Finite Group Databasebeta

← C2×#14542491 #17942264

Subgroups — C2×#14542491 #17942264

Order 4352.

7910 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542491C2132yesmaximal, direct product factor
2C2×#11763517C214yesmaximal, complemented
3C2×#11763506C214yesmaximal, complemented
4C2×#11763493C214yesmaximal, complemented
5C2×#11763459C214yesmaximal, complemented
6C2×#11763440C211yesmaximal, characteristic, complemented
7C2×#11763404C214yesmaximal, complemented
8C2×#11763319C214yesmaximal, complemented
9C2×#11763316C212yesmaximal, complemented
10C2×#11763292C212yesmaximal, complemented
11C22×C4⋊C4×D17C211yesmaximal, characteristic, complemented
12C2×C17×#537C211yesmaximal, characteristic, complemented
13C22×#63254C2214yes—
14C23×C4×D17C2211yescharacteristic
15C23×C4×D17C2212yes—
16#11763517C22164yescomplemented
17#11763506C22164yescomplemented
18#11763493C22164yescomplemented
19#11763459C22164yescomplemented
20#11763440C22116yescomplemented
21#11763404C22164yescomplemented
22C2×#10587574C2214yescomplemented
23C2×#10587565C2218yescomplemented
24C2×#10587576C2214yescomplemented
25C2×#10587575C2218yescomplemented
26#11763319C22164yescomplemented
27#11763316C22132yescomplemented
28C2×#10587569C2218yes—
29C2×#10587621C2214yes—
30C2×#10587621C2218yescomplemented
31C2×#10587583C2214yes—
32C2×#10587610C2214yes—
33C2×#10587564C2218yescomplemented
34C2×#10587632C2212yes—
35C2×#10587632C2212yes—
36C2×#10587624C2212yes—
37C2×#10587624C2212yes—
38C2×#10587624C2214yes—
39C2×#10587623C2214yes—
40#11763292C22132yescomplemented
41C2×Q8×Dic17C2214yescomplemented
42C2×D4×Dic17C2218yes—
43C2×D4×Dic17C2214yescomplemented
44C2×C4×C22⋊D17C2214yes—
45C2×C4×C22⋊D17C2214yes—
46C2×#10587567C2218yes—
47C2×#10587612C2212yes—
48C2×#10587612C2214yescomplemented
49C2×#10587578C2214yes—
50C22×C4⋊Dic17C2211yescharacteristic
51C22×C4⋊Dic17C2214yescomplemented
52C22×#63225C2212yes—
53C2×C4⋊C4×D17C2214yes—
54C2×C4⋊C4×D17C2214yes—
55C2×C4⋊C4×D17C22116yescomplemented
56C22×#41×C17C2212yes—
57C17×#537C22116yescomplemented
58C2×C17×C22⋊Q8C2214yescomplemented
59C2×C17×#121C2214yes—
60C2×C17×#121C2214yescomplemented
61C22×C4⋊C4×C17C2211yescharacteristic
62C24×D17C2311yescharacteristic
63C2×#41×C17—216no—
64C2×#41×C17C23116yes—
65C22×Q8×C17C2312yes—
66C22×D4×C17C2314yes—
67C22×D4×C17C2312yes—
68C22×D4×C17C2314yes—
69C23×C4×C17C2314yes—
70C23×C4×C17C2312yes—
71C23×C4×C17C2311yescharacteristic
72C22×C22⋊D17C2314yes—
73C22×C22⋊D17C2314yes—
74C23×Dic17C2311yescharacteristic
75C23×Dic17C2312yes—
76C23×Dic17C2318yes—
77C2×#63254—416no—
