Finite Group Databasebeta

← C2×#14533090 #17203376

Subgroups — C2×#14533090 #17203376

Order 4352.

3678 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×#10587580C212yesmaximal
2C22×C4×D68C211yesmaximal, characteristic
3C22×C4⋊C4×D17C211yesmaximal, characteristic, complemented
4#14533090C2116yesmaximal, direct product factor
5C2×#11762742C212yesmaximal, complemented
6C2×#11762738C212yesmaximal, complemented
7C2×#11762707C211yesmaximal, characteristic, complemented
8C2×#11762671C212yesmaximal, complemented
9C2×#11762666C211yesmaximal, characteristic, complemented
10C2×#11762666C211yesmaximal, characteristic, complemented
11C2×#11762666C211yesmaximal, characteristic, complemented
12C2×C17×#389C211yesmaximal, characteristic, complemented
13C23×D68C2211yescharacteristic
14C23×C4×D17C2211yescharacteristic
15C23×C4×D17C2212yes—
16C23×C4×D17C2211yescharacteristic
17C2×#10587580—216no—
18C2×#10587580C22116yes—
19C2×C4×D68—28no—
20C2×C4×D68C2218yes—
21C22×C4⋊Dic17C2211yescharacteristic
22C22×C4⋊Dic17C2211yescharacteristic
23C22×C4⋊Dic17C2211yescharacteristic
24C22×#63225C2212yes—
25C22×#63225C2212yes—
26C2×C4⋊C4×D17—28no—
27C2×C4⋊C4×D17C2218yescomplemented
28C22×#63227C2212yes—
29C22×#63227C2212yes—
30C22×#63227C2212yes—
31C22×#63227C2211yescharacteristic
32C22×#63227C2211yescharacteristic
33#11762742C22116yescomplemented
34#11762738C22116yescomplemented
35#11762707C2218yescomplemented
36#11762671C22116yescomplemented
37#11762666C2218yescomplemented
38#11762666C2218yescomplemented
39#11762666C2218yescomplemented
40C2×#10587502C2212yes—
41C2×#10587502C2212yes—
42C2×#10587502C2212yes—
43C2×#10587502C2212yes—
44C22×C4⋊C4×C17C2211yescharacteristic
45C22×C4⋊C4×C17C2211yescharacteristic
46C22×C4⋊C4×C17C2212yes—
47C22×C42×C17C2211yescharacteristic
48C17×#389C2218yescomplemented
49C2×C17×#95C2212yes—
50C24×D17C2311yescharacteristic
51C24×D17C2311yescharacteristic
52C23×C4×C17C2311yescharacteristic
53C23×C4×C17C2312yes—
54C23×C4×C17C2312yes—
55C23×C4×C17C2311yescharacteristic
56C23×C4×C17C2311yescharacteristic
57C23×Dic17C2312yes—
58C23×Dic17C2312yes—
59C23×Dic17C2311yescharacteristic
60C23×Dic17C2311yescharacteristic
61C22×D68—24no—
62C22×D68—24no—
63C22×D68D418yes—
64C22×D68C2314yes—
65C22×C4×D17—24no—
66C22×C4×D17—24no—
67C22×C4×D17—28no—
68C22×C4×D17—28no—
69C22×C4×D17—28no—
70C22×C4×D17—24no—
71C22×C4×D17—24no—
72C22×C4×D17—24no—
73C22×C4×D17D418yes—
74C22×C4×D17C2318yes—
75C22×C4×D17C2314yes—
76C22×C4×D17C2314yes—
77C2×C4⋊C4×C17—24no—
78C2×C4⋊C4×C17—28no—
79C2×C4⋊C4×C17—24no—
80C2×C4⋊C4×C17C2318yes—
81C2×C4⋊C4×C17C2314yes—
82C2×C4⋊C4×C17C2314yes—
83C2×C42×C17—24no—
84C2×C42×C17C2314yes—
85C2×#63227—24no—
86C2×#63227—28no—
87C2×#63227—28no—
88C2×#63227—28no—
89C2×#63227—24no—
90C2×#63227C2314yes—
91C2×#63227C2318yes—
92C2×#63227C2318yes—
93C2×#63227C2314yes—
94C2×#63227C2318yes—
95C2×C4⋊Dic17—24no—
96C2×C4⋊Dic17—24no—
