Finite Group Databasebeta

← #16961086

Subgroups — #16961086

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×#10587579C211yesmaximal, characteristic
2C2×#11762727C211yesmaximal, characteristic
3C2×#11762691C211yesmaximal, characteristic, complemented
4C2×#11762680C212yesmaximal, complemented
5#14527034C211yesmaximal, characteristic, complemented
6#14527032C212yesmaximal, complemented
7#14527026C212yesmaximal, complemented
8#14527024C211yesmaximal, characteristic
9#14527009C211yesmaximal, characteristic, complemented
10#14527005C212yesmaximal, complemented
11C17×#1444C211yesmaximal, characteristic, complemented
12C23×D68C2211yescharacteristic
13C23×C4×D17C2212yes—
14C2×#10587579—24no—
15C2×#10587579—28no—
16C22×#63225C2211yescharacteristic
17C22×#63225C2211yescharacteristic
18C22×#63227C2212yes—
19C22×#63227C2212yes—
20C22×#63227C2212yes—
21C22×#63227C2212yes—
22C22×#63227C2212yes—
23C22×C4×Dic17C2211yescharacteristic
24#11762727C418yescomplemented
25#11762691—24no—
26#11762680—28no—
27C2×#10587502C2211yescharacteristic
28C2×#10587502C2212yes—
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2212yes—
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2211yescharacteristic
40C24×D17C2312yes—
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
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—24no—
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17—24no—
66C2×C4⋊C4×C17—42no—
67C2×C4⋊C4×C17—42no—
68C2×#63227—24no—
69C2×#63227—24no—
70C2×#63227—24no—
71C2×#63227—24no—
72C2×#63227—24no—
73C2×#63227—24no—
74C2×#63227—24no—
75C2×#63227—42no—
76C2×#63227—42no—
77C2×#63227—42no—
78C2×#63227—42no—
79C2×#63227—42no—
80C2×#63227—42no—
81C2×#63227C2×C418yescomplemented
82C2×#63227C2×C418yescomplemented
83C2×#63225—22no—
84C2×#63225—22no—
85C2×#63225—22no—
86C2×#63225—24no—
87C2×#63225C2×C414yescomplemented
88C2×C4×Dic17—24no—
89C2×C4×Dic17D414yes—
90#10587579—232no—
91C17×#95—22no—
92C17×#95—42no—
93C17×#95C2×C414yescomplemented
94#10587502—24no—
95#10587502—22no—
96#10587502—22no—
97#10587502—22no—
98#10587502—22no—
99#10587502—24no—
100#10587502—42no—
101#10587502—42no—
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—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×C17—22no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—22no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17C22×C414yes—
143C22×C4×C17C22×C412yes—
144C22×C4×C17C2×D414yes—
145C22×C4×C17C2×D412yes—
146C22×C4×C17#4112yes—
147C22×Dic17—21no—
148C22×Dic17—22no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—22no—
163C22×Dic17—21no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17C22×C412yes—
172C22×Dic17C22×C412yes—
173C22×Dic17C2×D412yes—
174C22×Dic17C2×D412yes—
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—48no—
190#63227—216no—
191#63227—216no—
192#63227—48no—
193#63227—48no—
194#63227—48no—
195#63225—28no—
196#63225—44no—
197C4×Dic17—28no—
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)12yes—
206C23×C17H(5,𝔽2)11yescharacteristic
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—24no—
252C2×C4×C17—24no—
253C2×C4×C17—24no—
254C2×C4×C17—24no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17—42no—
258C2×C4×C17—42no—
