Finite Group Databasebeta

← C2×#14529984 #17061793

Subgroups — C2×#14529984 #17061793

Order 4352.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587478C211yesmaximal, characteristic, complemented
2C22×#10587506C211yesmaximal, characteristic, complemented
3C22×#10587506C211yesmaximal, characteristic, complemented
4C22×C22⋊#3622C211yesmaximal, characteristic
5C2×#11762669C211yesmaximal, characteristic
6#14529984C218yesmaximal, direct product factor
7C2×#11762571C211yesmaximal, characteristic, complemented
8C22×C17×C22⋊Q8C211yesmaximal, characteristic
9C23×#3622C2211yescharacteristic
10C23×#3622C2211yescharacteristic
11C22×C4⋊Dic17C2211yescharacteristic
12C22×C4⋊Dic17C2211yescharacteristic
13C22×C22⋊Dic17—21no—
14C2×#10587478—24no—
15C2×#10587478C2214yescomplemented
16C2×#10587506—24no—
17C2×#10587506—24no—
18C2×#10587506C2214yescomplemented
19C2×#10587506C2214yescomplemented
20C2×C22⋊#3622—24no—
21C2×C22⋊#3622C2214yes—
22#11762669—22no—
23#11762669C2214yes—
24C2×#10587502—21no—
25#11762571C2214yescomplemented
26C23×Q8×C17C2211yescharacteristic
27C24×C4×C17C2211yescharacteristic
28C2×C17×C22⋊Q8—24no—
29C2×C17×C22⋊Q8—24no—
30C2×C17×C22⋊Q8—22no—
31C2×C17×C22⋊Q8C2214yes—
32C22×C4⋊C4×C17—21no—
33C22×C4⋊C4×C17C2211yescharacteristic
34C22×C22⋊C4×C17—21no—
35C25×C17D411yescharacteristic
36C22×Q8×C17—22no—
37C22×Q8×C17—21no—
38C22×Q8×C17—22no—
39C22×Q8×C17—22no—
40C22×Q8×C17—22no—
41C22×Q8×C17—22no—
42C22×Q8×C17D414yes—
43C22×Q8×C17C2312yes—
44C23×C4×C17—21no—
45C23×C4×C17—21no—
46C23×C4×C17—22no—
47C23×C4×C17—22no—
48C23×C4×C17—22no—
49C23×C4×C17—22no—
50C23×C4×C17—22no—
51C23×C4×C17—22no—
52C23×C4×C17D411yescharacteristic
53C23×C4×C17D412yes—
54C23×C4×C17C2312yes—
55C23×C4×C17C2311yescharacteristic
56C23×Dic17—21no—
57C23×Dic17—21no—
58C17×C22⋊Q8—216no—
59C17×C22⋊Q8—28no—
60C17×C22⋊Q8—28no—
61C2×C4⋊C4×C17—28no—
62C2×C4⋊C4×C17—24no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17D414yes—
66C2×C4⋊C4×C17C2312yes—
67C2×C22⋊C4×C17—28no—
68C2×C22⋊C4×C17—24no—
69C2×C22⋊Dic17—24no—
70C2×C22⋊Dic17—44no—
71C2×C4⋊Dic17—21no—
72C2×C4⋊Dic17—21no—
73C2×C4⋊Dic17—22no—
74C2×C4⋊Dic17—22no—
75C2×C4⋊Dic17—22no—
76C2×C4⋊Dic17—22no—
77C2×C4⋊Dic17C2312yes—
78C2×C4⋊Dic17C2312yes—
79C22×#3622—22no—
80C22×#3622—22no—
81C22×#3622—22no—
82C22×#3622—22no—
83C22×#3622—22no—
84C22×#3622D414yes—
85C22×#3622C2312yes—
86C22×#3622C2312yes—
87#10587506—28no—
88#10587506—28no—
89#10587502—24no—
90C22⋊#3622—28no—
91#10587478—28no—
92C24×C17—22no—
93C24×C17—22no—
94C24×C17—22no—
95C24×C17—22no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17Q1614yes—
99C24×C17C2×D412yes—
100C24×C17C2×D411yescharacteristic
101C2×Q8×C17—24no—
102C2×Q8×C17—22no—
103C2×Q8×C17—28no—
104C2×Q8×C17—28no—
105C2×Q8×C17—22no—
106C2×Q8×C17—22no—
107C2×Q8×C17—22no—
108C2×Q8×C17—22no—
109C2×Q8×C17—44no—
110C2×Q8×C17—42no—
111C2×Q8×C17—42no—
112C2×Q8×C17—44no—
113C2×Q8×C17C2×D414yes—
114C22×C4×C17—22no—
115C22×C4×C17—24no—
116C22×C4×C17—24no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—24no—
