no—
599C2×C4—204no—
600C2×C4—208no—
601C2×C4—208no—
602C2×C4—204no—
603C2×C4—208no—
604C2×C4—202no—
605C2×C4—202no—
606C2×C4—204no—
607C2×C4C2×D4×D512yes—
608C2×C4C2×D4×D511yescharacteristic
609C2×C4#414914yes—
610C2×C4C24×D511yescharacteristic
611C8—24no—
612C8—104no—
613C5C2×#318911yescharacteristic, complemented, Sylow
614C22—22no—
615C22—22no—
616C22—28no—
617C22—54no—
618C22—54no—
619C22—102no—
620C22—102no—
621C22—102no—
622C22—102no—
623C22—108no—
624C22—108no—
625C22—2016no—
626C22#6598712yes—
627C22C2×D5×Q1614yes—
628C22C2×D5×Q1612yes—
629C22#6610712yes—
630C22C22×D4×D511yescharacteristic
631C4—22no—
632C4—42no—
633C4—44no—
634C4—42no—
635C4—102no—
636C4—102no—
637C4—202no—
638C4—204no—
639C4—202no—
640C4#6598712yes—
641C4C2×#414912yes—
642C2—24no—
643C2—104no—
644C2—104no—
645C2#1061321914yesdirect product factor
646C2C2×#6598711yescharacteristic
647C2C22×D5×Q1611yescharacteristic
648C2C2×#6610711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#484C2
socleC23×C5C22×D4
prime radicalC24×D5D4
hypercenterC2×#484D5
hypocenterC5C2×#3189
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
3. C2×C5×#484C2semidirect
4. C2×#65339C2semidirect
5. C2×#65259C2semidirect
6. #10613219C2direct (N × H)
7. C2×#65173C2semidirect
8. C2×#65176C2semidirect
9. C2×#65102C2semidirect
10. C2×#65097C2semidirect
11. C2×#65028C2semidirect
19. C22×#689C22semidirect
23. C22×Dic20C22semidirect
39. C5×#484C22semidirect
40. C2×C5×#107C22semidirect
41. C2×C5×#103C22semidirect
45. C2×#3948C22semidirect
46. #65339C22semidirect
47. #65259C22semidirect
48. C2×#3942C22semidirect
49. C2×#3942C22semidirect
50. C2×#3939C22semidirect
51. #65176C22semidirect
52. #65173C22semidirect
53. #65102C22semidirect
54. #65097C22semidirect
55. #65028C22semidirect
97. C2×#689C23semidirect
115. C2×Dic20C23semidirect
132. C5×#107C23semidirect
133. C5×#103C23semidirect
137. #3948C23semidirect
138. #3942C23semidirect
139. #3942C23semidirect
140. #3939C23semidirect
146. C2×#484D5semidirect
156. C23×D5Q16semidirect
268. #484C2×D5semidirect
275. C2×#107C2×D5semidirect
279. C2×#103C2×D5semidirect
298. C22×D5C2×Q16semidirect
421. #107C22×D5semidirect
425. #103C22×D5semidirect
545. C2×C5#3189semidirect
613. C5C2×#3189semidirect
645. C2#10613219direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×C22⋊Q82
C22×#40432
C2×C5×#4842
C2×#653392
C2×#652592
#106132192
C2×#651732
C2×#651762
C2×#651022
C2×#650972
C2×#650282
C2×D5×C22⋊C82
C2×#31895