78C2×#63254C23132yes—
79C22×C4×D17—28no—
80C22×C4×D17—48no—
81C22×C4×D17C2314yes—
82C22×C4×D17C2312yes—
83C22×C4×D17C2312yes—
84C22×C4×D17C2318yes—
85C22×C4×D17C2314yes—
86C22×C4×D17C23116yes—
87C22×Dic34C2314yes—
88C17×C22⋊Q8C23132yescomplemented
89C17×#121C23132yes—
90C17×#121C23132yescomplemented
91C2×C4⋊C4×C17C2314yes—
92C2×C4⋊C4×C17C2318yes—
93C2×C4⋊C4×C17C2314yes—
94C2×C22⋊C4×C17C2318yes—
95C2×C22⋊Dic17C2314yes—
96C2×C22⋊Dic17C2318yes—
97C2×C22⋊Dic17C2314yes—
98C2×C22⋊Dic17C2314yes—
99#10587632C23116yes—
100#10587632C23116yes—
101Q8×Dic17C23132yescomplemented
102#10587624C23116yes—
103#10587624C23116yes—
104#10587624C23132yes—
105#10587623C23132yes—
106#10587621C23132yes—
107#10587621C23164yescomplemented
108D4×Dic17C23164yes—
109D4×Dic17C23132yescomplemented
110C4×C22⋊D17C23132yes—
111C4×C22⋊D17C23132yes—
112C2×#63227C2314yes—
113C2×#63227C2314yes—
114#10587612C23116yes—
115#10587612C23132yescomplemented
116C2×C4⋊Dic17C2314yes—
117C2×C4⋊Dic17C2318yes—
118C2×C4⋊Dic17C2312yes—
119C2×C4⋊Dic17C2318yes—
120C2×C4⋊Dic17C2314yes—
121C2×C4⋊Dic17C2312yes—
122C2×C4⋊Dic17C23132yescomplemented
123#10587610C23132yes—
124C2×#63225C2318yes—
125C2×#63225C23116yes—
126C2×#63225C2318yes—
127C2×#63225C2314yes—
128C2×#63225C2314yes—
129C2×C4×Dic17C2318yes—
130C2×C4×Dic17C2314yes—
131#10587583C23132yes—
132#10587578C23132yes—
133C4⋊C4×D17C23132yes—
134C4⋊C4×D17C23132yes—
135#10587576C23132yescomplemented
136#10587575C23164yescomplemented
137#10587574C23132yescomplemented
138#10587569C23164yes—
139#10587567C23164yes—
140#10587565C23164yescomplemented
141#10587564C23164yescomplemented
142C24×C17C2414yes—
143C24×C17C2411yescharacteristic
144C23×D17—24no—
145C23×D17—24no—
146C23×D17—24no—
147C23×D17C2414yes—
148C23×D17C2411yescharacteristic
149C23×D17C2411yescharacteristic
150#41×C17—264no—
151C2×Q8×C17—28no—
152C2×Q8×C17C2418yes—
153C2×D4×C17—216no—
154C2×D4×C17—28no—
155C2×D4×C17—48no—
156C2×D4×C17C24116yes—
157C2×D4×C17C2418yes—
158C2×D4×C17C24116yes—
159C22×C4×C17—216no—
160C22×C4×C17—44no—
161C22×C4×C17#4118yescomplemented
162C22×C4×C17C24116yes—
163C22×C4×C17C2414yes—
164C22×C4×C17C2414yes—
165C22×C4×C17C2418yes—
166C22×C4×C17C2412yes—
167C22×C4×C17C2414yes—
168C2×C22⋊D17—216no—
169C2×C22⋊D17—216no—
170C2×C22⋊D17C24116yes—
171C2×C22⋊D17C24116yes—
172C22×Dic17—24no—
173C22×Dic17—44no—
174C22×Dic17—416no—
175C22×Dic17C2414yes—
176C22×Dic17C2414yes—
177C22×Dic17C2412yes—
178C22×Dic17C2414yes—
179C22×Dic17C2414yes—
180C22×Dic17C2418yes—
181C22×Dic17C24132yes—
182#63254—464no—
183C2×C4×D17—216no—
184C2×C4×D17—28no—
