Finite Group Databasebeta

← #17041957

Subgroups — #17041957

Order 4352.

988 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763087C211yesmaximal, characteristic
2#14531355C211yesmaximal, characteristic, complemented
3C4×D4×Dic17C211yesmaximal, characteristic
4#14529734C211yesmaximal, characteristic, complemented
5#14529544C211yesmaximal, characteristic, complemented
6#14529377C211yesmaximal, characteristic, complemented
7#14529373C211yesmaximal, characteristic, complemented
8#14529045C211yesmaximal, characteristic
9#14528782C211yesmaximal, characteristic
10#14528684C211yesmaximal, characteristic, complemented
11#14528680C211yesmaximal, characteristic, complemented
12#14528636C211yesmaximal, characteristic, complemented
13#14528611C211yesmaximal, characteristic
14#14528611C211yesmaximal, characteristic
15C17×#1864C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×#10587573—21no—
18C2×D4×Dic17C2211yescharacteristic
19C2×D4×Dic17C2211yescharacteristic
20C22×C4×Dic17—21no—
21#11763087C418yescomplemented
22C2×#10587489C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587485C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31#11762713—21no—
32#11762707C2211yescharacteristic
33#11762704—21no—
34#11762678C2211yescharacteristic
35C4⋊C4×Dic17C2211yescharacteristic
36C4⋊C4×Dic17C2211yescharacteristic
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40#11762660—21no—
41C22⋊C4×Dic17—21no—
42C22⋊C4×Dic17—21no—
43C4×C22⋊Dic17—21no—
44C42×Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762564C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×Dic17—21no—
63C23×Dic17—21no—
64C22×Dic34—21no—
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77D4×Dic17—22no—
78D4×Dic17—22no—
79D4×Dic17C2×C414yes—
80D4×Dic17Q814yescomplemented
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—44no—
95C2×C4×Dic17D411yescharacteristic
96C2×C4×Dic17D411yescharacteristic
97C2×C4×Dic17D411yescharacteristic
98C2×C4×Dic17D411yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C2×C4×Dic17C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103#10587574—22no—
104#10587574C2×C414yescomplemented
105#10587573—44no—
106C17×#106—22no—
107C17×#102—22no—
108C17×#102C2×C414yes—
109C17×#95—21no—
110#10587503—22no—
111#10587503C2×C414yes—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587489C2×C414yescomplemented
119#10587486—22no—
120#10587486—22no—
121#10587485—22no—
122#10587485C2×C414yescomplemented
123#10587479—22no—
124#10587478C2×C414yescomplemented
125C24×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17C22×C412yes—
130C2×D4×C17C2×Q812yes—
131C2×D4×C17#4112yes—
132C22×C4×C17—24no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—42no—
137C22×C4×C17—42no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—24no—
145C22×Dic17—24no—
146C22×Dic17—21no—
147C22×Dic17—42no—
148C22×Dic17—42no—
149C22×Dic17—42no—
150C22×Dic17—42no—
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C2×Dic34—24no—
156C2×Dic34—42no—
157C2×Dic34—42no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17C22×C412yes—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—42no—
169C4⋊C4×C17C22×C412yes—
170C4⋊C4×C17#4112yes—
171C22⋊C4×C17—24no—
172C42×C17—21no—
173C42×C17—21no—
174C22⋊Dic17—24no—
175C22⋊Dic17—24no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17C22×C412yes—
182C4⋊Dic17C22×C412yes—
183C4⋊Dic17C2×Q812yes—
184#63225—24no—
185#63225—42no—
186#63225—42no—
187#63225—42no—
188C4×Dic17—22no—
189C4×Dic17—22no—
190C4×Dic17—22no—
191C4×Dic17—41no—
192C4×Dic17—42no—
193C4×Dic17—42no—
194C4×Dic17—42no—
195C4×Dic17—42no—
196C4×Dic17—42no—
197C4×Dic17C22×C412yes—
198C4×Dic17C2×Q812yes—
199C4×Dic17#4112yes—
200C4×Dic17#4112yes—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622C22×C412yes—
207#21613—171nomaximal, Sylow
208C23×C17—22no—
209C23×C17—22no—
