Finite Group Databasebeta

← C2×#10617154 #13044841

Subgroups — C2×#10617154 #13044841

Order 1280.

2286 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#711C211yesmaximal, characteristic
2C2×#65773C212yesmaximal, complemented
3C2×#65754C211yesmaximal, characteristic, complemented
4C2×#65748C211yesmaximal, characteristic, complemented
5C2×C4×#704C211yesmaximal, characteristic, complemented
6C22×C4×#170C211yesmaximal, characteristic, complemented
7#10617154C2116yesmaximal, direct product factor
8C2×#65707C212yesmaximal, complemented
9C2×#65695C212yesmaximal, complemented
10C2×#65693C212yesmaximal, complemented
11C2×#65678C211yesmaximal, characteristic, complemented
12C2×C4×#700C211yesmaximal, characteristic, complemented
13C22×#711C411yescharacteristic
14C22×#704C2212yes—
15C23×#170C2212yes—
16C22×#700C2212yes—
17C2×C4×C5×D4C411yescharacteristic
18C2×D4×Dic5C2211yescharacteristic
19C2×C4×C22⋊D5C411yescharacteristic
20C2×C4×C22⋊D5C411yescharacteristic
21C22×C4×Dic5C2211yescharacteristic
22C22×C4×Dic5C2211yescharacteristic
23C4×#711C418yes—
24C4×#711C2218yes—
25C2×#4038C411yescharacteristic
26C2×#4033C2211yescharacteristic
27C2×#4027C2211yescharacteristic
28C2×#4027C2211yescharacteristic
29C2×#4023C411yescharacteristic
30C2×#4023C411yescharacteristic
31C2×C42×D5C2211yescharacteristic
32C2×C4×Dic10C411yescharacteristic
33C2×C22⋊#170C2212yes—
34C2×C22⋊#170C2212yes—
35#65773C22116yescomplemented
36#65754C2218yescomplemented
37C2×#4004C2212yes—
38C2×#4004C2212yes—
39C2×#4003C2212yes—
40C4×#704C2218yescomplemented
41C2×#4002C2212yes—
42C2×#4002C2212yes—
43#65748C2218yescomplemented
44C2×C4⋊#170C2212yes—
45C2×C4×#170C2211yescharacteristic
46C2×C4×#170C2211yescharacteristic
47C2×C4×#170C2212yes—
48C2×C4×#170C2211yescharacteristic
49C2×C4×#170C2211yescharacteristic
50C2×C4×#170C2218yescomplemented
51#65707C22116yescomplemented
52#65695C22116yescomplemented
53#65693C22116yescomplemented
54#65678C2218yescomplemented
55C4×#700C2218yescomplemented
56C2×#2965—51nomaximal, Sylow
57C22×C5×D4C2×C411yescharacteristic
58C23×C4×C5C2×C411yescharacteristic
59C23×C4×C5C2×C411yescharacteristic
60C22×C22⋊D5C2×C411yescharacteristic
61C22×C22⋊D5C2×C411yescharacteristic
62C23×Dic5C2×C411yescharacteristic
63C23×Dic5C2×C411yescharacteristic
64C23×Dic5C2311yescharacteristic
65C23×Dic5C2311yescharacteristic
66C2×#711—28no—
67C2×#711C2×C418yes—
68C22×C4×D5C2×C411yescharacteristic
69C22×C4×D5C2×C411yescharacteristic
70C22×C4×D5C2311yescharacteristic
71C22×Dic10C2×C411yescharacteristic
72C2×#704—28no—
73C2×#704C2318yes—
74C22×#170—28no—
75C22×#170C2312yes—
76C22×#170C2312yes—
77C22×#170C2312yes—
78C22×#170C2318yes—
79C22×#170C2312yes—
80C2×#700—28no—
81C2×#700C2318yes—
82C4×C5×D4C2×C418yes—
83C2×C5×C4⋊C4C2×C411yescharacteristic
84C2×C5×C22⋊C4C2×C411yescharacteristic
85C2×C5×C22⋊C4C2×C411yescharacteristic
86C2×C42×C5C2×C411yescharacteristic
87C2×C22⋊Dic5C2×C411yescharacteristic
88C2×C22⋊Dic5C2×C411yescharacteristic
