Finite Group Databasebeta

← #16961088

Subgroups — #16961088

Order 4352.

2242 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic17C211yesmaximal, characteristic
2C2×#11762723C211yesmaximal, characteristic
3C2×#11762691C211yesmaximal, characteristic, complemented
4C2×#11762681C212yesmaximal, complemented
5#14527048C211yesmaximal, characteristic, complemented
6#14527038C211yesmaximal, characteristic, complemented
7#14527031C212yesmaximal, complemented
8#14527027C212yesmaximal, complemented
9#14527009C211yesmaximal, characteristic, complemented
10#14527006C212yesmaximal, complemented
11C17×#1438C211yesmaximal, characteristic
12C24×Dic17C2212yes—
13C2×D4×Dic17—24no—
14C2×D4×Dic17—28no—
15C22×C4⋊Dic17C2211yescharacteristic
16C22×#63225C2211yescharacteristic
17C22×#63225C2211yescharacteristic
18C22×#63225C2212yes—
19C22×C22⋊Dic17C2212yes—
20C22×C22⋊Dic17C2212yes—
21C22×C22⋊Dic17C2212yes—
22C22×C4×Dic17C2211yescharacteristic
23#11762723C418yescomplemented
24#11762691—24no—
25#11762681—28no—
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2212yes—
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2211yescharacteristic
35C23×D4×C17C2211yescharacteristic
36C22×C22⋊C4×C17C2212yes—
37C22×C22⋊C4×C17C2212yes—
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2211yescharacteristic
40C25×C17C2312yes—
41C22×D4×C17—24no—
42C22×D4×C17—24no—
43C22×D4×C17—22no—
44C22×D4×C17—22no—
45C22×D4×C17C2×C414yes—
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2311yescharacteristic
48C23×C4×C17C2312yes—
49C23×C4×C17C2311yescharacteristic
50C23×Dic17—24no—
51C23×Dic17—24no—
52C23×Dic17—42no—
53C23×Dic17—42no—
54C23×Dic17—42no—
55C23×Dic17—42no—
56C23×Dic17D418yes—
57C23×Dic17C2311yescharacteristic
58C23×Dic17C2311yescharacteristic
59C23×Dic17C2312yes—
60C23×Dic17C2311yescharacteristic
61C23×Dic17C2312yes—
62C23×Dic17C2311yescharacteristic
63C2×C22⋊C4×C17—24no—
64C2×C22⋊C4×C17—24no—
65C2×C22⋊C4×C17—42no—
66C2×C22⋊C4×C17—42no—
67C2×C22⋊C4×C17—42no—
68C2×C22⋊C4×C17—42no—
69C2×C22⋊Dic17—24no—
70C2×C22⋊Dic17—24no—
71C2×C22⋊Dic17—24no—
72C2×C22⋊Dic17—24no—
73C2×C22⋊Dic17—24no—
74C2×C22⋊Dic17—42no—
75C2×C22⋊Dic17—42no—
76C2×C22⋊Dic17C2×C418yescomplemented
77C2×C22⋊Dic17C2×C418yescomplemented
78D4×Dic17—232no—
79C2×C4⋊Dic17—22no—
80C2×C4⋊Dic17—24no—
81C2×#63225—22no—
82C2×#63225—22no—
83C2×#63225—22no—
84C2×#63225—24no—
85C2×#63225—24no—
86C2×#63225—42no—
87C2×#63225—42no—
88C2×#63225C2×C414yescomplemented
89C2×C4×Dic17—24no—
90C2×C4×Dic17D414yes—
91C17×#95—22no—
92C17×#95—22no—
93#10587502—22no—
94#10587502—22no—
95#10587502—22no—
96#10587502—24no—
97#10587502—24no—
98#10587502—42no—
99#10587502—42no—
100#10587502—42no—
101#10587502C2×C414yescomplemented
102C24×C17—22no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17—22no—
