Finite Group Databasebeta

← #17056627

Subgroups — #17056627

Order 4352.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×Dic17C211yesmaximal, characteristic, complemented
2C2×SD16×Dic17C211yesmaximal, characteristic
3#14531859C211yesmaximal, characteristic
4#14531705C211yesmaximal, characteristic, complemented
5#14529596C211yesmaximal, characteristic
6#14529554C211yesmaximal, characteristic, complemented
7#14529469C211yesmaximal, characteristic
8#14529462C211yesmaximal, characteristic, complemented
9#14529141C211yesmaximal, characteristic
10#14529127C211yesmaximal, characteristic, complemented
11#14529054C211yesmaximal, characteristic, complemented
12#14529033C211yesmaximal, characteristic
13#14528649C211yesmaximal, characteristic, complemented
14C4×#10587506C211yesmaximal, characteristic, complemented
15C2×C17×#464C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×Q8×Dic17C2211yescharacteristic
19C2×D4×Dic17C2211yescharacteristic
20Q16×Dic17—24no—
21SD16×Dic17—24no—
22C2×#10587489C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32C2×C8×Dic17C2211yescharacteristic
33#11762716C2211yescharacteristic
34#11762712C2211yescharacteristic
35#11762708—21no—
36#11762708—21no—
37#11762708C2211yescharacteristic
38#11762708C2211yescharacteristic
39#11762702—21no—
40#11762702—21no—
41#11762684—21no—
42#11762675—21no—
43#11762675C2211yescharacteristic
44C4×C4⋊Dic17—21no—
45C4×C4⋊Dic17C2211yescharacteristic
46C4×C4⋊Dic17C2211yescharacteristic
47C22⋊C4×Dic17—21no—
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×Q16×C17C2211yescharacteristic
55C22×SD16×C17C2211yescharacteristic
56C2×C17×#124C2211yescharacteristic
57C2×C4×Q8×C17C2211yescharacteristic
58C17×#464C418yescomplemented
59C2×C17×C4⋊C8C2211yescharacteristic
60C2×C17×#103C2211yescharacteristic
61C2×C17×#102C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×Q8×C17C2311yescharacteristic
64C22×D4×C17C2311yescharacteristic
65C23×Dic17—21no—
66C2×Q16×C17—22no—
67C2×Q16×C17—22no—
68C2×Q16×C17C2×C414yescomplemented
69C2×SD16×C17—22no—
70C2×SD16×C17—22no—
71C2×SD16×C17C2×C414yes—
72C22×C8×C17C2311yescharacteristic
73C22×C8×C17C2311yescharacteristic
74C17×#124—22no—
75C17×#124C2×C414yes—
76C4×Q8×C17—22no—
77C4×Q8×C17C2×C414yescomplemented
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C42×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C22⋊Dic17—21no—
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87Q8×Dic17—22no—
88Q8×Dic17—22no—
89Q8×Dic17—22no—
90Q8×Dic17—41no—
91Q8×Dic17—41no—
92Q8×Dic17—41no—
93D4×Dic17—22no—
94D4×Dic17—41no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17—21no—
98C2×C4⋊Dic17—21no—
99C2×C4⋊Dic17—21no—
100C2×C4⋊Dic17D411yescharacteristic
101C2×C4⋊Dic17D411yescharacteristic
102C2×C4⋊Dic17C2311yescharacteristic
103C2×C4⋊Dic17C2311yescharacteristic
104C2×C4⋊Dic17C2311yescharacteristic
105C2×C4⋊Dic17C2311yescharacteristic
106C2×C4×Dic17—21no—
107C2×C4×Dic17—21no—
108C2×C4×Dic17—21no—
109C2×C4×Dic17—21no—
110C2×C4×Dic17C2311yescharacteristic
111C2×C4×Dic17C2311yescharacteristic
112C22×#3622C2311yescharacteristic
113C22×#3622C2311yescharacteristic
114C17×C4⋊C8C2×C414yescomplemented
115C17×#103C2×C414yescomplemented
116C17×#102C2×C414yes—
117#10587506—22no—
118#10587506—22no—
119#10587506—41no—
120#10587506D414yes—
121#10587503—22no—
122#10587503—41no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130#10587502D411yescharacteristic
131#10587502D411yescharacteristic
132#10587489—22no—
