Finite Group Databasebeta

← C2×#14531728 #17193668

Subgroups — C2×#14531728 #17193668

Order 4352.

3678 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D17C211yesmaximal, characteristic
2C22×#10587612C211yesmaximal, characteristic, complemented
3C22×#10587578C211yesmaximal, characteristic, complemented
4#14531728C2116yesmaximal, direct product factor
5C2×#11762709C212yesmaximal
6C2×#11762703C212yesmaximal, complemented
7C2×#11762698C212yesmaximal, complemented
8C2×#11762678C211yesmaximal, characteristic, complemented
9C2×#11762666C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×Dic17C211yesmaximal, characteristic, complemented
11C2×#11762660C212yesmaximal, complemented
12C2×C17×#380C211yesmaximal, characteristic, complemented
13C23×C22⋊D17C2211yescharacteristic
14C23×C4×D17C2211yescharacteristic
15C2×C4×C22⋊D17C4116yescomplemented
16C2×C4×C22⋊D17C2218yes—
17C2×#10587612—28no—
18C2×#10587612C2218yescomplemented
19C2×#10587578—28no—
20C2×#10587578C2218yescomplemented
21C22×C4⋊Dic17C2211yescharacteristic
22C22×C4⋊Dic17C2211yescharacteristic
23C22×#63225C2212yes—
24C22×#63225C2212yes—
25C22×#63225C2211yescharacteristic
26C22×C22⋊Dic17C2212yes—
27C22×C22⋊Dic17C2211yescharacteristic
28C22×#63227C2212yes—
29C22×#63227C2212yes—
30C22×#63227C2211yescharacteristic
31C22×C4×Dic17C2212yes—
32C22×C4×Dic17C2212yes—
33C22×C4×Dic17C2212yes—
34C22×C4×Dic17C2211yescharacteristic
35#11762709C22116yes—
36#11762703C22116yescomplemented
37#11762698C22116yescomplemented
38#11762678C2218yescomplemented
39#11762666C2218yescomplemented
40C4⋊C4×Dic17C2218yescomplemented
41#11762660C22116yescomplemented
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×C17C2×C418yes—
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⋊D17C2×C418yes—
63C22×C22⋊D17C2314yes—
64C23×Dic17C2312yes—
65C23×Dic17C2311yescharacteristic
66C23×Dic17C2311yescharacteristic
67C23×Dic17C2312yes—
68C23×Dic17C2311yescharacteristic
69C22×C4×D17—24no—
70C22×C4×D17—24no—
71C22×C4×D17C2×C418yes—
72C22×C4×D17C2314yes—
73C2×C4⋊C4×C17—24no—
74C2×C4⋊C4×C17—24no—
75C2×C4⋊C4×C17C2314yes—
76C2×C4⋊C4×C17C2314yes—
77C2×C22⋊C4×C17—44no—
78C2×C22⋊C4×C17C2318yes—
79C2×C22⋊Dic17—28no—
80C2×C22⋊Dic17C2×C418yes—
81C2×C22⋊Dic17C2318yes—
82C2×C22⋊Dic17C2314yes—
83C4×C22⋊D17C2×C4164yescomplemented
84C2×#63227—28no—
85C2×#63227—44no—
86C2×#63227C2×C418yes—
87C2×#63227C2318yes—
88C2×#63227C2318yes—
89C2×#63227C2314yes—
90#10587612—232no—
91C2×C4⋊Dic17—24no—
92C2×C4⋊Dic17D418yescomplemented
93C2×C4⋊Dic17C2314yes—
94C2×C4⋊Dic17C2314yes—
95C2×#63225—28no—
96C2×#63225—44no—
97C2×#63225C2×C418yes—
98C2×#63225C2318yes—
99C2×#63225C2318yes—
100C2×#63225C2314yes—
101C2×C4×Dic17—28no—
102C2×C4×Dic17—28no—
103C2×C4×Dic17—28no—
104C2×C4×Dic17C2×C418yes—
105C2×C4×Dic17C2318yes—
106C2×C4×Dic17C2318yes—
107C2×C4×Dic17C2318yes—
108C2×C4×Dic17C2314yes—
109#10587578—232no—
