Finite Group Databasebeta

← #16956644

Subgroups — #16956644

Order 4352.

1730 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762700C212yesmaximal
2C2×#11762673C211yesmaximal, characteristic, complemented
3C2×#11762668C211yesmaximal, characteristic
4C2×C4×#63227C211yesmaximal, characteristic
5#14527048C212yesmaximal, complemented
6#14527030C212yesmaximal
7#14527017C212yesmaximal, complemented
8#14527014C211yesmaximal, characteristic, complemented
9#14527008C211yesmaximal, characteristic, complemented
10#14527005C211yesmaximal, characteristic
11C4×C17×#95C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C22×C4⋊Dic17C2212yes—
14C22×#63225C2212yes—
15C22×#63225C2212yes—
16C22×#63227C2212yes—
17C22×#63227C2212yes—
18C22×#63227C2211yescharacteristic
19C22×#63227C2211yescharacteristic
20C22×C4×Dic17C2211yescharacteristic
21#11762700—28no—
22#11762673—24no—
23#11762668—24no—
24C4×#63227C418yescomplemented
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C2×#10587502C2211yescharacteristic
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2211yescharacteristic
34C22×C42×C17C2212yes—
35C22×C42×C17C2211yescharacteristic
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2212yes—
38C2×C17×#95C2211yescharacteristic
39C24×D17C2311yescharacteristic
40C23×C4×C17C2312yes—
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×Dic17C2312yes—
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C22×C4×D17—22no—
51C22×C4×D17—22no—
52C22×C4×D17—24no—
53C22×C4×D17—42no—
54C22×C4×D17C2×C414yes—
55C2×C42×C17—24no—
56C2×C42×C17—24no—
57C2×C42×C17—42no—
58C2×C42×C17—42no—
59C2×C42×C17C2×C414yes—
60C2×#63227—24no—
61C2×#63227—24no—
62C2×#63227—24no—
63C2×#63227—24no—
64C2×#63227—24no—
65C2×#63227—42no—
66C2×#63227—42no—
67C2×#63227C2×C414yes—
68C2×#63227C2×C414yes—
69C2×#63227D414yes—
70C2×#63227Q814yes—
71C2×C4⋊Dic17—24no—
72C2×C4⋊Dic17—42no—
73C2×C4⋊Dic17—42no—
74C2×#63225—24no—
75C2×#63225—24no—
76C2×#63225—24no—
77C2×#63225—24no—
78C2×#63225—42no—
79C2×#63225—42no—
80C2×C4×Dic17—22no—
81C2×C4×Dic17—22no—
82C2×C4×Dic17C2×C414yes—
83C17×#95—24no—
84C17×#95—22no—
85C17×#95D414yes—
86#10587502—24no—
87#10587502—22no—
88#10587502—24no—
89#10587502—22no—
90#10587502—24no—
91#10587502—24no—
92#10587502—24no—
93#10587502C2×C414yes—
94#10587502C2×C414yes—
95C24×C17C2411yescharacteristic
96C23×D17—21no—
97C23×D17—21no—
98C23×D17—22no—
99C23×D17—22no—
100C23×D17—22no—
101C23×D17—22no—
102C23×D17—21no—
103C23×D17—21no—
104C23×D17C22×C412yes—
105C23×D17C2×D412yes—
106C23×D17C2×Q812yes—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—21no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—21no—
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—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17C22×C412yes—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C2×D414yes—
130C22×C4×C17C2×D412yes—
131C22×C4×C17C2×D412yes—
132C22×C4×C17C2×Q812yes—
133C22×C4×C17#4114yes—
134C22×C4×C17#4114yes—
135C22×C4×C17#4112yes—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—22no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—22no—
