Finite Group Databasebeta

← C2×#14529998 #17061914

Subgroups — C2×#14529998 #17061914

Order 4352.

1434 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×#11762852C211yesmaximal, characteristic, complemented
2C22×#10587508C211yesmaximal, characteristic
3C2×#11762810C211yesmaximal, characteristic
4C22×#10587507C211yesmaximal, characteristic, complemented
5C2×#11762716C211yesmaximal, characteristic, complemented
6#14529998C218yesmaximal, direct product factor
7C2×#11762568C211yesmaximal, characteristic, complemented
8C2×C17×#536C211yesmaximal, characteristic
9C22×#63223C2211yescharacteristic
10C22×#63223C2211yescharacteristic
11C22×C4⋊Dic17C2211yescharacteristic
12C22×C4×Dic17C2211yescharacteristic
13#11762852—22no—
14#11762852C2214yescomplemented
15C2×#10587478—21no—
16C2×#10587479—21no—
17C2×#10587508—24no—
18C2×#10587508C2214yes—
19#11762810—22no—
20#11762810C2214yes—
21C2×#10587506—21no—
22C2×#10587503—21no—
23C2×#10587507—24no—
24C2×#10587507C2214yescomplemented
25#11762716—22no—
26#11762716C2214yescomplemented
27C2×#10587502—22no—
28#11762568C2214yescomplemented
29C22×#41×C17C2211yescharacteristic
30C23×Q8×C17C2211yescharacteristic
31C17×#536—22no—
32C17×#536—22no—
33C17×#536—22no—
34C17×#536C2214yes—
35C2×C17×C4⋊Q8—21no—
36C2×C17×#124—21no—
37C2×C17×#123—22no—
38C2×C17×C22⋊Q8—21no—
39C2×C17×C22⋊Q8—21no—
40C2×C17×#117C2211yescharacteristic
41C2×#41×C17—21no—
42C2×#41×C17—21no—
43C2×#41×C17—22no—
44C2×#41×C17—22no—
45C2×#41×C17—22no—
46C2×#41×C17—22no—
47C2×#41×C17D412yes—
48C2×#41×C17C2312yes—
49C22×Q8×C17—21no—
50C22×Q8×C17—21no—
51C22×Q8×C17—21no—
52C22×Q8×C17—21no—
53C22×Q8×C17—22no—
54C22×Q8×C17—22no—
55C22×Q8×C17—22no—
56C22×Q8×C17—22no—
57C22×Q8×C17D411yescharacteristic
58C22×Q8×C17D412yes—
59C22×Q8×C17C2312yes—
60C22×D4×C17—21no—
61C22×D4×C17D411yescharacteristic
62C23×C4×C17—21no—
63C23×C4×C17D412yes—
64C23×C4×C17C2311yescharacteristic
65C23×Dic17—21no—
66C23×Dic17—21no—
67C17×C4⋊Q8—24no—
68C17×C4⋊Q8—24no—
69C17×#124—24no—
70C17×#124—24no—
71C17×#123—28no—
72C17×#123—28no—
73C17×C22⋊Q8—24no—
74C17×C22⋊Q8—24no—
75C17×C22⋊Q8—24no—
76C17×C22⋊Q8—24no—
77C17×#117—21no—
78C17×#117—21no—
79C17×#117D412yes—
80C17×#117C2312yes—
81C2×C4⋊C4×C17—22no—
82C2×C4⋊C4×C17—22no—
83C2×C4⋊C4×C17D411yescharacteristic
84C2×C4⋊C4×C17D411yescharacteristic
85C2×C22⋊C4×C17—22no—
86C2×C22⋊C4×C17—22no—
87C2×C22⋊C4×C17—21no—
88C2×C42×C17—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17C2312yes—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—22no—
97C2×C4×Dic17—22no—
98C2×C4×Dic17C2312yes—
99C2×#63223—21no—
100C2×#63223—21no—
101C2×#63223—22no—
102C2×#63223—22no—
103C2×#63223—22no—
104C2×#63223D414yes—
105C2×#63223C2312yes—
106C2×#63223C2312yes—
107C22×#3622—21no—
108C22×#3622—21no—
109#10587508—28no—
110#10587507—28no—
111#10587506—24no—
112#10587503—24no—
113#10587502—28no—
114#10587479—24no—
115#10587478—24no—
116C24×C17—21no—
117C24×C17C2×D411yescharacteristic
118#41×C17—24no—
119#41×C17—24no—
120#41×C17—22no—
121#41×C17—22no—
122#41×C17—22no—
123#41×C17—22no—
124#41×C17—42no—
125#41×C17—42no—
126#41×C17—42no—
127#41×C17—42no—
128C2×Q8×C17—22no—
129C2×Q8×C17—22no—
130C2×Q8×C17—21no—
131C2×Q8×C17—24no—
132C2×Q8×C17—24no—
133C2×Q8×C17—22no—
134C2×Q8×C17—22no—
135C2×Q8×C17—22no—
136C2×Q8×C17—22no—
137C2×Q8×C17—22no—
138C2×Q8×C17—22no—