110C17×#95C2318yes—
111#10587502C2318yes—
112#10587502C2318yes—
113C24×C17—22no—
114C24×C17—22no—
115C24×C17—22no—
116C24×C17—22no—
117C24×C17—22no—
118C24×C17—22no—
119C24×C17C22×C414yes—
120C24×C17C2412yes—
121C24×C17C2411yescharacteristic
122C23×D17—22no—
123C23×D17—22no—
124C23×D17—22no—
125C23×D17—22no—
126C23×D17—22no—
127C23×D17—22no—
128C23×D17C22×C414yes—
129C23×D17C2412yes—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—232no—
133C22×C4×C17—28no—
134C22×C4×C17—28no—
135C22×C4×C17—24no—
136C22×C4×C17—24no—
137C22×C4×C17—24no—
138C22×C4×C17—24no—
139C22×C4×C17—24no—
140C22×C4×C17—24no—
141C22×C4×C17C22×C414yes—
142C22×C4×C17C22×C4116yes—
143C22×C4×C17C22×C414yes—
144C22×C4×C17C2×D414yes—
145C22×C4×C17#4114yes—
146C22×C4×C17C2414yes—
147C22×C4×C17C2414yes—
148C22×C4×C17C2412yes—
149C22×C4×C17C2412yes—
150C2×C22⋊D17—232no—
151C2×C22⋊D17—28no—
152C2×C22⋊D17—28no—
153C2×C22⋊D17C22×C4116yes—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—24no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—24no—
160C22×Dic17—24no—
161C22×Dic17—24no—
162C22×Dic17—24no—
163C22×Dic17C22×C414yes—
164C22×Dic17C22×C414yes—
165C22×Dic17C22×C414yes—
166C22×Dic17C2×D418yes—
167C22×Dic17#4114yes—
168C22×Dic17#4114yes—
169C22×Dic17C2414yes—
170C22×Dic17C2412yes—
171C22×Dic17C2412yes—
172C22×Dic17C2414yes—
173C22×Dic17C2412yes—
174C2×C4×D17—232no—
175C2×C4×D17—28no—
176C2×C4×D17—28no—
177C2×C4×D17C22×C4116yes—
178C4⋊C4×C17—28no—
179C4⋊C4×C17—28no—
180C22⋊C4×C17—48no—
181C22⋊Dic17—216no—
182C22⋊Dic17C22×C4116yes—
183#63227—216no—
184#63227—48no—
185#63227C22×C4116yes—
186C4⋊Dic17—28no—
187C4⋊Dic17C2×D4116yescomplemented
188#63225—216no—
189#63225—48no—
190#63225C22×C4116yes—
191C4×Dic17—216no—
192C4×Dic17—216no—
193C4×Dic17—216no—
194C4×Dic17C22×C4116yes—
195C2×#2354—171nomaximal, Sylow
196C23×C17—28no—
197C23×C17—28no—
198C23×C17—28no—
199C23×C17—22no—
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—44no—
206C23×C17—44no—
207C23×C17—44no—
208C23×C17—44no—
209C23×C17C23×C414yes—
210C23×C17C23×C412yes—
211C23×C17C22×D412yes—
212C23×C17C2×#4112yes—
213C23×C17C2×#4112yes—
214C23×C17C2×#4112yes—
215C23×C17H(5,𝔽2)12yes—
216C23×C172−1+412yes—
217C23×C17C2511yescharacteristic
218C22×D17—28no—
219C22×D17—28no—
220C22×D17—28no—
221C22×D17—22no—
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—22no—
225C22×D17—22no—
226C22×D17—22no—
227C22×D17—44no—
228C22×D17—44no—
229C22×D17—44no—
230C22×D17—44no—
231C22×D17C23×C414yes—
232C2×C4×C17—28no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—232no—
236C2×C4×C17—24no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—24no—
242C2×C4×C17—48no—
243C2×C4×C17—48no—
244C2×C4×C17C4×D4116yes—
245C2×C4×C17C23×C414yes—
246C2×C4×C17C23×C414yes—
247C2×C4×C17C22×D414yes—
248C2×C4×C17C2×#4114yes—
249C22⋊D17—232no—
250C2×Dic17—28no—
251C2×Dic17—28no—
