Finite Group Databasebeta

← C2×#10617144 #13044584

Subgroups — C2×#10617144 #13044584

Order 1280.

2178 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#711C211yesmaximal, characteristic
2C2×#65775C212yesmaximal
3C2×#65757C211yesmaximal, characteristic, complemented
4C2×#65754C211yesmaximal, characteristic, complemented
5C2×#65758C211yesmaximal, characteristic, complemented
6C22×#4004C211yesmaximal, characteristic, complemented
7#10617144C2116yesmaximal, direct product factor
8C2×#65706C212yesmaximal, complemented
9C2×#65697C212yesmaximal, complemented
10C2×#65691C212yesmaximal, complemented
11C2×#65684C211yesmaximal, characteristic, complemented
12C2×#65678C211yesmaximal, characteristic, complemented
13C22×#711C2211yescharacteristic
14C22×#704C2212yes—
15C23×#170C2212yes—
16C2×C4×C5×D4C411yescharacteristic
17C2×D4×Dic5C411yescharacteristic
18C2×C4×C22⋊D5C411yescharacteristic
19C2×C4×C22⋊D5C411yescharacteristic
20C22×C4×Dic5C2211yescharacteristic
21C22×C4×Dic5C2211yescharacteristic
22C4×#711C418yes—
23C4×#711C2218yes—
24C2×#4038C2211yescharacteristic
25C2×#4033C411yescharacteristic
26C2×#4027C411yescharacteristic
27C2×#4027C411yescharacteristic
28C2×#4023C2211yescharacteristic
29C2×#4023C2211yescharacteristic
30C2×C42×D5C2211yescharacteristic
31C2×C4×Dic10C411yescharacteristic
32C2×C22⋊#170C2212yes—
33C2×C22⋊#170C2212yes—
34#65775C22116yes—
35#65758C2218yescomplemented
36#65757C2218yescomplemented
37#65754C2218yescomplemented
38C2×#4004C2211yescharacteristic
39C2×#4004C2211yescharacteristic
40C2×#4004C2212yes—
41C2×#4004C2211yescharacteristic
42C2×#4004C2211yescharacteristic
43C2×#4004C2218yescomplemented
44C2×#4003C2212yes—
45C2×#4003C2212yes—
46C2×#4002C2212yes—
47C2×#4002C2212yes—
48C2×C4⋊#170C2212yes—
49C2×C4×#170C2212yes—
50C2×C4×#170C2212yes—
51#65706C22116yescomplemented
52#65697C22116yescomplemented
53#65691C22116yescomplemented
54#65684C2218yescomplemented
55#65678C2218yescomplemented
56C2×#2925—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—24no—
67C2×#711—24no—
68C2×#711C2×C414yes—
69C2×#711C2314yes—
70C22×C4×D5C2×C411yescharacteristic
71C22×C4×D5C2×C411yescharacteristic
72C22×C4×D5C2311yescharacteristic
73C22×Dic10C2×C411yescharacteristic
74C2×#704—28no—
75C2×#704C2318yes—
76C22×#170—28no—
77C22×#170C2312yes—
78C22×#170C2312yes—
79C22×#170C2312yes—
80C22×#170C2312yes—
81C22×#170C2318yes—
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×C418yes—
90C4×C22⋊D5C2×C418yes—
91C4×C22⋊D5C2×C418yes—
92C2×#685C2×C411yescharacteristic
93C2×#685C2×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×Dic5C2×C411yescharacteristic
102C2×C4×Dic5C2×C411yescharacteristic
103C2×C4×Dic5C2×C411yescharacteristic
104C2×C4×Dic5C2×C414yes—
105C2×C4×Dic5C2×C414yes—
106C2×C4×Dic5C2311yescharacteristic
107C2×C4×Dic5C2311yescharacteristic