122C22×C4×C17—24no—
123C22×C4×C17—24no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—42no—
131C22×C4×C17—42no—
132C22×C4×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17—44no—
135C22×C4×C17—44no—
136C22×C4×C17—44no—
137C22×C4×C17—44no—
138C22×C4×C17C2×D412yes—
139C22×C4×C17C2×D412yes—
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D412yes—
142C22×C4×C17C2×D412yes—
143C22×C4×C17#4112yes—
144C22×C4×C17#4112yes—
145C22×C4×C17#4112yes—
146C22×C4×C17C2411yescharacteristic
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—42no—
150C22×Dic17—42no—
151C22×Dic17—42no—
152C22×Dic17—42no—
153C22×Dic17—42no—
154C22×Dic17—42no—
155C4⋊C4×C17—216no—
156C4⋊C4×C17—24no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17C2×D414yes—
159C22⋊C4×C17—216no—
160C22⋊Dic17—48no—
161C4⋊Dic17—22no—
162C4⋊Dic17—22no—
163C4⋊Dic17—22no—
164C4⋊Dic17—22no—
165C4⋊Dic17—42no—
166C4⋊Dic17—42no—
167C2×#3622—28no—
168C2×#3622—22no—
169C2×#3622—22no—
170C2×#3622—22no—
171C2×#3622—22no—
172C2×#3622—22no—
173C2×#3622—44no—
174C2×#3622C2×D414yes—
175C2×#2036—171nomaximal, Sylow
176C23×C17—22no—
177C23×C17—22no—
178C23×C17—22no—
179C23×C17—22no—
180C23×C17—22no—
181C23×C17—22no—
182C23×C17—28no—
183C23×C17—28no—
184C23×C17—28no—
185C23×C17—44no—
186C23×C17—44no—
187C23×C17—44no—
188C23×C17—44no—
189C23×C17#12012yes—
190C23×C17#12112yes—
191C23×C17#12312yes—
192C23×C17C2×Q1614yes—
193C23×C17C2×Q1612yes—
194C23×C17#13512yes—
195C23×C17AGL(1,ℤ/8)12yes—
196C23×C17#13712yes—
197C23×C17C22×D411yescharacteristic
198Q8×C17—24no—
199Q8×C17—24no—
200Q8×C17—42no—
201Q8×C17—42no—
202Q8×C17—48no—
203C2×C4×C17—24no—
204C2×C4×C17—24no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—42no—
209C2×C4×C17—42no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—44no—
213C2×C4×C17—48no—
214C2×C4×C17—48no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—48no—
220C2×C4×C17#12012yes—
221C2×C4×C17#12114yes—
222C2×C4×C17#12114yes—
223C2×C4×C17#12112yes—
224C2×C4×C17#12314yes—
225C2×C4×C17#12312yes—
226C2×C4×C17#12414yes—
227C2×C4×C17C22×D411yescharacteristic
228C2×C4×C17C22×D411yescharacteristic
229C2×C4×C17C22×D411yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×Dic17—42no—
234C2×Dic17—42no—
235C2×Dic17—42no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—42no—
239C2×Dic17—84no—
240C2×Dic17—84no—
241#3622—24no—
242#3622—42no—
243C22×C22⋊Q8D1711yescharacteristic
244C22×#107—171no—
245C22×#103—171no—
246C22×#103—171no—
247C22×C22⋊C8—171no—
248C2×#380—171no—
249#2036—178no—
250C2×#340—171no—
251C22×C17—24no—
252C22×C17—24no—
253C22×C17—24no—
254C22×C17—42no—
255C22×C17—42no—
256C22×C17—42no—
257C22×C17—42no—
258C22×C17—416no—
259C22×C17#39414yes—
260C22×C17#44914yes—
261C22×C17#45114yes—
262C22×C17#46714yes—
263C22×C17#46814yes—
264C22×C17#48414yes—
265C22×C17#48514yes—
266C22×C17C2×#12011yescharacteristic
267C22×C17C2×#12111yescharacteristic
268C22×C17C2×#12311yescharacteristic
269C22×C17C22×Q1611yescharacteristic
