Finite Group Databasebeta

← #17045629

Subgroups — #17045629

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
1C2×#11762964C211yesmaximal, characteristic
2#14531368C211yesmaximal, characteristic
3#14529654C211yesmaximal, characteristic, complemented
4#14529644C211yesmaximal, characteristic, complemented
5#14529473C211yesmaximal, characteristic, complemented
6#14529459C211yesmaximal, characteristic, complemented
7#14529425C211yesmaximal, characteristic, complemented
8#14529418C211yesmaximal, characteristic, complemented
9#14529125C211yesmaximal, characteristic, complemented
10#14529125C211yesmaximal, characteristic, complemented
11#14529097C211yesmaximal, characteristic
12#14528981C211yesmaximal, characteristic
13#14528718C211yesmaximal, characteristic
14#14528550C211yesmaximal, characteristic
15C17×#2323C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C22×#63231C2211yescharacteristic
18C22×#63231C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20#11762964C418yescomplemented
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×C22⋊#3622C2211yescharacteristic
32C2×C22⋊#3622C2211yescharacteristic
33C2×#10587474C2211yescharacteristic
34C2×C4×#3622C2211yescharacteristic
35#11762715C2211yescharacteristic
36#11762715C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762715C2211yescharacteristic
39#11762714—21no—
40#11762714—21no—
41#11762670—21no—
42#11762670—21no—
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C17×C22⋊Q8C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C2×C17×#117C2211yescharacteristic
52C17×#391C2211yescharacteristic
53C17×#389—21no—
54C17×#389C2211yescharacteristic
55C17×#388—21no—
56C17×#388—21no—
57C17×#387—21no—
58C17×#380C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C22×Dic34—21no—
64C22×Dic34—22no—
65C22×Dic34—41no—
66C22×Dic34—41no—
67C22×Dic34—41no—
68C22×Dic34—41no—
69C22×Dic34C2×C414yes—
70C22×Dic34C2311yescharacteristic
71C22×Dic34C2311yescharacteristic
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8C2×C414yes—
74C4×Q8×C17—41no—
75C4×Q8×C17—41no—
76C17×#117—41no—
77C17×#117—41no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17D411yescharacteristic
82C2×C4⋊C4×C17D411yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C4⋊C4×C17C2311yescharacteristic
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17D411yescharacteristic
89C2×C22⋊C4×C17D411yescharacteristic
90C2×C42×C17—21no—
91C2×C42×C17—21no—
92C2×C42×C17C2311yescharacteristic
93C2×C42×C17C2311yescharacteristic
94C2×C22⋊Dic17—21no—
95C2×#63231—41no—
96C2×#63231—41no—
97C2×#63231—41no—
98C2×#63231—41no—
99C2×#63231C2×C414yes—
100C2×#63231C2×C414yes—
101#10587612—41no—
102#10587612—41no—
103C2×C4⋊Dic17C2311yescharacteristic
104C2×C4⋊Dic17C2311yescharacteristic
105C2×C4×Dic17—21no—
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95—21no—
115#10587506—41no—
116#10587506—41no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125C22⋊#3622—22no—
126C22⋊#3622C2×C414yes—
127#10587481—41no—
128#10587481—41no—
129#10587481—41no—
130#10587481—41no—
131#10587481C2×C414yes—
132#10587481C2×C414yes—
133#10587479—41no—
134#10587478—41no—
135#10587474—41no—
136C4×#3622—41no—
137C24×C17C2×D411yescharacteristic
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17C22×C412yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17C22×C412yes—
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17#4112yes—
159C22×C4×C17C2411yescharacteristic
160C22×Dic17—24no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—42no—
166C22×Dic17—42no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34—24no—
176C2×Dic34—21no—
177C2×Dic34—21no—
178C2×Dic34—44no—
179C2×Dic34—44no—
180C2×Dic34—42no—
181C2×Dic34—42no—
182C2×Dic34—44no—
183C2×Dic34—44no—
184C2×Dic34C22×C412yes—
185C2×Dic34C22×C412yes—
