Finite Group Databasebeta

← C2×#14533708 #17206097

Subgroups — C2×#14533708 #17206097

Order 4352.

3614 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587583C211yesmaximal, characteristic, complemented
2C22×#10587610C211yesmaximal, characteristic, complemented
3C22×#10587573C211yesmaximal, characteristic, complemented
4C22×C4×C22⋊D17C211yesmaximal, characteristic
5#14533708C2116yesmaximal, direct product factor
6C2×#11762733C212yesmaximal, complemented
7C2×#11762732C212yesmaximal, complemented
8C2×#11762709C212yesmaximal
9C2×#11762706C212yesmaximal, complemented
10C2×#11762678C211yesmaximal, characteristic, complemented
11C2×#11762666C211yesmaximal, characteristic, complemented
12C2×C17×#380C211yesmaximal, characteristic, complemented
13C23×Dic34C2211yescharacteristic
14C23×C22⋊D17C2211yescharacteristic
15C23×C4×D17C2211yescharacteristic
16C2×#10587583—28no—
17C2×#10587583C2218yescomplemented
18C2×#10587610—28no—
19C2×#10587610C2218yescomplemented
20C2×#10587573—28no—
21C2×#10587573C2218yescomplemented
22C2×C4×C22⋊D17—28no—
23C2×C4×C22⋊D17C2218yes—
24C22×C4⋊Dic17C2211yescharacteristic
25C22×C4⋊Dic17C2212yes—
26C22×#63225C2212yes—
27C22×#63225C2211yescharacteristic
28C22×#63225C2212yes—
29C22×#63225C2212yes—
30C22×#63225C2212yes—
31C22×C22⋊Dic17C2211yescharacteristic
32C22×C22⋊Dic17C2212yes—
33C22×#63227C2211yescharacteristic
34C22×#63227C2212yes—
35C22×#63227C2212yes—
36C22×C4×Dic17C2212yes—
37C22×C4×Dic17C2211yescharacteristic
38#11762733C22116yescomplemented
39#11762732C22116yescomplemented
40#11762709C22116yes—
41#11762706C22116yescomplemented
42#11762678C2218yescomplemented
43#11762666C2218yescomplemented
44C2×#10587502C2212yes—
45C24×C4×C17C2211yescharacteristic
46C22×C4⋊C4×C17C2211yescharacteristic
47C22×C4⋊C4×C17C2211yescharacteristic
48C22×C22⋊C4×C17C2212yes—
49C17×#380C2218yescomplemented
50C2×C17×#95C2212yes—
51C25×C17C2311yescharacteristic
52C24×D17C2311yescharacteristic
53C23×C4×C17—24no—
54C23×C4×C17—24no—
55C23×C4×C17—24no—
56C23×C4×C17C2312yes—
57C23×C4×C17C2312yes—
58C23×C4×C17C2311yescharacteristic
59C23×C4×C17C2314yes—
60C23×C4×C17C2311yescharacteristic
61C22×C22⋊D17—24no—
62C22×C22⋊D17—24no—
63C22×C22⋊D17Q818yescomplemented
64C22×C22⋊D17C2314yes—
65C23×Dic17C2312yes—
66C23×Dic17C2311yescharacteristic
67C23×Dic17C2311yescharacteristic
68C23×Dic17C2312yes—
69C23×Dic17C2311yescharacteristic
70C22×C4×D17—24no—
71C22×C4×D17—24no—
72C22×C4×D17—24no—
73C22×C4×D17C2314yes—
74C22×Dic34—24no—
75C22×Dic34—24no—
76C22×Dic34D418yescomplemented
77C22×Dic34C2314yes—
78C2×C4⋊C4×C17—24no—
79C2×C4⋊C4×C17—24no—
80C2×C4⋊C4×C17C2314yes—
81C2×C4⋊C4×C17C2314yes—
82C2×C22⋊C4×C17—44no—
83C2×C22⋊C4×C17C2318yes—
84C2×C22⋊Dic17—24no—
85C2×C22⋊Dic17—28no—
86C2×C22⋊Dic17C2318yes—
87C2×C22⋊Dic17C2314yes—
88C4×C22⋊D17—232no—
89C2×#63227—24no—
90C2×#63227—28no—
91C2×#63227—44no—
92C2×#63227C2318yes—
93C2×#63227C2314yes—
94C2×#63227C2318yes—
95C2×C4⋊Dic17—24no—
96C2×C4⋊Dic17—28no—
97C2×C4⋊Dic17C2318yes—