89D4×Dic5C2×C414yes—
90D4×Dic5C2314yes—
91C4×C22⋊D5C2×C418yes—
92C4×C22⋊D5C2×C418yes—
93C2×#685C2×C411yescharacteristic
94C2×#685C2×C411yescharacteristic
95C2×C4⋊Dic5C2×C411yescharacteristic
96C2×#683C2×C411yescharacteristic
97C2×#683C2×C411yescharacteristic
98C2×#683C2×C411yescharacteristic
99C2×#683C2×C411yescharacteristic
100C2×C4×Dic5C2×C411yescharacteristic
101C2×C4×Dic5C2×C411yescharacteristic
102C2×C4×Dic5C2×C411yescharacteristic
103C2×C4×Dic5C2×C411yescharacteristic
104C2×C4×Dic5C2×C411yescharacteristic
105C2×C4×Dic5C2×C414yes—
106C2×C4×Dic5C2×C414yes—
107C2×C4×Dic5C2311yescharacteristic
108C2×C4×Dic5C2311yescharacteristic
109C2×C4×Dic5C2311yescharacteristic
110C2×C4×Dic5C2311yescharacteristic
111C2×C4×Dic5C2314yes—
112C2×C4×Dic5C2314yes—
113#4038C2×C418yes—
114#4033C2×C414yes—
115#4033C2314yes—
116#4027C2×C414yes—
117#4027C2×C414yes—
118#4027C2314yes—
119#4027C2314yes—
120#4023C2×C418yes—
121#4023C2×C418yes—
122C42×D5C2×C414yes—
123C42×D5C2314yes—
124C4×Dic10C2×C418yes—
125C22⋊#170C2318yes—
126C22⋊#170C2318yes—
127#4004C2318yes—
128#4004C2318yes—
129#4003C2318yes—
130#4002C2318yes—
131#4002C2318yes—
132C4⋊#170C2318yes—
133C4×#170C2314yes—
134C4×#170C2314yes—
135C4×#170C2314yes—
136C4×#170C2314yes—
137C4×#170C2318yes—
138C2×C4×#41—51no—
139#2965—516no—
140C2×#446—52no—
141C2×#436—52no—
142C2×#435—52no—
143C2×#435—52no—
144C2×#431—51no—
145C2×#431—51no—
146C2×#431—51no—
147C2×C4×M16—51no—
148C2×C4×M16—51no—
149C22×C4×C8—51no—
150C24×C5C22×C411yescharacteristic
151C24×C5C22×C411yescharacteristic
152C23×D5C22×C411yescharacteristic
153C2×C5×D4—24no—
154C2×C5×D4C22×C414yes—
155C22×C4×C5—24no—
156C22×C4×C5—24no—
157C22×C4×C5C22×C411yescharacteristic
158C22×C4×C5C22×C411yescharacteristic
159C22×C4×C5C22×C411yescharacteristic
160C22×C4×C5C22×C411yescharacteristic
161C22×C4×C5C22×C414yes—
162C22×C4×C5C22×C414yes—
163C22×C4×C5C22×C411yescharacteristic
164C2×C22⋊D5—24no—
165C2×C22⋊D5—24no—
166C2×C22⋊D5C22×C414yes—
167C2×C22⋊D5C22×C414yes—
168C22×Dic5—24no—
169C22×Dic5—24no—
170C22×Dic5—24no—
171C22×Dic5M1618yescomplemented
172C22×Dic5C22×C411yescharacteristic
173C22×Dic5C22×C411yescharacteristic
174C22×Dic5C22×C411yescharacteristic
175C22×Dic5C22×C411yescharacteristic
176C22×Dic5C22×C414yes—
177C22×Dic5C22×C412yes—
178C22×Dic5C22×C412yes—
179C22×Dic5C22×C411yescharacteristic
180C22×Dic5C22×C411yescharacteristic
181C22×Dic5C22×C414yes—
182C22×Dic5C2412yes—
183C22×Dic5C2412yes—
184C22×Dic5C2411yescharacteristic
185#711—232no—
186C2×C4×D5—24no—
187C2×C4×D5—24no—
188C2×C4×D5—24no—
189C2×C4×D5C22×C414yes—
190C2×C4×D5C22×C414yes—
191C2×C4×D5C22×C412yes—
192C2×C4×D5C2412yes—
193C2×Dic10—24no—
194C2×Dic10C22×C414yes—
195#704—216no—
196C2×#170—24no—
197C2×#170—24no—
198C2×#170—24no—
199C2×#170—216no—
200C2×#170#4118yescomplemented
201C2×#170C2414yes—
202C2×#170C2414yes—
203C2×#170C2414yes—
204C2×#170C2414yes—
205#700—216no—
206C5×C4⋊C4C22×C414yes—
207C5×C22⋊C4C22×C414yes—
208C5×C22⋊C4C22×C414yes—
