Finite Group Databasebeta

← #17091840

Subgroups — #17091840

Order 4352.

920 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763118C211yesmaximal, characteristic, complemented
2#14532491C211yesmaximal, characteristic
3Q8×C22⋊Dic17C211yesmaximal, characteristic
4#14530320C211yesmaximal, characteristic, complemented
5#14530163C211yesmaximal, characteristic, complemented
6#14530161C211yesmaximal, characteristic, complemented
7#14529933C211yesmaximal, characteristic, complemented
8#14529054C211yesmaximal, characteristic, complemented
9#14529036C211yesmaximal, characteristic, complemented
10#14528983C211yesmaximal, characteristic
11#14528967C211yesmaximal, characteristic
12#14528817C211yesmaximal, characteristic
13#14528787C211yesmaximal, characteristic, complemented
14#14528719C211yesmaximal, characteristic
15C2×C17×#419C211yesmaximal, characteristic
16C2×#10587576C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×Q8×Dic17C2211yescharacteristic
19#11763118—24no—
20C2×#10587489C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×C22⋊#3622C2211yescharacteristic
31#11762745—21no—
32#11762732—21no—
33#11762716—21no—
34#11762716—21no—
35#11762716C2211yescharacteristic
36#11762716C2211yescharacteristic
37#11762708C2211yescharacteristic
38#11762707—21no—
39#11762705C2211yescharacteristic
40#11762693—21no—
41#11762678C2211yescharacteristic
42#11762669—21no—
43#11762669C2211yescharacteristic
44#11762669C2211yescharacteristic
45C4×C22⋊Dic17—21no—
46C4×C22⋊Dic17C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C23×Q8×C17C2211yescharacteristic
54C22×M16×C17C2211yescharacteristic
55C2×C17×#117C2211yescharacteristic
56C17×#419—24no—
57C2×C17×#103C2211yescharacteristic
58C2×C17×#103C2211yescharacteristic
59C2×C17×#103C2211yescharacteristic
60C2×C17×#103C2211yescharacteristic
61C22×Q8×C17—21no—
62C22×Q8×C17—22no—
63C22×Q8×C17—22no—
64C22×Q8×C17—22no—
65C22×Q8×C17—41no—
66C22×Q8×C17—41no—
67C22×Q8×C17Q814yes—
68C22×Q8×C17C2311yescharacteristic
69C22×Q8×C17C2311yescharacteristic
70C23×C4×C17—21no—
71C23×C4×C17C2311yescharacteristic
72C2×M16×C17—22no—
73C2×M16×C17—22no—
74C2×M16×C17—41no—
75C22×C8×C17C2311yescharacteristic
76C22×C8×C17C2311yescharacteristic
77C17×#117—22no—
78C17×#117—41no—
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C22⋊C4×C17—21no—
82C2×C42×C17—21no—
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85C2×C22⋊Dic17D411yescharacteristic
86C2×C22⋊Dic17D411yescharacteristic
87Q8×Dic17—22no—
88Q8×Dic17—22no—
89Q8×Dic17—41no—
90Q8×Dic17—41no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17—21no—
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17—21no—
105C2×C4×Dic17C2311yescharacteristic
106C2×C4×Dic17C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109#10587576—22no—
110#10587576—41no—
111C17×#103—22no—
112C17×#103—22no—
113C17×#103—22no—
114C17×#103D414yes—
115#10587506—22no—
116#10587506—22no—
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#10587502D411yescharacteristic
128#10587502D411yescharacteristic
129C22⋊#3622—22no—
130#10587489—22no—
131#10587488—41no—
132#10587485—22no—
133#10587485—41no—
134#10587479—22no—
135#10587478—41no—
136C24×C17C2×D411yescharacteristic
137C2×Q8×C17—24no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—28no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17—42no—
148C2×Q8×C17—42no—
149C2×Q8×C17—44no—
150C2×Q8×C17—44no—
151C2×Q8×C17—44no—
152C2×Q8×C17C2×D412yes—
153C2×Q8×C17C2×Q812yes—
154C2×Q8×C17C2×Q812yes—
155C2×Q8×C17#4112yes—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17—21no—
160C22×C4×C17—21no—
161C22×C4×C17—24no—
162C22×C4×C17—21no—
163C22×C4×C17—21no—
164C22×C4×C17—21no—
165C22×C4×C17—42no—
166C22×C4×C17—42no—
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17C2×Q812yes—
169C22×C4×C17#4112yes—
170C22×C4×C17C2411yescharacteristic
171C22×Dic17—21no—
172C22×Dic17—21no—
173C22×Dic17—21no—
174C22×Dic17—21no—
175C22×Dic17C2×D411yescharacteristic
176C22×Dic17C2×D411yescharacteristic
177C22×Dic17C2×D411yescharacteristic
178C22×Dic17C2×D411yescharacteristic
179M16×C17—44no—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C2×C8×C17—21no—