252C2×Dic17—216no—
253C2×Dic17—22no—
254C2×Dic17—22no—
255C2×Dic17—24no—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—48no—
263C2×Dic17#117116yescomplemented
264C2×Dic17C23×C414yes—
265C2×Dic17C23×C414yes—
266C2×Dic17C23×C414yes—
267C2×Dic17C22×D418yes—
268C2×Dic17C2×#4114yes—
269C2×Dic17C2×#4114yes—
270C4×D17—232no—
271C22×C4×D4—171no—
272C22×#117—171no—
273C22×#117—171no—
274#2354—1716no—
275C2×#390—172no—
276C2×#388—172no—
277C2×#388—172no—
278C2×#384—171no—
279C2×#380—171no—
280C2×#380D1711yescharacteristic, complemented
281C2×C4×C4⋊C4—172no—
282C2×C4×C4⋊C4—171no—
283C22×C17—24no—
284C22×C17—24no—
285C22×C17—24no—
286C22×C17—416no—
287C22×C17—42no—
288C22×C17—42no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17C2×#11714yes—
292C22×C17C2×C4×D414yes—
293C22×C17#52014yes—
294C22×C17#54714yes—
295C22×C17#55014yes—
296C22×C17#55514yes—
297C22×C17#55614yes—
298C22×C17C24×C411yescharacteristic
299C22×C17C23×D411yescharacteristic
300C22×C17C22×#4111yescharacteristic
301C22×C17C22×#4111yescharacteristic
302C22×C17C22×#4111yescharacteristic
303C22×C17C2×H(5,𝔽2)11yescharacteristic
304C22×C17C2×2−1+411yescharacteristic
305C2×D17—24no—
306C2×D17—24no—
307C2×D17—24no—
308C2×D17—416no—
309C2×D17—42no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—42no—
313C4×C17—24no—
314C4×C17—44no—
315C4×C17—44no—
316C4×C17C2×C4×D418yes—
317Dic17—24no—
318Dic17—24no—
319Dic17—28no—
320Dic17—44no—
321Dic17C2×#11718yescomplemented
322C23×D4—171no—
323C24×C4—171no—
324C24×C4C2×D1711yescharacteristic
325C2×C4×D4—1716no—
326C2×C4×D4—178no—
327C2×#117—178no—
328C2×#117—178no—
329C2×#117—348no—
330C2×#117—348no—
331C22×C4⋊C4—171no—
332C22×C4⋊C4—171no—
333C22×C4⋊C4—172no—
334C22×C4⋊C4—171no—
335C22×C4⋊C4—172no—
336C22×C4⋊C4C2×D1711yescharacteristic
337C22×C4⋊C4C2×D1711yescharacteristic
338C22×C22⋊C4—171no—
339C22×C22⋊C4—171no—
340C22×C22⋊C4—172no—
341C22×C22⋊C4—172no—
342C22×C22⋊C4—172no—
343C22×C22⋊C4C2×D1712yes—
344C22×C42—171no—
345C22×C42—172no—
346C22×C42—172no—
347C22×C42—172no—
348#390—1716no—
349#388—1716no—
350#388—1716no—
351#384—178no—
352#380—178no—
353#380C2×D1718yescomplemented
354C4×C4⋊C4—178no—
355C4×C4⋊C4—1716no—
356C2×#95—172no—
357C2×#95—172no—
358C2×#95C2×D1712yes—
359C2×C17—44no—
360C2×C17#235418yescomplemented
361C2×C17C22×#11711yescharacteristic
362C2×C17C22×C4×D411yescharacteristic
363C2×C17C2×#52011yescharacteristic
364C2×C17C2×#54711yescharacteristic
365C2×C17C2×#55011yescharacteristic
366C2×C17C2×#55511yescharacteristic
367C2×C17C2×#55611yescharacteristic
368D17—44no—
369C25—171no—
370C25C22×D1711yescharacteristic
371C22×D4—178no—
372C22×D4—174no—
373C22×D4—344no—
374C22×D4—344no—
375C23×C4—24no—
376C23×C4—24no—
377C23×C4—171no—
378C23×C4—171no—
379C23×C4—172no—
380C23×C4—171no—
381C23×C4—178no—
382C23×C4—174no—
383C23×C4—172no—
384C23×C4—344no—
385C23×C4—344no—
386C23×C4C4×D1718yes—
