Finite Group Databasebeta

← #17054158

Subgroups — #17054158

Order 4352.

886 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531757C211yesmaximal, characteristic, complemented
2#14531733C211yesmaximal, characteristic, complemented
3#14529661C211yesmaximal, characteristic
4#14529651C211yesmaximal, characteristic
5#14529413C211yesmaximal, characteristic, complemented
6#14529413C211yesmaximal, characteristic, complemented
7#14529406C211yesmaximal, characteristic, complemented
8#14529406C211yesmaximal, characteristic, complemented
9#14528952C211yesmaximal, characteristic, complemented
10#14528794C211yesmaximal, characteristic, complemented
11#14528784C211yesmaximal, characteristic, complemented
12#14528684C211yesmaximal, characteristic, complemented
13#106×Dic17C211yesmaximal, characteristic, complemented
14#14528656C211yesmaximal, characteristic, complemented
15C2×C17×#442C211yesmaximal, characteristic
16C2×#10587612—21no—
17C2×#10587489C2211yescharacteristic
18C2×#10587489C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587486C2211yescharacteristic
27C2×C8×Dic17C2211yescharacteristic
28#11762712C2211yescharacteristic
29#11762712C2211yescharacteristic
30#11762712C2211yescharacteristic
31#11762712C2211yescharacteristic
32#11762702—21no—
33#11762702—21no—
34#11762702—21no—
35#11762702—21no—
36#11762679C2211yescharacteristic
37#11762675C2211yescharacteristic
38#11762672—21no—
39#11762669—21no—
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45C4×C22⋊Dic17—21no—
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×D8×C17C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C2×C4×D4×C17C2211yescharacteristic
55C17×#442C418yescomplemented
56C2×C17×#106C2211yescharacteristic
57C2×C17×#102C2211yescharacteristic
58C2×C17×#102C2211yescharacteristic
59C2×C17×#97C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C2×D8×C17—22no—
65C2×D8×C17—22no—
66C2×D8×C17C2×C414yes—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C4×D4×C17C2×C414yes—
72C4×D4×C17C2×C414yes—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C42×C17C2311yescharacteristic
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85C2×C22⋊Dic17—21no—
86#10587612—44no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17D411yescharacteristic
90C2×C4⋊Dic17D411yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17C2311yescharacteristic
100C2×C4×Dic17C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C17×#106C2×C414yes—
104C17×#102C2×C414yes—
105C17×#102C2×C414yes—
106C17×#97C2×C414yes—
107#10587503—41no—
108#10587503—41no—
109#10587503—41no—
110#10587503—41no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502D411yescharacteristic
118#10587502D411yescharacteristic
119#10587489—22no—
120#10587489—41no—
121#10587488—22no—
122#10587488—41no—
123#10587486—22no—
124C8×Dic17—22no—
125C4⋊#3622—22no—
126C24×C17—21no—
127C24×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17C22×C412yes—
139C2×D4×C17C22×C412yes—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—24no—
143C22×C4×C17—24no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17—81no—
147C22×C4×C17—81no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17C2×D411yescharacteristic
156C22×Dic17C2×D411yescharacteristic
157C22×Dic17C2×D411yescharacteristic
158C22×Dic17C2×D411yescharacteristic
159D8×C17—28no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C22×C412yes—
163C2×C8×C17C22×C412yes—
164C2×C8×C17#4112yes—
165C2×C8×C17#4112yes—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17C22×C412yes—
169C4⋊C4×C17C22×C412yes—
170C22⋊C4×C17—24no—
171C22⋊C4×C17—24no—
172C42×C17C22×C412yes—
173C42×C17#4112yes—
174C22⋊Dic17—42no—
175C22⋊Dic17—42no—
176C22⋊Dic17—42no—
177C22⋊Dic17—42no—
178C22⋊Dic17—81no—
179C22⋊Dic17—81no—
180C22⋊Dic17—81no—
181C22⋊Dic17—81no—
182C4⋊Dic17—22no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—41no—
192C4⋊Dic17—42no—
193C4⋊Dic17—81no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C4×Dic17—42no—
