Finite Group Databasebeta

← #17070970

Subgroups — #17070970

Order 4352.

1072 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763113C211yesmaximal, characteristic, complemented
2C2×#11762909C211yesmaximal, characteristic
3#14534154C211yesmaximal, characteristic, complemented
4#14530119C211yesmaximal, characteristic, complemented
5#14530109C211yesmaximal, characteristic, complemented
6#14529932C211yesmaximal, characteristic, complemented
7#14529922C211yesmaximal, characteristic, complemented
8#14529754C211yesmaximal, characteristic, complemented
9#14529725C211yesmaximal, characteristic, complemented
10#14529118C211yesmaximal, characteristic
11#14529118C211yesmaximal, characteristic
12#14529045C211yesmaximal, characteristic
13#14529031C211yesmaximal, characteristic
14#14528784C211yesmaximal, characteristic
15C2×C17×#554C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18#11763113—24no—
19#11762909—24no—
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587474C2211yescharacteristic
31#11762742—21no—
32#11762742—21no—
33#11762716—21no—
34#11762715C2211yescharacteristic
35#11762715C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762705—21no—
39#11762679C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C17×#554—24no—
51C17×#554—24no—
52C2×C17×C4⋊Q8C2211yescharacteristic
53C2×C17×#125C2211yescharacteristic
54C2×C17×#125C2211yescharacteristic
55C2×C17×C22⋊Q8—21no—
56C2×C17×C22⋊Q8—21no—
57C2×C17×C22⋊Q8—21no—
58C2×C4×Q8×C17C2211yescharacteristic
59C2×C4×D4×C17C2211yescharacteristic
60C2×C4×D4×C17C2211yescharacteristic
61C2×C4×D4×C17C2211yescharacteristic
62C22×C4⋊C4×C17—21no—
63C22×Q8×C17—21no—
64C22×Q8×C17D411yescharacteristic
65C22×D4×C17C2311yescharacteristic
66C23×C4×C17—21no—
67C23×C4×C17—21no—
68C23×C4×C17—21no—
69C22×Dic34C2311yescharacteristic
70C17×C4⋊Q8—22no—
71C17×C4⋊Q8—22no—
72C17×#125—22no—
73C17×#125—22no—
74C17×#125—22no—
75C17×#125D414yes—
76C17×C22⋊Q8—28no—
77C17×C22⋊Q8—28no—
78C17×C22⋊Q8—28no—
79C4×Q8×C17—22no—
80C4×Q8×C17—22no—
81C4×D4×C17—22no—
82C4×D4×C17—22no—
83C4×D4×C17—22no—
84C4×D4×C17—22no—
85C4×D4×C17—22no—
86C4×D4×C17D414yes—
87C2×C4⋊C4×C17—28no—
88C2×C4⋊C4×C17—21no—
89C2×C4⋊C4×C17—21no—
90C2×C4⋊C4×C17—21no—
91C2×C4⋊C4×C17—21no—
92C2×C4⋊C4×C17—21no—
93C2×C4⋊C4×C17D411yescharacteristic
94C2×C4⋊C4×C17D411yescharacteristic
95C2×C4⋊C4×C17D411yescharacteristic
96C2×C4⋊C4×C17C2311yescharacteristic
97C2×C4⋊C4×C17C2311yescharacteristic
98C2×C4⋊C4×C17C2311yescharacteristic
99C2×C22⋊C4×C17—21no—
100C2×C22⋊C4×C17—21no—
101C2×C22⋊C4×C17—21no—
102C2×C42×C17C2311yescharacteristic
103C2×C42×C17C2311yescharacteristic
104C2×C42×C17C2311yescharacteristic
105C2×#63229—22no—
106C2×#63229—22no—
107C2×#63229Q814yescomplemented
108C2×C4⋊Dic17—21no—
109C2×C4⋊Dic17—21no—
110C2×C4⋊Dic17C2311yescharacteristic
111C2×C4⋊Dic17C2311yescharacteristic
112C2×C4⋊Dic17C2311yescharacteristic
113C2×#63225—21no—
114C2×#63225—21no—
115C2×#63225—21no—
116C2×C4×Dic17—21no—
117C22×#3622C2311yescharacteristic
118C22×#3622C2311yescharacteristic
119C22×#3622C2311yescharacteristic
120C22×#3622C2311yescharacteristic
121C4×Dic34—22no—
122C4×Dic34—22no—
123#10587503—22no—
124#10587503—22no—
125#10587503—22no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130#10587502—21no—
131#10587502—21no—
132#10587481—22no—
133#10587481—22no—
134#10587481—22no—
135#10587479—22no—
136#10587478—22no—
137#10587478—22no—
138C4⋊#3622—22no—
