Finite Group Databasebeta

← C2×#14532114 #17203010

Subgroups — C2×#14532114 #17203010

Order 4352.

5718 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#63254C211yesmaximal, characteristic
2C22×#10587569C212yesmaximal, complemented
3C22×C4⋊C4×D17C211yesmaximal, characteristic, complemented
4#14532114C2116yesmaximal, direct product factor
5C2×#11762730C211yesmaximal, characteristic, complemented
6C2×#11762729C211yesmaximal, characteristic, complemented
7C2×#11762723C211yesmaximal, characteristic, complemented
8C2×#11762685C212yesmaximal, complemented
9C2×#11762684C212yesmaximal, complemented
10C2×#11762681C212yesmaximal, complemented
11C2×#11762671C211yesmaximal, characteristic, complemented
12C2×C17×#394C211yesmaximal, characteristic, complemented
13C22×#63254—216no—
14C22×#63254—28no—
15C22×#63254C2218yes—
16C23×Dic34C2211yescharacteristic
17C23×C22⋊D17C2212yes—
18C24×Dic17C2212yes—
19C23×C4×D17C2211yescharacteristic
20C23×C4×D17C2211yescharacteristic
21C23×C4×D17C2211yescharacteristic
22C2×#10587569—216no—
23C2×#10587569C22116yescomplemented
24C22×C4⋊Dic17C2211yescharacteristic
25C22×C4⋊Dic17C2212yes—
26C22×#63225C2211yescharacteristic
27C22×#63225C2211yescharacteristic
28C22×#63225C2212yes—
29C2×C4⋊C4×D17—28no—
30C2×C4⋊C4×D17C2218yescomplemented
31C22×C22⋊Dic17C2212yes—
32C22×C22⋊Dic17C2212yes—
33C22×#63227C2212yes—
34#11762730C2218yescomplemented
35#11762729C2218yescomplemented
36#11762723C2218yescomplemented
37#11762685C22116yescomplemented
38#11762684C22116yescomplemented
39#11762681C22116yescomplemented
40#11762671C2218yescomplemented
41C2×#10587502C2212yes—
42C2×#10587502C2211yescharacteristic
43C2×#10587502C2212yes—
44C2×#10587502C2212yes—
45C23×D4×C17C2211yescharacteristic
46C22×C4⋊C4×C17C2211yescharacteristic
47C22×C22⋊C4×C17C2212yes—
48C22×C22⋊C4×C17C2212yes—
49C17×#394C2218yescomplemented
50C2×C17×#95C2211yescharacteristic
51C25×C17C2312yes—
52C24×D17C2311yescharacteristic
53C22×D4×C17—28no—
54C22×D4×C17—24no—
55C22×D4×C17C2314yes—
56C23×C4×C17C2311yescharacteristic
57C23×C4×C17C2312yes—
58C23×C4×C17C2311yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C22×C22⋊D17—28no—
61C22×C22⋊D17—28no—
62C22×C22⋊D17D4116yes—
63C22×C22⋊D17C2318yes—
64C23×Dic17—28no—
65C23×Dic17—28no—
66C23×Dic17—28no—
67C23×Dic17C2312yes—
68C23×Dic17C2311yescharacteristic
69C23×Dic17C2318yes—
70C23×Dic17C2311yescharacteristic
71C23×Dic17C2312yes—
72C23×Dic17C2311yescharacteristic
73C2×#63254—2128no—
74C2×#63254—264no—
75C2×#63254—232no—
76C22×C4×D17—24no—
77C22×C4×D17—28no—
78C22×C4×D17—24no—
79C22×C4×D17—24no—
80C22×C4×D17—24no—
81C22×C4×D17—28no—
82C22×C4×D17D418yes—
83C22×C4×D17C2314yes—
84C22×C4×D17C2314yes—
85C22×C4×D17C2314yes—
86C22×Dic34—24no—
87C22×Dic34—28no—
88C22×Dic34C2314yes—
89C2×C4⋊C4×C17—24no—
90C2×C4⋊C4×C17C2314yes—
91C2×C22⋊C4×C17—28no—
