Finite Group Databasebeta

← #17969070

Subgroups — #17969070

Order 4352.

990 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763843C211yesmaximal, characteristic
2C2×#11763839C211yesmaximal, characteristic
3#14544583C211yesmaximal, characteristic, complemented
4#14544576C211yesmaximal, characteristic, complemented
5#14544572C211yesmaximal, characteristic, complemented
6#14544525C211yesmaximal, characteristic, complemented
7#14544525C211yesmaximal, characteristic, complemented
8#14544525C211yesmaximal, characteristic, complemented
9#14544518C211yesmaximal, characteristic, complemented
10#14544513C211yesmaximal, characteristic, complemented
11C4×#10587543C211yesmaximal, characteristic, complemented
12#14544503C211yesmaximal, characteristic, complemented
13#14544451C211yesmaximal, characteristic, complemented
14#14544448C211yesmaximal, characteristic, complemented
15#14531222C211yesmaximal, characteristic
16C22×#63243C2211yescharacteristic
17C22×#63243C2211yescharacteristic
18C22×#63243C2211yescharacteristic
19C22×#63242C2211yescharacteristic
20#11763843C418yescomplemented
21#11763839C418yescomplemented
22C2×#10587544C2211yescharacteristic
23C2×C4⋊#3624C2211yescharacteristic
24C2×#10587543C2211yescharacteristic
25C2×#10587543C2211yescharacteristic
26C2×#10587543C2211yescharacteristic
27C2×#10587543C2211yescharacteristic
28C2×#10587542C2211yescharacteristic
29C2×#10587542C2211yescharacteristic
30C2×#10587542C2211yescharacteristic
31C2×C4×#3624C2211yescharacteristic
32C2×C4×#3624C2211yescharacteristic
33C2×C4×#3624C2211yescharacteristic
34C2×C4×#3624C2211yescharacteristic
35C2×C4×#3624C2211yescharacteristic
36#11763675C2211yescharacteristic
37#11763675C2211yescharacteristic
38#11763675C2211yescharacteristic
39#11763675C2211yescharacteristic
40#11763675C2211yescharacteristic
41#11763675C2211yescharacteristic
42#11763675C2211yescharacteristic
43#11763675C2211yescharacteristic
44#11763675C2211yescharacteristic
45#11763675C2211yescharacteristic
46C2×#10587579C2211yescharacteristic
47C2×#10587578C2211yescharacteristic
48C2×C42×D17C2211yescharacteristic
49#11762700C411yescharacteristic
50#11762698C411yescharacteristic
51#11762695C411yescharacteristic
52#11762687C411yescharacteristic
53#11762687C411yescharacteristic
54#11762683C411yescharacteristic
55#11762680C411yescharacteristic
56#11762660C2211yescharacteristic
57#11762656C2211yescharacteristic
58#11762656C2211yescharacteristic
59#11762654C2211yescharacteristic
60C17×#387C411yescharacteristic
61C22×D68C2×C411yescharacteristic
62C22×C4×D17C2×C411yescharacteristic
63C22×C4×D17C2×C411yescharacteristic
64C22×C4×D17C2311yescharacteristic
65C22×C4×D17C2311yescharacteristic
66C22×#3624C2311yescharacteristic
67C22×#3624C2311yescharacteristic
68C22×#3624C2311yescharacteristic
69C22×#3624C2311yescharacteristic
70C22×#3624C2311yescharacteristic
71C22×#3624C2311yescharacteristic
72C22×#3624C2311yescharacteristic
73C22×#3624C2311yescharacteristic
74C2×#63243—41no—
75C2×#63243—41no—
76C2×#63243—41no—
77C2×#63243—41no—
78C2×#63243—41no—
79C2×#63243—41no—
80C2×#63243C2×C414yes—
81C2×#63243C2×C414yes—
82C2×#63243C2×C414yescomplemented
83C2×#63242—41no—