199C4×Dic17—42no—
200C4×Dic17—81no—
201C4×Dic17—81no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208#22817—171nomaximal, Sylow
209C23×C17—22no—
210C23×C17—22no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17C22×D411yescharacteristic
224D4×C17—22no—
225D4×C17—22no—
226D4×C17—22no—
227D4×C17—22no—
228D4×C17—41no—
229D4×C17—41no—
230D4×C17—41no—
231D4×C17—41no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—48no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—84no—
244C2×C4×C17C4×D412yes—
245C2×C4×C17C4×D412yes—
246C2×C4×C17C23×C411yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C172−1+411yescharacteristic
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17AGL(1,ℤ/8)12yes—
276C8×C17—22no—
277C8×C17#11714yescomplemented
278#3622—41no—
279#3622—41no—
280C2×#442D1711yescharacteristic
281#2357—171no—
282#2354—171no—
283#1972—171no—
284#1970—171no—
285#1926—171no—
286#1926—171no—
287#1925—171no—
288#1925—171no—
289#1847—171no—
290#1804—171no—
291#1802—171no—
292#1776—171no—
293C4×#106—171no—
294#1771—171no—
295C22×C17—42no—
296C22×C17—42no—
297C22×C17—42no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—81no—
302C22×C17—81no—
303C22×C17—81no—
304C22×C17—81no—
305C22×C17—81no—
306C22×C17—81no—
307C22×C17—81no—
308C22×C17—81no—
309C22×C17C2×C4×D411yescharacteristic
310C22×C17#54711yescharacteristic
311C22×C17#55011yescharacteristic
312C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
313C22×C17#57511yescharacteristic
314C22×C17#57611yescharacteristic
315C22×C17#57711yescharacteristic
316C4×C17—22no—
317C4×C17—22no—
318C4×C17—42no—
319C4×C17—42no—
320C4×C17C2×#11711yescharacteristic
321C4×C17#52011yescharacteristic
322C4×C17#55511yescharacteristic
323C4×C17#55611yescharacteristic
324Dic17—22no—
325Dic17—41no—
326Dic17—41no—
327Dic17—41no—
328Dic17—81no—
329Dic17—81no—
330Dic17—81no—
331Dic17—81no—
332C22×D8C2×D1711yescharacteristic
333C2×C4×D4C2×D1711yescharacteristic
334C2×C4×D4C2×D1711yescharacteristic
335C2×#117—341no—
336#442Dic1718yescomplemented
337C2×#107—171no—
338C2×#107—171no—
339C2×#106—171no—
340C2×#106—171no—
341C2×#106C2×D1711yescharacteristic
342C2×C4⋊C8—171no—
343C2×#102—171no—
344C2×#102—171no—
345C2×#102—171no—
346C2×#102—171no—
347C2×#102C2×D1711yescharacteristic
348C2×#102C2×D1711yescharacteristic
349C2×#97—171no—
350C2×#97C2×D1711yescharacteristic
351C2×C4×C8—171no—
352#390—171no—
353#390—171no—
354#390—171no—
355#390—171no—
356#388—341no—
357#388—341no—
358#388—341no—
359#388—341no—
360#384—171no—
361#382—171no—
362#381—341no—
363#380—341no—
364C4×C4⋊C4—171no—
365C4×C4⋊C4—171no—
366C4×C4⋊C4—171no—
367C4×C4⋊C4—171no—
368C4×C4⋊C4—171no—
369C4×C22⋊C4—341no—
370#340—171no—
371#340—171no—
372#340—171no—
373#340—171no—
374#336—171no—
375#336—171no—
376C2×C17—82no—
377C2×C17—82no—
378C2×C17#235411yescharacteristic
379C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
380C2×C17#294611yescharacteristic
381C2×C17#334511yescharacteristic
382C2×C17#334711yescharacteristic
383C2×C17#339911yescharacteristic
384C2×C17#340111yescharacteristic
385C22×D4C22×D1711yescharacteristic
386C22×D4C22×D1711yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C2×D8—22no—
390C2×D8—22no—
391C2×D8C2×Dic1714yes—
392C22×C8—171no—
393C22×C8—171no—
394C22×C8C22×D1711yescharacteristic
395C22×C8C22×D1711yescharacteristic
396C4×D4—22no—
397C4×D4—22no—
398C4×D4C2×Dic1714yes—
399C4×D4C2×Dic1714yes—
400#117—684no—
401C2×C4⋊C4—171no—
402C2×C4⋊C4—171no—
403C2×C4⋊C4—171no—
404C2×C4⋊C4—171no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—171no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4—341no—
409C2×C4⋊C4C22×D1711yescharacteristic
410C2×C4⋊C4C22×D1711yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—341no—
414C2×C22⋊C4—341no—
415C2×C22⋊C4—341no—
416C2×C22⋊C4—341no—
417C2×C22⋊C4—341no—
418C2×C22⋊C4—341no—
419C2×C22⋊C4—341no—
420C2×C22⋊C4—341no—
421C2×C42—171no—
422C2×C42—171no—
423C2×C42—341no—
424C2×C42—341no—