98C2×C4⋊Dic17C2314yes—
99#10587610—232no—
100C2×#63225—24no—
101C2×#63225—28no—
102C2×#63225—28no—
103C2×#63225—28no—
104C2×#63225—44no—
105C2×#63225C2314yes—
106C2×#63225C2318yes—
107C2×#63225C2318yes—
108C2×#63225C2318yes—
109C2×#63225C2318yes—
110C2×C4×Dic17—28no—
111C2×C4×Dic17D418yes—
112C2×C4×Dic17C2318yes—
113C2×C4×Dic17C2314yes—
114#10587583—232no—
115#10587573—232no—
116C17×#95C2318yes—
117#10587502C2318yes—
118C24×C17—22no—
119C24×C17—22no—
120C24×C17—22no—
121C24×C17—22no—
122C24×C17—22no—
123C24×C17—22no—
124C24×C17C2×Q814yes—
125C24×C17C2412yes—
126C24×C17C2411yescharacteristic
127C23×D17—22no—
128C23×D17—22no—
129C23×D17—22no—
130C23×D17—22no—
131C23×D17—22no—
132C23×D17—22no—
133C23×D17C2×Q814yes—
134C23×D17C2412yes—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—24no—
138C22×C4×C17—24no—
139C22×C4×C17—22no—
140C22×C4×C17—24no—
141C22×C4×C17—24no—
142C22×C4×C17—24no—
143C22×C4×C17—24no—
144C22×C4×C17—28no—
145C22×C4×C17—28no—
146C22×C4×C17—28no—
147C22×C4×C17—416no—
148C22×C4×C17C2×D414yes—
149C22×C4×C17C2×D414yes—
150C22×C4×C17#4114yes—
151C22×C4×C17C2412yes—
152C22×C4×C17C2414yes—
153C22×C4×C17C2414yes—
154C22×C4×C17C2412yes—
155C2×C22⋊D17—232no—
156C2×C22⋊D17—28no—
157C2×C22⋊D17—28no—
158C2×C22⋊D17C2×Q8116yescomplemented
159C22×Dic17—24no—
160C22×Dic17—24no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17—24no—
166C22×Dic17—22no—
167C22×Dic17—24no—
168C22×Dic17—24no—
169C22×Dic17C2×D418yes—
170C22×Dic17C2×D414yes—
171C22×Dic17C2×D414yes—
172C22×Dic17C2×Q814yes—
173C22×Dic17#4114yes—
174C22×Dic17C2412yes—
175C22×Dic17C2414yes—
176C22×Dic17C2414yes—
177C22×Dic17C2412yes—
178C22×Dic17C2412yes—
179C2×C4×D17—28no—
180C2×C4×D17—28no—
181C2×C4×D17—28no—
182C2×C4×D17—416no—
183C2×Dic34—232no—
184C2×Dic34—28no—
185C2×Dic34—28no—
186C2×Dic34C2×D4116yescomplemented
187C4⋊C4×C17—28no—
188C4⋊C4×C17—28no—
189C22⋊C4×C17—48no—
190C22⋊Dic17—216no—
191C22⋊Dic17—28no—
192#63227—216no—
193#63227—28no—
194#63227—48no—
195C4⋊Dic17—28no—
196C4⋊Dic17—216no—
197#63225—216no—
198#63225—216no—
199#63225—28no—
200#63225—216no—
201#63225—48no—
202C4×Dic17—216no—
203C4×Dic17C2×D4116yes—
204C2×#2538—171nomaximal, Sylow
205C23×C17—28no—
206C23×C17—28no—
207C23×C17—28no—
208C23×C17—22no—
209C23×C17—22no—
210C23×C17—22no—
211C23×C17—22no—
212C23×C17—22no—
213C23×C17—22no—
214C23×C17—44no—
215C23×C17—44no—
216C23×C17—44no—
217C23×C17—44no—
218C23×C17C22×D412yes—
219C23×C17C22×D412yes—
220C23×C17C22×Q814yes—
221C23×C17C22×Q812yes—
222C23×C17C2×#4112yes—
223C23×C17C2×#4112yes—
224C23×C17H(5,𝔽2)12yes—
225C23×C172−1+412yes—
226C23×C17C2511yescharacteristic
227C22×D17—28no—
228C22×D17—28no—
229C22×D17—28no—
230C22×D17—22no—
231C22×D17—22no—
232C22×D17—22no—
233C22×D17—22no—
234C22×D17—22no—
235C22×D17—22no—
236C22×D17—44no—
237C22×D17—44no—
238C22×D17—44no—
239C22×D17—44no—
240C22×D17C22×Q814yes—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—24no—
247C2×C4×C17—24no—
248C2×C4×C17—24no—
