Finite Group Databasebeta

← #16961089

Subgroups — #16961089

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×C4×D68C211yesmaximal, characteristic
2C2×#11762726C211yesmaximal, characteristic
3C2×#11762680C212yesmaximal, complemented
4#14527048C211yesmaximal, characteristic, complemented
5#14527037C211yesmaximal, characteristic, complemented
6#14527032C212yesmaximal, complemented
7#14527028C212yesmaximal, complemented
8#14527024C211yesmaximal, characteristic
9#14527011C211yesmaximal, characteristic, complemented
10#14527005C212yesmaximal, complemented
11C2×C17×#387C211yesmaximal, characteristic, complemented
12C23×D68C2211yescharacteristic
13C23×C4×D17C2212yes—
14C2×C4×D68—24no—
15C2×C4×D68—28no—
16C22×C4⋊Dic17C2211yescharacteristic
17C22×#63225C2212yes—
18C22×#63227C2212yes—
19C22×#63227C2212yes—
20C22×#63227C2212yes—
21C22×#63227C2212yes—
22C22×#63227C2212yes—
23#11762726C418yescomplemented
24#11762680—28no—
25C2×#10587502C2212yes—
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2211yescharacteristic
32C22×C4⋊C4×C17C2211yescharacteristic
33C22×C4⋊C4×C17C2211yescharacteristic
34C22×C42×C17C2211yescharacteristic
35C17×#387—24no—
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2211yescharacteristic
40C24×D17C2312yes—
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2311yescharacteristic
48C23×Dic17C2312yes—
49C23×Dic17C2312yes—
50C23×Dic17C2312yes—
51C22×D68—24no—
52C22×D68—24no—
53C22×D68—22no—
54C22×D68—22no—
55C22×D68C2×C414yes—
56C22×C4×D17—24no—
57C22×C4×D17—24no—
58C22×C4×D17—42no—
59C22×C4×D17—42no—
60C22×C4×D17—42no—
61C22×C4×D17—42no—
62C22×C4×D17D418yes—
63C2×C4⋊C4×C17—22no—
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17—22no—
66C2×C4⋊C4×C17—24no—
67C2×C4⋊C4×C17C2×C414yescomplemented
68C2×C42×C17—24no—
69C2×C42×C17D414yes—
70C2×#63227—24no—
71C2×#63227—24no—
72C2×#63227—24no—
73C2×#63227—24no—
74C2×#63227—24no—
75C2×#63227—24no—
76C2×#63227—24no—
77C2×#63227—42no—
78C2×#63227—42no—
79C2×#63227—42no—
80C2×#63227—42no—
81C2×#63227—42no—
82C2×#63227—42no—
83C2×#63227C2×C418yescomplemented
84C2×#63227C2×C418yescomplemented
85C2×C4⋊Dic17—22no—
86C2×C4⋊Dic17—24no—
87C2×#63225—24no—
88C2×#63225—42no—
89C2×#63225—42no—
90C4×D68—232no—
91C17×#95—22no—
92C17×#95—22no—
93C17×#95—22no—
94C17×#95—22no—
95#10587502—24no—
96#10587502—22no—
97#10587502—24no—
98#10587502—42no—
99#10587502—42no—
100#10587502—42no—
101#10587502C2×C414yescomplemented
102C24×C17C2411yescharacteristic
103C23×D17—22no—
104C23×D17—22no—
105C23×D17—22no—
106C23×D17—22no—
107C23×D17—22no—
108C23×D17—22no—
109C23×D17—22no—
110C23×D17—22no—
111C23×D17—22no—
112C23×D17—22no—
113C23×D17—22no—
114C23×D17—22no—
115C23×D17C22×C414yes—
116C23×D17C2×D414yes—
117C23×D17#4114yes—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—22no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—21no—
143C22×C4×C17—22no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17C22×C412yes—
148C22×C4×C17C22×C412yes—
149C22×C4×C17C22×C412yes—
150C22×C4×C17C2×D414yes—
151C22×C4×C17C2×D412yes—
152C22×C4×C17C2×D412yes—
153C22×C4×C17C2×D412yes—
154C22×C4×C17#4112yes—
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×Dic17C22×C414yes—
175C22×Dic17C2×D414yes—
176C2×D68—28no—
177C2×D68—216no—
178C2×D68—28no—
179C2×D68—216no—
180C2×D68—28no—
181C2×C4×D17—216no—
182C2×C4×D17—48no—