84C2×#63242—41no—
85C2×#63242C2×C414yescomplemented
86C2×C4⋊C4×C17C2×C411yescharacteristic
87C2×C4⋊C4×C17C2×C411yescharacteristic
88C2×C42×C17C2×C411yescharacteristic
89C2×#63227C2×C411yescharacteristic
90C2×#63227C2×C411yescharacteristic
91C2×#63227C2×C411yescharacteristic
92C2×#63227C2×C411yescharacteristic
93C2×#63227C2×C411yescharacteristic
94C2×#63227C2×C411yescharacteristic
95C2×#63227C2×C411yescharacteristic
96C2×#63227C2×C411yescharacteristic
97C2×C4⋊Dic17C2×C411yescharacteristic
98C2×#63225C2×C411yescharacteristic
99C2×#63225C2×C411yescharacteristic
100C2×C4×Dic17C2×C411yescharacteristic
101C2×C4×Dic17C2311yescharacteristic
102C2×C4×Dic17C2311yescharacteristic
103C2×C4×Dic17C2311yescharacteristic
104C2×C4×Dic17C2311yescharacteristic
105C2×C4×Dic17C2311yescharacteristic
106#10587579C2×C414yes—
107#10587579C2×C414yes—
108#10587578—41no—
109#10587578—41no—
110C42×D17—41no—
111C42×D17—41no—
112#10587544C2×C414yescomplemented
113#10587543C2×C414yes—
114#10587543C2×C414yes—
115#10587543C2×C414yescomplemented
116#10587543C2×C414yescomplemented
117#10587542—41no—
118#10587542—41no—
119#10587542C2×C414yescomplemented
120C4⋊#3624C2×C414yescomplemented
121C4×#3624—41no—
122C4×#3624—41no—
123C4×#3624—41no—
124C4×#3624—41no—
125C4×#3624C2×C414yescomplemented
126C17×#95C2×C411yescharacteristic
127C17×#95C2×C411yescharacteristic
128C17×#95C2×C411yescharacteristic
129C17×#95C2×C411yescharacteristic
130#10587502C2×C411yescharacteristic
131#10587502C2×C411yescharacteristic
132#10587502C2×C411yescharacteristic
133#10587502C2×C411yescharacteristic
134#10587502C2×C411yescharacteristic
135#10587502C2×C411yescharacteristic
136C23×D17C22×C411yescharacteristic
137C23×D17C22×C411yescharacteristic
138C22×C4×C17C22×C411yescharacteristic
139C22×C4×C17C22×C411yescharacteristic
140C22×C4×C17C22×C411yescharacteristic
141C22×C4×C17C22×C411yescharacteristic
142C22×C4×C17C22×C411yescharacteristic
143C22×C4×C17C22×C411yescharacteristic
144C22×C4×C17C22×C411yescharacteristic
145C22×Dic17C22×C411yescharacteristic
146C22×Dic17C22×C411yescharacteristic
147C22×Dic17C22×C411yescharacteristic
148C22×Dic17C22×C411yescharacteristic
149C22×Dic17C22×C411yescharacteristic
150C22×Dic17C2411yescharacteristic
151C2×D68—41no—
152C2×D68—41no—
153C2×D68C4214yes—
154C2×C4×D17—21no—
155C2×C4×D17—21no—
156C2×C4×D17—21no—
157C2×C4×D17—21no—
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—21no—
162C2×C4×D17—22no—
163C2×C4×D17—22no—
164C2×C4×D17—41no—
165C2×C4×D17—41no—
166C2×C4×D17—41no—
167C2×C4×D17—41no—
168C2×C4×D17C22×C412yes—
169C2×C4×D17C22×C412yes—
170C2×C4×D17C22×C412yes—
171C2×C4×D17C22×C412yes—
172C2×#3624—21no—
173C2×#3624—21no—
174C2×#3624—21no—
175C2×#3624—21no—
176C2×#3624—21no—
177C2×#3624—21no—
178C2×#3624—21no—
179C2×#3624—21no—
180C2×#3624—21no—
181C2×#3624—21no—
182C2×#3624—21no—
183C2×#3624—21no—
184C2×#3624—21no—
185C2×#3624—21no—
186C2×#3624—21no—
187C2×#3624—21no—
188C2×#3624C22×C412yes—
189C2×#3624C22×C412yes—
190C2×#3624C22×C412yes—
191C2×#3624C22×C412yes—
192C2×#3624C22×C412yes—
193C2×#3624C22×C412yes—
194C2×#3624C22×C412yes—
195C2×#3624C22×C412yes—
196#63243—44no—
197#63243—44no—
198#63243—44no—
