Finite Group Databasebeta

← #17070822

Subgroups — #17070822

Order 4352.

1070 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763113C211yesmaximal, characteristic, complemented
2C2×C4×#63229C211yesmaximal, characteristic
3#14534150C211yesmaximal, characteristic, complemented
4#14530119C211yesmaximal, characteristic, complemented
5#14530114C211yesmaximal, characteristic, complemented
6#14529932C211yesmaximal, characteristic, complemented
7#14529922C211yesmaximal, characteristic, complemented
8#14529750C211yesmaximal, characteristic, complemented
9#14529721C211yesmaximal, characteristic, complemented
10#14529118C211yesmaximal, characteristic
11#14529118C211yesmaximal, characteristic
12#14529045C211yesmaximal, characteristic
13#14529031C211yesmaximal, characteristic
14#14528794C211yesmaximal, characteristic
15C2×C17×#554C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18#11763113—24no—
19C4×#63229—24no—
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31#11762747—21no—
32#11762728—21no—
33#11762715C2211yescharacteristic
34#11762715C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762690—21no—
38#11762690—21no—
39#11762675C2211yescharacteristic
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⋊Dic17C2311yescharacteristic
109C2×C4⋊Dic17C2311yescharacteristic
110C2×C4⋊Dic17C2311yescharacteristic
111C2×#63225—21no—
112C2×#63225—21no—
113C2×#63225—21no—
114C2×C4×Dic17—21no—
115C22×#3622C2311yescharacteristic
116C22×#3622C2311yescharacteristic
117C22×#3622C2311yescharacteristic
118C22×#3622C2311yescharacteristic
119C4×Dic34—22no—
120C4×Dic34—22no—
121#10587503—22no—
122#10587503—22no—
123#10587503—22no—
124#10587502—21no—
125#10587502—21no—
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#10587479—22no—
137#10587478—22no—
138C4⋊#3622—22no—
139C4×#3622—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—
212#63225—42no—
213#63225—42no—
214#63225—42no—
215C4×Dic17—42no—
216C2×#3622—21no—
217C2×#3622—21no—
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×#3622C2×Q812yes—
228#24337—171nomaximal, Sylow
229C23×C17—22no—
230C23×C17—22no—
231C23×C17—22no—
232C23×C17—41no—
233C23×C17—41no—
234C23×C17—41no—
235C23×C17—41no—
236C23×C17C22×D411yescharacteristic
237Q8×C17—44no—
238Q8×C17—84no—
239D4×C17—22no—
240D4×C17—22no—
241D4×C17—22no—
242D4×C17C22⋊Q814yes—
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—22no—
253C2×C4×C17—22no—
254C2×C4×C17—22no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—21no—
258C2×C4×C17—21no—
259C2×C4×C17—21no—
260C2×C4×C17—21no—
261C2×C4×C17—48no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—41no—
268C2×C4×C17—84no—
269C2×C4×C17—84no—
270C2×C4×C17#12012yes—
271C2×C4×C17#12012yes—
272C2×C4×C17#13512yes—
273C2×C4×C17C22×D411yescharacteristic
274C2×C4×C17C22×D411yescharacteristic
275C2×C4×C17C22×Q811yescharacteristic
276C2×C4×C17C2×#4111yescharacteristic
277C2×C4×C172−1+411yescharacteristic
278C2×C4×C172−1+411yescharacteristic
279C2×Dic17—22no—
280C2×Dic17—41no—
281C2×Dic17—41no—
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—
291Dic34—22no—
292Dic34—22no—
293Dic34—41no—
294Dic34—41no—
295#3622—22no—
296#3622—41no—
297#3622—41no—
298#3622—41no—
299C2×#554D1711yescharacteristic
300C2×#479—171no—
301C2×C4×SD16—171no—
302#2581—171no—
303#2059—171no—
304#2058—171no—
305#2026—171no—
