Finite Group Databasebeta

← #17054154

Subgroups — #17054154

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
1C2×#11762922C211yesmaximal, characteristic
2#14531755C211yesmaximal, characteristic, complemented
3#14531731C211yesmaximal, characteristic, complemented
4#14529660C211yesmaximal, characteristic
5#14529650C211yesmaximal, characteristic
6#14529412C211yesmaximal, characteristic, complemented
7#14529412C211yesmaximal, characteristic, complemented
8#14529405C211yesmaximal, characteristic, complemented
9#14529405C211yesmaximal, characteristic, complemented
10#14528952C211yesmaximal, characteristic, complemented
11#14528795C211yesmaximal, characteristic, complemented
12#14528785C211yesmaximal, characteristic, complemented
13#14528681C211yesmaximal, characteristic, complemented
14C4×#10587488C211yesmaximal, characteristic, complemented
15C17×#1771C211yesmaximal, characteristic, complemented
16C22×D136C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587578—21no—
20#11762922C418yescomplemented
21C2×#10587489C2211yescharacteristic
22C2×#10587489C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587488C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27C2×#10587492C2211yescharacteristic
28C2×#10587492C2211yescharacteristic
29C2×#10587492C2211yescharacteristic
30C2×#10587492C2211yescharacteristic
31C2×#10587492C2211yescharacteristic
32#11762711C2211yescharacteristic
33#11762711C2211yescharacteristic
34#11762711C2211yescharacteristic
35#11762711C2211yescharacteristic
36#11762701—21no—
37#11762701—21no—
38#11762699—21no—
39#11762699—21no—
40#11762679C2211yescharacteristic
41#11762675C2211yescharacteristic
42#11762671—21no—
43#11762666—21no—
44C4×C4⋊Dic17C2211yescharacteristic
45C4×C4⋊Dic17C2211yescharacteristic
46C4×C4⋊Dic17C2211yescharacteristic
47C4×C4⋊Dic17C2211yescharacteristic
48#11762656—21no—
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C17×#97C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C2×C4×C8×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D68C2311yescharacteristic
61C22×D68C2311yescharacteristic
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C2×C4⋊C4×C17D411yescharacteristic
69C2×C4⋊C4×C17D411yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×#63227—21no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×D136—22no—
96C2×D136—22no—
97C2×D136C2×C414yes—
98#10587578—44no—
99C4×D68—22no—
100C4×D68—22no—
101C4×D68C2×C414yes—
102C4×D68C2×C414yes—
103C17×C4⋊C8—22no—
104C17×#97—22no—
105C17×#97C2×C414yes—
106C4×C8×C17—22no—
107C17×#95D411yescharacteristic
108C17×#95D411yescharacteristic
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587492—41no—
116#10587492—41no—
117#10587492—41no—
118#10587492—41no—
119#10587492C2×C414yes—
120#10587492C2×C414yes—
121#10587489—22no—
122#10587489—41no—
123#10587488—22no—
124#10587488—41no—
125#10587488C2×C414yes—
126C23×D17—21no—
127C23×D17—21no—
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2411yescharacteristic
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68—21no—
147C2×D68—21no—
148C2×D68—21no—
149C2×D68—21no—
150C2×D68—21no—
151C2×D68C22×C412yes—
152C2×D68C22×C412yes—
153C2×C4×D17—24no—
154C2×C4×D17—24no—
155C2×C4×D17—42no—
156C2×C4×D17—42no—
157C2×C4×D17—81no—
158C2×C4×D17—81no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17C22×C412yes—
168C2×C8×C17C22×C412yes—
169C2×C8×C17#4112yes—
170C2×C8×C17#4112yes—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—41no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17C22×C412yes—
178C42×C17#4112yes—
179#63227—24no—
180#63227—24no—
181#63227—42no—
182#63227—42no—
183#63227—42no—
184#63227—42no—
185#63227—81no—
186#63227—81no—
187#63227—81no—
188#63227—81no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—42no—
200C4⋊Dic17—81no—
201C4⋊Dic17C22×C412yes—
202C4⋊Dic17C22×C412yes—
203C4×Dic17—42no—
204C4×Dic17—42no—
205C4×Dic17—81no—
206C4×Dic17—81no—
207D136—28no—
208#22817—171nomaximal, Sylow