210C23×C17—22no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17C22×D411yescharacteristic
216D4×C17—22no—
217D4×C17—22no—
218D4×C17—22no—
219D4×C17C4×Q814yescomplemented
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—21no—
224C2×C4×C17—22no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—48no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17#13712yes—
236C2×C4×C17C23×C411yescharacteristic
237C2×C4×C17C22×Q811yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—48no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—48no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17C4×D412yes—
263C2×Dic17C4×D412yes—
264C2×Dic17C22⋊Q812yes—
265C2×Dic17C22⋊Q812yes—
266C2×Dic17AGL(1,ℤ/8)12yes—
267C2×Dic17AGL(1,ℤ/8)12yes—
268Dic34—48no—
269C8×C17—22no—
270C8×C17—41no—
271#3622—22no—
272#3622—41no—
273C2×#474—171no—
274C4×C4⋊Q8—171no—
275C42×D4—171no—
276#1988—171no—
277#1948—171no—
278#1921—171no—
279#1921—171no—
280#1864—171no—
281#1864D1711yescharacteristic
282#1802—171no—
283#1776—171no—
284#1776—171no—
285#1767—171no—
286#1762—171no—
287#1762—171no—
288C22×C17—24no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17—42no—
292C22×C17—42no—
293C22×C17—42no—
294C22×C17—42no—
295C22×C17C2×C4×D411yescharacteristic
296C22×C17C2×C22⋊Q811yescharacteristic
297C22×C17#52511yescharacteristic
298C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C17C2×#13711yescharacteristic
301C22×C17#57511yescharacteristic
302C4×C17—22no—
303C4×C17—42no—
304C4×C17—41no—
305C4×C17—42no—
306C4×C17—81no—
307C4×C17C2×C4×Q811yescharacteristic
308C4×C17C4×#4111yescharacteristic
309C4×C17#52911yescharacteristic
310C4×C17#53311yescharacteristic
311Dic17—22no—
312Dic17—22no—
313Dic17—22no—
314Dic17—42no—
315Dic17—42no—
316Dic17—42no—
317Dic17—81no—
318Dic17#44214yes—
319C2×C4⋊Q8—171no—
320C2×C4×Q8—341no—
321C2×C4×D4—171no—
322C2×C4×D4—171no—
323C2×C4×D4C2×D1711yescharacteristic
324C22×C42—341no—
325#474—178no—
326C2×#107—171no—
327C2×#107—171no—
328C2×#106—171no—
329C2×#106C2×D1711yescharacteristic
330C2×C4⋊C8—171no—
331C2×C4⋊C8—171no—
332C2×#102—171no—
333C2×#102—171no—
334C2×#102C2×D1711yescharacteristic
335C2×#102C2×D1711yescharacteristic
336C2×#97—171no—
337C2×#97—171no—
338#391—341no—
339#389—171no—
340#389—341no—
341#389C2×D1711yescharacteristic
342#384—171no—
343C4×C4⋊C4—171no—
344C4×C4⋊C4—171no—
345C4×C4⋊C4—171no—
346C4×C4⋊C4—171no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—341no—
349C4×C22⋊C4—341no—
350C4×C22⋊C4—341no—
351C4×C22⋊C4—341no—
352C43—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340C2×D1711yescharacteristic
361#336—171no—
362#336C2×D1711yescharacteristic
363C2×C17—44no—
364C2×C17C4×C22⋊Q811yescharacteristic
365C2×C17C2×#44211yescharacteristic
366C2×C17C4×#13711yescharacteristic
367C2×C17#307711yescharacteristic
368C2×C17#307911yescharacteristic
369C2×C17#311011yescharacteristic
370C2×C17#311511yescharacteristic
371C22×Q8—341no—
372C22×D4C22×D1711yescharacteristic
373C23×C4—21no—
374C23×C4—341no—
375C23×C4—341no—
376C22×C8—171no—
377C22×C8—171no—
378C22×C8C22×D1711yescharacteristic
379C22×C8C22×D1711yescharacteristic
380C4⋊Q8—174no—
381C4⋊Q8—342no—
382C4×Q8—684no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—174no—
386C4×D4—174no—
387C4×D4—342no—
388C4×D4—342no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C4⋊C4C22×D1711yescharacteristic
400C2×C4⋊C4C22×D1711yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—341no—
403C2×C22⋊C4—341no—
404C2×C42—171no—
405C2×C42—171no—
406C2×C42—171no—
407C2×C42—171no—
408C2×C42—171no—
409C2×C42—171no—
410C2×C42—341no—
411C2×C42—341no—
412C2×C42—341no—
413C2×C42—341no—
414C2×C42—341no—
415C2×C42—684no—
416C2×C42C22×D1711yescharacteristic
417#107—174no—
418#107—174no—
419#106—22no—
420#106—342no—
421C4⋊C8—174no—
422C4⋊C8—342no—
423#102—22no—
424#102—174no—
425#102—342no—