139C2×Q8×C17—22no—
140C2×Q8×C17—22no—
141C2×Q8×C17—42no—
142C2×Q8×C17—42no—
143C2×Q8×C17—42no—
144C2×Q8×C17—42no—
145C2×Q8×C17—42no—
146C2×Q8×C17—42no—
147C2×Q8×C17—42no—
148C2×Q8×C17—42no—
149C2×Q8×C17C2×D412yes—
150C2×D4×C17—22no—
151C2×D4×C17—22no—
152C2×D4×C17—21no—
153C2×D4×C17—22no—
154C2×D4×C17—22no—
155C2×D4×C17—42no—
156C2×D4×C17—42no—
157C2×D4×C17C2×D412yes—
158C22×C4×C17—22no—
159C22×C4×C17—21no—
160C22×C4×C17—22no—
161C22×C4×C17—21no—
162C22×C4×C17—21no—
163C22×C4×C17—24no—
164C22×C4×C17—24no—
165C22×C4×C17—22no—
166C22×C4×C17—22no—
167C22×C4×C17—42no—
168C22×C4×C17—42no—
169C22×C4×C17C2×D412yes—
170C22×C4×C17C2×D414yes—
171C22×C4×C17C2×D411yescharacteristic
172C22×C4×C17C2×D411yescharacteristic
173C22×C4×C17C2×D411yescharacteristic
174C22×C4×C17C2×D411yescharacteristic
175C22×C4×C17C2×D411yescharacteristic
176C22×C4×C17#4112yes—
177C22×C4×C17#4112yes—
178C22×C4×C17#4112yes—
179C22×C4×C17C2411yescharacteristic
180C22×Dic17—24no—
181C22×Dic17—22no—
182C22×Dic17—22no—
183C22×Dic17—42no—
184C22×Dic17—42no—
185C22×Dic17—44no—
186C22×Dic17—41no—
187C22×Dic17—41no—
188C4⋊C4×C17—28no—
189C4⋊C4×C17—28no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17—21no—
192C4⋊C4×C17C2×D412yes—
193C4⋊C4×C17C2×D412yes—
194C22⋊C4×C17—28no—
195C22⋊C4×C17—28no—
196C22⋊C4×C17—22no—
197C22⋊C4×C17—22no—
198C42×C17—22no—
199C42×C17—22no—
200C4⋊Dic17—22no—
201C4⋊Dic17—22no—
202C4⋊Dic17—41no—
203C4⋊Dic17—41no—
204C4×Dic17—22no—
205C4×Dic17—22no—
206C4×Dic17—42no—
207#63223—24no—
208#63223—22no—
209#63223—22no—
210#63223—22no—
211#63223—42no—
212#63223C2×D414yes—
213C2×#3622—24no—
214C2×#3622—22no—
215C2×#3622—22no—
216C2×#3622—42no—
217C2×#2038—171nomaximal, Sylow
218C23×C17—22no—
219C23×C17—22no—
220C23×C17—22no—
221C23×C17—42no—
222C23×C17—42no—
223C23×C17—42no—
224C23×C17#12011yescharacteristic
225C23×C17#12011yescharacteristic
226C23×C17#12112yes—
227C23×C17#12312yes—
228C23×C17C22×D411yescharacteristic
229Q8×C17—22no—
230Q8×C17—22no—
231Q8×C17—42no—
232Q8×C17—42no—
233Q8×C17—42no—
234Q8×C17—42no—
235Q8×C17—44no—
236Q8×C17—44no—
237Q8×C17—42no—
238D4×C17—22no—
239D4×C17—22no—
240D4×C17—42no—
241D4×C17—42no—
242D4×C17—44no—
243D4×C17—42no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—24no—
247C2×C4×C17—22no—
248C2×C4×C17—24no—
249C2×C4×C17—24no—
250C2×C4×C17—44no—
251C2×C4×C17—44no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17—44no—
258C2×C4×C17#12011yescharacteristic
259C2×C4×C17#12011yescharacteristic
260C2×C4×C17#12112yes—
261C2×C4×C17#12112yes—
262C2×C4×C17#12112yes—
263C2×C4×C17#12112yes—
264C2×C4×C17#12112yes—
265C2×C4×C17#12312yes—
266C2×C4×C17#12312yes—
267C2×C4×C17#12312yes—
268C2×C4×C17#12412yes—
269C2×C4×C17#12412yes—
270C2×C4×C17C22×D411yescharacteristic
271C2×C4×C17C22×D411yescharacteristic
272C2×C4×C17C22×D411yescharacteristic
273C2×C4×C17C2×#4111yescharacteristic
274C2×C4×C17C2×#4111yescharacteristic
275C2×C4×C17C2×#4111yescharacteristic
276C2×Dic17—24no—
277C2×Dic17—42no—
278C2×Dic17—48no—
279C2×Dic17—42no—
280C2×Dic17—44no—
281C2×Dic17—42no—
282C2×Dic17—82no—
283C2×Dic17—82no—
284#3622—24no—
285#3622—42no—
286C2×#536D1711yescharacteristic
287C2×#428—171no—
288C22×#104—171no—
289C2×#417—171no—
290C22×#101—171no—
291C2×#391—171no—
292#2038—178no—
293C2×#339—171no—
294C22×C17—21no—
295C22×C17—21no—