249C2×C4×C17—24no—
250C2×C4×C17—44no—
251C2×C4×C17—48no—
252C2×C4×C17—48no—
253C2×C4×C17—416no—
254C2×C4×C17#121116yes—
255C2×C4×C17C22×D414yes—
256C2×C4×C17C22×D414yes—
257C2×C4×C17C2×#4114yes—
258C22⋊D17—232no—
259C2×Dic17—216no—
260C2×Dic17—28no—
261C2×Dic17—24no—
262C2×Dic17—24no—
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—24no—
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—24no—
270C2×Dic17—24no—
271C2×Dic17—44no—
272C2×Dic17—48no—
273C2×Dic17C22⋊Q8116yescomplemented
274C2×Dic17C22×D418yes—
275C2×Dic17C22×D414yes—
276C2×Dic17C22×D414yes—
277C2×Dic17C22×Q814yes—
278C2×Dic17C2×#4114yes—
279C4×D17—416no—
280Dic34—232no—
281C22×C22⋊Q8—171no—
282C22×C22⋊Q8—171no—
283C22×C4×Q8—171no—
284C22×C4×D4—171no—
285#2538—1716no—
286C2×#396—172no—
287C2×#396—172no—
288C2×#390—172no—
289C2×#389—172no—
290C2×#384—171no—
291C2×#380—171no—
292C2×#380D1711yescharacteristic, complemented
293C22×C17—24no—
294C22×C17—24no—
295C22×C17—24no—
296C22×C17—42no—
297C22×C17—42no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—416no—
301C22×C17C2×#12114yes—
302C22×C17C2×C22⋊Q814yes—
303C22×C17#53714yes—
304C22×C17D4×Q814yes—
305C22×C17#55014yes—
306C22×C17#55414yes—
307C22×C17#55614yes—
308C22×C17C23×D411yescharacteristic
309C22×C17C23×D411yescharacteristic
310C22×C17C23×Q811yescharacteristic
311C22×C17C22×#4111yescharacteristic
312C22×C17C22×#4111yescharacteristic
313C22×C17C2×H(5,𝔽2)11yescharacteristic
314C22×C17C2×2−1+411yescharacteristic
315C2×D17—24no—
316C2×D17—24no—
317C2×D17—24no—
318C2×D17—42no—
319C2×D17—42no—
320C2×D17—42no—
321C2×D17—42no—
322C2×D17—416no—
323C4×C17—42no—
324C4×C17—44no—
325C4×C17—44no—
326C4×C17C2×#12118yes—
327Dic17—28no—
328Dic17—24no—
329Dic17—42no—
330Dic17—44no—
331Dic17C2×C22⋊Q818yescomplemented
332C23×Q8—171no—
333C23×D4—171no—
334C24×C4—171no—
335C24×C4C2×D1711yescharacteristic
336C2×C22⋊Q8—178no—
337C2×C22⋊Q8—178no—
338C2×C22⋊Q8—348no—
339C2×C22⋊Q8—348no—
340C2×C4×Q8—178no—
341C2×C4×Q8—348no—
342C2×C4×D4—178no—
343C2×C4×D4—348no—
344C22×C4⋊C4—172no—
345C22×C4⋊C4—172no—
346C22×C4⋊C4—171no—
347C22×C4⋊C4—172no—
348C22×C4⋊C4—172no—
349C22×C4⋊C4—171no—
350C22×C4⋊C4—172no—
351C22×C4⋊C4C2×D1711yescharacteristic
352C22×C4⋊C4C2×D1711yescharacteristic
353C22×C22⋊C4—171no—
354C22×C22⋊C4—172no—
355C22×C22⋊C4—171no—
356C22×C22⋊C4—172no—
357C22×C22⋊C4—172no—
358C22×C22⋊C4C2×D1712yes—
359C22×C42—171no—
360C22×C42—172no—
361#396—1716no—
362#396—1716no—
363#390—1716no—
364#389—1716no—
365#384—178no—
366#380—178no—
367#380C2×D1718yescomplemented
368C2×#95—172no—
369C2×#95C2×D1712yes—
370C2×C17—44no—
371C2×C17#253818yescomplemented
372C2×C17C22×#12111yescharacteristic
373C2×C17C22×C22⋊Q811yescharacteristic
374C2×C17C2×#53711yescharacteristic
375C2×C17C2×D4×Q811yescharacteristic
376C2×C17C2×#55011yescharacteristic
377C2×C17C2×#55411yescharacteristic
378C2×C17C2×#55611yescharacteristic
379D17—44no—
380C25—171no—
381C25C22×D1711yescharacteristic
382C22×Q8—178no—
383C22×Q8—174no—