97C2×C4⋊Dic17—24no—
98C2×C4⋊Dic17C2314yes—
99C2×C4⋊Dic17C2314yes—
100C2×C4⋊Dic17C2314yes—
101C2×#63225—28no—
102C2×#63225—28no—
103C2×#63225C2318yes—
104C2×#63225C2318yes—
105#10587580—264no—
106C4⋊C4×D17—232no—
107C4×D68—232no—
108C17×#95C2318yes—
109#10587502C2318yes—
110#10587502C2318yes—
111#10587502C2318yes—
112#10587502C2318yes—
113C24×C17C2411yescharacteristic
114C23×D17—22no—
115C23×D17—22no—
116C23×D17—22no—
117C23×D17—22no—
118C23×D17—22no—
119C23×D17—22no—
120C23×D17—22no—
121C23×D17—22no—
122C23×D17—22no—
123C23×D17—22no—
124C23×D17C2×D414yes—
125C23×D17C2×D414yes—
126C23×D17C2×D414yes—
127C23×D17#4114yes—
128C23×D17C2412yes—
129C23×D17C2412yes—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—24no—
133C22×C4×C17—24no—
134C22×C4×C17—22no—
135C22×C4×C17—24no—
136C22×C4×C17—24no—
137C22×C4×C17—24no—
138C22×C4×C17—22no—
139C22×C4×C17—22no—
140C22×C4×C17—24no—
141C22×C4×C17C2×D414yes—
142C22×C4×C17C2×D414yes—
143C22×C4×C17#4114yes—
144C22×C4×C17#4114yes—
145C22×C4×C17C2414yes—
146C22×C4×C17C2412yes—
147C22×C4×C17C2414yes—
148C22×C4×C17C2412yes—
149C22×C4×C17C2412yes—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—24no—
153C22×Dic17—24no—
154C22×Dic17—22no—
155C22×Dic17—24no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—24no—
159C22×Dic17—24no—
160C22×Dic17—24no—
161C22×Dic17C2×D414yes—
162C22×Dic17C2412yes—
163C22×Dic17C2414yes—
164C22×Dic17C2414yes—
165C22×Dic17C2412yes—
166C2×D68—232no—
167C2×D68—28no—
168C2×D68—28no—
169C2×D68C2×D4116yes—
170C2×C4×D17—232no—
171C2×C4×D17—28no—
172C2×C4×D17—28no—
173C2×C4×D17—216no—
174C2×C4×D17—216no—
175C2×C4×D17—216no—
176C2×C4×D17—28no—
177C2×C4×D17—28no—
178C2×C4×D17—28no—
179C2×C4×D17—432no—
180C2×C4×D17—416no—
181C2×C4×D17C2×D4116yes—
182C4⋊C4×C17—28no—
183C4⋊C4×C17—216no—
184C4⋊C4×C17—28no—
185C42×C17—28no—
186#63227—28no—
187#63227—216no—
188#63227—216no—
189#63227—216no—
190#63227—28no—
191C4⋊Dic17—28no—
192C4⋊Dic17—28no—
193C4⋊Dic17—28no—
194#63225—216no—
195#63225—216no—
196C2×#2486—171nomaximal, Sylow
197C23×C17C22×D412yes—
198C23×C17C22×D412yes—
199C23×C17C2×#4112yes—
200C23×C17C2×#4112yes—
201C23×C17C2×#4112yes—
202C23×C172−1+412yes—
203C23×C172−1+412yes—
204C23×C17C2511yescharacteristic
205C22×D17—28no—
206C22×D17—28no—
207C22×D17—28no—
208C22×D17—28no—
209C22×D17—28no—
210C22×D17—28no—
211C22×D17—22no—
212C22×D17—22no—
213C22×D17—22no—
214C22×D17—22no—
215C22×D17—28no—
216C22×D17—28no—
217C22×D17—28no—
218C22×D17—22no—
219C22×D17—22no—
220C22×D17—22no—
221C22×D17—22no—
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—44no—
225C22×D17—44no—
226C22×D17—44no—
227C22×D17—44no—
228C22×D17#121116yes—
229C22×D17C22×D414yes—
230C22×D17C22×D414yes—
231C22×D17C22×D414yes—
232C22×D17C2×#4114yes—
233C2×C4×C17—28no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—24no—
242C2×C4×C17—22no—
243C2×C4×C17—24no—
244C2×C4×C17—22no—
245C2×C4×C17—48no—