425C2×C42—341no—
426C2×C42—341no—
427C2×C42C22×D1711yescharacteristic
428#107—342no—
429#107—681no—
430#106—342no—
431#106—681no—
432#106C2×Dic1714yes—
433C4⋊C8—342no—
434#102—681no—
435#102—681no—
436#102—681no—
437#102—681no—
438#102C2×Dic1714yes—
439#102C2×Dic1714yes—
440#97—342no—
441#97C2×Dic1714yes—
442C4×C8—342no—
443#95—171no—
444#95—171no—
445#95—341no—
446#95—341no—
447#95—341no—
448#95—341no—
449#95—341no—
450#95—341no—
451C17#2281711yescharacteristic, complemented, Sylow
452C24—21no—
453C24—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4C22×Dic1712yes—
465C2×D4C22×Dic1712yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—171no—
475C22×C4—171no—
476C22×C4—171no—
477C22×C4—171no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4D4×D1711yescharacteristic
483C22×C4D4×D1711yescharacteristic
484C22×C4C23×D1711yescharacteristic
485D8—28no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8#6325212yes—
495C2×C8#6325212yes—
496C2×C8C22×Dic1712yes—
497C2×C8C22×Dic1712yes—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—342no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—681no—
510C4⋊C4—682no—
511C4⋊C4—1361no—
512C4⋊C4C22×Dic1712yes—
513C4⋊C4C22×Dic1712yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—682no—
517C22⋊C4—682no—
518C22⋊C4—682no—
519C22⋊C4—682no—
520C22⋊C4—1361no—
521C22⋊C4—1361no—
522C22⋊C4—1361no—
523C22⋊C4—1361no—
524C42—341no—
525C42—341no—
526C42—341no—
527C42—341no—
528C42—682no—
529C42—682no—
530C42—1361no—
531C42—1361no—
532C42#6325412yes—
533C42C22×Dic1712yes—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×D4×D1711yescharacteristic
549D4—22no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
569C2×C4—172no—
570C2×C4—341no—
571C2×C4—341no—
572C2×C4—341no—
573C2×C4—341no—
574C2×C4—341no—
575C2×C4—341no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4D4×Dic1712yes—
594C2×C4D4×Dic1712yes—
595C2×C4C2×#6325211yescharacteristic
596C2×C4C2×#6325211yescharacteristic
597C2×C4C2×#6325411yescharacteristic
598C2×C4#1058769611yescharacteristic
599C2×C4#1058769711yescharacteristic
600C2×C4C23×Dic1711yescharacteristic
601C8—22no—
602C8—681no—
603C8—681no—
604C8#1058761214yescomplemented
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×D4×Dic1711yescharacteristic
620C22#1176346411yescharacteristic
621C22#1176348011yescharacteristic
622C22AGL(1,ℤ/8)×D1711yescharacteristic
623C22#1176360511yescharacteristic
624C22#1176361311yescharacteristic
625C22#1176361911yescharacteristic
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—342no—
631C4—681no—
632C4—681no—
633C4—681no—
634C4—1361no—
635C4—1361no—
636C4—1361no—
637C4—1361no—
638C4C2×#1058761211yescharacteristic
639C4#1176329211yescharacteristic
640C4#1176352411yescharacteristic
641C4#1176352911yescharacteristic
642C2—82no—
643C2—82no—
644C2#1453173311yescharacteristic
645C2AGL(1,ℤ/8)×Dic1711yescharacteristic
646C2#1453761711yescharacteristic
647C2#1454125911yescharacteristic
648C2#1454128911yescharacteristic
649C2#1454182711yescharacteristic
650C2#1454185511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#442C2
socleC23×C17C22×D4
prime radicalC22×D8×C17C22
cosocleC22×C4×C17C24
hypercenterC2×#442D17
hypocenterC17#22817
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. #14531757C2semidirect
2. #14531733C2semidirect
5. #14529413C2semidirect
6. #14529413C2semidirect
7. #14529406C2semidirect
8. #14529406C2semidirect
9. #14528952C2semidirect
10. #14528794C2semidirect
11. #14528784C2semidirect
12. #14528684C2semidirect
13. #106×Dic17C2semidirect
14. #14528656C2semidirect
55. C17×#442C4semidirect
277. C8×C17#117semidirect
336. #442Dic17semidirect
451. C17#22817semidirect
604. C8#10587612semidirect

Maximal subgroups

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

subgroupindex
#145317572
#145317332
#145296612
#145296512
#145294132
#145294062
#145289522
#145287942
#145287842
#145286842
#106×Dic172
#145286562
C2×C17×#4422
#2281717