Finite Group Databasebeta

← C2×#14532233 #17202279

Subgroups — C2×#14532233 #17202279

Order 4352.

3830 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587583C211yesmaximal, characteristic, complemented
2C22×#10587583C211yesmaximal, characteristic, complemented
3C22×C4×D68C211yesmaximal, characteristic
4C22×C4⋊C4×D17C211yesmaximal, characteristic, complemented
5#14532233C2116yesmaximal, direct product factor
6C2×#11762741C212yesmaximal, complemented
7C2×#11762731C212yesmaximal, complemented
8C2×#11762715C211yesmaximal, characteristic, complemented
9C2×#11762687C212yesmaximal
10C2×#11762666C211yesmaximal, characteristic, complemented
11C2×#11762666C212yesmaximal, complemented
12C2×C17×#389C211yesmaximal, characteristic, complemented
13C23×Dic34C2211yescharacteristic
14C23×D68C2211yescharacteristic
15C23×C4×D17C2211yescharacteristic
16C23×C4×D17C2212yes—
17C23×C4×D17C2211yescharacteristic
18C2×#10587583—28no—
19C2×#10587583—28no—
20C2×#10587583C2218yescomplemented
21C2×#10587583C2218yescomplemented
22C2×C4×D68—28no—
23C2×C4×D68C2218yes—
24C22×C4⋊Dic17C2211yescharacteristic
25C22×C4⋊Dic17C2212yes—
26C22×C4⋊Dic17C2211yescharacteristic
27C22×C4⋊Dic17C2212yes—
28C22×#63225C2212yes—
29C2×C4⋊C4×D17—28no—
30C2×C4⋊C4×D17C2218yescomplemented
31C22×#63227C2211yescharacteristic
32C22×#63227C2212yes—
33C22×#63227C2212yes—
34C22×#63227C2211yescharacteristic
35C22×#63227C2212yes—
36#11762741C22116yescomplemented
37#11762731C22116yescomplemented
38#11762715C2218yescomplemented
39#11762687C22116yes—
40#11762666C2218yescomplemented
41#11762666C22116yescomplemented
42C2×#10587502C2212yes—
43C2×#10587502C2212yes—
44C2×#10587502C2212yes—
45C22×C4⋊C4×C17C2211yescharacteristic
46C22×C4⋊C4×C17C2212yes—
47C22×C4⋊C4×C17C2211yescharacteristic
48C22×C42×C17C2211yescharacteristic
49C17×#389C2218yescomplemented
50C2×C17×#95C2212yes—
51C24×D17C2311yescharacteristic
52C24×D17C2311yescharacteristic
53C23×C4×C17C2311yescharacteristic
54C23×C4×C17C2312yes—
55C23×C4×C17C2312yes—
56C23×C4×C17C2311yescharacteristic
57C23×C4×C17C2311yescharacteristic
58C23×Dic17C2312yes—
59C23×Dic17C2312yes—
60C23×Dic17C2311yescharacteristic
61C23×Dic17C2311yescharacteristic
62C22×D68—24no—
63C22×D68—24no—
64C22×D68Q818yescomplemented
65C22×D68C2314yes—
66C22×C4×D17—24no—
67C22×C4×D17—24no—
68C22×C4×D17—28no—
69C22×C4×D17—28no—
70C22×C4×D17—24no—
71C22×C4×D17—24no—
72C22×C4×D17—24no—
73C22×C4×D17—44no—
74C22×C4×D17D418yes—
75C22×C4×D17C2314yes—
76C22×C4×D17C2318yes—
77C22×C4×D17C2314yes—
78C22×Dic34—24no—
79C22×Dic34—24no—