426#102C4×D1714yes—
427#97—342no—
428#97—342no—
429#95—21no—
430#95—341no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435#95—341no—
436C17#2161311yescharacteristic, complemented, Sylow
437C24—21no—
438C2×Q8—344no—
439C2×Q8—682no—
440C2×Q8—682no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C4×D1712yes—
445C2×D4#6325412yes—
446C2×D4Q8×D1712yes—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—171no—
454C22×C4—171no—
455C22×C4—171no—
456C22×C4—171no—
457C22×C4—344no—
458C22×C4—344no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—682no—
464C22×C4—682no—
465C22×C4—682no—
466C22×C4—682no—
467C22×C4D4×D1711yescharacteristic
468C22×C4D4×D1711yescharacteristic
469C22×C4C23×D1711yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—172no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8C2×C4×D1712yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—172no—
488C4⋊C4—172no—
489C4⋊C4—172no—
490C4⋊C4—344no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—682no—
497C4⋊C4—682no—
498C4⋊C4—682no—
499C4⋊C4C2×C4×D1712yes—
500C4⋊C4#6325412yes—
501C22⋊C4—24no—
502C22⋊C4—344no—
503C22⋊C4—344no—
504C42—21no—
505C42—21no—
506C42—172no—
507C42—172no—
508C42—172no—
509C42—172no—
510C42—342no—
511C42—342no—
512C42—342no—
513C42—681no—
514C42—682no—
515C42—682no—
516C42—682no—
517C42—682no—
518C42—682no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D4×D1711yescharacteristic
527Q8—688no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4#1058757314yescomplemented
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—172no—
548C2×C4—172no—
549C2×C4—172no—
550C2×C4—172no—
551C2×C4—172no—
552C2×C4—172no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—342no—
560C2×C4—342no—
561C2×C4—342no—
562C2×C4—688no—
563C2×C4—688no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4#1058760112yes—
574C2×C4C22×C4×D1711yescharacteristic
575C2×C4C2×#6325411yescharacteristic
576C2×C4C2×#6325411yescharacteristic
577C2×C4C2×Q8×D1711yescharacteristic
578C2×C4#41×D1711yescharacteristic
579C2×C4#41×D1711yescharacteristic
580C8—22no—
581C8—41no—
582C8—342no—
583C8—681no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22C4×D4×D1711yescharacteristic
592C22C22⋊Q8×D1711yescharacteristic
593C22#1176331511yescharacteristic
594C22AGL(1,ℤ/8)×D1711yescharacteristic
595C22AGL(1,ℤ/8)×D1711yescharacteristic
596C22C2×#1058760111yescharacteristic
597C22#1176360611yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—174no—
604C4—342no—
605C4—342no—
606C4—342no—
607C4—682no—
608C4—682no—
609C4—682no—
610C4—1361no—
611C4C2×#1058757311yescharacteristic
612C4C4×#6325411yescharacteristic
613C4#1176333611yescharacteristic
614C4#1176336111yescharacteristic
615C2—44no—
616C2#1453131611yescharacteristic
617C2D17×#44211yescharacteristic
618C2C4×#1058760111yescharacteristic
619C2#1453857911yescharacteristic
620C2#1453859611yescharacteristic
621C2#1453887811yescharacteristic
622C2#1453892811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1864C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1864D17
hypocenterC17#21613
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #14531355C2semidirect
4. #14529734C2semidirect
5. #14529544C2semidirect
6. #14529377C2semidirect
7. #14529373C2semidirect
10. #14528684C2semidirect
11. #14528680C2semidirect
12. #14528636C2semidirect
21. #11763087C4semidirect
80. D4×Dic17Q8semidirect
104. #10587574C2×C4semidirect
118. #10587489C2×C4semidirect
122. #10587485C2×C4semidirect
124. #10587478C2×C4semidirect
219. D4×C17C4×Q8semidirect
436. C17#21613semidirect
531. D4#10587573semidirect

Maximal subgroups

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

subgroupindex
C2×#117630872
#145313552
C4×D4×Dic172
#145297342
#145295442
#145293772
#145293732
#145290452
#145287822
#145286842
#145286802
#145286362
#145286112
C17×#18642
#2161317