Finite Group Databasebeta

← #17051594

Subgroups — #17051594

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×#11762818C211yesmaximal, characteristic
2#14531675C211yesmaximal, characteristic
3#14529477C211yesmaximal, characteristic, complemented
4#14529463C211yesmaximal, characteristic, complemented
5#14529422C211yesmaximal, characteristic, complemented
6#14529415C211yesmaximal, characteristic, complemented
7#14529052C211yesmaximal, characteristic, complemented
8#14529038C211yesmaximal, characteristic, complemented
9#14528876C211yesmaximal, characteristic
10#14528815C211yesmaximal, characteristic
11#14528719C211yesmaximal, characteristic
12#14528614C211yesmaximal, characteristic, complemented
13C4×#10587506C211yesmaximal, characteristic, complemented
14#14528568C211yesmaximal, characteristic
15Q8×C22⋊C4×C17C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587612C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20#11762818C418yescomplemented
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30C2×#10587474C2211yescharacteristic
31C2×C4×#3622C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762703—21no—
35#11762703—21no—
36#11762676—21no—
37#11762669C2211yescharacteristic
38C4×C4⋊Dic17C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40#11762660—21no—
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C23×Q8×C17C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C17×#391—21no—
53C17×#391—21no—
54C17×#391C2211yescharacteristic
55C17×#391C2211yescharacteristic
56C17×#380—21no—
57C17×#380C2211yescharacteristic
58C4×C22⋊C4×C17—21no—
59C4×C22⋊C4×C17C2211yescharacteristic
60C22×Q8×C17—21no—
61C22×Q8×C17—22no—
62C22×Q8×C17—22no—
63C22×Q8×C17—22no—
64C22×Q8×C17—41no—
65C22×Q8×C17—41no—
66C22×Q8×C17C2×C414yes—
67C22×Q8×C17C2311yescharacteristic
68C22×Q8×C17C2311yescharacteristic
69C23×C4×C17—21no—
70C23×C4×C17C2311yescharacteristic
71C4×Q8×C17—22no—
72C4×Q8×C17—22no—
73C4×Q8×C17—41no—
74C4×Q8×C17—41no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17D411yescharacteristic
81C2×C22⋊C4×C17D411yescharacteristic
82C2×C42×C17—21no—
83C2×C42×C17—21no—
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×C22⋊Dic17—21no—
87C2×C22⋊Dic17—21no—
88#10587612—22no—
89#10587612C2×C414yes—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×#63223—41no—
100C2×#63223—41no—
101C2×#63223C2×C414yes—
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C17×#95—21no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95D411yescharacteristic
112C17×#95D411yescharacteristic
113#10587506—22no—
114#10587506—22no—
115#10587506—41no—
116#10587506—41no—
117#10587506C2×C414yes—
118#10587506C2×C414yes—
119#10587506C2×C414yes—
120#10587506C2×C414yes—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126C22⋊#3622—22no—
127#10587479—41no—
128#10587478—41no—
129#10587474—41no—
130C4×#3622—41no—
131C24×C17C2×D411yescharacteristic
132C2×Q8×C17—24no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—28no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—42no—
143C2×Q8×C17—42no—
144C2×Q8×C17—44no—
145C2×Q8×C17—44no—
146C2×Q8×C17—44no—
147C2×Q8×C17C22×C412yes—
148C2×Q8×C17C22×C412yes—
149C2×Q8×C17#4112yes—
150C2×Q8×C17#4112yes—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—24no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17—21no—
160C22×C4×C17—42no—
161C22×C4×C17—42no—
162C22×C4×C17C22×C412yes—
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2×D411yescharacteristic
165C22×C4×C17C2×D411yescharacteristic
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17#4112yes—
169C22×C4×C17C2411yescharacteristic
170C22×Dic17—21no—
171C22×Dic17—21no—
172C22×Dic17—21no—
173C22×Dic17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—42no—
179C4⋊C4×C17—42no—
180C22⋊C4×C17—22no—
181C22⋊C4×C17—41no—
182C22⋊C4×C17—42no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—42no—
188C42×C17—42no—
189C22⋊Dic17—24no—
190C22⋊Dic17—81no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17C22×C412yes—
198C4⋊Dic17C22×C412yes—
199#63225—42no—
200#63225—42no—
201C4×Dic17—24no—
202C4×Dic17—81no—
203C4×Dic17—81no—
204#63223—44no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622C22×C412yes—
216C2×#3622C22×C412yes—
217#22581—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
226Q8×C17—22no—
227Q8×C17—22no—
228Q8×C17—22no—
229Q8×C17—22no—
230Q8×C17—22no—
231Q8×C17—48no—
232Q8×C17—41no—
233Q8×C17—41no—
234Q8×C17#11714yes—
235C2×C4×C17—22no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—48no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17C4×D412yes—
266C2×C4×C17#13712yes—
267C2×C4×C17C23×C411yescharacteristic
268C2×C4×C17C2×#4111yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×C4×C17C2×#4111yescharacteristic
271C2×C4×C17H(5,𝔽2)11yescharacteristic
272C2×C4×C17H(5,𝔽2)11yescharacteristic
273C2×Dic17—22no—
274C2×Dic17—22no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285#3622—22no—
286#3622—22no—
287#3622—41no—
288#3622—41no—
289C2×#419—171no—
290#2347—171no—
291Q8×C22⋊C4D1711yescharacteristic
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#1753—171no—
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—24no—
320C4×C17—22no—
321C4×C17—41no—
322C4×C17—42no—
