Finite Group Databasebeta

← #17054649

Subgroups — #17054649

Order 4352.

1086 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762967C211yesmaximal, characteristic
2C2×#11762915C211yesmaximal, characteristic, complemented
3C22×#10587481C211yesmaximal, characteristic
4#14531789C211yesmaximal, characteristic
5#14531616C211yesmaximal, characteristic
6#14529653C211yesmaximal, characteristic, complemented
7#14529472C211yesmaximal, characteristic, complemented
8#14529417C211yesmaximal, characteristic, complemented
9#14529205C211yesmaximal, characteristic
10#14529126C211yesmaximal, characteristic, complemented
11#14529126C211yesmaximal, characteristic, complemented
12#14529049C211yesmaximal, characteristic, complemented
13#14528613C211yesmaximal, characteristic, complemented
14#14528563C211yesmaximal, characteristic
15C17×#1853C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C22×Dic68C2211yescharacteristic
18C22×Dic68C2211yescharacteristic
19C23×#3622C2211yescharacteristic
20C2×#10587610C2211yescharacteristic
21C2×#10587573C2211yescharacteristic
22C2×#10587573C2211yescharacteristic
23C2×#10587573C2211yescharacteristic
24#11762967C418yescomplemented
25#11762915—24no—
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587481—24no—
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×#10587481C2211yescharacteristic
35C2×#10587486C2211yescharacteristic
36C2×#10587486C2211yescharacteristic
37#11762714C2211yescharacteristic
38#11762714C2211yescharacteristic
39#11762714C2211yescharacteristic
40#11762713—21no—
41#11762713—21no—
42#11762705C2211yescharacteristic
43#11762704—21no—
44#11762690—21no—
45#11762682—21no—
46#11762667—21no—
47C4⋊C4×Dic17C2211yescharacteristic
48C22⋊C4×Dic17—21no—
49C4×C22⋊Dic17—21no—
50C4×C22⋊Dic17C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762563C2211yescharacteristic
54#11762563C2211yescharacteristic
55C22×C4⋊C4×C17C2211yescharacteristic
56C2×C17×#106C2211yescharacteristic
57C2×C17×#106C2211yescharacteristic
58C2×C17×C22⋊C8C2211yescharacteristic
59C17×#380C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C23×C4×C17—21no—
63C23×C4×C17C2311yescharacteristic
64C23×Dic17—21no—
65C22×Dic34—22no—
66C22×Dic34—22no—
67C22×Dic34—21no—
68C22×Dic34—22no—
69C22×Dic34—41no—
70C22×Dic34—41no—
71C22×Dic34C2×C414yes—
72C22×Dic34C2311yescharacteristic
73C22×Dic34C2311yescharacteristic
74C22×Dic34C2311yescharacteristic
75C22×C8×C17C2311yescharacteristic
76C22×C8×C17C2311yescharacteristic
77C2×C4⋊C4×C17—22no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17—41no—
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C22⋊C4×C17—21no—
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17D411yescharacteristic
87C2×C22⋊Dic17D411yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89#10587610—22no—
90#10587610C2×C414yes—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17C2311yescharacteristic
101C2×C4×Dic17C2311yescharacteristic
102C22×#3622—22no—