186C4⋊C4×C17—24no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17—21no—
192C4⋊C4×C17—41no—
193C4⋊C4×C17—41no—
194C4⋊C4×C17—42no—
195C4⋊C4×C17—81no—
196C4⋊C4×C17C22×C412yes—
197C22⋊C4×C17—24no—
198C22⋊C4×C17—41no—
199C22⋊C4×C17—41no—
200C22⋊C4×C17—81no—
201C42×C17—21no—
202C42×C17—21no—
203C42×C17—21no—
204C42×C17—21no—
205C42×C17—42no—
206C42×C17—81no—
207C22⋊Dic17—42no—
208#63231—44no—
209#63231—44no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—21no—
213C4⋊Dic17—21no—
214C4×Dic17—42no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622C22×C412yes—
226C2×#3622C22×C412yes—
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—41no—
239Q8×C17—41no—
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—22no—
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—21no—
257C2×C4×C17—21no—
258C2×C4×C17—21no—
259C2×C4×C17—21no—
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×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17C4×D412yes—
271C2×C4×C17C23×C411yescharacteristic
272C2×C4×C17C2×#4111yescharacteristic
273C2×C4×C17H(5,𝔽2)11yescharacteristic
274C2×C4×C17H(5,𝔽2)11yescharacteristic
275C2×C4×C17H(5,𝔽2)11yescharacteristic
276C2×C4×C17H(5,𝔽2)11yescharacteristic
277C2×Dic17—22no—
278C2×Dic17—22no—
279C2×Dic17—48no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290Dic34—22no—
291Dic34—22no—
292Dic34—22no—
293Dic34—22no—
294Dic34—48no—
295Dic34—41no—
296Dic34—41no—
297Dic34—41no—
298Dic34—41no—
299#3622—22no—
300#3622—22no—
301#3622—41no—
302#3622—41no—
303C2×#451—171no—
304#2323D1711yescharacteristic
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—41no—
335C4×C17—41no—
336C4×C17—41no—
337C4×C17—41no—
338C4×C17—42no—
339C4×C17—81no—
340C4×C17—81no—
341C4×C17#51811yescharacteristic
342C4×C17#51811yescharacteristic
343C4×C17#53811yescharacteristic
344C4×C17#53811yescharacteristic
345Dic17—42no—
346Dic17—42no—
347Dic17—81no—
348Dic17—81no—
349C23×Q8—171no—
350C22×Q16—171no—
351C22×Q16—171no—
352C2×C22⋊Q8C2×D1711yescharacteristic
353C2×C4×Q8C2×D1711yescharacteristic
354C2×#117—171no—
355C2×#117C2×D1711yescharacteristic
356#451—178no—
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×#103—171no—
366C2×#103—171no—
367C2×C22⋊C8—171no—
368C2×C22⋊C8—171no—
369C2×#97—171no—
370C2×C4×C8—171no—
371#391—171no—
372#391—171no—
373#391—171no—
374#391—171no—
375#391—341no—
376#391—341no—
377#391C2×D1711yescharacteristic
378#389—21no—
379#389C2×D1711yescharacteristic
380#388—21no—
381#388—21no—
382#387—21no—
383#381—341no—
384#381—341no—
385#380C2×D1711yescharacteristic
386C4×C4⋊C4C2×D1711yescharacteristic
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—174no—
402C22×Q8—171no—
403C22×Q8—171no—
404C22×Q8—342no—
405C22×Q8—341no—
406C22×Q8—681no—
407C22×Q8—681no—
408C22×Q8—681no—
409C22×Q8—681no—
410C22×Q8C22×D1711yescharacteristic
411C23×C4—341no—
412C23×C4C22×D1711yescharacteristic
413C2×Q16—174no—
414C2×Q16—174no—
415C2×Q16—681no—
416C2×Q16—681no—
417C2×Q16—681no—
418C2×Q16—681no—
419C22×C8—171no—
420C22×C8—171no—
421C22×C8—171no—
422C22×C8—171no—
423C22⋊Q8—22no—
424C22⋊Q8C4×D1714yes—
425C4×Q8—41no—
426C4×Q8—41no—
427#117—41no—
428#117—41no—
429#117—681no—
430#117—681no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—171no—
435C2×C4⋊C4—171no—
436C2×C4⋊C4C22⋊D1711yescharacteristic
437C2×C4⋊C4C22⋊D1711yescharacteristic
438C2×C4⋊C4C22×D1711yescharacteristic
439C2×C4⋊C4C22×D1711yescharacteristic
440C2×C4⋊C4C22×D1711yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—341no—
444C2×C22⋊C4C22⋊D1711yescharacteristic
445C2×C22⋊C4C22⋊D1711yescharacteristic
446C2×C42—21no—
447C2×C42—21no—
448C2×C42—341no—
449C2×C42C22×D1711yescharacteristic
450C2×C42C22×D1711yescharacteristic
451#107—681no—
452#106—681no—
453#103—174no—
454#103—174no—
455#103—681no—
456#103—681no—
457#103—681no—
458#103—681no—
459#103—681no—
460#103—681no—
461C22⋊C8—174no—