184C2×C8×C17—21no—
185C2×C8×C17C2×D412yes—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17C2×D412yes—
190C22⋊C4×C17—42no—
191C42×C17—42no—
192C22⋊Dic17—22no—
193C22⋊Dic17—41no—
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—42no—
205C4⋊Dic17—42no—
206C4⋊Dic17—42no—
207C4⋊Dic17—42no—
208#63225—42no—
209#63225—42no—
210#63225—81no—
211#63225—81no—
212C4×Dic17—21no—
213C4×Dic17—21no—
214C4×Dic17—21no—
215C4×Dic17—21no—
216C4×Dic17—42no—
217C4×Dic17—42no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224#25981—171nomaximal, Sylow
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17C22⋊Q812yes—
232C23×C17C22×D411yescharacteristic
233Q8×C17—22no—
234Q8×C17—22no—
235Q8×C17—22no—
236Q8×C17—22no—
237Q8×C17—22no—
238Q8×C17—48no—
239Q8×C17—41no—
240Q8×C17—41no—
241Q8×C17C22⋊Q814yes—
242C2×C4×C17—22no—
243C2×C4×C17—21no—
244C2×C4×C17—22no—
245C2×C4×C17—21no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—48no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17C22⋊Q812yes—
262C2×C4×C17C22×D411yescharacteristic
263C2×C4×C17C22×Q811yescharacteristic
264C2×C4×C17C2×#4111yescharacteristic
265C2×C4×C17C2×#4111yescharacteristic
266C2×C4×C17H(5,𝔽2)11yescharacteristic
267C2×C4×C17H(5,𝔽2)11yescharacteristic
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—21no—
276C2×Dic17—21no—
277C2×Dic17—21no—
278C2×Dic17—21no—
279C2×Dic17—41no—
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—
290C2×Dic17—41no—
291C2×Dic17#13712yes—
292C8×C17—22no—
293C8×C17—41no—
294#3622—41no—
295#3622—41no—
296C2×#479—171no—
297C2×#419D1711yescharacteristic
298#2441—171no—
299Q8×C22⋊C4—171no—
300#2095—171no—
301#2066—171no—
302#2066—171no—
303#2026—171no—
304#1865—171no—
305#1863—171no—
306#1854—171no—
307#1851—171no—
308#1809—171no—
309#1803—171no—
310#1787—171no—
311C22×C17—22no—
312C22×C17—22no—
313C22×C17—22no—
314C22×C17—41no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17C2×C22⋊Q811yescharacteristic
319C22×C17D4211yescharacteristic
320C22×C17#54611yescharacteristic
321C22×C17C2×#13711yescharacteristic
322C22×C17#57511yescharacteristic
323C22×C17#57711yescharacteristic
324C22×C17#57811yescharacteristic
325C4×C17—22no—
326C4×C17—24no—
327C4×C17—42no—
328C4×C17—42no—
329C4×C17—81no—
330C4×C17C2×C22⋊Q811yescharacteristic
331C4×C17#54311yescharacteristic
332C4×C17#54611yescharacteristic
333C4×C17#55111yescharacteristic
334Dic17—22no—
335Dic17—41no—
336Dic17—41no—
337Dic17—41no—
338Dic17—81no—
339Dic17—81no—
340Dic17—81no—
341Dic17—81no—
342C23×Q8C2×D1711yescharacteristic
343C22×M16C2×D1711yescharacteristic
344C2×#125—171no—
345C2×C4×Q8—171no—
346C2×C4×Q8—171no—
347C2×#117C2×D1711yescharacteristic
348#479—344no—
349C2×#107—171no—
350C2×#107—171no—
351C2×#106—171no—
352C2×#106—171no—
353C2×C4⋊C8—171no—
354C2×C4⋊C8—171no—
355#419—24no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#103C2×D1711yescharacteristic
361C2×#103C2×D1711yescharacteristic
362C2×#103C2×D1711yescharacteristic
363C2×#103C2×D1711yescharacteristic
364C2×C22⋊C8—171no—
365#399—341no—
366#396—341no—
367#391—171no—
368#391—171no—
369#391—341no—
370#391—341no—
371#389—171no—
372#389—171no—
373#389—341no—
374#387—341no—
375#384—171no—
376#380—171no—
377#380—171no—
378#380—341no—
379C4×C22⋊C4—171no—
380C4×C22⋊C4—341no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#336—171no—
386#336—171no—
387C2×C17—42no—
388C2×C17#243711yescharacteristic
389C2×C17#307811yescharacteristic
390C2×C17#308211yescharacteristic
391C2×C17#327111yescharacteristic
392C2×C17#327611yescharacteristic
393C2×C17#332011yescharacteristic
394C2×C17#332711yescharacteristic
395C22×Q8—21no—
396C22×Q8—22no—
397C22×Q8—22no—
398C22×Q8—22no—
399C22×Q8—41no—
400C22×Q8—41no—
401C22×Q8Dic3414yes—
402C22×Q8C22×D1711yescharacteristic
403C22×Q8C22×D1711yescharacteristic
404C23×C4—21no—
405C23×C4C22×D1711yescharacteristic
406C2×M16—22no—
407C2×M16—22no—
408C2×M16—41no—
409C22×C8—171no—
410C22×C8—171no—
411C22×C8C22×D1711yescharacteristic
412C22×C8C22×D1711yescharacteristic
413#125—342no—
414#125—681no—
415C4×Q8—342no—
416C4×Q8—342no—
417C4×Q8—681no—
418C4×Q8—681no—
419#117—22no—