108C2×C4×Dic5C2311yescharacteristic
109C2×C4×Dic5C2311yescharacteristic
110C2×C4×Dic5C2314yes—
111C2×C4×Dic5C2314yes—
112#4038C2×C414yes—
113#4038C2314yes—
114#4033C2×C418yes—
115#4027C2×C418yes—
116#4027C2×C418yes—
117#4023C2×C414yes—
118#4023C2×C414yes—
119#4023C2314yes—
120#4023C2314yes—
121C42×D5C2×C414yes—
122C42×D5C2314yes—
123C4×Dic10C2×C418yes—
124C22⋊#170C2318yes—
125C22⋊#170C2318yes—
126#4004C2314yes—
127#4004C2314yes—
128#4004C2314yes—
129#4004C2314yes—
130#4004C2318yes—
131#4003C2318yes—
132#4003C2318yes—
133#4002C2318yes—
134#4002C2318yes—
135C4⋊#170C2318yes—
136C4×#170C2318yes—
137C4×#170C2318yes—
138C2×C4×#41—51no—
139#2925—516no—
140C2×#433—52no—
141C2×#433—52no—
142C2×#432—51no—
143C2×#432—51no—
144C2×#431—51no—
145C2×#431—51no—
146C2×#423—51no—
147C22×C4⋊C8—51no—
148C2×#408—52no—
149C2×#405—52no—
150C24×C5C22×C411yescharacteristic
151C24×C5C22×C411yescharacteristic
152C23×D5C22×C411yescharacteristic
153C2×C5×D4—24no—
154C2×C5×D4C22×C414yes—
155C22×C4×C5—42no—
156C22×C4×C5—42no—
157C22×C4×C5C22×C411yescharacteristic
158C22×C4×C5C22×C411yescharacteristic
159C22×C4×C5C22×C411yescharacteristic
160C22×C4×C5C22×C411yescharacteristic
161C22×C4×C5C22×C411yescharacteristic
162C22×C4×C5C22×C414yes—
163C22×C4×C5C22×C414yes—
164C2×C22⋊D5—24no—
165C2×C22⋊D5—24no—
166C2×C22⋊D5C22×C414yes—
167C2×C22⋊D5C22×C414yes—
168C22×Dic5—24no—
169C22×Dic5—24no—
170C22×Dic5—22no—
171C22×Dic5—22no—
172C22×Dic5C22×C411yescharacteristic
173C22×Dic5C22×C411yescharacteristic
174C22×Dic5C22×C411yescharacteristic
175C22×Dic5C22×C411yescharacteristic
176C22×Dic5C22×C412yes—
177C22×Dic5C22×C414yes—
178C22×Dic5C22×C414yes—
179C22×Dic5C22×C411yescharacteristic
180C22×Dic5C22×C412yes—
181C22×Dic5C22×C411yescharacteristic
182C22×Dic5#4114yes—
183C22×Dic5#4114yes—
184C22×Dic5C2412yes—
185C22×Dic5C2411yescharacteristic
186C22×Dic5C2412yes—
187#711—216no—
188#711—28no—
189#711—28no—
190C2×C4×D5—24no—
191C2×C4×D5—22no—
192C2×C4×D5—22no—
193C2×C4×D5—42no—
194C2×C4×D5C22×C414yes—
195C2×C4×D5C22×C412yes—
196C2×C4×D5C22×C414yes—
197C2×C4×D5C2412yes—
198C2×Dic10—24no—
199C2×Dic10C22×C414yes—
200#704—216no—
201C2×#170—24no—
202C2×#170—24no—
203C2×#170—216no—
204C2×#170—24no—
205C2×#170—24no—
206C2×#170C2414yes—
207C2×#170C2414yes—
208C2×#170C2414yes—
209C2×#170C2414yes—
210C5×C4⋊C4C22×C414yes—
211C5×C22⋊C4C22×C414yes—
212C5×C22⋊C4C22×C414yes—
213C42×C5C22×C414yes—
214C22⋊Dic5C22×C414yes—
215C22⋊Dic5C22×C414yes—
216#685C22×C414yes—
217#685C22×C414yes—
218C4⋊Dic5C22×C414yes—
219#683C22×C414yes—
220#683C22×C414yes—
221#683C22×C414yes—
222#683C22×C414yes—
223C4×Dic5C22×C414yes—
224C4×Dic5C22×C414yes—
225C4×Dic5C22×C412yes—
226C4×Dic5C22×C412yes—
227C4×Dic5C22×C414yes—
228C4×Dic5C22×C412yes—
229C4×Dic5C22×C412yes—
230C4×Dic5C22×C414yes—
231C4×Dic5C22×C414yes—
232C4×Dic5C2412yes—
233C4×Dic5C2412yes—
234C4×Dic5C2412yes—
