Finite Group Databasebeta

← C2×#14543180 #17942116

Subgroups — C2×#14543180 #17942116

Order 4352.

7810 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14543180C2132yesmaximal, direct product factor
2C2×#11763557C211yesmaximal, characteristic, complemented
3C2×#11763553C212yesmaximal, complemented
4C2×#11763513C214yesmaximal, complemented
5C2×#11763477C214yesmaximal, complemented
6C2×#11763457C214yesmaximal, complemented
7C2×#11763411C214yesmaximal, complemented
8C2×#11763389C214yesmaximal, complemented
9C2×#11763386C214yesmaximal, complemented
10C2×#11763365C211yesmaximal, characteristic, complemented
11C2×#11763358C212yesmaximal, complemented
12C2×C17×#562C211yesmaximal, characteristic, complemented
13C22×#63256C2212yes—
14C22×#63252C2214yes—
15C22×#63254C2212yes—
16C22×#63254C2214yes—
17#11763557C22116yescomplemented
18#11763553C22132yescomplemented
19#11763513C22164yescomplemented
20#11763477C22164yescomplemented
21#11763457C22164yescomplemented
22#11763411C22164yescomplemented
23#11763389C22164yescomplemented
24#11763386C22164yescomplemented
25#11763365C22116yescomplemented
26#11763358C22132yescomplemented
27C2×#10587574C2212yes—
28C2×#10587630C2212yes—
29C2×#10587626C2214yes—
30C2×#10587626C2212yes—
31C2×#10587556C2211yescharacteristic
32C2×#10587575C2214yes—
33C2×#10587561C2212yes—
34C2×#10587580C2214yes—
35C2×#10587572C2214yes—
36C2×#10587569C2214yes—
37C2×#10587582C2214yes—
38C2×#10587582C2214yes—
39C2×#10587582C2214yes—
40C2×#10587564C2214yes—
41C2×#10587564C2214yes—
42C2×#10587632C2214yes—
43C2×#10587581C2214yes—
44C2×#10587581C2214yes—
45C2×#10587616C2214yes—
46C2×#10587570C2214yes—
47C2×#10587570C2214yes—
48C2×#10587570C2214yes—
49C2×#10587570C2214yes—
50C2×#10587570C2214yes—
51C2×#10587625C2214yes—
52C2×#10587625C2214yes—
53C2×#10587625C2214yes—
54C2×#10587573C2214yes—
55C2×C4×C22⋊D17C2214yes—
56C2×#10587567C2214yes—
57C2×#10587567C2214yes—
58C2×#10587567C2214yes—
59C2×#10587578C2212yes—
60C22×#63225C2214yes—
61C2×C4⋊C4×D17C2214yes—
62C2×C42×D17C2211yescharacteristic
63C17×#562C22116yescomplemented
64C2×C17×#125C2211yescharacteristic
65C2×C17×#124C2212yes—
66C2×C17×#123C2214yes—
67C2×C17×C22⋊Q8C2214yes—
68C2×C17×#121C2214yes—
69C22×Q8×C17C2312yes—
70C22×D4×C17C2314yes—
71C22×D4×C17C2312yes—
72C23×C4×C17C2314yes—
73C22×C22⋊D17C2314yes—
74C22×C22⋊D17C2314yes—
75C22×C22⋊D17C2314yes—
76C22×C22⋊D17C2314yes—
77C22×C22⋊D17C2314yes—
78C23×Dic17C2314yes—
79C23×Dic17C2314yes—