103C22×#3622—22no—
104C22×#3622—41no—
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C2×Dic68—22no—
108C2×Dic68—22no—
109C2×Dic68—41no—
110C2×Dic68—41no—
111C2×Dic68C2×C414yescomplemented
112C2×Dic68C2×C414yescomplemented
113#10587573—22no—
114#10587573—22no—
115#10587573—22no—
116#10587573—41no—
117#10587573—41no—
118#10587573—41no—
119C17×#106—22no—
120C17×#106—41no—
121C17×C22⋊C8C2×C414yes—
122C17×#95—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130#10587502—21no—
131#10587502D411yescharacteristic
132#10587502D411yescharacteristic
133#10587487C2×C414yescomplemented
134#10587487C2×C414yescomplemented
135#10587486—22no—
136#10587486—41no—
137#10587481—22no—
138#10587481—22no—
139#10587481—22no—
140#10587481—22no—
141#10587481—41no—
142#10587481D414yes—
143C24×C17C2×D411yescharacteristic
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—42no—
153C22×C4×C17—42no—
154C22×C4×C17—81no—
155C22×C4×C17C22×C412yes—
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17#4112yes—
158C22×C4×C17C2411yescharacteristic
159C22×Dic17—21no—
160C22×Dic17—24no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—42no—
165C22×Dic17—42no—
166C22×Dic17C2×D411yescharacteristic
167C22×Dic17C2×D411yescharacteristic
168C22×Dic17C2×D411yescharacteristic
169C22×Dic17C2×D411yescharacteristic
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34—21no—
176C2×Dic34—21no—
177C2×Dic34—21no—
178C2×Dic34—21no—
179C2×Dic34—21no—
180C2×Dic34—21no—
181C2×Dic34—28no—
182C2×Dic34—24no—
183C2×Dic34—21no—
184C2×Dic34—21no—
185C2×Dic34—44no—
186C2×Dic34—44no—
187C2×Dic34—44no—
188C2×Dic34—42no—
189C2×Dic34—42no—
190C2×Dic34C22×C412yes—
191C2×Dic34C22×C412yes—
192C2×Dic34C22×C412yes—
193C2×Dic34C2×D412yes—
194C2×Dic34#4112yes—
195C2×C8×C17—21no—
196C2×C8×C17—21no—
197C2×C8×C17—21no—
198C2×C8×C17—21no—
199C2×C8×C17C22×C412yes—
200C2×C8×C17C22×C412yes—
201C4⋊C4×C17—21no—
202C4⋊C4×C17—21no—
203C4⋊C4×C17—21no—
204C4⋊C4×C17—21no—
205C4⋊C4×C17—21no—
206C4⋊C4×C17—21no—
207C4⋊C4×C17—21no—
208C4⋊C4×C17—42no—
209C4⋊C4×C17C2×D412yes—
210C22⋊C4×C17—42no—
211C22⋊Dic17—24no—
212C22⋊Dic17—22no—
213C22⋊Dic17—22no—
214C4⋊Dic17—21no—
215C4⋊Dic17C22×C412yes—
216#63225—24no—
217#63225—42no—
218#63225—42no—
219#63225—42no—
220#63225—81no—
221C4×Dic17—21no—
222C4×Dic17—21no—
223C4×Dic17—21no—
224C4×Dic17—21no—
225C4×Dic17—42no—
226C4×Dic17—42no—
227C4×Dic17—42no—
228C4×Dic17—81no—
229C2×#3622—21no—
230C2×#3622—21no—
231C2×#3622—21no—
232C2×#3622—21no—
233C2×#3622—21no—
234C2×#3622—44no—
235C2×#3622C2×D412yes—
236Dic68—28no—
237Dic68—44no—
238#22859—171nomaximal, Sylow
239C23×C17—21no—
240C23×C17—21no—
241C23×C17—21no—
242C23×C17—21no—
243C23×C17C4×D412yes—
244C23×C17#13712yes—
245C23×C17#13712yes—
246C23×C17C22×D411yescharacteristic
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—84no—
266C2×C4×C17C4×D412yes—
267C2×C4×C17C23×C411yescharacteristic
268C2×C4×C17C22×D411yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×C4×C17C2×#4111yescharacteristic