80C22×Dic34—24no—
81C22×Dic34C2314yes—
82C2×C4⋊C4×C17—24no—
83C2×C4⋊C4×C17—44no—
84C2×C4⋊C4×C17D418yescomplemented
85C2×C4⋊C4×C17C2314yes—
86C2×C4⋊C4×C17C2318yes—
87C2×C4⋊C4×C17C2314yes—
88C2×C42×C17—24no—
89C2×C42×C17C2314yes—
90C2×#63227—24no—
91C2×#63227—28no—
92C2×#63227—28no—
93C2×#63227—24no—
94C2×#63227—44no—
95C2×#63227C2314yes—
96C2×#63227C2318yes—
97C2×#63227C2314yes—
98C2×#63227C2318yes—
99C2×#63227C2318yes—
100C2×C4⋊Dic17—24no—
101C2×C4⋊Dic17—28no—
102C2×C4⋊Dic17—24no—
103C2×C4⋊Dic17—28no—
104C2×C4⋊Dic17C2314yes—
105C2×C4⋊Dic17C2318yes—
106C2×C4⋊Dic17C2314yes—
107C2×C4⋊Dic17C2318yes—
108C2×#63225—28no—
109C2×#63225C2318yes—
110#10587583—232no—
111#10587583—232no—
112C4⋊C4×D17—232no—
113C4×D68—232no—
114C17×#95C2318yes—
115#10587502C2318yes—
116#10587502C2318yes—
117#10587502C2318yes—
118C24×C17C2411yescharacteristic
119C23×D17—22no—
120C23×D17—22no—
121C23×D17—22no—
122C23×D17—22no—
123C23×D17—22no—
124C23×D17—22no—
125C23×D17—22no—
126C23×D17—22no—
127C23×D17—22no—
128C23×D17—22no—
129C23×D17C2×D414yes—
130C23×D17C2×Q814yes—
131C23×D17C2×Q814yes—
132C23×D17#4114yes—
133C23×D17C2412yes—
134C23×D17C2412yes—
135C22×C4×C17—22no—
136C22×C4×C17—24no—
137C22×C4×C17—22no—
138C22×C4×C17—24no—
139C22×C4×C17—24no—
140C22×C4×C17—22no—
141C22×C4×C17—24no—
142C22×C4×C17—22no—
143C22×C4×C17—22no—
144C22×C4×C17—24no—
145C22×C4×C17C2×D418yes—
146C22×C4×C17C2×D414yes—
147C22×C4×C17C2×D414yes—
148C22×C4×C17C2×Q814yes—
149C22×C4×C17#4114yes—
150C22×C4×C17C2412yes—
151C22×C4×C17C2414yes—
152C22×C4×C17C2414yes—
153C22×C4×C17C2412yes—
154C22×C4×C17C2412yes—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—24no—
158C22×Dic17—24no—
159C22×Dic17—24no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—24no—
163C22×Dic17—22no—
164C22×Dic17—24no—
165C22×Dic17—24no—
166C22×Dic17C2×D414yes—
167C22×Dic17C2414yes—
168C22×Dic17C2414yes—
169C22×Dic17C2412yes—
170C22×Dic17C2412yes—
171C2×D68—232no—
172C2×D68—28no—
173C2×D68—28no—
174C2×D68C2×Q8116yescomplemented
175C2×C4×D17—232no—
176C2×C4×D17—28no—
177C2×C4×D17—28no—
178C2×C4×D17—216no—
179C2×C4×D17—28no—
180C2×C4×D17—216no—
181C2×C4×D17—28no—
182C2×C4×D17—28no—
183C2×C4×D17—48no—
184C2×C4×D17—432no—
185C2×C4×D17—416no—
186C2×C4×D17C2×D4116yes—
187C2×Dic34—28no—
188C2×Dic34—28no—
189C2×Dic34—28no—
190C2×Dic34—416no—
191C4⋊C4×C17—28no—
192C4⋊C4×C17—48no—
193C4⋊C4×C17C2×D4116yescomplemented
194C42×C17—28no—
195#63227—28no—
196#63227—216no—
197#63227—216no—
198#63227—28no—
199#63227—48no—
200C4⋊Dic17—28no—
201C4⋊Dic17—216no—
