Finite Group Databasebeta

← #17053521

Subgroups — #17053521

Order 4352.

902 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic17C211yesmaximal, characteristic
2#14531740C211yesmaximal, characteristic, complemented
3D4×#63225C211yesmaximal, characteristic
4#14529477C211yesmaximal, characteristic, complemented
5#14529471C211yesmaximal, characteristic, complemented
6#14529370C211yesmaximal, characteristic, complemented
7#14529368C211yesmaximal, characteristic, complemented
8#14529120C211yesmaximal, characteristic
9#14528910C211yesmaximal, characteristic, complemented
10#14528787C211yesmaximal, characteristic, complemented
11#14528782C211yesmaximal, characteristic
12#14528650C211yesmaximal, characteristic, complemented
13#14528611C211yesmaximal, characteristic
14#14528609C211yesmaximal, characteristic
15C17×#1877C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×D4×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19C22×#63225—21no—
20SD16×Dic17C418yescomplemented
21C2×#10587488C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587474C2211yescharacteristic
28C2×#10587486C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30#11762716C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762705C2211yescharacteristic
33#11762694—21no—
34#11762692—21no—
35#11762682—21no—
36#11762681—21no—
37#11762678C2211yescharacteristic
38#11762678C2211yescharacteristic
39#11762673—21no—
40#11762667—21no—
41C4⋊C4×Dic17C2211yescharacteristic
42C4×#63225C2211yescharacteristic
43C4×C4⋊Dic17C2211yescharacteristic
44#11762660—21no—
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×SD16×C17C2211yescharacteristic
54C2×C4×D4×C17C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C17×#391C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×Dic17—21no—
64C23×Dic17—21no—
65C2×SD16×C17—22no—
66C2×SD16×C17—22no—
67C2×SD16×C17C2×C414yes—
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C4×D4×C17—22no—
71C4×D4×C17—41no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C22⋊C4×C17—21no—
75C2×C42×C17C2311yescharacteristic
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78Q8×Dic17—22no—
79Q8×Dic17C2×C414yescomplemented
80D4×Dic17—22no—
81D4×Dic17—22no—
82D4×Dic17—41no—
83D4×Dic17C2×C414yes—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—44no—
96C2×#63225D411yescharacteristic
97C2×#63225D411yescharacteristic
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17C2311yescharacteristic
102C2×C4×Dic17C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C17×#103—41no—
106C17×#102—22no—
107C17×#95—21no—
108C17×#95—21no—
109#10587506—41no—
110#10587506C2×C414yescomplemented
111#10587503—22no—
112#10587503C2×C414yes—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502D411yescharacteristic
120#10587502D411yescharacteristic
121#10587488C2×C414yescomplemented
122#10587486—41no—
123#10587485—22no—
124C8×Dic17C2×C414yescomplemented
125#10587474—41no—
126C24×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17C22×C412yes—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17C22×C412yes—
137C2×D4×C17#4112yes—
138C2×D4×C17#4112yes—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17—81no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2411yescharacteristic
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—24no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—42no—
154C22×Dic17—42no—
155C22×Dic17—42no—
156C22×Dic17—42no—
157C22×Dic17—81no—
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162SD16×C17—28no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17C22×C412yes—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C22⋊C4×C17—42no—
174C42×C17—21no—
175C42×C17—21no—
176C22⋊Dic17—24no—
177C22⋊Dic17—42no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—24no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17C22×C412yes—
186C4⋊Dic17C22×C412yes—