259C2×C4×C17—42no—
260C2×C4×C17—42no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17—41no—
268C2×C4×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17—41no—
271C2×C4×C17—42no—
272C2×C4×C17—42no—
273C2×C4×C17C4×D414yes—
274C2×Dic17—24no—
275C2×Dic17—24no—
276C2×Dic17—22no—
277C2×Dic17—22no—
278C2×Dic17—22no—
279C2×Dic17—22no—
280C2×Dic17—24no—
281C2×Dic17—22no—
282C2×Dic17—22no—
283C2×Dic17—22no—
284C2×Dic17—22no—
285C2×Dic17—22no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—42no—
289C2×Dic17—41no—
290C2×Dic17—42no—
291C2×Dic17—41no—
292C2×Dic17—42no—
293C2×Dic17—41no—
294C2×Dic17—41no—
295C2×Dic17—42no—
296C2×Dic17—42no—
297C2×Dic17—42no—
298C2×Dic17—42no—
299C2×Dic17—42no—
300C2×Dic17—42no—
301C2×Dic17—42no—
302C2×Dic17#12014yes—
303D68—232no—
304C4×D17—432no—
305C22×C4×D4—171no—
306C2×#394—171no—
307C2×#387—171no—
308C2×#385—172no—
309#1444D1711yescharacteristic, complemented
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#52012yes—
320C22×C17#52011yescharacteristic
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—44no—
356C4×C17—44no—
357C4×C17—81no—
358C4×C17—81no—
359Dic17—24no—
360Dic17—42no—
361Dic17—42no—
362Dic17—42no—
363Dic17—82no—
364C23×D4—171no—
365C24×C4—172no—
366C2×C4×D4—344no—
367C2×C4×D4—348no—
368C22×C4⋊C4—171no—
369C22×C4⋊C4—171no—
370C22×C4⋊C4C2×D1711yescharacteristic
371C22×C4⋊C4C2×D1712yes—
372C22×C22⋊C4—172no—
373C22×C22⋊C4—172no—
374C22×C22⋊C4—172no—
375C22×C22⋊C4—172no—
376C22×C22⋊C4—172no—
377C22×C42—171no—
378#394—178no—
379#387—344no—
380#385—348no—
381C2×#95—171no—
382C2×#95—172no—
383C2×#95—171no—
384C2×#95—171no—
385C2×#95—171no—
386C2×#95—172no—
387C2×#95—172no—
388C2×#95—172no—
389C2×#95C2×D1711yescharacteristic
390C2×#95C2×D1712yes—
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—171no—
411C23×C4—171no—
412C23×C4—171no—
413C23×C4—171no—
414C23×C4—178no—
415C23×C4—172no—
416C23×C4—344no—
417C23×C4—344no—
418C23×C4—682no—
419C23×C4—682no—
420C23×C4—682no—
421C23×C4—682no—
422C23×C4C22×D1711yescharacteristic
423C23×C4C22×D1711yescharacteristic
424C23×C4C22×D1712yes—
425C23×C4C22×D1712yes—
426C23×C4C22×D1711yescharacteristic
427C4×D4—3432no—
428C2×C4⋊C4—24no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—24no—
431C2×C4⋊C4—42no—
432C2×C4⋊C4—42no—
433C2×C4⋊C4—174no—
434C2×C4⋊C4—342no—
435C2×C4⋊C4—342no—
436C2×C4⋊C4—344no—
437C2×C4⋊C4—342no—
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—174no—
454C2×C42—344no—
455#95—22no—
456#95—42no—
457#95—344no—
458#95—342no—
459#95—342no—
460#95—342no—
461#95—344no—
462#95—342no—
463#95—682no—
464#95—682no—
465#95C4×D1714yescomplemented
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—22no—
491C22×C4—22no—
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—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—172no—
513C22×C4—174no—
514C22×C4—172no—
515C22×C4—172no—
516C22×C4—172no—
517C22×C4—3416no—
518C22×C4—342no—
519C22×C4—341no—
520C22×C4—341no—
521C22×C4—341no—
522C22×C4—342no—
523C22×C4—341no—
524C22×C4—342no—
525C22×C4—341no—
526C22×C4—341no—
527C22×C4—341no—
528C22×C4—341no—
529C22×C4—342no—
530C22×C4—341no—
531C22×C4—342no—
532C22×C4—341no—
533C22×C4—341no—
534C22×C4—341no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—342no—
