Finite Group Databasebeta

← #17051663

Subgroups — #17051663

Order 4352.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762819C211yesmaximal, characteristic
2#14531667C211yesmaximal, characteristic
3C22⋊C4×Dic34C211yesmaximal, characteristic
4#14529474C211yesmaximal, characteristic, complemented
5#14529460C211yesmaximal, characteristic, complemented
6#14529426C211yesmaximal, characteristic, complemented
7#14529419C211yesmaximal, characteristic, complemented
8#14529055C211yesmaximal, characteristic, complemented
9#14529041C211yesmaximal, characteristic, complemented
10#14528880C211yesmaximal, characteristic
11#14528817C211yesmaximal, characteristic
12#14528720C211yesmaximal, characteristic
13#14528615C211yesmaximal, characteristic, complemented
14C4×#10587487C211yesmaximal, characteristic, complemented
15C17×#1753C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20C2×#10587489C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22#11762819C418yescomplemented
23C2×#10587487C2211yescharacteristic
24C2×#10587487C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587487C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31#11762715C2211yescharacteristic
32#11762715C2211yescharacteristic
33#11762713—21no—
34#11762713—21no—
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762697—21no—
38#11762697—21no—
39#11762673—21no—
40#11762669C2211yescharacteristic
41#11762669C2211yescharacteristic
42#11762667—21no—
43C4×C4⋊Dic17C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45C22⋊C4×Dic17—21no—
46#11762654—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C2×C17×C22⋊C8C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C2×C4×C8×C17C2211yescharacteristic
57C4×C22⋊C4×C17C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17—21no—
62C22×Dic34—22no—
63C22×Dic34—22no—
64C22×Dic34—21no—
65C22×Dic34—22no—
66C22×Dic34—41no—
67C22×Dic34—41no—
68C22×Dic34C2×C414yes—
69C22×Dic34C2311yescharacteristic
70C22×Dic34C2311yescharacteristic
71C2×M16×C17—41no—
72C2×M16×C17—41no—
73C2×M16×C17C2×C414yes—
74C22×C8×C17C2311yescharacteristic
75C22×C8×C17C2311yescharacteristic
76C22×C8×C17C2311yescharacteristic
77C22×C8×C17C2311yescharacteristic
78C2×C22⋊C4×C17D411yescharacteristic
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85#10587612—22no—
86#10587612C2×C414yes—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C4×Dic34—22no—
103C4×Dic34—22no—
104C4×Dic34—41no—
105C4×Dic34—41no—
106C17×C22⋊C8—22no—
107C17×#97—41no—
108C4×C8×C17—41no—
109C17×#95D411yescharacteristic
110C17×#95D411yescharacteristic
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587489—41no—
122#10587488—41no—
123#10587487—22no—
124#10587487—22no—
125#10587487—41no—
126#10587487—41no—
127#10587487C2×C414yes—
128#10587487C2×C414yes—
129#10587487C2×C414yes—
130#10587487C2×C414yes—
131C24×C17C2×D411yescharacteristic
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17C22×C412yes—
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17#4112yes—
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—24no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—42no—
154C22×Dic17—42no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C2×Dic34—28no—
163C2×Dic34—24no—
164C2×Dic34—21no—
165C2×Dic34—44no—
166C2×Dic34—44no—
167C2×Dic34—44no—
168C2×Dic34—42no—
169C2×Dic34—42no—
170C2×Dic34C22×C412yes—
171C2×Dic34C22×C412yes—
172C2×Dic34#4112yes—
173C2×Dic34#4112yes—
174M16×C17—44no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—21no—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C2×C8×C17—21no—
184C2×C8×C17—21no—
185C2×C8×C17C22×C412yes—
186C2×C8×C17C22×C412yes—
187C22⋊C4×C17—22no—
188C22⋊C4×C17—41no—
189C42×C17—21no—
190C42×C17—21no—
191C42×C17—21no—
192C42×C17—21no—
193C22⋊Dic17—24no—
194C22⋊Dic17—42no—
195C22⋊Dic17—81no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17C22×C412yes—
207C4⋊Dic17C22×C412yes—
