Finite Group Databasebeta

← #17044360

Subgroups — #17044360

Order 4352.

1094 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#63227C211yesmaximal, characteristic
2#14529398C212yesmaximal, complemented
3#14529396C212yesmaximal, complemented
4#14529391C212yesmaximal, complemented
5#14529389C212yesmaximal, complemented
6#14528764C211yesmaximal, characteristic, complemented
7#14528693C211yesmaximal, characteristic
8#14528693C211yesmaximal, characteristic
9#14528548C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic17C211yesmaximal, characteristic, complemented
11C17×#1797C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C22×#63227C411yescharacteristic
14C22×#63227C411yescharacteristic
15C22×C4×Dic17C2211yescharacteristic
16C2×C4⋊#3622C2212yescomplemented
17C2×C22⋊#3622C2211yescharacteristic
18C2×C22⋊#3622C2211yescharacteristic
19C2×#10587491C2212yes—
20C2×#10587491C2212yes—
21C2×#10587491C2212yes—
22C2×#10587491C2212yes—
23C2×#10587486C2212yescomplemented
24C2×C8×Dic17C2212yescomplemented
25C4×#63227C411yescharacteristic
26C4×#63227C411yescharacteristic
27C4×#63227C411yescharacteristic
28C4×#63227C411yescharacteristic
29C4×#63227C2211yescharacteristic
30C4×#63227C2211yescharacteristic
31C4×#63227C2211yescharacteristic
32C4×#63227C2211yescharacteristic
33C2×#10587502C411yescharacteristic
34C2×#10587502C411yescharacteristic
35#11762564C2212yescomplemented
36#11762563C2212yescomplemented
37#11762563C2212yescomplemented
38C22×C42×C17C2211yescharacteristic
39C2×C17×C4⋊C8C2212yescomplemented
40C2×C17×C22⋊C8C2211yescharacteristic
41C2×C17×C22⋊C8C2211yescharacteristic
42C17×#336C2212yescomplemented
43C24×D17C2×C411yescharacteristic
44C23×C4×C17C2×C411yescharacteristic
45C23×C4×C17C2×C411yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×Dic17C2×C411yescharacteristic
48C23×Dic17C2×C411yescharacteristic
49C23×Dic17C2×C411yescharacteristic
50C22×C4×D17—22no—
51C22×C4×D17—22no—
52C22×C4×D17—42no—
53C22×C4×D17—42no—
54C22×C4×D17C2×C411yescharacteristic
55C22×C4×D17C2×C411yescharacteristic
56C22×C4×D17C2311yescharacteristic
57C22×C4×D17C2311yescharacteristic
58C22×C8×C17C2312yes—
59C22×C8×C17C2312yes—
60C2×C42×C17—42no—
61C2×C42×C17C2×C411yescharacteristic
62C2×C42×C17C2×C411yescharacteristic
63C2×C42×C17C2311yescharacteristic
64C2×C42×C17C2311yescharacteristic
65C2×#63227—24no—
66C2×#63227—24no—
67C2×#63227C2×C411yescharacteristic
68C2×#63227C2×C411yescharacteristic
69C2×#63227C2×C411yescharacteristic
70C2×#63227C2×C411yescharacteristic
71C2×#63227C2×C411yescharacteristic
72C2×#63227C2×C411yescharacteristic
73C2×#63227C2×C411yescharacteristic
74C2×#63227C2×C411yescharacteristic
75C2×C4×Dic17—42no—
76C2×C4×Dic17C2×C411yescharacteristic
77C2×C4×Dic17C2×C411yescharacteristic
78C2×C4×Dic17C2311yescharacteristic
79C2×C4×Dic17C2311yescharacteristic
80C22×#3622C2312yes—
81C22×#3622C2312yes—
82C17×C4⋊C8—42no—
83C17×C22⋊C8—22no—
84C17×C22⋊C8—22no—
85#10587502C2×C411yescharacteristic
86#10587502C2×C411yescharacteristic
87#10587502C2×C411yescharacteristic
88#10587502C2×C411yescharacteristic
89#10587502C2×C411yescharacteristic
90#10587502C2×C411yescharacteristic
91#10587502C2×C411yescharacteristic
92#10587502C2×C411yescharacteristic
93C22⋊#3622—22no—
94C22⋊#3622—22no—
95#10587491—42no—
96#10587491—42no—
97#10587491—42no—
98#10587491—42no—
99#10587486—42no—
100C8×Dic17—42no—
101C4⋊#3622—42no—
102C24×C17C22×C411yescharacteristic