542C22×C4—688no—
543C22×C4—688no—
544C22×C4—688no—
545C22×C4—688no—
546C22×C4—688no—
547C22×C4—688no—
548C22×C4C2×C4×D1714yes—
549C22×C4C2×C4×D1712yes—
550C22×C4D4×D1714yes—
551C22×C4D4×D1712yes—
552C22×C4#6325612yes—
553C4⋊C4—28no—
554C4⋊C4—48no—
555C4⋊C4—348no—
556C4⋊C4—684no—
557C22⋊C4—3416no—
558C22⋊C4—3416no—
559C22⋊C4—688no—
560C22⋊C4—688no—
561C22⋊C4—688no—
562C42—348no—
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×D4×D1712yes—
600C23C2×D4×D1711yescharacteristic
601C23#1058769111yescharacteristic
602C23#1058769112yes—
603C23C2×#6325611yescharacteristic
604C23#1058769412yes—
605C23#1058769411yescharacteristic
606C23#41×D1712yes—
607C23#1058769611yescharacteristic
608C23#1058769611yescharacteristic
609D4—3432no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—174no—
639C2×C4—344no—
640C2×C4—344no—
641C2×C4—342no—
642C2×C4—342no—
643C2×C4—344no—
644C2×C4—342no—
645C2×C4—342no—
646C2×C4—342no—
647C2×C4—342no—
648C2×C4—342no—
649C2×C4—342no—
650C2×C4—342no—
651C2×C4—6832no—
652C2×C4—681no—
653C2×C4—681no—
654C2×C4—682no—
655C2×C4—681no—
656C2×C4—682no—
657C2×C4—682no—
658C2×C4—681no—
659C2×C4—681no—
660C2×C4—681no—
661C2×C4—682no—
662C2×C4—682no—
663C2×C4—682no—
664C2×C4—682no—
665C2×C4—682no—
666C2×C4—682no—
667C2×C4—682no—
668C2×C4#1058757914yes—
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×#1058757911yescharacteristic
686C22#1176328111yescharacteristic
687C22#1176329012yes—
688C22#1176329111yescharacteristic
689C22#120×D1711yescharacteristic
690C22#1176331711yescharacteristic
691C22#1176332212yes—
692C22#1176336712yes—
693C22#1176336912yes—
694C22#1176337211yescharacteristic
695C22#1176337511yescharacteristic
696C22#1176338412yes—
697C22#1176338411yescharacteristic
698C22#1176338911yescharacteristic
699C22#1176338912yes—
700C22#1176339512yes—
701C22#1176340512yes—
702C22#1176340512yes—
703C22#1176340712yes—
704C22#1176341511yescharacteristic
705C22#1176343311yescharacteristic
706C22#1176343312yes—
707C4—24no—
708C4—44no—
709C4—44no—
710C4—81no—
711C4—81no—
712C4—344no—
713C4—682no—
714C4—682no—
715C4—682no—
716C4—1362no—
717C2—688no—
718C2#1453128511yescharacteristic
719C2#1453144812yes—
720C2#1453147511yescharacteristic
721C2#1453148112yes—
722C2#1453149211yescharacteristic
723C2#1453449511yescharacteristic
724C2#1453451611yescharacteristic
725C2#1453463412yes—
726C2#1453465411yescharacteristic
727C2#1453468211yescharacteristic
728C2#1453471012yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1444C2
socle, cosocleC24×C17C24
prime radicalC23×D68C22
hypercenter#1444D17
hypocenterC17#14300
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
3. C2×#11762691C2semidirect
4. C2×#11762680C2semidirect
5. #14527034C2semidirect
6. #14527032C2semidirect
7. #14527026C2semidirect
9. #14527009C2semidirect
10. #14527005C2semidirect
11. C17×#1444C2semidirect
24. #11762727C4semidirect
81. C2×#63227C2×C4semidirect
82. C2×#63227C2×C4semidirect
87. C2×#63225C2×C4semidirect
93. C17×#95C2×C4semidirect
309. #1444D17semidirect
465. #95C4×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×#105875792
C2×#117627272
C2×#117626912
C2×#117626802
#145270342
#145270322
#145270262
#145270242
#145270092
#145270052
C17×#14442
#1430017