92C2×C22⋊C4×C17—28no—
93C2×C22⋊C4×C17C2318yes—
94C2×C22⋊C4×C17C2318yes—
95C2×C22⋊Dic17—28no—
96C2×C22⋊Dic17—28no—
97C2×C22⋊Dic17C2318yes—
98C2×C22⋊Dic17C2318yes—
99C2×#63227—28no—
100C2×#63227C2318yes—
101C2×C4⋊Dic17—24no—
102C2×C4⋊Dic17—28no—
103C2×C4⋊Dic17C2318yes—
104C2×C4⋊Dic17C2314yes—
105C2×#63225—24no—
106C2×#63225—24no—
107C2×#63225—28no—
108C2×#63225C2314yes—
109C2×#63225C2318yes—
110C2×#63225C2314yes—
111C4⋊C4×D17—232no—
112#10587569—264no—
113C17×#95C2314yes—
114#10587502C2314yes—
115#10587502C2318yes—
116#10587502C2318yes—
117#10587502C2318yes—
118C24×C17—24no—
119C24×C17—24no—
120C24×C17—24no—
121C24×C17—24no—
122C24×C17—24no—
123C24×C17—24no—
124C24×C17C2×D418yes—
125C24×C17C2414yes—
126C24×C17C2411yescharacteristic
127C23×D17—22no—
128C23×D17—22no—
129C23×D17—24no—
130C23×D17C2×D414yes—
131C23×D17C2×D418yes—
132C23×D17C2412yes—
133C2×D4×C17—232no—
134C2×D4×C17—216no—
135C2×D4×C17—28no—
136C22×C4×C17—24no—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—24no—
140C22×C4×C17—22no—
141C22×C4×C17—24no—
142C22×C4×C17—24no—
143C22×C4×C17—24no—
144C22×C4×C17C2×D414yes—
145C22×C4×C17#4114yescomplemented
146C22×C4×C17C2412yes—
147C22×C4×C17C2414yes—
148C22×C4×C17C2412yes—
149C22×C4×C17C2412yes—
150C2×C22⋊D17—264no—
151C2×C22⋊D17—216no—
152C2×C22⋊D17—216no—
153C2×C22⋊D17C2×D4132yes—
154C22×Dic17—264no—
155C22×Dic17—24no—
156C22×Dic17—24no—
157C22×Dic17—24no—
158C22×Dic17—22no—
159C22×Dic17—216no—
160C22×Dic17—22no—
161C22×Dic17—216no—
162C22×Dic17—216no—
163C22×Dic17—22no—
164C22×Dic17—24no—
165C22×Dic17—24no—
166C22×Dic17—24no—
167C22×Dic17—24no—
168C22×Dic17C2×D418yes—
169C22×Dic17C2×D414yes—
170C22×Dic17#4114yescomplemented
171C22×Dic17C2412yes—
172C22×Dic17C2414yes—
173C22×Dic17C2414yes—
174C22×Dic17C2412yes—
175C22×Dic17C2412yes—
176#63254—2256no—
177C2×C4×D17—232no—
178C2×C4×D17—232no—
179C2×C4×D17—216no—
180C2×C4×D17—28no—
181C2×C4×D17—28no—
182C2×C4×D17—28no—
183C2×C4×D17—216no—
184C2×C4×D17—28no—
185C2×C4×D17—416no—
186C2×C4×D17C2×D4116yes—
187C2×Dic34—232no—
188C2×Dic34—28no—
189C2×Dic34—216no—
190C4⋊C4×C17—28no—
191C22⋊C4×C17—216no—
192C22⋊C4×C17—216no—
193C22⋊Dic17—216no—
194C22⋊Dic17—216no—
195#63227—216no—
196C4⋊Dic17—216no—
197C4⋊Dic17—28no—
198#63225—28no—
199#63225—216no—
200#63225—28no—
201C2×#2406—171nomaximal, Sylow
202C23×C17—216no—
203C23×C17—216no—
204C23×C17—216no—
205C23×C17—24no—
206C23×C17—24no—
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—24no—
211C23×C17—48no—
212C23×C17—48no—
213C23×C17—48no—
214C23×C17—48no—
215C23×C17C22×D418yes—
216C23×C17C22×D412yes—
217C23×C17C22×D414yes—
218C23×C17C2×#4112yes—
219C23×C17C2×#4112yes—
220C23×C172−1+414yes—
221C23×C17C2511yescharacteristic
222C22×D17—28no—
223C22×D17—28no—
224C22×D17—216no—
225C22×D17—216no—
226C22×D17—22no—
227C22×D17—22no—