185C2×C4×D17—28no—
186C2×C4×D17—232no—
187C2×C4×D17—432no—
188C2×C4×D17—48no—
189C2×C4×D17C24116yes—
190C2×C4×D17C2418yes—
191C2×C4×D17C2418yes—
192C2×C4×D17C24116yes—
193C2×Dic34—48no—
194C2×Dic34C24116yes—
195C4⋊C4×C17C24116yes—
196C4⋊C4×C17C24116yes—
197C22⋊C4×C17C24132yes—
198C22⋊Dic17C24116yes—
199C22⋊Dic17C24116yes—
200C22⋊Dic17C24132yes—
201C22⋊Dic17C24116yes—
202#63227C24116yes—
203#63227C24116yes—
204C4⋊Dic17C24132yes—
205C4⋊Dic17C24116yes—
206C4⋊Dic17C2418yes—
207C4⋊Dic17C2418yes—
208C4⋊Dic17C24116yes—
209#63225C24116yes—
210#63225C24132yes—
211#63225C24116yes—
212#63225C24132yes—
213C4×Dic17C24132yes—
214C4×Dic17C24116yes—
215C2×#3492—171nomaximal, Sylow
216C23×C17—22no—
217C23×C17—22no—
218C23×C17—28no—
219C23×C17—28no—
220C23×C17—28no—
221C23×C17C2×#4114yes—
222C23×C17C2518yes—
223C23×C17C2512yes—
224C23×C17C2511yescharacteristic
225C22×D17—232no—
226C22×D17—28no—
227C22×D17—22no—
228C22×D17—22no—
229C22×D17—22no—
230C22×D17—22no—
231C22×D17—28no—
232C22×D17—28no—
233C22×D17H(5,𝔽2)14yes—
234C22×D172−1+414yes—
235C22×D17C2512yes—
236C22×D17C2512yes—
237Q8×C17—216no—
238D4×C17—232no—
239D4×C17—216no—
240D4×C17—416no—
241C2×C4×C17—28no—
242C2×C4×C17—28no—
243C2×C4×C17—232no—
244C2×C4×C17—48no—
245C2×C4×C17C2×#4118yes—
246C2×C4×C17C2×#41116yescomplemented
247C2×C4×C17C2518yes—
248C2×C4×C17C2518yes—
249C2×C4×C17C2514yes—
250C22⋊D17—232no—
251C22⋊D17—232no—
252C2×Dic17—28no—
253C2×Dic17—24no—
254C2×Dic17—28no—
255C2×Dic17—28no—
256C2×Dic17—28no—
257C2×Dic17—48no—
258C2×Dic17—432no—
259C2×Dic17C2514yes—
260C2×Dic17C2518yes—
261C2×Dic17C2518yes—
262C2×Dic17C2518yes—
263C4×D17—232no—
264C4×D17—216no—
265C4×D17—216no—
266C4×D17—416no—
267Dic34—416no—
268#3492—1732no—
269C2×#554—174no—
270C2×#553—174no—
271C2×#552—174no—
272C2×#547—174no—
273C2×#544—171no—
274C2×#539—174no—
275C2×#537D1711yescharacteristic, complemented
276C2×#525—172no—
277C2×#525—174no—
278C2×#520—172no—
279C23×C4⋊C4—171no—
280C22×C17—22no—
281C22×C17—22no—
282C22×C17—28no—
283C22×C17—28no—
284C22×C17—28no—
285C22×C17—28no—
286C22×C17—416no—
287C22×C17C22×#4114yes—
288C22×C17C22×#4112yes—
289C22×C17C2×H(5,𝔽2)12yes—
290C22×C17C2×2−1+412yes—
291C22×C17C2611yescharacteristic
292C2×D17—232no—
293C2×D17—22no—
294C2×D17—22no—
295C2×D17—22no—
296C2×D17—22no—
297C2×D17—28no—
298C2×D17—28no—
299C2×D17C2×H(5,𝔽2)14yes—
300C2×D17C2×2−1+414yes—
301C4×C17—28no—
302C4×C17—28no—
303C4×C17C22×#4118yes—
304Dic17—24no—
305Dic17—28no—
306Dic17—28no—
307Dic17—28no—
308C22×#41—174no—
309C22×#41C2×D1712yes—
310C24×C4—171no—
311C24×C4—172no—
312#554—1764no—
