Finite Group Databasebeta

← C2×#10617111 #13044577

Subgroups — C2×#10617111 #13044577

Order 1280.

2254 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×#117C211yesmaximal, characteristic
2C2×#65755C211yesmaximal, characteristic
3C22×#4003C211yesmaximal, characteristic
4#10617111C2116yesmaximal, direct product factor
5C2×#65706C212yesmaximal, complemented
6C2×#65706C212yesmaximal, complemented
7C2×#65696C212yesmaximal, complemented
8C2×#65691C212yesmaximal, complemented
9C2×#65684C211yesmaximal, characteristic, complemented
10C2×#65683C211yesmaximal, characteristic, complemented
11C2×#65680C211yesmaximal, characteristic, complemented
12C2×#65678C211yesmaximal, characteristic, complemented
13C23×C4×D5C2211yescharacteristic
14C22×#704C2212yes—
15C23×#170C2212yes—
16C2×C5×#117C411yescharacteristic
17C2×#4072C411yescharacteristic
18D5×#117C418yes—
19D5×#117C2218yes—
20C2×#4038C2211yescharacteristic
21C2×#4038C2211yescharacteristic
22C2×D5×C4⋊C4C411yescharacteristic
23C2×D5×C4⋊C4C411yescharacteristic
24C2×D5×C22⋊C4C411yescharacteristic
25C2×D5×C22⋊C4C411yescharacteristic
26C2×#4023C2211yescharacteristic
27C2×#4023C2211yescharacteristic
28C2×#4018C411yescharacteristic
29C2×#4018C411yescharacteristic
30C2×C42×D5C2211yescharacteristic
31C2×C42×D5C2211yescharacteristic
32#65755C2218yes—
33C2×#4004C2212yescomplemented
34C2×#4004C2212yescomplemented
35C2×#4003C2218yes—
36C2×#4003C2211yescharacteristic
37C2×#4003C2211yescharacteristic
38C2×#4003C2212yes—
39C2×#4003C2211yescharacteristic
40C2×#4003C2211yescharacteristic
41C2×#4003C2212yes—
42C2×#4002C2212yescomplemented
43C2×#4002C2212yescomplemented
44C2×C4⋊#170C2212yescomplemented
45C2×C4⋊#170C2212yescomplemented
46C2×C4×#170C2212yescomplemented
47C2×C4×#170C2212yescomplemented
48#65706C22116yescomplemented
49#65706C22116yescomplemented
50#65696C22116yescomplemented
51#65691C22116yescomplemented
52#65684C2218yescomplemented
53#65683C2218yescomplemented
54#65680C2218yescomplemented
55#65678C2218yescomplemented
56C2×#2924—51nomaximal, Sylow
57C24×D5C2×C411yescharacteristic
58C23×C4×C5C2×C411yescharacteristic
59C23×Dic5C2311yescharacteristic
60C22×C4×D5—28no—
61C22×C4×D5C2×C411yescharacteristic
62C22×C4×D5C2×C411yescharacteristic
63C22×C4×D5C2×C411yescharacteristic
64C22×C4×D5C2×C411yescharacteristic
65C22×C4×D5C2×C414yes—
66C22×C4×D5C2×C411yescharacteristic
67C22×C4×D5C2×C411yescharacteristic
68C22×C4×D5C2311yescharacteristic
69C22×C4×D5C2311yescharacteristic
70C22×C4×D5C2314yes—
71C2×#704—28no—
72C2×#704C2318yes—
