Finite Group Databasebeta

← #16963272

Subgroups — #16963272

Order 4352.

1918 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762712C211yesmaximal, characteristic
2C2×#11762689C211yesmaximal, characteristic
3#14527036C212yesmaximal, complemented
4#14527027C212yesmaximal, complemented
5#14527027C212yesmaximal, complemented
6#14527010C212yesmaximal, complemented
7#14527007C212yesmaximal, complemented
8#14527006C212yesmaximal, complemented
9C2×C17×#386C211yesmaximal, characteristic
10C24×Dic17C2211yescharacteristic
11C22×C4⋊Dic17C2211yescharacteristic
12C22×#63225C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2211yescharacteristic
15C22×C22⋊Dic17C2212yes—
16C22×C22⋊Dic17C2212yes—
17C22×C4×Dic17C2211yescharacteristic
18#11762712C418yescomplemented
19#11762689—24no—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C23×D4×C17C2211yescharacteristic
29C24×C4×C17C2211yescharacteristic
30C22×C22⋊C4×C17C2212yes—
31C22×C22⋊C4×C17C2211yescharacteristic
32C17×#386—24no—
33C2×C17×#95C2212yes—
34C25×C17C2311yescharacteristic
35C25×C17C2311yescharacteristic
36C22×D4×C17—41no—
37C22×D4×C17—41no—
38C22×D4×C17—41no—
39C22×D4×C17—41no—
40C22×D4×C17C2×C414yes—
41C22×D4×C17D414yes—
42C22×D4×C17Q814yes—
43C23×C4×C17—22no—
44C23×C4×C17—24no—
45C23×C4×C17—41no—
46C23×C4×C17—41no—
47C23×C4×C17—42no—
48C23×C4×C17C2312yes—
49C23×C4×C17C2312yes—
50C23×C4×C17C2311yescharacteristic
51C23×Dic17—22no—
52C23×Dic17—24no—
53C23×Dic17—41no—
54C23×Dic17—41no—
55C23×Dic17—42no—
56C23×Dic17C2312yes—
57C23×Dic17C2312yes—
58C23×Dic17C2312yes—
59C23×Dic17C2312yes—
60C2×C22⋊C4×C17—24no—
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17—24no—
63C2×C22⋊C4×C17—42no—
64C2×C22⋊C4×C17—41no—
65C2×C22⋊C4×C17—41no—
66C2×C22⋊Dic17—24no—
67C2×C22⋊Dic17—22no—
68C2×C22⋊Dic17—24no—
69C2×C22⋊Dic17—42no—
70C2×C22⋊Dic17—42no—
71C2×C22⋊Dic17—42no—
72C2×C22⋊Dic17—42no—
73C2×C22⋊Dic17—42no—
74C2×C22⋊Dic17—41no—
75C2×C22⋊Dic17—41no—
76C2×C22⋊Dic17C2×C418yescomplemented
77C2×C22⋊Dic17C2×C418yescomplemented
78C2×C4⋊Dic17—41no—
79C2×C4⋊Dic17—41no—
80C2×C4⋊Dic17C2×C414yescomplemented
81C2×#63225—42no—
82C2×#63225—42no—
83C2×#63225—42no—
84C2×C4×Dic17—41no—
85C2×C4×Dic17—41no—
86C2×C4×Dic17C2×C414yescomplemented
87C17×#95—24no—
88#10587502—24no—
89#10587502—24no—
90#10587502—24no—
91#10587502—42no—
92#10587502—42no—
93#10587502—42no—
94#10587502—42no—
95#10587502—42no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17—22no—
99C24×C17—22no—
100C24×C17—21no—
101C24×C17—21no—
102C24×C17—22no—
103C24×C17—21no—
104C24×C17—21no—
105C24×C17—22no—
106C24×C17—21no—
107C24×C17—21no—
108C24×C17—21no—
109C24×C17—21no—
110C24×C17—22no—
111C24×C17—22no—
112C24×C17C22×C412yes—
113C24×C17C22×C412yes—
114C24×C17C2×D412yes—
115C24×C17C2×D412yes—
116C24×C17C2×Q812yes—
117C24×C17C2×Q812yes—
118C24×C17C2411yescharacteristic
119C2×D4×C17—44no—
120C2×D4×C17—44no—
121C2×D4×C17—44no—
122C2×D4×C17—44no—
123C2×D4×C17—44no—
124C2×D4×C17—44no—
125C2×D4×C17C4⋊C4116yescomplemented
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
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×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—22no—
143C22×C4×C17—44no—
144C22×C4×C17—48no—
145C22×C4×C17—48no—
146C22×C4×C17—44no—
147C22×C4×C17—44no—
148C22×C4×C17C22×C412yes—
149C22×C4×C17C2×D412yes—
150C22×C4×C17C2×Q812yes—
151C22×C4×C17#4114yes—
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×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17—22no—