199#63242—44no—
200C4⋊C4×C17—41no—
201C4⋊C4×C17C4214yes—
202C42×C17—41no—
203#63227—22no—
204#63227—22no—
205#63227—41no—
206#63227—41no—
207#63227—41no—
208#63227—41no—
209#63227C4214yes—
210#63227C4214yes—
211C4⋊Dic17—41no—
212#63225—22no—
213#63225—22no—
214C4×Dic17—21no—
215C4×Dic17—21no—
216C4×Dic17—21no—
217C4×Dic17—21no—
218C4×Dic17—21no—
219C4×Dic17—21no—
220C4×Dic17—21no—
221C4×Dic17—21no—
222C4×Dic17—41no—
223C4×Dic17C22×C412yes—
224C4×Dic17C22×C412yes—
225#20561—171nomaximal, Sylow
226C23×C17C23×C411yescharacteristic
227C22×D17—21no—
228C22×D17—21no—
229C22×D17—21no—
230C22×D17—21no—
231C22×D17—21no—
232C22×D17—21no—
233C22×D17—21no—
234C22×D17—21no—
235C22×D17—21no—
236C22×D17—21no—
237C22×D17—21no—
238C22×D17—21no—
239C22×D17C2×C4212yes—
240C22×D17C2×C4212yes—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—21no—
258C2×C4×C17C2×C4212yes—
259C2×C4×C17C2×C4212yes—
260C2×C4×C17C2×C4212yes—
261C2×C4×C17#13112yes—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17C2×C4212yes—
276C2×Dic17#13112yes—
277C2×Dic17C23×C411yescharacteristic
278C2×Dic17C23×C411yescharacteristic
279C2×Dic17H(5,𝔽2)11yescharacteristic
280C2×Dic17H(5,𝔽2)11yescharacteristic
281C2×Dic17H(5,𝔽2)11yescharacteristic
282C2×Dic172−1+411yescharacteristic
283D68—44no—
284C4×D17—22no—
285C4×D17—22no—
286C4×D17—22no—
287C4×D17—22no—
288C4×D17—22no—
289C4×D17—22no—
290C4×D17—22no—
291C4×D17—22no—
292C4×D17—44no—
293C4×D17—44no—
294#3624—22no—
295#3624—22no—
296#3624—22no—
297#3624—22no—
298#3624—22no—
299#3624—22no—
300#3624—22no—
301#3624—22no—
302C2×#436—171no—
303C2×#435—171no—
304#2290—171no—
305#1771—171no—
306C8×C4⋊C4—171no—
307#1768—171no—
308#1753—171no—
309#1753—171no—
310#1753—171no—
311C8×C22⋊C4—171no—
312#1750—171no—
313C4×C22⋊C8—171no—
314#1746—171no—
315#1729—171no—
316#1728—171no—
317C22×C17C22×C4211yescharacteristic
318C22×C17#51811yescharacteristic
319C22×C17#52011yescharacteristic
320C22×C17C2×#13111yescharacteristic
321C22×C17C2×#13111yescharacteristic
322C22×C17#56811yescharacteristic
323C22×C17#56811yescharacteristic
324C2×D17—22no—
325C2×D17—22no—
326C2×D17—22no—
327C2×D17—22no—
328C2×D17—22no—
329C2×D17—22no—
330C2×D17—41no—
331C2×D17—41no—
332C2×D17—41no—
333C2×D17—41no—
334C2×D17—41no—
335C2×D17—41no—
336C2×D17—41no—
337C2×D17—41no—
338C4×C17—22no—
339C4×C17—22no—
340C4×C17—22no—
341C4×C17—22no—
342C4×C17—41no—
343C4×C17—41no—
344C4×C17—41no—
345Dic17—22no—
346Dic17—22no—
347Dic17—41no—
348Dic17—41no—
349Dic17—41no—
350Dic17#51811yescharacteristic
351Dic17#51811yescharacteristic
352Dic17#52011yescharacteristic
353Dic17#52011yescharacteristic
354C22×M16—171no—
355C22×M16—171no—
356C22×M16—171no—
357C23×C8—171no—
358C2×C4×D4—171no—
359C2×#117—171no—
360C22×C42—171no—
361#436—178no—
362#435—178no—
363C2×C4⋊C8—171no—
364C2×C4⋊C8—171no—
365C2×C22⋊C8—171no—
366C2×C22⋊C8—171no—
367C2×C22⋊C8—171no—
368C2×C22⋊C8—171no—
369C2×#97—171no—
370C2×#97—171no—
371C2×#97—171no—
372C2×C4×C8—171no—
373C2×C4×C8—171no—
374C2×C4×C8—171no—
375C2×C4×C8—171no—
376C2×C4×C8—171no—
377#388—171no—
378#388—171no—
379#387—171no—
380#387#59511yescharacteristic
381#386—171no—
382#386—171no—
383#385—171no—
384#385—171no—