202C4⋊Dic17—28no—
203C4⋊Dic17—216no—
204#63225—216no—
205C2×#2417—171nomaximal, Sylow
206C23×C17C22×D412yes—
207C23×C17C22×D412yes—
208C23×C17C22×Q812yes—
209C23×C17C2×#4112yes—
210C23×C17C2×#4112yes—
211C23×C17H(5,𝔽2)12yes—
212C23×C172−1+412yes—
213C23×C17C2511yescharacteristic
214C22×D17—28no—
215C22×D17—28no—
216C22×D17—28no—
217C22×D17—28no—
218C22×D17—28no—
219C22×D17—28no—
220C22×D17—22no—
221C22×D17—22no—
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—28no—
225C22×D17—28no—
226C22×D17—28no—
227C22×D17—22no—
228C22×D17—22no—
229C22×D17—22no—
230C22×D17—22no—
231C22×D17—22no—
232C22×D17—22no—
233C22×D17—44no—
234C22×D17—44no—
235C22×D17—44no—
236C22×D17—44no—
237C22×D17C22⋊Q8116yescomplemented
238C22×D17C22×D414yes—
239C22×D17C22×Q814yes—
240C22×D17C22×Q814yes—
241C22×D17C2×#4114yes—
242C2×C4×C17—28no—
243C2×C4×C17—216no—
244C2×C4×C17—22no—
245C2×C4×C17—24no—
246C2×C4×C17—22no—
247C2×C4×C17—24no—
248C2×C4×C17—22no—
249C2×C4×C17—24no—
250C2×C4×C17—22no—
251C2×C4×C17—24no—
252C2×C4×C17—22no—
253C2×C4×C17—24no—
254C2×C4×C17—48no—
255C2×C4×C17—44no—
256C2×C4×C17#121116yes—
257C2×C4×C17C22×D418yes—
258C2×C4×C17C22×D414yes—
259C2×C4×C17C22×D414yes—
260C2×C4×C17C22×Q814yes—
261C2×C4×C17C2×#4114yes—
262C2×Dic17—28no—
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—24no—
266C2×Dic17—24no—
267C2×Dic17—24no—
268C2×Dic17—22no—
269C2×Dic17—22no—
270C2×Dic17—24no—
271C2×Dic17—24no—
272C2×Dic17—22no—
273C2×Dic17—24no—
274C2×Dic17—48no—
275C2×Dic17—44no—
276C2×Dic17—48no—
277C2×Dic17C22×D414yes—
278D68—232no—
279C4×D17—232no—
280C4×D17—432no—
281C4×D17—416no—
282Dic34—416no—
283C22×C22⋊Q8—171no—
284C22×C22⋊Q8—171no—
285C22×C4×D4—171no—
286C23×C4⋊C4—171no—
287#2417—1716no—
288C2×#398—172no—
289C2×#396—172no—
290C2×#391—171no—
291C2×#389D1711yescharacteristic, complemented
292C2×#386—172no—
293C2×#380—172no—
294C2×#380—171no—
295C22×C17C2×#12114yes—
296C22×C17C2×C22⋊Q814yes—
297C22×C17#53714yes—
298C22×C17#54614yes—
299C22×C17D4×Q814yes—
300C22×C17#55214yes—
301C22×C17#55414yes—
302C22×C17C23×D411yescharacteristic
303C22×C17C23×D411yescharacteristic
304C22×C17C23×Q811yescharacteristic
305C22×C17C22×#4111yescharacteristic
306C22×C17C22×#4111yescharacteristic
307C22×C17C2×H(5,𝔽2)11yescharacteristic
308C22×C17C2×2−1+411yescharacteristic
309C2×D17—24no—
310C2×D17—232no—
311C2×D17—24no—
312C2×D17—24no—
313C2×D17—24no—
314C2×D17—24no—
315C2×D17—24no—
316C2×D17—24no—
317C2×D17—24no—
318C2×D17—24no—
319C2×D17—42no—
320C2×D17—416no—
321C2×D17—42no—
322C2×D17—42no—
323C2×D17—42no—