209C23×C17C22×D411yescharacteristic
210C22×D17—22no—
211C22×D17—22no—
212C22×D17—41no—
213C22×D17—41no—
214C22×D17—41no—
215C22×D17—41no—
216C22×D17—41no—
217C22×D17—41no—
218C22×D17—41no—
219C22×D17—41no—
220C22×D17—41no—
221C22×D17—41no—
222C22×D17—41no—
223C22×D17—41no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
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—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17C4×D412yes—
240C2×C4×C17C4×D412yes—
241C2×C4×C17AGL(1,ℤ/8)12yes—
242C2×C4×C17C23×C411yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17H(5,𝔽2)11yescharacteristic
247C2×C4×C172−1+411yescharacteristic
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266D68—22no—
267D68—22no—
268D68—22no—
269D68—22no—
270D68—41no—
271D68—41no—
272D68—41no—
273D68—41no—
274C4×D17—48no—
275C4×D17—84no—
276C8×C17—22no—
277C8×C17—41no—
278C8×C17—41no—
279C8×C17#11714yescomplemented
280C2×#442—171no—
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#1771D1711yescharacteristic, complemented
295C22×C17C2×C4×D411yescharacteristic
296C22×C17#54711yescharacteristic
297C22×C17#55011yescharacteristic
298C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C17#57511yescharacteristic
300C22×C17#57611yescharacteristic
301C22×C17#57711yescharacteristic
302C2×D17—42no—
303C2×D17—42no—
304C2×D17—42no—
305C2×D17—42no—
306C2×D17—42no—
307C2×D17—42no—
308C2×D17—81no—
309C2×D17—81no—
310C2×D17—81no—
311C2×D17—81no—
312C2×D17—81no—
313C2×D17—81no—
314C2×D17—81no—
315C2×D17—81no—
316C4×C17—22no—
317C4×C17—22no—
318C4×C17—22no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17—41no—
322C4×C17C2×#11711yescharacteristic
323C4×C17#52011yescharacteristic
324C4×C17#55511yescharacteristic
325C4×C17#55611yescharacteristic
326Dic17—42no—
327Dic17—42no—
328Dic17—81no—
329Dic17—81no—
330Dic17—81no—
331Dic17—81no—
332C22×D8—171no—
333C2×C4×D4—171no—
334C2×C4×D4—171no—
335C2×#117—341no—
336#442—178no—
337C2×#107—171no—
338C2×#107—171no—
339C2×#106—171no—
340C2×#106—171no—
341C2×#106—171no—
342C2×C4⋊C8C2×D1711yescharacteristic
343C2×#102—171no—
344C2×#102—171no—
345C2×#102—171no—
346C2×#102—171no—
347C2×#102—171no—
348C2×#102—171no—
349C2×#97C2×D1711yescharacteristic
350C2×#97C2×D1711yescharacteristic
351C2×C4×C8C2×D1711yescharacteristic
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⋊C4C2×D1711yescharacteristic
369C4×C22⋊C4—341no—
370#340—171no—
371#340—171no—
372#340—171no—
373#340—171no—
374#336C2×D1711yescharacteristic
375#336C2×D1711yescharacteristic
376C2×C17#235411yescharacteristic
377C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
378C2×C17#294611yescharacteristic
379C2×C17#334511yescharacteristic
380C2×C17#334711yescharacteristic
381C2×C17#339911yescharacteristic
382C2×C17#340111yescharacteristic
383D17—82no—
384D17—82no—
385C22×D4—171no—
386C22×D4—171no—
387C23×C4—341no—
388C23×C4—341no—
389C2×D8—174no—
390C2×D8—342no—
391C2×D8—342no—
392C22×C8C22×D1711yescharacteristic
393C22×C8C22×D1711yescharacteristic
394C22×C8C22×D1711yescharacteristic
395C22×C8C22×D1711yescharacteristic
396C4×D4—174no—
397C4×D4—174no—
398C4×D4—342no—
399C4×D4—342no—
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⋊C4D6811yescharacteristic
410C2×C4⋊C4D6811yescharacteristic
411C2×C22⋊C4—341no—
412C2×C22⋊C4—341no—
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—341no—
422C2×C42—341no—
423C2×C42—341no—
424C2×C42—341no—
425C2×C42C22×D1711yescharacteristic
426C2×C42C22×D1711yescharacteristic
427C2×C42C22×D1711yescharacteristic
428#107—342no—
429#107—681no—
430#106—174no—
431#106—342no—
432#106—681no—
433C4⋊C8—22no—
434#102—174no—
435#102—174no—
436#102—681no—
437#102—681no—
438#102—681no—
439#102—681no—
440#97—22no—
441#97C4×D1714yes—
442C4×C8—22no—
443#95—341no—
444#95—341no—
445#95—341no—
446#95—341no—
447#95—341no—
448#95—341no—
449#95D6811yescharacteristic
450#95D6811yescharacteristic
451C17#2281711yescharacteristic, complemented, Sylow
452C24—341no—
