Finite Group Databasebeta

← #16965377

Subgroups — #16965377

Order 4352.

1530 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587565C211yesmaximal, characteristic
2C2×#11762678C211yesmaximal, characteristic, complemented
3C2×#11762676C211yesmaximal, characteristic, complemented
4#14527038C211yesmaximal, characteristic, complemented
5#14527036C212yesmaximal, complemented
6#14527035C212yesmaximal, complemented
7#14527034C211yesmaximal, characteristic, complemented
8#14527029C211yesmaximal, characteristic
9#14527027C212yesmaximal
10#14527027C212yesmaximal
11C17×#1439C211yesmaximal, characteristic
12C2×#10587565—24no—
13C2×#10587565—24no—
14C2×#10587565C418yescomplemented
15C22×C4⋊Dic17C2211yescharacteristic
16C22×C4⋊Dic17C2211yescharacteristic
17C22×#63225C2211yescharacteristic
18C22×#63225C2211yescharacteristic
19C22×#63225C2212yes—
20C22×#63225C2212yes—
21C22×C22⋊Dic17C2211yescharacteristic
22C22×C22⋊Dic17C2211yescharacteristic
23C22×C22⋊Dic17C2212yes—
24C22×C4×Dic17C2211yescharacteristic
25#11762678—24no—
26#11762676—24no—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17C2212yes—
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×Dic17C2312yes—
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C2×C4⋊C4×C17—22no—
52C2×C4⋊C4×C17—42no—
53C2×C22⋊C4×C17—22no—
54C2×C22⋊C4×C17—22no—
55C2×C22⋊C4×C17—42no—
56C2×C22⋊C4×C17—42no—
57C2×C22⋊C4×C17—42no—
58C2×C22⋊C4×C17C2×C414yes—
59C2×C22⋊Dic17—22no—
60C2×C22⋊Dic17—22no—
61C2×C22⋊Dic17—22no—
62C2×C22⋊Dic17—22no—
63C2×C22⋊Dic17—42no—
64C2×C22⋊Dic17—42no—
65C2×C22⋊Dic17—42no—
66C2×C22⋊Dic17C2×C414yes—
67C2×C22⋊Dic17C2×C414yes—
68C2×C4⋊Dic17—22no—
69C2×C4⋊Dic17—22no—
70C2×C4⋊Dic17—22no—
71C2×C4⋊Dic17—42no—
72C2×C4⋊Dic17C2×C414yes—
73C2×#63225—22no—
74C2×#63225—22no—
75C2×#63225—22no—
76C2×#63225—22no—
77C2×#63225—24no—
78C2×#63225—24no—
79C2×#63225—42no—
80C2×#63225—42no—
81C2×#63225—42no—
82C2×#63225—42no—
83C2×#63225C2×C414yes—
84C2×#63225C2×C414yes—
85C2×C4×Dic17C2×C414yes—
86C2×C4×Dic17D414yes—
87C2×C4×Dic17D414yes—
88#10587565—232no—
89C17×#95—22no—
90C17×#95—42no—
91#10587502—22no—
92#10587502—24no—
93#10587502—24no—
94#10587502—42no—
95#10587502—42no—
96#10587502—42no—
97#10587502—42no—
98#10587502—42no—
99C24×C17—21no—
100C24×C17—21no—
101C24×C17—21no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17—21no—
105C24×C17—21no—
106C24×C17—22no—
107C24×C17—22no—
108C24×C17—21no—
109C24×C17C22×C412yes—
110C24×C17C2411yescharacteristic
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—21no—
114C22×C4×C17—21no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17C22×C412yes—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C2×D412yes—
134C22×C4×C17C2×D412yes—
135C22×Dic17—22no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—22no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—22no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17C22×C412yes—
166C22×Dic17C22×C412yes—
167C22×Dic17C22×C412yes—
168C22×Dic17C22×C412yes—
169C22×Dic17C2×D412yes—
170C22×Dic17C2×D412yes—
171C22×Dic17C2×D412yes—
172C22×Dic17C2×D412yes—
173C4⋊C4×C17—44no—
174C22⋊C4×C17—28no—
175C22⋊C4×C17—48no—
176C22⋊Dic17—28no—
177C22⋊Dic17—28no—
178C22⋊Dic17—48no—
179C4⋊Dic17—28no—
180C4⋊Dic17—44no—
181#63225—28no—
182#63225—28no—
183#63225—48no—
184#63225—48no—
185C4×Dic17C22⋊C4116yescomplemented
186#14665—171nomaximal, Sylow
187C23×C17—22no—
188C23×C17—24no—
189C23×C17—24no—
190C23×C17—22no—
191C23×C17—22no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17—42no—
209C23×C17—42no—
210C23×C17C23×C411yescharacteristic
211C23×C17C22×D411yescharacteristic
212C23×C17C22×D411yescharacteristic
213C23×C17H(5,𝔽2)12yes—
214C23×C17H(5,𝔽2)12yes—
215C23×C17H(5,𝔽2)11yescharacteristic
216C23×C17H(5,𝔽2)11yescharacteristic
217C23×C172−1+411yescharacteristic
218C23×C172−1+412yes—
219C23×C172−1+412yes—
220C23×C172−1+411yescharacteristic