73C22×#170—28no—
74C22×#170C2318yes—
75C22×#170C2312yes—
76C22×#170C2312yes—
77C22×#170C2312yes—
78C22×#170C2312yes—
79C5×#117C2×C418yes—
80C2×C5×C4⋊C4C2×C411yescharacteristic
81C2×C5×C4⋊C4C2×C411yescharacteristic
82C2×C5×C22⋊C4C2×C411yescharacteristic
83C2×C5×C22⋊C4C2×C411yescharacteristic
84C2×C42×C5C2×C411yescharacteristic
85C2×C42×C5C2×C411yescharacteristic
86C2×C22⋊Dic5C2×C411yescharacteristic
87C2×C22⋊Dic5C2×C411yescharacteristic
88C2×#685C2×C411yescharacteristic
89C2×#685C2×C411yescharacteristic
90C2×#685C2×C411yescharacteristic
91C2×#685C2×C411yescharacteristic
92#4072C2×C418yes—
93C2×C4⋊Dic5C2×C411yescharacteristic
94C2×C4⋊Dic5C2×C411yescharacteristic
95C2×#683C2×C411yescharacteristic
96C2×#683C2×C411yescharacteristic
97C2×#683C2×C411yescharacteristic
98C2×#683C2×C411yescharacteristic
99C2×C4×Dic5C2×C411yescharacteristic
100C2×C4×Dic5C2×C411yescharacteristic
101C2×C4×Dic5C2311yescharacteristic
102C2×C4×Dic5C2311yescharacteristic
103C2×C4×Dic5C2311yescharacteristic
104C2×C4×Dic5C2311yescharacteristic
105#4038C2×C414yes—
106#4038C2×C414yes—
107#4038C2314yes—
108#4038C2314yes—
109D5×C4⋊C4C2×C418yes—
110D5×C4⋊C4C2×C418yes—
111D5×C22⋊C4C2×C418yes—
112D5×C22⋊C4C2×C418yes—
113#4023C2×C414yes—
114#4023C2×C414yes—
115#4023C2314yes—
116#4023C2314yes—
117#4018C2×C418yes—
118#4018C2×C418yes—
119C42×D5C2×C414yes—
120C42×D5C2×C414yes—
121C42×D5C2314yes—
122C42×D5C2314yes—
123#4004C2318yescomplemented
124#4004C2318yescomplemented
125#4003C2314yes—
126#4003C2314yes—
127#4003C2314yes—
128#4003C2314yes—
129#4003C2318yes—
130#4003C2318yes—
131#4002C2318yescomplemented
132#4002C2318yescomplemented
133C4⋊#170C2318yescomplemented
134C4⋊#170C2318yescomplemented
135C4×#170C2318yescomplemented
136C4×#170C2318yescomplemented
137C22×#117—51no—
138#2924—516no—
139C2×#433—52no—
140C2×#433—52no—
141C2×#432—51no—
142C2×#432—51no—
143C2×#431—51no—
144C2×#431—51no—
145C2×#424—52no—
146C2×#407—51no—
147C22×C22⋊C8—51no—
148C2×#405—52no—
149C24×C5C22×C411yescharacteristic
150C23×D5—24no—
151C23×D5—24no—
152C23×D5—42no—
153C23×D5C22×C411yescharacteristic
154C23×D5C22×C414yes—
155C23×D5C22×C411yescharacteristic
156C22×C4×C5—24no—
157C22×C4×C5C22×C411yescharacteristic
158C22×C4×C5C22×C411yescharacteristic
159C22×C4×C5C22×C411yescharacteristic
160C22×C4×C5C22×C411yescharacteristic
161C22×C4×C5C22×C411yescharacteristic
162C22×C4×C5C22×C411yescharacteristic
163C22×C4×C5C22×C414yes—
164C22×Dic5C22×C411yescharacteristic
165C22×Dic5C22×C411yescharacteristic
166C22×Dic5C22×C411yescharacteristic