453C24—341no—
454C2×D4—172no—
455C2×D4—172no—
456C2×D4—341no—
457C2×D4—341no—
458C2×D4—341no—
459C2×D4—341no—
460C2×D4—341no—
461C2×D4—341no—
462C2×D4—341no—
463C2×D4—341no—
464C2×D4—341no—
465C2×D4—341no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—344no—
469C22×C4—344no—
470C22×C4—341no—
471C22×C4—341no—
472C22×C4—341no—
473C22×C4—341no—
474C22×C4—682no—
475C22×C4—682no—
476C22×C4—1361no—
477C22×C4—1361no—
478C22×C4C2×D6811yescharacteristic
479C22×C4C2×D6811yescharacteristic
480C22×C4C2×D6811yescharacteristic
481C22×C4C2×D6811yescharacteristic
482C22×C4C2×D6811yescharacteristic
483C22×C4C2×D6811yescharacteristic
484C22×C4C23×D1711yescharacteristic
485D8—348no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8C2×C4×D1712yes—
495C2×C8C2×C4×D1712yes—
496C2×C8#6325412yes—
497C2×C8#6325612yes—
498C4⋊C4—22no—
499C4⋊C4—41no—
500C4⋊C4—172no—
501C4⋊C4—172no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—341no—
512C4⋊C4—682no—
513C4⋊C4—1361no—
514C22⋊C4—344no—
515C22⋊C4—344no—
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—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—682no—
529C42—682no—
530C42—1361no—
531C42—1361no—
532C42C2×C4×D1712yes—
533C42#6325212yes—
534C23—342no—
535C23—342no—
536C23—681no—
537C23—681no—
538C23—681no—
539C23—681no—
540C23—681no—
541C23—681no—
542C23—681no—
543C23—681no—
544C23—681no—
545C23—681no—
546C23—681no—
547C23—681no—
548C23C22×D6811yescharacteristic
549D4—342no—
550D4—342no—
551D4—342no—
552D4—342no—
553D4—681no—
554D4—681no—
555D4—681no—
556D4—681no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—688no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—681no—
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—1364no—
592C2×C4C4×D6812yes—
593C2×C4C4×D6812yes—
594C2×C4#1058759412yes—
595C2×C4C22×C4×D1711yescharacteristic
596C2×C4C2×#6325211yescharacteristic
597C2×C4C2×#6325411yescharacteristic
598C2×C4C2×#6325611yescharacteristic
599C2×C4#1058769611yescharacteristic
600C2×C4#1058769711yescharacteristic
601C8—22no—
602C8—41no—
603C8—41no—
604C8#1058757814yescomplemented
605C22—682no—
606C22—682no—
607C22—682no—
608C22—682no—
609C22—682no—
610C22—682no—
611C22—1361no—
612C22—1361no—
613C22—1361no—
614C22—1361no—
615C22—1361no—
616C22—1361no—
617C22—1361no—
618C22—1361no—
619C22C2×C4×D6811yescharacteristic
620C22#1176346311yescharacteristic
621C22#1176348111yescharacteristic
622C22C2×#1058759411yescharacteristic
623C22#1176360911yescharacteristic
624C22#1176361411yescharacteristic
625C22#1176362111yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—682no—
633C4—682no—
634C4—1361no—
635C4—1361no—
636C4—1361no—
637C4—1361no—
638C4C2×#1058757811yescharacteristic
639C4#1176328911yescharacteristic
640C4#1176352411yescharacteristic
641C4#1176353211yescharacteristic
642C2—1362no—
643C2—1362no—
644C2#1453173111yescharacteristic
645C2C4×#1058759411yescharacteristic
646C2#1453761811yescharacteristic
647C2#1454125611yescharacteristic
648C2#1454128611yescharacteristic
649C2#1454182811yescharacteristic
650C2#1454185211yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1771C2
socleC23×C17C22×D4
prime radicalC22×D136C22
cosocleC22×C4×C17C24
hypercenter#1771D17
hypocenterC17#22817
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #14531755C2semidirect
3. #14531731C2semidirect
6. #14529412C2semidirect
7. #14529412C2semidirect
8. #14529405C2semidirect
9. #14529405C2semidirect
10. #14528952C2semidirect
11. #14528795C2semidirect
12. #14528785C2semidirect
13. #14528681C2semidirect
14. C4×#10587488C2semidirect
15. C17×#1771C2semidirect
20. #11762922C4semidirect
279. C8×C17#117semidirect
294. #1771D17semidirect
451. C17#22817semidirect
604. C8#10587578semidirect

Maximal subgroups

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

subgroupindex
C2×#117629222
#145317552
#145317312
#145296602
#145296502
#145294122
#145294052
#145289522
#145287952
#145287852
#145286812
C4×#105874882
C17×#17712
#2281717