Finite Group Databasebeta

← C2×#10617141 #13044569

Subgroups — C2×#10617141 #13044569

Order 1280.

2134 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#711C211yesmaximal, characteristic
2C2×#65775C212yesmaximal
3C2×#65759C211yesmaximal, characteristic, complemented
4C2×#65757C211yesmaximal, characteristic, complemented
5C2×C4×#704C211yesmaximal, characteristic, complemented
6C22×#4002C211yesmaximal, characteristic, complemented
7#10617141C2116yesmaximal, direct product factor
8C2×#65706C212yesmaximal, complemented
9C2×#65697C212yesmaximal, complemented
10C2×#65696C212yesmaximal, complemented
11C2×#65684C211yesmaximal, characteristic, complemented
12C2×#65683C211yesmaximal, 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#65759C2218yescomplemented
36#65757C2218yescomplemented
37C2×#4004C2212yes—
38C2×#4004C2212yes—
39C2×#4003C2212yes—
40C2×#4003C2212yes—
41C4×#704C2218yescomplemented
42C2×#4002C2211yescharacteristic
43C2×#4002C2211yescharacteristic
44C2×#4002C2212yes—
45C2×#4002C2211yescharacteristic
46C2×#4002C2211yescharacteristic
47C2×#4002C2218yescomplemented
48C2×C4⋊#170C2212yes—
49C2×C4⋊#170C2212yes—
50C2×C4×#170C2212yes—
51#65706C22116yescomplemented
52#65697C22116yescomplemented
53#65696C22116yescomplemented
54#65684C2218yescomplemented
55#65683C2218yescomplemented
56C2×#2923—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×C414yes—
68C2×#711C2314yes—
69C22×C4×D5C2×C411yescharacteristic
70C22×C4×D5C2×C411yescharacteristic
71C22×C4×D5C2311yescharacteristic
72C22×Dic10C2×C411yescharacteristic
73C2×#704—28no—
74C2×#704C2318yes—
75C22×#170—28no—
76C22×#170C2312yes—
77C22×#170C2312yes—
78C22×#170C2312yes—
79C22×#170C2312yes—
80C22×#170C2318yes—
81C4×C5×D4C2×C418yes—
82C2×C5×C4⋊C4C2×C411yescharacteristic
83C2×C5×C22⋊C4C2×C411yescharacteristic
84C2×C5×C22⋊C4C2×C411yescharacteristic
85C2×C42×C5C2×C411yescharacteristic
86C2×C22⋊Dic5C2×C411yescharacteristic
87C2×C22⋊Dic5C2×C411yescharacteristic
88D4×Dic5C2×C418yes—
89C4×C22⋊D5C2×C418yes—
90C4×C22⋊D5C2×C418yes—
91C2×#685C2×C411yescharacteristic
92C2×#685C2×C411yescharacteristic
93C2×C4⋊Dic5C2×C411yescharacteristic
94C2×#683C2×C411yescharacteristic
95C2×#683C2×C411yescharacteristic
96C2×#683C2×C411yescharacteristic
97C2×#683C2×C411yescharacteristic
98C2×C4×Dic5C2×C411yescharacteristic
99C2×C4×Dic5C2×C411yescharacteristic
100C2×C4×Dic5C2×C411yescharacteristic
101C2×C4×Dic5C2×C411yescharacteristic
102C2×C4×Dic5C2×C411yescharacteristic
103C2×C4×Dic5C2×C414yes—
104C2×C4×Dic5C2×C414yes—
105C2×C4×Dic5C2311yescharacteristic
