Finite Group Databasebeta

← #17045639

Subgroups — #17045639

Order 4352.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531468C211yesmaximal, characteristic
2#14531370C211yesmaximal, characteristic
3#14529656C211yesmaximal, characteristic, complemented
4#14529646C211yesmaximal, characteristic, complemented
5#14529476C211yesmaximal, characteristic, complemented
6#14529462C211yesmaximal, characteristic, complemented
7#14529427C211yesmaximal, characteristic, complemented
8#14529420C211yesmaximal, characteristic, complemented
9#14529127C211yesmaximal, characteristic, complemented
10#14529127C211yesmaximal, characteristic, complemented
11#14529097C211yesmaximal, characteristic
12#14528985C211yesmaximal, characteristic
13#14528719C211yesmaximal, characteristic
14#14528552C211yesmaximal, characteristic
15C2×C17×#451C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×#10587612C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19C2×#10587489C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×C22⋊#3622C2211yescharacteristic
28C2×#10587486C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30#11762716—21no—
31#11762716—21no—
32#11762716C2211yescharacteristic
33#11762716C2211yescharacteristic
34#11762716C2211yescharacteristic
35#11762716C2211yescharacteristic
36#11762716C2211yescharacteristic
37#11762708—21no—
38#11762708C2211yescharacteristic
39#11762702—21no—
40#11762702—21no—
41#11762695—21no—
42#11762672—21no—
43#11762672—21no—
44#11762669C2211yescharacteristic
45C4⋊C4×Dic17C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C23×Q8×C17C2211yescharacteristic
53C22×Q16×C17C2211yescharacteristic
54C22×Q16×C17C2211yescharacteristic
55C2×C17×C22⋊Q8C2211yescharacteristic
56C17×#451C418yescomplemented
57C2×C17×#103C2211yescharacteristic
58C2×C17×#103C2211yescharacteristic
59C2×C17×C22⋊C8C2211yescharacteristic
60C22×Q8×C17—21no—
61C22×Q8×C17—22no—
62C22×Q8×C17—41no—
63C22×Q8×C17—41no—
64C22×Q8×C17—41no—
65C22×Q8×C17—41no—
66C22×Q8×C17C2×C414yes—
67C22×Q8×C17C2311yescharacteristic
68C22×Q8×C17C2311yescharacteristic
69C22×Q8×C17C2311yescharacteristic
70C23×C4×C17—21no—
71C23×C4×C17C2311yescharacteristic
72C2×Q16×C17—41no—
73C2×Q16×C17—41no—
74C2×Q16×C17—41no—
75C2×Q16×C17—41no—
76C2×Q16×C17C2×C414yes—
77C2×Q16×C17C2×C414yes—
78C22×C8×C17C2311yescharacteristic
79C22×C8×C17C2311yescharacteristic
80C17×C22⋊Q8—22no—
81C17×C22⋊Q8C2×C414yes—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C22⋊C4×C17—21no—
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87C2×C22⋊Dic17D411yescharacteristic
88C2×C22⋊Dic17D411yescharacteristic
89Q8×Dic17—41no—
90Q8×Dic17—41no—
91#10587612—41no—
92#10587612—41no—
93#10587612—41no—
94#10587612—41no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17D411yescharacteristic
98C2×C4⋊Dic17D411yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4⋊Dic17C2311yescharacteristic
101C2×C4⋊Dic17C2311yescharacteristic
102C2×C4⋊Dic17C2311yescharacteristic
103C2×C4×Dic17—21no—
104C2×C4×Dic17—21no—
105C2×C4×Dic17—21no—
106C2×C4×Dic17C2311yescharacteristic
107C2×C4×Dic17C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C17×#103C2×C414yes—
111C17×#103C2×C414yes—
112C17×C22⋊C8C2×C414yes—
113#10587506—41no—
114#10587506—41no—
115#10587506—41no—
116#10587506—41no—
117#10587506—41no—
118#10587506—41no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130#10587502—21no—
131#10587502—21no—
132C22⋊#3622—22no—