139#10587474—22no—
140C24×C17—21no—
141C2×Q8×C17—24no—
142C2×Q8×C17—22no—
143C2×Q8×C17—22no—
144C2×Q8×C17—41no—
145C2×Q8×C17—81no—
146C2×Q8×C17—81no—
147C2×D4×C17—21no—
148C2×D4×C17—21no—
149C2×D4×C17—21no—
150C2×D4×C17C2×D412yes—
151C2×D4×C17C2×Q812yes—
152C2×D4×C17#4112yes—
153C22×C4×C17—24no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—24no—
158C22×C4×C17—24no—
159C22×C4×C17—42no—
160C22×C4×C17—42no—
161C22×C4×C17—81no—
162C22×C4×C17—81no—
163C22×C4×C17—81no—
164C22×C4×C17—81no—
165C22×C4×C17C2×D411yescharacteristic
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17C2×D411yescharacteristic
169C22×C4×C17C2×D411yescharacteristic
170C22×C4×C17C2×D411yescharacteristic
171C22×C4×C17C2411yescharacteristic
172C22×Dic17—21no—
173C22×Dic17—21no—
174C22×Dic17—21no—
175C22×Dic17—21no—
176C2×Dic34—21no—
177C2×Dic34—21no—
178C2×Dic34—21no—
179C2×Dic34—21no—
180C2×Dic34—21no—
181C2×Dic34C2×Q812yes—
182C4⋊C4×C17—24no—
183C4⋊C4×C17—24no—
184C4⋊C4×C17—24no—
185C4⋊C4×C17—24no—
186C4⋊C4×C17—22no—
187C4⋊C4×C17—22no—
188C4⋊C4×C17—24no—
189C4⋊C4×C17—22no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17—21no—
192C4⋊C4×C17—21no—
193C4⋊C4×C17C2×D412yes—
194C4⋊C4×C17C2×D412yes—
195C4⋊C4×C17C2×D412yes—
196C22⋊C4×C17—24no—
197C22⋊C4×C17—24no—
198C22⋊C4×C17—24no—
199C42×C17—21no—
200C42×C17—21no—
201C42×C17—21no—
202C42×C17—21no—
203C42×C17C2×D412yes—
204C42×C17C2×D412yes—
205#63229—28no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—42no—
213C4⋊Dic17—81no—
214#63225—42no—
215#63225—81no—
216#63225—81no—
217C4×Dic17—42no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—21no—
228C2×#3622—21no—
229C2×#3622C2×Q812yes—
230#24349—171nomaximal, Sylow
231C23×C17—22no—
232C23×C17—22no—
233C23×C17—22no—
234C23×C17—41no—
235C23×C17—41no—
236C23×C17—41no—
237C23×C17—41no—
238C23×C17C22×D411yescharacteristic
239Q8×C17—44no—
240Q8×C17—84no—
241D4×C17—22no—
242D4×C17—22no—
243D4×C17—22no—
244D4×C17C22⋊Q814yes—
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—22no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—21no—
258C2×C4×C17—21no—
259C2×C4×C17—21no—
260C2×C4×C17—21no—
261C2×C4×C17—21no—
262C2×C4×C17—21no—
263C2×C4×C17—48no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—41no—
268C2×C4×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17—84no—
271C2×C4×C17—84no—
272C2×C4×C17#12012yes—
273C2×C4×C17#12012yes—
274C2×C4×C17#13712yes—
275C2×C4×C17C22×D411yescharacteristic
276C2×C4×C17C22×D411yescharacteristic
277C2×C4×C17C22×Q811yescharacteristic
278C2×C4×C17C2×#4111yescharacteristic
279C2×C4×C172−1+411yescharacteristic
280C2×C4×C172−1+411yescharacteristic
281C2×Dic17—22no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290C2×Dic17—41no—
291C2×Dic17—41no—
292C2×Dic17—41no—
293Dic34—22no—
294Dic34—22no—
295Dic34—41no—
296Dic34—41no—
297#3622—22no—
298#3622—41no—
299#3622—41no—
300#3622—41no—
301C2×#554D1711yescharacteristic
302C2×#479—171no—
303C2×#440—171no—
304#2582—171no—
305#2059—171no—
306#2057—171no—
307#2026—171no—
308#2024—171no—
309#1992—171no—
310#1985—171no—
311#1877—171no—
312#1877—171no—
313#1864—171no—
314#1862—171no—
315#1802—171no—
316C22×C17—24no—
317C22×C17—42no—
318C22×C17—42no—
319C22×C17—42no—
320C22×C17—42no—
321C22×C17—42no—
322C22×C17—42no—
323C22×C17C2×#12011yescharacteristic
324C22×C17#54811yescharacteristic
325C22×C17D4×Q811yescharacteristic
326C22×C17C2×#13711yescharacteristic