106C2×C4×Dic5C2311yescharacteristic
107C2×C4×Dic5C2311yescharacteristic
108C2×C4×Dic5C2311yescharacteristic
109C2×C4×Dic5C2314yes—
110C2×C4×Dic5C2314yes—
111#4038C2×C414yes—
112#4038C2314yes—
113#4033C2×C418yes—
114#4027C2×C418yes—
115#4027C2×C418yes—
116#4023C2×C414yes—
117#4023C2×C414yes—
118#4023C2314yes—
119#4023C2314yes—
120C42×D5C2×C414yes—
121C42×D5C2314yes—
122C4×Dic10C2×C418yes—
123C22⋊#170C2318yes—
124C22⋊#170C2318yes—
125#4004C2318yes—
126#4004C2318yes—
127#4003C2318yes—
128#4003C2318yes—
129#4002C2314yes—
130#4002C2314yes—
131#4002C2314yes—
132#4002C2314yes—
133#4002C2318yes—
134C4⋊#170C2318yes—
135C4⋊#170C2318yes—
136C4×#170C2318yes—
137C2×C4×#41—51no—
138#2923—516no—
139C2×#433—52no—
140C2×#433—52no—
141C2×#432—51no—
142C2×#432—51no—
143C2×#432—51no—
144C2×#432—51no—
145C2×#424—52no—
146C2×#408—52no—
147C2×C4×M16—51no—
148C22×#97—51no—
149C24×C5C22×C411yescharacteristic
150C24×C5C22×C411yescharacteristic
151C23×D5C22×C411yescharacteristic
152C2×C5×D4—24no—
153C2×C5×D4C22×C414yes—
154C22×C4×C5—42no—
155C22×C4×C5—42no—
156C22×C4×C5C22×C411yescharacteristic
157C22×C4×C5C22×C411yescharacteristic
158C22×C4×C5C22×C411yescharacteristic
159C22×C4×C5C22×C411yescharacteristic
160C22×C4×C5C22×C411yescharacteristic
161C22×C4×C5C22×C414yes—
162C22×C4×C5C22×C414yes—
163C2×C22⋊D5—24no—
164C2×C22⋊D5—24no—
165C2×C22⋊D5C22×C414yes—
166C2×C22⋊D5C22×C414yes—
167C22×Dic5—24no—
168C22×Dic5—42no—
169C22×Dic5—42no—
170C22×Dic5C22×C411yescharacteristic
171C22×Dic5C22×C411yescharacteristic
172C22×Dic5C22×C411yescharacteristic
173C22×Dic5C22×C411yescharacteristic
174C22×Dic5C22×C411yescharacteristic
175C22×Dic5C22×C412yes—
176C22×Dic5C22×C414yes—
177C22×Dic5C22×C414yes—
178C22×Dic5C22×C411yescharacteristic
179C22×Dic5C22×C412yes—
180C22×Dic5#4118yes—
181C22×Dic5C2412yes—
182C22×Dic5C2411yescharacteristic
183C22×Dic5C2412yes—
184#711—216no—
185#711—216no—
186C2×C4×D5—24no—
187C2×C4×D5—42no—
188C2×C4×D5—42no—
189C2×C4×D5C22×C412yes—
190C2×C4×D5C22×C414yes—
191C2×C4×D5C22×C414yes—
192C2×C4×D5C2412yes—
193C2×Dic10—24no—
194C2×Dic10C22×C414yes—
195#704—216no—
196C2×#170—24no—
197C2×#170—24no—
198C2×#170—216no—
199C2×#170—24no—
200C2×#170—24no—
201C2×#170C2414yes—
202C2×#170C2414yes—
203C2×#170C2414yes—
204C2×#170C2414yes—
205C5×C4⋊C4C22×C414yes—
206C5×C22⋊C4C22×C414yes—
207C5×C22⋊C4C22×C414yes—
208C42×C5C22×C414yes—
209C22⋊Dic5C22×C414yes—
210C22⋊Dic5C22×C414yes—
211#685C22×C414yes—
212#685C22×C414yes—
213C4⋊Dic5C22×C414yes—
214#683C22×C414yes—
215#683C22×C414yes—
216#683C22×C414yes—