296C22×C17—21no—
297C22×C17—21no—
298C22×C17—24no—
299C22×C17—24no—
300C22×C17—42no—
301C22×C17—42no—
302C22×C17—42no—
303C22×C17—84no—
304C22×C17#39412yes—
305C22×C17#39412yes—
306C22×C17#45412yes—
307C22×C17#45612yes—
308C22×C17C2×#12011yescharacteristic
309C22×C17C2×#12111yescharacteristic
310C22×C17C2×#12311yescharacteristic
311C4×C17—44no—
312C4×C17—42no—
313C4×C17—42no—
314C4×C17—42no—
315C4×C17#39412yes—
316C4×C17#39412yes—
317C4×C17C2×#12111yescharacteristic
318C4×C17C2×#12111yescharacteristic
319C4×C17C2×#12311yescharacteristic
320C4×C17C2×#12411yescharacteristic
321Dic17—44no—
322Dic17—82no—
323C22×#41C2×D1711yescharacteristic
324C23×Q8C2×D1711yescharacteristic
325C22×M16—171no—
326C22×M16—171no—
327#536—22no—
328#536—22no—
329#536—22no—
330#536C2×D1714yes—
331C2×C4⋊Q8—21no—
332C2×#124—21no—
333C2×#123—22no—
334C2×C22⋊Q8—21no—
335C2×C22⋊Q8—21no—
336C2×#117C2×D1711yescharacteristic
337C22×C4⋊C4—171no—
338C22×C42—171no—
339#428—174no—
340#428—342no—
341C2×#107—341no—
342C2×#106—341no—
343C2×#104—174no—
344C2×#104—344no—
345#417—174no—
346#417—342no—
347C2×#103—341no—
348C2×#102—341no—
349C2×#101—174no—
350C2×#101—344no—
351#391—174no—
352#391—342no—
353C2×#95—342no—
354#339—174no—
355C2×C17—21no—
356C2×C17—21no—
357C2×C17—22no—
358C2×C17—82no—
359C2×C17#203814yescomplemented
360C2×C17C2×#39411yescharacteristic
361C2×C17C2×#45411yescharacteristic
362C2×C17C2×#45611yescharacteristic
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—22no—
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41C22⋊D1712yes—
370C2×#41C22×D1712yes—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—21no—
376C22×Q8—21no—
377C22×Q8—21no—
378C22×Q8—21no—
379C22×Q8C22⋊D1711yescharacteristic
380C22×Q8C22⋊D1712yes—
381C22×Q8C22×D1712yes—
382C22×D4—21no—
383C22×D4C22⋊D1711yescharacteristic
384C23×C4—21no—
385C23×C4—341no—
386C23×C4—341no—
387C23×C4C22⋊D1712yes—
388C23×C4C22×D1711yescharacteristic
389C2×M16—174no—
390C2×M16—172no—
391C2×M16—172no—
392C2×M16—342no—
393C2×M16—341no—
394C2×M16—341no—
395C2×M16—342no—
396C2×M16—342no—
397C22×C8—341no—
398C22×C8—341no—
399C4⋊Q8—24no—
400C4⋊Q8—24no—
401#124—24no—
402#124—24no—
403#123—28no—
404#123—28no—
405C22⋊Q8—24no—
406C22⋊Q8—24no—
407C22⋊Q8—24no—
408C22⋊Q8—24no—
409#117—21no—
410#117—21no—
411#117C22⋊D1712yes—
412#117C22×D1712yes—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—172no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4C22⋊D1711yescharacteristic
422C2×C4⋊C4C22⋊D1711yescharacteristic
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—21no—
426C2×C42—21no—
427C2×C42—172no—
428C2×C42—342no—
429C2×C42—342no—
430C2×C42—341no—
431#107—344no—
432#106—344no—
433#104—348no—
434#103—344no—
435#102—344no—
436#101—348no—
437#95—348no—
438C17C2×#203811yescharacteristic, complemented, Sylow
439C24—21no—
440C24C2×C22⋊D1711yescharacteristic
441#41—22no—
442#41—22no—
443#41—22no—
444#41—22no—
445#41—24no—
446#41—24no—
447#41—42no—
448#41—42no—
449#41—42no—
450#41—42no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8C2×C22⋊D1712yes—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4C2×C22⋊D1712yes—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—344no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—682no—
496C22×C4—682no—
497C22×C4—684no—
498C22×C4—681no—
499C22×C4—681no—
