Finite Group Databasebeta

← #17038639

Subgroups — #17038639

Order 4352.

950 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762919C211yesmaximal, characteristic
2#14531249C211yesmaximal, characteristic, complemented
3#14531245C211yesmaximal, characteristic, complemented
4#14529657C211yesmaximal, characteristic
5#14529647C211yesmaximal, characteristic
6#14529526C211yesmaximal, characteristic, complemented
7#14529511C211yesmaximal, characteristic, complemented
8#14529482C211yesmaximal, characteristic, complemented
9#14529478C211yesmaximal, characteristic, complemented
10#14529468C211yesmaximal, characteristic, complemented
11#14529464C211yesmaximal, characteristic, complemented
12#14528682C211yesmaximal, characteristic, complemented
13#14528610C211yesmaximal, characteristic, complemented
14#97×Dic17C211yesmaximal, characteristic, complemented
15C17×#1762C211yesmaximal, characteristic, complemented
16C22×#63228C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×#10587612—21no—
19C2×#10587558—21no—
20#11762919C418yescomplemented
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587492C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32C2×C8×Dic17C2211yescharacteristic
33#11762712C2211yescharacteristic
34#11762712C2211yescharacteristic
35#11762710C2211yescharacteristic
36#11762710C2211yescharacteristic
37#11762708C2211yescharacteristic
38#11762708C2211yescharacteristic
39#11762705C2211yescharacteristic
40#11762705C2211yescharacteristic
41#11762702—21no—
42#11762702—21no—
43#11762698—21no—
44#11762698—21no—
45C4×C4⋊Dic17C2211yescharacteristic
46C4×#63227—21no—
47C4×C22⋊Dic17—21no—
48C42×Dic17C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C17×#97C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C4×D4×C17—22no—
67C4×D4×C17C2×C414yes—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C42×C17—21no—
73C2×C42×C17C2311yescharacteristic
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78C2×#63228—22no—
79C2×#63228—22no—
80C2×#63228C2×C414yes—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×#63227—21no—
85C2×#63227—21no—
86#10587612—44no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×#63225—21no—
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17D411yescharacteristic
96C2×C4×Dic17D411yescharacteristic
97C2×C4×Dic17D411yescharacteristic
98C2×C4×Dic17D411yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C2×C4×Dic17C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C4×D68—22no—
104C4×D68C2×C414yes—
105#10587558—44no—
106C17×#102—22no—
107C17×#102—22no—
108C17×#97—22no—
109C17×#95—21no—
110#10587503C2×C414yes—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587492—22no—
117#10587492—22no—
118#10587486—22no—
119#10587486—22no—
120C8×Dic17—22no—
121C8×Dic17—22no—
122#10587480C2×C414yes—
123#10587479—22no—
124#10587479C2×C414yes—
125#10587478—22no—
126#10587474C2×C414yes—
127C24×C17—21no—
128C23×D17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17C22×C412yes—
135C22×C4×C17—21no—
136C22×C4×C17—24no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—42no—
140C22×C4×C17—42no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C22×Dic17C2×D411yescharacteristic
151C2×D68—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68—21no—
155C2×D68—21no—
156C2×D68C22×C412yes—
157C2×C4×D17—24no—
158C2×C4×D17—42no—
159C2×C4×D17—42no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17#4112yes—
165C2×C8×C17#4112yes—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17C22×C412yes—
172C22⋊C4×C17—24no—
173C42×C17—42no—
174C42×C17C22×C412yes—
175C42×C17#4112yes—
176C22⋊Dic17—42no—
177C22⋊Dic17—42no—
178C22⋊Dic17—42no—
179C22⋊Dic17—42no—
180#63228—28no—
181#63227—24no—
182#63227—42no—
183#63227—42no—
184#63227—42no—
185#63227—42no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—42no—
192C4⋊Dic17—42no—
193C4⋊Dic17C22×C412yes—
194#63225—42no—
195#63225—42no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C4×Dic17—21no—
199C4×Dic17—21no—
200C4×Dic17—22no—
201C4×Dic17—22no—
202C4×Dic17—41no—
203C4×Dic17—41no—
204C4×Dic17—42no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622C22×C412yes—
208C2×#3622C22×C412yes—
209C2×#3622#4112yes—
210C2×#3622#4112yes—