103C23×D17—22no—
104C23×D17—22no—
105C23×D17—22no—
106C23×D17—22no—
107C23×D17—22no—
108C23×D17—41no—
109C23×D17—41no—
110C23×D17C22×C411yescharacteristic
111C23×D17C22×C411yescharacteristic
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—41no—
119C22×C4×C17—41no—
120C22×C4×C17C22×C411yescharacteristic
121C22×C4×C17C22×C411yescharacteristic
122C22×C4×C17C22×C411yescharacteristic
123C22×C4×C17C22×C411yescharacteristic
124C22×C4×C17C22×C411yescharacteristic
125C22×C4×C17#4112yes—
126C22×C4×C17#4112yes—
127C22×C4×C17C2411yescharacteristic
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—41no—
134C22×Dic17—41no—
135C22×Dic17—41no—
136C22×Dic17—41no—
137C22×Dic17C22×C411yescharacteristic
138C22×Dic17C22×C411yescharacteristic
139C22×Dic17C22×C411yescharacteristic
140C22×Dic17C22×C411yescharacteristic
141C22×Dic17C22×C411yescharacteristic
142C22×Dic17C22×C411yescharacteristic
143C2×C4×D17—22no—
144C2×C4×D17—22no—
145C2×C4×D17—21no—
146C2×C4×D17—21no—
147C2×C4×D17—21no—
148C2×C4×D17—21no—
149C2×C4×D17—22no—
150C2×C4×D17—22no—
151C2×C4×D17—22no—
152C2×C4×D17—22no—
153C2×C4×D17—41no—
154C2×C4×D17—41no—
155C2×C4×D17—41no—
156C2×C4×D17—41no—
157C2×C4×D17—48no—
158C2×C4×D17—48no—
159C2×C8×C17—22no—
160C2×C8×C17—22no—
161C2×C8×C17—22no—
162C2×C8×C17—22no—
163C2×C8×C17—22no—
164C2×C8×C17—22no—
165C42×C17—22no—
166C42×C17—22no—
167C42×C17—41no—
168C42×C17—41no—
169#63227—22no—
170#63227—22no—
171#63227—22no—
172#63227—22no—
173#63227—22no—
174#63227—22no—
175#63227M1614yes—
176#63227M1614yes—
177C4×Dic17—22no—
178C4×Dic17—22no—
179C4×Dic17—41no—
180C4×Dic17—41no—
181C2×#3622—22no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622—22no—
185C2×#3622—22no—
186C2×#3622—22no—
187#21904—171nomaximal, Sylow
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—21no—
191C23×C17—21no—
192C23×C17#11712yes—
193C23×C17#13112yes—
194C23×C17#13112yes—
195C23×C17C23×C411yescharacteristic
196C22×D17—28no—
197C22×D17—21no—
198C22×D17—21no—
199C22×D17—21no—
200C22×D17—21no—
201C22×D17—21no—
202C22×D17—21no—
203C22×D17—21no—
204C22×D17—21no—
205C22×D17—21no—
206C22×D17—21no—
207C22×D17—21no—
208C22×D17—21no—
209C22×D17—44no—
210C22×D17—44no—
211C22×D17—44no—
212C22×D17—44no—
213C22×D17—44no—
214C22×D17—44no—
215C22×D17C2×M1612yes—
216C22×D17C2×M1612yes—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—24no—
224C2×C4×C17—24no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—44no—
232C2×C4×C17—44no—
233C2×C4×C17#11712yes—
234C2×C4×C17C2×M1612yes—
235C2×C4×C17C2×M1612yes—
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17H(5,𝔽2)12yes—
239C2×C4×C17H(5,𝔽2)12yes—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—44no—
257C2×Dic17—44no—
258C2×Dic17—44no—
259C2×Dic17C2×M1612yes—
260C2×Dic17C2×M1612yes—
261C4×D17—44no—
262C4×D17—44no—
263C4×D17—42no—
264C4×D17—42no—
265C4×D17—42no—
266C4×D17—42no—
267C8×C17—42no—
268C8×C17—42no—
269#3622—42no—
270#3622—42no—
271C2×C4×C22⋊C4—171no—
272#1924—172no—
273#1924—172no—
274#1923—172no—
275#1923—172no—
276#1797—171no—
277#1797D1711yescharacteristic, complemented
278#1779—171no—
279#1779—171no—
280#1749—171no—
281C4×C22⋊C8—171no—
282C22×C17—22no—
283C22×C17—22no—
284C22×C17—22no—
285C22×C17—22no—
286C22×C17—22no—
287C22×C17—22no—
288C22×C17#43314yes—
289C22×C17C2×#11711yescharacteristic
290C22×C17#51811yescharacteristic
291C22×C17#51811yescharacteristic
292C22×C17C22×M1611yescharacteristic