271C2×C4×C17H(5,𝔽2)11yescharacteristic
272C2×C4×C17H(5,𝔽2)11yescharacteristic
273C2×Dic17—22no—
274C2×Dic17—22no—
275C2×Dic17—22no—
276C2×Dic17—22no—
277C2×Dic17—22no—
278C2×Dic17—22no—
279C2×Dic17—21no—
280C2×Dic17—21no—
281C2×Dic17—21no—
282C2×Dic17—21no—
283C2×Dic17—21no—
284C2×Dic17—21no—
285C2×Dic17—21no—
286C2×Dic17—21no—
287C2×Dic17—21no—
288C2×Dic17—21no—
289C2×Dic17—21no—
290C2×Dic17—21no—
291C2×Dic17—48no—
292C2×Dic17—41no—
293C2×Dic17—41no—
294C2×Dic17—41no—
295C2×Dic17—41no—
296C2×Dic17—41no—
297C2×Dic17—41no—
298Dic34—22no—
299Dic34—22no—
300Dic34—22no—
301Dic34—22no—
302Dic34—22no—
303Dic34—22no—
304Dic34—22no—
305Dic34—48no—
306Dic34—41no—
307Dic34—41no—
308Dic34—41no—
309Dic34—41no—
310Dic34C4×D414yes—
311C8×C17—22no—
312C8×C17—22no—
313#3622—22no—
314#3622—41no—
315C2×#451—171no—
316C2×#441—171no—
317C22×#103—171no—
318#2360—171no—
319Q8×C22⋊C4—171no—
320#1971—171no—
321#1935—171no—
322#1927—171no—
323#1892—171no—
324#1878—171no—
325#1878—171no—
326#1865—171no—
327#1853D1711yescharacteristic
328#1763—171no—
329#1752—171no—
330C22×C17—22no—
331C22×C17—22no—
332C22×C17—22no—
333C22×C17—22no—
334C22×C17—22no—
335C22×C17—22no—
336C22×C17#44114yescomplemented
337C22×C17C2×C4×D411yescharacteristic
338C22×C17D4211yescharacteristic
339C22×C17#54611yescharacteristic
340C22×C17C2×#13711yescharacteristic
341C22×C17C2×#13711yescharacteristic
342C22×C17#57711yescharacteristic
343C22×C17#57811yescharacteristic
344C4×C17—22no—
345C4×C17—42no—
346C4×C17—81no—
347C4×C17—81no—
348C4×C17—81no—
349C4×C17C2×C4×D411yescharacteristic
350C4×C17#51811yescharacteristic
351C4×C17#54611yescharacteristic
352C4×C17#55111yescharacteristic
353Dic17—24no—
354Dic17—42no—
355Dic17—42no—
356Dic17—42no—
357Dic17—41no—
358Dic17—41no—
359Dic17—41no—
360Dic17—41no—
361C23×Q8—171no—
362C22×Q16—171no—
363C22×Q16—171no—
364C23×C8—171no—
365C2×C22⋊Q8—171no—
366C2×C4×Q8—171no—
367C2×C4×Q8—171no—
368C2×C4×Q8—171no—
369C22×C4⋊C4C2×D1711yescharacteristic
370#451—178no—
371#441—344no—
372C2×#106C2×D1711yescharacteristic
373C2×#106C2×D1711yescharacteristic
374C2×#103—171no—
375C2×#103—171no—
376C2×#103—171no—
377C2×#103—171no—
378C2×#103—171no—
379C2×#103—171no—
380C2×#103—171no—
381C2×#103—171no—
382C2×#103—344no—
383C2×C22⋊C8C2×D1711yescharacteristic
384C2×#97—171no—
385C2×#97—171no—
386#391—171no—
387#391—171no—
388#391—171no—
389#391—341no—
390#391—341no—
391#389—171no—
392#389—341no—
393#387—341no—
394#385—341no—
395#380—341no—
396#380C2×D1711yescharacteristic
397C4×C4⋊C4—171no—
398C4×C22⋊C4—171no—
399C4×C22⋊C4—341no—
400C4×C22⋊C4—341no—
401#340—171no—
402#340—171no—
403#340C2×D1711yescharacteristic
404#340C2×D1711yescharacteristic
405#336—171no—
406#336—171no—
407C2×C17—24no—
408C2×C17#235811yescharacteristic
409C2×C17C2×#44111yescharacteristic
410C2×C17#294711yescharacteristic
411C2×C17#326611yescharacteristic
412C2×C17#327011yescharacteristic
413C2×C17#331211yescharacteristic
