Finite Group Databasebeta

← #16961338

Subgroups — #16961338

Order 4352.

1590 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762744C211yesmaximal, characteristic
2C2×#11762704C212yesmaximal, complemented
3C2×#11762700C211yesmaximal, characteristic
4C2×#11762699C212yesmaximal
5#14527052C211yesmaximal, characteristic, complemented
6#14527050C212yesmaximal, complemented
7#14527047C211yesmaximal, characteristic, complemented
8#14527028C212yesmaximal
9#14527026C211yesmaximal, characteristic
10C4×#10587502C211yesmaximal, characteristic, complemented
11C2×C17×#389C211yesmaximal, characteristic, complemented
12C22×#63225C2211yescharacteristic
13C22×#63225C2212yes—
14C22×#63225C2212yes—
15C22×#63227C2211yescharacteristic
16C22×#63227C2211yescharacteristic
17C22×#63227C2212yes—
18C22×#63227C2212yes—
19C22×#63227C2211yescharacteristic
20C22×C4×Dic17C2212yes—
21#11762744C418yescomplemented
22#11762704—28no—
23#11762700—24no—
24#11762699—28no—
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2212yes—
33C2×#10587502C2211yescharacteristic
34C22×C4⋊C4×C17C2212yes—
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C22×C42×C17C2211yescharacteristic
38C17×#389—24no—
39C2×C17×#95C2212yes—
40C24×D17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2312yes—
50C23×Dic17C2312yes—
51C2×C4⋊C4×C17—22no—
52C2×C4⋊C4×C17—22no—
53C2×C4⋊C4×C17—24no—
54C2×C4⋊C4×C17—24no—
55C2×C4⋊C4×C17—22no—
56C2×C4⋊C4×C17—24no—
57C2×C4⋊C4×C17—42no—
58C2×C4⋊C4×C17C2×C414yes—
59C2×C42×C17—42no—
60C2×C42×C17D414yes—
61C2×#63227—22no—
62C2×#63227—24no—
63C2×#63227—22no—
64C2×#63227—22no—
65C2×#63227—22no—
66C2×#63227—24no—
67C2×#63227—24no—
68C2×#63227—24no—
69C2×#63227—42no—
70C2×#63227—42no—
71C2×#63227—42no—
72C2×#63227C2×C414yes—
73C2×#63227C2×C418yes—
74C2×#63225—22no—
75C2×#63225—24no—
76C2×#63225—24no—
77C2×#63225—24no—
78C2×#63225—24no—
79C2×#63225—42no—
80C2×#63225—42no—
81C2×#63225C2×C418yes—
82C2×C4×Dic17—42no—
83C2×C4×Dic17—42no—
84C2×C4×Dic17D418yes—
85C17×#95—24no—
86#10587502—22no—
87#10587502—22no—
88#10587502—22no—
89#10587502—24no—
90#10587502—24no—
91#10587502—24no—
92#10587502—24no—
93#10587502—24no—
94#10587502C2×C414yes—
95C24×C17C2411yescharacteristic
96C23×D17—22no—
97C23×D17—21no—
98C23×D17—21no—
99C23×D17—22no—
100C23×D17—22no—
101C23×D17—21no—
102C23×D17—21no—
103C23×D17—22no—
104C23×D17—21no—
105C23×D17—21no—
106C23×D17C22×C412yes—
107C22×C4×C17—21no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
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—21no—
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—22no—
128C22×C4×C17—22no—
129C22×C4×C17C22×C414yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C2×D414yes—
132C22×C4×C17C2×D412yes—
133C22×C4×C17C2×D412yes—
134C22×C4×C17C2×D412yes—
135C22×C4×C17#4112yes—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—21no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—21no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—22no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—22no—
156C22×Dic17—21no—
157C22×Dic17—22no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17C22×C412yes—
161C22×Dic17C22×C414yes—
162C22×Dic17C2×D414yes—
163C22×Dic17C2×D414yes—
164C22×Dic17#4114yes—
165C4⋊C4×C17—28no—
166C4⋊C4×C17—44no—
167C4⋊C4×C17—48no—
168C42×C17—44no—
169#63227—28no—
170#63227—216no—
171#63227—48no—
172#63227—44no—
173#63227—44no—
174#63225—216no—
175#63225—48no—
176#63225—44no—
177C4×Dic17—48no—
178#14322—171nomaximal, Sylow
179C23×C17C23×C411yescharacteristic
180C23×C17C22×D412yes—
181C23×C17C22×D411yescharacteristic
182C23×C17C2×#4111yescharacteristic
183C23×C17C2×#4112yes—
184C23×C17H(5,𝔽2)12yes—
185C23×C17H(5,𝔽2)11yescharacteristic
186C23×C172−1+411yescharacteristic
187C23×C172−1+412yes—
188C23×C172−1+411yescharacteristic
189C23×C172−1+411yescharacteristic
190C22×D17—24no—
191C22×D17—22no—
192C22×D17—22no—
193C22×D17—24no—
194C22×D17—22no—
195C22×D17—41no—
196C22×D17—41no—