324C2×D17C2×C22⋊Q818yescomplemented
325C4×C17—24no—
326C4×C17—28no—
327C4×C17—42no—
328C4×C17—44no—
329C4×C17C2×#12118yes—
330Dic17—24no—
331Dic17—44no—
332Dic17—42no—
333Dic17—44no—
334C23×Q8—171no—
335C23×D4—171no—
336C24×C4—171no—
337C24×C4—171no—
338C24×C4—172no—
339C2×C22⋊Q8—178no—
340C2×C22⋊Q8—178no—
341C2×C22⋊Q8—348no—
342C2×C22⋊Q8—348no—
343C2×C4×D4—178no—
344C2×C4×D4—348no—
345C22×C4⋊C4—172no—
346C22×C4⋊C4—172no—
347C22×C4⋊C4—171no—
348C22×C4⋊C4—172no—
349C22×C4⋊C4—171no—
350C22×C4⋊C4—178no—
351C22×C4⋊C4—348no—
352C22×C4⋊C4C2×D1711yescharacteristic
353C22×C4⋊C4C2×D1712yes—
354C22×C4⋊C4C2×D1711yescharacteristic
355C22×C22⋊C4—171no—
356C22×C22⋊C4—172no—
357C22×C22⋊C4—171no—
358C22×C22⋊C4—172no—
359C22×C22⋊C4—172no—
360C22×C42C2×D1711yescharacteristic
361#398—1716no—
362#396—1716no—
363#391—178no—
364#389C2×D1718yescomplemented
365#386—1716no—
366#380—178no—
367#380—1716no—
368C2×#95—172no—
369C2×#95—172no—
370C2×#95—172no—
371C2×#95C2×D1712yes—
372C2×C17#241718yescomplemented
373C2×C17C22×#12111yescharacteristic
374C2×C17C22×C22⋊Q811yescharacteristic
375C2×C17C2×#53711yescharacteristic
376C2×C17C2×#54611yescharacteristic
377C2×C17C2×D4×Q811yescharacteristic
378C2×C17C2×#55211yescharacteristic
379C2×C17C2×#55411yescharacteristic
380D17—28no—
381D17—44no—
382C25—171no—
383C25—171no—
384C22×Q8—174no—
385C22×Q8—344no—
386C22×Q8—344no—
387C22×Q8—344no—
388C22×D4—178no—
389C22×D4—174no—
390C22×D4—344no—
391C22×D4—344no—
392C23×C4—171no—
393C23×C4—178no—
394C23×C4—174no—
395C23×C4—172no—
396C23×C4—171no—
397C23×C4—172no—
398C23×C4—174no—
399C23×C4—178no—
400C23×C4—348no—
401C23×C4—344no—
402C23×C4—344no—
403C23×C4—348no—
404C23×C4—344no—
405C23×C4—344no—
406C23×C4—344no—
407C23×C4—684no—
408C23×C4C22×D1712yes—
409C23×C4C22×D1711yescharacteristic
410C23×C4C22×D1712yes—
411C23×C4C22×D1711yescharacteristic
412C23×C4C22×D1711yescharacteristic
413C22⋊Q8—3432no—
414C22⋊Q8—3432no—
415C4×D4—3432no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—44no—
418C2×C4⋊C4—174no—
419C2×C4⋊C4—178no—
420C2×C4⋊C4—174no—
421C2×C4⋊C4—178no—
422C2×C4⋊C4—178no—
423C2×C4⋊C4—344no—
424C2×C4⋊C4—348no—
425C2×C4⋊C4—348no—
426C2×C4⋊C4—344no—
427C2×C4⋊C4—348no—
428C2×C4⋊C4—3432no—
429C2×C4⋊C4D6818yescomplemented
430C2×C4⋊C4C22×D1714yes—
431C2×C4⋊C4C22×D1718yes—
432C2×C4⋊C4C22×D1714yes—
433C2×C22⋊C4—174no—
434C2×C22⋊C4—174no—
435C2×C22⋊C4—178no—
436C2×C22⋊C4—178no—
437C2×C22⋊C4—178no—
438C2×C22⋊C4—344no—
439C2×C22⋊C4—344no—
440C2×C22⋊C4—348no—
441C2×C22⋊C4—348no—
442C2×C22⋊C4—684no—
443C2×C42—24no—