217#683C22×C414yes—
218C4×Dic5C22×C414yes—
219C4×Dic5C22×C414yes—
220C4×Dic5C22×C412yes—
221C4×Dic5C22×C412yes—
222C4×Dic5C22×C414yes—
223C4×Dic5C22×C412yes—
224C4×Dic5C22×C412yes—
225C4×Dic5C22×C414yes—
226C4×Dic5C22×C414yes—
227C4×Dic5C2412yes—
228C4×Dic5C2412yes—
229C4×Dic5C2412yes—
230C4×Dic5C2412yes—
231C22×#41—51no—
232C22×M16—52no—
233C23×C8—52no—
234C4×#41—58no—
235C4×#41—58no—
236C2×C4×Q8—51no—
237C2×C4×Q8—51no—
238C2×C4×D4—51no—
239C2×C4×D4—51no—
240C2×C4×D4—51no—
241C2×C4×D4—51no—
242C2×C4×D4—51no—
243C2×C4×D4AGL(1,𝔽5)11yescharacteristic
244C2×#117—51no—
245C2×#117—51no—
246C2×#117—51no—
247C22×C42—51no—
248C22×C42—51no—
249C22×C42—51no—
250#433—516no—
251#433—516no—
252#432—58no—
253#432—58no—
254#432—58no—
255#432—58no—
256#424—516no—
257C2×C4⋊C8—52no—
258C2×C4⋊C8—52no—
259C2×C4⋊C8—52no—
260C2×C4⋊C8—52no—
261#408—516no—
262C2×C22⋊C8—52no—
263C2×C22⋊C8—52no—
264C2×C22⋊C8—52no—
265C2×C22⋊C8—52no—
266C4×M16—58no—
267C2×#97—51no—
268C2×#97—51no—
269C2×#97—52no—
270C2×#97—58no—
271C2×#97—51no—
272C2×#97—51no—
273C2×C4×C8—52no—
274C23×C5—22no—
275C23×C5—22no—
276C23×C5—22no—
277C23×C5—22no—
278C23×C5—22no—
279C23×C5#11714yes—
280C23×C5C23×C412yes—
281C23×C5C23×C412yes—
282C23×C5C23×C411yescharacteristic
283C22×D5—22no—
284C22×D5—22no—
285C22×D5—22no—
286C22×D5C23×C412yes—
287C5×D4—28no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—44no—
294C2×C4×C5—44no—
295C2×C4×C5C23×C412yes—
296C2×C4×C5C23×C412yes—
297C2×C4×C5C23×C412yes—
298C2×C4×C5C23×C412yes—
299C2×C4×C5C23×C412yes—
300C22⋊D5—28no—
301C22⋊D5—28no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—24no—
308C2×Dic5—24no—
309C2×Dic5—44no—
310C2×Dic5—44no—
311C2×Dic5#11714yes—
312C2×Dic5#11718yes—
313C2×Dic5C23×C412yes—
314C2×Dic5C23×C412yes—
315C2×Dic5C23×C412yes—
316C2×Dic5C23×C412yes—
317C2×Dic5C23×C412yes—
318C2×Dic5C23×C411yescharacteristic
319C2×Dic5C23×C412yes—
320C2×Dic5C2×#4114yes—
321C2×Dic5C2×#4118yes—
322C2×Dic5C2511yescharacteristic
323C4×D5—24no—
324C4×D5—24no—
325C4×D5—44no—
326C4×D5—44no—
327Dic10—28no—
328#170—24no—
329#170—24no—
330#170—24no—
331#170—24no—
332C2×#41—54no—
333C2×#41—54no—
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—108no—
350C22×C8—52no—
351C22×C8—52no—
352C22×C8—58no—
353C22×C8—52no—
354C22×C8—52no—
355C22×C8—108no—
356C4×Q8—58no—
357C4×Q8—58no—
358C4×D4—58no—
359C4×D4—58no—
360C4×D4—58no—
361C4×D4—58no—
362C4×D4—58no—
363C4×D4C2×AGL(1,𝔽5)18yes—
364#117—54no—
365#117—54no—
366#117—54no—
367#117—54no—
368#117—54no—
369#117—54no—
370C2×C4⋊C4—51no—
371C2×C4⋊C4—51no—
