Finite Group Databasebeta

← #16956848

Subgroups — #16956848

Order 4352.

1862 conjugacy classes of subgroups.

The subgroup lattice has 694 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⋊C4×Dic17C211yesmaximal, characteristic, complemented
4#14527037C211yesmaximal, characteristic, complemented
5#14527030C211yesmaximal, characteristic
6#14527026C211yesmaximal, characteristic
7#14527020C212yesmaximal
8#14527018C212yesmaximal, complemented
9#14527009C212yesmaximal, complemented
10#14527005C212yesmaximal
11C17×#1441C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C2×#10587578—24no—
14C2×#10587578—24no—
15C2×#10587578C418yescomplemented
16C22×C4⋊Dic17C2211yescharacteristic
17C22×C4⋊Dic17C2211yescharacteristic
18C22×#63227C2212yes—
19C22×#63227C2212yes—
20C22×#63227C2211yescharacteristic
21C22×#63227C2211yescharacteristic
22C22×C4×Dic17C2211yescharacteristic
23C22×C4×Dic17C2212yes—
24C22×C4×Dic17C2211yescharacteristic
25#11762676—24no—
26C4⋊C4×Dic17—24no—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C2×C17×#95C2212yes—
38C2×C17×#95C2212yes—
39C2×C17×#95C2211yescharacteristic
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—22no—
52C22×C4×D17—22no—
53C22×C4×D17—22no—
54C22×C4×D17—22no—
55C22×C4×D17—22no—
56C22×C4×D17—22no—
57C22×C4×D17C2×C414yes—
58C2×C4⋊C4×C17—22no—
59C2×C4⋊C4×C17—22no—
60C2×C4⋊C4×C17—22no—
61C2×C4⋊C4×C17—24no—
62C2×C4⋊C4×C17C2×C414yes—
63C2×#63227—24no—
64C2×#63227—24no—
65C2×#63227—22no—
66C2×#63227—22no—
67C2×#63227—24no—
68C2×#63227—22no—
69C2×#63227—22no—
70C2×#63227—24no—
71C2×#63227—42no—
72C2×#63227—42no—
73C2×#63227C2×C414yes—
74C2×#63227C2×C414yes—
75C2×C4⋊Dic17—22no—
76C2×C4⋊Dic17—22no—
77C2×C4⋊Dic17—22no—
78C2×C4⋊Dic17—24no—
79C2×C4⋊Dic17C2×C414yes—
80C2×C4×Dic17—22no—
81C2×C4×Dic17—22no—
82C2×C4×Dic17—24no—
83C2×C4×Dic17—42no—
84C2×C4×Dic17—42no—
85C2×C4×Dic17C2×C414yes—
86C2×C4×Dic17C2×C414yes—
87C2×C4×Dic17D414yes—
88C2×C4×Dic17D414yes—
89#10587578—232no—
90C17×#95—22no—
91C17×#95—24no—
92C17×#95—24no—
93#10587502—22no—
94#10587502—24no—
95#10587502—24no—
96#10587502—24no—
97#10587502—24no—
98#10587502—24no—
99#10587502—42no—
100#10587502—42no—
101C24×C17C2411yescharacteristic
102C23×D17—21no—
103C23×D17—21no—
104C23×D17—21no—
105C23×D17—21no—
106C23×D17—22no—
107C23×D17—22no—
108C23×D17—22no—
109C23×D17—22no—
110C23×D17C22×C412yes—
111C23×D17#4112yes—
112C23×D17#4112yes—
113C22×C4×C17—21no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—21no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
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×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17C22×C412yes—
137C22×C4×C17C22×C412yes—
138C22×C4×C17C22×C412yes—
139C22×C4×C17C2×D412yes—
140C22×C4×C17C2×D412yes—
141C22×C4×C17#4114yes—
142C22×C4×C17#4112yes—
143C22×C4×C17#4112yes—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—21no—
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×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17C22×C412yes—
168C22×Dic17C22×C412yes—
169C22×Dic17C22×C412yes—
170C22×Dic17C2×D412yes—
171C22×Dic17C2×D412yes—
172C22×Dic17C2×D412yes—