228C22×D17—24no—
229C22×D17#120116yes—
230C22×D17C22×D414yes—
231C22×D17C22×D418yes—
232D4×C17—232no—
233C2×C4×C17—28no—
234C2×C4×C17—28no—
235C2×C4×C17—22no—
236C2×C4×C17—24no—
237C2×C4×C17—22no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—22no—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—44no—
244C2×C4×C17—48no—
245C2×C4×C17C22×D414yes—
246C2×C4×C17C2×#4114yescomplemented
247C22⋊D17—264no—
248C2×Dic17—264no—
249C2×Dic17—216no—
250C2×Dic17—28no—
251C2×Dic17—28no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—24no—
258C2×Dic17—24no—
259C2×Dic17—22no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—44no—
263C2×Dic17—48no—
264C2×Dic17C22×D418yes—
265C2×Dic17C22×D414yes—
266C2×Dic17C2×#4114yescomplemented
267C4×D17—232no—
268C4×D17—232no—
269C4×D17—416no—
270Dic34—232no—
271C23×#41—171no—
272C22×#123—172no—
273C23×C4⋊C4—171no—
274#2406—1716no—
275C2×#395—171no—
276C2×#395—171no—
277C2×#394—171no—
278C2×#394D1711yescharacteristic, complemented
279C2×#386—172no—
280C2×#385—172no—
281C2×#385—172no—
282C2×#381—171no—
283C22×C17—28no—
284C22×C17—28no—
285C22×C17—28no—
286C22×C17—432no—
287C22×C17—44no—
288C22×C17—44no—
289C22×C17—44no—
290C22×C17—44no—
291C22×C17C2×#12014yes—
292C22×C17#52514yes—
293C22×C17#53614yes—
294C22×C17#54718yes—
295C22×C17#54718yes—
296C22×C17C23×D411yescharacteristic
297C22×C17C23×D412yes—
298C22×C17C22×#4111yescharacteristic
299C22×C17C22×#4111yescharacteristic
300C22×C17C2×2−1+412yes—
301C2×D17—232no—
302C2×D17—24no—
303C2×D17—24no—
304C2×D17—28no—
305C2×D17—28no—
306C2×D17C2×#12018yes—
307C4×C17—24no—
308C4×C17—24no—
309C4×C17—42no—
310C4×C17—44no—
311Dic17—28no—
312Dic17—24no—
313Dic17—24no—
314Dic17—42no—
315Dic17—44no—
316C22×#41—178no—
317C22×#41—348no—
318C22×#41—3416no—
319C23×Q8—171no—
320C23×D4—172no—
321C23×D4C2×D1711yescharacteristic
322C24×C4—171no—
323C24×C4—172no—
324C24×C4—171no—
325C24×C4—171no—
326C2×#123—1716no—
327C2×#123—3416no—
328C22×C4⋊C4—171no—
329C22×C4⋊C4—172no—
330C22×C4⋊C4—171no—
331C22×C4⋊C4—172no—
332C22×C4⋊C4—171no—
333C22×C4⋊C4—178no—
334C22×C4⋊C4—348no—
335C22×C4⋊C4C2×D1711yescharacteristic
336C22×C22⋊C4—172no—
337C22×C22⋊C4—172no—
338C22×C22⋊C4—172no—
339C22×C22⋊C4C2×D1712yes—
340C22×C22⋊C4C2×D1712yes—
341#395—178no—
342#395—178no—
343#394—178no—
344#394C2×D1718yescomplemented
345#386—1716no—
346#385—1716no—
347#385—1716no—
348#381—178no—
349C2×#95—171no—
350C2×#95—172no—
351C2×#95—172no—
352C2×#95—172no—
353C2×#95C2×D1711yescharacteristic
354C2×C17—48no—
355C2×C17#240618yescomplemented
356C2×C17C22×#12011yescharacteristic
357C2×C17C2×#52511yescharacteristic
358C2×C17C2×#53611yescharacteristic
359C2×C17C2×#54712yes—
360C2×C17C2×#54712yes—
361D17—28no—
362C25—171no—
363C25C22×D1712yes—
364C2×#41—3432no—
365C2×#41—3464no—
366C2×#41—34128no—
367C22×Q8—174no—
368C22×Q8—348no—