372C2×C4⋊C4—51no—
373C2×C4⋊C4—51no—
374C2×C4⋊C4—51no—
375C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
376C2×C22⋊C4—51no—
377C2×C22⋊C4—51no—
378C2×C22⋊C4—51no—
379C2×C22⋊C4—51no—
380C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
381C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
382C2×C42—51no—
383C2×C42—51no—
384C2×C42—51no—
385C2×C42—51no—
386C2×C42—51no—
387C2×C42—51no—
388C2×C42—51no—
389C2×C42—51no—
390C2×C42—54no—
391C2×C42—54no—
392C2×C42—54no—
393C2×C42—54no—
394C2×C42—51no—
395C2×C42—54no—
396C2×C42—54no—
397C2×C42C2×AGL(1,𝔽5)11yescharacteristic
398C4⋊C8—58no—
399C4⋊C8—58no—
400C4⋊C8—58no—
401C4⋊C8—58no—
402C22⋊C8—58no—
403C22⋊C8—58no—
404C22⋊C8—58no—
405C22⋊C8—58no—
406#97—54no—
407#97—54no—
408#97—58no—
409#97—54no—
410#97—54no—
411C4×C8—58no—
412C22×C5—28no—
413C22×C5—22no—
414C22×C5—22no—
415C22×C5—22no—
416C22×C5—22no—
417C22×C5—22no—
418C22×C5—44no—
419C22×C5C2×#11714yes—
420C22×C5C2×#11712yes—
421C22×C5#56812yes—
422C22×C5#56812yes—
423C22×C5C24×C411yescharacteristic
424C2×D5—22no—
425C2×D5—22no—
426C2×D5—22no—
427C2×D5—44no—
428C4×C5—22no—
429C4×C5—22no—
430C4×C5—22no—
431C4×C5—22no—
432C4×C5—22no—
433Dic5—22no—
434Dic5—22no—
435Dic5—22no—
436Dic5—22no—
437Dic5—22no—
438Dic5C2×#11714yes—
439Dic5C2×#11712yes—
440Dic5C22×#4112yes—
441C24—51no—
442C24C22×AGL(1,𝔽5)11yescharacteristic
443C24C22×AGL(1,𝔽5)11yescharacteristic
444#41—1016no—
445#41—1016no—
446C2×Q8—54no—
447C2×Q8—104no—
448C2×D4—24no—
449C2×D4—54no—
450C2×D4—54no—
451C2×D4—104no—
452C2×D4—104no—
453C2×D4C22×AGL(1,𝔽5)14yes—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—58no—
457C22×C4—52no—
458C22×C4—52no—
459C22×C4—51no—
460C22×C4—51no—
461C22×C4—52no—
462C22×C4—52no—
463C22×C4—51no—
464C22×C4—51no—
465C22×C4—51no—
466C22×C4—51no—
467C22×C4—51no—
468C22×C4—54no—
469C22×C4—54no—
470C22×C4—52no—
471C22×C4—52no—
472C22×C4—54no—
473C22×C4—54no—
474C22×C4—104no—
475C22×C4—104no—
476C22×C4—202no—
477C22×C4—202no—
478C22×C4—202no—
479C22×C4—202no—
480C22×C4C22×AGL(1,𝔽5)11yescharacteristic
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)14yes—
486C22×C4C22×AGL(1,𝔽5)14yes—
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⋊C4C22×AGL(1,𝔽5)14yes—
503C22⋊C4—54no—
504C22⋊C4—54no—
505C22⋊C4—54no—
506C22⋊C4—54no—
507C22⋊C4C22×AGL(1,𝔽5)14yes—
508C22⋊C4C22×AGL(1,𝔽5)14yes—
509C42—52no—
510C42—52no—
511C42—52no—
512C42—52no—
513C42—52no—
514C42—52no—
515C42—52no—
516C42—52no—
517C42—54no—
518C42—54no—
519C42—54no—
520C42—54no—
521C42—54no—
522C42C22×AGL(1,𝔽5)14yes—
523C2×C5—24no—
