Finite Group Databasebeta

← #16955608

Subgroups — #16955608

Order 4352.

2102 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762683C212yesmaximal
2C2×C4×#63227C211yesmaximal, characteristic
3C2×#11762656C211yesmaximal, characteristic
4C2×#95×D17C211yesmaximal, characteristic
5#14527018C212yesmaximal, complemented
6#14527015C212yesmaximal, complemented
7#14527014C211yesmaximal, characteristic, complemented
8#14527012C211yesmaximal, characteristic, complemented
9#14527010C211yesmaximal, characteristic, complemented
10#14527005C212yesmaximal
11C17×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D17C2212yes—
13C23×C4×D17C2211yescharacteristic
14C22×C4⋊Dic17C2212yes—
15C22×#63225C2212yes—
16C22×#63227C2211yescharacteristic
17C22×#63227C2212yes—
18C22×#63227C2211yescharacteristic
19C22×C4×Dic17C2212yes—
20C22×C4×Dic17C2211yescharacteristic
21#11762683C4116yescomplemented
22C4×#63227C418yescomplemented
23#11762656—24no—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2211yescharacteristic
33#95×D17—24no—
34C22×C42×C17C2211yescharacteristic
35C2×C17×#95C2211yescharacteristic
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2212yes—
38C2×C17×#95C2212yes—
39C24×D17C2311yescharacteristic
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×Dic17C2312yes—
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C22×C4×D17—24no—
51C22×C4×D17—24no—
52C22×C4×D17—42no—
53C22×C4×D17—42no—
54C22×C4×D17—42no—
55C22×C4×D17—42no—
56C22×C4×D17—42no—
57C22×C4×D17—42no—
58C22×C4×D17C2×C418yes—
59C22×C4×D17C2×C414yes—
60C22×C4×D17D414yes—
61C22×C4×D17Q814yes—
62C2×C42×C17—42no—
63C2×C42×C17C2×C414yescomplemented
64C2×#63227—24no—
65C2×#63227—24no—
66C2×#63227—24no—
67C2×#63227C2×C414yes—
68C2×#63227C2×C418yes—
69C2×#63227C2×C414yes—
70C2×#63227C2×C418yes—
71C2×C4⋊Dic17—42no—
72C2×C4⋊Dic17—42no—
73C2×C4⋊Dic17C2×C418yescomplemented
74C2×#63225—42no—
75C2×#63225—42no—
76C2×#63225C2×C418yescomplemented
77C2×C4×Dic17—24no—
78C2×C4×Dic17—42no—
79C2×C4×Dic17—42no—
80C2×C4×Dic17—42no—
81C2×C4×Dic17C2×C414yescomplemented
82C17×#95—22no—
83C17×#95—22no—
84C17×#95—42no—
85C17×#95C2×C418yescomplemented
86#10587502—22no—
87#10587502—22no—
88#10587502—24no—
89#10587502—42no—
90#10587502—42no—
91#10587502—42no—
92#10587502C2×C414yescomplemented
93#10587502C2×C414yescomplemented
94#10587502C2×C418yescomplemented
95C24×C17C2411yescharacteristic
96C23×D17—22no—
97C23×D17—22no—
98C23×D17—21no—
99C23×D17—21no—
100C23×D17—21no—
101C23×D17—21no—
102C23×D17C22×C412yes—
103C23×D17C22×C414yes—
104C23×D17C2×D412yes—
105C23×D17C2×Q812yes—
106C23×D17#4114yes—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—21no—
110C22×C4×C17—22no—
111C22×C4×C17—21no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
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—21no—
128C22×C4×C17C22×C414yes—
129C22×C4×C17C22×C414yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C22×C412yes—
132C22×C4×C17C22×C414yes—
133C22×C4×C17C22×C412yes—
134C22×C4×C17C2×D412yes—
135C22×C4×C17C2×Q812yes—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—21no—
157C22×Dic17C22×C414yes—
158C22×Dic17C22×C412yes—
159C22×Dic17C22×C414yes—
160C22×Dic17C22×C412yes—
161C22×Dic17C22×C414yes—
162C22×Dic17C22×C412yes—
163C22×Dic17C2×D412yes—