323C4×C17—41no—
324C4×C17—41no—
325C4×C17—81no—
326C4×C17—81no—
327C4×C17C2×#11711yescharacteristic
328C4×C17#51811yescharacteristic
329C4×C17#55111yescharacteristic
330C4×C17#55111yescharacteristic
331Dic17—42no—
332Dic17—42no—
333Dic17—81no—
334Dic17—81no—
335C23×Q8C2×D1711yescharacteristic
336C22×M16—171no—
337C2×C4×Q8C2×D1711yescharacteristic
338C2×C4×Q8C2×D1711yescharacteristic
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⋊C8—171no—
352C2×#97—171no—
353C2×C4×C8—171no—
354#391—21no—
355#391—21no—
356#391C2×D1711yescharacteristic
357#391C2×D1711yescharacteristic
358#389—171no—
359#389—171no—
360#388—341no—
361#388—341no—
362#383—341no—
363#380—21no—
364#380—171no—
365#380C2×D1711yescharacteristic
366C4×C4⋊C4—171no—
367C4×C4⋊C4—171no—
368C4×C4⋊C4—341no—
369C4×C22⋊C4—21no—
370C4×C22⋊C4C2×D1711yescharacteristic
371#340—171no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#336—171no—
378#336—171no—
379C2×C17—42no—
380C2×C17#234211yescharacteristic
381C2×C17C4×#13711yescharacteristic
382C2×C17#294711yescharacteristic
383C2×C17#331311yescharacteristic
384C2×C17#332011yescharacteristic
385C2×C17#332611yescharacteristic
386C2×C17#332811yescharacteristic
387C22×Q8—22no—
388C22×Q8—21no—
389C22×Q8—22no—
390C22×Q8—22no—
391C22×Q8—41no—
392C22×Q8—41no—
393C22×Q8C4×D1714yes—
394C22×Q8C22×D1711yescharacteristic
395C22×Q8C22×D1711yescharacteristic
396C23×C4—21no—
397C23×C4C22×D1711yescharacteristic
398C2×M16—174no—
399C2×M16—681no—
400C2×M16—681no—
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8—171no—
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—41no—
408C4×Q8—41no—
409#117—174no—
410#117—342no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
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⋊C4C22×D1711yescharacteristic
420C2×C4⋊C4C22×D1711yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—341no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4C22⋊D1711yescharacteristic
425C2×C22⋊C4C22⋊D1711yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
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—342no—
444#97—681no—
445C4×C8—681no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95C22⋊D1711yescharacteristic
457#95C22⋊D1711yescharacteristic
458C17#2258111yescharacteristic, complemented, Sylow
459C24C2×C22⋊D1711yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—24no—
465C2×Q8—28no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8C2×C4×D1712yes—
476C2×Q8C2×C4×D1712yes—
477C2×Q8#6325412yes—
478C2×Q8#6325412yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4C2×C4×D1712yes—
495C22×C4#6325212yes—
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C23×D1711yescharacteristic
502M16—684no—
503C2×C8—172no—
504C2×C8—172no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—172no—
522C4⋊C4—172no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—682no—
530C4⋊C4—682no—
531C22⋊C4—22no—
532C22⋊C4—41no—
533C22⋊C4—42no—
534C22⋊C4—344no—
535C22⋊C4—1361no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—42no—
542C42—344no—
543C42—1361no—
544C42—1361no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C4×C22⋊D1712yes—
552C23C22×C22⋊D1711yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#1058756314yes—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—342no—
593C2×C4—342no—
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×C22⋊D1712yes—
605C2×C4#1058762812yes—
606C2×C4C22×C4×D1711yescharacteristic
607C2×C4C2×#6325211yescharacteristic
608C2×C4C2×#6325411yescharacteristic
609C2×C4C2×#6325411yescharacteristic
610C2×C4#1058769111yescharacteristic
611C2×C4#1058769111yescharacteristic
612C8—342no—
613C8—342no—
614C8—681no—
615C8—681no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D1711yescharacteristic
624C22#1176345311yescharacteristic
625C22#1176345311yescharacteristic
626C22C2×#1058762811yescharacteristic
627C22#1176360711yescharacteristic
628C22#1176361711yescharacteristic
629C22#1176362411yescharacteristic
630C4—22no—
631C4—24no—
632C4—22no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—682no—
640C4—682no—
641C4—1361no—
642C4—1361no—
643C4C2×#1058756311yescharacteristic
644C4#1176328211yescharacteristic
645C4#1176348511yescharacteristic
646C4#1176348511yescharacteristic
647C2—42no—
648C2#1453162911yescharacteristic
649C2C4×#1058762811yescharacteristic
650C2#1453762711yescharacteristic
651C2#1454085011yescharacteristic
652C2#1454095511yescharacteristic
653C2#1454103711yescharacteristic
654C2#1454106711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingQ8×C22⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenterQ8×C22⋊C4D17
hypocenterC17#22581
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #14529477C2semidirect
4. #14529463C2semidirect
5. #14529422C2semidirect
6. #14529415C2semidirect
7. #14529052C2semidirect
8. #14529038C2semidirect
12. #14528614C2semidirect
13. C4×#10587506C2semidirect
20. #11762818C4semidirect
458. C17#22581semidirect

Maximal subgroups

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

subgroupindex
C2×#117628182
#145316752
#145294772
#145294632
#145294222
#145294152
#145290522
#145290382
#145288762
#145288152
#145287192
#145286142
C4×#105875062
#145285682
Q8×C22⋊C4×C172
#2258117