167C22×Dic5C22×C411yescharacteristic
168C22×Dic5C22×C412yes—
169C22×Dic5C22×C411yescharacteristic
170C22×Dic5#4118yes—
171C22×Dic5C2411yescharacteristic
172C22×Dic5C2412yes—
173C2×C4×D5—24no—
174C2×C4×D5—24no—
175C2×C4×D5—24no—
176C2×C4×D5—24no—
177C2×C4×D5—216no—
178C2×C4×D5—24no—
179C2×C4×D5—24no—
180C2×C4×D5—216no—
181C2×C4×D5—42no—
182C2×C4×D5—42no—
183C2×C4×D5C22×C412yes—
184C2×C4×D5C22×C414yes—
185C2×C4×D5C22×C414yes—
186C2×C4×D5C22×C412yes—
187C2×C4×D5C22×C414yes—
188C2×C4×D5C22×C414yes—
189C2×C4×D5C22×C414yes—
190C2×C4×D5C22×C414yes—
191C2×C4×D5C2412yes—
192C2×C4×D5C2412yes—
193#704—216no—
194C2×#170—24no—
195C2×#170—24no—
196C2×#170—216no—
197C2×#170—24no—
198C2×#170—24no—
199C2×#170C2414yes—
200C2×#170C2414yes—
201C2×#170C2414yes—
202C2×#170C2414yes—
203C5×C4⋊C4C22×C414yes—
204C5×C4⋊C4C22×C414yes—
205C5×C22⋊C4C22×C414yes—
206C5×C22⋊C4C22×C414yes—
207C42×C5C22×C414yes—
208C42×C5C22×C414yes—
209C22⋊Dic5C22×C414yes—
210C22⋊Dic5C22×C414yes—
211#685C22×C414yes—
212#685C22×C414yes—
213#685C22×C414yes—
214#685C22×C414yes—
215C4⋊Dic5C22×C414yes—
216C4⋊Dic5C22×C414yes—
217#683C22×C414yes—
218#683C22×C414yes—
219#683C22×C414yes—
220#683C22×C414yes—
221C4×Dic5C22×C412yes—
222C4×Dic5C22×C412yes—
223C4×Dic5C22×C412yes—
224C4×Dic5C22×C412yes—
225C4×Dic5C22×C414yes—
226C4×Dic5C22×C414yes—
227C4×Dic5C2412yes—
228C4×Dic5C2412yes—
229C4×Dic5C2412yes—
230C4×Dic5C2412yes—
231C24×C4—51no—
232C22×M16—52no—
233C23×C8—52no—
234C2×#117—51no—
235C2×#117—58no—
236C2×#117—58no—
237C2×#117—51no—
238C2×#117—51no—
239C2×#117—51no—
240C2×#117—51no—
241C2×#117—51no—
242C2×#117—51no—
243C2×#117AGL(1,𝔽5)11yescharacteristic
244C22×C4⋊C4—51no—
245C22×C4⋊C4—51no—
246C22×C22⋊C4—51no—
247C22×C22⋊C4—51no—
248C22×C42—51no—
249C22×C42—51no—
250#433—516no—
251#433—516no—
252#432—58no—
253#432—58no—
254#431—58no—
255#431—58no—
256#424—516no—
257C2×C4⋊C8—52no—
258C2×C4⋊C8—52no—
259C2×C4⋊C8—52no—
260C2×C4⋊C8—52no—
261#407—58no—
262C2×C22⋊C8—51no—
263C2×C22⋊C8—51no—
264C2×C22⋊C8—52no—
265C2×C22⋊C8—58no—
266C2×C22⋊C8—51no—
267C2×C22⋊C8—51no—
268C2×C22⋊C8—52no—
269#405—516no—
270C2×#97—52no—
271C2×#97—52no—
272C2×C4×C8—52no—
273C2×C4×C8—52no—
274C23×C5—22no—
275C23×C5—22no—
276C23×C5#11714yes—
277C23×C5C23×C412yes—
278C23×C5C23×C411yescharacteristic
279C22×D5—22no—
280C22×D5—22no—
281C22×D5—28no—
282C22×D5—22no—
283C22×D5—22no—
284C22×D5—22no—
285C22×D5—28no—
286C22×D5—44no—
287C22×D5—416no—
288C22×D5#13114yes—
289C22×D5C23×C412yes—
290C22×D5C23×C412yes—