306#2024—171no—
307#1991—171no—
308#1984—171no—
309#1877—171no—
310#1877—171no—
311#1864—171no—
312#1862—171no—
313#1804—171no—
314C22×C17—24no—
315C22×C17—42no—
316C22×C17—42no—
317C22×C17—42no—
318C22×C17—42no—
319C22×C17—42no—
320C22×C17—42no—
321C22×C17C2×#12011yescharacteristic
322C22×C17#54811yescharacteristic
323C22×C17D4×Q811yescharacteristic
324C22×C17C2×#13511yescharacteristic
325C22×C17#57511yescharacteristic
326C22×C17#57711yescharacteristic
327C22×C17#57711yescharacteristic
328C4×C17—22no—
329C4×C17—22no—
330C4×C17—22no—
331C4×C17—41no—
332C4×C17—42no—
333C4×C17—41no—
334C4×C17—41no—
335C4×C17—42no—
336C4×C17—42no—
337C4×C17C2×C22⋊Q811yescharacteristic
338C4×C17#53611yescharacteristic
339C4×C17#54811yescharacteristic
340C4×C17D4×Q811yescharacteristic
341Dic17—42no—
342Dic17—81no—
343Dic17—81no—
344Dic17—81no—
345C22×SD16—171no—
346#554—24no—
347#554—24no—
348C2×C4⋊Q8C2×D1711yescharacteristic
349C2×#125C2×D1711yescharacteristic
350C2×#125C2×D1711yescharacteristic
351C2×C22⋊Q8—21no—
352C2×C22⋊Q8—21no—
353C2×C22⋊Q8—21no—
354C2×C4×Q8—171no—
355C2×C4×Q8C2×D1711yescharacteristic
356C2×C4×D4C2×D1711yescharacteristic
357C2×C4×D4C2×D1711yescharacteristic
358C2×C4×D4C2×D1711yescharacteristic
359C22×C4⋊C4—21no—
360#479—344no—
361C4×SD16—344no—
362C2×#107—171no—
363C2×#106—171no—
364C2×#106—171no—
365C2×C4⋊C8—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#103—171no—
369C2×#102—171no—
370C2×#102—171no—
371C2×#102—171no—
372C2×C4×C8—171no—
373#400—341no—
374#395—341no—
375#391—171no—
376#391—171no—
377#389—171no—
378#389—171no—
379#387—341no—
380#387—341no—
381#382—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340—171no—
386#340—171no—
387#340—171no—
388#336—171no—
389#336—171no—
390#336—171no—
391#336—171no—
392C2×C17—44no—
393C2×C17#240911yescharacteristic
394C2×C17#300211yescharacteristic
395C2×C17#300911yescharacteristic
396C2×C17#336011yescharacteristic
397C2×C17#337611yescharacteristic
398C2×C17#338111yescharacteristic
399C2×C17#338811yescharacteristic
400C22×Q8—21no—
401C22×Q8—171no—
402C22×Q8C22⋊D1711yescharacteristic
403C22×D4C22×D1711yescharacteristic
404C23×C4—21no—
405C23×C4—21no—
406C23×C4—21no—
407C2×SD16—174no—
408C2×SD16—342no—
409C2×SD16—342no—
410C22×C8—171no—
411C22×C8—171no—
412C22×C8—171no—
413C22×C8—171no—
414C4⋊Q8—22no—
415C4⋊Q8—22no—
416#125—22no—
417#125—22no—
418#125—22no—
419#125C22⋊D1714yes—
420C22⋊Q8—28no—
421C22⋊Q8—28no—
422C22⋊Q8—28no—
423C4×Q8—22no—
424C4×Q8—22no—
425C4×Q8—342no—
426C4×Q8—342no—
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4C22⋊D1714yes—
433C2×C4⋊C4—28no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—171no—
440C2×C4⋊C4—171no—
441C2×C4⋊C4—171no—
442C2×C4⋊C4—341no—
443C2×C4⋊C4—341no—
444C2×C4⋊C4—341no—
445C2×C4⋊C4C22⋊D1711yescharacteristic
446C2×C4⋊C4C22⋊D1711yescharacteristic
447C2×C4⋊C4C22⋊D1711yescharacteristic
448C2×C4⋊C4C22×D1711yescharacteristic
449C2×C4⋊C4C22×D1711yescharacteristic
450C2×C4⋊C4C22×D1711yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C42—341no—
455C2×C42C22×D1711yescharacteristic
456C2×C42C22×D1711yescharacteristic
457C2×C42C22×D1711yescharacteristic
458#107—342no—
459#106—342no—
460#106—342no—
461C4⋊C8—342no—
462#103—342no—
463#103—342no—
464#103—342no—
465#102—342no—
466#102—342no—
467#102—342no—
468C4×C8—342no—
469#95—341no—