164C22×Dic17C2×Q812yes—
165C2×C4×D17—216no—
166C2×C4×D17—44no—
167C2×C4×D17—44no—
168C2×C4×D17—48no—
169C2×C4×D17—48no—
170C2×C4×D17—48no—
171C2×C4×D17—48no—
172C2×C4×D17—48no—
173C2×C4×D17—48no—
174C2×C4×D17—48no—
175C2×C4×D17—48no—
176C2×C4×D17C4⋊C4116yescomplemented
177C42×C17—44no—
178#63227—28no—
179#63227—216no—
180#63227C42116yescomplemented
181C4⋊Dic17—48no—
182#63225—48no—
183C4×Dic17—44no—
184C4×Dic17—48no—
185#13782—171nomaximal, Sylow
186C23×C17C23×C411yescharacteristic
187C23×C17C23×C412yes—
188C23×C17C22×D411yescharacteristic
189C23×C17C22×Q811yescharacteristic
190C23×C17C2×#4112yes—
191C23×C17H(5,𝔽2)12yes—
192C23×C17H(5,𝔽2)12yes—
193C23×C17H(5,𝔽2)11yescharacteristic
194C23×C17H(5,𝔽2)11yescharacteristic
195C23×C172−1+411yescharacteristic
196C23×C172−1+411yescharacteristic
197C22×D17—24no—
198C22×D17—22no—
199C22×D17—22no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—24no—
205C22×D17—24no—
206C22×D17—22no—
207C22×D17—22no—
208C22×D17—24no—
209C22×D17—24no—
210C22×D17—24no—
211C22×D17—24no—
212C22×D17—24no—
213C22×D17—22no—
214C22×D17—22no—
215C22×D17C2×C4214yes—
216C22×D17C2×C4⋊C414yes—
217C22×D17#11714yes—
218C22×D17C4×D418yes—
219C22×D17C4×Q818yes—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—24no—
224C2×C4×C17—24no—
225C2×C4×C17—22no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—22no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—22no—
232C2×C4×C17—24no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—22no—
236C2×C4×C17—24no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17C2×C4214yes—
248C2×C4×C17C2×C4⋊C414yes—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—24no—
253C2×Dic17—22no—
254C2×Dic17—24no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—22no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—22no—
263C2×Dic17—24no—
264C2×Dic17—24no—
265C2×Dic17—22no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17C2×C4214yes—
277C2×Dic17C2×C4⋊C414yes—
278C4×D17—416no—
279C4×D17—432no—
280C2×#385—172no—
281C2×C4×C22⋊C4—171no—
282C2×C4×C22⋊C4—171no—
283C22×#95—171no—
284#1436—172no—
285#1436—172no—
286C4⋊#95—171no—
287C4⋊#95—171no—
288#1434—171no—
289#1434D1711yescharacteristic, complemented
290C22⋊#95—172no—
291C22×C17C22×C4211yescharacteristic
292C22×C17C22×C4⋊C411yescharacteristic
293C22×C17C2×#11711yescharacteristic
294C22×C17C2×C4×D412yes—
295C22×C17C2×C4×Q812yes—
296C22×C17#51812yes—
297C22×C17#51811yescharacteristic
298C22×C17#51811yescharacteristic
299C22×C17#51812yes—
300C22×C17#52012yes—
301C22×C17#52011yescharacteristic
302C22×C17#52011yescharacteristic
303C22×C17#52012yes—
304C22×C17#53412yes—
305C22×C17#53511yescharacteristic
306C22×C17#53611yescharacteristic
307C22×C17#53812yes—
308C22×C17#53812yes—
309C22×C17#53912yes—
310C22×C17#53912yes—
311C22×C17#54312yes—
312C22×C17#54411yescharacteristic
313C22×C17#54411yescharacteristic
314C2×D17—24no—
315C2×D17—24no—
316C2×D17—24no—
317C2×D17—24no—
318C2×D17—28no—
319C2×D17—28no—
320C2×D17—28no—
321C2×D17—28no—
322C2×D17—24no—
323C2×D17—24no—
324C2×D17C4×C4⋊C418yes—
325C4×C17—24no—
326C4×C17—24no—
327C4×C17—42no—
328C4×C17—44no—
329C4×C17—44no—
330Dic17—24no—
331Dic17—24no—
332Dic17—42no—
333Dic17—44no—
334Dic17—44no—
335C24×C4—171no—