209C42×C5C22×C414yes—
210C22⋊Dic5C22×C414yes—
211C22⋊Dic5C22×C414yes—
212#685C22×C414yes—
213#685C22×C414yes—
214C4⋊Dic5C22×C414yes—
215#683C22×C414yes—
216#683C22×C414yes—
217#683C22×C414yes—
218#683C22×C414yes—
219C4×Dic5C22×C414yes—
220C4×Dic5C22×C414yes—
221C4×Dic5C22×C412yes—
222C4×Dic5C22×C412yes—
223C4×Dic5C22×C414yes—
224C4×Dic5C22×C414yes—
225C4×Dic5C22×C414yes—
226C4×Dic5C22×C412yes—
227C4×Dic5C22×C412yes—
228C4×Dic5C2412yes—
229C4×Dic5C2412yes—
230C4×Dic5C2412yes—
231C4×Dic5C2412yes—
232C22×#41—51no—
233C22×M16—52no—
234C22×M16—52no—
235C23×C8—52no—
236C4×#41—58no—
237C4×#41—58no—
238C2×C4×Q8—51no—
239C2×C4×Q8—51no—
240C2×C4×D4—51no—
241C2×C4×D4—51no—
242C2×C4×D4—51no—
243C2×C4×D4—51no—
244C2×C4×D4—51no—
245C2×C4×D4AGL(1,𝔽5)11yescharacteristic
246C2×#117—51no—
247C2×#117—51no—
248C2×#117—51no—
249C22×C42—51no—
250C22×C42—51no—
251C22×C42—51no—
252#446—516no—
253#436—516no—
254#435—516no—
255#435—516no—
256#431—58no—
257#431—58no—
258#431—58no—
259C2×C4⋊C8—52no—
260C2×C4⋊C8—52no—
261C2×C4⋊C8—52no—
262C2×C22⋊C8—52no—
263C2×C22⋊C8—52no—
264C2×C22⋊C8—52no—
265C4×M16—58no—
266C4×M16—58no—
267C2×#97—52no—
268C2×#97—52no—
269C2×C4×C8—51no—
270C2×C4×C8—51no—
271C2×C4×C8—52no—
272C2×C4×C8—58no—
273C2×C4×C8—51no—
274C2×C4×C8—51no—
275C23×C5—22no—
276C23×C5—22no—
277C23×C5—22no—
278C23×C5—22no—
279C23×C5—22no—
280C23×C5C2×M1614yes—
281C23×C5C23×C412yes—
282C23×C5C23×C412yes—
283C23×C5C23×C411yescharacteristic
284C22×D5—22no—
285C22×D5—22no—
286C22×D5—22no—
287C22×D5C23×C412yes—
288C5×D4—28no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—28no—
294C2×C4×C5—28no—
295C2×C4×C5#13114yes—
296C2×C4×C5C23×C412yes—
297C2×C4×C5C23×C412yes—
298C2×C4×C5C23×C412yes—
299C2×C4×C5C23×C412yes—
300C2×C4×C5C23×C412yes—
301C22⋊D5—28no—
302C22⋊D5—28no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—24no—
308C2×Dic5—24no—
309C2×Dic5—28no—
310C2×Dic5—24no—
311C2×Dic5—24no—
312C2×Dic5C2×M1614yes—
313C2×Dic5C2×M1614yes—
314C2×Dic5C2×M16116yescomplemented
315C2×Dic5C23×C412yes—
316C2×Dic5C23×C412yes—
317C2×Dic5C23×C412yes—
318C2×Dic5C23×C412yes—
319C2×Dic5C23×C411yescharacteristic
320C2×Dic5C23×C412yes—
321C2×Dic5C23×C412yes—
322C2×Dic5C2×#4114yes—
323C2×Dic5C2511yescharacteristic
324C4×D5—28no—
325C4×D5—24no—
326C4×D5—28no—
327C4×D5—24no—
328Dic10—28no—
329#170—24no—
330#170—24no—
331#170—24no—
332#170C2×#4118yescomplemented
333C2×#41—58no—
334C2×#41—108no—
335C22×Q8—51no—
336C22×D4—51no—
337C22×D4—51no—
338C22×D4C2×AGL(1,𝔽5)11yescharacteristic
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—51no—
342C23×C4—51no—
343C23×C4—51no—
344C23×C4—51no—
345C23×C4—51no—
346C23×C4C2×AGL(1,𝔽5)11yescharacteristic
347C23×C4C2×AGL(1,𝔽5)11yescharacteristic
348C2×M16—58no—
349C2×M16—58no—
350C2×M16—108no—
351C2×M16—108no—
352C22×C8—52no—
353C22×C8—52no—
354C22×C8—58no—
355C22×C8—52no—
356C22×C8—52no—
357C22×C8—108no—
358C4×Q8—54no—
359C4×Q8—54no—
360C4×Q8—58no—
361C4×D4—54no—
362C4×D4—54no—