106C24×C17—22no—
107C24×C17—22no—
108C24×C17—22no—
109C24×C17—22no—
110C24×C17—22no—
111C24×C17—22no—
112C24×C17—22no—
113C24×C17—22no—
114C24×C17C22×C414yes—
115C24×C17C2×D414yes—
116C24×C17#4114yes—
117C24×C17C2411yescharacteristic
118C2×D4×C17—28no—
119C2×D4×C17—216no—
120C2×D4×C17—28no—
121C2×D4×C17—216no—
122C2×D4×C17—28no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—22no—
127C22×C4×C17—21no—
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—21no—
136C22×C4×C17—22no—
137C22×C4×C17—21no—
138C22×C4×C17—22no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—22no—
142C22×C4×C17C22×C412yes—
143C22×C4×C17C2×D412yes—
144C22×C4×C17#4112yes—
145C22×Dic17—22no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C22×Dic17—22no—
171C22×Dic17—21no—
172C22×Dic17—22no—
173C22×Dic17—21no—
174C22×Dic17—216no—
175C22×Dic17—48no—
176C22×Dic17—48no—
177C22×Dic17—48no—
178C22×Dic17—48no—
179C22×Dic17—48no—
180C22×Dic17—48no—
181C22×Dic17C22×C414yes—
182C22×Dic17C22×C412yes—
183C22×Dic17C22×C412yes—
184C22×Dic17C2×D414yes—
185C22×Dic17C2×D412yes—
186C22×Dic17C2×D412yes—
187C22×Dic17C2×D414yes—
188C22⋊C4×C17—48no—
189C22⋊C4×C17—48no—
190C22⋊Dic17—216no—
191C22⋊Dic17—216no—
192C22⋊Dic17—48no—
193C4⋊Dic17—28no—
194#63225—28no—
195#63225—48no—
196#63225—44no—
197C4×Dic17—28no—
198#14300—171nomaximal, Sylow
199C23×C17—24no—
200C23×C17—24no—
201C23×C17—24no—
202C23×C17—24no—
203C23×C17—24no—
204C23×C17—24no—
205C23×C17—24no—
206C23×C17—24no—
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—24no—
211C23×C17—24no—
212C23×C17—24no—
213C23×C17—24no—
214C23×C17—24no—
215C23×C17—24no—
216C23×C17—24no—
217C23×C17—42no—
218C23×C17—42no—
219C23×C17—42no—
220C23×C17—42no—
221C23×C17—42no—
222C23×C17—42no—
223C23×C17—42no—
224C23×C17—42no—
225C23×C17—42no—
226C23×C17—42no—
227C23×C17—42no—
228C23×C17—42no—
229C23×C17—42no—
230C23×C17—42no—
231C23×C17—42no—
232C23×C17—42no—
233C23×C17C4×D418yes—
234C23×C17C23×C411yescharacteristic
235C23×C17C22×D412yes—
236C23×C17C22×D411yescharacteristic
237C23×C17C2×#4112yes—
238C23×C17C2×#4111yescharacteristic
239C23×C17H(5,𝔽2)11yescharacteristic
240C23×C17H(5,𝔽2)12yes—
241C23×C17H(5,𝔽2)11yescharacteristic
242C23×C17H(5,𝔽2)11yescharacteristic
243C23×C172−1+412yes—
244C23×C172−1+411yescharacteristic
245D4×C17—232no—
246C2×C4×C17—24no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—24no—
250C2×C4×C17—41no—
251C2×C4×C17—42no—
252C2×C4×C17—41no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—42no—
260C2×C4×C17—41no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
263C2×C4×C17—42no—
264C2×C4×C17—42no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17C4×D414yes—
268C2×Dic17—24no—
269C2×Dic17—24no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—24no—
273C2×Dic17—24no—