246C2×C4×C17—44no—
247C2×C4×C17—48no—
248C2×C4×C17#123116yescomplemented
249C2×C4×C17C22×D414yes—
250C2×C4×C17C22×D414yes—
251C2×C4×C17C2×#4114yes—
252C2×C4×C17C2×#4114yes—
253C2×Dic17—28no—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—24no—
257C2×Dic17—24no—
258C2×Dic17—22no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17—24no—
264C2×Dic17—24no—
265C2×Dic17—48no—
266C2×Dic17—44no—
267C2×Dic17—48no—
268C2×Dic17C22×D414yes—
269D68—232no—
270C4×D17—232no—
271C4×D17—432no—
272C4×D17—416no—
273C22×#123—172no—
274C22×C4×D4—171no—
275C23×C4⋊C4—171no—
276#2486—1716no—
277C2×#398—172no—
278C2×#397—172no—
279C2×#389—171no—
280C2×#389D1711yescharacteristic, complemented
281C2×#381—172no—
282C2×#380—171no—
283C2×#380—171no—
284C2×#380—171no—
285C22×C17C2×#12114yes—
286C22×C17C2×#12314yes—
287C22×C17#53614yes—
288C22×C17#54714yes—
289C22×C17#54714yes—
290C22×C17#55214yes—
291C22×C17#55214yes—
292C22×C17C23×D411yescharacteristic
293C22×C17C23×D411yescharacteristic
294C22×C17C22×#4111yescharacteristic
295C22×C17C22×#4111yescharacteristic
296C22×C17C22×#4111yescharacteristic
297C22×C17C2×2−1+411yescharacteristic
298C22×C17C2×2−1+411yescharacteristic
299C2×D17—24no—
300C2×D17—24no—
301C2×D17—24no—
302C2×D17—24no—
303C2×D17—24no—
304C2×D17—24no—
305C2×D17—232no—
306C2×D17—24no—
307C2×D17—24no—
308C2×D17—24no—
309C2×D17—42no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—42no—
313C2×D17—416no—
314C2×D17C2×#12118yes—
315C4×C17—24no—
316C4×C17—44no—
317C4×C17—42no—
318C4×C17—44no—
319C4×C17C2×#12318yescomplemented
320Dic17—24no—
321Dic17—44no—
322Dic17—42no—
323Dic17—44no—
324C23×D4—171no—
325C24×C4—171no—
326C24×C4—172no—
327C24×C4—171no—
328C2×#123—1716no—
329C2×#123—3416no—
330C2×C4×D4—178no—
331C2×C4×D4—348no—
332C22×C4⋊C4—172no—
333C22×C4⋊C4—172no—
334C22×C4⋊C4—171no—
335C22×C4⋊C4—171no—
336C22×C4⋊C4—171no—
337C22×C4⋊C4—178no—
338C22×C4⋊C4—348no—
339C22×C4⋊C4C2×D1712yes—
340C22×C4⋊C4C2×D1711yescharacteristic
341C22×C4⋊C4C2×D1711yescharacteristic
342C22×C22⋊C4—171no—
343C22×C22⋊C4—172no—
344C22×C22⋊C4—172no—
345C22×C22⋊C4—171no—
346C22×C22⋊C4—172no—
347C22×C42C2×D1711yescharacteristic
348#398—1716no—
349#397—1716no—
350#389—178no—
351#389C2×D1718yescomplemented
352#381—1716no—
353#380—178no—
354#380—178no—
355#380—178no—
356C2×#95—172no—
357C2×#95—172no—
358C2×#95—172no—
359C2×#95—172no—
360C2×#95C2×D1712yes—
361C2×C17#248618yescomplemented
362C2×C17C22×#12111yescharacteristic
363C2×C17C22×#12311yescharacteristic
364C2×C17C2×#53611yescharacteristic
365C2×C17C2×#54711yescharacteristic
366C2×C17C2×#54711yescharacteristic
367C2×C17C2×#55211yescharacteristic
368C2×C17C2×#55211yescharacteristic
369D17—28no—
370D17—44no—
371C25—171no—
372C25—171no—
373C22×D4—178no—
374C22×D4—174no—
375C22×D4—344no—
376C22×D4—344no—
377C23×C4—171no—
378C23×C4—178no—
379C23×C4—174no—
380C23×C4—172no—
381C23×C4—172no—
382C23×C4—171no—