270C22×C17C2×#13511yescharacteristic
271C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C17C2×#13711yescharacteristic
273C4×C17—42no—
274C4×C17—44no—
275C4×C17—44no—
276C4×C17#39414yes—
277C4×C17C2×#12111yescharacteristic
278C4×C17C2×#12111yescharacteristic
279C4×C17C2×#12311yescharacteristic
280C4×C17C2×#12411yescharacteristic
281Dic17—82no—
282Dic17—82no—
283C23×Q8C2×D1711yescharacteristic
284C24×C4C2×D1711yescharacteristic
285C23×C8—171no—
286C23×C8—171no—
287C2×C22⋊Q8—24no—
288C2×C22⋊Q8—22no—
289C2×C22⋊Q8—24no—
290C2×C22⋊Q8C2×D1714yes—
291C22×C4⋊C4—21no—
292C22×C4⋊C4—171no—
293C22×C4⋊C4—171no—
294C22×C4⋊C4C2×D1711yescharacteristic
295C22×C22⋊C4—21no—
296C22×C22⋊C4—341no—
297C2×#107—174no—
298C2×#107—344no—
299C2×#103—174no—
300C2×#103—174no—
301C2×#103—344no—
302C2×#103—344no—
303C2×C22⋊C8—174no—
304C2×C22⋊C8—344no—
305#380—174no—
306#380—342no—
307C2×#95—341no—
308#340—174no—
309C2×C17—44no—
310C2×C17#203618yescomplemented
311C2×C17C2×#39411yescharacteristic
312C2×C17C2×#45111yescharacteristic
313C2×C17C2×#44911yescharacteristic
314C2×C17C2×#46711yescharacteristic
315C2×C17C2×#46811yescharacteristic
316C2×C17C2×#48511yescharacteristic
317C2×C17C2×#48411yescharacteristic
318C25C22⋊D1711yescharacteristic
319C22×Q8—21no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8C22⋊D1714yes—
326C22×Q8C22×D1712yes—
327C23×C4—21no—
328C23×C4—21no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—341no—
336C23×C4—341no—
337C23×C4C22⋊D1712yes—
338C23×C4C22⋊D1711yescharacteristic
339C23×C4C22×D1712yes—
340C23×C4C22×D1711yescharacteristic
341C22×C8—174no—
342C22×C8—172no—
343C22×C8—172no—
344C22×C8—342no—
345C22×C8—342no—
346C22×C8—342no—
347C22×C8—342no—
348C22×C8—342no—
349C22⋊Q8—216no—
350C22⋊Q8—28no—
351C22⋊Q8—28no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—172no—
357C2×C4⋊C4—172no—
358C2×C4⋊C4—342no—
359C2×C4⋊C4—342no—
360C2×C4⋊C4—342no—
361C2×C4⋊C4—342no—
362C2×C4⋊C4—341no—
363C2×C4⋊C4—341no—
364C2×C4⋊C4C22⋊D1714yes—
365C2×C4⋊C4C22×D1712yes—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—344no—
369C2×C22⋊C4—684no—
370#107—348no—
371#103—348no—
372#103—348no—
373C22⋊C8—348no—
374#95—344no—
375C17C2×#203611yescharacteristic, complemented, Sylow
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24#6323114yes—
383C24C2×C22⋊D1712yes—
384C24C2×C22⋊D1711yescharacteristic
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8C2×C22⋊D1714yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—342no—
423C22×C4—342no—
424C22×C4—682no—
425C22×C4—682no—
426C22×C4—682no—
427C22×C4—682no—
428C22×C4—682no—
429C22×C4—682no—
430C22×C4D4×D1712yes—
431C22×C4#6325412yes—
432C22×C4#6325412yes—
433C22×C4#6325412yes—
434C22×C4C2×C22⋊D1712yes—
435C22×C4C2×C22⋊D1712yes—
436C22×C4C2×C22⋊D1712yes—
437C22×C4C2×C22⋊D1711yescharacteristic
438C22×C4C23×D1711yescharacteristic
439C2×C8—174no—
440C2×C8—342no—
441C2×C8—342no—
442C2×C8—348no—
443C2×C8—342no—
444C2×C8—342no—
445C2×C8—342no—
446C2×C8—684no—