336C24×C4—172no—
337C22×C4⋊C4—172no—
338C22×C4⋊C4—172no—
339C22×C22⋊C4—171no—
340C22×C22⋊C4—171no—
341C22×C22⋊C4—172no—
342C22×C42—171no—
343C22×C42—172no—
344C22×C42C2×D1711yescharacteristic
345#385—1716no—
346C4×C22⋊C4—178no—
347C4×C22⋊C4—344no—
348C2×#95—172no—
349C2×#95—171no—
350C2×#95—171no—
351C2×#95—171no—
352C2×#95—171no—
353C2×#95—172no—
354C2×#95—172no—
355C2×#95—172no—
356C2×#95—172no—
357C2×#95—344no—
358C2×#95C2×D1711yescharacteristic
359C2×#95C2×D1711yescharacteristic
360C2×#95C2×D1712yes—
361C2×#95C2×D1712yes—
362C2×C17C2×C4×C4⋊C411yescharacteristic
363C2×C17#227511yescharacteristic
364C2×C17#227511yescharacteristic
365C2×C17#231511yescharacteristic
366C2×C17#231711yescharacteristic
367C2×C17#231811yescharacteristic
368C2×C17#231811yescharacteristic
369C2×C17#232112yes—
370C2×C17#232212yes—
371C2×C17#232912yes—
372C2×C17#232912yes—
373D17—28no—
374C25—171no—
375C23×C4—171no—
376C23×C4—171no—
377C23×C4—174no—
378C23×C4—174no—
379C23×C4—174no—
380C23×C4—172no—
381C23×C4—171no—
382C23×C4—172no—
383C23×C4—178no—
384C23×C4—344no—
385C23×C4—344no—
386C23×C4—682no—
387C23×C4—682no—
388C23×C4—682no—
389C23×C4—682no—
390C23×C4—682no—
391C23×C4—682no—
392C23×C4C22×D1711yescharacteristic
393C23×C4C22×D1712yes—
394C23×C4C22×D1712yes—
395C23×C4C22×D1711yescharacteristic
396C23×C4C22×D1711yescharacteristic
397C2×C4⋊C4—178no—
398C2×C4⋊C4—178no—
399C2×C4⋊C4—682no—
400C2×C4⋊C4—682no—
401C2×C4⋊C4—682no—
402C2×C4⋊C4—682no—
403C2×C22⋊C4—174no—
404C2×C22⋊C4—178no—
405C2×C22⋊C4—174no—
406C2×C22⋊C4—178no—
407C2×C22⋊C4—344no—
408C2×C22⋊C4—344no—
409C2×C22⋊C4—344no—
410C2×C42—42no—
411C2×C42—174no—
412C2×C42—344no—
413C2×C42—682no—
414C2×C42—682no—
415C2×C42—682no—
416C2×C42C4×D1714yescomplemented
417#95—22no—
418#95—22no—
419#95—42no—
420#95—174no—
421#95—174no—
422#95—178no—
423#95—344no—
424#95—342no—
425#95—342no—
426#95—682no—
427#95—682no—
428#95—682no—
429#95C4×D1718yescomplemented
430C17#1378211yescharacteristic, complemented, Sylow
431C24—172no—
432C24—174no—
433C24—172no—
434C24—172no—
435C24—174no—
436C24—342no—
437C24—341no—
438C24—342no—
439C24—341no—
440C24—341no—
441C24—341no—
442C24C23×D1711yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—172no—
465C22×C4—172no—
466C22×C4—174no—
467C22×C4—172no—
468C22×C4—172no—
469C22×C4—1716no—
470C22×C4—172no—
471C22×C4—174no—
472C22×C4—174no—
473C22×C4—3416no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—341no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—341no—
495C22×C4—684no—
496C22×C4—684no—
497C22×C4—688no—
498C22×C4—688no—
499C22×C4—688no—
500C22×C4—688no—
501C22×C4—688no—
502C22×C4—688no—
503C22×C4—688no—
504C22×C4—688no—
505C22×C4C2×C4×D1714yes—
506C22×C4C2×C4×D1712yes—
507C22×C4C2×C4×D1714yes—
508C22×C4C2×C4×D1712yes—
509C22×C4C2×C4×D1714yes—
510C22×C4C2×C4×D1712yes—
511C22×C4D4×D1712yes—
512C22×C4Q8×D1712yes—
513C4⋊C4—688no—
514C4⋊C4—688no—
515C22⋊C4—1716no—
516C22⋊C4—348no—
517C22⋊C4—3416no—
518C42—44no—
519C42—684no—
520C42—688no—
521C23—174no—
522C23—174no—
523C23—174no—
524C23—178no—
525C23—178no—
526C23—344no—
527C23—342no—
528C23—342no—
529C23—344no—
530C23—344no—
531C23—342no—
532C23—342no—
533C23—344no—
534C23—344no—
535C23—342no—
536C23—342no—
537C23—344no—
538C23—344no—
539C23—344no—