145C22×Dic17—21no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
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×Dic17C22×C412yes—
162C22×Dic17C22×C412yes—
163C22×Dic17C22×C412yes—
164C22×Dic17C2×D412yes—
165C22×Dic17C2×Q812yes—
166C22×Dic17#4114yes—
167C2×C4×D17—28no—
168C2×C4×D17—28no—
169C2×C4×D17—44no—
170C2×C4×D17—48no—
171C2×C4×D17—48no—
172C42×C17—44no—
173C42×C17—48no—
174#63227—28no—
175#63227—48no—
176#63227—48no—
177#63227C4⋊C4116yescomplemented
178C4⋊Dic17—48no—
179#63225—48no—
180#63225—48no—
181C4×Dic17—28no—
182#13915—171nomaximal, Sylow
183C23×C17C23×C411yescharacteristic
184C23×C17C22×D411yescharacteristic
185C23×C17C22×D411yescharacteristic
186C23×C17C22×Q811yescharacteristic
187C23×C17C2×#4112yes—
188C23×C17C2×#4111yescharacteristic
189C23×C17C2×#4112yes—
190C23×C17C2×#4112yes—
191C23×C17H(5,𝔽2)12yes—
192C23×C17H(5,𝔽2)11yescharacteristic
193C23×C172−1+411yescharacteristic
194C22×D17—22no—
195C22×D17—22no—
196C22×D17—22no—
197C22×D17—22no—
198C22×D17—22no—
199C22×D17—22no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—22no—
203C22×D17—24no—
204C22×D17—22no—
205C22×D17—22no—
206C22×D17—24no—
207C22×D17—22no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—42no—
213C22×D17—42no—
214C22×D17—42no—
215C22×D17—42no—
216C22×D17C2×C4⋊C414yes—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—24no—
224C2×C4×C17—24no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
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×C17C2×C4⋊C414yes—
244C2×C4×C17#11714yes—
245C2×C4×C17C4×D414yes—
246C2×Dic17—22no—
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—24no—
252C2×Dic17—22no—
253C2×Dic17—22no—
254C2×Dic17—24no—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
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×Dic17C2×C4⋊C414yes—
276C2×Dic17#11714yes—
277C4×D17—416no—
278C2×#388—172no—
279C2×#382—171no—
280C2×#380—171no—
281C2×C4×C22⋊C4—171no—
282#1444—172no—
283#1439—172no—
284#1436—172no—
285C4⋊#95—171no—
286#1434—171no—
287C22⋊#95—171no—
288C4×#95D1711yescharacteristic, complemented
289C22×C17C22×C4⋊C411yescharacteristic
290C22×C17C2×#11711yescharacteristic
291C22×C17C2×#11711yescharacteristic
292C22×C17C2×C4×D411yescharacteristic
293C22×C17C4×#4111yescharacteristic
294C22×C17#51811yescharacteristic
295C22×C17#52011yescharacteristic
296C22×C17#52912yes—
297C22×C17#53212yes—
298C22×C17#53212yes—
299C22×C17#53312yes—
300C22×C17#53812yes—
301C22×C17#53912yes—
302C22×C17D4211yescharacteristic
303C22×C17#54612yes—
304C22×C17#54612yes—
305C22×C17#54711yescharacteristic
306C22×C17D4×Q811yescharacteristic
307C22×C17#55112yes—
308C22×C17#55212yes—
309C22×C17#55412yes—
310C22×C17#55411yescharacteristic
311C22×C17#55512yes—
312C2×D17—24no—
313C2×D17—24no—
314C2×D17—24no—
315C2×D17—42no—
316C2×D17—44no—
317C2×D17—42no—
318C2×D17—42no—
319C2×D17—42no—
320C2×D17—44no—
321C2×D17—44no—
322C2×D17—44no—
323C4×C17—24no—
324C4×C17—24no—
325C4×C17—24no—
326C4×C17—44no—
327C4×C17—44no—
328Dic17—24no—
329Dic17—24no—
330Dic17—42no—
331Dic17—82no—
332Dic17—82no—
333C24×C4—171no—
334C22×C4⋊C4—172no—
335C22×C4⋊C4—172no—
336C22×C4⋊C4—172no—