327C22×C17#57511yescharacteristic
328C22×C17#57611yescharacteristic
329C22×C17#57711yescharacteristic
330C4×C17—22no—
331C4×C17—22no—
332C4×C17—22no—
333C4×C17—41no—
334C4×C17—42no—
335C4×C17—41no—
336C4×C17—41no—
337C4×C17—42no—
338C4×C17—42no—
339C4×C17C2×C22⋊Q811yescharacteristic
340C4×C17#53611yescharacteristic
341C4×C17#54811yescharacteristic
342C4×C17D4×Q811yescharacteristic
343Dic17—42no—
344Dic17—81no—
345Dic17—81no—
346Dic17—81no—
347C22×SD16—171no—
348#554—24no—
349#554—24no—
350C2×C4⋊Q8C2×D1711yescharacteristic
351C2×#125C2×D1711yescharacteristic
352C2×#125C2×D1711yescharacteristic
353C2×C22⋊Q8—21no—
354C2×C22⋊Q8—21no—
355C2×C22⋊Q8—21no—
356C2×C4×Q8—171no—
357C2×C4×Q8C2×D1711yescharacteristic
358C2×C4×D4C2×D1711yescharacteristic
359C2×C4×D4C2×D1711yescharacteristic
360C2×C4×D4C2×D1711yescharacteristic
361C22×C4⋊C4—21no—
362#479—344no—
363#440—344no—
364C2×#107—171no—
365C2×#107—171no—
366C2×#106—171no—
367C2×C4⋊C8—171no—
368C2×#103—171no—
369C2×#103—171no—
370C2×#103—171no—
371C2×#102—171no—
372C2×#102—171no—
373C2×#102—171no—
374C2×#97—171no—
375#398—341no—
376#398—341no—
377#391—171no—
378#391—171no—
379#391—341no—
380#389—171no—
381#389—171no—
382#389—341no—
383#384—171no—
384#340—171no—
385#340—171no—
386#340—171no—
387#340—171no—
388#340—171no—
389#340—171no—
390#336—171no—
391#336—171no—
392#336—171no—
393#336—171no—
394C2×C17—44no—
395C2×C17#240911yescharacteristic
396C2×C17#300711yescharacteristic
397C2×C17#301211yescharacteristic
398C2×C17#337011yescharacteristic
399C2×C17#337211yescharacteristic
400C2×C17#338511yescharacteristic
401C2×C17#338811yescharacteristic
402C22×Q8—21no—
403C22×Q8—171no—
404C22×Q8C22⋊D1711yescharacteristic
405C22×D4C22×D1711yescharacteristic
406C23×C4—21no—
407C23×C4—21no—
408C23×C4—21no—
409C2×SD16—174no—
410C2×SD16—342no—
411C2×SD16—342no—
412C22×C8—171no—
413C22×C8—171no—
414C22×C8—171no—
415C22×C8—171no—
416C4⋊Q8—22no—
417C4⋊Q8—22no—
418#125—22no—
419#125—22no—
420#125—22no—
421#125C22⋊D1714yes—
422C22⋊Q8—28no—
423C22⋊Q8—28no—
424C22⋊Q8—28no—
425C4×Q8—22no—
426C4×Q8—22no—
427C4×Q8—342no—
428C4×Q8—342no—
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4C22⋊D1714yes—
435C2×C4⋊C4—28no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—171no—
442C2×C4⋊C4—171no—
443C2×C4⋊C4—171no—
444C2×C4⋊C4—341no—
445C2×C4⋊C4—341no—
446C2×C4⋊C4—341no—
447C2×C4⋊C4—341no—
448C2×C4⋊C4—341no—
449C2×C4⋊C4C22⋊D1711yescharacteristic
450C2×C4⋊C4C22⋊D1711yescharacteristic
451C2×C4⋊C4C22⋊D1711yescharacteristic
452C2×C4⋊C4C22×D1711yescharacteristic
453C2×C4⋊C4C22×D1711yescharacteristic
454C2×C4⋊C4C22×D1711yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—21no—
458C2×C42—341no—
459C2×C42C22×D1711yescharacteristic
460C2×C42C22×D1711yescharacteristic
461C2×C42C22×D1711yescharacteristic
462#107—342no—
463#107—342no—
464#106—342no—
465C4⋊C8—342no—
466#103—342no—
467#103—342no—
468#103—342no—
469#102—342no—
470#102—342no—
471#102—342no—
472#97—342no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477#95—341no—
478#95—341no—
479C17#2434911yescharacteristic, complemented, Sylow
480C24—21no—
481C2×Q8—24no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—41no—
485C2×Q8—81no—
486C2×Q8—81no—
487C2×Q8—172no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4Q8×D1712yes—
497C2×D4#6325612yes—
498C2×D4C2×C22⋊D1712yes—
499C22×C4—24no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4—81no—