369C22×Q8—344no—
370C22×D4—28no—
371C22×D4—24no—
372C22×D4—1716no—
373C22×D4—178no—
374C22×D4—348no—
375C22×D4—348no—
376C22×D4C22×D1714yes—
377C23×C4—171no—
378C23×C4—171no—
379C23×C4—172no—
380C23×C4—178no—
381C23×C4—174no—
382C23×C4—178no—
383C23×C4—172no—
384C23×C4—171no—
385C23×C4—174no—
386C23×C4—174no—
387C23×C4—344no—
388C23×C4—344no—
389C23×C4—348no—
390C23×C4—344no—
391C23×C4—348no—
392C23×C4—348no—
393C23×C4—348no—
394C23×C4—348no—
395C23×C4—344no—
396C23×C4C22×D1712yes—
397C23×C4C22×D1711yescharacteristic
398C23×C4C22×D1711yescharacteristic
399C23×C4C22×D1711yescharacteristic
400#123—3464no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—174no—
403C2×C4⋊C4—174no—
404C2×C4⋊C4—174no—
405C2×C4⋊C4—178no—
406C2×C4⋊C4—178no—
407C2×C4⋊C4—3432no—
408C2×C4⋊C4—344no—
409C2×C4⋊C4—344no—
410C2×C4⋊C4—348no—
411C2×C4⋊C4—348no—
412C2×C4⋊C4—344no—
413C2×C4⋊C4C22×D1714yes—
414C2×C22⋊C4—28no—
415C2×C22⋊C4—28no—
416C2×C22⋊C4—178no—
417C2×C22⋊C4—178no—
418C2×C22⋊C4—178no—
419C2×C22⋊C4—348no—
420C2×C22⋊C4—348no—
421C2×C22⋊C4—348no—
422C2×C22⋊C4C22×D1718yes—
423C2×C22⋊C4C22×D1718yes—
424#95—178no—
425#95—174no—
426#95—178no—
427#95—178no—
428#95C22×D1714yes—
429C17C2×#240611yescharacteristic, complemented, Sylow
430C24—24no—
431C24—24no—
432C24—24no—
433C24—24no—
434C24—24no—
435C24—24no—
436C24—178no—
437C24—174no—
438C24—172no—
439C24—342no—
440C24—342no—
441C24—344no—
442C24D4×D1718yes—
443C24C23×D1714yes—
444C24C23×D1711yescharacteristic
445#41—34256no—
446C2×Q8—3416no—
447C2×Q8—3432no—
448C2×Q8—348no—
449C2×D4—232no—
450C2×D4—216no—
451C2×D4—28no—
452C2×D4—1732no—
453C2×D4—3464no—
454C2×D4—3416no—
455C2×D4—3416no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—174no—
465C22×C4—178no—
466C22×C4—174no—
467C22×C4—172no—
468C22×C4—172no—
469C22×C4—174no—
470C22×C4—1716no—
471C22×C4—174no—
472C22×C4—172no—
473C22×C4—348no—
474C22×C4—348no—
475C22×C4—3432no—
476C22×C4—3416no—
477C22×C4—3432no—
478C22×C4—348no—
479C22×C4—3416no—
480C22×C4—3416no—
481C22×C4—3416no—
482C22×C4—3464no—
483C22×C4—342no—
484C22×C4—344no—
485C22×C4—344no—
486C22×C4—344no—
487C22×C4—3416no—
488C22×C4—342no—
489C22×C4—348no—
490C22×C4—344no—
491C22×C4—344no—
492C22×C4—342no—
493C22×C4—344no—
494C22×C4—344no—
495C22×C4—6816no—
496C22×C4D4×D1714yes—
497C22×C4#6325414yescomplemented
498C22×C4C23×D1714yes—
499C22×C4C23×D1712yes—
500C22×C4C23×D1712yes—
501C22×C4C23×D1712yes—
502C4⋊C4—28no—
503C4⋊C4—348no—
504C4⋊C4—348no—
505C4⋊C4—3416no—
506C4⋊C4—3416no—
507C4⋊C4—348no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—3416no—
511C22⋊C4—3416no—
512C22⋊C4—3416no—
513C23—216no—
514C23—216no—
515C23—216no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—1716no—
527C23—178no—
528C23—174no—
529C23—3416no—
530C23—3416no—
531C23—348no—
532C23—342no—
533C23—348no—
534C23—342no—