313#553—1764no—
314#552—1764no—
315#547—1764no—
316#544—1716no—
317#539—1764no—
318#537C2×D17116yescomplemented
319C2×C4⋊Q8—174no—
320C2×#126—178no—
321C2×#125—178no—
322C2×#125—174no—
323#525—1732no—
324#525—1764no—
325C2×#123—174no—
326C2×#123—178no—
327C2×#123—178no—
328C2×C22⋊Q8—172no—
329C2×C22⋊Q8—172no—
330C2×C22⋊Q8—174no—
331C2×C22⋊Q8—174no—
332C2×C22⋊Q8—178no—
333C2×C22⋊Q8C2×D1714yescomplemented
334C2×#121—172no—
335C2×#121—174no—
336C2×#121—172no—
337C2×#121C2×D1714yes—
338C2×#121C2×D1714yescomplemented
339C2×#120—174no—
340#520—1732no—
341C2×C4×Q8—174no—
342C2×C4×D4—178no—
343C2×C4×D4—174no—
344C2×C4×D4—174no—
345C2×C4×D4—174no—
346C2×C4×D4—178no—
347C2×#117—172no—
348C2×#117—174no—
349C2×#117—174no—
350C22×C4⋊C4—172no—
351C22×C4⋊C4—171no—
352C22×C4⋊C4—174no—
353C22×C4⋊C4—174no—
354C22×C4⋊C4—174no—
355C22×C4⋊C4—1716no—
356C22×C4⋊C4C2×D1711yescharacteristic
357C2×C17—24no—
358C2×C17—48no—
359C2×C17#349214yescomplemented
360C2×C17C23×#4111yescharacteristic
361C2×C17C22×H(5,𝔽2)11yescharacteristic
362C2×C17C22×2−1+411yescharacteristic
363D17—24no—
364D17—24no—
365C25—171no—
366C2×#41—216no—
367C2×#41—1732no—
368C2×#41—6816no—
369C2×#41C22×D17116yes—
370C22×Q8—174no—
371C22×Q8C22×D1712yes—
372C22×D4—174no—
373C22×D4—174no—
374C22×D4C22×D1714yes—
375C22×D4C22×D1712yes—
376C22×D4C22×D1714yes—
377C23×C4—171no—
378C23×C4—172no—
379C23×C4—178no—
380C23×C4—174no—
381C23×C4—172no—
382C23×C4—172no—
383C23×C4—178no—
384C23×C4—174no—
385C23×C4—1716no—
386C23×C4—348no—
387C23×C4—688no—
388C23×C4C22×D1714yes—
389C23×C4C22×D1712yes—
390C23×C4C22×D1711yescharacteristic
391C4⋊Q8—1732no—
392#126—1764no—
393#125—1764no—
394#125—1732no—
395#123—1732no—
396#123—1764no—
397#123—1764no—
398C22⋊Q8—1716no—
399C22⋊Q8—1716no—
400C22⋊Q8—1732no—
401C22⋊Q8—1732no—
402C22⋊Q8—1764no—
403C22⋊Q8C22×D17132yescomplemented
404#121—1716no—
405#121—1732no—
406#121—1716no—
407#121C22×D17132yes—
408#121C22×D17132yescomplemented
409#120—1732no—
410C4×Q8—1732no—
411C4×D4—1764no—
412C4×D4—1732no—
413C4×D4—1732no—
414C4×D4—1732no—
415C4×D4—1764no—
416#117—1716no—
417#117—1732no—
418#117—1732no—
419C2×C4⋊C4—174no—
420C2×C4⋊C4—178no—
421C2×C4⋊C4—178no—
422C2×C4⋊C4—1716no—
423C2×C4⋊C4—172no—
424C2×C4⋊C4—178no—
425C2×C4⋊C4—1732no—
426C2×C4⋊C4—172no—
427C2×C4⋊C4—174no—
428C2×C4⋊C4—174no—
429C2×C4⋊C4—178no—
430C2×C4⋊C4—174no—
431C2×C4⋊C4—1732no—
432C2×C4⋊C4—1732no—
433C2×C4⋊C4C22×D1714yes—
434C2×C4⋊C4C22×D1718yes—
435C2×C4⋊C4C22×D1714yes—
436C2×C22⋊C4—174no—
437C2×C22⋊C4—178no—
438C2×C22⋊C4—174no—
439C2×C22⋊C4—174no—
440C2×C22⋊C4—174no—
