Finite Group Databasebeta

← C2×#14528770 #17033717

Subgroups — C2×#14528770 #17033717

Order 4352.

2066 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
1C22×D4×Dic17C211yesmaximal, characteristic
2C22×#10587505C211yesmaximal, characteristic
3C2×#11762667C212yesmaximal, complemented
4#14528770C218yesmaximal, direct product factor
5C2×#11762575C212yesmaximal, complemented
6C22×C17×#99C211yesmaximal, characteristic
7C24×Dic17C411yescharacteristic, complemented
8C24×Dic17C2211yescharacteristic
9C2×D4×Dic17—22no—
10C2×D4×Dic17—22no—
11C2×D4×Dic17—24no—
12C2×D4×Dic17C414yescomplemented
13C2×D4×Dic17C2214yes—
14C22×C4⋊Dic17C411yescharacteristic, complemented
15C22×#63225—22no—
16C22×C22⋊Dic17C411yescharacteristic, complemented
17C22×C22⋊Dic17C2212yes—
18C22×C22⋊Dic17C2211yescharacteristic
19C22×C4×Dic17C411yescharacteristic, complemented
20C2×#10587505—24no—
21C2×#10587505C2214yes—
22#11762667—24no—
23#11762667C2218yescomplemented
24C2×#10587502—22no—
25#11762575C2218yescomplemented
26C23×D4×C17C2211yescharacteristic
27C22×C22⋊C4×C17C2212yes—
28C2×C17×#99—24no—
29C2×C17×#99C2214yes—
30C25×C17C2×C411yescharacteristic
31C25×C17C2311yescharacteristic
32C22×D4×C17—22no—
33C22×D4×C17—22no—
34C22×D4×C17—24no—
35C22×D4×C17C2×C412yes—
36C22×D4×C17D414yes—
37C22×D4×C17Q814yes—
38C22×D4×C17C2312yes—
39C23×C4×C17—22no—
40C23×C4×C17C2×C411yescharacteristic
41C23×Dic17—22no—
42C23×Dic17—21no—
43C23×Dic17—21no—
44C23×Dic17—22no—
45C23×Dic17—44no—
46C23×Dic17—44no—
47C23×Dic17C2×C411yescharacteristic
48C23×Dic17C2×C411yescharacteristic
49C23×Dic17C2×C411yescharacteristic
50C23×Dic17C2×C411yescharacteristic
51C23×Dic17C2×C412yes—
52C23×Dic17C2×C414yescomplemented
53C23×Dic17D412yes—
54C23×Dic17D412yes—
55C23×Dic17D414yes—
56C23×Dic17C2312yes—
57C2×C22⋊C4×C17—24no—
58C2×C22⋊C4×C17—24no—
59C2×C22⋊C4×C17—22no—
60C2×C22⋊C4×C17C2314yes—
61C2×C22⋊Dic17—22no—
62C2×C22⋊Dic17—22no—
63C2×C22⋊Dic17—22no—
64C2×C22⋊Dic17—22no—
65C2×C22⋊Dic17—24no—
66C2×C22⋊Dic17—24no—
67C2×C22⋊Dic17—22no—
68C2×C22⋊Dic17C2×C412yes—
69C2×C22⋊Dic17C2×C414yescomplemented
70C2×C22⋊Dic17C2314yes—
71C2×C22⋊Dic17C2312yes—
72D4×Dic17—28no—
73D4×Dic17—28no—
74D4×Dic17—28no—
75D4×Dic17—28no—
76C2×C4⋊Dic17—22no—
77C2×C4⋊Dic17—22no—
78C2×C4⋊Dic17C2×C414yescomplemented
79C2×#63225—28no—
80C2×#63225—48no—
81C2×C4×Dic17—22no—
82C2×C4×Dic17—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17C2×C414yescomplemented
85C17×#99—28no—
86#10587505—28no—
87#10587502—28no—
88C24×C17—21no—
89C24×C17—21no—
90C24×C17—22no—
91C24×C17—42no—
92C24×C17—42no—
93C24×C17—42no—
94C24×C17—42no—
95C24×C17C4⋊C414yescomplemented
96C24×C17C22×C411yescharacteristic
97C24×C17C22×C412yes—
98C24×C17C22×C411yescharacteristic
99C24×C17C2×D412yes—
100C24×C17C2×D412yes—
101C24×C17C2×D412yes—