384C22×Q8—344no—
385C22×Q8—344no—
386C22×D4—178no—
387C22×D4—174no—
388C22×D4—344no—
389C22×D4—344no—
390C23×C4—24no—
391C23×C4—24no—
392C23×C4—24no—
393C23×C4—171no—
394C23×C4—171no—
395C23×C4—172no—
396C23×C4—172no—
397C23×C4—171no—
398C23×C4—174no—
399C23×C4—344no—
400C23×C4—344no—
401C23×C4—344no—
402C23×C4C22×D1714yes—
403C23×C4C22×D1712yes—
404C23×C4C22×D1712yes—
405C23×C4C22×D1711yescharacteristic
406C23×C4C22×D1711yescharacteristic
407C22⋊Q8—3432no—
408C22⋊Q8—3432no—
409C4×Q8—3432no—
410C4×D4—3432no—
411C2×C4⋊C4—24no—
412C2×C4⋊C4—24no—
413C2×C4⋊C4—174no—
414C2×C4⋊C4—178no—
415C2×C4⋊C4—174no—
416C2×C4⋊C4—178no—
417C2×C4⋊C4—178no—
418C2×C4⋊C4—178no—
419C2×C4⋊C4—178no—
420C2×C4⋊C4—344no—
421C2×C4⋊C4—348no—
422C2×C4⋊C4—348no—
423C2×C4⋊C4—348no—
424C2×C4⋊C4—348no—
425C2×C4⋊C4—344no—
426C2×C4⋊C4—684no—
427C2×C4⋊C4C22×D1714yes—
428C2×C4⋊C4C22×D1714yes—
429C2×C22⋊C4—44no—
430C2×C22⋊C4—174no—
431C2×C22⋊C4—178no—
432C2×C22⋊C4—174no—
433C2×C22⋊C4—178no—
434C2×C22⋊C4—178no—
435C2×C22⋊C4—344no—
436C2×C22⋊C4—348no—
437C2×C22⋊C4—344no—
438C2×C22⋊C4—348no—
439C2×C22⋊C4—684no—
440C2×C22⋊C4C22×D1718yes—
441C2×C42—178no—
442C2×C42—174no—
443C2×C42—178no—
444C2×C42—348no—
445#95—178no—
446#95C22×D1718yes—
447C17C2×#253811yescharacteristic, complemented, Sylow
448C24—22no—
449C24—22no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24—174no—
455C24—172no—
456C24—342no—
457C24—342no—
458C24—342no—
459C24—342no—
460C24—342no—
461C24—342no—
462C24Q8×D1714yes—
463C24C23×D1712yes—
464C24C23×D1711yescharacteristic
465C2×Q8—1716no—
466C2×Q8—348no—
467C2×Q8—348no—
468C2×Q8—3432no—
469C2×D4—1716no—
470C2×D4—348no—
471C2×D4—3432no—
472C2×D4—348no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—416no—
486C22×C4—174no—
487C22×C4—174no—
488C22×C4—174no—
489C22×C4—172no—
490C22×C4—174no—
491C22×C4—178no—
492C22×C4—172no—
493C22×C4—174no—
494C22×C4—172no—
495C22×C4—174no—
496C22×C4—344no—
497C22×C4—344no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—344no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—344no—
505C22×C4—344no—
506C22×C4—348no—
507C22×C4—348no—
508C22×C4—348no—
509C22×C4—6816no—
510C22×C4#6325214yes—
511C22×C4D4×D1714yes—
512C22×C4D4×D1714yes—
513C22×C4C23×D1714yes—
514C22×C4C23×D1712yes—
515C22×C4C23×D1714yes—
516C22×C4C23×D1712yes—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—348no—
520C4⋊C4—3416no—
521C4⋊C4—3416no—
522C4⋊C4—3416no—
523C4⋊C4—3416no—
524C4⋊C4—348no—
525C4⋊C4—688no—
526C22⋊C4—48no—
527C22⋊C4—348no—
528C22⋊C4—3416no—
529C22⋊C4—348no—
530C22⋊C4—3416no—
531C22⋊C4—688no—
532C42—1716no—
533C42—3416no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—174no—
548C23—348no—
549C23—348no—
550C23—348no—
551C23—342no—
552C23—342no—
553C23—342no—
554C23—342no—
555C23—342no—
556C23—342no—
557C23—684no—
558C23—684no—
559C23—684no—
560C23—684no—
561C23C2×#6325212yes—
562C23C2×D4×D1712yes—
563C23C2×D4×D1712yes—
564C23#1058769112yes—
565C23C2×Q8×D1714yes—
566C23C2×Q8×D1712yes—