274C2×Dic17—22no—
275C2×Dic17—22no—
276C2×Dic17—22no—
277C2×Dic17—24no—
278C2×Dic17—22no—
279C2×Dic17—24no—
280C2×Dic17—22no—
281C2×Dic17—24no—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—22no—
285C2×Dic17—22no—
286C2×Dic17—432no—
287C2×Dic17—41no—
288C2×Dic17—42no—
289C2×Dic17—41no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17—42no—
293C2×Dic17—41no—
294C2×Dic17—42no—
295C2×Dic17—42no—
296C2×Dic17—42no—
297C2×Dic17—42no—
298C2×Dic17—42no—
299C2×Dic17—41no—
300C2×Dic17—42no—
301C2×Dic17—42no—
302C2×Dic17—41no—
303C2×Dic17—41no—
304C2×Dic17#12014yes—
305C22×C4×D4—171no—
306C2×#394—171no—
307C2×#387—171no—
308C2×#385—172no—
309#1444—171no—
310#1441—171no—
311#1440—172no—
312#1439—172no—
313#1438D1711yescharacteristic
314#1434—171no—
315C22⋊#95—172no—
316C22×C17—28no—
317C22×C17—28no—
318C22×C17—28no—
319C22×C17—44no—
320C22×C17—44no—
321C22×C17—44no—
322C22×C17—44no—
323C22×C17—44no—
324C22×C17—44no—
325C22×C17—44no—
326C22×C17—44no—
327C22×C17—44no—
328C22×C17—44no—
329C22×C17—44no—
330C22×C17—44no—
331C22×C17C2×C4×D412yes—
332C22×C17C2×C4×D411yescharacteristic
333C22×C17#51811yescharacteristic
334C22×C17#52011yescharacteristic
335C22×C17#52012yes—
336C22×C17C2×#12011yescharacteristic
337C22×C17#52512yes—
338C22×C17#52511yescharacteristic
339C22×C17#53412yes—
340C22×C17#53412yes—
341C22×C17#53511yescharacteristic
342C22×C17#53511yescharacteristic
343C22×C17#53611yescharacteristic
344C22×C17#53612yes—
345C22×C17#53712yes—
346C22×C17#53711yescharacteristic
347C22×C17#53812yes—
348C22×C17#53912yes—
349C22×C17#53912yes—
350C22×C17#53912yes—
351C22×C17#54011yescharacteristic
352C22×C17#54212yes—
353C22×C17#54211yescharacteristic
354C4×C17—24no—
355C4×C17—81no—
356C4×C17—82no—
357C4×C17—81no—
358Dic17—24no—
359Dic17—42no—
360Dic17—44no—
361Dic17—44no—
362Dic17—42no—
363Dic17—42no—
364C23×D4C2×D1711yescharacteristic
365C24×C4—172no—
366C2×C4×D4—344no—
367C2×C4×D4—348no—
368C22×C4⋊C4—171no—
369C22×C4⋊C4—171no—
370C22×C4⋊C4—171no—
371C22×C4⋊C4—172no—
372C22×C22⋊C4—172no—
373C22×C22⋊C4—172no—
374C22×C22⋊C4—172no—
375C22×C22⋊C4C2×D1712yes—
376C22×C22⋊C4C2×D1712yes—
377C22×C42—171no—
378#394—178no—
379#387—344no—
380#385—348no—
381C2×#95—171no—
382C2×#95—171no—
383C2×#95—172no—
384C2×#95—171no—
385C2×#95—171no—
386C2×#95—172no—
387C2×#95—172no—
388C2×#95—172no—
389C2×#95—172no—
390C2×#95C2×D1711yescharacteristic
391C2×#95C2×D1711yescharacteristic
392C2×C17—48no—
393C2×C17C4×#12011yescharacteristic
394C2×C17#232212yes—
395C2×C17#232411yescharacteristic
396C2×C17#232512yes—
397C2×C17#232611yescharacteristic
398C2×C17#261311yescharacteristic
399C2×C17#261611yescharacteristic
400C2×C17#263012yes—
401C2×C17#263311yescharacteristic
402C2×C17#263611yescharacteristic
403C2×C17#263812yes—
404C25C22×D1712yes—
405C22×D4—24no—