133#10587489—41no—
134#10587488—41no—
135#10587486—41no—
136C8×Dic17—41no—
137C24×C17C2×D411yescharacteristic
138C2×Q8×C17—24no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17—21no—
148C2×Q8×C17—21no—
149C2×Q8×C17—21no—
150C2×Q8×C17—21no—
151C2×Q8×C17—21no—
152C2×Q8×C17—21no—
153C2×Q8×C17—21no—
154C2×Q8×C17—44no—
155C2×Q8×C17—44no—
156C2×Q8×C17—42no—
157C2×Q8×C17—42no—
158C2×Q8×C17—44no—
159C2×Q8×C17—44no—
160C2×Q8×C17C22×C412yes—
161C2×Q8×C17C22×C412yes—
162C2×Q8×C17C22×C412yes—
163C22×C4×C17—21no—
164C22×C4×C17—21no—
165C22×C4×C17—21no—
166C22×C4×C17—21no—
167C22×C4×C17—24no—
168C22×C4×C17—21no—
169C22×C4×C17—21no—
170C22×C4×C17—21no—
171C22×C4×C17—21no—
172C22×C4×C17—42no—
173C22×C4×C17—42no—
174C22×C4×C17C22×C412yes—
175C22×C4×C17C2×D411yescharacteristic
176C22×C4×C17#4112yes—
177C22×C4×C17C2411yescharacteristic
178C22×Dic17—21no—
179C22×Dic17—21no—
180C22×Dic17—21no—
181C22×Dic17—21no—
182C22×Dic17C2×D411yescharacteristic
183C22×Dic17C2×D411yescharacteristic
184C22×Dic17C2×D411yescharacteristic
185C22×Dic17C2×D411yescharacteristic
186Q16×C17—44no—
187Q16×C17—44no—
188C2×C8×C17—21no—
189C2×C8×C17—21no—
190C2×C8×C17—21no—
191C2×C8×C17—21no—
192C2×C8×C17C22×C412yes—
193C2×C8×C17C22×C412yes—
194C4⋊C4×C17—24no—
195C4⋊C4×C17—21no—
196C4⋊C4×C17C22×C412yes—
197C22⋊C4×C17—24no—
198C22⋊Dic17—41no—
199C22⋊Dic17—41no—
200C22⋊Dic17—42no—
201C22⋊Dic17—81no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—41no—
211C4⋊Dic17—41no—
212C4⋊Dic17—42no—
213C4⋊Dic17—81no—
214C4×Dic17—21no—
215C4×Dic17—21no—
216C4×Dic17—21no—
217C4×Dic17—21no—
218C4×Dic17—42no—
219C4×Dic17—42no—
220C4×Dic17—81no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227#22054—171nomaximal, Sylow
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17—21no—
234C23×C17C4×D412yes—
235C23×C17C22×D411yescharacteristic
236Q8×C17—22no—
237Q8×C17—22no—
238Q8×C17—22no—
239Q8×C17—22no—
240Q8×C17—22no—
241Q8×C17—22no—
242Q8×C17—48no—
243Q8×C17—41no—
244Q8×C17—41no—
245Q8×C17—41no—
246Q8×C17—41no—
247Q8×C17—41no—
248Q8×C17—41no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—22no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—22no—
258C2×C4×C17—22no—
259C2×C4×C17—48no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—41no—
268C2×C4×C17C4×D412yes—
269C2×C4×C17C23×C411yescharacteristic
270C2×C4×C17C2×#4111yescharacteristic
271C2×C4×C17H(5,𝔽2)11yescharacteristic
272C2×C4×C17H(5,𝔽2)11yescharacteristic
273C2×C4×C17H(5,𝔽2)11yescharacteristic
274C2×C4×C17H(5,𝔽2)11yescharacteristic
275C2×Dic17—21no—
276C2×Dic17—21no—
277C2×Dic17—21no—
278C2×Dic17—21no—
279C2×Dic17—21no—
280C2×Dic17—21no—
281C2×Dic17—21no—
282C2×Dic17—21no—
283C2×Dic17—21no—
284C2×Dic17—21no—
285C2×Dic17—21no—
286C2×Dic17—21no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290C2×Dic17—41no—
291C2×Dic17—41no—
292C2×Dic17—41no—
293C2×Dic17—41no—
294C2×Dic17—41no—
295C2×Dic17—41no—
296C2×Dic17—41no—
297C2×Dic17—41no—
298C2×Dic17—41no—
299C8×C17—22no—
300C8×C17—22no—
301#3622—41no—
302#3622—41no—
303C2×#451D1711yescharacteristic
304#2323—171no—
305#2310—171no—
306#1971—171no—
307#1969—171no—
308#1935—171no—
309#1933—171no—
310#1928—171no—
311#1927—171no—
312#1878—171no—
313#1878—171no—
314#1873—171no—
315#1854—171no—
316#1787—171no—
317#1750—171no—
318C22×C17—22no—
319C22×C17—22no—
320C22×C17—22no—
321C22×C17—41no—
322C22×C17—41no—
323C22×C17—41no—
324C22×C17—41no—
325C22×C17C2×C4×D411yescharacteristic