462C22⋊C8—342no—
463#97—681no—
464C4×C8—681no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
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
479C24C2×C22⋊D1711yescharacteristic
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—172no—
486C2×Q8—172no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—344no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×Q8—341no—
494C2×Q8—341no—
495C2×Q8—341no—
496C2×Q8—341no—
497C2×Q8—341no—
498C2×Q8—684no—
499C2×Q8—684no—
500C2×Q8—682no—
501C2×Q8—682no—
502C2×Q8—684no—
503C2×Q8—684no—
504C2×Q8C2×C4×D1712yes—
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—341no—
514C22×C4—341no—
515C22×C4—344no—
516C22×C4—341no—
517C22×C4—341no—
518C22×C4—682no—
519C22×C4—682no—
520C22×C4C2×C4×D1712yes—
521C22×C4#6325212yes—
522C22×C4C2×C22⋊D1711yescharacteristic
523C22×C4C2×C22⋊D1711yescharacteristic
524C22×C4C2×C22⋊D1711yescharacteristic
525C22×C4C2×C22⋊D1711yescharacteristic
526C22×C4C2×C22⋊D1711yescharacteristic
527C22×C4C23×D1711yescharacteristic
528Q16—684no—
529Q16—684no—
530C2×C8—172no—
531C2×C8—172no—
532C2×C8—341no—
533C2×C8—341no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—341no—
540C2×C8—341no—
541C2×C8—341no—
542C4⋊C4—21no—
543C4⋊C4—24no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—42no—
551C4⋊C4—81no—
552C4⋊C4—341no—
553C4⋊C4—341no—
554C4⋊C4—341no—
555C4⋊C4—341no—
556C4⋊C4C2×C4×D1712yes—
557C22⋊C4—24no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—81no—
561C22⋊C4—682no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—81no—
568C42—682no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23C4×C22⋊D1712yes—
576C23C22×C22⋊D1711yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581Q8—342no—
582Q8—342no—
583Q8—342no—
584Q8—342no—
585Q8—688no—
586Q8—681no—
587Q8—681no—
588Q8—681no—
589Q8—681no—
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—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—342no—
621C2×C4—342no—
622C2×C4—688no—
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×C4C4×C22⋊D1712yes—
634C2×C4C22×C4×D1711yescharacteristic
635C2×C4C2×#6325211yescharacteristic
636C2×C4#1058769111yescharacteristic
637C2×C4#1058769111yescharacteristic
638C2×C4#1058769611yescharacteristic
639C2×C4#1058769611yescharacteristic
640C8—342no—
641C8—342no—
642C8—681no—
643C8—681no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×C4×C22⋊D1711yescharacteristic
652C22#1176336811yescharacteristic
653C22#1176337311yescharacteristic
654C22#1176362011yescharacteristic
655C22#1176362411yescharacteristic
656C22#1176362411yescharacteristic
657C22#1176362411yescharacteristic
658C4—22no—
659C4—42no—
660C4—41no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—682no—
668C4—682no—
669C4—1361no—
670C4—1361no—
671C4#1176328111yescharacteristic
672C4#1176328111yescharacteristic
673C4#1176339211yescharacteristic
674C4#1176339211yescharacteristic
675C2—42no—
676C2#1453143811yescharacteristic
677C2#1453762311yescharacteristic
678C2#1453765011yescharacteristic
679C2#1453966011yescharacteristic
680C2#1453967511yescharacteristic
681C2#1453986111yescharacteristic
682C2#1453988711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2323C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#2323D17
hypocenterC17#22054
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
3. #14529654C2semidirect
4. #14529644C2semidirect
5. #14529473C2semidirect
6. #14529459C2semidirect
7. #14529425C2semidirect
8. #14529418C2semidirect
9. #14529125C2semidirect
10. #14529125C2semidirect
20. #11762964C4semidirect
478. C17#22054semidirect

Maximal subgroups

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

subgroupindex
C2×#117629642
#145313682
#145296542
#145296442
#145294732
#145294592
#145294252
#145294182
#145291252
#145290972
#145289812
#145287182
#145285502
C17×#23232
#2205417