80C2×#63256—48no—
81C2×#63256C23116yes—
82C2×#63254—416no—
83C2×#63254—48no—
84C2×#63254C23132yes—
85C2×#63254C23116yes—
86C2×#63252—416no—
87C2×#63252C23132yes—
88C22×D68C2314yes—
89C22×D68C2312yes—
90C22×D68C2314yes—
91C22×C4×D17C2311yescharacteristic
92C22×C4×D17C2312yes—
93C22×C4×D17C2314yes—
94C22×C4×D17C2314yes—
95C22×C4×D17C2314yes—
96C22×Dic34C2314yes—
97C22×Dic34C2312yes—
98C17×#125C2318yes—
99C17×#124C23116yes—
100C17×#123C23132yes—
101C17×C22⋊Q8C23132yes—
102C17×#121C23132yes—
103C2×C4⋊C4×C17C2314yes—
104C2×C4⋊C4×C17C2314yes—
105C2×C4⋊C4×C17C2312yes—
106C2×C22⋊C4×C17C2314yes—
107C2×C22⋊C4×C17C2314yes—
108C2×C22⋊C4×C17C2314yes—
109C2×C42×C17C2311yescharacteristic
110C2×C22⋊Dic17C2314yes—
111#10587632C23132yes—
112#10587630C23116yes—
113#10587626C23116yes—
114#10587626C23132yes—
115#10587625C23132yes—
116#10587625C23132yes—
117#10587625C23132yes—
118#10587616C23132yes—
119C4×C22⋊D17C23132yes—
120C2×#63227C2314yes—
121C2×#63227C2314yes—
122C2×#63227C2314yes—
123C2×#63227C2314yes—
124C2×#63227C2314yes—
125C2×C4⋊Dic17C2314yes—
126C2×C4⋊Dic17C2312yes—
127C2×#63225C23132yes—
128C2×#63225C2314yes—
129C2×#63225C2314yes—
130C2×#63225C2314yes—
131C2×#63225C2314yes—
132C2×#63225C2314yes—
133C2×#63225C2314yes—
134C2×C4×Dic17C2311yescharacteristic
135C2×C4×Dic17C2312yes—
136C2×C4×Dic17C2314yes—
137#10587582C23132yes—
138#10587582C23132yes—
139#10587582C23132yes—
140#10587581C23132yes—
141#10587581C23132yes—
142#10587580C23132yes—
143#10587578C23116yes—
144C4⋊C4×D17C23132yes—
145#10587575C23132yes—
146#10587574C23116yes—
147#10587573C23132yes—
148#10587572C23132yes—
149#10587570C23132yes—
150#10587570C23132yes—
151#10587570C23132yes—
152#10587570C23132yes—
153#10587570C23132yes—
154#10587569C23132yes—
155#10587567C23132yes—
156#10587567C23132yes—
157#10587567C23132yes—
158#10587564C23132yes—
159#10587564C23132yes—
160#10587561C23116yes—
161C42×D17C2318yes—
162#10587556C2318yes—
163C24×C17C2414yes—
164C23×D17C2414yes—
165C23×D17C2411yescharacteristic
166C2×Q8×C17—44no—
167C2×Q8×C17C2418yes—
168C2×D4×C17—48no—
169C2×D4×C17—44no—
170C2×D4×C17C24116yes—
171C2×D4×C17C2418yes—
172C22×C4×C17—48no—
173C22×C4×C17C2414yes—
174C22×C4×C17C2412yes—
175C22×C4×C17C2414yes—
176C22×C4×C17C2411yescharacteristic
177C22×C4×C17C24116yes—
178C2×C22⋊D17—216no—
179C2×C22⋊D17—216no—
180C2×C22⋊D17—48no—
181C2×C22⋊D17—48no—
182C2×C22⋊D17—48no—
183C2×C22⋊D17C24116yes—
184C2×C22⋊D17C24116yes—
185C2×C22⋊D17C24116yes—
186C2×C22⋊D17C24116yes—