183C2×C4×D17—48no—
184C2×C4×D17—48no—
185C2×C4×D17—48no—
186C2×C4×D17—48no—
187C2×C4×D17—48no—
188C4⋊C4×C17—28no—
189C4⋊C4×C17—44no—
190C42×C17—28no—
191#63227—216no—
192#63227—216no—
193#63227—48no—
194#63227—48no—
195#63227—48no—
196C4⋊Dic17—28no—
197#63225—48no—
198#14300—171nomaximal, Sylow
199C23×C17C23×C411yescharacteristic
200C23×C17C22×D412yes—
201C23×C17C22×D411yescharacteristic
202C23×C17C2×#4111yescharacteristic
203C23×C17C2×#4112yes—
204C23×C17H(5,𝔽2)11yescharacteristic
205C23×C17H(5,𝔽2)11yescharacteristic
206C23×C17H(5,𝔽2)12yes—
207C23×C17H(5,𝔽2)11yescharacteristic
208C23×C172−1+412yes—
209C23×C172−1+411yescharacteristic
210C22×D17—24no—
211C22×D17—24no—
212C22×D17—24no—
213C22×D17—24no—
214C22×D17—24no—
215C22×D17—24no—
216C22×D17—24no—
217C22×D17—24no—
218C22×D17—24no—
219C22×D17—24no—
220C22×D17—24no—
221C22×D17—24no—
222C22×D17—24no—
223C22×D17—24no—
224C22×D17—24no—
225C22×D17—24no—
226C22×D17—24no—
227C22×D17—24no—
228C22×D17—42no—
229C22×D17—42no—
230C22×D17—42no—
231C22×D17—42no—
232C22×D17—42no—
233C22×D17—42no—
234C22×D17—42no—
235C22×D17—42no—
236C22×D17—42no—
237C22×D17—42no—
238C22×D17—42no—
239C22×D17—42no—
240C22×D17—42no—
241C22×D17—42no—
242C22×D17—42no—
243C22×D17—42no—
244C22×D17C4×D418yes—
245C2×C4×C17—24no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—24no—
249C2×C4×C17—24no—
250C2×C4×C17—24no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—22no—
255C2×C4×C17—24no—
256C2×C4×C17—22no—
257C2×C4×C17—22no—
258C2×C4×C17—22no—
259C2×C4×C17—22no—
260C2×C4×C17—22no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—42no—
264C2×C4×C17—41no—
265C2×C4×C17—42no—
266C2×C4×C17—41no—
267C2×C4×C17—42no—
268C2×C4×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17—41no—
271C2×C4×C17—41no—
272C2×C4×C17—42no—
273C2×C4×C17—42no—
274C2×C4×C17—41no—
275C2×C4×C17—41no—
276C2×C4×C17—41no—
277C2×C4×C17—41no—
278C2×C4×C17—42no—
279C2×C4×C17—42no—
280C2×C4×C17C4×D414yes—
281C2×C4×C17#12014yes—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—24no—
285C2×Dic17—24no—
286C2×Dic17—24no—
287C2×Dic17—24no—
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—
303D68—232no—
304C4×D17—432no—
305C22×C4×D4—171no—
306C2×#394—171no—
307C2×#387D1711yescharacteristic, complemented
308C2×#385—172no—
309#1444—171no—
310#1441—171no—
311#1440—172no—
312#1439—172no—
313#1438—171no—
314#1434—171no—
315C22⋊#95—172no—
316C22×C17C2×C4×D412yes—
317C22×C17C2×C4×D411yescharacteristic
318C22×C17#51811yescharacteristic
319C22×C17#52011yescharacteristic
320C22×C17#52012yes—
321C22×C17C2×#12011yescharacteristic
322C22×C17#52511yescharacteristic
323C22×C17#52512yes—
324C22×C17#53412yes—
325C22×C17#53412yes—
326C22×C17#53511yescharacteristic
327C22×C17#53511yescharacteristic
328C22×C17#53612yes—
329C22×C17#53611yescharacteristic
330C22×C17#53711yescharacteristic
331C22×C17#53712yes—
332C22×C17#53812yes—
333C22×C17#53912yes—
334C22×C17#53912yes—
335C22×C17#53912yes—
336C22×C17#54011yescharacteristic
337C22×C17#54211yescharacteristic
338C22×C17#54212yes—
339C2×D17—28no—
340C2×D17—28no—
341C2×D17—28no—
342C2×D17—44no—
343C2×D17—44no—
344C2×D17—44no—
345C2×D17—44no—
346C2×D17—44no—
347C2×D17—44no—
348C2×D17—44no—
349C2×D17—44no—
350C2×D17—44no—
351C2×D17—44no—
352C2×D17—44no—
353C2×D17—44no—
354C4×C17—24no—
355C4×C17—24no—
356C4×C17—42no—