235C4×Dic5C2412yes—
236C22×#41—51no—
237C22×M16—52no—
238C23×C8—52no—
239C4×#41—58no—
240C4×#41—58no—
241C2×C4×Q8—51no—
242C2×C4×Q8—51no—
243C2×C4×D4—51no—
244C2×C4×D4—51no—
245C2×C4×D4—51no—
246C2×C4×D4—51no—
247C2×C4×D4—51no—
248C2×C4×D4AGL(1,𝔽5)11yescharacteristic
249C2×#117—51no—
250C2×#117—51no—
251C2×#117—51no—
252C22×C42—51no—
253C22×C42—51no—
254C22×C42—51no—
255#433—516no—
256#433—516no—
257#432—58no—
258#432—58no—
259#431—58no—
260#431—58no—
261#423—58no—
262C2×C4⋊C8—51no—
263C2×C4⋊C8—51no—
264C2×C4⋊C8—52no—
265C2×C4⋊C8—58no—
266C2×C4⋊C8—51no—
267C2×C4⋊C8—51no—
268C2×C4⋊C8—52no—
269#408—516no—
270C2×C22⋊C8—52no—
271C2×C22⋊C8—52no—
272C2×C22⋊C8—52no—
273C2×C22⋊C8—52no—
274#405—516no—
275C2×#97—52no—
276C2×#97—52no—
277C2×C4×C8—52no—
278C2×C4×C8—52no—
279C23×C5—22no—
280C23×C5—22no—
281C23×C5—22no—
282C23×C5—22no—
283C23×C5—22no—
284C23×C5#11714yes—
285C23×C5C23×C412yes—
286C23×C5C23×C412yes—
287C23×C5C23×C411yescharacteristic
288C22×D5—22no—
289C22×D5—22no—
290C22×D5—22no—
291C22×D5C23×C412yes—
292C5×D4—28no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—22no—
296C2×C4×C5—22no—
297C2×C4×C5—22no—
298C2×C4×C5—44no—
299C2×C4×C5—44no—
300C2×C4×C5C23×C412yes—
301C2×C4×C5C23×C412yes—
302C2×C4×C5C23×C412yes—
303C2×C4×C5C23×C412yes—
304C2×C4×C5C23×C412yes—
305C22⋊D5—28no—
306C22⋊D5—28no—
307C2×Dic5—22no—
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—24no—
312C2×Dic5—28no—
313C2×Dic5—28no—
314C2×Dic5—22no—
315C2×Dic5—22no—
316C2×Dic5#11714yes—
317C2×Dic5#11718yes—
318C2×Dic5#13114yes—
319C2×Dic5C23×C412yes—
320C2×Dic5C23×C412yes—
321C2×Dic5C23×C412yes—
322C2×Dic5C23×C412yes—
323C2×Dic5C23×C411yescharacteristic
324C2×Dic5C23×C412yes—
325C2×Dic5C23×C412yes—
326C2×Dic5C2×#4112yes—
327C2×Dic5C2×#4112yes—
328C2×Dic5C2×#4114yes—
329C2×Dic5C2×#4114yes—
330C2×Dic5C2511yescharacteristic
331C4×D5—24no—
332C4×D5—28no—
333C4×D5—22no—
334C4×D5—22no—
335C4×D5—44no—
336Dic10—28no—
337#170—24no—
338#170—24no—
339#170—24no—
340#170—24no—
341C2×#41—54no—
342C2×#41—54no—
343C2×#41—104no—
344C2×#41—104no—
345C22×Q8—51no—
346C22×D4—51no—
347C22×D4—51no—
348C22×D4C2×AGL(1,𝔽5)11yescharacteristic
349C23×C4—51no—
350C23×C4—51no—
351C23×C4—51no—
352C23×C4—51no—
353C23×C4—51no—
354C23×C4—51no—
355C23×C4—51no—
356C23×C4C2×AGL(1,𝔽5)11yescharacteristic
357C23×C4C2×AGL(1,𝔽5)11yescharacteristic
358C2×M16—58no—
359C2×M16—108no—
360C22×C8—52no—
361C22×C8—52no—
362C22×C8—58no—
363C22×C8—52no—
364C22×C8—52no—
365C22×C8—108no—
366C4×Q8—58no—
367C4×Q8—58no—
368C4×D4—58no—
369C4×D4—58no—
370C4×D4—58no—
371C4×D4—58no—
372C4×D4—58no—
373C4×D4C2×AGL(1,𝔽5)18yes—
374#117—54no—
375#117—54no—
376#117—54no—
377#117—54no—
378#117—54no—
379#117—54no—
380C2×C4⋊C4—51no—
381C2×C4⋊C4—51no—
382C2×C4⋊C4—51no—
383C2×C4⋊C4—51no—
384C2×C4⋊C4—51no—
385C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