524C2×C5—42no—
525C2×C5#292314yescomplemented
526C2×C5C22×#11711yescharacteristic
527C2×C5C2×#56811yescharacteristic
528C2×C5C2×#56811yescharacteristic
529D5—42no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—52no—
536C23—102no—
537C23—102no—
538C23—102no—
539C23#413014yes—
540C23C23×AGL(1,𝔽5)12yes—
541C23C23×AGL(1,𝔽5)12yes—
542C23C23×AGL(1,𝔽5)11yescharacteristic
543Q8—108no—
544D4—28no—
545D4—108no—
546D4—108no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—54no—
555C2×C4—58no—
556C2×C4—54no—
557C2×C4—58no—
558C2×C4—51no—
559C2×C4—51no—
560C2×C4—52no—
561C2×C4—52no—
562C2×C4—52no—
563C2×C4—52no—
564C2×C4—52no—
565C2×C4—52no—
566C2×C4—104no—
567C2×C4—104no—
568C2×C4—104no—
569C2×C4—104no—
570C2×C4—102no—
571C2×C4—102no—
572C2×C4—102no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—204no—
576C2×C4—204no—
577C2×C4—204no—
578C2×C4—204no—
579C2×C4C23×AGL(1,𝔽5)12yes—
580C2×C4C23×AGL(1,𝔽5)12yes—
581C2×C4C23×AGL(1,𝔽5)12yes—
582C2×C4C23×AGL(1,𝔽5)12yes—
583C2×C4C23×AGL(1,𝔽5)12yes—
584C8—104no—
585C8—104no—
586C8—104no—
587C8—104no—
588C5C2×#292311yescharacteristic, complemented, Sylow
589C22—22no—
590C22—22no—
591C22—28no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—44no—
596C22—102no—
597C22—102no—
598C22—102no—
599C22—204no—
600C22C2×#413014yes—
601C22C2×#413012yes—
602C22#6625612yes—
603C22#6626412yes—
604C22C24×AGL(1,𝔽5)11yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—52no—
611C4—52no—
612C4—54no—
613C4—102no—
614C4—102no—
615C4—102no—
616C4—102no—
617C4—102no—
618C2—24no—
619C2—42no—
620C2—202no—
621C2#1061714114yesdirect product factor
622C2C22×#413011yescharacteristic
623C2C2×#6625611yescharacteristic
624C2C2×#6626411yescharacteristic
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×#2923
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×#65759C2semidirect
4. C2×#65757C2semidirect
5. C2×C4×#704C2semidirect
6. C22×#4002C2semidirect
7. #10617141C2direct (N × H)
8. C2×#65706C2semidirect
9. C2×#65697C2semidirect
10. C2×#65696C2semidirect
11. C2×#65684C2semidirect
12. C2×#65683C2semidirect
35. #65759C22semidirect
36. #65757C22semidirect
41. C4×#704C22semidirect
47. C2×#4002C22semidirect
51. #65706C22semidirect
52. #65697C22semidirect
53. #65696C22semidirect
54. #65684C22semidirect
55. #65683C22semidirect
525. C2×C5#2923semidirect
588. C5C2×#2923semidirect
621. C2#10617141direct (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×#657592
C2×#657572
C2×C4×#7042
C22×#40022
#106171412
C2×#657062
C2×#656972
C2×#656962
C2×#656842
C2×#656832
C2×#29235