441C2×C22⋊C4—174no—
442C2×C22⋊C4C22×D1718yes—
443C2×C42—178no—
444C2×C42—174no—
445C17C2×#349211yescharacteristic, complemented, Sylow
446C24—171no—
447C24—171no—
448C24—174no—
449C24—344no—
450C24—344no—
451C24—344no—
452C24C23×D1714yes—
453C24C23×D1711yescharacteristic
454#41—264no—
455#41—6864no—
456C2×Q8—28no—
457C2×Q8—1716no—
458C2×Q8—688no—
459C2×Q8C23×D1718yes—
460C2×D4—216no—
461C2×D4—28no—
462C2×D4—48no—
463C2×D4—1716no—
464C2×D4—1716no—
465C2×D4—3416no—
466C2×D4—3416no—
467C2×D4C23×D17116yes—
468C2×D4C23×D1718yes—
469C2×D4C23×D17116yes—
470C22×C4—216no—
471C22×C4—44no—
472C22×C4—174no—
473C22×C4—174no—
474C22×C4—174no—
475C22×C4—174no—
476C22×C4—172no—
477C22×C4—178no—
478C22×C4—1732no—
479C22×C4—1716no—
480C22×C4—178no—
481C22×C4—178no—
482C22×C4—1716no—
483C22×C4—3416no—
484C22×C4—344no—
485C22×C4—348no—
486C22×C4—348no—
487C22×C4—3432no—
488C22×C4—688no—
489C22×C4—6832no—
490C22×C4—684no—
491C22×C4—6816no—
492C22×C4#6325418yescomplemented
493C22×C4C23×D17116yes—
494C22×C4C23×D1714yes—
495C22×C4C23×D1714yes—
496C22×C4C23×D1718yes—
497C22×C4C23×D1712yes—
498C22×C4C23×D1714yes—
499C4⋊C4—1716no—
500C4⋊C4—1732no—
501C4⋊C4—178no—
502C4⋊C4—1732no—
503C4⋊C4—178no—
504C4⋊C4—1716no—
505C4⋊C4—1716no—
506C4⋊C4—1732no—
507C4⋊C4—1716no—
508C4⋊C4C23×D17116yes—
509C4⋊C4C23×D17116yes—
510C22⋊C4—1716no—
511C22⋊C4—1732no—
512C22⋊C4—1716no—
513C22⋊C4—1716no—
514C22⋊C4—1716no—
515C22⋊C4—1716no—
516C22⋊C4C23×D17132yes—
517C42—1732no—
518C42—1716no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—174no—
525C23—174no—
526C23—172no—
527C23—172no—
528C23—348no—
529C23—348no—
530C23—3432no—
531C23—348no—
532C23—342no—
533C23—342no—
534C23—342no—
535C23—342no—
536C23C2×#6325414yes—
537C23C24×D1718yes—
538C23C24×D1712yes—
539C23C24×D1711yescharacteristic
540Q8—216no—
541Q8—6816no—
542D4—232no—
543D4—216no—
544D4—416no—
545D4—3432no—
546D4—3432no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—232no—
550C2×C4—48no—
551C2×C4—178no—
552C2×C4—178no—
553C2×C4—178no—
554C2×C4—174no—
555C2×C4—3432no—
556C2×C4—348no—
557C2×C4—344no—
558C2×C4—348no—
559C2×C4—348no—
560C2×C4—348no—
561C2×C4—3416no—
562C2×C4—3416no—
563C2×C4—6816no—
564C2×C4—688no—
565C2×C4—6832no—
566C2×C4C2×#6325418yes—
567C2×C4C2×#63254116yescomplemented
568C2×C4C24×D1718yes—
569C2×C4C24×D1718yes—
570C2×C4C24×D1714yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—416no—
578C22—174no—
579C22—174no—
580C22—3432no—
581C22—348no—
582C22—348no—
583C22—342no—
584C22—342no—
585C22—342no—
586C22—342no—
587C22C22×#6325414yes—
588C22C22×#6325412yes—