337C22×C22⋊C4—171no—
338C22×C22⋊C4—171no—
339C22×C22⋊C4—172no—
340C22×C22⋊C4—172no—
341C22×C42—171no—
342C22×C42C2×D1712yes—
343C22×C42C2×D1711yescharacteristic
344#388—348no—
345#382—344no—
346#380—344no—
347C4×C22⋊C4—178no—
348C2×#95—171no—
349C2×#95—171no—
350C2×#95—171no—
351C2×#95—172no—
352C2×#95—171no—
353C2×#95—172no—
354C2×#95—172no—
355C2×#95—172no—
356C2×#95—172no—
357C2×#95C2×D1711yescharacteristic
358C2×#95C2×D1712yes—
359C2×#95C2×D1711yescharacteristic
360C2×C17#228511yescharacteristic
361C2×C17#229311yescharacteristic
362C2×C17#231811yescharacteristic
363C2×C17#234211yescharacteristic
364C2×C17#234611yescharacteristic
365C2×C17D4×C4⋊C411yescharacteristic
366C2×C17#234911yescharacteristic
367C2×C17#255312yes—
368C2×C17#255512yes—
369C2×C17#255612yes—
370C2×C17#257412yes—
371D17—44no—
372C25—171no—
373C23×C4—171no—
374C23×C4—171no—
375C23×C4—171no—
376C23×C4—174no—
377C23×C4—172no—
378C23×C4—172no—
379C23×C4—344no—
380C23×C4—342no—
381C23×C4—342no—
382C23×C4—682no—
383C23×C4C22×D1712yes—
384C23×C4C22×D1712yes—
385C23×C4C22×D1711yescharacteristic
386C23×C4C22×D1711yescharacteristic
387C23×C4C22×D1711yescharacteristic
388C2×C4⋊C4—344no—
389C2×C4⋊C4—344no—
390C2×C4⋊C4—344no—
391C2×C4⋊C4—344no—
392C2×C4⋊C4—344no—
393C2×C4⋊C4—682no—
394C2×C4⋊C4—682no—
395C2×C4⋊C4—682no—
396C2×C4⋊C4—682no—
397C2×C22⋊C4—174no—
398C2×C22⋊C4—174no—
399C2×C22⋊C4—174no—
400C2×C22⋊C4—174no—
401C2×C22⋊C4—344no—
402C2×C22⋊C4—344no—
403C2×C22⋊C4—344no—
404C2×C22⋊C4—344no—
405C2×C22⋊C4—344no—
406C2×C22⋊C4—682no—
407C2×C22⋊C4—682no—
408C2×C42—24no—
409C2×C42—24no—
410C2×C42—42no—
411C2×C42—42no—
412C2×C42—174no—
413C2×C42—342no—
414C2×C42—342no—
415C2×C42C4×D1714yes—
416#95—24no—
417#95—22no—
418#95—174no—
419#95—174no—
420#95—344no—
421#95—344no—
422#95—342no—
423#95—344no—
424#95—342no—
425#95—344no—
426#95—344no—
427#95D6814yes—
428C17#1391511yescharacteristic, complemented, Sylow
429C24—172no—
430C24—172no—
431C24—172no—
432C24—341no—
433C24—342no—
434C24—342no—
435C24—341no—
436C24—342no—
437C24—342no—
438C24—341no—
439C24—341no—
440C24C23×D1711yescharacteristic
441C22×C4—22no—
442C22×C4—22no—
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—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—172no—
461C22×C4—172no—
462C22×C4—172no—
463C22×C4—172no—
464C22×C4—174no—
465C22×C4—172no—
466C22×C4—348no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4—342no—
472C22×C4—341no—
473C22×C4—341no—
474C22×C4—341no—
475C22×C4—342no—
476C22×C4—341no—
477C22×C4—342no—
478C22×C4—348no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—684no—
494C22×C4—688no—
495C22×C4—688no—
496C22×C4C2×C4×D1712yes—
497C22×C4C2×C4×D1712yes—
498C22×C4C2×C4×D1712yes—
499C22×C4C2×D6814yes—
500C22×C4C2×D6812yes—
501C22×C4#6325214yes—
502C22×C4#6325214yes—
503C22×C4#6325212yes—
504C22×C4D4×D1712yes—
505C22×C4Q8×D1712yes—
506C4⋊C4—688no—
507C4⋊C4—688no—
508C4⋊C4—688no—
509C22⋊C4—1716no—
510C22⋊C4—348no—
511C22⋊C4—688no—
512C22⋊C4—688no—
513C42—44no—
514C42—48no—
515C42—348no—
516C23—174no—
517C23—342no—
518C23—342no—
519C23—344no—
520C23—342no—
521C23—342no—
522C23—342no—