187C2×C22⋊D17C24116yes—
188C22×Dic17—216no—
189C22×Dic17—48no—
190C22×Dic17C2414yes—
191C22×Dic17C2414yes—
192C22×Dic17C24116yes—
193C22×Dic17C24116yes—
194C22×Dic17C2412yes—
195C22×Dic17C2411yescharacteristic
196#63256—432no—
197#63254—464no—
198#63254—432no—
199#63252—464no—
200C2×D68—48no—
201C2×D68—44no—
202C2×D68—48no—
203C2×D68C24116yes—
204C2×D68C2418yes—
205C2×D68C24116yes—
206C2×C4×D17—216no—
207C2×C4×D17—28no—
208C2×C4×D17—24no—
209C2×C4×D17—48no—
210C2×C4×D17—48no—
211C2×C4×D17C24116yes—
212C2×C4×D17C2418yes—
213C2×C4×D17C2414yes—
214C2×C4×D17C24116yes—
215C2×C4×D17C24116yes—
216C2×Dic34—216no—
217C2×Dic34—44no—
218C2×Dic34C24116yes—
219C2×Dic34C2418yes—
220C4⋊C4×C17C24116yes—
221C4⋊C4×C17C24116yes—
222C4⋊C4×C17C2418yes—
223C22⋊C4×C17C24116yes—
224C22⋊C4×C17C24116yes—
225C22⋊C4×C17C24116yes—
226C42×C17C2414yes—
227C22⋊Dic17C24116yes—
228#63227C24116yes—
229#63227C24116yes—
230#63227C24116yes—
231#63227C24116yes—
232#63227C24116yes—
233C4⋊Dic17C2418yes—
234C4⋊Dic17C24116yes—
235#63225C24116yes—
236#63225C24116yes—
237#63225C24116yes—
238#63225C24116yes—
239#63225C24116yes—
240#63225C24116yes—
241C4×Dic17C2414yes—
242C4×Dic17C2418yes—
243C4×Dic17C24116yes—
244C2×#3548—171nomaximal, Sylow
245C23×C17—28no—
246C23×C17—28no—
247C23×C17—28no—
248C23×C17C2518yes—
249C23×C17C2511yescharacteristic
250C22×D17—28no—
251C22×D17—24no—
252C22×D17—28no—
253C22×D17—22no—
254C22×D17—28no—
255C22×D17C2518yes—
256C22×D17C2512yes—
257Q8×C17—48no—
258D4×C17—416no—
259D4×C17—48no—
260C2×C4×C17—28no—
261C2×C4×C17—24no—
262C2×C4×C17—28no—
263C2×C4×C17—22no—
264C2×C4×C17—416no—
265C2×C4×C17C2512yes—
266C2×C4×C17C2514yes—
267C2×C4×C17C2518yes—
268C2×C4×C17C2518yes—
269C22⋊D17—232no—
270C22⋊D17—232no—
271C22⋊D17—416no—
272C22⋊D17—416no—
273C22⋊D17—416no—
274C2×Dic17—28no—
275C2×Dic17—28no—
276C2×Dic17—232no—
277C2×Dic17—416no—
278C2×Dic17H(5,𝔽2)18yes—
279C2×Dic172−1+414yes—
280C2×Dic17C2518yes—
281C2×Dic17C2518yes—
282C2×Dic17C2514yes—
283C2×Dic17C2512yes—
284D68—416no—
285D68—48no—
286D68—416no—
287C4×D17—232no—
288C4×D17—216no—
289C4×D17—28no—
290C4×D17—416no—
291C4×D17—416no—
292Dic34—232no—
293Dic34—48no—
294#3548—1732no—
295C2×#562—171no—
296C2×#562—172no—
297C2×#562D1711yescharacteristic, complemented
298C2×#554—174no—
299C2×#550—174no—
300C2×#547—174no—
301C2×#540—174no—
302C2×#537—174no—
303C2×#537—174no—
304C2×#533—171no—
305C2×#532—172no—
306C22×C17—28no—
307C22×C17—28no—
308C22×C17—28no—
