Finite Group Databasebeta

← C2×#14533778 #17206134

Subgroups — C2×#14533778 #17206134

Order 4352.

3526 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587574C211yesmaximal, characteristic, complemented
2C22×#10587571C212yesmaximal
3C22×C4×C22⋊D17C211yesmaximal, characteristic
4#14533778C2116yesmaximal, direct product factor
5C2×#11762738C212yesmaximal, complemented
6C2×#11762737C212yesmaximal, complemented
7C2×#11762713C212yesmaximal, complemented
8C2×#11762669C211yesmaximal, characteristic, complemented
9C2×#11762666C211yesmaximal, characteristic, complemented
10C2×#11762666C211yesmaximal, characteristic, complemented
11C2×C4⋊C4×Dic17C211yesmaximal, characteristic, complemented
12C2×C17×#380C211yesmaximal, characteristic, complemented
13C23×Dic34C2212yes—
14C23×C22⋊D17C2211yescharacteristic
15C23×C4×D17C2211yescharacteristic
16C2×#10587574—28no—
17C2×#10587574C2218yescomplemented
18C2×#10587571—216no—
19C2×#10587571C22116yes—
20C2×C4×C22⋊D17—28no—
21C2×C4×C22⋊D17C2218yes—
22C22×C4⋊Dic17C2211yescharacteristic
23C22×C4⋊Dic17C2211yescharacteristic
24C22×#63225C2211yescharacteristic
25C22×#63225C2212yes—
26C22×C22⋊Dic17C2212yes—
27C22×C22⋊Dic17C2211yescharacteristic
28C22×#63227C2212yes—
29C22×#63227C2212yes—
30C22×#63227C2211yescharacteristic
31C22×C4×Dic17C2212yes—
32C22×C4×Dic17C2211yescharacteristic
33#11762738C22116yescomplemented
34#11762737C22116yescomplemented
35#11762713C22116yescomplemented
36#11762669C2218yescomplemented
37#11762666C2218yescomplemented
38#11762666C2218yescomplemented
39C4⋊C4×Dic17C2218yescomplemented
40C2×#10587502C2212yes—
41C2×#10587502C2212yes—
42C2×#10587502C2212yes—
43C2×#10587502C2212yes—
44C24×C4×C17C2211yescharacteristic
45C22×C4⋊C4×C17C2211yescharacteristic
46C22×C4⋊C4×C17C2211yescharacteristic
47C22×C22⋊C4×C17C2212yes—
48C17×#380C2218yescomplemented
49C2×C17×#95C2212yes—
50C25×C17C2311yescharacteristic
51C24×D17C2311yescharacteristic
52C23×C4×C17—24no—
53C23×C4×C17—24no—
54C23×C4×C17—24no—
55C23×C4×C17C2312yes—
56C23×C4×C17C2312yes—
57C23×C4×C17C2311yescharacteristic
58C23×C4×C17C2314yes—
59C23×C4×C17C2311yescharacteristic
60C22×C22⋊D17—24no—
61C22×C22⋊D17—24no—
62C22×C22⋊D17D418yes—
63C22×C22⋊D17C2314yes—
64C23×Dic17C2312yes—
65C23×Dic17C2312yes—
66C23×Dic17C2311yescharacteristic
67C23×Dic17C2311yescharacteristic
68C23×Dic17C2311yescharacteristic
69C22×C4×D17—24no—
70C22×C4×D17—24no—
71C22×C4×D17—24no—
72C22×C4×D17C2314yes—
73C22×Dic34—28no—
74C22×Dic34—28no—
75C22×Dic34—28no—
76C22×Dic34C2318yes—
77C2×C4⋊C4×C17—24no—
78C2×C4⋊C4×C17—24no—
79C2×C4⋊C4×C17C2314yes—
80C2×C4⋊C4×C17C2314yes—
81C2×C22⋊C4×C17—28no—
82C2×C22⋊C4×C17C2318yes—
83C2×C22⋊Dic17—28no—
84C2×C22⋊Dic17—24no—
85C2×C22⋊Dic17C2314yes—
86C2×C22⋊Dic17C2318yes—
87C4×C22⋊D17—232no—
88C2×#63227—28no—
89C2×#63227—28no—
90C2×#63227—24no—
91C2×#63227C2314yes—
92C2×#63227C2318yes—
93C2×#63227C2318yes—