197C22×D17—42no—
198C22×D17—42no—
199C22×D17—41no—
200C22×D17—41no—
201C22×D17—41no—
202C22×D17—42no—
203C22×D17—42no—
204C22×D17—42no—
205C22×D17—41no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—41no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—41no—
213C2×C4×C17—24no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—24no—
217C2×C4×C17—22no—
218C2×C4×C17—24no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—24no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17C4×D414yes—
241C2×C4×C17#12714yes—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—22no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—42no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—41no—
270C2×Dic17C4×D418yes—
271C2×#399—171no—
272C2×#389—172no—
273C2×#389D1711yescharacteristic, complemented
274C2×#388—171no—
275C2×#388—172no—
276#1445—172no—
277#1445—171no—
278#1444—171no—
279#1439—172no—
280#1439—171no—
281C4×#95—171no—
282C22×C17C2×C4×D412yes—
283C22×C17C2×C4×D411yescharacteristic
284C22×C17#51911yescharacteristic
285C22×C17#52012yes—
286C22×C17#52011yescharacteristic
287C22×C17C2×#12711yescharacteristic
288C22×C17#53211yescharacteristic
289C22×C17#53212yes—
290C22×C17#53412yes—
291C22×C17#53411yescharacteristic
292C22×C17#53611yescharacteristic
293C22×C17#53611yescharacteristic
294C22×C17#53612yes—
295C22×C17#53612yes—
296C22×C17#53712yes—
297C22×C17#53711yescharacteristic
298C22×C17#53912yes—
299C22×C17#53912yes—
300C22×C17#53912yes—
301C22×C17#54111yescharacteristic
302C22×C17#54112yes—
303C22×C17#54212yes—
304C22×C17#54211yescharacteristic
305C2×D17—42no—
306C2×D17—44no—
307C2×D17—42no—
308C2×D17—42no—
309C2×D17—44no—
310C2×D17—81no—
311C2×D17—81no—
312C2×D17—81no—
313C2×D17—82no—
314C2×D17—81no—
315C2×D17—82no—
316C4×C17—24no—
317C4×C17—24no—
318C4×C17—44no—
319C4×C17—42no—
320C4×C17—82no—
321Dic17—28no—
322Dic17—42no—
323Dic17—44no—
324Dic17—81no—
325Dic17—81no—
326C22×C4⋊C4—172no—
327C22×C4⋊C4—171no—
328C22×C4⋊C4—172no—
329C22×C4⋊C4C2×D1712yes—
330C22×C4⋊C4C2×D1711yescharacteristic
331C22×C4⋊C4C2×D1711yescharacteristic
332C22×C22⋊C4—172no—
333C22×C22⋊C4—171no—
334C22×C22⋊C4—171no—
335C22×C22⋊C4—172no—
336C22×C22⋊C4—171no—
337C22×C42—172no—
338C22×C42C2×D1711yescharacteristic
339#399—178no—
340#389—24no—
341#389—348no—
342#388—348no—
343#388—344no—
344C2×#95—171no—
345C2×#95—172no—
346C2×#95—171no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—171no—
351C2×#95—172no—
352C2×#95—171no—
353C2×#95C2×D1712yes—
354C2×C17C4×#12711yescharacteristic
355C2×C17#232212yes—
356C2×C17#232211yescharacteristic
357C2×C17#232311yescharacteristic
358C2×C17#232312yes—
359C2×C17#265611yescharacteristic
360C2×C17#265811yescharacteristic
361C2×C17#266311yescharacteristic
362C2×C17#266312yes—
363C2×C17#266411yescharacteristic
364C2×C17#266412yes—
365D17—82no—
366C25—171no—
367C23×C4—172no—
368C23×C4—171no—
369C23×C4—172no—
370C23×C4—171no—
371C23×C4—171no—
372C23×C4C22×D1712yes—
373C23×C4C22×D1712yes—
374C23×C4C22×D1711yescharacteristic
375C23×C4C22×D1711yescharacteristic
376C23×C4C22×D1711yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4—178no—
385C2×C4⋊C4—344no—
386C2×C4⋊C4—344no—
387C2×C4⋊C4—344no—
388C2×C4⋊C4—344no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—682no—
391C2×C4⋊C4—682no—
392C2×C4⋊C4C4×D1714yes—
393C2×C22⋊C4—178no—
394C2×C22⋊C4—174no—
395C2×C22⋊C4—344no—
396C2×C22⋊C4—344no—
397C2×C22⋊C4—342no—
398C2×C22⋊C4—342no—
399C2×C22⋊C4—344no—
400C2×C22⋊C4—344no—
401C2×C22⋊C4—342no—
402C2×C22⋊C4—342no—
403C2×C22⋊C4—682no—
404C2×C22⋊C4—682no—
405C2×C22⋊C4—682no—
406C2×C42—42no—
407C2×C42—178no—
408C2×C42—682no—
409C2×C42—682no—
410C2×C42C22⋊D1714yes—
411#95—24no—
412#95—174no—
413#95—344no—
414#95—344no—
415#95—344no—
416#95—344no—
417#95—344no—
418#95—342no—
419#95—342no—
420#95—342no—
421C17#1432211yescharacteristic, complemented, Sylow
422C24—172no—
423C24—341no—
424C24—342no—
425C24—342no—
426C24—341no—
