Finite Group Databasebeta

← #16956852

Subgroups — #16956852

Order 4352.

1862 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
1C22×#10587578C211yesmaximal, characteristic
2C2×#11762676C211yesmaximal, characteristic, complemented
3C2×C4×#63225C211yesmaximal, characteristic, complemented
4#14527036C212yesmaximal, complemented
5#14527028C212yesmaximal
6#14527022C211yesmaximal, characteristic
7#14527020C211yesmaximal, characteristic
8#14527016C211yesmaximal, characteristic, complemented
9#14527010C212yesmaximal, complemented
10#14527005C212yesmaximal
11C17×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C2×#10587578—24no—
14C2×#10587578—24no—
15C2×#10587578C418yescomplemented
16C22×C4⋊Dic17C2211yescharacteristic
17C22×#63225C2212yes—
18C22×#63227C2212yes—
19C22×#63227C2212yes—
20C22×#63227C2211yescharacteristic
21C22×#63227C2211yescharacteristic
22C22×C4×Dic17C2211yescharacteristic
23C22×C4×Dic17C2211yescharacteristic
24C22×C4×Dic17C2211yescharacteristic
25#11762676—24no—
26C4×#63225—24no—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C42×C17C2211yescharacteristic
37C2×C17×#95C2212yes—
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2212yes—
40C24×D17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2312yes—
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C22×C4×D17—24no—
52C22×C4×D17—24no—
53C22×C4×D17—22no—
54C22×C4×D17—22no—
55C22×C4×D17C2×C414yes—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17C2×C414yes—
59C2×C42×C17—22no—
60C2×C42×C17—42no—
61C2×#63227—24no—
62C2×#63227—24no—
63C2×#63227—22no—
64C2×#63227—22no—
65C2×#63227—24no—
66C2×#63227—22no—
67C2×#63227—22no—
68C2×#63227—24no—
69C2×#63227—42no—
70C2×#63227—42no—
71C2×#63227C2×C414yes—
72C2×#63227C2×C414yes—
73C2×C4⋊Dic17—22no—
74C2×C4⋊Dic17—22no—
75C2×C4⋊Dic17C2×C414yes—
76C2×#63225—24no—
77C2×#63225—24no—
78C2×#63225—24no—
79C2×C4×Dic17—22no—
80C2×C4×Dic17—22no—
81C2×C4×Dic17—22no—
82C2×C4×Dic17—42no—
83C2×C4×Dic17C2×C414yes—
84C2×C4×Dic17C2×C414yes—
85C2×C4×Dic17D414yes—
86C2×C4×Dic17D414yes—
87#10587578—232no—
88C17×#95—22no—
89C17×#95—24no—
90C17×#95—42no—
91#10587502—22no—
92#10587502—24no—
93#10587502—24no—
94#10587502—24no—
95#10587502—24no—
96#10587502—24no—
97#10587502—24no—
98#10587502—42no—
99C24×C17C2411yescharacteristic
100C23×D17—21no—
101C23×D17—21no—
102C23×D17—22no—
103C23×D17—22no—
104C23×D17—22no—
105C23×D17—22no—
106C23×D17—22no—
107C23×D17C22×C412yes—
108C23×D17#4114yes—
109C22×C4×C17—21no—
110C22×C4×C17—21no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—21no—
116C22×C4×C17—21no—
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×C17C22×C412yes—
134C22×C4×C17C2×D412yes—
135C22×C4×C17C2×D412yes—
136C22×C4×C17#4114yes—
137C22×C4×C17#4114yes—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
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×Dic17C22×C412yes—
160C22×Dic17C22×C412yes—
161C22×Dic17C22×C412yes—
162C22×Dic17C2×D412yes—
163C22×Dic17C2×D412yes—
164C22×Dic17C2×D412yes—
165C22×Dic17C2×D412yes—
166C22×Dic17#4114yes—
167C2×C4×D17—216no—
168C2×C4×D17—216no—
169C2×C4×D17—28no—
170C2×C4×D17—28no—
171C2×C4×D17—28no—
172C4⋊C4×C17—28no—
173C42×C17—44no—
174#63227—28no—
175#63227—28no—
176#63227—48no—
177#63227—48no—
178C4⋊Dic17—28no—
179#63225—216no—
180C4×Dic17—28no—
181C4×Dic17—44no—
182C4×Dic17C22⋊C4116yescomplemented
183#13938—171nomaximal, Sylow
184C23×C17C23×C411yescharacteristic
185C23×C17C22×D411yescharacteristic
186C23×C17C22×D411yescharacteristic
187C23×C17C2×#4112yes—
188C23×C17C2×#4112yes—
189C23×C17C2×#4112yes—
190C23×C17C2×#4112yes—
191C23×C17H(5,𝔽2)12yes—
192C23×C172−1+412yes—
193C22×D17—22no—
194C22×D17—22no—
195C22×D17—24no—
196C22×D17—24no—