175C22×Dic17—22no—
176C22×Dic17—22no—
177C22×Dic17—44no—
178C22×Dic17—48no—
179C22×Dic17—48no—
180C22×Dic17—44no—
181C22×Dic17—44no—
182C22×Dic17C22×C414yes—
183C22×Dic17C22×C414yes—
184C22×Dic17#4114yes—
185C22⋊C4×C17—48no—
186C22⋊C4×C17—44no—
187C22⋊Dic17—48no—
188C22⋊Dic17—48no—
189C22⋊Dic17—48no—
190C22⋊Dic17—44no—
191C4⋊Dic17—44no—
192#63225—48no—
193C4×Dic17—44no—
194#14486—171nomaximal, Sylow
195C23×C17—24no—
196C23×C17—24no—
197C23×C17—24no—
198C23×C17—24no—
199C23×C17—22no—
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—22no—
206C23×C17—22no—
207C23×C17—22no—
208C23×C17—22no—
209C23×C17—22no—
210C23×C17—22no—
211C23×C17—24no—
212C23×C17—24no—
213C23×C17—24no—
214C23×C17—24no—
215C23×C17—24no—
216C23×C17—24no—
217C23×C17—24no—
218C23×C17—24no—
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—41no—
228C23×C17—41no—
229C23×C17—41no—
230C23×C17—41no—
231C23×C17—42no—
232C23×C17—42no—
233C23×C17—41no—
234C23×C17—41no—
235C23×C17—42no—
236C23×C17—42no—
237C23×C17—41no—
238C23×C17—41no—
239C23×C17C2×C4⋊C414yes—
240C23×C17C2×C4⋊C414yes—
241C23×C17C23×C411yescharacteristic
242C23×C17C22×D411yescharacteristic
243C23×C17C22×Q811yescharacteristic
244C23×C17C2×#4112yes—
245C23×C17C2×#4112yes—
246C23×C17H(5,𝔽2)11yescharacteristic
247C23×C17H(5,𝔽2)11yescharacteristic
248C23×C17H(5,𝔽2)11yescharacteristic
249C23×C17H(5,𝔽2)11yescharacteristic
250C23×C17H(5,𝔽2)12yes—
251C23×C17H(5,𝔽2)12yes—
252D4×C17—416no—
253C2×C4×C17—22no—
254C2×C4×C17—22no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—22no—
258C2×C4×C17—22no—
259C2×C4×C17—24no—
260C2×C4×C17—24no—
261C2×C4×C17—24no—
262C2×C4×C17—416no—
263C2×C4×C17—42no—
264C2×C4×C17—42no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17—42no—
268C2×C4×C17—42no—
269C2×C4×C17—42no—
270C2×C4×C17—42no—
271C2×C4×C17—42no—
272C2×C4×C17—42no—
273C2×C4×C17—42no—
274C2×C4×C17C2×C4⋊C414yes—
275C2×Dic17—24no—
276C2×Dic17—24no—
277C2×Dic17—24no—
278C2×Dic17—24no—
279C2×Dic17—24no—
280C2×Dic17—24no—
281C2×Dic17—24no—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—416no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17—42no—
289C2×Dic17—42no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17—42no—
293C2×Dic17—42no—
294C2×Dic17—42no—
295C2×Dic17—42no—
296C2×Dic17—42no—
297C2×Dic17—42no—
298C2×Dic17—42no—
299C2×Dic17—42no—
300C2×Dic17—42no—
301C2×Dic17—42no—
302C2×Dic17—42no—
303C2×Dic17—42no—
304C2×#390—171no—
305C2×#386—171no—
306C2×#386D1711yescharacteristic
307#1441—172no—
308#1439—172no—
309#1439—172no—
310#1434—172no—
311C22⋊#95—172no—
312C22⋊#95—172no—
313C22×C17—24no—
314C22×C17—24no—
315C22×C17—28no—
316C22×C17—28no—
317C22×C17—44no—
318C22×C17—44no—
319C22×C17—42no—
320C22×C17—42no—
321C22×C17—42no—
322C22×C17—42no—
323C22×C17—42no—
324C22×C17—42no—
325C22×C17—42no—
326C22×C17—42no—
327C22×C17—44no—
328C22×C17—44no—
329C22×C17—42no—
330C22×C17—42no—
331C22×C17—42no—
332C22×C17—42no—
333C22×C17—44no—
334C22×C17—44no—
335C22×C17C22×C4⋊C411yescharacteristic
336C22×C17C4×#4112yes—
337C22×C17#51812yes—
338C22×C17#51811yescharacteristic
339C22×C17#51811yescharacteristic
340C22×C17#53411yescharacteristic
341C22×C17#53411yescharacteristic
342C22×C17#53812yes—
343C22×C17#53812yes—
344C22×C17#54012yes—
345C22×C17#54012yes—
346C22×C17#54311yescharacteristic
347C22×C17#54311yescharacteristic
348C22×C17#54612yes—
349C22×C17#54612yes—