500C22×C4D4×D1712yes—
501C22×C4#6325412yes—
502C22×C4#6325412yes—
503C22×C4#6325412yes—
504C22×C4C2×C22⋊D1714yes—
505C22×C4C2×C22⋊D1711yescharacteristic
506C22×C4C2×C22⋊D1711yescharacteristic
507C22×C4C2×C22⋊D1711yescharacteristic
508C22×C4C2×C22⋊D1711yescharacteristic
509C22×C4C2×C22⋊D1711yescharacteristic
510C22×C4C23×D1711yescharacteristic
511M16—174no—
512M16—342no—
513M16—344no—
514M16—342no—
515M16—342no—
516M16—682no—
517C2×C8—342no—
518C2×C8—344no—
519C2×C8—342no—
520C2×C8—682no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—342no—
526C4⋊C4—342no—
527C4⋊C4—681no—
528C4⋊C4—681no—
529C4⋊C4C2×C22⋊D1712yes—
530C4⋊C4C2×C22⋊D1712yes—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C42—22no—
536C42—22no—
537C42—342no—
538C42—342no—
539C42—682no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23#1058762112yes—
547C23#1058762512yes—
548C23#1058763911yescharacteristic
549C23#1058763911yescharacteristic
550C23C22×C22⋊D1711yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—44no—
554Q8—44no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560D4—22no—
561D4—22no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4—42no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—344no—
581C2×C4—682no—
582C2×C4—688no—
583C2×C4—682no—
584C2×C4—684no—
585C2×C4—682no—
586C2×C4—1362no—
587C2×C4—1362no—
588C2×C4#1058762112yes—
589C2×C4#1058762112yes—
590C2×C4#1058762112yes—
591C2×C4#1058762212yes—
592C2×C4#1058762212yes—
593C2×C4#1058762412yes—
594C2×C4#1058762412yes—
595C2×C4#1058762512yes—
596C2×C4#1058762512yes—
597C2×C4#1058762512yes—
598C2×C4#1058763911yescharacteristic
599C2×C4#1058763911yescharacteristic
600C2×C4C2×D4×D1711yescharacteristic
601C2×C4C2×#6325411yescharacteristic
602C2×C4C2×#6325411yescharacteristic
603C2×C4C2×#6325411yescharacteristic
604C2×C4C22×C22⋊D1711yescharacteristic
605C2×C4C22×C22⋊D1711yescharacteristic
606C8—344no—
607C8—682no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—84no—
618C22#1176272312yes—
619C22#1176272312yes—
620C22#1176298212yes—
621C22#1176299212yes—
622C22C2×#1058763911yescharacteristic
623C22C2×#1058762511yescharacteristic
624C22C2×#1058762111yescharacteristic
625C4—42no—
626C4—42no—
627C4—44no—
628C4—42no—
629C4—684no—
630C4—1362no—
631C4#1176272312yes—
632C4#1176272312yes—
633C4C2×#1058762411yescharacteristic
634C4C2×#1058762511yescharacteristic
635C4C2×#1058762111yescharacteristic
636C4C2×#1058762211yescharacteristic
637C2—21no—
638C2—22no—
639C2—21no—
640C2—82no—
641C2#1452999814yesdirect product factor
642C2C2×#1176272311yescharacteristic
643C2C2×#1176298211yescharacteristic
644C2C2×#1176299211yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#536C2
socleC23×C17#120
prime radicalC22×#41×C17C22
hypercenterC2×#536D17
hypocenterC17C2×#2038
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#11762852C2semidirect
4. C22×#10587507C2semidirect
5. C2×#11762716C2semidirect
6. #14529998C2direct (N × H)
7. C2×#11762568C2semidirect
14. #11762852C22semidirect
24. C2×#10587507C22semidirect
26. #11762716C22semidirect
28. #11762568C22semidirect
359. C2×C17#2038semidirect
438. C17C2×#2038semidirect
641. C2#14529998direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117628522
C22×#105875082
C2×#117628102
C22×#105875072
C2×#117627162
#145299982
C2×#117625682
C2×C17×#5362
C2×#203817