420#117—41no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—341no—
431C2×C4⋊C4—341no—
432C2×C4⋊C4—341no—
433C2×C4⋊C4C22×D1711yescharacteristic
434C2×C4⋊C4C22×D1711yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—171no—
437C2×C22⋊C4—171no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4—341no—
440C2×C42—21no—
441C2×C42—171no—
442C2×C42—171no—
443C2×C42—341no—
444C2×C42—341no—
445#107—342no—
446#107—681no—
447#106—342no—
448#106—681no—
449C4⋊C8—342no—
450C4⋊C8—681no—
451#103—22no—
452#103—22no—
453#103—22no—
454#103—342no—
455#103—342no—
456#103—681no—
457#103—681no—
458#103C22⋊D1714yes—
459C22⋊C8—342no—
460#95—171no—
461#95—171no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469#95—341no—
470C17#2598111yescharacteristic, complemented, Sylow
471C24D4×D1711yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—24no—
477C2×Q8—28no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8—44no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8C2×Dic3412yes—
488C2×Q8C2×Dic3412yes—
489C2×Q8#6325212yes—
490C2×Q8C2×C22⋊D1712yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—171no—
503C22×C4—171no—
504C22×C4—171no—
505C22×C4—171no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—341no—
509C22×C4—341no—
510C22×C4C2×Dic3412yes—
511C22×C4D4×D1711yescharacteristic
512C22×C4#6325412yes—
513C22×C4C23×D1711yescharacteristic
514M16—44no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8C2×C22⋊D1712yes—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—682no—
539C4⋊C4—682no—
540C4⋊C4—682no—
541C4⋊C4—682no—
542C4⋊C4—682no—
543C4⋊C4—682no—
544C4⋊C4—1361no—
545C4⋊C4—1361no—
546C4⋊C4C2×C22⋊D1712yes—
547C22⋊C4—42no—
548C22⋊C4—342no—
549C22⋊C4—681no—
550C22⋊C4—682no—
551C22⋊C4—1361no—
552C42—42no—
553C42—341no—
554C42—341no—
555C42—341no—
556C42—341no—
557C42—682no—
558C42—682no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#1058756412yes—
566C23C2×D4×D1711yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—48no—
573Q8—41no—
574Q8—41no—
575Q8#1058761014yes—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—172no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—341no—
602C2×C4—341no—
603C2×C4—341no—
604C2×C4—341no—
605C2×C4—341no—
606C2×C4—341no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4#1058756412yes—
620C2×C4C22×Dic3411yescharacteristic
621C2×C4C2×#6325211yescharacteristic
622C2×C4C2×#6325411yescharacteristic
623C2×C4#1058769111yescharacteristic
624C2×C4#1058769111yescharacteristic
625C2×C4C22×C22⋊D1711yescharacteristic
626C8—22no—
627C8—41no—
628C8—681no—
629C8—681no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#1058756411yescharacteristic
638C22D4×C22⋊D1711yescharacteristic
639C22#1176344911yescharacteristic
640C22#137×D1711yescharacteristic
641C22#1176360811yescharacteristic
642C22#1176361811yescharacteristic
643C22#1176362511yescharacteristic
644C4—22no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—81no—
649C4—342no—
650C4—681no—
651C4—681no—
652C4—681no—
653C4—1361no—
654C4—1361no—
655C4—1361no—
656C4—1361no—
657C4C2×#1058761011yescharacteristic
658C4#1176343511yescharacteristic
659C4#1176345311yescharacteristic
660C4#1176348511yescharacteristic
661C2—42no—
662C2#1453244511yescharacteristic
663C2#1453858611yescharacteristic
664C2#1453862811yescharacteristic
665C2#1454036311yescharacteristic
666C2#1454042411yescharacteristic
667C2#1454095611yescharacteristic
668C2#1454104811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#419C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenterC2×#419D17
hypocenterC17#25981
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
1. C2×#11763118C2semidirect
4. #14530320C2semidirect
5. #14530163C2semidirect
6. #14530161C2semidirect
7. #14529933C2semidirect
8. #14529054C2semidirect
9. #14529036C2semidirect
13. #14528787C2semidirect
470. C17#25981semidirect

Maximal subgroups

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

subgroupindex
C2×#117631182
#145324912
Q8×C22⋊Dic172
#145303202
#145301632
#145301612
#145299332
#145290542
#145290362
#145289832
#145289672
#145288172
#145287872
#145287192
C2×C17×#4192
#2598117