589C22H(5,𝔽2)×D1712yes—
590C222−1+4×D1712yes—
591C22C25×D1711yescharacteristic
592C4—28no—
593C4—28no—
594C4—344no—
595C4—348no—
596C4—348no—
597C4—348no—
598C4C22×#6325418yes—
599C2—24no—
600C2—48no—
601C2—344no—
602C2—344no—
603C2#1454249114yesdirect product factor
604C2C23×#6325411yescharacteristic
605C2C2×H(5,𝔽2)×D1711yescharacteristic
606C2C2×2−1+4×D1711yescharacteristic
subgroupG/H
centerC23C24×D17
commutator, cosocleC22×C17C26
FrattiniC22C25×D17
FittingC2×C17×#537C2
socleC23×C17C25
hypercenterC2×#537D17
hypocenterC17C2×#3492
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #14542491C2direct (N × H)
2. C2×#11763517C2semidirect
3. C2×#11763506C2semidirect
4. C2×#11763493C2semidirect
5. C2×#11763459C2semidirect
6. C2×#11763440C2semidirect
7. C2×#11763404C2semidirect
8. C2×#11763319C2semidirect
9. C2×#11763316C2semidirect
10. C2×#11763292C2semidirect
11. C22×C4⋊C4×D17C2semidirect
12. C2×C17×#537C2semidirect
16. #11763517C22semidirect
17. #11763506C22semidirect
18. #11763493C22semidirect
19. #11763459C22semidirect
20. #11763440C22semidirect
21. #11763404C22semidirect
22. C2×#10587574C22semidirect
23. C2×#10587565C22semidirect
24. C2×#10587576C22semidirect
25. C2×#10587575C22semidirect
26. #11763319C22semidirect
27. #11763316C22semidirect
30. C2×#10587621C22semidirect
33. C2×#10587564C22semidirect
40. #11763292C22semidirect
41. C2×Q8×Dic17C22semidirect
43. C2×D4×Dic17C22semidirect
48. C2×#10587612C22semidirect
51. C22×C4⋊Dic17C22semidirect
55. C2×C4⋊C4×D17C22semidirect
57. C17×#537C22semidirect
58. C2×C17×C22⋊Q8C22semidirect
60. C2×C17×#121C22semidirect
88. C17×C22⋊Q8C23semidirect
90. C17×#121C23semidirect
101. Q8×Dic17C23semidirect
107. #10587621C23semidirect
109. D4×Dic17C23semidirect
115. #10587612C23semidirect
122. C2×C4⋊Dic17C23semidirect
135. #10587576C23semidirect
136. #10587575C23semidirect
137. #10587574C23semidirect
140. #10587565C23semidirect
141. #10587564C23semidirect
161. C22×C4×C17#41semidirect
246. C2×C4×C17C2×#41semidirect
275. C2×#537D17semidirect
318. #537C2×D17semidirect
333. C2×C22⋊Q8C2×D17semidirect
338. C2×#121C2×D17semidirect
359. C2×C17#3492semidirect
403. C22⋊Q8C22×D17semidirect
408. #121C22×D17semidirect
445. C17C2×#3492semidirect
492. C22×C4#63254semidirect
567. C2×C4C2×#63254semidirect
603. C2#14542491direct (N × H)

Maximal subgroups

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

subgroupindex
#145424912
C2×#117635172
C2×#117635062
C2×#117634932
C2×#117634592
C2×#117634402
C2×#117634042
C2×#117633192
C2×#117633162
C2×#117632922
C22×C4⋊C4×D172
C2×C17×#5372
C2×#349217