211#21286—171nomaximal, Sylow
212C23×C17—22no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17C22×D411yescharacteristic
220C22×D17—22no—
221C22×D17—41no—
222C22×D17—41no—
223C22×D17—41no—
224C22×D17—41no—
225C22×D17—41no—
226C22×D17—41no—
227D4×C17—22no—
228D4×C17—22no—
229D4×C17—41no—
230D4×C17—41no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—22no—
236C2×C4×C17—48no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17C4×D412yes—
246C2×C4×C17C4×D412yes—
247C2×C4×C17C23×C411yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—22no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
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×Dic17AGL(1,ℤ/8)12yes—
273C2×Dic17AGL(1,ℤ/8)12yes—
274D68—22no—
275D68—22no—
276D68—41no—
277D68—41no—
278C4×D17—48no—
279C8×C17—22no—
280C8×C17—22no—
281#3622—22no—
282#3622#11714yescomplemented
283C2×#442—171no—
284#2293—171no—
285#2293—171no—
286#1972—171no—
287#1970—171no—
288#1944—171no—
289#1941—171no—
290#1936—171no—
291#1936—171no—
292#1934—171no—
293#1934—171no—
294#1776—171no—
295#1762—171no—
296#1762D1711yescharacteristic, complemented
297C4×#97—171no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—81no—
302C22×C17—81no—
303C22×C17—81no—
304C22×C17—81no—
305C22×C17C2×C4×D411yescharacteristic
306C22×C17#53211yescharacteristic
307C22×C17#53211yescharacteristic
308C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C17#57511yescharacteristic
311C22×C17#57511yescharacteristic
312C2×D17—42no—
313C2×D17—42no—
314C2×D17—42no—
315C2×D17—81no—
316C2×D17—81no—
317C2×D17—81no—
318C2×D17—81no—
319C4×C17—22no—
320C4×C17—22no—
321C4×C17—42no—
322C4×C17—81no—
323C4×C17—81no—
324C4×C17C2×#11711yescharacteristic
325C4×C17C4×#4111yescharacteristic
326C4×C17#52911yescharacteristic
327C4×C17#52911yescharacteristic
328Dic17—22no—
329Dic17—22no—
330Dic17—41no—
331Dic17—41no—
332Dic17—42no—
333Dic17—81no—
334Dic17—81no—
335C22×D8—171no—
336C2×C4×D4—171no—
337C2×C4×D4C2×D1711yescharacteristic
338C2×#117—341no—
339C2×#117—341no—
340#442—178no—
341C2×#107—171no—
342C2×#106—171no—
343C2×#106—171no—
344C2×#102—171no—
345C2×#102—171no—
346C2×#102—171no—
347C2×#102—171no—
348C2×#102C2×D1711yescharacteristic
349C2×#102C2×D1711yescharacteristic
350C2×#97—171no—
351C2×#97—171no—
352C2×#97—171no—
353C2×#97C2×D1711yescharacteristic
354C2×C4×C8—171no—
355C2×C4×C8—171no—
356#390—171no—
357#390—171no—
358#390—171no—
359#390—171no—
360#389—171no—
361#389—171no—
362#389—171no—
363#389—171no—
364#388—341no—
365#388—341no—
366#388—341no—
367#388—341no—
368C4×C4⋊C4—171no—
369C4×C4⋊C4C2×D1711yescharacteristic
370C4×C22⋊C4—341no—
371C4×C22⋊C4—341no—
372C43—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#340C2×D1711yescharacteristic
378#340C2×D1711yescharacteristic
379C2×C17—82no—
380C2×C17#229311yescharacteristic
381C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
382C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
383C2×C17#318011yescharacteristic
384C2×C17#318011yescharacteristic
385C2×C17#318311yescharacteristic
386C2×C17#318411yescharacteristic
387D17—82no—
388C22×D4—171no—
389C22×D4C22×D1711yescharacteristic
390C23×C4—21no—
391C23×C4—341no—
392C2×D8—174no—
393C2×D8—342no—
394C2×D8—342no—
395C22×C8—171no—
396C22×C8—171no—
397C22×C8C22×D1711yescharacteristic
398C22×C8C22×D1711yescharacteristic
399C4×D4—22no—
400C4×D4—174no—
401C4×D4—342no—
402C4×D4C4×D1714yes—
403#117—684no—
404#117—684no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—171no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—341no—
409C2×C4⋊C4—341no—
410C2×C4⋊C4—341no—
411C2×C4⋊C4—341no—
412C2×C4⋊C4C22×D1711yescharacteristic
413C2×C4⋊C4C22×D1711yescharacteristic
414C2×C4⋊C4C22×D1711yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—341no—
417C2×C22⋊C4—341no—
418C2×C22⋊C4—341no—
419C2×C22⋊C4—341no—
420C2×C22⋊C4—341no—
421C2×C22⋊C4—341no—
422C2×C22⋊C4—341no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4—341no—
425C2×C42—21no—
426C2×C42—171no—
427C2×C42—171no—
428C2×C42—171no—
429C2×C42—171no—
430C2×C42—171no—
431C2×C42—171no—
432C2×C42—341no—
433C2×C42C22×D1711yescharacteristic
434#107—342no—
435#106—174no—
436#106—342no—
437#102—22no—
438#102—22no—
439#102—174no—
440#102—174no—