187#63225—22no—
188#63225—22no—
189#63225—42no—
190#63225—42no—
191#63225—42no—
192#63225—42no—
193#63225—81no—
194#63225—81no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—24no—
198C4×Dic17—81no—
199C4×Dic17—81no—
200C4×Dic17C22×C412yes—
201C4×Dic17#4112yes—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622C22×C412yes—
208#22762—171nomaximal, Sylow
209C23×C17—22no—
210C23×C17—22no—
211C23×C17—22no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17C22×D411yescharacteristic
217Q8×C17—22no—
218Q8×C17—22no—
219Q8×C17—41no—
220Q8×C17—41no—
221D4×C17—22no—
222D4×C17—22no—
223D4×C17—22no—
224D4×C17#11714yes—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—84no—
241C2×C4×C17C23×C411yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17H(5,𝔽2)11yescharacteristic
246C2×C4×C172−1+411yescharacteristic
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—48no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—84no—
269C2×Dic17C4×D412yes—
270C2×Dic17C4×D412yes—
271C2×Dic17AGL(1,ℤ/8)12yes—
272C2×Dic17#13712yes—
273C8×C17—22no—
274C8×C17—41no—
275#3622—22no—
276#3622—41no—
277C2×C4×SD16—171no—
278#2355—171no—
279D4×C4⋊C4—171no—
280#1935—171no—
281#1935—171no—
282#1920—171no—
283#1920—171no—
284#1877—171no—
285#1877D1711yescharacteristic
286#1834—171no—
287#1803—171no—
288#1802—171no—
289#1770—171no—
290#1762—171no—
291#1762—171no—
292C22×C17—24no—
293C22×C17—42no—
294C22×C17—42no—
295C22×C17—42no—
296C22×C17—42no—
297C22×C17—42no—
298C22×C17—42no—
299C22×C17C2×C4×D411yescharacteristic
300C22×C17#54611yescharacteristic
301C22×C17#54811yescharacteristic
302C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C17C2×#13711yescharacteristic
304C22×C17#57711yescharacteristic
305C22×C17#57711yescharacteristic
306C4×C17—41no—
307C4×C17—42no—
308C4×C17—41no—
309C4×C17—81no—
310C4×C17—81no—
311C4×C17C2×#11711yescharacteristic
312C4×C17#52011yescharacteristic
313C4×C17#55211yescharacteristic
314C4×C17#55311yescharacteristic
315Dic17—22no—
316Dic17—41no—
317Dic17—42no—
318Dic17—42no—
319Dic17—41no—
320Dic17—81no—
321Dic17—81no—
322Dic17#44014yes—
323C22×SD16C2×D1711yescharacteristic
324C2×C4×Q8—171no—
325C2×C4×D4—171no—
326C2×C4×D4—171no—
327C2×C4×D4C2×D1711yescharacteristic
328C22×C4⋊C4—341no—
329C4×SD16—178no—
330C2×#106—171no—
331C2×C4⋊C8—171no—
332C2×#103—171no—
333C2×#103—171no—
334C2×#103C2×D1711yescharacteristic
335C2×#102—171no—
336C2×#102—171no—
337C2×#102C2×D1711yescharacteristic
338C2×#97—171no—
339C2×#97—171no—
340C2×C4×C8—171no—
341#391—171no—
342#391C2×D1711yescharacteristic
343#389—171no—
344#389—171no—
345#387—341no—
346#387—341no—
347#385—341no—
348#385—341no—
349#384—171no—
350#384—171no—
351#382—341no—
352#380—341no—
353C4×C4⋊C4—171no—
354C4×C4⋊C4—171no—
355C4×C4⋊C4—171no—
356C4×C4⋊C4—341no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#340C2×D1711yescharacteristic
365#336—171no—
366#336C2×D1711yescharacteristic
367C2×C17—44no—
368C2×C17#235211yescharacteristic
369C2×C17C2×#44011yescharacteristic
370C2×C17#295211yescharacteristic
371C2×C17#332411yescharacteristic
372C2×C17#333111yescharacteristic
373C2×C17#337211yescharacteristic
374C2×C17#337311yescharacteristic
375C22×Q8C22×D1711yescharacteristic
376C22×D4C22×D1711yescharacteristic
377C23×C4—21no—
378C23×C4—341no—
379C23×C4—341no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16C4×D1714yes—
383C22×C8—171no—
384C22×C8—171no—
385C22×C8C22×D1711yescharacteristic
386C22×C8C22×D1711yescharacteristic
387C4×Q8—174no—
388C4×Q8—342no—
389C4×D4—22no—
390C4×D4—41no—
391C4×D4—174no—
392C4×D4—342no—
393C4×D4—342no—
394C4×D4—681no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4—684no—
409C2×C4⋊C4C22×D1711yescharacteristic
410C2×C4⋊C4C22×D1711yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—341no—
413C2×C22⋊C4—341no—
414C2×C42—171no—
415C2×C42—171no—
416C2×C42—341no—
417C2×C42—341no—