309C22×C17—416no—
310C22×C17C2×H(5,𝔽2)14yes—
311C22×C17C2×2−1+412yes—
312C22×C17C2611yescharacteristic
313C2×D17—28no—
314C2×D17—24no—
315C2×D17—28no—
316C2×D17—22no—
317C2×D17—28no—
318C2×D17—416no—
319C2×D17—44no—
320C4×C17—28no—
321C4×C17—24no—
322C4×C17—28no—
323C4×C17—22no—
324Dic17—28no—
325Dic17—28no—
326Dic17C2×H(5,𝔽2)18yes—
327Dic17C2×2−1+414yes—
328C22×#41—172no—
329C22×#41—172no—
330C22×#41—174no—
331C22×#41—174no—
332#562—1716no—
333#562—1732no—
334#562C2×D17116yescomplemented
335#554—1764no—
336#550—1764no—
337#547—1764no—
338#540—1764no—
339#537—1764no—
340#537—1764no—
341#533—1716no—
342#532—1732no—
343C2×C4⋊Q8—172no—
344C2×C4⋊Q8—172no—
345C2×#127—172no—
346C2×#127—174no—
347C2×#125—174no—
348C2×#125—171no—
349C2×#125C2×D1711yescharacteristic
350C2×#124—172no—
351C2×#124C2×D1712yes—
352C2×#123—174no—
353C2×#123—174no—
354C2×#123—174no—
355C2×#123C2×D1714yes—
356C2×C22⋊Q8—174no—
357C2×C22⋊Q8—174no—
358C2×C22⋊Q8—174no—
359C2×C22⋊Q8—174no—
360C2×C22⋊Q8—174no—
361C2×C22⋊Q8—174no—
362C2×C22⋊Q8C2×D1714yes—
363C2×#121—174no—
364C2×#121—174no—
365C2×#121—174no—
366C2×#121—174no—
367C2×#121—174no—
368C2×#121—174no—
369C2×#121—174no—
370C2×#121—174no—
371C2×#121—174no—
372C2×#121—174no—
373C2×#121—174no—
374C2×#121C2×D1714yes—
375C2×C4×Q8—174no—
376C2×C4×D4—174no—
377C2×C4×D4—174no—
378C2×C4×D4—174no—
379C2×C4×D4—174no—
380C2×#117—172no—
381C22×C4⋊C4—174no—
382C22×C4⋊C4—174no—
383C22×C42—171no—
384C2×C17—48no—
385C2×C17#354814yescomplemented
386C2×C17C22×H(5,𝔽2)12yes—
387C2×C17C22×2−1+411yescharacteristic
388D17—48no—
389D17—42no—
390C2×#41—1716no—
391C2×#41—1716no—
392C2×#41—1732no—
393C2×#41—1732no—
394C2×#41—6816no—
395C2×#41—688no—
396C2×#41—688no—
397C2×#41—6816no—
398C22×Q8—174no—
399C22×Q8—172no—
400C22×Q8C22×D1712yes—
401C22×D4—174no—
402C22×D4—174no—
403C22×D4—174no—
404C22×D4—174no—
405C22×D4—174no—
406C22×D4—174no—
407C22×D4—174no—
408C22×D4—172no—
409C22×D4C22×D1714yes—
410C22×D4C22×D1712yes—
411C23×C4—174no—
412C23×C4—174no—
413C23×C4—174no—
414C23×C4—171no—
415C23×C4—172no—
416C23×C4—174no—
417C23×C4—174no—
418C23×C4C22×D1714yes—
419C4⋊Q8—1716no—
420C4⋊Q8—1716no—
421#127—1716no—
422#127—1732no—
423#125—1732no—
424#125—178no—
425#125C22×D1718yes—
426#124—1716no—
427#124C22×D17116yes—
428#123—1732no—
429#123—1732no—
430#123—1732no—
431#123C22×D17132yes—
432C22⋊Q8—1732no—
433C22⋊Q8—1732no—
434C22⋊Q8—1732no—
435C22⋊Q8—1732no—
436C22⋊Q8—1732no—
437C22⋊Q8—1732no—
438C22⋊Q8C22×D17132yes—