427C24—342no—
428C24—341no—
429C24—342no—
430C24—341no—
431C24—341no—
432C24—341no—
433C24C23×D1711yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
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—174no—
457C22×C4—174no—
458C22×C4—174no—
459C22×C4—172no—
460C22×C4—174no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—341no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—341no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—341no—
479C22×C4—342no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4C2×C4×D1714yes—
486C22×C4C2×C4×D1712yes—
487C22×C4#6325212yes—
488C22×C4D4×D1714yes—
489C22×C4C2×C22⋊D1712yes—
490C22×C4C2×C22⋊D1712yes—
491C22×C4C2×C22⋊D1712yes—
492C4⋊C4—28no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—3416no—
496C4⋊C4—688no—
497C4⋊C4—684no—
498C22⋊C4—3416no—
499C22⋊C4—348no—
500C22⋊C4—684no—
501C22⋊C4—688no—
502C22⋊C4—684no—
503C42—44no—
504C42—688no—
505C23—344no—
506C23—342no—
507C23—342no—
508C23—344no—
509C23—342no—
510C23—681no—
511C23—681no—
512C23—682no—
513C23—682no—
514C23—681no—
515C23—681no—
516C23—681no—
517C23—682no—
518C23—682no—
519C23—682no—
520C23—681no—
521C23—682no—
522C23—682no—
523C23—682no—
524C23—681no—
525C23—682no—
526C23—682no—
527C23—681no—
528C23C22×C4×D1711yescharacteristic
529C23C2×#6325211yescharacteristic
530C23C2×D4×D1712yes—
531C23#1058769112yes—
532C23#1058769411yescharacteristic
533C23#1058769412yes—
534C23#1058769411yescharacteristic
535C23#41×D1712yes—
536C23#1058769611yescharacteristic
537C23#1058769711yescharacteristic
538C23C22×C22⋊D1711yescharacteristic
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—178no—
567C2×C4—344no—
568C2×C4—344no—
569C2×C4—344no—
570C2×C4—344no—
571C2×C4—344no—
572C2×C4—344no—
573C2×C4—344no—
574C2×C4—342no—
575C2×C4—344no—
576C2×C4—344no—
577C2×C4—344no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—682no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—681no—
595C2×C4C4×C22⋊D1714yes—
596C2×C4#1058762614yes—
597C22—682no—
598C22—684no—
599C22—682no—
600C22—682no—
601C22—684no—
602C22—1361no—
603C22—1362no—
604C22—1361no—
605C22—1362no—
606C22—1361no—
607C22—1361no—
608C22C4×D4×D1712yes—
609C22C2×C4×C22⋊D1711yescharacteristic
610C22#1176328511yescharacteristic
611C22#1176329012yes—
612C22#1176329111yescharacteristic
613C22C2×#1058762611yescharacteristic
614C22#1176335511yescharacteristic
615C22#1176335612yes—
616C22#1176336712yes—
617C22#1176336811yescharacteristic
618C22#1176338311yescharacteristic
619C22#1176338311yescharacteristic
620C22#1176338412yes—
621C22#1176338412yes—
622C22#1176338612yes—
623C22#1176338711yescharacteristic
624C22#1176340512yes—
625C22#1176340612yes—
626C22#1176340712yes—
627C22#1176341711yescharacteristic
628C22#1176341812yes—
629C22#1176342912yes—
630C22#1176343111yescharacteristic
631C4—24no—
632C4—24no—
633C4—44no—
634C4—42no—
635C4—82no—
636C4—348no—
637C4—682no—
638C4—684no—
639C4—1361no—
640C4—1361no—
641C2—1362no—
642C2C4×#1058762611yescharacteristic
643C2#1453144812yes—
644C2#1453145311yescharacteristic
645C2#1453146011yescharacteristic
646C2#1453146112yes—
647C2#1453488811yescharacteristic
648C2#1453489611yescharacteristic
649C2#1453493511yescharacteristic
650C2#1453493812yes—
651C2#1453494511yescharacteristic
652C2#1453494712yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#389C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC2×#389D17
hypocenterC17#14322
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×#11762704C2semidirect
5. #14527052C2semidirect
6. #14527050C2semidirect
7. #14527047C2semidirect
10. C4×#10587502C2semidirect
11. C2×C17×#389C2semidirect
21. #11762744C4semidirect
273. C2×#389D17semidirect
421. C17#14322semidirect

Maximal subgroups

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

subgroupindex
C2×#117627442
C2×#117627042
C2×#117627002
C2×#117626992
#145270522
#145270502
#145270472
#145270282
#145270262
C4×#105875022
C2×C17×#3892
#1432217