363C4×D4—58no—
364C4×D4—58no—
365C4×D4—54no—
366C4×D4—54no—
367C4×D4—54no—
368C4×D4—54no—
369C4×D4C2×AGL(1,𝔽5)18yes—
370#117—58no—
371#117—58no—
372#117—58no—
373C2×C4⋊C4—51no—
374C2×C4⋊C4—51no—
375C2×C4⋊C4—51no—
376C2×C4⋊C4—51no—
377C2×C4⋊C4—51no—
378C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
379C2×C22⋊C4—51no—
380C2×C22⋊C4—51no—
381C2×C22⋊C4—51no—
382C2×C22⋊C4—51no—
383C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
384C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
385C2×C42—51no—
386C2×C42—51no—
387C2×C42—51no—
388C2×C42—51no—
389C2×C42—51no—
390C2×C42—51no—
391C2×C42—54no—
392C2×C42—54no—
393C2×C42—54no—
394C2×C42—54no—
395C2×C42—51no—
396C2×C42—51no—
397C2×C42—51no—
398C2×C42—54no—
399C2×C42—54no—
400C2×C42C2×AGL(1,𝔽5)11yescharacteristic
401C4⋊C8—58no—
402C4⋊C8—58no—
403C4⋊C8—58no—
404C22⋊C8—58no—
405C22⋊C8—58no—
406C22⋊C8—58no—
407#97—58no—
408#97—58no—
409C4×C8—54no—
410C4×C8—54no—
411C4×C8—58no—
412C4×C8—54no—
413C4×C8—54no—
414C22×C5—22no—
415C22×C5—28no—
416C22×C5—22no—
417C22×C5—22no—
418C22×C5—22no—
419C22×C5—22no—
420C22×C5—44no—
421C22×C5C4×#4112yes—
422C22×C5C22×M1614yes—
423C22×C5C22×M1612yes—
424C22×C5C2×#13112yes—
425C22×C5C24×C411yescharacteristic
426C2×D5—22no—
427C2×D5—22no—
428C2×D5—22no—
429C2×D5—44no—
430C4×C5—22no—
431C4×C5—22no—
432C4×C5—22no—
433C4×C5—22no—
434C4×C5C2×#13114yes—
435Dic5—22no—
436Dic5—22no—
437Dic5—22no—
438Dic5—22no—
439Dic5C4×#4112yes—
440Dic5C22×M1614yes—
441Dic5C22×M1614yes—
442Dic5C22×#4112yes—
443C24—51no—
444C24C22×AGL(1,𝔽5)11yescharacteristic
445C24C22×AGL(1,𝔽5)11yescharacteristic
446#41—1032no—
447C2×Q8—54no—
448C2×Q8—104no—
449C2×D4—24no—
450C2×D4—54no—
451C2×D4—54no—
452C2×D4—104no—
453C2×D4—104no—
454C2×D4C22×AGL(1,𝔽5)14yes—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—58no—
458C22×C4—51no—
459C22×C4—51no—
460C22×C4—51no—
461C22×C4—54no—
462C22×C4—52no—
463C22×C4—52no—
464C22×C4—51no—
465C22×C4—51no—
466C22×C4—51no—
467C22×C4—51no—
468C22×C4—54no—
469C22×C4—52no—
470C22×C4—52no—
471C22×C4—52no—
472C22×C4—54no—
473C22×C4—52no—
474C22×C4—54no—
475C22×C4—104no—
476C22×C4—104no—
477C22×C4—104no—
478C22×C4—104no—
479C22×C4—104no—
480C22×C4—104no—
481C22×C4C22×AGL(1,𝔽5)11yescharacteristic
482C22×C4C22×AGL(1,𝔽5)11yescharacteristic
483C22×C4C22×AGL(1,𝔽5)14yes—
484C22×C4C22×AGL(1,𝔽5)14yes—
485C22×C4C22×AGL(1,𝔽5)11yescharacteristic
486C22×C4C22×AGL(1,𝔽5)11yescharacteristic
487C22×C4C22×AGL(1,𝔽5)11yescharacteristic
488M16—1016no—
489M16—1016no—
490C2×C8—54no—
491C2×C8—58no—
492C2×C8—54no—
493C2×C8—54no—
494C2×C8—54no—
495C2×C8—1016no—
496C2×C8—104no—
497C2×C8—104no—
498C2×C8—104no—
499C4⋊C4—54no—
500C4⋊C4—54no—
501C4⋊C4—54no—
502C4⋊C4—54no—
503C4⋊C4—54no—
504C4⋊C4C22×AGL(1,𝔽5)14yes—
505C22⋊C4—54no—
506C22⋊C4—54no—
507C22⋊C4—54no—
508C22⋊C4—54no—
509C22⋊C4C22×AGL(1,𝔽5)14yes—
510C22⋊C4C22×AGL(1,𝔽5)14yes—
511C42—52no—
512C42—52no—
513C42—52no—
514C42—52no—
515C42—52no—
516C42—54no—
517C42—54no—
518C42—54no—