386C2×C22⋊C4—51no—
387C2×C22⋊C4—51no—
388C2×C22⋊C4—51no—
389C2×C22⋊C4—51no—
390C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
391C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
392C2×C42—51no—
393C2×C42—51no—
394C2×C42—51no—
395C2×C42—51no—
396C2×C42—51no—
397C2×C42—51no—
398C2×C42—51no—
399C2×C42—51no—
400C2×C42—54no—
401C2×C42—54no—
402C2×C42—54no—
403C2×C42—54no—
404C2×C42—51no—
405C2×C42—54no—
406C2×C42—54no—
407C2×C42C2×AGL(1,𝔽5)11yescharacteristic
408C4⋊C8—58no—
409C4⋊C8—54no—
410C4⋊C8—54no—
411C4⋊C8—58no—
412C4⋊C8—54no—
413C4⋊C8—54no—
414C22⋊C8—58no—
415C22⋊C8—58no—
416C22⋊C8—58no—
417C22⋊C8—58no—
418#97—58no—
419#97—58no—
420C4×C8—58no—
421C4×C8—58no—
422C22×C5—28no—
423C22×C5—22no—
424C22×C5—22no—
425C22×C5—22no—
426C22×C5—22no—
427C22×C5—22no—
428C22×C5—44no—
429C22×C5C2×#11714yes—
430C22×C5C2×#11712yes—
431C22×C5C2×#13112yes—
432C22×C5#56812yes—
433C22×C5C24×C411yescharacteristic
434C2×D5—22no—
435C2×D5—22no—
436C2×D5—22no—
437C2×D5—44no—
438C4×C5—22no—
439C4×C5—22no—
440C4×C5—22no—
441C4×C5—22no—
442C4×C5—22no—
443Dic5—22no—
444Dic5—22no—
445Dic5—22no—
446Dic5—22no—
447Dic5C2×#11714yes—
448Dic5C2×#11712yes—
449Dic5C2×#13114yes—
450Dic5C22×#4111yescharacteristic
451Dic5C22×#4111yescharacteristic
452C24—51no—
453C24C22×AGL(1,𝔽5)11yescharacteristic
454C24C22×AGL(1,𝔽5)11yescharacteristic
455#41—108no—
456#41—108no—
457#41—1016no—
458C2×Q8—54no—
459C2×Q8—104no—
460C2×D4—24no—
461C2×D4—54no—
462C2×D4—54no—
463C2×D4—104no—
464C2×D4—104no—
465C2×D4C22×AGL(1,𝔽5)14yes—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—54no—
469C22×C4—54no—
470C22×C4—52no—
471C22×C4—52no—
472C22×C4—51no—
473C22×C4—51no—
474C22×C4—51no—
475C22×C4—52no—
476C22×C4—52no—
477C22×C4—54no—
478C22×C4—54no—
479C22×C4—51no—
480C22×C4—51no—
481C22×C4—51no—
482C22×C4—51no—
483C22×C4—52no—
484C22×C4—52no—
485C22×C4—54no—
486C22×C4—54no—
487C22×C4—102no—
488C22×C4—102no—
489C22×C4—104no—
490C22×C4—102no—
491C22×C4—102no—
492C22×C4—104no—
493C22×C4—104no—
494C22×C4—202no—
495C22×C4C22×AGL(1,𝔽5)11yescharacteristic
496C22×C4C22×AGL(1,𝔽5)11yescharacteristic
497C22×C4C22×AGL(1,𝔽5)11yescharacteristic
498C22×C4C22×AGL(1,𝔽5)11yescharacteristic
499C22×C4C22×AGL(1,𝔽5)11yescharacteristic
500C22×C4C22×AGL(1,𝔽5)14yes—
501C22×C4C22×AGL(1,𝔽5)14yes—
502M16—1016no—
503C2×C8—54no—
504C2×C8—54no—
505C2×C8—54no—
506C2×C8—54no—
507C2×C8—104no—
508C2×C8—104no—
509C2×C8—1016no—
510C2×C8—104no—
511C2×C8—104no—
512C4⋊C4—54no—
513C4⋊C4—54no—
514C4⋊C4—54no—
515C4⋊C4—54no—
516C4⋊C4—54no—
517C4⋊C4C22×AGL(1,𝔽5)14yes—
518C22⋊C4—54no—
519C22⋊C4—54no—
520C22⋊C4—54no—
521C22⋊C4—54no—
522C22⋊C4C22×AGL(1,𝔽5)14yes—
523C22⋊C4C22×AGL(1,𝔽5)14yes—
524C42—52no—
525C42—52no—
526C42—52no—
527C42—52no—
528C42—52no—
529C42—52no—
530C42—52no—
531C42—52no—
532C42—54no—
533C42—54no—
534C42—54no—
535C42—54no—