173C22×Dic17C2×D412yes—
174C22×Dic17#4114yes—
175C2×C4×D17—28no—
176C2×C4×D17—28no—
177C2×C4×D17—28no—
178C2×C4×D17—28no—
179C2×C4×D17—28no—
180C2×C4×D17—28no—
181C2×C4×D17—28no—
182C4⋊C4×C17—28no—
183C4⋊C4×C17—28no—
184#63227—28no—
185#63227—28no—
186#63227—48no—
187#63227—48no—
188C4⋊Dic17—28no—
189C4⋊Dic17—28no—
190C4×Dic17—28no—
191C4×Dic17—48no—
192C4×Dic17C22⋊C4116yescomplemented
193#13938—171nomaximal, Sylow
194C23×C17C23×C411yescharacteristic
195C23×C17C22×D411yescharacteristic
196C23×C17C22×D411yescharacteristic
197C23×C17C2×#4111yescharacteristic
198C23×C17C2×#4112yes—
199C23×C17C2×#4111yescharacteristic
200C23×C17C2×#4111yescharacteristic
201C23×C17C2×#4111yescharacteristic
202C23×C17C2×#4112yes—
203C23×C17H(5,𝔽2)12yes—
204C23×C172−1+412yes—
205C22×D17—22no—
206C22×D17—22no—
207C22×D17—24no—
208C22×D17—24no—
209C22×D17—22no—
210C22×D17—22no—
211C22×D17—22no—
212C22×D17—22no—
213C22×D17—24no—
214C22×D17—24no—
215C22×D17—24no—
216C22×D17—24no—
217C22×D17—41no—
218C22×D17—41no—
219C22×D17—42no—
220C22×D17—41no—
221C22×D17—41no—
222C22×D17—42no—
223C22×D17—42no—
224C22×D17—42no—
225C22×D17—42no—
226C22×D17—42no—
227C22×D17#11714yes—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—24no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—22no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17C2×C22⋊C414yes—
258C2×C4×C17#11714yes—
259C2×Dic17—22no—
260C2×Dic17—22no—
261C2×Dic17—24no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—24no—
269C2×Dic17—22no—
270C2×Dic17—24no—
271C2×Dic17—24no—
272C2×Dic17—24no—
273C2×Dic17—24no—
274C2×Dic17—42no—
275C2×Dic17—41no—
276C2×Dic17—42no—
277C2×Dic17—41no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17—42no—
282C2×Dic17—42no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17C2×C22⋊C414yes—
289C2×Dic17C2×C22⋊C414yes—
290C4×D17—232no—
291C22×#117—171no—
292C2×#383—171no—
293C2×C4×C4⋊C4—171no—
294#1441—171no—
295#1441D1711yescharacteristic, complemented
296#1439—171no—
297#1439—171no—
298#1437—172no—
299#1436—172no—
300#1434—172no—
301C22⋊#95—172no—
302C22×C17C22×C22⋊C411yescharacteristic
303C22×C17C2×#11711yescharacteristic
304C22×C17C2×#11711yescharacteristic
305C22×C17C4×#4112yes—
306C22×C17#52012yes—
307C22×C17#52511yescharacteristic
308C22×C17#52511yescharacteristic
309C22×C17#52511yescharacteristic
310C22×C17#52511yescharacteristic
311C22×C17#52511yescharacteristic
312C22×C17#52511yescharacteristic
313C22×C17#53212yes—
314C22×C17#53511yescharacteristic
315C22×C17#53611yescharacteristic
316C22×C17#53712yes—
317C22×C17#55112yes—
318C22×C17#55112yes—
319C22×C17#55212yes—
320C22×C17#55212yes—
321C22×C17#55312yes—
322C22×C17#55312yes—
323C22×C17#55612yes—
324C22×C17#55612yes—
325C2×D17—28no—
326C2×D17—24no—
327C2×D17—42no—
328C2×D17—42no—
329C2×D17—42no—
330C2×D17—42no—
331C2×D17—42no—
332C2×D17—44no—
333C2×D17—44no—
334C2×D17—44no—
335C2×D17—42no—
336C4×C17—24no—
337C4×C17—24no—
338C4×C17—44no—
339C4×C17—42no—
340C4×C17—82no—
341Dic17—24no—
342Dic17—24no—
343Dic17—42no—
344Dic17—44no—
345Dic17—82no—
346C24×C4—171no—
347C2×#117—178no—
348C2×#117—344no—
349C2×#117—344no—