385C4×C4⋊C4—171no—
386C4×C22⋊C4—171no—
387C4×C22⋊C4—171no—
388#376—171no—
389#336—171no—
390#336—171no—
391#336—171no—
392#336—171no—
393#336—171no—
394#336—171no—
395#336—171no—
396#336—171no—
397#336—171no—
398#336—171no—
399C2×C17#227511yescharacteristic
400C2×C17#287311yescharacteristic
401C2×C17C4×#13111yescharacteristic
402C2×C17#297611yescharacteristic
403C2×C17#297611yescharacteristic
404C2×C17#299311yescharacteristic
405C2×C17#299311yescharacteristic
406D17—42no—
407D17—42no—
408C22×D4—171no—
409C23×C4—171no—
410C23×C4—171no—
411C23×C4—171no—
412C23×C4—171no—
413C2×M16—174no—
414C2×M16—174no—
415C2×M16—174no—
416C2×M16—681no—
417C2×M16—681no—
418C2×M16—681no—
419C2×M16—681no—
420C2×M16—681no—
421C2×M16—681no—
422C22×C8—171no—
423C22×C8—171no—
424C22×C8—171no—
425C22×C8—171no—
426C22×C8—171no—
427C22×C8—171no—
428C22×C8—171no—
429C22×C8—171no—
430C22×C8—174no—
431C22×C8—681no—
432C22×C8—681no—
433C4×D4—174no—
434C4×D4—174no—
435#117—681no—
436#117—681no—
437C2×C4⋊C4—171no—
438C2×C4⋊C4—171no—
439C2×C4⋊C4—171no—
440C2×C4⋊C4C2×#59511yescharacteristic
441C2×C4⋊C4C2×#59511yescharacteristic
442C2×C22⋊C4—171no—
443C2×C22⋊C4—171no—
444C2×C22⋊C4—171no—
445C2×C22⋊C4—171no—
446C2×C22⋊C4—171no—
447C2×C22⋊C4—171no—
448C2×C22⋊C4—171no—
449C2×C22⋊C4—171no—
450C2×C42—171no—
451C2×C42—171no—
452C2×C42—171no—
453C2×C42—171no—
454C2×C42—171no—
455C2×C42—171no—
456C2×C42—681no—
457C2×C42—681no—
458C2×C42C2×#59511yescharacteristic
459C4⋊C8—174no—
460C4⋊C8—174no—
461C22⋊C8—174no—
462C22⋊C8—174no—
463C22⋊C8—174no—
464C22⋊C8—174no—
465#97—174no—
466#97—681no—
467#97—681no—
468C4×C8—174no—
469C4×C8—681no—
470C4×C8—681no—
471C4×C8—681no—
472C4×C8—681no—
473#95—171no—
474#95—171no—
475#95—171no—
476#95—171no—
477#95—171no—
478#95—171no—
479#95C2×#59511yescharacteristic
480#95C2×#59511yescharacteristic
481#95C2×#59511yescharacteristic
482#95C2×#59511yescharacteristic
483C17#2056111yescharacteristic, complemented, Sylow
484C24—171no—
485C24—171no—
486C2×D4—174no—
487C2×D4—681no—
488C2×D4—681no—
489C22×C4—171no—
490C22×C4—171no—
491C22×C4—171no—
492C22×C4—172no—
493C22×C4—172no—
494C22×C4—172no—
495C22×C4—172no—
496C22×C4—171no—
497C22×C4—171no—
498C22×C4—171no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—681no—
510C22×C4—681no—
511C22×C4—681no—
512C22×C4—681no—
513C22×C4C22×#59511yescharacteristic
514C22×C4C22×#59511yescharacteristic
515C22×C4C22×#59511yescharacteristic
516C22×C4C22×#59511yescharacteristic
517C22×C4C22×#59511yescharacteristic
518C22×C4C22×#59511yescharacteristic
519C22×C4C22×#59511yescharacteristic
520M16—684no—
521M16—684no—
522M16—684no—
523C2×C8—172no—
524C2×C8—172no—
525C2×C8—172no—
526C2×C8—172no—
527C2×C8—172no—
528C2×C8—172no—
529C2×C8—172no—
530C2×C8—172no—
531C2×C8—341no—
532C2×C8—341no—
533C2×C8—341no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—341no—
540C2×C8—341no—
541C2×C8—341no—
542C2×C8—341no—
543C2×C8—341no—
544C2×C8—341no—
545C2×C8—341no—
546C2×C8—341no—
547C2×C8—684no—
548C4⋊C4—41no—
549C4⋊C4—342no—
550C4⋊C4—342no—
551C4⋊C4—681no—
552C4⋊C4C4×#59514yes—
553C22⋊C4—174no—
554C22⋊C4—174no—
555C22⋊C4—342no—
556C22⋊C4—342no—
557C22⋊C4—681no—
558C22⋊C4—681no—