208#63225—42no—
209#63225—42no—
210#63225—81no—
211#63225—81no—
212C4×Dic17—24no—
213C4×Dic17—42no—
214C4×Dic17—42no—
215C4×Dic17—42no—
216C4×Dic17—42no—
217#22585—171nomaximal, Sylow
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—21no—
224C23×C17C4×D412yes—
225C23×C17C22×D411yescharacteristic
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17C4×D412yes—
244C2×C4×C17#13712yes—
245C2×C4×C17C23×C411yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17H(5,𝔽2)11yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×Dic17—22no—
252C2×Dic17—22no—
253C2×Dic17—22no—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—48no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
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×Dic17—41no—
276Dic34—22no—
277Dic34—22no—
278Dic34—22no—
279Dic34—22no—
280Dic34—22no—
281Dic34—48no—
282Dic34—41no—
283Dic34—41no—
284Dic34#11714yes—
285C8×C17—22no—
286C8×C17—22no—
287C8×C17—41no—
288C8×C17—41no—
289C2×#419—171no—
290#2346—171no—
291Q8×C22⋊C4—171no—
292#1935—171no—
293#1933—171no—
294#1928—171no—
295#1927—171no—
296#1865—171no—
297#1863—171no—
298#1825—171no—
299#1809—171no—
300#1787—171no—
301#1763—171no—
302C4×#103—171no—
303#1753D1711yescharacteristic
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17C2×C4×D411yescharacteristic
312C22×C17#54611yescharacteristic
313C22×C17#54611yescharacteristic
314C22×C17C2×#13711yescharacteristic
315C22×C17#57511yescharacteristic
316C22×C17#57711yescharacteristic
317C22×C17#57811yescharacteristic
318C4×C17—22no—
319C4×C17—22no—
320C4×C17—41no—
321C4×C17—41no—
322C4×C17—41no—
323C4×C17C2×#11711yescharacteristic
324C4×C17#51811yescharacteristic
325C4×C17#55111yescharacteristic
326C4×C17#55111yescharacteristic
327Dic17—24no—
328Dic17—42no—
329Dic17—42no—
330Dic17—42no—
331Dic17—81no—
332Dic17—81no—
333Dic17—81no—
334Dic17—81no—
335C23×Q8—171no—
336C22×M16C2×D1711yescharacteristic
337C2×C4×Q8—171no—
338C2×C4×Q8—171no—
339C2×#117—171no—
340C2×#107—171no—
341C2×#106—171no—
342#419—178no—
343C2×#103—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#103—171no—
347C2×#103—171no—
348C2×#103—171no—
349C2×#103—171no—
350C2×#103—171no—
351C2×C22⋊C8C2×D1711yescharacteristic
352C2×#97C2×D1711yescharacteristic
353C2×C4×C8C2×D1711yescharacteristic
354#391—171no—
355#391—171no—
356#391—341no—
357#391—341no—
358#389—171no—
359#389—171no—
360#388—341no—
361#388—341no—
362#382—341no—
363#380—171no—
364#380—171no—
365#380—341no—
366C4×C4⋊C4—171no—
367C4×C4⋊C4—171no—
368C4×C22⋊C4—341no—
369C4×C22⋊C4C2×D1711yescharacteristic
370#376—341no—
371#340—171no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#336C2×D1711yescharacteristic
378#336C2×D1711yescharacteristic
379C2×C17—42no—
380C2×C17#234211yescharacteristic
381C2×C17C4×#13711yescharacteristic
382C2×C17#294711yescharacteristic
383C2×C17#331211yescharacteristic
384C2×C17#332111yescharacteristic
385C2×C17#332711yescharacteristic
386C2×C17#332911yescharacteristic
387C22×Q8—174no—
388C22×Q8—171no—
389C22×Q8—171no—
390C22×Q8—342no—
391C22×Q8—342no—
392C22×Q8—341no—
393C22×Q8—342no—
394C22×Q8—681no—
395C22×Q8—681no—
396C23×C4—341no—
397C23×C4C22×D1711yescharacteristic
398C2×M16—41no—
399C2×M16—41no—
400C2×M16C4×D1714yes—
401C22×C8C22×D1711yescharacteristic
402C22×C8C22×D1711yescharacteristic
403C22×C8C22×D1711yescharacteristic
404C22×C8C22×D1711yescharacteristic
405C4×Q8—342no—
406C4×Q8—342no—
407C4×Q8—681no—
408C4×Q8—681no—
409#117—174no—
410#117—342no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C22⋊C4—341no—
422C2×C22⋊C4—341no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4D6811yescharacteristic
425C2×C22⋊C4D6811yescharacteristic
426C2×C42—341no—
427C2×C42—341no—
428C2×C42—341no—
429C2×C42—341no—
430C2×C42—341no—
431C2×C42C22×D1711yescharacteristic
432C2×C42C22×D1711yescharacteristic
433#107—681no—
434#106—681no—
435#103—174no—
436#103—174no—
437#103—174no—