350C22×C17#55111yescharacteristic
351C22×C17#55111yescharacteristic
352C22×C17#55412yes—
353C22×C17#55412yes—
354C22×C17#55512yes—
355C22×C17#56012yes—
356C22×C17#56011yescharacteristic
357C22×C17#56011yescharacteristic
358C4×C17—24no—
359C4×C17—44no—
360C4×C17—82no—
361Dic17—44no—
362Dic17—44no—
363Dic17—44no—
364Dic17—82no—
365C23×D4C2×D1711yescharacteristic
366C24×C4—171no—
367C24×C4C2×D1711yescharacteristic
368C22×C4⋊C4—171no—
369C22×C4⋊C4—172no—
370C22×C22⋊C4—172no—
371C22×C22⋊C4—172no—
372C22×C22⋊C4—172no—
373C22×C22⋊C4—171no—
374C22×C22⋊C4C2×D1712yes—
375C22×C22⋊C4C2×D1711yescharacteristic
376C22×C42—171no—
377#390—178no—
378#386—24no—
379#386—344no—
380C2×#95—172no—
381C2×#95—172no—
382C2×#95—172no—
383C2×#95—172no—
384C2×#95—172no—
385C2×#95—172no—
386C2×#95—172no—
387C2×#95—172no—
388C2×#95C2×D1712yes—
389C2×C17—44no—
390C2×C17—44no—
391C2×C17#231511yescharacteristic
392C2×C17#233312yes—
393C2×C17#234912yes—
394C2×C17#237211yescharacteristic
395C2×C17#238011yescharacteristic
396C2×C17#269711yescharacteristic
397C2×C17#269711yescharacteristic
398C2×C17#269912yes—
399C2×C17#274811yescharacteristic
400C2×C17#274811yescharacteristic
401C2×C17#278412yes—
402C25C22×D1711yescharacteristic
403C25C22×D1711yescharacteristic
404C22×D4—41no—
405C22×D4—41no—
406C22×D4—41no—
407C22×D4—41no—
408C22×D4Dic3414yes—
409C22×D4C4×D1714yes—
410C22×D4C22⋊D1714yes—
411C23×C4—22no—
412C23×C4—24no—
413C23×C4—41no—
414C23×C4—41no—
415C23×C4—42no—
416C23×C4—172no—
417C23×C4—172no—
418C23×C4—172no—
419C23×C4—172no—
420C23×C4—344no—
421C23×C4—342no—
422C23×C4—681no—
423C23×C4—681no—
424C23×C4—682no—
425C23×C4C22×D1712yes—
426C23×C4C22×D1712yes—
427C23×C4C22×D1711yescharacteristic
428C2×C4⋊C4—174no—
429C2×C4⋊C4—682no—
430C2×C4⋊C4—682no—
431C2×C4⋊C4—682no—
432C2×C4⋊C4—681no—
433C2×C4⋊C4—681no—
434C2×C22⋊C4—24no—
435C2×C22⋊C4—22no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—42no—
438C2×C22⋊C4—41no—
439C2×C22⋊C4—41no—
440C2×C22⋊C4—178no—
441C2×C22⋊C4—178no—
442C2×C22⋊C4—344no—
443C2×C22⋊C4—344no—
444C2×C22⋊C4—342no—
445C2×C22⋊C4—682no—
446C2×C22⋊C4—682no—
447C2×C22⋊C4—682no—
448C2×C22⋊C4—682no—
449C2×C22⋊C4—682no—
450C2×C22⋊C4—681no—
451C2×C22⋊C4—681no—
452C2×C42—174no—
453C2×C42—681no—
454C2×C42—681no—
455#95—24no—
456#95—344no—
457#95—344no—
458#95—344no—
459#95—682no—
460#95—682no—
461#95—682no—
462#95—682no—
463#95—682no—
464C17#1448611yescharacteristic, complemented, Sylow
465C24—22no—
466C24—22no—
467C24—22no—
468C24—21no—
469C24—21no—
470C24—22no—
471C24—21no—
472C24—21no—
473C24—21no—
474C24—21no—
475C24—21no—
476C24—21no—
477C24—22no—
478C24—22no—
479C24—22no—
480C24—22no—
481C24C2×Dic3412yes—
482C24C2×Dic3412yes—
483C24C2×C4×D1712yes—
484C24C2×C4×D1712yes—
485C24C2×C22⋊D1712yes—
486C24C2×C22⋊D1712yes—
487C24C23×D1711yescharacteristic
488C2×D4—44no—
489C2×D4—44no—
490C2×D4—44no—
491C2×D4—44no—
492C2×D4—44no—
493C2×D4—44no—
494C2×D4#63225116yescomplemented
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—44no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—44no—
516C22×C4—44no—
517C22×C4—174no—
518C22×C4—174no—
519C22×C4—174no—
520C22×C4—342no—
521C22×C4—342no—
522C22×C4—342no—
523C22×C4—342no—
524C22×C4—342no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—342no—
531C22×C4—342no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—342no—
542C22×C4—342no—
543C22×C4—342no—
544C22×C4—342no—
545C22×C4—684no—
546C22×C4—688no—