102C24×C17C2×Q812yes—
103C24×C17#4114yes—
104C24×C17C2411yescharacteristic
105C2×D4×C17—24no—
106C2×D4×C17—24no—
107C2×D4×C17—28no—
108C2×D4×C17—28no—
109C2×D4×C17—28no—
110C2×D4×C17—28no—
111C2×D4×C17—24no—
112C2×D4×C17—24no—
113C2×D4×C17C4⋊C418yescomplemented
114C2×D4×C17C2×D414yes—
115C2×D4×C17C2×Q814yes—
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17—22no—
119C22×C4×C17—24no—
120C22×C4×C17—44no—
121C22×C4×C17—44no—
122C22×C4×C17—44no—
123C22×C4×C17C4⋊C414yescomplemented
124C22×C4×C17C22×C412yes—
125C22×Dic17—24no—
126C22×Dic17—28no—
127C22×Dic17—21no—
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—24no—
132C22×Dic17—24no—
133C22×Dic17—24no—
134C22×Dic17—44no—
135C22×Dic17—48no—
136C22×Dic17—44no—
137C22×Dic17—42no—
138C22×Dic17—44no—
139C22×Dic17—44no—
140C22×Dic17—48no—
141C22×Dic17—48no—
142C22×Dic17—44no—
143C22×Dic17—42no—
144C22×Dic17—42no—
145C22×Dic17—44no—
146C22×Dic17—44no—
147C22×Dic17—44no—
148C22×Dic17C22⋊C412yes—
149C22×Dic17C22⋊C411yescharacteristic
150C22×Dic17C22⋊C412yes—
151C22×Dic17C22⋊C414yes—
152C22×Dic17C22⋊C412yes—
153C22×Dic17C22⋊C414yes—
154C22×Dic17C22⋊C411yescharacteristic
155C22×Dic17C22×C412yes—
156C22×Dic17C22×C412yes—
157C22×Dic17C22×C412yes—
158C22×Dic17C22×C412yes—
159C22×Dic17C2×D414yes—
160C22×Dic17C2×D414yes—
161C22⋊C4×C17—24no—
162C22⋊C4×C17—24no—
163C22⋊C4×C17—44no—
164C22⋊Dic17—24no—
165C22⋊Dic17—24no—
166C22⋊Dic17—28no—
167C22⋊Dic17—24no—
168C22⋊Dic17—24no—
169C22⋊Dic17—44no—
170C4⋊Dic17—28no—
171#63225—416no—
172C4×Dic17—24no—
173C4×Dic17—24no—
174C2×#1799—171nomaximal, Sylow
175C23×C17—24no—
176C23×C17—22no—
177C23×C17—24no—
178C23×C17—22no—
179C23×C17—22no—
180C23×C17—22no—
181C23×C17—24no—
182C23×C17—24no—
183C23×C17—41no—
184C23×C17—42no—
185C23×C17—41no—
186C23×C17—41no—
187C23×C17—42no—
188C23×C17—44no—
189C23×C17—44no—
190C23×C17—48no—
191C23×C17—41no—
192C23×C17—42no—
193C23×C17—48no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—48no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17C2×C22⋊C412yes—
201C23×C17C2×C22⋊C411yescharacteristic
202C23×C17C2×C22⋊C411yescharacteristic
203C23×C17C2×C4⋊C412yes—
204C23×C17C2×C4⋊C412yes—
205C23×C17C2×C4⋊C414yescomplemented
206C23×C17#11712yes—
207C23×C17#11712yes—
208C23×C17#12114yes—
209C23×C17#12114yes—
210C23×C17C22⋊Q814yes—
211C23×C17C22⋊Q814yes—
212C23×C17C23×C411yescharacteristic
213C23×C17C22×D412yes—
214C23×C17C22×D411yescharacteristic
215C23×C17C22×Q811yescharacteristic
216C23×C17C2×#4112yes—
217D4×C17—28no—
218D4×C17—28no—
219D4×C17—216no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—24no—
223C2×C4×C17—24no—
224C2×C4×C17—44no—
225C2×C4×C17—44no—
226C2×C4×C17—44no—
227C2×C4×C17—88no—
228C2×C4×C17C2×C4⋊C414yescomplemented
229C2×Dic17—22no—
230C2×Dic17—24no—