418C2×C42—341no—
419C2×C42C22×D1711yescharacteristic
420#106—174no—
421C4⋊C8—342no—
422#103—41no—
423#103—174no—
424#103—681no—
425#102—22no—
426#102—174no—
427#102—342no—
428#97—681no—
429#97—681no—
430C4×C8—174no—
431#95—21no—
432#95—21no—
433#95—171no—
434#95—171no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441C17#2276211yescharacteristic, complemented, Sylow
442C24—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×C4×D1712yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D1712yes—
453C2×D4#6325412yes—
454C2×D4#6325412yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—171no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—171no—
465C22×C4—344no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—682no—
471C22×C4—682no—
472C22×C4—682no—
473C22×C4—682no—
474C22×C4—1361no—
475C22×C4D4×D1711yescharacteristic
476C22×C4D4×D1711yescharacteristic
477C22×C4C23×D1711yescharacteristic
478SD16—28no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—172no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8C2×C4×D1712yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—172no—
496C4⋊C4—172no—
497C4⋊C4—342no—
498C4⋊C4—342no—
499C4⋊C4—344no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—682no—
507C4⋊C4—682no—
508C4⋊C4—682no—
509C4⋊C4—682no—
510C4⋊C4—1361no—
511C4⋊C4—1361no—
512C22⋊C4—42no—
513C22⋊C4—344no—
514C22⋊C4—682no—
515C42—21no—
516C42—21no—
517C42—172no—
518C42—172no—
519C42—344no—
520C42—341no—
521C42—341no—
522C42—1361no—
523C42—1361no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D1711yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1058756314yes—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4—172no—
557C2×C4—172no—
558C2×C4—172no—
559C2×C4—172no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—342no—
563C2×C4—342no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—341no—
569C2×C4—341no—
570C2×C4—688no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—1364no—
582C2×C4C22×C4×D1711yescharacteristic
583C2×C4C2×#6325411yescharacteristic
584C2×C4C2×#6325411yescharacteristic
585C2×C4#1058769111yescharacteristic
586C2×C4#41×D1711yescharacteristic
587C2×C4#1058769711yescharacteristic
588C8—22no—
589C8—41no—
590C8—342no—
591C8—681no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C4×D4×D1711yescharacteristic
600C22#1176344611yescharacteristic
601C22#1176346511yescharacteristic
602C22AGL(1,ℤ/8)×D1711yescharacteristic
603C22#137×D1711yescharacteristic
604C22#1176361511yescharacteristic
605C22#1176361611yescharacteristic
606C4—41no—
607C4—42no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—174no—
612C4—342no—
613C4—681no—
614C4—682no—
615C4—682no—
616C4—681no—
617C4—1361no—
618C4—1361no—
619C4C2×#1058756311yescharacteristic
620C4#1176328811yescharacteristic
621C4#1176349611yescharacteristic
622C4#1176350211yescharacteristic
623C2—44no—
624C2#1453171111yescharacteristic
625C2D17×#44011yescharacteristic
626C2#1453766211yescharacteristic
627C2#1454100811yescharacteristic
628C2#1454109311yescharacteristic
629C2#1454156011yescharacteristic
630C2#1454157111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1877C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1877D17
hypocenterC17#22762
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #14531740C2semidirect
4. #14529477C2semidirect
5. #14529471C2semidirect
6. #14529370C2semidirect
7. #14529368C2semidirect
9. #14528910C2semidirect
10. #14528787C2semidirect
12. #14528650C2semidirect
20. SD16×Dic17C4semidirect
79. Q8×Dic17C2×C4semidirect
110. #10587506C2×C4semidirect
121. #10587488C2×C4semidirect
124. C8×Dic17C2×C4semidirect
441. C17#22762semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic172
#145317402
D4×#632252
#145294772
#145294712
#145293702
#145293682
#145291202
#145289102
#145287872
#145287822
#145286502
#145286112
#145286092
C17×#18772
#2276217