439#121—1732no—
440#121—1732no—
441#121—1732no—
442#121—1732no—
443#121—1732no—
444#121—1732no—
445#121—1732no—
446#121—1732no—
447#121—1732no—
448#121—1732no—
449#121—1732no—
450#121C22×D17132yes—
451C4×Q8—1732no—
452C4×D4—1732no—
453C4×D4—1732no—
454C4×D4—1732no—
455C4×D4—1732no—
456#117—1716no—
457C2×C4⋊C4—174no—
458C2×C4⋊C4—174no—
459C2×C4⋊C4—174no—
460C2×C4⋊C4—174no—
461C2×C4⋊C4—174no—
462C2×C4⋊C4—1732no—
463C2×C4⋊C4—172no—
464C2×C4⋊C4—174no—
465C2×C4⋊C4—174no—
466C2×C4⋊C4—1732no—
467C2×C4⋊C4C22×D1714yes—
468C2×C4⋊C4C22×D1712yes—
469C2×C4⋊C4C22×D1714yes—
470C2×C22⋊C4—174no—
471C2×C22⋊C4—174no—
472C2×C22⋊C4—174no—
473C2×C22⋊C4—174no—
474C2×C22⋊C4—174no—
475C2×C22⋊C4—174no—
476C2×C22⋊C4C22×D1714yes—
477C2×C22⋊C4C22×D1714yes—
478C2×C22⋊C4C22×D1714yes—
479C2×C42—174no—
480C2×C42—171no—
481C2×C42—172no—
482C2×C42—178no—
483C2×C42C22×D1711yescharacteristic
484C17C2×#354811yescharacteristic, complemented, Sylow
485C24—171no—
486C24—174no—
487C24C23×D1714yes—
488#41—6864no—
489#41—6832no—
490#41—6832no—
491#41—6864no—
492C2×Q8—44no—
493C2×Q8—1716no—
494C2×Q8—178no—
495C2×Q8—3416no—
496C2×Q8—684no—
497C2×Q8C23×D1718yes—
498C2×D4—48no—
499C2×D4—44no—
500C2×D4—1716no—
501C2×D4—1716no—
502C2×D4—1716no—
503C2×D4—1716no—
504C2×D4—1716no—
505C2×D4—1716no—
506C2×D4—1716no—
507C2×D4—178no—
508C2×D4—3416no—
509C2×D4—3416no—
510C2×D4—688no—
511C2×D4—688no—
512C2×D4—684no—
513C2×D4—688no—
514C2×D4—688no—
515C2×D4—688no—
516C2×D4C23×D17116yes—
517C2×D4C23×D1718yes—
518C22×C4—48no—
519C22×C4—171no—
520C22×C4—172no—
521C22×C4—1716no—
522C22×C4—174no—
523C22×C4—174no—
524C22×C4—1716no—
525C22×C4—174no—
526C22×C4—178no—
527C22×C4—1716no—
528C22×C4—1716no—
529C22×C4—1716no—
530C22×C4—344no—
531C22×C4—3416no—
532C22×C4—348no—
533C22×C4—3416no—
534C22×C4—688no—
535C22×C4—688no—
536C22×C4—688no—
537C22×C4C23×D1711yescharacteristic
538C22×C4C23×D1712yes—
539C22×C4C23×D1714yes—
540C22×C4C23×D1714yes—
541C22×C4C23×D17116yes—
542C4⋊C4—1716no—
543C4⋊C4—1716no—
544C4⋊C4—1716no—
545C4⋊C4—1716no—
546C4⋊C4—1716no—
547C4⋊C4—178no—
548C4⋊C4—1716no—
549C4⋊C4—1716no—
550C4⋊C4C23×D17116yes—
551C4⋊C4C23×D1718yes—
552C4⋊C4C23×D17116yes—
553C22⋊C4—1716no—
554C22⋊C4—1716no—
555C22⋊C4—1716no—
556C22⋊C4—1716no—
557C22⋊C4—1716no—
558C22⋊C4—1716no—
559C22⋊C4C23×D17116yes—
560C22⋊C4C23×D17116yes—
561C22⋊C4C23×D17116yes—
562C42—174no—
563C42—178no—
564C42—1716no—
565C42C23×D1714yes—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—172no—