387C23×C4C22×D1712yes—
388C23×C4C22×D1712yes—
389C23×C4C22×D1714yes—
390C23×C4C22×D1711yescharacteristic
391C23×C4C22×D1711yescharacteristic
392C4×D4—1764no—
393#117—3432no—
394#117—3432no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—178no—
398C2×C4⋊C4—178no—
399C2×C4⋊C4—174no—
400C2×C4⋊C4—174no—
401C2×C4⋊C4—174no—
402C2×C4⋊C4—178no—
403C2×C4⋊C4—178no—
404C2×C4⋊C4—348no—
405C2×C4⋊C4—344no—
406C2×C4⋊C4—684no—
407C2×C4⋊C4C22×D1714yes—
408C2×C4⋊C4C22×D1714yes—
409C2×C22⋊C4—44no—
410C2×C22⋊C4—178no—
411C2×C22⋊C4—174no—
412C2×C22⋊C4—178no—
413C2×C22⋊C4—174no—
414C2×C22⋊C4—178no—
415C2×C22⋊C4—178no—
416C2×C22⋊C4—178no—
417C2×C22⋊C4—348no—
418C2×C22⋊C4—348no—
419C2×C22⋊C4—684no—
420C2×C22⋊C4C22×D1718yes—
421C2×C42—178no—
422C2×C42—174no—
423C2×C42—178no—
424C2×C42—178no—
425C2×C42—178no—
426C2×C42—348no—
427C2×C42—348no—
428C2×C42—348no—
429#95—178no—
430#95—178no—
431#95C22×D1718yes—
432C17C2×#235411yescharacteristic, complemented, Sylow
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—174no—
440C24—172no—
441C24—342no—
442C24—342no—
443C24—342no—
444C24—342no—
445C24—342no—
446C24—342no—
447C24C2×C4×D1714yes—
448C24C23×D1712yes—
449C24C23×D1711yescharacteristic
450C2×D4—1716no—
451C2×D4—3432no—
452C2×D4—348no—
453C2×D4—348no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—232no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—174no—
466C22×C4—174no—
467C22×C4—174no—
468C22×C4—172no—
469C22×C4—174no—
470C22×C4—174no—
471C22×C4—178no—
472C22×C4—172no—
473C22×C4—172no—
474C22×C4—174no—
475C22×C4—1716no—
476C22×C4—174no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—344no—
481C22×C4—342no—
482C22×C4—344no—
483C22×C4—348no—
484C22×C4—348no—
485C22×C4—3432no—
486C22×C4—344no—
487C22×C4—344no—
488C22×C4—344no—
489C22×C4C2×C4×D1714yes—
490C22×C4C2×C4×D17116yes—
491C22×C4C2×C4×D1714yes—
492C22×C4D4×D1714yes—
493C22×C4#6325414yes—
494C22×C4C23×D1714yes—
495C22×C4C23×D1714yes—
496C22×C4C23×D1712yes—
497C22×C4C23×D1712yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—1716no—
501C4⋊C4—1716no—
502C4⋊C4—3416no—
503C4⋊C4—348no—
504C4⋊C4—688no—
505C22⋊C4—48no—
506C22⋊C4—1716no—
507C22⋊C4—1716no—
508C22⋊C4—3416no—
509C22⋊C4—3416no—
510C22⋊C4—688no—
511C42—1716no—
512C42—3416no—
513C42—3416no—
514C42—3416no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—174no—
529C23—348no—
530C23—348no—
531C23—348no—
532C23—342no—
533C23—342no—
534C23—342no—
535C23—342no—
536C23—342no—
537C23—342no—
538C23—684no—
539C23—684no—
540C23—684no—
541C23—684no—
542C23C22×C4×D1714yes—
543C23C22×C4×D1712yes—
544C23C2×D4×D1712yes—
545C23C2×#6325412yes—
546C23#1058769112yes—
547C23#1058769412yes—
548C23#41×D1712yes—
549C23#41×D1712yes—
550C23C24×D1711yescharacteristic
551D4—3432no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—232no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—1716no—