540C23—344no—
541C23—344no—
542C23—342no—
543C23—342no—
544C23C22×C4×D1711yescharacteristic
545C23C22×C4×D1712yes—
546C23C2×D4×D1711yescharacteristic
547C23#1058769112yes—
548C23C2×Q8×D1711yescharacteristic
549C23#41×D1712yes—
550C23#1058769611yescharacteristic
551C23#1058769612yes—
552C23#1058769611yescharacteristic
553C23#1058769711yescharacteristic
554C23#1058769711yescharacteristic
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—174no—
583C2×C4—174no—
584C2×C4—342no—
585C2×C4—342no—
586C2×C4—344no—
587C2×C4—342no—
588C2×C4—342no—
589C2×C4—344no—
590C2×C4—344no—
591C2×C4—344no—
592C2×C4—342no—
593C2×C4—344no—
594C2×C4—344no—
595C2×C4—344no—
596C2×C4—344no—
597C2×C4—342no—
598C2×C4—344no—
599C2×C4—344no—
600C2×C4—342no—
601C2×C4—6816no—
602C2×C4—6832no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4—682no—
606C2×C4—682no—
607C2×C4—682no—
608C2×C4—682no—
609C2×C4—682no—
610C2×C4—682no—
611C2×C4—682no—
612C2×C4—682no—
613C2×C4C42×D1714yes—
614C2×C4C4⋊C4×D1714yes—
615C22—178no—
616C22—344no—
617C22—344no—
618C22—344no—
619C22—344no—
620C22—344no—
621C22—344no—
622C22—348no—
623C22—348no—
624C22—348no—
625C22—348no—
626C22C2×C42×D1711yescharacteristic
627C22C2×C4⋊C4×D1711yescharacteristic
628C22#117×D1711yescharacteristic
629C22C4×D4×D1712yes—
630C22C4×Q8×D1712yes—
631C22#1176328112yes—
632C22#1176328211yescharacteristic
633C22#1176328311yescharacteristic
634C22#1176328312yes—
635C22#1176328812yes—
636C22#1176328911yescharacteristic
637C22#1176328911yescharacteristic
638C22#1176328912yes—
639C22#1176336712yes—
640C22#1176337711yescharacteristic
641C22#1176337911yescharacteristic
642C22#1176339512yes—
643C22#1176339812yes—
644C22#1176340012yes—
645C22#1176340212yes—
646C22#1176343412yes—
647C22#1176343911yescharacteristic
648C22#1176343911yescharacteristic
649C4—24no—
650C4—24no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—344no—
655C4—344no—
656C4—682no—
657C4—684no—
658C4—684no—
659C2—348no—
660C2C4×C4⋊C4×D1711yescharacteristic
661C2#1453114311yescharacteristic
662C2#1453114611yescharacteristic
663C2#1453139711yescharacteristic
664C2#1453140911yescharacteristic
665C2#1453141311yescharacteristic
666C2#1453141511yescharacteristic
667C2#1453143712yes—
668C2#1453145412yes—
669C2#1453151112yes—
670C2#1453151312yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1434D17
hypocenterC17#13782
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
5. #14527018C2semidirect
6. #14527015C2semidirect
7. #14527014C2semidirect
8. #14527012C2semidirect
9. #14527010C2semidirect
11. C17×#1434C2semidirect
21. #11762683C4semidirect
22. C4×#63227C4semidirect
63. C2×C42×C17C2×C4semidirect
73. C2×C4⋊Dic17C2×C4semidirect
76. C2×#63225C2×C4semidirect
81. C2×C4×Dic17C2×C4semidirect
85. C17×#95C2×C4semidirect
92. #10587502C2×C4semidirect
93. #10587502C2×C4semidirect
94. #10587502C2×C4semidirect
176. C2×C4×D17C4⋊C4semidirect
180. #63227C42semidirect
289. #1434D17semidirect
416. C2×C42C4×D17semidirect
429. #95C4×D17semidirect
430. C17#13782semidirect

Maximal subgroups

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

subgroupindex
C2×#117626832
C2×C4×#632272
C2×#117626562
C2×#95×D172
#145270182
#145270152
#145270142
#145270122
#145270102
#145270052
C17×#14342
#1378217