291C2×C4×C5—28no—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—22no—
296C2×C4×C5—22no—
297C2×C4×C5—22no—
298C2×C4×C5C23×C412yes—
299C2×C4×C5C23×C412yes—
300C2×C4×C5C23×C412yes—
301C2×C4×C5C23×C412yes—
302C2×C4×C5C23×C412yes—
303C2×C4×C5C23×C412yes—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—22no—
308C2×Dic5#11718yes—
309C2×Dic5#11714yes—
310C2×Dic5C23×C412yes—
311C2×Dic5C23×C412yes—
312C2×Dic5C23×C412yes—
313C2×Dic5C23×C412yes—
314C2×Dic5C23×C411yescharacteristic
315C2×Dic5C23×C412yes—
316C2×Dic5C2×#4114yes—
317C2×Dic5C2×#4118yes—
318C2×Dic5C2511yescharacteristic
319C4×D5—28no—
320C4×D5—28no—
321C4×D5—24no—
322C4×D5—28no—
323C4×D5—28no—
324C4×D5—24no—
325C4×D5—24no—
326C4×D5—24no—
327C4×D5—44no—
328C4×D5—44no—
329#170—24no—
330#170—24no—
331#170—24no—
332#170—24no—
333C25—51no—
334C23×C4—51no—
335C23×C4—51no—
336C23×C4—51no—
337C23×C4—51no—
338C23×C4—54no—
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—54no—
342C23×C4—51no—
343C23×C4—51no—
344C23×C4—51no—
345C23×C4—108no—
346C23×C4C2×AGL(1,𝔽5)11yescharacteristic
347C2×M16—58no—
348C2×M16—108no—
349C22×C8—52no—
350C22×C8—52no—
351C22×C8—58no—
352C22×C8—52no—
353C22×C8—52no—
354C22×C8—108no—
355#117—58no—
356#117—54no—
357#117—54no—
358#117—54no—
359#117—54no—
360#117—54no—
361#117—54no—
362#117—54no—
363#117—54no—
364#117—58no—
365#117—58no—
366#117C2×AGL(1,𝔽5)18yes—
367C2×C4⋊C4—51no—
368C2×C4⋊C4—51no—
369C2×C4⋊C4—51no—
370C2×C4⋊C4—51no—
371C2×C4⋊C4—51no—
372C2×C4⋊C4—51no—
373C2×C4⋊C4—58no—
374C2×C4⋊C4—58no—
375C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
376C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
377C2×C22⋊C4—51no—
378C2×C22⋊C4—51no—
379C2×C22⋊C4—51no—
380C2×C22⋊C4—51no—
381C2×C22⋊C4—51no—
382C2×C22⋊C4—51no—
383C2×C22⋊C4—58no—
384C2×C22⋊C4—58no—
385C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
386C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
387C2×C42—51no—
388C2×C42—51no—
389C2×C42—51no—
390C2×C42—51no—
391C2×C42—51no—
392C2×C42—51no—
393C2×C42—54no—
394C2×C42—54no—
395C2×C42—54no—
396C2×C42—54no—
397C2×C42C2×AGL(1,𝔽5)11yescharacteristic
398C2×C42C2×AGL(1,𝔽5)11yescharacteristic
399C4⋊C8—58no—
400C4⋊C8—58no—
401C4⋊C8—58no—
402C4⋊C8—58no—
403C22⋊C8—54no—
404C22⋊C8—54no—
405C22⋊C8—58no—
406C22⋊C8—54no—
407C22⋊C8—54no—
408C22⋊C8—58no—
409#97—58no—
410#97—58no—
411C4×C8—58no—
412C4×C8—58no—
413C22×C5—28no—
414C22×C5—22no—
415C22×C5—22no—
416C22×C5C2×#11714yes—
417C22×C5C2×#11712yes—
418C22×C5C2×#13112yes—
419C22×C5#56812yes—
420C22×C5C24×C411yescharacteristic