565C2×C4—174no—
566C2×C4—174no—
567C2×C4—174no—
568C2×C4—174no—
569C2×C4—174no—
570C2×C4—178no—
571C2×C4—3416no—
572C2×C4—348no—
573C2×C4—348no—
574C2×C4—342no—
575C2×C4—344no—
576C2×C4—342no—
577C2×C4—342no—
578C2×C4—342no—
579C2×C4—344no—
580C2×C4—3432no—
581C2×C4—344no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—688no—
585C2×C4#10587567116yes—
586C2×C4C22×C4×D1714yes—
587C2×C4C22×C4×D1714yes—
588C2×C4C2×D4×D1714yes—
589C2×C4C2×#6325414yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—416no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—344no—
599C22—344no—
600C22—344no—
601C22—6816no—
602C22—682no—
603C22—682no—
604C22—682no—
605C22—682no—
606C22#117×D1714yes—
607C22C2×#1058756714yes—
608C22#1176329014yes—
609C22#1176346114yes—
610C22#1176347714yes—
611C22#1176352314yes—
612C22#1176353014yes—
613C22C23×C4×D1711yescharacteristic
614C22C22×D4×D1711yescharacteristic
615C22C2×#41×D1711yescharacteristic
616C22C2×#41×D1711yescharacteristic
617C22C22×#6325411yescharacteristic
618C22C2×#1058769111yescharacteristic
619C22C2×#1058769411yescharacteristic
620C4—24no—
621C4—44no—
622C4—44no—
623C4—178no—
624C4—344no—
625C4—344no—
626C4—348no—
627C4—684no—
628C4C2×#1058756718yes—
629C2—44no—
630C2—684no—
631C2#1453172818yesdirect product factor
632C2C2×#117×D1711yescharacteristic
633C2C22×#1058756711yescharacteristic
634C2C2×#1176329011yescharacteristic
635C2C2×#1176346111yescharacteristic
636C2C2×#1176347711yescharacteristic
637C2C2×#1176352311yescharacteristic
638C2C2×#1176353011yescharacteristic
subgroupG/H
centerC24C23×D17
commutatorC22×C17C24×C4
FrattiniC23C24×D17
FittingC2×C17×#380C2
socleC24×C17C24
prime radicalC23×C22⋊D17C22
cosocleC23×C17C25
hypercenterC2×#380D17
hypocenterC17C2×#2354
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#10587612C2semidirect
3. C22×#10587578C2semidirect
4. #14531728C2direct (N × H)
6. C2×#11762703C2semidirect
7. C2×#11762698C2semidirect
8. C2×#11762678C2semidirect
9. C2×#11762666C2semidirect
10. C2×C4⋊C4×Dic17C2semidirect
11. C2×#11762660C2semidirect
12. C2×C17×#380C2semidirect
15. C2×C4×C22⋊D17C4semidirect
18. C2×#10587612C22semidirect
20. C2×#10587578C22semidirect
36. #11762703C22semidirect
37. #11762698C22semidirect
38. #11762678C22semidirect
39. #11762666C22semidirect
40. C4⋊C4×Dic17C22semidirect
41. #11762660C22semidirect
48. C17×#380C22semidirect
83. C4×C22⋊D17C2×C4semidirect
92. C2×C4⋊Dic17D4semidirect
187. C4⋊Dic17C2×D4semidirect
263. C2×Dic17#117semidirect
280. C2×#380D17semidirect
321. Dic17C2×#117semidirect
353. #380C2×D17semidirect
360. C2×C17#2354semidirect
432. C17C2×#2354semidirect
631. C2#14531728direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D172
C22×#105876122
C22×#105875782
#145317282
C2×#117627092
C2×#117627032
C2×#117626982
C2×#117626782
C2×#117626662
C2×C4⋊C4×Dic172
C2×#117626602
C2×C17×#3802
C2×#235417