326C22×C17#53411yescharacteristic
327C22×C17#53511yescharacteristic
328C22×C17#57711yescharacteristic
329C22×C17#57811yescharacteristic
330C22×C17#57811yescharacteristic
331C22×C17#57811yescharacteristic
332C4×C17—22no—
333C4×C17—42no—
334C4×C17—42no—
335C4×C17—42no—
336C4×C17—42no—
337C4×C17#51811yescharacteristic
338C4×C17#51811yescharacteristic
339C4×C17#53811yescharacteristic
340C4×C17#53811yescharacteristic
341Dic17—41no—
342Dic17—41no—
343Dic17—41no—
344Dic17—41no—
345Dic17—81no—
346Dic17—81no—
347Dic17—81no—
348Dic17—81no—
349C23×Q8C2×D1711yescharacteristic
350C22×Q16C2×D1711yescharacteristic
351C22×Q16C2×D1711yescharacteristic
352C2×C22⋊Q8C2×D1711yescharacteristic
353C2×C4×Q8—171no—
354C2×#117—171no—
355C2×#117—171no—
356#451Dic1718yescomplemented
357C2×#107—171no—
358C2×#106—171no—
359C2×#103—171no—
360C2×#103—171no—
361C2×#103—171no—
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103C2×D1711yescharacteristic
366C2×#103C2×D1711yescharacteristic
367C2×C22⋊C8—171no—
368C2×C22⋊C8C2×D1711yescharacteristic
369C2×#97—171no—
370C2×C4×C8—171no—
371#391—171no—
372#391—171no—
373#391—171no—
374#391—171no—
375#391—171no—
376#391—341no—
377#391—341no—
378#389—171no—
379#389—341no—
380#388—341no—
381#388—341no—
382#387—341no—
383#381—341no—
384#381—341no—
385#380—171no—
386C4×C4⋊C4—171no—
387#340—171no—
388#340—171no—
389#340—171no—
390#340—171no—
391#336—171no—
392#336—171no—
393C2×C17—42no—
394C2×C17#232111yescharacteristic
395C2×C17#294711yescharacteristic
396C2×C17#295011yescharacteristic
397C2×C17#320111yescharacteristic
398C2×C17#320211yescharacteristic
399C2×C17#322311yescharacteristic
400C2×C17#322511yescharacteristic
401C22×Q8—21no—
402C22×Q8—22no—
403C22×Q8—41no—
404C22×Q8—41no—
405C22×Q8—41no—
406C22×Q8—41no—
407C22×Q8C2×Dic1714yes—
408C22×Q8C22×D1711yescharacteristic
409C22×Q8C22×D1711yescharacteristic
410C22×Q8C22×D1711yescharacteristic
411C23×C4—21no—
412C23×C4C22×D1711yescharacteristic
413C2×Q16—41no—
414C2×Q16—41no—
415C2×Q16—41no—
416C2×Q16—41no—
417C2×Q16C2×Dic1714yes—
418C2×Q16C2×Dic1714yes—
419C22×C8—171no—
420C22×C8—171no—
421C22×C8C22×D1711yescharacteristic
422C22×C8C22×D1711yescharacteristic
423C22⋊Q8—22no—
424C22⋊Q8C2×Dic1714yes—
425C4×Q8—681no—
426C4×Q8—681no—
427#117—681no—
428#117—681no—
429#117—681no—
430#117—681no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—171no—
434C2×C4⋊C4—171no—
435C2×C4⋊C4—171no—
436C2×C4⋊C4—171no—
437C2×C4⋊C4—171no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4C22×D1711yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—171no—
443C2×C22⋊C4—171no—
444C2×C22⋊C4—341no—
445C2×C22⋊C4—341no—
446C2×C42—171no—
447C2×C42—171no—
448C2×C42—341no—
449C2×C42—341no—
450C2×C42—341no—
451#107—681no—
452#106—681no—
453#103—681no—
454#103—681no—
455#103—681no—
456#103—681no—
457#103—681no—
458#103—681no—
459#103C2×Dic1714yes—
460#103C2×Dic1714yes—
461C22⋊C8—342no—
462C22⋊C8C2×Dic1714yes—
463#97—681no—
464C4×C8—681no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469#95—341no—
470#95—341no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477#95—341no—
478C17#2205411yescharacteristic, complemented, Sylow
479C24D4×D1711yescharacteristic
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—24no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—44no—
497C2×Q8—44no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—44no—
501C2×Q8—44no—
502C2×Q8C22×Dic1712yes—