406C22×D4—24no—
407C22×D4—22no—
408C22×D4—22no—
409C22×D4C4×D1714yes—
410C23×C4—171no—
411C23×C4—171no—
412C23×C4—172no—
413C23×C4—171no—
414C23×C4—171no—
415C23×C4—172no—
416C23×C4—178no—
417C23×C4—344no—
418C23×C4—344no—
419C23×C4—682no—
420C23×C4—682no—
421C23×C4—682no—
422C23×C4—682no—
423C23×C4C22×D1711yescharacteristic
424C23×C4C22×D1711yescharacteristic
425C23×C4C22×D1712yes—
426C23×C4C22×D1711yescharacteristic
427C4×D4—3432no—
428C2×C4⋊C4—174no—
429C2×C4⋊C4—344no—
430C2×C4⋊C4—342no—
431C2×C4⋊C4—342no—
432C2×C4⋊C4—342no—
433C2×C4⋊C4—344no—
434C2×C4⋊C4—344no—
435C2×C4⋊C4—342no—
436C2×C4⋊C4—682no—
437C2×C4⋊C4—682no—
438C2×C22⋊C4—24no—
439C2×C22⋊C4—24no—
440C2×C22⋊C4—42no—
441C2×C22⋊C4—42no—
442C2×C22⋊C4—42no—
443C2×C22⋊C4—42no—
444C2×C22⋊C4—178no—
445C2×C22⋊C4—178no—
446C2×C22⋊C4—344no—
447C2×C22⋊C4—344no—
448C2×C22⋊C4—344no—
449C2×C22⋊C4—344no—
450C2×C22⋊C4—344no—
451C2×C22⋊C4—682no—
452C2×C22⋊C4—682no—
453C2×C42—174no—
454C2×C42—344no—
455#95—22no—
456#95—22no—
457#95—174no—
458#95—344no—
459#95—344no—
460#95—342no—
461#95—342no—
462#95—342no—
463#95—682no—
464#95—682no—
465#95—682no—
466C17#1430011yescharacteristic, complemented, Sylow
467C24—22no—
468C24—22no—
469C24—22no—
470C24—22no—
471C24—22no—
472C24—22no—
473C24—22no—
474C24—22no—
475C24—22no—
476C24—22no—
477C24—22no—
478C24—22no—
479C24C2×C4×D1714yes—
480C24D4×D1714yes—
481C24#6325414yes—
482C24C23×D1711yescharacteristic
483C2×D4—28no—
484C2×D4—28no—
485C2×D4—216no—
486C2×D4—216no—
487C2×D4—28no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—172no—
508C22×C4—174no—
509C22×C4—174no—
510C22×C4—172no—
511C22×C4—172no—
512C22×C4—172no—
513C22×C4—174no—
514C22×C4—3416no—
515C22×C4—342no—
516C22×C4—341no—
517C22×C4—341no—
518C22×C4—341no—
519C22×C4—342no—
520C22×C4—342no—
521C22×C4—341no—
522C22×C4—342no—
523C22×C4—342no—
524C22×C4—342no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—341no—
528C22×C4—341no—
529C22×C4—341no—
530C22×C4—341no—
531C22×C4—342no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—341no—
536C22×C4—342no—
537C22×C4—341no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—341no—
542C22×C4—342no—
543C22×C4—341no—
544C22×C4—688no—
545C22×C4—688no—
546C22×C4—688no—
547C22×C4—688no—
548C22×C4—688no—
549C22×C4—688no—
550C22×C4C2×C4×D1712yes—
551C22×C4D4×D1712yes—
552C22×C4#6325412yes—
553C4⋊C4—348no—
554C4⋊C4—348no—
555C4⋊C4—684no—
556C4⋊C4—688no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—3416no—
560C22⋊C4—3416no—
561C22⋊C4—688no—
562C42—348no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23#1058756718yes—
598C23C22×C4×D1711yescharacteristic
599C23C2×D4×D1712yes—
600C23C2×D4×D1711yescharacteristic
601C23C2×#6325412yes—
602C23C2×#6325411yescharacteristic
603C23#1058769111yescharacteristic
604C23#1058769112yes—