567C23#41×D1712yes—
568C23#1058769712yes—
569C23C24×D1711yescharacteristic
570Q8—3432no—
571D4—3432no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—416no—
585C2×C4—1716no—
586C2×C4—174no—
587C2×C4—174no—
588C2×C4—174no—
589C2×C4—174no—
590C2×C4—178no—
591C2×C4—3416no—
592C2×C4—344no—
593C2×C4—344no—
594C2×C4—342no—
595C2×C4—342no—
596C2×C4—348no—
597C2×C4—344no—
598C2×C4—342no—
599C2×C4—342no—
600C2×C4—344no—
601C2×C4—342no—
602C2×C4—344no—
603C2×C4—688no—
604C2×C4—684no—
605C2×C4—6816no—
606C2×C4#10587570116yes—
607C2×C4C2×#6325214yes—
608C2×C4C2×D4×D1714yes—
609C2×C4C2×D4×D1714yes—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—416no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—344no—
619C22—344no—
620C22—344no—
621C22—682no—
622C22—6816no—
623C22—682no—
624C22—682no—
625C22—682no—
626C22C2×#1058757014yes—
627C22C22⋊Q8×D1714yes—
628C22#1176338814yes—
629C22#1176347214yes—
630C22#1176347714yes—
631C22#1176351414yes—
632C22#1176352714yes—
633C22C22×D4×D1711yescharacteristic
634C22C22×D4×D1711yescharacteristic
635C22C22×Q8×D1711yescharacteristic
636C22C2×#41×D1711yescharacteristic
637C22C22×#6325211yescharacteristic
638C22C2×#1058769111yescharacteristic
639C22C2×#1058769711yescharacteristic
640C4—44no—
641C4—42no—
642C4—44no—
643C4—178no—
644C4—344no—
645C4—348no—
646C4—682no—
647C4—684no—
648C4C2×#1058757018yes—
649C2—44no—
650C2—684no—
651C2#1453370818yesdirect product factor
652C2C22×#1058757011yescharacteristic
653C2C2×C22⋊Q8×D1711yescharacteristic
654C2C2×#1176338811yescharacteristic
655C2C2×#1176347211yescharacteristic
656C2C2×#1176347711yescharacteristic
657C2C2×#1176351411yescharacteristic
658C2C2×#1176352711yescharacteristic
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC2×C17×#380C2
socleC24×C17C24
prime radicalC23×C22⋊D17C22
hypercenterC2×#380D17
hypocenterC17C2×#2538
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#10587583C2semidirect
2. C22×#10587610C2semidirect
3. C22×#10587573C2semidirect
5. #14533708C2direct (N × H)
6. C2×#11762733C2semidirect
7. C2×#11762732C2semidirect
9. C2×#11762706C2semidirect
10. C2×#11762678C2semidirect
11. C2×#11762666C2semidirect
12. C2×C17×#380C2semidirect
17. C2×#10587583C22semidirect
19. C2×#10587610C22semidirect
21. C2×#10587573C22semidirect
38. #11762733C22semidirect
39. #11762732C22semidirect
41. #11762706C22semidirect
42. #11762678C22semidirect
43. #11762666C22semidirect
49. C17×#380C22semidirect
63. C22×C22⋊D17Q8semidirect
76. C22×Dic34D4semidirect
158. C2×C22⋊D17C2×Q8semidirect
186. C2×Dic34C2×D4semidirect
273. C2×Dic17C22⋊Q8semidirect
292. C2×#380D17semidirect
331. Dic17C2×C22⋊Q8semidirect
367. #380C2×D17semidirect
371. C2×C17#2538semidirect
447. C17C2×#2538semidirect
651. C2#14533708direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875832
C22×#105876102
C22×#105875732
C22×C4×C22⋊D172
#145337082
C2×#117627332
C2×#117627322
C2×#117627092
C2×#117627062
C2×#117626782
C2×#117626662
C2×C17×#3802
C2×#253817