221C2×C4×C17—24no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—24no—
225C2×C4×C17—22no—
226C2×C4×C17—24no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17C2×C22⋊C414yes—
245C2×Dic17—24no—
246C2×Dic17—22no—
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—24no—
253C2×Dic17—22no—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—24no—
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—42no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17—42no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17C2×C22⋊C414yes—
282C2×Dic17C2×C22⋊C414yes—
283C22×#126—171no—
284C2×#384—171no—
285C2×#383—171no—
286#1441—172no—
287#1441—172no—
288#1441—171no—
289#1441—171no—
290#1439—172no—
291#1439—172no—
292#1439—171no—
293#1439D1711yescharacteristic
294C22×C17—42no—
295C22×C17—42no—
296C22×C17—42no—
297C22×C17—44no—
298C22×C17—44no—
299C22×C17—81no—
300C22×C17—81no—
301C22×C17—81no—
302C22×C17—81no—
303C22×C17—82no—
304C22×C17—82no—
305C22×C17C22×C22⋊C411yescharacteristic
306C22×C17#52012yes—
307C22×C17#52012yes—
308C22×C17#52011yescharacteristic
309C22×C17#52011yescharacteristic
310C22×C17#53511yescharacteristic
311C22×C17#53511yescharacteristic
312C22×C17#53511yescharacteristic
313C22×C17#53611yescharacteristic
314C22×C17#53611yescharacteristic
315C22×C17#53611yescharacteristic
316C22×C17#53712yes—
317C22×C17#53711yescharacteristic
318C22×C17#53712yes—
319C22×C17#53711yescharacteristic
320C22×C17#56012yes—
321C22×C17#56012yes—
322C22×C17#56312yes—
323C22×C17#56312yes—
324C22×C17#56312yes—
325C22×C17#56312yes—
326C22×C17#56312yes—
327C22×C17#56312yes—
328C4×C17—24no—
329C4×C17—42no—
330C4×C17—82no—
331C4×C17—82no—
332Dic17—24no—
333Dic17—24no—
334Dic17—42no—
335Dic17—42no—
336Dic17—82no—
337Dic17—82no—
338C2×#126—178no—
339C2×#126—344no—
340C2×#126—344no—
341C22×C4⋊C4—171no—
342C22×C4⋊C4—171no—
343C22×C4⋊C4—171no—
344C22×C4⋊C4—172no—
345C22×C4⋊C4—172no—
346C22×C4⋊C4—171no—
347C22×C4⋊C4C2×D1711yescharacteristic
348C22×C22⋊C4—171no—
349C22×C22⋊C4—171no—
350C22×C22⋊C4—172no—
351C22×C22⋊C4C2×D1711yescharacteristic
352C22×C22⋊C4C2×D1712yes—
353C22×C42—171no—
354#384—344no—
355#383—344no—
356C2×#95—172no—
357C2×#95—172no—
358C2×#95—171no—
359C2×#95—172no—
360C2×#95—172no—
361C2×#95—172no—
362C2×#95—172no—
363C2×#95—172no—
364C2×#95C2×D1711yescharacteristic
365C2×#95C2×D1712yes—
366C2×C17—82no—
367C2×C17#231111yescharacteristic
368C2×C17#231111yescharacteristic
369C2×C17#231111yescharacteristic
370C2×C17#238312yes—
371C2×C17#238312yes—
372C2×C17#281811yescharacteristic
373C2×C17#282111yescharacteristic
374C2×C17#282111yescharacteristic
375C2×C17#282111yescharacteristic
376C2×C17#282312yes—
377C2×C17#282312yes—
378C25C22×D1711yescharacteristic
379C23×C4—171no—
380C23×C4—171no—
381C23×C4—171no—
382C23×C4—171no—
383C23×C4—172no—
384C23×C4—172no—
385C23×C4C22×D1712yes—
386C23×C4C22×D1712yes—
387C23×C4C22×D1711yescharacteristic
388C23×C4C22×D1711yescharacteristic
389#126—3432no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—42no—
392C2×C4⋊C4—174no—
393C2×C4⋊C4—174no—
394C2×C4⋊C4—174no—
395C2×C4⋊C4—342no—
396C2×C4⋊C4—342no—
397C2×C4⋊C4—342no—
398C2×C4⋊C4—342no—
399C2×C4⋊C4—342no—
400C2×C4⋊C4—342no—
401C2×C4⋊C4—344no—
402C2×C4⋊C4—344no—
403C2×C4⋊C4—342no—
404C2×C4⋊C4—682no—
405C2×C4⋊C4—682no—
406C2×C4⋊C4—682no—
407C2×C4⋊C4—682no—
408C2×C4⋊C4—682no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—42no—
412C2×C22⋊C4—42no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4—174no—
415C2×C22⋊C4—174no—
416C2×C22⋊C4—342no—
417C2×C22⋊C4—342no—
418C2×C22⋊C4—342no—
419C2×C22⋊C4—342no—
420C2×C22⋊C4—682no—
421C2×C22⋊C4—682no—
422C2×C22⋊C4—682no—
423C2×C22⋊C4C4×D1714yes—
424C2×C42—174no—
425C2×C42—174no—
426C2×C42—174no—
427#95—22no—
428#95—42no—
429#95—344no—
430#95—344no—
431#95—342no—
432#95—682no—
433#95—682no—
434#95—682no—
435#95—682no—
436#95—682no—
437C17#1466511yescharacteristic, complemented, Sylow
438C24—21no—