547C22×C4—688no—
548C22×C4—684no—
549C22×C4—684no—
550C22×C4C2×Dic3412yes—
551C22×C4C2×C4×D1712yes—
552C22×C4#6325414yes—
553C22×C4C2×C22⋊D1712yes—
554C4⋊C4—684no—
555C4⋊C4—688no—
556C22⋊C4—48no—
557C22⋊C4—44no—
558C22⋊C4—688no—
559C22⋊C4—688no—
560C22⋊C4—688no—
561C22⋊C4—684no—
562C42—684no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—24no—
584C23—24no—
585C23—24no—
586C23—24no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23C2×#6322514yes—
608C23C2×#6322514yes—
609C23C22×Dic3411yescharacteristic
610C23C22×C4×D1711yescharacteristic
611C23C2×#6325412yes—
612C23#1058769111yescharacteristic
613C23#1058769112yes—
614C23#1058769111yescharacteristic
615C23#1058769111yescharacteristic
616C23#1058769111yescharacteristic
617C23#41×D1712yes—
618C23#1058769612yes—
619C23C22×C22⋊D1711yescharacteristic
620D4—416no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—416no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—344no—
643C2×C4—344no—
644C2×C4—344no—
645C2×C4—344no—
646C2×C4—344no—
647C2×C4—344no—
648C2×C4—344no—
649C2×C4—344no—
650C2×C4—344no—
651C2×C4—6816no—
652C2×C4—682no—
653C2×C4—682no—
654C2×C4—682no—
655C2×C4—682no—
656C2×C4—682no—
657C2×C4—682no—
658C2×C4—682no—
659C2×C4—682no—
660C2×C4—682no—
661C2×C4—682no—
662C2×C4—682no—
663C2×C4—682no—
664C2×C4—682no—
665C2×C4—682no—
666C2×C4—682no—
667C2×C4—682no—
668C2×C4—682no—
669C2×C4—682no—
670C2×C4—682no—
671C2×C4C2×#6322514yes—
672C22—24no—
673C22—24no—
674C22—28no—
675C22—28no—
676C22—44no—
677C22—44no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22C22×#6322511yescharacteristic
695C22C4×#6325412yes—
696C22#1176328112yes—
697C22#1176328211yescharacteristic
698C22#1176328211yescharacteristic
699C22#1176336811yescharacteristic
700C22#1176336811yescharacteristic
701C22#1176339412yes—
702C22#1176339512yes—
703C22#1176340912yes—
704C22#1176341212yes—
705C22#1176343511yescharacteristic
706C22#1176343511yescharacteristic
707C22#1176344712yes—
708C22#1176345312yes—
709C22#1176348711yescharacteristic
710C22#1176349011yescharacteristic
711C22#1176351612yes—
712C22#1176351812yes—
713C22#1176352312yes—
714C22#1176354712yes—
715C22#1176354811yescharacteristic
716C22#1176354911yescharacteristic
717C4—24no—
718C4—44no—
719C4—82no—
720C4—684no—
721C4—684no—
722C4—684no—
723C4—1362no—
724C2—44no—
725C2—44no—
726C2#1453139611yescharacteristic
727C2#1453154512yes—
728C2#1453168912yes—
729C2#1453189311yescharacteristic
730C2#1453195911yescharacteristic
731C2#1453534311yescharacteristic
732C2#1453534511yescharacteristic
733C2#1453536712yes—
734C2#1453599511yescharacteristic
735C2#1453599911yescharacteristic
736C2#1453648212yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#386C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#386D17
hypocenterC17#14486
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
3. #14527036C2semidirect
4. #14527027C2semidirect
5. #14527027C2semidirect
6. #14527010C2semidirect
7. #14527007C2semidirect
8. #14527006C2semidirect
18. #11762712C4semidirect
76. C2×C22⋊Dic17C2×C4semidirect
77. C2×C22⋊Dic17C2×C4semidirect
80. C2×C4⋊Dic17C2×C4semidirect
86. C2×C4×Dic17C2×C4semidirect
125. C2×D4×C17C4⋊C4semidirect
464. C17#14486semidirect
494. C2×D4#63225semidirect

Maximal subgroups

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

subgroupindex
C2×#117627122
C2×#117626892
#145270362
#145270272
#145270102
#145270072
#145270062
C2×C17×#3862
#1448617