293C22×C17C22×M1611yescharacteristic
294C22×C17C2×#13111yescharacteristic
295C22×C17C2×#13111yescharacteristic
296C2×D17—22no—
297C2×D17—22no—
298C2×D17—22no—
299C2×D17—22no—
300C2×D17—22no—
301C2×D17—22no—
302C2×D17—416no—
303C2×D17—41no—
304C2×D17—41no—
305C2×D17—41no—
306C2×D17—41no—
307C2×D17—41no—
308C2×D17—41no—
309C2×D17—41no—
310C2×D17—41no—
311C4×C17—22no—
312C4×C17—22no—
313C4×C17—22no—
314C4×C17—41no—
315C4×C17—41no—
316C4×C17#53812yes—
317C4×C17#53812yes—
318Dic17—22no—
319Dic17—22no—
320Dic17—41no—
321Dic17—41no—
322Dic17—41no—
323Dic17—41no—
324C24×C4—171no—
325C22×C22⋊C4—171no—
326C22×C22⋊C4—171no—
327C22×C42—171no—
328C22×C42C2×D1711yescharacteristic
329C2×C4⋊C8—172no—
330C2×C4⋊C8C2×D1712yescomplemented
331C2×C22⋊C8—171no—
332C2×C22⋊C8—171no—
333C2×C22⋊C8—172no—
334C2×C22⋊C8—172no—
335C2×C22⋊C8—172no—
336C2×C22⋊C8—172no—
337C2×C22⋊C8C2×D1711yescharacteristic
338C2×C22⋊C8C2×D1711yescharacteristic
339C2×#97—172no—
340C2×C4×C8—172no—
341C4×C22⋊C4—171no—
342C4×C22⋊C4—171no—
343C4×C22⋊C4—171no—
344C4×C22⋊C4—171no—
345C4×C22⋊C4—171no—
346C4×C22⋊C4—171no—
347C4×C22⋊C4—171no—
348C4×C22⋊C4—171no—
349C2×#95—171no—
350C2×#95—171no—
351#336—172no—
352#336—172no—
353#336—172no—
354#336C2×D1712yescomplemented
355C2×C17—24no—
356C2×C17#231211yescharacteristic
357C2×C17C2×#43311yescharacteristic
358C2×C17#292011yescharacteristic
359C2×C17#297411yescharacteristic
360C2×C17#297411yescharacteristic
361C2×C17#297411yescharacteristic
362C2×C17#297411yescharacteristic
363D17—42no—
364D17—42no—
365C25—171no—
366C23×C4—171no—
367C23×C4—171no—
368C23×C4—171no—
369C23×C4—171no—
370C23×C4—171no—
371C23×C4—171no—
372C23×C4—171no—
373C23×C4—342no—
374C23×C4—342no—
375C23×C4—682no—
376C23×C4—682no—
377C23×C4C4×D1711yescharacteristic
378C23×C4C4×D1711yescharacteristic
379C23×C4C22×D1711yescharacteristic
380C22×C8—172no—
381C22×C8—172no—
382C22×C8C22×D1712yes—
383C22×C8C22×D1712yes—
384C2×C22⋊C4—171no—
385C2×C22⋊C4—171no—
386C2×C22⋊C4—171no—
387C2×C22⋊C4—171no—
388C2×C22⋊C4—171no—
389C2×C22⋊C4—171no—
390C2×C22⋊C4—171no—
391C2×C22⋊C4—171no—
392C2×C22⋊C4—344no—
393C2×C22⋊C4—344no—
394C2×C42—42no—
395C2×C42—171no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—682no—
400C2×C42C4×D1711yescharacteristic
401C2×C42C4×D1711yescharacteristic
402C2×C42C22×D1711yescharacteristic
403C2×C42C22×D1711yescharacteristic
404C4⋊C8—42no—
405C4⋊C8—682no—
406C22⋊C8—22no—
407C22⋊C8—22no—
408C22⋊C8—342no—
409C22⋊C8—342no—
410C22⋊C8—682no—
411C22⋊C8—682no—
412C22⋊C8—682no—
413C22⋊C8—682no—
414#97—682no—
415C4×C8—682no—
416#95—171no—
417#95—171no—
418#95—171no—
419#95—171no—
420#95—171no—
421#95—171no—
422#95—171no—
423#95—171no—
424C17#2190411yescharacteristic, complemented, Sylow
425C24—171no—
426C24—171no—
427C24—342no—
428C24—342no—
429C24—342no—
430C24—342no—
431C24—342no—
432C24—681no—
433C24—681no—
434C24C2×C4×D1711yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—171no—
444C22×C4—171no—
445C22×C4—171no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—171no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4—342no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—681no—
465C22×C4—681no—
466C22×C4—681no—
467C22×C4—681no—
468C22×C4—688no—
469C22×C4—688no—
470C22×C4—681no—
471C22×C4—681no—
472C22×C4—681no—
473C22×C4—681no—
474C22×C4C2×C4×D1711yescharacteristic
475C22×C4C2×C4×D1711yescharacteristic