383C23×C4—178no—
384C23×C4—174no—
385C23×C4—348no—
386C23×C4—348no—
387C23×C4—348no—
388C23×C4—344no—
389C23×C4—344no—
390C23×C4—344no—
391C23×C4—344no—
392C23×C4—344no—
393C23×C4C22×D1712yes—
394C23×C4C22×D1711yescharacteristic
395C23×C4C22×D1712yes—
396C23×C4C22×D1711yescharacteristic
397C23×C4C22×D1711yescharacteristic
398#123—3464no—
399C4×D4—3432no—
400C2×C4⋊C4—24no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—28no—
403C2×C4⋊C4—174no—
404C2×C4⋊C4—178no—
405C2×C4⋊C4—174no—
406C2×C4⋊C4—178no—
407C2×C4⋊C4—174no—
408C2×C4⋊C4—3432no—
409C2×C4⋊C4—348no—
410C2×C4⋊C4—344no—
411C2×C4⋊C4—344no—
412C2×C4⋊C4—348no—
413C2×C4⋊C4—344no—
414C2×C4⋊C4C22×D1714yes—
415C2×C4⋊C4C22×D1718yes—
416C2×C4⋊C4C22×D1714yes—
417C2×C22⋊C4—174no—
418C2×C22⋊C4—174no—
419C2×C22⋊C4—178no—
420C2×C22⋊C4—178no—
421C2×C22⋊C4—178no—
422C2×C22⋊C4—344no—
423C2×C22⋊C4—344no—
424C2×C22⋊C4—348no—
425C2×C22⋊C4—348no—
426C2×C22⋊C4—348no—
427C2×C42—24no—
428C2×C42C22×D1714yes—
429#95—178no—
430#95—178no—
431#95—178no—
432#95—178no—
433#95C22×D1718yes—
434C17C2×#248611yescharacteristic, complemented, Sylow
435C24—174no—
436C24—174no—
437C24—174no—
438C24—172no—
439C24—174no—
440C24—172no—
441C24—342no—
442C24—342no—
443C24—342no—
444C24—342no—
445C24—342no—
446C24—342no—
447C24—342no—
448C24—342no—
449C24—342no—
450C24—342no—
451C24C23×D1711yescharacteristic
452C2×D4—1716no—
453C2×D4—348no—
454C2×D4—3432no—
455C2×D4—348no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—174no—
468C22×C4—172no—
469C22×C4—1716no—
470C22×C4—172no—
471C22×C4—174no—
472C22×C4—174no—
473C22×C4—3416no—
474C22×C4—3416no—
475C22×C4—3416no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—348no—
479C22×C4—348no—
480C22×C4—3432no—
481C22×C4—342no—
482C22×C4—344no—
483C22×C4—344no—
484C22×C4—342no—
485C22×C4—344no—
486C22×C4—344no—
487C22×C4—342no—
488C22×C4—344no—
489C22×C4—344no—
490C22×C4—348no—
491C22×C4—348no—
492C22×C4—348no—
493C22×C4—6832no—
494C22×C4—6816no—
495C22×C4#6325214yes—
496C22×C4D4×D1714yes—
497C22×C4D4×D1714yes—
498C22×C4#6325614yes—
499C22×C4C23×D1712yes—
500C22×C4C23×D1714yes—
501C22×C4C23×D1714yes—
502C22×C4C23×D1712yes—
503C22×C4C23×D1712yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—3416no—
508C4⋊C4—348no—
509C4⋊C4—348no—
510C4⋊C4—3416no—
511C4⋊C4—348no—
512C22⋊C4—348no—
513C22⋊C4—348no—
514C22⋊C4—3416no—
515C22⋊C4—3416no—
516C22⋊C4—3416no—
517C42—28no—
518C23—1716no—
519C23—174no—
520C23—174no—
521C23—174no—
522C23—174no—
523C23—348no—
524C23—348no—
525C23—348no—
526C23—348no—
527C23—348no—
528C23—342no—
529C23—342no—
530C23—348no—
531C23—342no—
532C23—342no—
533C23—348no—
534C23—348no—
535C23—348no—
536C23—342no—
537C23—342no—
538C23—342no—
539C23—342no—
540C23—342no—
541C23—342no—
542C23—684no—
543C23—684no—
544C23—684no—
545C23—684no—
546C23C2×#6325212yes—
547C23C2×D4×D1712yes—
548C23C2×D4×D1712yes—
549C23C2×#6325612yes—