421C2×D5—22no—
422C2×D5—28no—
423C2×D5—22no—
424C2×D5—22no—
425C2×D5—22no—
426C2×D5—22no—
427C2×D5—416no—
428C2×D5—44no—
429C2×D5C2×#13114yes—
430C4×C5—22no—
431C4×C5—22no—
432C4×C5—22no—
433C4×C5—22no—
434C4×C5—22no—
435C4×C5—22no—
436Dic5—22no—
437Dic5—22no—
438Dic5—22no—
439Dic5—22no—
440Dic5C2×#11712yes—
441Dic5C2×#11714yes—
442Dic5C22×#4112yes—
443C24—51no—
444C24—51no—
445C24—54no—
446C24—104no—
447C24—104no—
448C24—202no—
449C24C22×AGL(1,𝔽5)11yescharacteristic
450C22×C4—24no—
451C22×C4—58no—
452C22×C4—52no—
453C22×C4—52no—
454C22×C4—51no—
455C22×C4—51no—
456C22×C4—51no—
457C22×C4—51no—
458C22×C4—51no—
459C22×C4—51no—
460C22×C4—52no—
461C22×C4—52no—
462C22×C4—52no—
463C22×C4—54no—
464C22×C4—54no—
465C22×C4—54no—
466C22×C4—52no—
467C22×C4—54no—
468C22×C4—54no—
469C22×C4—54no—
470C22×C4—104no—
471C22×C4—104no—
472C22×C4—1016no—
473C22×C4—104no—
474C22×C4—1016no—
475C22×C4—104no—
476C22×C4—104no—
477C22×C4—104no—
478C22×C4—202no—
479C22×C4—202no—
480C22×C4C22×AGL(1,𝔽5)14yes—
481C22×C4C22×AGL(1,𝔽5)11yescharacteristic
482C22×C4C22×AGL(1,𝔽5)11yescharacteristic
483C22×C4C22×AGL(1,𝔽5)11yescharacteristic
484C22×C4C22×AGL(1,𝔽5)11yescharacteristic
485C22×C4C22×AGL(1,𝔽5)11yescharacteristic
486C22×C4C22×AGL(1,𝔽5)11yescharacteristic
487M16—1016no—
488C2×C8—54no—
489C2×C8—54no—
490C2×C8—54no—
491C2×C8—54no—
492C2×C8—104no—
493C2×C8—104no—
494C2×C8—1016no—
495C2×C8—104no—
496C2×C8—104no—
497C4⋊C4—54no—
498C4⋊C4—54no—
499C4⋊C4—54no—
500C4⋊C4—54no—
501C4⋊C4—54no—
502C4⋊C4—54no—
503C4⋊C4C22×AGL(1,𝔽5)14yes—
504C4⋊C4C22×AGL(1,𝔽5)14yes—
505C22⋊C4—54no—
506C22⋊C4—54no—
507C22⋊C4—54no—
508C22⋊C4—54no—
509C22⋊C4—54no—
510C22⋊C4—54no—
511C22⋊C4C22×AGL(1,𝔽5)14yes—
512C22⋊C4C22×AGL(1,𝔽5)14yes—
513C42—52no—
514C42—52no—
515C42—52no—
516C42—52no—
517C42—52no—
518C42—52no—
519C42—52no—
520C42—52no—
521C42—54no—
522C42—54no—
523C42C22×AGL(1,𝔽5)14yes—
524C42C22×AGL(1,𝔽5)14yes—
525C2×C5—24no—
526C2×C5#292414yescomplemented
527C2×C5C22×#11711yescharacteristic
528C2×C5C22×#13111yescharacteristic
529C2×C5C2×#56811yescharacteristic
530D5—24no—
531D5—42no—
532C23—22no—
533C23—22no—
534C23—54no—
535C23—52no—
536C23—52no—
537C23—108no—
538C23—102no—
539C23—102no—
540C23—108no—
541C23—102no—
542C23—102no—
543C23—102no—
544C23—204no—
545C23—2016no—
546C23#413014yes—
547C23C23×AGL(1,𝔽5)12yes—
548C23C23×AGL(1,𝔽5)11yescharacteristic
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—54no—
557C2×C4—58no—
558C2×C4—58no—
559C2×C4—54no—
560C2×C4—51no—
561C2×C4—51no—
562C2×C4—52no—