511C22×C4—341no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—341no—
515C22×C4C2×C22⋊D1711yescharacteristic
516C22×C4C2×C22⋊D1711yescharacteristic
517C22×C4C2×C22⋊D1711yescharacteristic
518C22×C4C2×C22⋊D1711yescharacteristic
519C22×C4C2×C22⋊D1711yescharacteristic
520C22×C4C2×C22⋊D1711yescharacteristic
521C22×C4C23×D1711yescharacteristic
522SD16—348no—
523C2×C8—172no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C2×C8—341no—
528C2×C8—341no—
529C2×C8—341no—
530C2×C8—341no—
531C2×C8—341no—
532C2×C8—341no—
533C2×C8—341no—
534C2×C8—341no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—341no—
547C4⋊C4—341no—
548C4⋊C4—341no—
549C4⋊C4—341no—
550C4⋊C4—341no—
551C4⋊C4—341no—
552C4⋊C4—682no—
553C4⋊C4—682no—
554C4⋊C4—1361no—
555C4⋊C4—1361no—
556C4⋊C4—1361no—
557C4⋊C4C2×C22⋊D1712yes—
558C4⋊C4C2×C22⋊D1712yes—
559C4⋊C4C2×C22⋊D1712yes—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—682no—
568C42C2×C22⋊D1712yes—
569C42C2×C22⋊D1712yes—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C22×C22⋊D1711yescharacteristic
578Q8—44no—
579Q8—84no—
580Q8—342no—
581Q8—342no—
582Q8—681no—
583Q8—681no—
584D4—22no—
585D4—22no—
586D4—22no—
587D4#1058763214yes—
588C2×C4—21no—
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—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—48no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—342no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—681no—
624C2×C4—681no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4#1058763712yes—
628C2×C4#1058763912yes—
629C2×C4#1058763912yes—
630C2×C4C2×Q8×D1711yescharacteristic
631C2×C4C2×#6325611yescharacteristic
632C2×C4#1058769411yescharacteristic
633C2×C4#1058769411yescharacteristic
634C2×C4C22×C22⋊D1711yescharacteristic
635C2×C4C22×C22⋊D1711yescharacteristic
636C8—342no—
637C8—681no—
638C8—681no—
639C8—681no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22C2×#1058763911yescharacteristic
648C22#1176347011yescharacteristic
649C22Q8×C22⋊D1711yescharacteristic
650C22C2×#1058763711yescharacteristic
651C22#1176360711yescharacteristic
652C22#1176361211yescharacteristic
653C22#1176362011yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—41no—
658C4—42no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—42no—
663C4—682no—
664C4—1361no—
665C4—1361no—
666C4—1361no—
667C4C2×#1058763211yescharacteristic
668C4#1176338311yescharacteristic
669C4#1176347011yescharacteristic
670C4Q8×C22⋊D1711yescharacteristic
671C2—44no—
672C2#1453214311yescharacteristic
673C2#1453804411yescharacteristic
674C2#1453808811yescharacteristic
675C2#1454153711yescharacteristic
676C2#1454156511yescharacteristic
677C2#1454169411yescharacteristic
678C2#1454172911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#554C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenterC2×#554D17
hypocenterC17#24349
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#11763113C2semidirect
3. #14534154C2semidirect
4. #14530119C2semidirect
5. #14530109C2semidirect
6. #14529932C2semidirect
7. #14529922C2semidirect
8. #14529754C2semidirect
9. #14529725C2semidirect
107. C2×#63229Q8semidirect
479. C17#24349semidirect

Maximal subgroups

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

subgroupindex
C2×#117631132
C2×#117629092
#145341542
#145301192
#145301092
#145299322
#145299222
#145297542
#145297252
#145291182
#145290452
#145290312
#145287842
C2×C17×#5542
#2434917