447C4⋊C4—216no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—342no—
451C4⋊C4—342no—
452C4⋊C4—342no—
453C4⋊C4—342no—
454C4⋊C4—682no—
455C4⋊C4—682no—
456C4⋊C4C2×C22⋊D1714yes—
457C22⋊C4—216no—
458C22⋊C4—688no—
459C23—28no—
460C23—22no—
461C23—22no—
462C23—28no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—28no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23#1058762112yes—
473C23#1058762512yes—
474C23#1058762812yes—
475C23C2×#6323114yes—
476C23C2×#6323112yes—
477C23#1058763512yes—
478C23#1058763612yes—
479C23#1058763912yes—
480C23C22×C22⋊D1711yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—48no—
484Q8—42no—
485Q8—42no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—682no—
504C2×C4—682no—
505C2×C4—682no—
506C2×C4—682no—
507C2×C4—682no—
508C2×C4—682no—
509C2×C4—1364no—
510C2×C4—1364no—
511C2×C4#1058762112yes—
512C2×C4#1058762114yes—
513C2×C4#1058762214yes—
514C2×C4#1058762414yes—
515C2×C4#1058762512yes—
516C2×C4#1058762514yes—
517C2×C4#1058763912yes—
518C2×C4C2×D4×D1711yescharacteristic
519C2×C4C2×#6325411yescharacteristic
520C2×C4C2×#6325411yescharacteristic
521C2×C4C2×#6325411yescharacteristic
522C2×C4C22×C22⋊D1711yescharacteristic
523C2×C4C22×C22⋊D1711yescharacteristic
524C8—344no—
525C8—682no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—416no—
534C22#1176272314yes—
535C22#1176295414yes—
536C22#1176296514yes—
537C22#1176304814yes—
538C22#1176305714yes—
539C22#1176314214yes—
540C22#1176314714yes—
541C22C2×#1058763911yescharacteristic
542C22C2×#1058762511yescharacteristic
543C22C2×#1058762111yescharacteristic
544C22C22×#6323111yescharacteristic
545C22C2×#1058763611yescharacteristic
546C22C2×#1058763511yescharacteristic
547C22C2×#1058762811yescharacteristic
548C4—44no—
549C4—42no—
550C4—44no—
551C4—1362no—
552C4—1362no—
553C4#1176272314yes—
554C4C2×#1058762411yescharacteristic
555C4C2×#1058762511yescharacteristic
556C4C2×#1058762111yescharacteristic
557C4C2×#1058762211yescharacteristic
558C2—44no—
559C2#1452998418yesdirect product factor
560C2C2×#1176272311yescharacteristic
561C2C2×#1176296511yescharacteristic
562C2C2×#1176295411yescharacteristic
563C2C2×#1176304811yescharacteristic
564C2C2×#1176305711yescharacteristic
565C2C2×#1176314711yescharacteristic
566C2C2×#1176314211yescharacteristic
subgroupG/H
centerC24C2×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC22×C17×C22⋊Q8C2
socleC24×C17C2×D4
prime radicalC25×C17D4
hypercenterC22×C22⋊Q8D17
hypocenterC17C2×#2036
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#10587478C2semidirect
2. C22×#10587506C2semidirect
3. C22×#10587506C2semidirect
6. #14529984C2direct (N × H)
7. C2×#11762571C2semidirect
15. C2×#10587478C22semidirect
18. C2×#10587506C22semidirect
19. C2×#10587506C22semidirect
25. #11762571C22semidirect
310. C2×C17#2036semidirect
375. C17C2×#2036semidirect
559. C2#14529984direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105874782
C22×#105875062
C22×C22⋊#36222
C2×#117626692
#145299842
C2×#117625712
C22×C17×C22⋊Q82
C2×#203617