470#95—341no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477C17#2433711yescharacteristic, complemented, Sylow
478C24—21no—
479C2×Q8—24no—
480C2×Q8—22no—
481C2×Q8—22no—
482C2×Q8—41no—
483C2×Q8—81no—
484C2×Q8—81no—
485C2×Q8—172no—
486C2×Q8—341no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4Q8×D1712yes—
495C2×D4#6325612yes—
496C2×D4C2×C22⋊D1712yes—
497C22×C4—24no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—81no—
506C22×C4—81no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—341no—
510C22×C4—341no—
511C22×C4—341no—
512C22×C4—341no—
513C22×C4C2×C22⋊D1711yescharacteristic
514C22×C4C2×C22⋊D1711yescharacteristic
515C22×C4C2×C22⋊D1711yescharacteristic
516C22×C4C2×C22⋊D1711yescharacteristic
517C22×C4C2×C22⋊D1711yescharacteristic
518C22×C4C2×C22⋊D1711yescharacteristic
519C22×C4C23×D1711yescharacteristic
520SD16—348no—
521C2×C8—172no—
522C2×C8—341no—
523C2×C8—341no—
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—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—341no—
545C4⋊C4—341no—
546C4⋊C4—341no—
547C4⋊C4—341no—
548C4⋊C4—341no—
549C4⋊C4—341no—
550C4⋊C4—682no—
551C4⋊C4—682no—
552C4⋊C4—682no—
553C4⋊C4C2×C22⋊D1712yes—
554C4⋊C4C2×C22⋊D1712yes—
555C4⋊C4C2×C22⋊D1712yes—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—682no—
564C42C2×C22⋊D1712yes—
565C42C2×C22⋊D1712yes—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23C22×C22⋊D1711yescharacteristic
574Q8—44no—
575Q8—84no—
576Q8—342no—
577Q8—342no—
578Q8—681no—
579Q8—681no—
580D4—22no—
581D4—22no—
582D4—22no—
583D4#1058763214yes—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—342no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4#1058763612yes—
624C2×C4#1058763912yes—
625C2×C4#1058763912yes—
626C2×C4C2×Q8×D1711yescharacteristic
627C2×C4C2×#6325611yescharacteristic
628C2×C4#1058769411yescharacteristic
629C2×C4#1058769411yescharacteristic
630C2×C4C22×C22⋊D1711yescharacteristic
631C2×C4C22×C22⋊D1711yescharacteristic
632C8—342no—
633C8—681no—
634C8—681no—
635C8—681no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22C2×#1058763911yescharacteristic
644C22#1176347011yescharacteristic
645C22Q8×C22⋊D1711yescharacteristic
646C22C2×#1058763611yescharacteristic
647C22#1176360711yescharacteristic
648C22#1176362011yescharacteristic
649C22#1176362011yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—682no—
660C4—1361no—
661C4—1361no—
662C4—1361no—
663C4C2×#1058763211yescharacteristic
664C4#1176338311yescharacteristic
665C4#1176347011yescharacteristic
666C4Q8×C22⋊D1711yescharacteristic
667C2—44no—
668C2#1453214311yescharacteristic
669C2#1453800211yescharacteristic
670C2#1453806111yescharacteristic
671C2#1454143311yescharacteristic
672C2#1454161111yescharacteristic
673C2#1454165911yescharacteristic
674C2#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#24337
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. #14534150C2semidirect
4. #14530119C2semidirect
5. #14530114C2semidirect
6. #14529932C2semidirect
7. #14529922C2semidirect
8. #14529750C2semidirect
9. #14529721C2semidirect
107. C2×#63229Q8semidirect
477. C17#24337semidirect

Maximal subgroups

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

subgroupindex
C2×#117631132
C2×C4×#632292
#145341502
#145301192
#145301142
#145299322
#145299222
#145297502
#145297212
#145291182
#145290452
#145290312
#145287942
C2×C17×#5542
#2433717