350C22×C4⋊C4—171no—
351C22×C4⋊C4—171no—
352C22×C4⋊C4C2×D1711yescharacteristic
353C22×C4⋊C4C2×D1711yescharacteristic
354C22×C22⋊C4—171no—
355C22×C22⋊C4—171no—
356C22×C22⋊C4—172no—
357C22×C22⋊C4—172no—
358C22×C42—171no—
359C22×C42—171no—
360C22×C42—172no—
361#383—344no—
362C4×C4⋊C4—344no—
363C2×#95—172no—
364C2×#95—172no—
365C2×#95—172no—
366C2×#95—172no—
367C2×#95—172no—
368C2×#95—171no—
369C2×#95—172no—
370C2×#95—172no—
371C2×#95C2×D1712yes—
372C2×#95C2×D1712yes—
373C2×#95C2×D1711yescharacteristic
374C2×C17#228011yescharacteristic
375C2×C17#229711yescharacteristic
376C2×C17#231111yescharacteristic
377C2×C17#234312yes—
378C2×C17#234712yes—
379C2×C17#259811yescharacteristic
380C2×C17#259811yescharacteristic
381C2×C17#259911yescharacteristic
382C2×C17#259911yescharacteristic
383C2×C17#260112yes—
384C2×C17#260112yes—
385D17—44no—
386C25—171no—
387C23×C4—171no—
388C23×C4—171no—
389C23×C4—171no—
390C23×C4—172no—
391C23×C4—174no—
392C23×C4—172no—
393C23×C4—342no—
394C23×C4—342no—
395C23×C4—342no—
396C23×C4—342no—
397C23×C4—342no—
398C23×C4—342no—
399C23×C4C22×D1712yes—
400C23×C4C22×D1711yescharacteristic
401C23×C4C22×D1712yes—
402C23×C4C22×D1711yescharacteristic
403C23×C4C22×D1711yescharacteristic
404#117—3432no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—174no—
410C2×C4⋊C4—342no—
411C2×C4⋊C4—342no—
412C2×C4⋊C4—344no—
413C2×C4⋊C4—342no—
414C2×C4⋊C4C4×D1714yes—
415C2×C22⋊C4—174no—
416C2×C22⋊C4—174no—
417C2×C22⋊C4—342no—
418C2×C22⋊C4—342no—
419C2×C22⋊C4—342no—
420C2×C22⋊C4—342no—
421C2×C22⋊C4—344no—
422C2×C22⋊C4—344no—
423C2×C22⋊C4—344no—
424C2×C22⋊C4—344no—
425C2×C22⋊C4—682no—
426C2×C22⋊C4—682no—
427C2×C42—174no—
428C2×C42—174no—
429C2×C42—174no—
430C2×C42—174no—
431C2×C42—342no—
432C2×C42—342no—
433C2×C42—344no—
434C2×C42—682no—
435C2×C42—682no—
436#95—22no—
437#95—24no—
438#95—24no—
439#95—344no—
440#95—344no—
441#95—344no—
442#95—344no—
443#95—342no—
444#95—344no—
445#95—682no—
446#95—682no—
447C17#1393811yescharacteristic, complemented, Sylow
448C24—172no—
449C24—172no—
450C24—172no—
451C24—341no—
452C24—341no—
453C24—342no—
454C24—341no—
455C24—341no—
456C24—342no—
457C24—342no—
458C24—342no—
459C24C23×D1711yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—172no—
484C22×C4—172no—
485C22×C4—172no—
486C22×C4—172no—
487C22×C4—172no—
488C22×C4—172no—
489C22×C4—172no—
490C22×C4—174no—
491C22×C4—348no—
492C22×C4—348no—
493C22×C4—348no—
494C22×C4—348no—
495C22×C4—348no—
496C22×C4—348no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—342no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—342no—
503C22×C4—348no—
504C22×C4—342no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—342no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—342no—
519C22×C4—342no—
520C22×C4—342no—
521C22×C4C2×C4×D1712yes—
522C22×C4C2×C4×D1712yes—
523C22×C4C2×C4×D1712yes—
524C22×C4D4×D1712yes—
525C22×C4D4×D1712yes—
526C22×C4#6325414yes—
527C22×C4#6325412yes—
528C22×C4#6325612yes—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—348no—
532C4⋊C4—348no—
533C22⋊C4—348no—
534C22⋊C4—348no—
535C22⋊C4—688no—
536C22⋊C4—688no—
537C42—1716no—
538C42—348no—