231C2×Dic17—24no—
232C2×Dic17—24no—
233C2×Dic17—28no—
234C2×Dic17—28no—
235C2×Dic17—22no—
236C2×Dic17—22no—
237C2×Dic17—44no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240C2×Dic17—44no—
241C2×Dic17—48no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—48no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—44no—
250C2×Dic17—44no—
251C2×Dic17—44no—
252C2×Dic17—88no—
253C2×Dic17#9914yescomplemented
254C2×Dic17C2×C22⋊C414yes—
255C2×Dic17C2×C22⋊C412yes—
256C2×Dic17C2×C22⋊C412yes—
257C22×C4×D4—171no—
258C22×#99—171no—
259C22×#99D1711yescharacteristic
260C2×#380—172no—
261#1799—178no—
262C2×#342—172no—
263C22×C17—28no—
264C22×C17—24no—
265C22×C17—24no—
266C22×C17—28no—
267C22×C17—22no—
268C22×C17—21no—
269C22×C17—21no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—21no—
273C22×C17—21no—
274C22×C17—22no—
275C22×C17—24no—
276C22×C17—42no—
277C22×C17—42no—
278C22×C17—44no—
279C22×C17—44no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—42no—
283C22×C17—44no—
284C22×C17—44no—
285C22×C17—48no—
286C22×C17—48no—
287C22×C17—42no—
288C22×C17—42no—
289C22×C17#38012yes—
290C22×C17#38014yes—
291C22×C17#38012yes—
292C22×C17C2×#9912yes—
293C22×C17#41012yes—
294C22×C17C22×C22⋊C411yescharacteristic
295C22×C17C22×C4⋊C411yescharacteristic
296C22×C17C2×#11711yescharacteristic
297C22×C17C2×#12112yes—
298C22×C17C2×C22⋊Q812yes—
299C4×C17—24no—
300C4×C17—84no—
301Dic17—22no—
302Dic17—24no—
303Dic17—42no—
304Dic17—42no—
305Dic17—84no—
306Dic17C2×#9914yescomplemented
307C23×D4C2×D1711yescharacteristic
308C24×C4—171no—
309C24×C4—171no—
310C2×C4×D4—174no—
311C2×C4×D4—174no—
312C2×C4×D4—342no—
313C2×C4×D4—342no—
314C2×C4×D4—344no—
315C22×C4⋊C4—171no—
316C22×C4⋊C4—342no—
317C22×C22⋊C4—171no—
318C22×C22⋊C4—171no—
319C22×C22⋊C4—172no—
320C22×C22⋊C4C2×D1712yes—
321C22×C42—171no—
322C2×#99—24no—
323C2×#99—174no—
324C2×#99—344no—
325C2×#99C2×D1714yes—
326#380—178no—
327#380—344no—
328C2×#95—342no—
329#342—178no—
330C2×C17—22no—
331C2×C17—21no—
332C2×C17—21no—
333C2×C17—22no—
334C2×C17—22no—
335C2×C17—24no—
336C2×C17—44no—
337C2×C17#179914yescomplemented
338C2×C17C2×#38011yescharacteristic
339C2×C17C22×#9911yescharacteristic
340C2×C17C2×#41011yescharacteristic
341C25C4×D1711yescharacteristic
342C25C22×D1711yescharacteristic
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—24no—
346C22×D4Dic3414yes—
347C22×D4C4×D1712yes—
348C22×D4C22⋊D1714yes—
349C22×D4C22×D1712yes—
350C23×C4—22no—
351C23×C4—171no—
352C23×C4—172no—
353C23×C4—171no—
354C23×C4—172no—
355C23×C4—172no—
356C23×C4—174no—
357C23×C4—172no—
358C23×C4—171no—
359C23×C4—171no—
360C23×C4—174no—
361C23×C4—341no—
362C23×C4—342no—
363C23×C4—341no—
364C23×C4—342no—
365C23×C4—684no—
366C23×C4—684no—
367C23×C4C4×D1711yescharacteristic
368C4×D4—348no—