563C2×C4—52no—
564C2×C4—52no—
565C2×C4—52no—
566C2×C4—52no—
567C2×C4—104no—
568C2×C4—104no—
569C2×C4—104no—
570C2×C4—108no—
571C2×C4—108no—
572C2×C4—104no—
573C2×C4—108no—
574C2×C4—108no—
575C2×C4—102no—
576C2×C4—102no—
577C2×C4—102no—
578C2×C4—102no—
579C2×C4—204no—
580C2×C4—204no—
581C2×C4C23×AGL(1,𝔽5)12yes—
582C2×C4C23×AGL(1,𝔽5)12yes—
583C2×C4C23×AGL(1,𝔽5)12yes—
584C2×C4C23×AGL(1,𝔽5)12yes—
585C2×C4C23×AGL(1,𝔽5)12yes—
586C2×C4C23×AGL(1,𝔽5)12yes—
587C8—104no—
588C8—104no—
589C8—104no—
590C8—104no—
591C5C2×#292411yescharacteristic, complemented, Sylow
592C22—22no—
593C22—22no—
594C22—28no—
595C22—54no—
596C22—102no—
597C22—102no—
598C22—108no—
599C22—102no—
600C22—102no—
601C22—102no—
602C22—2016no—
603C22—204no—
604C22C2×#413014yes—
605C22C2×#413012yes—
606C22D5⋊#13112yes—
607C22#6626412yes—
608C22C24×AGL(1,𝔽5)11yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—52no—
616C4—52no—
617C4—54no—
618C4—102no—
619C4—102no—
620C4—102no—
621C4—102no—
622C2—24no—
623C2—104no—
624C2—202no—
625C2#1061711114yesdirect product factor
626C2C22×#413011yescharacteristic
627C2C2×D5⋊#13111yescharacteristic
628C2C2×#6626411yescharacteristic
subgroupG/H
centerC23C23×AGL(1,𝔽5)
commutatorC22×C5C24×C4
FrattiniC22C24×AGL(1,𝔽5)
FittingC2×C5×#117C4
socleC23×C5C23×C4
prime radicalC24×D5C2×C4
cosocleC2×Dic5C25
hypercenterC2×#117AGL(1,𝔽5)
hypocenterC5C2×#2924
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
4. #10617111C2direct (N × H)
5. C2×#65706C2semidirect
6. C2×#65706C2semidirect
7. C2×#65696C2semidirect
8. C2×#65691C2semidirect
9. C2×#65684C2semidirect
10. C2×#65683C2semidirect
11. C2×#65680C2semidirect
12. C2×#65678C2semidirect
33. C2×#4004C22semidirect
34. C2×#4004C22semidirect
42. C2×#4002C22semidirect
43. C2×#4002C22semidirect
44. C2×C4⋊#170C22semidirect
45. C2×C4⋊#170C22semidirect
46. C2×C4×#170C22semidirect
47. C2×C4×#170C22semidirect
48. #65706C22semidirect
49. #65706C22semidirect
50. #65696C22semidirect
51. #65691C22semidirect
52. #65684C22semidirect
53. #65683C22semidirect
54. #65680C22semidirect
55. #65678C22semidirect
123. #4004C23semidirect
124. #4004C23semidirect
131. #4002C23semidirect
132. #4002C23semidirect
133. C4⋊#170C23semidirect
134. C4⋊#170C23semidirect
135. C4×#170C23semidirect
136. C4×#170C23semidirect
526. C2×C5#2924semidirect
591. C5C2×#2924semidirect
625. C2#10617111direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×#1172
C2×#657552
C22×#40032
#106171112
C2×#657062
C2×#656962
C2×#656912
C2×#656842
C2×#656832
C2×#656802
C2×#656782
C2×#29245