441#102—342no—
442#102—342no—
443#97—22no—
444#97—174no—
445#97—342no—
446#97—342no—
447C4×C8—342no—
448C4×C8—342no—
449#95—21no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455C17#2128611yescharacteristic, complemented, Sylow
456C24—21no—
457C24—341no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—172no—
464C2×D4—341no—
465C2×D4—341no—
466C2×D4—341no—
467C2×D4—341no—
468C2×D4—341no—
469C2×D4C2×C4×D1712yes—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—171no—
477C22×C4—171no—
478C22×C4—171no—
479C22×C4—171no—
480C22×C4—341no—
481C22×C4—344no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—682no—
485C22×C4—682no—
486C22×C4D4×D1711yescharacteristic
487C22×C4D4×D1711yescharacteristic
488C22×C4C23×D1711yescharacteristic
489D8—348no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—172no—
495C2×C8—172no—
496C2×C8—172no—
497C2×C8—172no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8#6325212yes—
501C2×C8#6325212yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—172no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—341no—
512C4⋊C4—341no—
513C4⋊C4—682no—
514C4⋊C4—682no—
515C4⋊C4—682no—
516C4⋊C4—682no—
517C4⋊C4C2×C4×D1712yes—
518C22⋊C4—24no—
519C22⋊C4—344no—
520C22⋊C4—682no—
521C22⋊C4—682no—
522C22⋊C4—682no—
523C22⋊C4—682no—
524C22⋊C4—682no—
525C22⋊C4—682no—
526C22⋊C4—682no—
527C22⋊C4—682no—
528C42—42no—
529C42—342no—
530C42—342no—
531C42—341no—
532C42—341no—
533C42—341no—
534C42—341no—
535C42—681no—
536C42—681no—
537C42—682no—
538C42C2×C4×D1712yes—
539C42#6325412yes—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—342no—
548C23—681no—
549C23—681no—
550C23—681no—
551C23—681no—
552C23—681no—
553C23—681no—
554C23C2×D4×D1711yescharacteristic
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—342no—
560D4—342no—
561D4—681no—
562D4—681no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—172no—
578C2×C4—172no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—341no—
583C2×C4—342no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—688no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4#1058756712yes—
600C2×C4#1058756712yes—
601C2×C4C22×C4×D1711yescharacteristic
602C2×C4C2×#6325211yescharacteristic
603C2×C4C2×#6325211yescharacteristic
604C2×C4C2×#6325411yescharacteristic
605C2×C4#41×D1711yescharacteristic
606C2×C4#41×D1711yescharacteristic
607C8—22no—
608C8—22no—
609C8—174no—
610C8—342no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—682no—
619C22—682no—
620C22—682no—
621C22—1361no—
622C22—1361no—
623C22—1361no—
624C22—1361no—
625C22C2×#1058756711yescharacteristic
626C22#1176335611yescharacteristic
627C22#1176335611yescharacteristic
628C22AGL(1,ℤ/8)×D1711yescharacteristic
629C22AGL(1,ℤ/8)×D1711yescharacteristic
630C22#1176360511yescharacteristic
631C22#1176360811yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—342no—
638C4—342no—
639C4—682no—
640C4—681no—
641C4—681no—
642C4—1361no—
643C4—1361no—
644C4#117×D1711yescharacteristic
645C4C4×#6325211yescharacteristic
646C4#1176333911yescharacteristic
647C4#1176333911yescharacteristic
648C2—82no—
649C2—1362no—
650C2#1453124711yescharacteristic
651C2#1453758811yescharacteristic
652C2#1453759711yescharacteristic
653C2#1453947811yescharacteristic
654C2#1453948011yescharacteristic
655C2#1453950911yescharacteristic
656C2#1453951911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1762C2
socleC23×C17C22×D4
prime radicalC22×#63228C22
cosocleC22×C4×C17C24
hypercenter#1762D17
hypocenterC17#21286
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #14531249C2semidirect
3. #14531245C2semidirect
6. #14529526C2semidirect
7. #14529511C2semidirect
8. #14529482C2semidirect
9. #14529478C2semidirect
10. #14529468C2semidirect
11. #14529464C2semidirect
12. #14528682C2semidirect
13. #14528610C2semidirect
14. #97×Dic17C2semidirect
15. C17×#1762C2semidirect
20. #11762919C4semidirect
282. #3622#117semidirect
296. #1762D17semidirect
455. C17#21286semidirect

Maximal subgroups

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

subgroupindex
C2×#117629192
#145312492
#145312452
#145296572
#145296472
#145295262
#145295112
#145294822
#145294782
#145294682
#145294642
#145286822
#145286102
#97×Dic172
C17×#17622
#2128617