414C2×C17#331611yescharacteristic
415C22×Q8—174no—
416C22×Q8—171no—
417C22×Q8—171no—
418C22×Q8—171no—
419C22×Q8—342no—
420C22×Q8—342no—
421C22×Q8—341no—
422C22×Q8—342no—
423C22×Q8—681no—
424C22×Q8—681no—
425C23×C4—21no—
426C23×C4—341no—
427C23×C4C22×D1711yescharacteristic
428C2×Q16—174no—
429C2×Q16—174no—
430C2×Q16—342no—
431C2×Q16—342no—
432C2×Q16—681no—
433C2×Q16—681no—
434C22×C8—171no—
435C22×C8—171no—
436C22×C8—342no—
437C22×C8—342no—
438C22×C8—681no—
439C22×C8C22×D1711yescharacteristic
440C22×C8C22×D1711yescharacteristic
441C22⋊Q8—174no—
442C22⋊Q8—342no—
443C4×Q8—342no—
444C4×Q8—342no—
445C4×Q8—342no—
446C4×Q8—681no—
447C4×Q8—681no—
448C4×Q8—681no—
449C2×C4⋊C4—22no—
450C2×C4⋊C4—21no—
451C2×C4⋊C4—41no—
452C2×C4⋊C4—171no—
453C2×C4⋊C4—341no—
454C2×C4⋊C4—341no—
455C2×C4⋊C4—341no—
456C2×C4⋊C4—341no—
457C2×C4⋊C4—341no—
458C2×C4⋊C4C22×D1711yescharacteristic
459C2×C4⋊C4C22×D1711yescharacteristic
460C2×C4⋊C4C22×D1711yescharacteristic
461C2×C4⋊C4C22×D1711yescharacteristic
462C2×C22⋊C4—21no—
463C2×C22⋊C4—171no—
464C2×C22⋊C4—171no—
465C2×C22⋊C4—341no—
466C2×C42—171no—
467C2×C42—171no—
468C2×C42—341no—
469C2×C42—341no—
470C2×C42—341no—
471C2×C42—341no—
472#106—22no—
473#106—41no—
474#103—174no—
475#103—174no—
476#103—174no—
477#103—342no—
478#103—342no—
479#103—342no—
480#103—342no—
481#103—681no—
482C22⋊C8C4×D1714yes—
483#97—342no—
484#97—681no—
485#95—21no—
486#95—171no—
487#95—171no—
488#95—341no—
489#95—341no—
490#95—341no—
491#95—341no—
492#95—341no—
493#95—341no—
494#95—341no—
495#95—341no—
496C17#2285911yescharacteristic, complemented, Sylow
497C24D4×D1711yescharacteristic
498C2×Q8—172no—
499C2×Q8—172no—
500C2×Q8—172no—
501C2×Q8—172no—
502C2×Q8—172no—
503C2×Q8—348no—
504C2×Q8—341no—
505C2×Q8—341no—
506C2×Q8—341no—
507C2×Q8—341no—
508C2×Q8—341no—
509C2×Q8—344no—
510C2×Q8—341no—
511C2×Q8—341no—
512C2×Q8—341no—
513C2×Q8—341no—
514C2×Q8—341no—
515C2×Q8—341no—
516C2×Q8—341no—
517C2×Q8—341no—
518C2×Q8—682no—
519C2×Q8—682no—
520C2×Q8—684no—
521C2×Q8—684no—
522C2×Q8—684no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—42no—
532C22×C4—42no—
533C22×C4—81no—
534C22×C4—171no—
535C22×C4—171no—
536C22×C4—171no—
537C22×C4—171no—
538C22×C4—341no—
539C22×C4—341no—
540C22×C4—341no—
541C22×C4—341no—
542C22×C4—344no—
543C22×C4—682no—
544C22×C4—682no—
545C22×C4C2×C4×D1712yes—
546C22×C4D4×D1711yescharacteristic
547C22×C4#6325612yes—
548C22×C4C23×D1711yescharacteristic
549Q16—348no—
550Q16—684no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—21no—
554C2×C8—21no—
555C2×C8—172no—
556C2×C8—341no—
557C2×C8—341no—
558C2×C8—341no—
559C2×C8—341no—
560C2×C8—341no—
561C2×C8—684no—
562C2×C8C2×C4×D1712yes—
563C2×C8C2×C4×D1712yes—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—42no—
572C4⋊C4—172no—
573C4⋊C4—341no—
574C4⋊C4—344no—
575C4⋊C4—682no—
576C4⋊C4—682no—
577C4⋊C4—682no—
578C4⋊C4—1361no—
579C4⋊C4D4×D1712yes—
580C22⋊C4—42no—
581C22⋊C4—344no—
582C22⋊C4—342no—
583C22⋊C4—342no—
584C42—341no—