94C2×C4⋊Dic17—24no—
95C2×C4⋊Dic17—24no—
96C2×C4⋊Dic17C2314yes—
97C2×C4⋊Dic17C2314yes—
98C2×#63225—24no—
99C2×#63225—28no—
100C2×#63225C2314yes—
101C2×#63225C2318yes—
102C2×C4×Dic17—28no—
103C2×C4×Dic17D418yes—
104C2×C4×Dic17C2318yes—
105C2×C4×Dic17C2314yes—
106#10587574—232no—
107#10587571—264no—
108C17×#95C2318yes—
109#10587502C2318yes—
110#10587502C2318yes—
111#10587502C2318yes—
112#10587502C2318yes—
113C24×C17—22no—
114C24×C17—22no—
115C24×C17—22no—
116C24×C17—22no—
117C24×C17—22no—
118C24×C17—22no—
119C24×C17C2×D414yes—
120C24×C17C2412yes—
121C24×C17C2411yescharacteristic
122C23×D17—22no—
123C23×D17—22no—
124C23×D17—22no—
125C23×D17—22no—
126C23×D17—22no—
127C23×D17—22no—
128C23×D17C2×D414yes—
129C23×D17C2412yes—
130C22×C4×C17—24no—
131C22×C4×C17—24no—
132C22×C4×C17—22no—
133C22×C4×C17—24no—
134C22×C4×C17—24no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—24no—
138C22×C4×C17—24no—
139C22×C4×C17—28no—
140C22×C4×C17—28no—
141C22×C4×C17—28no—
142C22×C4×C17—416no—
143C22×C4×C17C2×D414yes—
144C22×C4×C17#4114yes—
145C22×C4×C17#4114yes—
146C22×C4×C17C2414yes—
147C22×C4×C17C2412yes—
148C22×C4×C17C2414yes—
149C22×C4×C17C2412yes—
150C2×C22⋊D17—232no—
151C2×C22⋊D17—28no—
152C2×C22⋊D17—28no—
153C2×C22⋊D17C2×D4116yes—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—24no—
157C22×Dic17—24no—
158C22×Dic17—22no—
159C22×Dic17—24no—
160C22×Dic17—24no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—24no—
164C22×Dic17—24no—
165C22×Dic17C2×D414yes—
166C22×Dic17C2×D414yes—
167C22×Dic17C2×D414yes—
168C22×Dic17#4114yes—
169C22×Dic17C2414yes—
170C22×Dic17C2412yes—
171C22×Dic17C2414yes—
172C22×Dic17C2412yes—
173C22×Dic17C2412yes—
174C2×C4×D17—28no—
175C2×C4×D17—28no—
176C2×C4×D17—28no—
177C2×C4×D17—416no—
178C2×Dic34—216no—
179C2×Dic34—216no—
180C2×Dic34—216no—
181C2×Dic34—432no—
182C4⋊C4×C17—28no—
183C4⋊C4×C17—28no—
184C22⋊C4×C17—216no—
185C22⋊Dic17—216no—
186C22⋊Dic17—28no—
187#63227—216no—
188#63227—216no—
189#63227—28no—
190C4⋊Dic17—28no—
191C4⋊Dic17—28no—
192#63225—216no—
193#63225—28no—
194C4×Dic17—216no—
195C4×Dic17C2×D4116yes—
196C2×#2547—171nomaximal, Sylow
197C23×C17—28no—
198C23×C17—28no—
199C23×C17—28no—
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—22no—
206C23×C17—44no—
207C23×C17—44no—
208C23×C17—44no—
209C23×C17—44no—
210C23×C17C22×D414yes—
211C23×C17C22×D412yes—
212C23×C17C22×D412yes—
213C23×C17C2×#4112yes—
214C23×C17C2×#4112yes—
215C23×C17C2×#4112yes—
216C23×C172−1+412yes—
217C23×C172−1+412yes—
218C23×C17C2511yescharacteristic
219C22×D17—28no—
220C22×D17—28no—
221C22×D17—28no—
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—22no—
225C22×D17—22no—
226C22×D17—22no—
227C22×D17—22no—
228C22×D17—44no—
229C22×D17—44no—
230C22×D17—44no—
231C22×D17—44no—
232C22×D17C22×D414yes—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—24no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—24no—