439C24—21no—
440C24—21no—
441C24—22no—
442C24—22no—
443C24—21no—
444C24—21no—
445C24—22no—
446C24—22no—
447C24—21no—
448C24C2×C4×D1712yes—
449C24C23×D1711yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—172no—
471C22×C4—172no—
472C22×C4—172no—
473C22×C4—172no—
474C22×C4—172no—
475C22×C4—172no—
476C22×C4—172no—
477C22×C4—172no—
478C22×C4—342no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—342no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—342no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—342no—
493C22×C4—341no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—342no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—342no—
507C22×C4—342no—
508C22×C4C2×C4×D1712yes—
509C22×C4C2×C4×D1712yes—
510C22×C4D4×D1712yes—
511C22×C4D4×D1712yes—
512C4⋊C4—44no—
513C4⋊C4—348no—
514C4⋊C4—348no—
515C4⋊C4—348no—
516C4⋊C4—684no—
517C4⋊C4—688no—
518C4⋊C4—688no—
519C22⋊C4—28no—
520C22⋊C4—48no—
521C22⋊C4—348no—
522C22⋊C4—348no—
523C22⋊C4—688no—
524C42—1716no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23C22×C4×D1711yescharacteristic
549C23C2×D4×D1711yescharacteristic
550C23C2×D4×D1711yescharacteristic
551C23#1058769112yes—
552C23#1058769112yes—
553C23#1058769111yescharacteristic
554C23#1058769411yescharacteristic
555C23#1058769611yescharacteristic
556C23#1058769712yes—
557C23#1058769712yes—
558C23#1058769711yescharacteristic
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—174no—
583C2×C4—174no—
584C2×C4—344no—
585C2×C4—342no—
586C2×C4—342no—
587C2×C4—342no—
588C2×C4—342no—
589C2×C4—342no—
590C2×C4—342no—
591C2×C4—342no—
592C2×C4—344no—
593C2×C4—342no—
594C2×C4—342no—
595C2×C4—342no—
596C2×C4—344no—
597C2×C4—342no—
598C2×C4—342no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—682no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—682no—
607C2×C4—682no—
608C2×C4—682no—
609C2×C4—682no—
610C2×C4—682no—
611C2×C4—682no—
612C2×C4—682no—
613C2×C4—682no—
614C2×C4—682no—
615C2×C4—682no—
616C2×C4—682no—
617C2×C4—682no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4C22⋊C4×D1714yes—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22C2×C22⋊C4×D1711yescharacteristic
633C22#1176328812yes—
634C22#1176328812yes—
635C22#1176328911yescharacteristic
636C22#1176329011yescharacteristic
637C22#1176337211yescharacteristic
638C22#1176337411yescharacteristic
639C22#1176337511yescharacteristic
640C22#1176337911yescharacteristic
641C22#1176338111yescharacteristic
642C22#1176338211yescharacteristic
643C22#1176338612yes—
644C22#1176338811yescharacteristic
645C22#1176338812yes—
646C22#1176338911yescharacteristic
647C22#1176354712yes—
648C22#1176354812yes—
649C22#1176355912yes—
650C22#1176356012yes—
651C22#1176356012yes—
652C22#1176356112yes—
653C22#1176356212yes—
654C22#1176356212yes—
655C4—24no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—344no—
660C4—344no—
661C4—682no—
662C4—682no—
663C4—1362no—
664C4—1362no—
665C2—82no—
666C2#1453137111yescharacteristic
667C2#1453137311yescharacteristic
668C2#1453137411yescharacteristic
669C2#1453197112yes—
670C2#1453197212yes—
671C2#1453682311yescharacteristic
672C2#1453684111yescharacteristic
673C2#1453684611yescharacteristic
674C2#1453684711yescharacteristic
675C2#1453685512yes—
676C2#1453685712yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1439C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenter#1439D17
hypocenterC17#14665
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#11762678C2semidirect
3. C2×#11762676C2semidirect
4. #14527038C2semidirect
5. #14527036C2semidirect
6. #14527035C2semidirect
7. #14527034C2semidirect
14. C2×#10587565C4semidirect
185. C4×Dic17C22⋊C4semidirect
437. C17#14665semidirect

Maximal subgroups

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

subgroupindex
C22×#105875652
C2×#117626782
C2×#117626762
#145270382
#145270362
#145270352
#145270342
#145270292
#145270272
C17×#14392
#1466517