550C23#1058769412yes—
551C23#41×D1712yes—
552C23#1058769712yes—
553C23C24×D1711yescharacteristic
554D4—3432no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—174no—
571C2×C4—348no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—3432no—
575C2×C4—344no—
576C2×C4—344no—
577C2×C4—342no—
578C2×C4—344no—
579C2×C4—344no—
580C2×C4—342no—
581C2×C4—344no—
582C2×C4—342no—
583C2×C4—344no—
584C2×C4—6832no—
585C2×C4—688no—
586C2×C4—688no—
587C2×C4—684no—
588C2×C4—6816no—
589C2×C4#10587580116yescomplemented
590C2×C4C2×#6325214yes—
591C2×C4C2×D4×D1714yes—
592C2×C4C2×D4×D1714yes—
593C2×C4C2×#6325614yes—
594C22—178no—
595C22—344no—
596C22—344no—
597C22—344no—
598C22—344no—
599C22—344no—
600C22—344no—
601C22—3432no—
602C22—344no—
603C22—344no—
604C22—344no—
605C22—682no—
606C22—682no—
607C22—682no—
608C22—682no—
609C22—6816no—
610C22#121×D1714yes—
611C22C2×#1058758014yes—
612C22#1176338214yes—
613C22#1176346214yes—
614C22#1176346414yes—
615C22#1176349714yes—
616C22#1176350014yes—
617C22C22×D4×D1711yescharacteristic
618C22C22×D4×D1711yescharacteristic
619C22C2×#41×D1711yescharacteristic
620C22C22×#6325211yescharacteristic
621C22C22×#6325611yescharacteristic
622C22C2×#1058769711yescharacteristic
623C22C2×#1058769411yescharacteristic
624C4—24no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—344no—
629C4—684no—
630C4—684no—
631C4—682no—
632C4C2×#1058758018yescomplemented
633C2—348no—
634C2—684no—
635C2#1453309018yesdirect product factor
636C2C2×#121×D1711yescharacteristic
637C2C22×#1058758011yescharacteristic
638C2C2×#1176338211yescharacteristic
639C2C2×#1176346211yescharacteristic
640C2C2×#1176346411yescharacteristic
641C2C2×#1176349711yescharacteristic
642C2C2×#1176350011yescharacteristic
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC2×C17×#389C2
socleC24×C17C24
prime radicalC23×D68C22
hypercenterC2×#389D17
hypocenterC17C2×#2486
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊C4×D17C2semidirect
4. #14533090C2direct (N × H)
5. C2×#11762742C2semidirect
6. C2×#11762738C2semidirect
7. C2×#11762707C2semidirect
8. C2×#11762671C2semidirect
9. C2×#11762666C2semidirect
10. C2×#11762666C2semidirect
11. C2×#11762666C2semidirect
12. C2×C17×#389C2semidirect
27. C2×C4⋊C4×D17C22semidirect
33. #11762742C22semidirect
34. #11762738C22semidirect
35. #11762707C22semidirect
36. #11762671C22semidirect
37. #11762666C22semidirect
38. #11762666C22semidirect
39. #11762666C22semidirect
48. C17×#389C22semidirect
248. C2×C4×C17#123semidirect
280. C2×#389D17semidirect
319. C4×C17C2×#123semidirect
351. #389C2×D17semidirect
361. C2×C17#2486semidirect
434. C17C2×#2486semidirect
589. C2×C4#10587580semidirect
632. C4C2×#10587580semidirect
635. C2#14533090direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875802
C22×C4×D682
C22×C4⋊C4×D172
#145330902
C2×#117627422
C2×#117627382
C2×#117627072
C2×#117626712
C2×#117626662
C2×C17×#3892
C2×#248617