438#103—174no—
439#103—342no—
440#103—342no—
441#103—681no—
442#103—681no—
443C22⋊C8—22no—
444#97—41no—
445C4×C8—41no—
446#95—341no—
447#95—341no—
448#95—341no—
449#95—341no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95D6811yescharacteristic
457#95D6811yescharacteristic
458C17#2258511yescharacteristic, complemented, Sylow
459C24C2×D6811yescharacteristic
460C2×Q8—172no—
461C2×Q8—172no—
462C2×Q8—172no—
463C2×Q8—172no—
464C2×Q8—348no—
465C2×Q8—341no—
466C2×Q8—341no—
467C2×Q8—341no—
468C2×Q8—341no—
469C2×Q8—344no—
470C2×Q8—341no—
471C2×Q8—341no—
472C2×Q8—341no—
473C2×Q8—341no—
474C2×Q8—682no—
475C2×Q8—682no—
476C2×Q8—684no—
477C2×Q8—684no—
478C2×Q8—684no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—344no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—682no—
493C22×C4—682no—
494C22×C4C2×C4×D1712yes—
495C22×C4C2×D6811yescharacteristic
496C22×C4C2×D6811yescharacteristic
497C22×C4C2×D6811yescharacteristic
498C22×C4C2×D6811yescharacteristic
499C22×C4C2×D6811yescharacteristic
500C22×C4#6325212yes—
501C22×C4C23×D1711yescharacteristic
502M16—44no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8C2×C4×D1712yes—
514C2×C8C2×C4×D1712yes—
515C4⋊C4—172no—
516C4⋊C4—172no—
517C4⋊C4—341no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—682no—
528C4⋊C4—682no—
529C4⋊C4—1361no—
530C4⋊C4—1361no—
531C22⋊C4—22no—
532C22⋊C4—41no—
533C22⋊C4—344no—
534C22⋊C4—682no—
535C22⋊C4—1361no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—344no—
541C42—682no—
542C42—682no—
543C42—682no—
544C42—682no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C4×D6812yes—
552C23C22×D6811yescharacteristic
553Q8—174no—
554Q8—342no—
555Q8—342no—
556Q8—342no—
557Q8—342no—
558Q8—342no—
559Q8—688no—
560Q8—681no—
561Q8—681no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—342no—
580C2×C4—342no—
581C2×C4—342no—
582C2×C4—342no—
583C2×C4—342no—
584C2×C4—342no—
585C2×C4—688no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
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—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4C4×D6812yes—
605C2×C4#1058759512yes—
606C2×C4C22×C4×D1711yescharacteristic
607C2×C4C2×#6325211yescharacteristic
608C2×C4#1058769111yescharacteristic
609C2×C4#1058769111yescharacteristic
610C2×C4#41×D1711yescharacteristic
611C2×C4#41×D1711yescharacteristic
612C8—22no—
613C8—22no—
614C8—41no—
615C8—41no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×D6811yescharacteristic
624C22#1176345411yescharacteristic
625C22#1176345411yescharacteristic
626C22C2×#1058759511yescharacteristic
627C22#1176360911yescharacteristic
628C22#1176362111yescharacteristic
629C22#1176362611yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—344no—
636C4—682no—
637C4—682no—
638C4—682no—
639C4—1361no—
640C4—1361no—
641C4—1361no—
642C4—1361no—
643C4#117×D1711yescharacteristic
644C4#1176328211yescharacteristic
645C4#1176348811yescharacteristic
646C4#1176348811yescharacteristic
647C2—42no—
648C2#1453163611yescharacteristic
649C2C4×#1058759511yescharacteristic
650C2#1453762911yescharacteristic
651C2#1454083711yescharacteristic
652C2#1454097211yescharacteristic
653C2#1454105411yescharacteristic
654C2#1454107911yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1753C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#1753D17
hypocenterC17#22585
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #14529474C2semidirect
5. #14529460C2semidirect
6. #14529426C2semidirect
7. #14529419C2semidirect
8. #14529055C2semidirect
9. #14529041C2semidirect
13. #14528615C2semidirect
14. C4×#10587487C2semidirect
22. #11762819C4semidirect
458. C17#22585semidirect

Maximal subgroups

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

subgroupindex
C2×#117628192
#145316672
C22⋊C4×Dic342
#145294742
#145294602
#145294262
#145294192
#145290552
#145290412
#145288802
#145288172
#145287202
#145286152
C4×#105874872
C17×#17532
#2258517