536C42—54no—
537C42C22×AGL(1,𝔽5)14yes—
538C2×C5—24no—
539C2×C5—42no—
540C2×C5#292514yescomplemented
541C2×C5C22×#11711yescharacteristic
542C2×C5C22×#13111yescharacteristic
543C2×C5C2×#56811yescharacteristic
544D5—42no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—52no—
551C23—102no—
552C23—102no—
553C23—102no—
554C23#413014yes—
555C23C23×AGL(1,𝔽5)12yes—
556C23C23×AGL(1,𝔽5)12yes—
557C23C23×AGL(1,𝔽5)11yescharacteristic
558Q8—108no—
559D4—28no—
560D4—108no—
561D4—108no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—52no—
570C2×C4—52no—
571C2×C4—54no—
572C2×C4—54no—
573C2×C4—54no—
574C2×C4—58no—
575C2×C4—54no—
576C2×C4—51no—
577C2×C4—51no—
578C2×C4—52no—
579C2×C4—52no—
580C2×C4—52no—
581C2×C4—52no—
582C2×C4—52no—
583C2×C4—52no—
584C2×C4—102no—
585C2×C4—102no—
586C2×C4—104no—
587C2×C4—108no—
588C2×C4—102no—
589C2×C4—102no—
590C2×C4—104no—
591C2×C4—108no—
592C2×C4—108no—
593C2×C4—102no—
594C2×C4—102no—
595C2×C4—102no—
596C2×C4—102no—
597C2×C4—204no—
598C2×C4C23×AGL(1,𝔽5)12yes—
599C2×C4C23×AGL(1,𝔽5)12yes—
600C2×C4C23×AGL(1,𝔽5)12yes—
601C2×C4C23×AGL(1,𝔽5)12yes—
602C2×C4C23×AGL(1,𝔽5)12yes—
603C8—104no—
604C8—104no—
605C8—104no—
606C8—104no—
607C5C2×#292511yescharacteristic, complemented, Sylow
608C22—22no—
609C22—22no—
610C22—28no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—44no—
615C22—102no—
616C22—102no—
617C22—102no—
618C22—204no—
619C22C2×#413014yes—
620C22C2×#413012yes—
621C22#6625612yes—
622C22D5⋊#13112yes—
623C22C24×AGL(1,𝔽5)11yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—51no—
630C4—51no—
631C4—52no—
632C4—54no—
633C4—54no—
634C4—102no—
635C4—102no—
636C4—102no—
637C4—102no—
638C2—24no—
639C2—42no—
640C2—202no—
641C2#1061714414yesdirect product factor
642C2C22×#413011yescharacteristic
643C2C2×#6625611yescharacteristic
644C2C2×D5⋊#13111yescharacteristic
subgroupG/H
centerC23C23×AGL(1,𝔽5)
commutatorC22×C5C24×C4
FrattiniC22C24×AGL(1,𝔽5)
FittingC2×C4×C5×D4C4
socleC23×C5C23×C4
prime radicalC22×#711C22
cosocleC2×Dic5C25
hypercenterC2×C4×D4AGL(1,𝔽5)
hypocenterC5C2×#2925
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
3. C2×#65757C2semidirect
4. C2×#65754C2semidirect
5. C2×#65758C2semidirect
6. C22×#4004C2semidirect
7. #10617144C2direct (N × H)
8. C2×#65706C2semidirect
9. C2×#65697C2semidirect
10. C2×#65691C2semidirect
11. C2×#65684C2semidirect
12. C2×#65678C2semidirect
35. #65758C22semidirect
36. #65757C22semidirect
37. #65754C22semidirect
43. C2×#4004C22semidirect
51. #65706C22semidirect
52. #65697C22semidirect
53. #65691C22semidirect
54. #65684C22semidirect
55. #65678C22semidirect
540. C2×C5#2925semidirect
607. C5C2×#2925semidirect
641. C2#10617144direct (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×#657752
C2×#657572
C2×#657542
C2×#657582
C22×#40042
#106171442
C2×#657062
C2×#656972
C2×#656912
C2×#656842
C2×#656782
C2×#29255