585C42—341no—
586C42—341no—
587C42—341no—
588C42—682no—
589C42—682no—
590C42—682no—
591C42—1361no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23#1058757912yes—
597C23#1058760112yes—
598C23#1058760112yes—
599C23C2×D4×D1711yescharacteristic
600Q8—174no—
601Q8—342no—
602Q8—342no—
603Q8—342no—
604Q8—342no—
605Q8—342no—
606Q8—342no—
607Q8—342no—
608Q8—688no—
609Q8—681no—
610Q8—681no—
611Q8—681no—
612Q8—681no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—84no—
632C2×C4—342no—
633C2×C4—342no—
634C2×C4—342no—
635C2×C4—342no—
636C2×C4—342no—
637C2×C4—342no—
638C2×C4—341no—
639C2×C4—341no—
640C2×C4—341no—
641C2×C4—341no—
642C2×C4—341no—
643C2×C4—341no—
644C2×C4—341no—
645C2×C4—341no—
646C2×C4—341no—
647C2×C4—341no—
648C2×C4—341no—
649C2×C4—341no—
650C2×C4—688no—
651C2×C4—681no—
652C2×C4—681no—
653C2×C4—681no—
654C2×C4—681no—
655C2×C4—681no—
656C2×C4—681no—
657C2×C4#1058757912yes—
658C2×C4C22×C4×D1711yescharacteristic
659C2×C4C2×D4×D1711yescharacteristic
660C2×C4#1058769111yescharacteristic
661C2×C4C2×#6325611yescharacteristic
662C2×C4#41×D1711yescharacteristic
663C2×C4#1058769611yescharacteristic
664C8—22no—
665C8—22no—
666C8—342no—
667C8—681no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22#1176291514yescomplemented
675C22C2×#1058757911yescharacteristic
676C22#1176344111yescharacteristic
677C22#1176345111yescharacteristic
678C22C2×#1058760111yescharacteristic
679C22C2×#1058760111yescharacteristic
680C22#1176361911yescharacteristic
681C22#1176362511yescharacteristic
682C4—22no—
683C4—42no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—344no—
688C4—681no—
689C4—681no—
690C4—681no—
691C4—681no—
692C4—682no—
693C4—682no—
694C4—682no—
695C4C4×D4×D1711yescharacteristic
696C4#1176328111yescharacteristic
697C4#1176345011yescharacteristic
698C4#1176348611yescharacteristic
699C2—24no—
700C2#1453176011yescharacteristic
701C2C2×#1176291511yescharacteristic
702C2#1453762511yescharacteristic
703C2#1454029811yescharacteristic
704C2#1454034711yescharacteristic
705C2#1454083311yescharacteristic
706C2#1454089311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1853C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#1853D17
hypocenterC17#22859
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#11762915C2semidirect
6. #14529653C2semidirect
7. #14529472C2semidirect
8. #14529417C2semidirect
10. #14529126C2semidirect
11. #14529126C2semidirect
12. #14529049C2semidirect
13. #14528613C2semidirect
24. #11762967C4semidirect
111. C2×Dic68C2×C4semidirect
112. C2×Dic68C2×C4semidirect
133. #10587487C2×C4semidirect
134. #10587487C2×C4semidirect
336. C22×C17#441semidirect
496. C17#22859semidirect
674. C22#11762915semidirect

Maximal subgroups

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

subgroupindex
C2×#117629672
C2×#117629152
C22×#105874812
#145317892
#145316162
#145296532
#145294722
#145294172
#145292052
#145291262
#145290492
#145286132
#145285632
C17×#18532
#2285917