444C2×C42C22×D1714yes—
445#95—178no—
446#95—178no—
447#95—178no—
448#95C22×D1718yes—
449C17C2×#241711yescharacteristic, complemented, Sylow
450C24—174no—
451C24—174no—
452C24—174no—
453C24—172no—
454C24—174no—
455C24—172no—
456C24—342no—
457C24—342no—
458C24—342no—
459C24—342no—
460C24—342no—
461C24—342no—
462C24—342no—
463C24—342no—
464C24—342no—
465C24—342no—
466C24C23×D1711yescharacteristic
467C2×Q8—348no—
468C2×Q8—348no—
469C2×Q8—348no—
470C2×Q8—6816no—
471C2×D4—1716no—
472C2×D4—348no—
473C2×D4—3432no—
474C2×D4—348no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—174no—
486C22×C4—172no—
487C22×C4—1716no—
488C22×C4—174no—
489C22×C4—174no—
490C22×C4—172no—
491C22×C4—3416no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—348no—
495C22×C4—348no—
496C22×C4—3432no—
497C22×C4—344no—
498C22×C4—342no—
499C22×C4—344no—
500C22×C4—342no—
501C22×C4—344no—
502C22×C4—344no—
503C22×C4—342no—
504C22×C4—344no—
505C22×C4—344no—
506C22×C4—3416no—
507C22×C4—348no—
508C22×C4—348no—
509C22×C4—348no—
510C22×C4—688no—
511C22×C4—6832no—
512C22×C4—6816no—
513C22×C4C2×D6818yes—
514C22×C4C2×D6814yes—
515C22×C4D4×D1714yes—
516C22×C4Q8×D1714yes—
517C22×C4#6325614yes—
518C22×C4C23×D1712yes—
519C22×C4C23×D1714yes—
520C22×C4C23×D1714yes—
521C22×C4C23×D1712yes—
522C22×C4C23×D1712yes—
523C4⋊C4—28no—
524C4⋊C4—48no—
525C4⋊C4—348no—
526C4⋊C4—3416no—
527C4⋊C4—3416no—
528C4⋊C4—348no—
529C4⋊C4—3416no—
530C4⋊C4C2×D68116yescomplemented
531C22⋊C4—348no—
532C22⋊C4—348no—
533C22⋊C4—3416no—
534C22⋊C4—3416no—
535C22⋊C4—688no—
536C42—28no—
537C23—1716no—
538C23—174no—
539C23—174no—
540C23—174no—
541C23—174no—
542C23—348no—
543C23—348no—
544C23—348no—
545C23—348no—
546C23—342no—
547C23—342no—
548C23—342no—
549C23—342no—
550C23—348no—
551C23—348no—
552C23—348no—
553C23—348no—
554C23—348no—
555C23—342no—
556C23—342no—
557C23—342no—
558C23—342no—
559C23—342no—
560C23—342no—
561C23—684no—
562C23—684no—
563C23—684no—
564C23—684no—
565C23C22×D6812yes—
566C23C2×D4×D1712yes—
567C23#1058769112yes—
568C23C2×Q8×D1712yes—
569C23C2×#6325612yes—
570C23#41×D1712yes—
571C23#1058769712yes—
572C23C24×D1711yescharacteristic
573Q8—6816no—
574D4—3432no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—216no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—174no—
590C2×C4—348no—
591C2×C4—342no—
592C2×C4—342no—
593C2×C4—3432no—
594C2×C4—342no—
595C2×C4—344no—
596C2×C4—344no—
597C2×C4—344no—
598C2×C4—342no—
599C2×C4—344no—
600C2×C4—342no—
601C2×C4—344no—
602C2×C4—344no—
603C2×C4—6832no—
604C2×C4—684no—
605C2×C4—688no—
606C2×C4—688no—