535C23—344no—
536C23C2×D4×D1718yes—
537C23C2×D4×D1714yes—
538C23C2×D4×D1712yes—
539C23C2×#6325412yes—
540C23#41×D1712yes—
541C23#1058769714yes—
542C23C24×D1711yescharacteristic
543Q8—3432no—
544D4—232no—
545D4—3464no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—174no—
559C2×C4—178no—
560C2×C4—174no—
561C2×C4—3432no—
562C2×C4—3432no—
563C2×C4—3464no—
564C2×C4—348no—
565C2×C4—3416no—
566C2×C4—348no—
567C2×C4—342no—
568C2×C4—344no—
569C2×C4—344no—
570C2×C4—344no—
571C2×C4—342no—
572C2×C4—344no—
573C2×C4—344no—
574C2×C4—342no—
575C2×C4—344no—
576C2×C4—344no—
577C2×C4—6816no—
578C2×C4—688no—
579C2×C4—684no—
580C2×C4C2×D4×D1714yes—
581C2×C4C2×#6325414yescomplemented
582C22—28no—
583C22—28no—
584C22—28no—
585C22—44no—
586C22—432no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—178no—
591C22—344no—
592C22—3432no—
593C22—348no—
594C22—348no—
595C22—344no—
596C22#120×D1714yes—
597C22#1176331914yes—
598C22#1176337914yes—
599C22#1176345718yes—
600C22#1176345918yes—
601C22C22×D4×D1711yescharacteristic
602C22C22×D4×D1712yes—
603C22C2×#41×D1711yescharacteristic
604C22C22×#6325411yescharacteristic
605C22C2×#1058769712yes—
606C4—24no—
607C4—24no—
608C4—44no—
609C4—42no—
610C4—344no—
611C4—348no—
612C4—344no—
613C4—684no—
614C4—682no—
615C2—48no—
616C2—348no—
617C2#1453211418yesdirect product factor
618C2C2×#120×D1711yescharacteristic
619C2C2×#1176331911yescharacteristic
620C2C2×#1176337911yescharacteristic
621C2C2×#1176345712yes—
622C2C2×#1176345912yes—
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC2×C17×#394C2
socleC24×C17C24
prime radicalC23×#63254C2
hypercenterC2×#394D17
hypocenterC17C2×#2406
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×#10587569C2semidirect
3. C22×C4⋊C4×D17C2semidirect
4. #14532114C2direct (N × H)
5. C2×#11762730C2semidirect
6. C2×#11762729C2semidirect
7. C2×#11762723C2semidirect
8. C2×#11762685C2semidirect
9. C2×#11762684C2semidirect
10. C2×#11762681C2semidirect
11. C2×#11762671C2semidirect
12. C2×C17×#394C2semidirect
23. C2×#10587569C22semidirect
30. C2×C4⋊C4×D17C22semidirect
34. #11762730C22semidirect
35. #11762729C22semidirect
36. #11762723C22semidirect
37. #11762685C22semidirect
38. #11762684C22semidirect
39. #11762681C22semidirect
40. #11762671C22semidirect
49. C17×#394C22semidirect
145. C22×C4×C17#41semidirect
170. C22×Dic17#41semidirect
246. C2×C4×C17C2×#41semidirect
266. C2×Dic17C2×#41semidirect
278. C2×#394D17semidirect
344. #394C2×D17semidirect
355. C2×C17#2406semidirect
429. C17C2×#2406semidirect
497. C22×C4#63254semidirect
581. C2×C4C2×#63254semidirect
617. C2#14532114direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#632542
C22×#105875692
C22×C4⋊C4×D172
#145321142
C2×#117627302
C2×#117627292
C2×#117627232
C2×#117626852
C2×#117626842
C2×#117626812
C2×#117626712
C2×C17×#3942
C2×#240617