197C22×D17—24no—
198C22×D17—24no—
199C22×D17—24no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—24no—
203C22×D17—42no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—42no—
213C22×D17#11714yes—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—24no—
217C2×C4×C17—24no—
218C2×C4×C17—24no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—24no—
222C2×C4×C17—22no—
223C2×C4×C17—24no—
224C2×C4×C17—24no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
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×C17C2×C22⋊C414yes—
242C2×C4×C17#11714yes—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—24no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17C2×C22⋊C414yes—
271C2×Dic17C2×C22⋊C414yes—
272C4×D17—232no—
273C22×#117—171no—
274C2×#383—171no—
275C2×C4×C4⋊C4—171no—
276#1441—172no—
277#1439—172no—
278#1437—171no—
279#1437—171no—
280#1436—171no—
281#1436D1711yescharacteristic, complemented
282#1434—172no—
283C22⋊#95—172no—
284C22×C17C22×C22⋊C411yescharacteristic
285C22×C17C2×#11711yescharacteristic
286C22×C17C2×#11711yescharacteristic
287C22×C17C4×#4112yes—
288C22×C17#52012yes—
289C22×C17#52512yes—
290C22×C17#52512yes—
291C22×C17#52512yes—
292C22×C17#53211yescharacteristic
293C22×C17#53211yescharacteristic
294C22×C17#53511yescharacteristic
295C22×C17#53611yescharacteristic
296C22×C17#53712yes—
297C22×C17#55112yes—
298C22×C17#55112yes—
299C22×C17#55212yes—
300C22×C17#55212yes—
301C22×C17#55312yes—
302C22×C17#55312yes—
303C22×C17#55612yes—
304C22×C17#55612yes—
305C2×D17—28no—
306C2×D17—24no—
307C2×D17—42no—
308C2×D17—44no—
309C2×D17—44no—
310C2×D17—44no—
311C2×D17—44no—
312C2×D17—44no—
313C2×D17—42no—
314C4×C17—24no—
315C4×C17—24no—
316C4×C17—42no—
317C4×C17—44no—
318C4×C17—82no—
319Dic17—24no—
320Dic17—24no—
321Dic17—44no—
322Dic17—42no—
323Dic17—82no—
324C24×C4—171no—
325C2×#117—178no—
326C2×#117—344no—
327C2×#117—344no—
328C22×C4⋊C4—171no—
329C22×C4⋊C4—172no—
330C22×C4⋊C4C2×D1711yescharacteristic
331C22×C22⋊C4—171no—
332C22×C22⋊C4—171no—
333C22×C22⋊C4—172no—
334C22×C22⋊C4—172no—
335C22×C42—171no—
336C22×C42—171no—
337C22×C42—171no—
338C22×C42C2×D1711yescharacteristic
339#383—344no—
340C4×C4⋊C4—344no—
341C2×#95—172no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—171no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95C2×D1712yes—
350C2×#95C2×D1711yescharacteristic
351C2×#95C2×D1712yes—
352C2×C17#228011yescharacteristic
353C2×C17#229711yescharacteristic
354C2×C17#231111yescharacteristic
355C2×C17#234312yes—
356C2×C17#234712yes—
357C2×C17#259812yes—
358C2×C17#259912yes—
359C2×C17#260112yes—
360C2×C17#260112yes—
361D17—44no—
362C25—171no—
363C23×C4—171no—
364C23×C4—171no—
365C23×C4—171no—
366C23×C4—172no—
367C23×C4—174no—
368C23×C4—172no—
369C23×C4—342no—
370C23×C4—342no—
371C23×C4—344no—
372C23×C4—344no—
373C23×C4C22×D1712yes—
374C23×C4C22×D1711yescharacteristic
375C23×C4C22×D1712yes—
376C23×C4C22×D1711yescharacteristic
377C23×C4C22×D1711yescharacteristic
378#117—3432no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—174no—
382C2×C4⋊C4—342no—
383C2×C4⋊C4—342no—
384C2×C4⋊C4—344no—
385C2×C4⋊C4—344no—
386C2×C4⋊C4—344no—
387C2×C4⋊C4C4×D1714yes—
388C2×C22⋊C4—174no—
389C2×C22⋊C4—174no—
390C2×C22⋊C4—342no—
391C2×C22⋊C4—342no—
392C2×C22⋊C4—342no—
393C2×C22⋊C4—342no—
394C2×C22⋊C4—344no—
395C2×C22⋊C4—344no—
396C2×C22⋊C4—344no—
397C2×C22⋊C4—344no—
398C2×C22⋊C4—682no—
399C2×C22⋊C4—682no—
400C2×C42—22no—
401C2×C42—42no—
402C2×C42—174no—
403C2×C42—174no—
404C2×C42—174no—
405C2×C42—174no—
406C2×C42—342no—
407C2×C42—342no—
408C2×C42—342no—
409C2×C42—682no—
410#95—24no—
411#95—22no—
412#95—42no—
413#95—344no—
414#95—344no—
415#95—344no—
416#95—344no—
417#95—344no—
418#95—342no—
419#95—344no—