539C42—688no—
540C23—174no—
541C23—344no—
542C23—344no—
543C23—344no—
544C23—344no—
545C23—342no—
546C23—342no—
547C23—344no—
548C23—344no—
549C23—342no—
550C23—342no—
551C23—342no—
552C23—342no—
553C23—682no—
554C23—682no—
555C23—682no—
556C23—682no—
557C23—681no—
558C23—681no—
559C23—682no—
560C23—681no—
561C23—681no—
562C23—682no—
563C23C22×C4×D1711yescharacteristic
564C23C2×D4×D1711yescharacteristic
565C23C2×D4×D1711yescharacteristic
566C23C2×#6325412yes—
567C23C2×#6325411yescharacteristic
568C23C2×#6325611yescharacteristic
569C23#1058769412yes—
570C23#41×D1711yescharacteristic
571C23#41×D1711yescharacteristic
572C23#41×D1712yes—
573C23#1058769612yes—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—174no—
604C2×C4—174no—
605C2×C4—3432no—
606C2×C4—342no—
607C2×C4—342no—
608C2×C4—344no—
609C2×C4—342no—
610C2×C4—342no—
611C2×C4—342no—
612C2×C4—342no—
613C2×C4—342no—
614C2×C4—342no—
615C2×C4—344no—
616C2×C4—344no—
617C2×C4—342no—
618C2×C4—344no—
619C2×C4—344no—
620C2×C4—344no—
621C2×C4—682no—
622C2×C4—682no—
623C2×C4—682no—
624C2×C4—682no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—682no—
628C2×C4—682no—
629C2×C4—682no—
630C2×C4—682no—
631C2×C4—682no—
632C2×C4—682no—
633C2×C4—682no—
634C2×C4—682no—
635C2×C4C22⋊C4×D1714yes—
636C2×C4#1058757814yes—
637C22—348no—
638C22—344no—
639C22—682no—
640C22—684no—
641C22—682no—
642C22—682no—
643C22—682no—
644C22—682no—
645C22—682no—
646C22—684no—
647C22—684no—
648C22C2×C22⋊C4×D1711yescharacteristic
649C22#117×D1711yescharacteristic
650C22C2×#1058757811yescharacteristic
651C22C4×#6325412yes—
652C22#1176329112yes—
653C22#1176331511yescharacteristic
654C22#1176331711yescharacteristic
655C22#1176331911yescharacteristic
656C22#1176331911yescharacteristic
657C22#1176332211yescharacteristic
658C22#1176332211yescharacteristic
659C22#1176335412yes—
660C22#1176337711yescharacteristic
661C22#1176338411yescharacteristic
662C22#1176338912yes—
663C22#1176348212yes—
664C22#1176348412yes—
665C22#1176349112yes—
666C22#1176349312yes—
667C22#1176351112yes—
668C22#1176351212yes—
669C22#1176353012yes—
670C22#1176353312yes—
671C4—24no—
672C4—24no—
673C4—42no—
674C4—44no—
675C4—82no—
676C4—344no—
677C4—344no—
678C4—682no—
679C4—684no—
680C4—1362no—
681C2—684no—
682C2#1453116611yescharacteristic
683C2#1453126911yescharacteristic
684C2#1453137511yescharacteristic
685C2#1453163912yes—
686C2#1453167612yes—
687C2#1453434511yescharacteristic
688C2#1453435311yescharacteristic
689C2#1453435611yescharacteristic
690C2#1453436411yescharacteristic
691C2#1453437612yes—
692C2#1453438412yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1441D17
hypocenterC17#13938
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
2. C2×#11762676C2semidirect
3. C2×C4⋊C4×Dic17C2semidirect
4. #14527037C2semidirect
8. #14527018C2semidirect
9. #14527009C2semidirect
11. C17×#1441C2semidirect
15. C2×#10587578C4semidirect
192. C4×Dic17C22⋊C4semidirect
295. #1441D17semidirect
447. 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⋊C4×Dic172
#145270372
#145270302
#145270262
#145270202
#145270182
#145270092
#145270052
C17×#14412
#1393817