476C22×C4C2×C4×D1711yescharacteristic
477C22×C4C2×C4×D1711yescharacteristic
478C22×C4C2×C4×D1711yescharacteristic
479C22×C4#6325412yes—
480C22×C4#6325612yes—
481C22×C4C23×D1711yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—342no—
489C2×C8—342no—
490C2×C8—342no—
491C2×C8—342no—
492C2×C8—342no—
493C2×C8—342no—
494C22⋊C4—174no—
495C22⋊C4—174no—
496C22⋊C4—342no—
497C22⋊C4—342no—
498C22⋊C4—342no—
499C22⋊C4—342no—
500C22⋊C4—342no—
501C22⋊C4—342no—
502C42—22no—
503C42—22no—
504C42—41no—
505C42—41no—
506C42—342no—
507C42—342no—
508C42—681no—
509C42—681no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—172no—
515C23—172no—
516C23—348no—
517C23—341no—
518C23—341no—
519C23—341no—
520C23—341no—
521C23—341no—
522C23—341no—
523C23—341no—
524C23—341no—
525C23—341no—
526C23—341no—
527C23—341no—
528C23—341no—
529C23—684no—
530C23—684no—
531C23—684no—
532C23—684no—
533C23—684no—
534C23—684no—
535C23#1058757812yes—
536C23#1058758712yes—
537C23#1058759312yes—
538C23C22×C4×D1711yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—172no—
556C2×C4—172no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—341no—
569C2×C4—341no—
570C2×C4—341no—
571C2×C4—341no—
572C2×C4—341no—
573C2×C4—684no—
574C2×C4—684no—
575C2×C4—684no—
576C2×C4—684no—
577C2×C4—684no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4#1058757812yes—
583C2×C4M16×D1712yes—
584C2×C4M16×D1712yes—
585C2×C4C2×#6325411yescharacteristic
586C2×C4#1058769112yes—
587C2×C4C2×#6325611yescharacteristic
588C2×C4#1058769612yes—
589C8—42no—
590C8—42no—
591C8—682no—
592C8—682no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—342no—
600C22—342no—
601C22—342no—
602C22—342no—
603C22—342no—
604C22—342no—
605C22—6816no—
606C22—681no—
607C22—681no—
608C22—681no—
609C22—681no—
610C22—681no—
611C22—681no—
612C22—681no—
613C22—681no—
614C22#1176286914yes—
615C22C2×#1058757811yescharacteristic
616C22#1176328211yescharacteristic
617C22#1176328311yescharacteristic
618C22C2×M16×D1711yescharacteristic
619C22C2×M16×D1711yescharacteristic
620C22C2×#1058758711yescharacteristic
621C22C2×#1058759311yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—342no—
628C4—342no—
629C4—681no—
630C4—681no—
631C4—681no—
632C4—681no—
633C4#1176339612yes—
634C4#1176339912yes—
635C2—24no—
636C2—682no—
637C2—682no—
638C2#1453138111yescharacteristic
639C2C2×#1176286911yescharacteristic
640C2#1453741611yescharacteristic
641C2#1453781511yescharacteristic
642C2#1453781811yescharacteristic
643C2#1453782011yescharacteristic
644C2#1453782111yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1797C2
prime radicalC24×D17C2×C4
cosocleC22×C4×C17C24
hypercenter#1797D17
hypocenterC17#21904
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #14529398C2semidirect
3. #14529396C2semidirect
4. #14529391C2semidirect
5. #14529389C2semidirect
6. #14528764C2semidirect
9. #14528548C2semidirect
10. C22⋊C8×Dic17C2semidirect
11. C17×#1797C2semidirect
16. C2×C4⋊#3622C22semidirect
23. C2×#10587486C22semidirect
24. C2×C8×Dic17C22semidirect
35. #11762564C22semidirect
36. #11762563C22semidirect
37. #11762563C22semidirect
39. C2×C17×C4⋊C8C22semidirect
42. C17×#336C22semidirect
277. #1797D17semidirect
330. C2×C4⋊C8C2×D17semidirect
354. #336C2×D17semidirect
424. C17#21904semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#632272
#145293982
#145293962
#145293912
#145293892
#145287642
#145286932
#145285482
C22⋊C8×Dic172
C17×#17972
#2190417