357C4×C17—42no—
358C4×C17—42no—
359C4×C17—81no—
360C4×C17—81no—
361Dic17—44no—
362Dic17—44no—
363Dic17—82no—
364C23×D4—171no—
365C24×C4—172no—
366C2×C4×D4—348no—
367C2×C4×D4—344no—
368C22×C4⋊C4—171no—
369C22×C4⋊C4—172no—
370C22×C4⋊C4C2×D1711yescharacteristic
371C22×C4⋊C4C2×D1711yescharacteristic
372C22×C22⋊C4—172no—
373C22×C22⋊C4—172no—
374C22×C22⋊C4—172no—
375C22×C22⋊C4—172no—
376C22×C22⋊C4—172no—
377C22×C42C2×D1711yescharacteristic
378#394—178no—
379#387—24no—
380#385—348no—
381C2×#95—171no—
382C2×#95—172no—
383C2×#95—171no—
384C2×#95—172no—
385C2×#95—172no—
386C2×#95—172no—
387C2×#95—172no—
388C2×#95C2×D1711yescharacteristic
389C2×#95C2×D1711yescharacteristic
390C2×#95C2×D1711yescharacteristic
391C2×#95C2×D1711yescharacteristic
392C2×C17C4×#12011yescharacteristic
393C2×C17#232212yes—
394C2×C17#232411yescharacteristic
395C2×C17#232512yes—
396C2×C17#232611yescharacteristic
397C2×C17#261311yescharacteristic
398C2×C17#261611yescharacteristic
399C2×C17#263012yes—
400C2×C17#263311yescharacteristic
401C2×C17#263611yescharacteristic
402C2×C17#263812yes—
403D17—48no—
404C25—172no—
405C22×D4—174no—
406C22×D4—344no—
407C22×D4—344no—
408C22×D4—342no—
409C22×D4—342no—
410C23×C4—172no—
411C23×C4—172no—
412C23×C4—178no—
413C23×C4—172no—
414C23×C4—344no—
415C23×C4—344no—
416C23×C4—682no—
417C23×C4—682no—
418C23×C4—682no—
419C23×C4—682no—
420C23×C4C22×D1711yescharacteristic
421C23×C4C22×D1711yescharacteristic
422C23×C4C22×D1711yescharacteristic
423C23×C4C22×D1711yescharacteristic
424C23×C4C22×D1711yescharacteristic
425C23×C4C22×D1711yescharacteristic
426C23×C4C22×D1711yescharacteristic
427C4×D4—3432no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—24no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—344no—
433C2×C4⋊C4—342no—
434C2×C4⋊C4—344no—
435C2×C4⋊C4—682no—
436C2×C4⋊C4—682no—
437C2×C4⋊C4C4×D1714yescomplemented
438C2×C22⋊C4—178no—
439C2×C22⋊C4—178no—
440C2×C22⋊C4—344no—
441C2×C22⋊C4—344no—
442C2×C22⋊C4—344no—
443C2×C22⋊C4—344no—
444C2×C22⋊C4—344no—
445C2×C22⋊C4—344no—
446C2×C22⋊C4—344no—
447C2×C22⋊C4—682no—
448C2×C22⋊C4—682no—
449C2×C22⋊C4—682no—
450C2×C22⋊C4—682no—
451C2×C22⋊C4—682no—
452C2×C22⋊C4—682no—
453C2×C42—24no—
454C2×C42C22⋊D1714yes—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—174no—
460#95—344no—
461#95—342no—
462#95—344no—
463#95—682no—
464#95—682no—
465#95—682no—
466C17#1430011yescharacteristic, complemented, Sylow
467C24—174no—
468C24—174no—
469C24—174no—
470C24—342no—
471C24—342no—
472C24—342no—
473C24—342no—
474C24—342no—
475C24—342no—
476C24—342no—
477C24—342no—
478C24—342no—
479C24—342no—
480C24—342no—
481C24—342no—
482C24C23×D1711yescharacteristic
483C2×D4—3416no—
484C2×D4—348no—
485C2×D4—348no—
486C2×D4—3416no—
487C2×D4—348no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—174no—
518C22×C4—174no—
519C22×C4—342no—
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—3416no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—688no—
540C22×C4—688no—
541C22×C4—688no—
542C22×C4—688no—
543C22×C4—688no—
544C22×C4—688no—
545C22×C4C2×C4×D1712yes—
546C22×C4C2×C4×D1712yes—
547C22×C4C2×C4×D1712yes—
548C22×C4#6325212yes—
549C22×C4D4×D1714yes—
550C22×C4C2×C22⋊D1712yes—
551C22×C4C2×C22⋊D1712yes—
552C22×C4C2×C22⋊D1712yes—
553C4⋊C4—28no—
554C4⋊C4—44no—
555C4⋊C4—348no—
556C4⋊C4—688no—
557C22⋊C4—3416no—
558C22⋊C4—3416no—