242C2×C4×C17—48no—
243C2×C4×C17—44no—
244C2×C4×C17—48no—
245C2×C4×C17—416no—
246C2×C4×C17#124116yescomplemented
247C2×C4×C17C22×D414yes—
248C2×C4×C17C2×#4114yes—
249C2×C4×C17C2×#4114yes—
250C22⋊D17—232no—
251C2×Dic17—28no—
252C2×Dic17—22no—
253C2×Dic17—22no—
254C2×Dic17—24no—
255C2×Dic17—22no—
256C2×Dic17—24no—
257C2×Dic17—24no—
258C2×Dic17—24no—
259C2×Dic17—22no—
260C2×Dic17—22no—
261C2×Dic17—24no—
262C2×Dic17—24no—
263C2×Dic17—48no—
264C2×Dic17—44no—
265C2×Dic17—48no—
266C2×Dic17#121116yes—
267C2×Dic17C22×D414yes—
268C2×Dic17C22×D414yes—
269C2×Dic17C22×D414yes—
270C2×Dic17C2×#4114yes—
271C4×D17—416no—
272Dic34—432no—
273C22×C4⋊Q8—171no—
274C22×#124—172no—
275C22×C4×D4—171no—
276#2547—1716no—
277C2×#397—172no—
278C2×#397—172no—
279C2×#391—172no—
280C2×#380—171no—
281C2×#380—171no—
282C2×#380—171no—
283C2×#380D1711yescharacteristic, complemented
284C2×C4×C4⋊C4—171no—
285C22×C17—24no—
286C22×C17—24no—
287C22×C17—24no—
288C22×C17—42no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17—42no—
292C22×C17—416no—
293C22×C17C2×#12114yes—
294C22×C17C2×#12414yes—
295C22×C17#53614yes—
296C22×C17#54714yes—
297C22×C17#54714yes—
298C22×C17#55614yes—
299C22×C17#55614yes—
300C22×C17C23×D411yescharacteristic
301C22×C17C23×D411yescharacteristic
302C22×C17C22×#4111yescharacteristic
303C22×C17C22×#4111yescharacteristic
304C22×C17C22×#4111yescharacteristic
305C22×C17C2×2−1+411yescharacteristic
306C22×C17C2×2−1+411yescharacteristic
307C2×D17—24no—
308C2×D17—24no—
309C2×D17—24no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—42no—
313C2×D17—42no—
314C2×D17—416no—
315C4×C17—44no—
316C4×C17—42no—
317C4×C17—44no—
318C4×C17C2×#12418yescomplemented
319Dic17—24no—
320Dic17—44no—
321Dic17—42no—
322Dic17—44no—
323Dic17C2×#12118yes—
324C23×Q8—172no—
325C23×D4—171no—
326C24×C4—171no—
327C24×C4C2×D1711yescharacteristic
328C2×C4⋊Q8—178no—
329C2×C4⋊Q8—348no—
330C2×#124—1716no—
331C2×#124—3416no—
332C2×C4×D4—178no—
333C2×C4×D4—348no—
334C22×C4⋊C4—172no—
335C22×C4⋊C4—171no—
336C22×C4⋊C4—171no—
337C22×C4⋊C4—171no—
338C22×C4⋊C4C2×D1711yescharacteristic
339C22×C4⋊C4C2×D1711yescharacteristic
340C22×C22⋊C4—171no—
341C22×C22⋊C4—172no—
342C22×C22⋊C4—171no—
343C22×C22⋊C4—172no—
344C22×C22⋊C4—172no—
345C22×C22⋊C4C2×D1712yes—
346C22×C42—171no—
347C22×C42—172no—
348#397—1716no—
349#397—1716no—
350#391—1716no—
351#380—178no—
352#380—178no—
353#380—178no—
354#380C2×D1718yescomplemented
355C4×C4⋊C4—178no—
356C2×#95—172no—
357C2×#95—172no—
358C2×#95—172no—
359C2×#95—172no—
360C2×#95C2×D1712yes—
361C2×C17—44no—
362C2×C17#254718yescomplemented
363C2×C17C22×#12111yescharacteristic
364C2×C17C22×#12411yescharacteristic
365C2×C17C2×#53611yescharacteristic
366C2×C17C2×#54711yescharacteristic
367C2×C17C2×#54711yescharacteristic
368C2×C17C2×#55611yescharacteristic
369C2×C17C2×#55611yescharacteristic
370D17—44no—
371C25—171no—
372C25C22×D1711yescharacteristic