420#95—682no—
421C17#1393811yescharacteristic, complemented, Sylow
422C24—172no—
423C24—174no—
424C24—342no—
425C24—341no—
426C24—341no—
427C24—342no—
428C24—342no—
429C24—342no—
430C24—342no—
431C24C23×D1711yescharacteristic
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—172no—
455C22×C4—172no—
456C22×C4—172no—
457C22×C4—172no—
458C22×C4—172no—
459C22×C4—172no—
460C22×C4—172no—
461C22×C4—174no—
462C22×C4—348no—
463C22×C4—348no—
464C22×C4—3416no—
465C22×C4—3416no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—348no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—341no—
474C22×C4—341no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4C2×C4×D1712yes—
489C22×C4C2×C4×D1712yes—
490C22×C4C2×C4×D1712yes—
491C22×C4#6325214yes—
492C22×C4D4×D1712yes—
493C22×C4D4×D1712yes—
494C22×C4#6325614yes—
495C4⋊C4—28no—
496C4⋊C4—348no—
497C4⋊C4—3416no—
498C22⋊C4—348no—
499C22⋊C4—348no—
500C22⋊C4—688no—
501C22⋊C4—688no—
502C42—44no—
503C42—1716no—
504C42—348no—
505C42—684no—
506C23—174no—
507C23—344no—
508C23—344no—
509C23—344no—
510C23—344no—
511C23—342no—
512C23—342no—
513C23—344no—
514C23—344no—
515C23—344no—
516C23—344no—
517C23—682no—
518C23—682no—
519C23—682no—
520C23—682no—
521C23—682no—
522C23—681no—
523C23—681no—
524C23—681no—
525C23—681no—
526C23—682no—
527C23C22×C4×D1711yescharacteristic
528C23C2×#6325212yes—
529C23C2×D4×D1711yescharacteristic
530C23C2×D4×D1711yescharacteristic
531C23#1058769112yes—
532C23C2×#6325612yes—
533C23#41×D1712yes—
534C23#41×D1712yes—
535C23#1058769712yes—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—174no—
564C2×C4—174no—
565C2×C4—3432no—
566C2×C4—342no—
567C2×C4—342no—
568C2×C4—344no—
569C2×C4—344no—
570C2×C4—344no—
571C2×C4—344no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—344no—
575C2×C4—344no—
576C2×C4—342no—
577C2×C4—344no—
578C2×C4—344no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4#1058755814yes—
594C2×C4C22⋊C4×D1714yes—
595C22—348no—
596C22—344no—
597C22—682no—
598C22—684no—
599C22—682no—
600C22—684no—
601C22—684no—
602C22—684no—
603C22—684no—
604C22C2×C22⋊C4×D1711yescharacteristic
605C22#117×D1711yescharacteristic
606C22C2×#1058755811yescharacteristic
607C22C4×#6325612yes—
608C22#1176328812yes—
609C22#1176331812yes—
610C22#1176331812yes—
611C22#1176332212yes—
612C22#1176335811yescharacteristic
613C22#1176336011yescharacteristic
614C22#1176337411yescharacteristic
615C22#1176338111yescharacteristic
616C22#1176338612yes—
617C22#1176348612yes—
618C22#1176348712yes—
619C22#1176349512yes—
620C22#1176349712yes—
621C22#1176350712yes—
622C22#1176350912yes—
623C22#1176352712yes—
624C22#1176353212yes—
625C4—24no—
626C4—24no—
627C4—42no—
628C4—44no—
629C4—82no—
630C4—344no—
631C4—344no—
632C4—682no—
633C4—684no—
634C4—1362no—
635C2—684no—
636C2#1453116511yescharacteristic
637C2#1453127311yescharacteristic
638C2#1453137111yescharacteristic
639C2#1453164312yes—
640C2#1453167312yes—
641C2#1453434912yes—
642C2#1453436012yes—
643C2#1453438012yes—
644C2#1453438812yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1436C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1436D17
hypocenterC17#13938
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
2. C2×#11762676C2semidirect
3. C2×C4×#63225C2semidirect
4. #14527036C2semidirect
8. #14527016C2semidirect
9. #14527010C2semidirect
11. C17×#1436C2semidirect
15. C2×#10587578C4semidirect
182. C4×Dic17C22⋊C4semidirect
281. #1436D17semidirect
421. C17#13938semidirect

Maximal subgroups

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

subgroupindex
C22×#105875782
C2×#117626762
C2×C4×#632252
#145270362
#145270282
#145270222
#145270202
#145270162
#145270102
#145270052
C17×#14362
#1393817