605C23#1058769111yescharacteristic
606C23#1058769611yescharacteristic
607C23#1058769712yes—
608C23#1058769711yescharacteristic
609D4—232no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—174no—
632C2×C4—344no—
633C2×C4—344no—
634C2×C4—342no—
635C2×C4—342no—
636C2×C4—344no—
637C2×C4—344no—
638C2×C4—342no—
639C2×C4—342no—
640C2×C4—342no—
641C2×C4—344no—
642C2×C4—342no—
643C2×C4—344no—
644C2×C4—342no—
645C2×C4—344no—
646C2×C4—344no—
647C2×C4—344no—
648C2×C4—342no—
649C2×C4—342no—
650C2×C4—6832no—
651C2×C4—681no—
652C2×C4—681no—
653C2×C4—682no—
654C2×C4—682no—
655C2×C4—682no—
656C2×C4—682no—
657C2×C4—682no—
658C2×C4—682no—
659C2×C4—681no—
660C2×C4—682no—
661C2×C4—682no—
662C2×C4—682no—
663C2×C4—681no—
664C2×C4—682no—
665C2×C4—682no—
666C2×C4—681no—
667C2×C4—681no—
668C2×C4#1058756714yes—
669C22—28no—
670C22—28no—
671C22—28no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22C2×#1058756712yes—
685C22C2×#1058756711yescharacteristic
686C22#1176328111yescharacteristic
687C22#1176328811yescharacteristic
688C22#1176328812yes—
689C22#120×D1711yescharacteristic
690C22#1176331511yescharacteristic
691C22#1176331512yes—
692C22#1176336712yes—
693C22#1176336912yes—
694C22#1176337211yescharacteristic
695C22#1176337511yescharacteristic
696C22#1176337911yescharacteristic
697C22#1176338112yes—
698C22#1176338612yes—
699C22#1176338811yescharacteristic
700C22#1176339412yes—
701C22#1176340312yes—
702C22#1176340412yes—
703C22#1176340412yes—
704C22#1176341311yescharacteristic
705C22#1176342612yes—
706C22#1176342811yescharacteristic
707C4—24no—
708C4—81no—
709C4—81no—
710C4—82no—
711C4—344no—
712C4—684no—
713C4—682no—
714C4—682no—
715C4—684no—
716C4—682no—
717C2—48no—
718C2#1453128311yescharacteristic
719C2#1453145812yes—
720C2#1453148011yescharacteristic
721C2#1453148512yes—
722C2#1453149411yescharacteristic
723C2#1453449511yescharacteristic
724C2#1453451211yescharacteristic
725C2#1453462012yes—
726C2#1453465111yescharacteristic
727C2#1453467811yescharacteristic
728C2#1453470012yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1438C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenter#1438D17
hypocenterC17#14300
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
3. C2×#11762691C2semidirect
4. C2×#11762681C2semidirect
5. #14527048C2semidirect
6. #14527038C2semidirect
7. #14527031C2semidirect
8. #14527027C2semidirect
9. #14527009C2semidirect
10. #14527006C2semidirect
23. #11762723C4semidirect
76. C2×C22⋊Dic17C2×C4semidirect
77. C2×C22⋊Dic17C2×C4semidirect
88. C2×#63225C2×C4semidirect
101. #10587502C2×C4semidirect
466. C17#14300semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic172
C2×#117627232
C2×#117626912
C2×#117626812
#145270482
#145270382
#145270312
#145270272
#145270092
#145270062
C17×#14382
#1430017