373C22×Q8—178no—
374C22×Q8—348no—
375C22×Q8—348no—
376C22×Q8—348no—
377C22×D4—178no—
378C22×D4—174no—
379C22×D4—344no—
380C22×D4—344no—
381C23×C4—24no—
382C23×C4—24no—
383C23×C4—24no—
384C23×C4—171no—
385C23×C4—171no—
386C23×C4—172no—
387C23×C4—172no—
388C23×C4—171no—
389C23×C4—174no—
390C23×C4—344no—
391C23×C4—344no—
392C23×C4—344no—
393C23×C4C22×D1714yes—
394C23×C4C22×D1712yes—
395C23×C4C22×D1712yes—
396C23×C4C22×D1711yescharacteristic
397C23×C4C22×D1711yescharacteristic
398C4⋊Q8—3432no—
399#124—3464no—
400C4×D4—3432no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—174no—
404C2×C4⋊C4—174no—
405C2×C4⋊C4—178no—
406C2×C4⋊C4—174no—
407C2×C4⋊C4—344no—
408C2×C4⋊C4—348no—
409C2×C4⋊C4—344no—
410C2×C4⋊C4—344no—
411C2×C4⋊C4C22×D1714yes—
412C2×C4⋊C4C22×D1714yes—
413C2×C22⋊C4—28no—
414C2×C22⋊C4—174no—
415C2×C22⋊C4—178no—
416C2×C22⋊C4—174no—
417C2×C22⋊C4—178no—
418C2×C22⋊C4—178no—
419C2×C22⋊C4—344no—
420C2×C22⋊C4—348no—
421C2×C22⋊C4—348no—
422C2×C22⋊C4—344no—
423C2×C22⋊C4—348no—
424C2×C22⋊C4C22×D1718yes—
425C2×C42—178no—
426C2×C42—174no—
427C2×C42—178no—
428C2×C42—348no—
429#95—178no—
430#95—178no—
431#95—178no—
432#95—178no—
433#95C22×D1718yes—
434C17C2×#254711yescharacteristic, complemented, Sylow
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—174no—
442C24—172no—
443C24—342no—
444C24—342no—
445C24—342no—
446C24—342no—
447C24—342no—
448C24—342no—
449C24D4×D1714yes—
450C24C23×D1712yes—
451C24C23×D1711yescharacteristic
452C2×Q8—3416no—
453C2×Q8—3416no—
454C2×Q8—3416no—
455C2×Q8—6832no—
456C2×D4—1716no—
457C2×D4—348no—
458C2×D4—3432no—
459C2×D4—348no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—416no—
473C22×C4—174no—
474C22×C4—174no—
475C22×C4—174no—
476C22×C4—172no—
477C22×C4—174no—
478C22×C4—172no—
479C22×C4—172no—
480C22×C4—174no—
481C22×C4—174no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—344no—
485C22×C4—342no—
486C22×C4—344no—
487C22×C4—344no—
488C22×C4—342no—
489C22×C4—344no—
490C22×C4—342no—
491C22×C4—344no—
492C22×C4—344no—
493C22×C4—348no—
494C22×C4—348no—
495C22×C4—348no—
496C22×C4—6816no—
497C22×C4#6325214yes—
498C22×C4D4×D1714yes—
499C22×C4#6325414yes—
500C22×C4C23×D1712yes—
501C22×C4C23×D1714yes—
502C22×C4C23×D1714yes—
503C22×C4C23×D1712yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—348no—
507C4⋊C4—3416no—
508C4⋊C4—348no—
509C4⋊C4—348no—
510C22⋊C4—216no—
511C22⋊C4—348no—
512C22⋊C4—3416no—
513C22⋊C4—3416no—
514C22⋊C4—348no—
515C22⋊C4—3416no—
516C42—1716no—
517C42—3416no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—174no—
532C23—348no—
533C23—348no—
534C23—348no—
535C23—342no—
536C23—342no—
537C23—342no—
538C23—342no—
539C23—342no—
540C23—342no—
541C23—684no—
542C23—684no—
543C23—684no—
544C23—684no—
545C23C2×#6325212yes—
546C23C2×D4×D1714yes—
547C23C2×D4×D1712yes—
548C23C2×D4×D1712yes—
549C23C2×#6325412yes—
550C23#1058769412yes—
551C23#41×D1712yes—