570C23—178no—
571C23—348no—
572C23—344no—
573C23—348no—
574C23—342no—
575C23—348no—
576C23C24×D1718yes—
577C23C24×D1711yescharacteristic
578Q8—48no—
579Q8—3432no—
580Q8—688no—
581D4—416no—
582D4—48no—
583D4—3432no—
584D4—3432no—
585D4—6816no—
586D4—6816no—
587D4—688no—
588D4—6816no—
589D4—6816no—
590D4—6816no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—416no—
596C2×C4—178no—
597C2×C4—174no—
598C2×C4—172no—
599C2×C4—174no—
600C2×C4—178no—
601C2×C4—178no—
602C2×C4—3432no—
603C2×C4—3416no—
604C2×C4—348no—
605C2×C4—3432no—
606C2×C4—348no—
607C2×C4—348no—
608C2×C4—6816no—
609C2×C4—6816no—
610C2×C4—6816no—
611C2×C4C24×D1712yes—
612C2×C4C24×D1714yes—
613C2×C4C24×D1718yes—
614C2×C4C24×D1718yes—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—416no—
619C22—348no—
620C22—344no—
621C22—348no—
622C22—342no—
623C22—348no—
624C22—684no—
625C22—6816no—
626C22H(5,𝔽2)×D1714yes—
627C222−1+4×D1712yes—
628C22C25×D1711yescharacteristic
629C4—28no—
630C4—28no—
631C4—24no—
632C4—22no—
633C4—178no—
634C4—174no—
635C4—348no—
636C4—348no—
637C2—48no—
638C2—682no—
639C2—688no—
640C2#1454318014yesdirect product factor
641C2C2×H(5,𝔽2)×D1712yes—
642C2C2×2−1+4×D1711yescharacteristic
subgroupG/H
centerC23C24×D17
commutator, cosocleC22×C17C26
FrattiniC22C25×D17
FittingC2×C17×#562C2
socleC23×C17C25
hypercenterC2×#562D17
hypocenterC17C2×#3548
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #14543180C2direct (N × H)
2. C2×#11763557C2semidirect
3. C2×#11763553C2semidirect
4. C2×#11763513C2semidirect
5. C2×#11763477C2semidirect
6. C2×#11763457C2semidirect
7. C2×#11763411C2semidirect
8. C2×#11763389C2semidirect
9. C2×#11763386C2semidirect
10. C2×#11763365C2semidirect
11. C2×#11763358C2semidirect
12. C2×C17×#562C2semidirect
17. #11763557C22semidirect
18. #11763553C22semidirect
19. #11763513C22semidirect
20. #11763477C22semidirect
21. #11763457C22semidirect
22. #11763411C22semidirect
23. #11763389C22semidirect
24. #11763386C22semidirect
25. #11763365C22semidirect
26. #11763358C22semidirect
63. C17×#562C22semidirect
297. C2×#562D17semidirect
334. #562C2×D17semidirect
385. C2×C17#3548semidirect
484. C17C2×#3548semidirect
640. C2#14543180direct (N × H)

Maximal subgroups

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

subgroupindex
#145431802
C2×#117635572
C2×#117635532
C2×#117635132
C2×#117634772
C2×#117634572
C2×#117634112
C2×#117633892
C2×#117633862
C2×#117633652
C2×#117633582
C2×C17×#5622
C2×#354817