369C4×D4—348no—
370C4×D4—348no—
371C4×D4—348no—
372C2×C4⋊C4—174no—
373C2×C4⋊C4—342no—
374C2×C4⋊C4—342no—
375C2×C4⋊C4—348no—
376C2×C4⋊C4—688no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—172no—
381C2×C22⋊C4—174no—
382C2×C22⋊C4—172no—
383C2×C22⋊C4—174no—
384C2×C22⋊C4—342no—
385C2×C22⋊C4—342no—
386C2×C22⋊C4—342no—
387C2×C22⋊C4—342no—
388C2×C22⋊C4—344no—
389C2×C22⋊C4—344no—
390C2×C22⋊C4—342no—
391C2×C22⋊C4C22×D1714yes—
392C2×C42—174no—
393C2×C42—342no—
394C2×C42—341no—
395C2×C42—341no—
396#99—28no—
397#99—348no—
398#95—348no—
399C17C2×#179911yescharacteristic, complemented, Sylow
400C24—22no—
401C24—21no—
402C24—21no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24#6322514yescomplemented
408C24C2×Dic3412yes—
409C24C2×C4×D1711yescharacteristic
410C24C2×C4×D1712yes—
411C24C2×C4×D1711yescharacteristic
412C24D4×D1712yes—
413C24D4×D1712yes—
414C24#6325414yes—
415C24C2×C22⋊D1712yes—
416C24C23×D1711yescharacteristic
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—24no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4#6322518yescomplemented
426C2×D4C2×Dic3414yes—
427C2×D4C2×C22⋊D1714yes—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—171no—
436C22×C4—172no—
437C22×C4—174no—
438C22×C4—172no—
439C22×C4—174no—
440C22×C4—171no—
441C22×C4—172no—
442C22×C4—172no—
443C22×C4—174no—
444C22×C4—172no—
445C22×C4—174no—
446C22×C4—172no—
447C22×C4—172no—
448C22×C4—348no—
449C22×C4—344no—
450C22×C4—344no—
451C22×C4—344no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—344no—
457C22×C4—684no—
458C22×C4—684no—
459C22×C4—688no—
460C22×C4—684no—
461C22×C4—684no—
462C22×C4—682no—
463C22×C4—688no—
464C22×C4—684no—
465C22×C4—688no—
466C22×C4—682no—
467C22×C4—682no—
468C22×C4—684no—
469C22×C4—684no—
470C22×C4—684no—
471C22×C4#6322514yescomplemented
472C22×C4C2×C4×D1712yes—
473C4⋊C4—348no—
474C4⋊C4—6816no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—44no—
478C22⋊C4—348no—
479C22⋊C4—344no—
480C22⋊C4—344no—
481C22⋊C4—344no—
482C22⋊C4—344no—
483C22⋊C4—684no—
484C42—344no—
485C42—344no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—44no—
498C23—41no—
499C23—42no—
500C23—48no—
501C23—44no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—48no—
506C23—42no—
507C23—48no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23#1058756312yes—
512C23#1058756312yes—
513C23#1058756414yes—
514C23#1058756414yes—
515C23C22⋊C4×D1712yes—
516C23C22⋊C4×D1711yescharacteristic
517C23C22⋊C4×D1711yescharacteristic
518C23C2×#6322512yes—
519C23C2×#6322512yes—
520C23C2×#6322514yescomplemented
521C23#1058762514yes—
522C23#1058762514yes—
523C23C22×Dic3411yescharacteristic
524C23C22×C4×D1711yescharacteristic
525C23C2×D4×D1712yes—
526C23C2×#6325412yes—
527C23C22×C22⋊D1711yescharacteristic
528D4—28no—
529D4—28no—