559C22⋊C4—688no—
560C22⋊C4—688no—
561C22⋊C4—688no—
562C42—28no—
563C23—178no—
564C23—344no—
565C23—344no—
566C23—344no—
567C23—344no—
568C23—344no—
569C23—344no—
570C23—344no—
571C23—344no—
572C23—344no—
573C23—344no—
574C23—344no—
575C23—344no—
576C23—344no—
577C23—344no—
578C23—344no—
579C23—344no—
580C23—344no—
581C23—344no—
582C23—682no—
583C23—682no—
584C23—682no—
585C23—682no—
586C23—682no—
587C23—682no—
588C23—682no—
589C23—682no—
590C23—682no—
591C23—682no—
592C23—682no—
593C23—682no—
594C23—682no—
595C23—682no—
596C23—682no—
597C23—682no—
598C23C22×C4×D1711yescharacteristic
599C23C2×#6325211yescharacteristic
600C23C2×D4×D1712yes—
601C23#1058769412yes—
602C23#41×D1712yes—
603C23#1058769611yescharacteristic
604C23#1058769611yescharacteristic
605C23#1058769611yescharacteristic
606C23#1058769612yes—
607C23#1058769711yescharacteristic
608C23C22×C22⋊D1711yescharacteristic
609D4—3432no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—344no—
646C2×C4—344no—
647C2×C4—344no—
648C2×C4—344no—
649C2×C4—344no—
650C2×C4—344no—
651C2×C4—6832no—
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×C4C4×C22⋊D1714yes—
668C2×C4#1058762314yes—
669C22—348no—
670C22—348no—
671C22—348no—
672C22—684no—
673C22—684no—
674C22—684no—
675C22—684no—
676C22—684no—
677C22—684no—
678C22—684no—
679C22—684no—
680C22—684no—
681C22—684no—
682C22—684no—
683C22—684no—
684C22C4×D4×D1712yes—
685C22C2×C4×C22⋊D1711yescharacteristic
686C22#1176328311yescharacteristic
687C22#1176328911yescharacteristic
688C22#1176329112yes—
689C22C2×#1058762311yescharacteristic
690C22#1176331611yescharacteristic
691C22#1176331812yes—
692C22#1176337012yes—
693C22#1176337012yes—
694C22#1176337311yescharacteristic
695C22#1176337311yescharacteristic
696C22#1176338212yes—
697C22#1176338311yescharacteristic
698C22#1176338711yescharacteristic
699C22#1176338912yes—
700C22#1176339812yes—
701C22#1176340012yes—
702C22#1176340712yes—
703C22#1176340712yes—
704C22#1176341411yescharacteristic
705C22#1176343011yescharacteristic
706C22#1176343112yes—
707C4—24no—
708C4—24no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—81no—
713C4—81no—
714C4—684no—
715C4—684no—
716C4—1362no—
717C2—688no—
718C2C4×#1058762311yescharacteristic
719C2#1453145712yes—
720C2#1453147711yescharacteristic
721C2#1453148312yes—
722C2#1453149111yescharacteristic
723C2#1453450011yescharacteristic
724C2#1453451411yescharacteristic
725C2#1453463312yes—
726C2#1453465311yescharacteristic
727C2#1453468111yescharacteristic
728C2#1453470812yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#387C2
socle, cosocleC24×C17C24
prime radicalC23×D68C22
hypercenterC2×#387D17
hypocenterC17#14300
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
3. C2×#11762680C2semidirect
4. #14527048C2semidirect
5. #14527037C2semidirect
6. #14527032C2semidirect
7. #14527028C2semidirect
9. #14527011C2semidirect
10. #14527005C2semidirect
11. C2×C17×#387C2semidirect
23. #11762726C4semidirect
67. C2×C4⋊C4×C17C2×C4semidirect
83. C2×#63227C2×C4semidirect
84. C2×#63227C2×C4semidirect
101. #10587502C2×C4semidirect
307. C2×#387D17semidirect
437. C2×C4⋊C4C4×D17semidirect
466. C17#14300semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D682
C2×#117627262
C2×#117626802
#145270482
#145270372
#145270322
#145270282
#145270242
#145270112
#145270052
C2×C17×#3872
#1430017