519C42—52no—
520C42—52no—
521C42—52no—
522C42—54no—
523C42—54no—
524C42C22×AGL(1,𝔽5)14yes—
525C2×C5—24no—
526C2×C5—42no—
527C2×C5#296514yescomplemented
528C2×C5C2×C4×#4111yescharacteristic
529C2×C5C23×M1611yescharacteristic
530C2×C5C22×#13111yescharacteristic
531D5—42no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—52no—
538C23—102no—
539C23—102no—
540C23—102no—
541C23D5⋊M1614yes—
542C23C23×AGL(1,𝔽5)12yes—
543C23C23×AGL(1,𝔽5)12yes—
544C23C23×AGL(1,𝔽5)11yescharacteristic
545Q8—108no—
546D4—28no—
547D4—108no—
548D4—108no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—54no—
556C2×C4—54no—
557C2×C4—516no—
558C2×C4—54no—
559C2×C4—51no—
560C2×C4—52no—
561C2×C4—51no—
562C2×C4—52no—
563C2×C4—52no—
564C2×C4—52no—
565C2×C4—52no—
566C2×C4—52no—
567C2×C4—104no—
568C2×C4—108no—
569C2×C4—104no—
570C2×C4—104no—
571C2×C4—104no—
572C2×C4—108no—
573C2×C4—104no—
574C2×C4—104no—
575C2×C4—102no—
576C2×C4—102no—
577C2×C4—102no—
578C2×C4—102no—
579C2×C4—108no—
580C2×C4#413114yes—
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—
586C8—58no—
587C8—104no—
588C8—104no—
589C8—104no—
590C5C2×#296511yescharacteristic, complemented, Sylow
591C22—22no—
592C22—22no—
593C22—28no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—44no—
598C22—102no—
599C22—102no—
600C22—102no—
601C22—204no—
602C22C2×D5⋊M1614yes—
603C22C2×D5⋊M1612yes—
604C22C2×#413112yes—
605C22#41×AGL(1,𝔽5)12yes—
606C22C24×AGL(1,𝔽5)11yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—54no—
612C4—54no—
613C4—52no—
614C4—52no—
615C4—102no—
616C4—102no—
617C4—102no—
618C4—102no—
619C4C2×#413114yes—
620C2—24no—
621C2—42no—
622C2—202no—
623C2#1061715414yesdirect product factor
624C2C22×D5⋊M1611yescharacteristic
625C2C22×#413111yescharacteristic
626C2C2×#41×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C4C22×AGL(1,𝔽5)
commutatorC22×C5C24×C4
FrattiniC22C24×AGL(1,𝔽5)
FittingC2×C4×C5×D4C4
socleC23×C5C23×C4
prime radicalC22×#711C4
cosocleC2×Dic5C25
hypercenterC2×C4×D4AGL(1,𝔽5)
hypocenterC5C2×#2965
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#65773C2semidirect
3. C2×#65754C2semidirect
4. C2×#65748C2semidirect
5. C2×C4×#704C2semidirect
6. C22×C4×#170C2semidirect
7. #10617154C2direct (N × H)
8. C2×#65707C2semidirect
9. C2×#65695C2semidirect
10. C2×#65693C2semidirect
11. C2×#65678C2semidirect
12. C2×C4×#700C2semidirect
35. #65773C22semidirect
36. #65754C22semidirect
40. C4×#704C22semidirect
43. #65748C22semidirect
50. C2×C4×#170C22semidirect
51. #65707C22semidirect
52. #65695C22semidirect
53. #65693C22semidirect
54. #65678C22semidirect
55. C4×#700C22semidirect
171. C22×Dic5M16semidirect
200. C2×#170#41semidirect
314. C2×Dic5C2×M16semidirect
332. #170C2×#41semidirect
527. C2×C5#2965semidirect
590. C5C2×#2965semidirect
623. C2#10617154direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#7112
C2×#657732
C2×#657542
C2×#657482
C2×C4×#7042
C22×C4×#1702
#106171542
C2×#657072
C2×#656952
C2×#656932
C2×#656782
C2×C4×#7002
C2×#29655