503C2×Q8C22×Dic1712yes—
504C2×Q8C22×Dic1712yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—24no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—171no—
517C22×C4—171no—
518C22×C4—171no—
519C22×C4—171no—
520C22×C4—341no—
521C22×C4—341no—
522C22×C4—341no—
523C22×C4—341no—
524C22×C4D4×D1711yescharacteristic
525C22×C4#6325412yes—
526C22×C4C22×Dic1712yes—
527C22×C4C23×D1711yescharacteristic
528Q16—44no—
529Q16—44no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—341no—
540C2×C8C22×Dic1712yes—
541C2×C8C22×Dic1712yes—
542C4⋊C4—21no—
543C4⋊C4—24no—
544C4⋊C4—341no—
545C4⋊C4—341no—
546C4⋊C4—341no—
547C4⋊C4—341no—
548C4⋊C4—341no—
549C4⋊C4—341no—
550C4⋊C4—341no—
551C4⋊C4—341no—
552C4⋊C4—681no—
553C4⋊C4—681no—
554C4⋊C4—682no—
555C4⋊C4—1361no—
556C4⋊C4C22×Dic1712yes—
557C22⋊C4—24no—
558C22⋊C4—681no—
559C22⋊C4—681no—
560C22⋊C4—682no—
561C22⋊C4—1361no—
562C42—341no—
563C42—341no—
564C42—341no—
565C42—341no—
566C42—682no—
567C42—682no—
568C42—1361no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23D4×Dic1712yes—
576C23C2×D4×D1711yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—48no—
584Q8—41no—
585Q8—41no—
586Q8—41no—
587Q8—41no—
588Q8—41no—
589Q8—41no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—341no—
610C2×C4—341no—
611C2×C4—341no—
612C2×C4—341no—
613C2×C4—341no—
614C2×C4—341no—
615C2×C4—341no—
616C2×C4—341no—
617C2×C4—341no—
618C2×C4—341no—
619C2×C4—341no—
620C2×C4—341no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—681no—
624C2×C4—681no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4D4×Dic1712yes—
634C2×C4C2×#6325411yescharacteristic
635C2×C4#1058769111yescharacteristic
636C2×C4#1058769111yescharacteristic
637C2×C4#1058769111yescharacteristic
638C2×C4#1058769111yescharacteristic
639C2×C4C23×Dic1711yescharacteristic
640C8—22no—
641C8—22no—
642C8—681no—
643C8—681no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×D4×Dic1711yescharacteristic
652C22#1176336911yescharacteristic
653C22#1176337411yescharacteristic
654C22#1176361811yescharacteristic
655C22#1176362211yescharacteristic
656C22#1176362211yescharacteristic
657C22#1176362511yescharacteristic
658C4—22no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—681no—
664C4—681no—
665C4—681no—
666C4—681no—
667C4—1361no—
668C4—1361no—
669C4—1361no—
670C4—1361no—
671C4#1176328411yescharacteristic
672C4#1176328411yescharacteristic
673C4#1176339411yescharacteristic
674C4#1176339411yescharacteristic
675C2—42no—
676C2#1453144511yescharacteristic
677C2#1453762811yescharacteristic
678C2#1453765211yescharacteristic
679C2#1453967011yescharacteristic
680C2#1453967711yescharacteristic
681C2#1453986311yescharacteristic
682C2#1453989911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#451C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenterC2×#451D17
hypocenterC17#22054
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
3. #14529656C2semidirect
4. #14529646C2semidirect
5. #14529476C2semidirect
6. #14529462C2semidirect
7. #14529427C2semidirect
8. #14529420C2semidirect
9. #14529127C2semidirect
10. #14529127C2semidirect
56. C17×#451C4semidirect
356. #451Dic17semidirect
478. C17#22054semidirect

Maximal subgroups

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

subgroupindex
#145314682
#145313702
#145296562
#145296462
#145294762
#145294622
#145294272
#145294202
#145291272
#145290972
#145289852
#145287192
#145285522
C2×C17×#4512
#2205417