530D4—216no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—88no—
539C2×C4—172no—
540C2×C4—174no—
541C2×C4—172no—
542C2×C4—174no—
543C2×C4—348no—
544C2×C4—344no—
545C2×C4—344no—
546C2×C4—348no—
547C2×C4—342no—
548C2×C4—344no—
549C2×C4—342no—
550C2×C4—342no—
551C2×C4—688no—
552C2×C4—684no—
553C2×C4—684no—
554C2×C4—682no—
555C2×C4—682no—
556C2×C4—688no—
557C2×C4—682no—
558C2×C4—682no—
559C2×C4—682no—
560C2×C4—682no—
561C2×C4—682no—
562C2×C4—682no—
563C2×C4—684no—
564C2×C4—684no—
565C2×C4—684no—
566C2×C4—1368no—
567C2×C4C2×#6322514yescomplemented
568C22—28no—
569C22—24no—
570C22—24no—
571C22—28no—
572C22—22no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—22no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—24no—
581C22—41no—
582C22—42no—
583C22—41no—
584C22—42no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—48no—
592C22—42no—
593C22—42no—
594C22#1176266712yes—
595C22#1176266714yes—
596C22#1176266712yes—
597C22#99×D1712yes—
598C22#1176278112yes—
599C22C2×C22⋊C4×D1711yescharacteristic
600C22C22×#6322511yescharacteristic
601C22C2×#1058756311yescharacteristic
602C22C2×#1058762512yes—
603C22C2×#1058756412yes—
604C4—24no—
605C4—84no—
606C4—174no—
607C4—342no—
608C4—344no—
609C4—682no—
610C4—682no—
611C4—1364no—
612C2—22no—
613C2—21no—
614C2—21no—
615C2—22no—
616C2—22no—
617C2—24no—
618C2—44no—
619C2#1452877014yesdirect product factor
620C2C2×#1176266711yescharacteristic
621C2C2×#99×D1711yescharacteristic
622C2C2×#1176278111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC23×C17C23×C4
FrattiniC24C23×D17
FittingC22×C17×#99C2
socleC23×C17C2×C22⋊C4
prime radicalC23×D4×C17C22
cosocleC24×C17C24
hypercenterC22×#99D17
hypocenterC17C2×#1799
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C25 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
3. C2×#11762667C2semidirect
4. #14528770C2direct (N × H)
5. C2×#11762575C2semidirect
7. C24×Dic17C4semidirect
12. C2×D4×Dic17C4semidirect
14. C22×C4⋊Dic17C4semidirect
16. C22×C22⋊Dic17C4semidirect
19. C22×C4×Dic17C4semidirect
23. #11762667C22semidirect
25. #11762575C22semidirect
52. C23×Dic17C2×C4semidirect
69. C2×C22⋊Dic17C2×C4semidirect
78. C2×C4⋊Dic17C2×C4semidirect
84. C2×C4×Dic17C2×C4semidirect
95. C24×C17C4⋊C4semidirect
113. C2×D4×C17C4⋊C4semidirect
123. C22×C4×C17C4⋊C4semidirect
205. C23×C17C2×C4⋊C4semidirect
228. C2×C4×C17C2×C4⋊C4semidirect
253. C2×Dic17#99semidirect
306. Dic17C2×#99semidirect
337. C2×C17#1799semidirect
399. C17C2×#1799semidirect
407. C24#63225semidirect
425. C2×D4#63225semidirect
471. C22×C4#63225semidirect
520. C23C2×#63225semidirect
567. C2×C4C2×#63225semidirect
619. C2#14528770direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Dic172
C22×#105875052
C2×#117626672
#145287702
C2×#117625752
C22×C17×#992
C2×#179917