552C23#1058769712yes—
553C23C24×D1711yescharacteristic
554Q8—6832no—
555D4—3432no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—1716no—
570C2×C4—174no—
571C2×C4—174no—
572C2×C4—174no—
573C2×C4—174no—
574C2×C4—348no—
575C2×C4—342no—
576C2×C4—342no—
577C2×C4—344no—
578C2×C4—344no—
579C2×C4—342no—
580C2×C4—344no—
581C2×C4—344no—
582C2×C4—342no—
583C2×C4—344no—
584C2×C4—342no—
585C2×C4—344no—
586C2×C4—688no—
587C2×C4—688no—
588C2×C4—684no—
589C2×C4—6816no—
590C2×C4#10587571116yescomplemented
591C2×C4C2×#6325214yes—
592C2×C4C2×D4×D1714yes—
593C2×C4C2×#6325414yes—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—42no—
598C22—416no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—344no—
603C22—344no—
604C22—344no—
605C22—682no—
606C22—6816no—
607C22—682no—
608C22—682no—
609C22—682no—
610C22#121×D1714yes—
611C22C2×#1058757114yes—
612C22#1176338214yes—
613C22#1176345914yes—
614C22#1176346014yes—
615C22#1176352714yes—
616C22#1176353014yes—
617C22C22×D4×D1711yescharacteristic
618C22C22×D4×D1711yescharacteristic
619C22C2×#41×D1711yescharacteristic
620C22C22×#6325411yescharacteristic
621C22C22×#6325211yescharacteristic
622C22C2×#1058769711yescharacteristic
623C22C2×#1058769411yescharacteristic
624C4—44no—
625C4—44no—
626C4—42no—
627C4—178no—
628C4—344no—
629C4—684no—
630C4—684no—
631C4—682no—
632C4C2×#1058757118yescomplemented
633C2—44no—
634C2—684no—
635C2#1453377818yesdirect product factor
636C2C2×#121×D1711yescharacteristic
637C2C22×#1058757111yescharacteristic
638C2C2×#1176338211yescharacteristic
639C2C2×#1176345911yescharacteristic
640C2C2×#1176346011yescharacteristic
641C2C2×#1176352711yescharacteristic
642C2C2×#1176353011yescharacteristic
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC2×C17×#380C2
socleC24×C17C24
prime radicalC23×C22⋊D17C22
hypercenterC2×#380D17
hypocenterC17C2×#2547
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#10587574C2semidirect
4. #14533778C2direct (N × H)
5. C2×#11762738C2semidirect
6. C2×#11762737C2semidirect
7. C2×#11762713C2semidirect
8. C2×#11762669C2semidirect
9. C2×#11762666C2semidirect
10. C2×#11762666C2semidirect
11. C2×C4⋊C4×Dic17C2semidirect
12. C2×C17×#380C2semidirect
17. C2×#10587574C22semidirect
33. #11762738C22semidirect
34. #11762737C22semidirect
35. #11762713C22semidirect
36. #11762669C22semidirect
37. #11762666C22semidirect
38. #11762666C22semidirect
39. C4⋊C4×Dic17C22semidirect
48. C17×#380C22semidirect
246. C2×C4×C17#124semidirect
283. C2×#380D17semidirect
318. C4×C17C2×#124semidirect
354. #380C2×D17semidirect
362. C2×C17#2547semidirect
434. C17C2×#2547semidirect
590. C2×C4#10587571semidirect
632. C4C2×#10587571semidirect
635. C2#14533778direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875742
C22×#105875712
C22×C4×C22⋊D172
#145337782
C2×#117627382
C2×#117627372
C2×#117627132
C2×#117626692
C2×#117626662
C2×C4⋊C4×Dic172
C2×C17×#3802
C2×#254717