607C2×C4—6816no—
608C2×C4#10587581116yes—
609C2×C4C22×D6818yes—
610C2×C4C22×D6814yes—
611C2×C4C2×D4×D1714yes—
612C2×C4C2×Q8×D1714yes—
613C2×C4C2×#6325614yes—
614C22—178no—
615C22—344no—
616C22—3432no—
617C22—344no—
618C22—344no—
619C22—344no—
620C22—344no—
621C22—344no—
622C22—344no—
623C22—344no—
624C22—344no—
625C22—682no—
626C22—6816no—
627C22—682no—
628C22—682no—
629C22—682no—
630C22C2×#1058758114yes—
631C22C22⋊Q8×D1714yes—
632C22#1176338814yes—
633C22#1176345414yes—
634C22Q8×D6814yes—
635C22#1176349714yes—
636C22#1176352014yes—
637C22C22×D4×D1711yescharacteristic
638C22C23×D6811yescharacteristic
639C22C22×Q8×D1711yescharacteristic
640C22C2×#41×D1711yescharacteristic
641C22C22×#6325611yescharacteristic
642C22C2×#1058769111yescharacteristic
643C22C2×#1058769711yescharacteristic
644C4—28no—
645C4—24no—
646C4—42no—
647C4—44no—
648C4—344no—
649C4—684no—
650C4—682no—
651C4—684no—
652C4C2×#1058758118yes—
653C2—348no—
654C2—684no—
655C2#1453223318yesdirect product factor
656C2C22×#1058758111yescharacteristic
657C2C2×C22⋊Q8×D1711yescharacteristic
658C2C2×#1176338811yescharacteristic
659C2C2×#1176345411yescharacteristic
660C2C2×Q8×D6811yescharacteristic
661C2C2×#1176349711yescharacteristic
662C2C2×#1176352011yescharacteristic
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC2×C17×#389C2
socleC24×C17C24
prime radicalC23×D68C22
hypercenterC2×#389D17
hypocenterC17C2×#2417
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#10587583C2semidirect
2. C22×#10587583C2semidirect
4. C22×C4⋊C4×D17C2semidirect
5. #14532233C2direct (N × H)
6. C2×#11762741C2semidirect
7. C2×#11762731C2semidirect
8. C2×#11762715C2semidirect
10. C2×#11762666C2semidirect
11. C2×#11762666C2semidirect
12. C2×C17×#389C2semidirect
20. C2×#10587583C22semidirect
21. C2×#10587583C22semidirect
30. C2×C4⋊C4×D17C22semidirect
36. #11762741C22semidirect
37. #11762731C22semidirect
38. #11762715C22semidirect
40. #11762666C22semidirect
41. #11762666C22semidirect
49. C17×#389C22semidirect
64. C22×D68Q8semidirect
84. C2×C4⋊C4×C17D4semidirect
174. C2×D68C2×Q8semidirect
193. C4⋊C4×C17C2×D4semidirect
237. C22×D17C22⋊Q8semidirect
291. C2×#389D17semidirect
324. C2×D17C2×C22⋊Q8semidirect
364. #389C2×D17semidirect
372. C2×C17#2417semidirect
429. C2×C4⋊C4D68semidirect
449. C17C2×#2417semidirect
530. C4⋊C4C2×D68semidirect
655. C2#14532233direct (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×C4×D682
C22×C4⋊C4×D172
#145322332
C2×#117627412
C2×#117627312
C2×#117627152
C2×#117626872
C2×#117626662
C2×C17×#3892
C2×#241717