523C23—342no—
524C23—342no—
525C23—342no—
526C23—342no—
527C23—344no—
528C23—344no—
529C23—342no—
530C23—344no—
531C23—682no—
532C23—682no—
533C23—682no—
534C23—682no—
535C23—682no—
536C23—682no—
537C23—682no—
538C23—682no—
539C23C22×C4×D1711yescharacteristic
540C23C22×D6811yescharacteristic
541C23C2×#6325211yescharacteristic
542C23C2×#6325212yes—
543C23C2×#6325212yes—
544C23C2×D4×D1711yescharacteristic
545C23#1058769112yes—
546C23C2×Q8×D1711yescharacteristic
547C23#41×D1712yes—
548C23#1058769611yescharacteristic
549C23#1058769711yescharacteristic
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—174no—
577C2×C4—174no—
578C2×C4—342no—
579C2×C4—342no—
580C2×C4—344no—
581C2×C4—342no—
582C2×C4—342no—
583C2×C4—342no—
584C2×C4—342no—
585C2×C4—342no—
586C2×C4—342no—
587C2×C4—344no—
588C2×C4—342no—
589C2×C4—342no—
590C2×C4—342no—
591C2×C4—6816no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—682no—
599C2×C4—682no—
600C2×C4—682no—
601C2×C4—682no—
602C2×C4—682no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4—682no—
606C2×C4—682no—
607C2×C4—682no—
608C2×C4#1058755814yes—
609C2×C4C4×D6814yes—
610C2×C4C4⋊C4×D1714yes—
611C22—344no—
612C22—344no—
613C22—344no—
614C22—682no—
615C22—682no—
616C22—682no—
617C22—684no—
618C22—682no—
619C22—684no—
620C22—684no—
621C22—684no—
622C22C2×C4⋊C4×D1711yescharacteristic
623C22#117×D1711yescharacteristic
624C22C2×#1058755811yescharacteristic
625C22C2×C4×D6811yescharacteristic
626C22C4×#6325211yescharacteristic
627C22#1176328211yescharacteristic
628C22#1176328911yescharacteristic
629C22#1176334312yes—
630C22#1176335612yes—
631C22#1176335912yes—
632C22#1176336312yes—
633C22#1176339212yes—
634C22#1176340112yes—
635C22D4×D6811yescharacteristic
636C22#1176344912yes—
637C22#1176345412yes—
638C22#1176346411yescharacteristic
639C22Q8×D6811yescharacteristic
640C22#1176348812yes—
641C22#1176349212yes—
642C22#1176351412yes—
643C22#1176351911yescharacteristic
644C22#1176352212yes—
645C4—24no—
646C4—24no—
647C4—24no—
648C4—44no—
649C4—44no—
650C4—344no—
651C4—344no—
652C4—682no—
653C4—1362no—
654C4—1362no—
655C2—684no—
656C2#1453118811yescharacteristic
657C2#1453124811yescharacteristic
658C2#1453141111yescharacteristic
659C2#1453163611yescharacteristic
660C2#1453167011yescharacteristic
661C2C4⋊C4×D6811yescharacteristic
662C2#1453169111yescharacteristic
663C2#1453382912yes—
664C2#1453385312yes—
665C2#1453386212yes—
666C2#1453405212yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC4×C17×#95C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC4×#95D17
hypocenterC17#13915
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
2. C2×#11762673C2semidirect
5. #14527048C2semidirect
7. #14527017C2semidirect
8. #14527014C2semidirect
9. #14527008C2semidirect
11. C4×C17×#95C2semidirect
24. C4×#63227C4semidirect
177. #63227C4⋊C4semidirect
288. C4×#95D17semidirect
428. C17#13915semidirect

Maximal subgroups

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

subgroupindex
C2×#117627002
C2×#117626732
C2×#117626682
C2×C4×#632272
#145270482
#145270302
#145270172
#145270142
#145270082
#145270052
C4×C17×#952
#1391517