559C22⋊C4—681no—
560C22⋊C4—681no—
561C42—41no—
562C42—172no—
563C42—172no—
564C42—341no—
565C42—341no—
566C42—341no—
567C42—341no—
568C42—341no—
569C42—341no—
570C42—341no—
571C42—341no—
572C42—681no—
573C23—172no—
574C23—172no—
575C23—341no—
576C23—341no—
577C23—341no—
578C23—341no—
579C23—341no—
580C23—341no—
581C23—341no—
582C23—341no—
583C23—341no—
584C23—341no—
585C23—341no—
586C23—341no—
587C23C23×#59511yescharacteristic
588D4—684no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—171no—
607C2×C4—171no—
608C2×C4—171no—
609C2×C4—171no—
610C2×C4—171no—
611C2×C4—171no—
612C2×C4—172no—
613C2×C4—172no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—342no—
618C2×C4—342no—
619C2×C4—342no—
620C2×C4—342no—
621C2×C4—342no—
622C2×C4—341no—
623C2×C4—341no—
624C2×C4—341no—
625C2×C4—341no—
626C2×C4—341no—
627C2×C4—341no—
628C2×C4—341no—
629C2×C4—341no—
630C2×C4—341no—
631C2×C4—341no—
632C2×C4—341no—
633C2×C4—341no—
634C2×C4—341no—
635C2×C4—684no—
636C2×C4—684no—
637C2×C4C2×C4×#59512yes—
638C2×C4C2×C4×#59512yes—
639C2×C4C2×C4×#59512yes—
640C2×C4#1058768012yes—
641C8—342no—
642C8—342no—
643C8—342no—
644C8—342no—
645C8—342no—
646C8—342no—
647C8—342no—
648C8—342no—
649C22—342no—
650C22—342no—
651C22—342no—
652C22—342no—
653C22—342no—
654C22—342no—
655C22—681no—
656C22—681no—
657C22—681no—
658C22—681no—
659C22—681no—
660C22—681no—
661C22—681no—
662C22—681no—
663C22C22×C4×#59511yescharacteristic
664C22#1176387611yescharacteristic
665C22#1176387811yescharacteristic
666C22D17⋊#13111yescharacteristic
667C22C2×#1058768011yescharacteristic
668C22#1176388811yescharacteristic
669C22#1176388811yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—171no—
678C4—171no—
679C4—171no—
680C4—171no—
681C4—342no—
682C4—342no—
683C4—681no—
684C4—681no—
685C4—681no—
686C2—682no—
687C2—682no—
688C2#1454522311yescharacteristic
689C2#1454534911yescharacteristic
690C2C4×#1058768011yescharacteristic
691C2#1454565211yescharacteristic
692C2#1454565411yescharacteristic
693C2#1454572811yescharacteristic
694C2#1454573011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#595
commutatorC22×C17C22×C42
FittingC17×#387C4
socleC23×C17C23×C4
prime radicalC22×D68C2×C4
cosocleC22×Dic17C24
hypercenter#387#595
hypocenterC17#20561
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #14544583C2semidirect
4. #14544576C2semidirect
5. #14544572C2semidirect
6. #14544525C2semidirect
7. #14544525C2semidirect
8. #14544525C2semidirect
9. #14544518C2semidirect
10. #14544513C2semidirect
11. C4×#10587543C2semidirect
12. #14544503C2semidirect
13. #14544451C2semidirect
14. #14544448C2semidirect
20. #11763843C4semidirect
21. #11763839C4semidirect
82. C2×#63243C2×C4semidirect
85. C2×#63242C2×C4semidirect
112. #10587544C2×C4semidirect
115. #10587543C2×C4semidirect
116. #10587543C2×C4semidirect
119. #10587542C2×C4semidirect
120. C4⋊#3624C2×C4semidirect
125. C4×#3624C2×C4semidirect
483. C17#20561semidirect

Maximal subgroups

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

subgroupindex
C2×#117638432
C2×#117638392
#145445832
#145445762
#145445722
#145445252
#145445182
#145445132
C4×#105875432
#145445032
#145444512
#145444482
#145312222
#2056117