133#10587488—22no—
134C8×Dic17—22no—
135C8×Dic17—22no—
136C4×#3622—22no—
137C24×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17C22×C412yes—
147C2×Q8×C17C22×C412yes—
148C2×Q8×C17C2×D412yes—
149C2×Q8×C17#4112yes—
150C2×D4×C17—21no—
151C2×D4×C17—21no—
152C2×D4×C17—21no—
153C2×D4×C17—21no—
154C2×D4×C17—21no—
155C2×D4×C17C22×C412yes—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2411yescharacteristic
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—42no—
167C22×Dic17—42no—
168C22×Dic17—81no—
169C22×Dic17C2×D411yescharacteristic
170C22×Dic17C2×D411yescharacteristic
171C22×Dic17C2×D411yescharacteristic
172C22×Dic17C2×D411yescharacteristic
173Q16×C17—28no—
174SD16×C17—28no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17C22×C412yes—
180C2×C8×C17C22×C412yes—
181C4⋊C4×C17—24no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17C22×C412yes—
184C22⋊C4×C17—24no—
185C22⋊C4×C17—24no—
186C42×C17—24no—
187C42×C17C22×C412yes—
188C42×C17#4112yes—
189C22⋊Dic17—42no—
190C22⋊Dic17—42no—
191C22⋊Dic17—81no—
192C4⋊Dic17—21no—
193C4⋊Dic17—22no—
194C4⋊Dic17—22no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—42no—
202C4⋊Dic17—42no—
203C4⋊Dic17—42no—
204C4⋊Dic17—42no—
205C4⋊Dic17—42no—
206C4⋊Dic17C2×D412yes—
207C4×Dic17—21no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—42no—
212C4×Dic17—42no—
213C4×Dic17—42no—
214C4×Dic17—81no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622C2×D412yes—
221#23000—171nomaximal, Sylow
222C23×C17—22no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—41no—
229C23×C17C22×D411yescharacteristic
230Q8×C17—22no—
231Q8×C17—22no—
232Q8×C17—22no—
233Q8×C17—22no—
234Q8×C17—22no—
235Q8×C17—41no—
236Q8×C17—41no—
237Q8×C17C4×D414yes—
238D4×C17—22no—
239D4×C17—22no—
240D4×C17—41no—
241D4×C17—41no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17C4×D412yes—
256C2×C4×C17C4×D412yes—
257C2×C4×C17C23×C411yescharacteristic
258C2×C4×C17C22×D411yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17H(5,𝔽2)11yescharacteristic
262C2×C4×C172−1+411yescharacteristic
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—22no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—84no—
286C2×Dic17#13512yes—
287C2×Dic17#13512yes—
288C8×C17—22no—
289C8×C17—22no—
290#3622—22no—
291#3622—41no—
292C2×#464D1711yescharacteristic
293C2×C4×Q16—171no—
294C2×C4×SD16—171no—
295#2367—171no—
296#2351—171no—
297#1958—171no—
298#1949—171no—
299#1934—171no—
300#1933—171no—
301#1880—171no—
302#1878—171no—
303#1865—171no—
304#1862—171no—
305C8×C4⋊C4—171no—
306C4×#103—171no—
307C22×C17—42no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—81no—
311C22×C17—81no—
312C22×C17—81no—
313C22×C17—81no—
314C22×C17C2×C4×D411yescharacteristic
315C22×C17D4211yescharacteristic
316C22×C17#54811yescharacteristic
317C22×C17C2×#13511yescharacteristic
318C22×C17C2×#13511yescharacteristic
319C22×C17#57711yescharacteristic
320C22×C17#57811yescharacteristic
321C4×C17—24no—
322C4×C17—22no—
323C4×C17—22no—
324C4×C17—42no—
325C4×C17—42no—
326C4×C17C2×C4×D411yescharacteristic
327C4×C17#52011yescharacteristic
328C4×C17#54711yescharacteristic
329C4×C17#55311yescharacteristic
330Dic17—22no—
331Dic17—22no—
332Dic17—41no—
333Dic17—42no—
334Dic17—41no—
335Dic17—81no—
336Dic17—81no—
337Dic17—81no—
338C22×Q16C2×D1711yescharacteristic
339C22×SD16C2×D1711yescharacteristic
340C2×#124C2×D1711yescharacteristic
341C2×C4×Q8—171no—
342C2×C4×Q8—171no—
343C2×C4×Q8—171no—
344C2×C4×Q8C2×D1711yescharacteristic
345C2×C4×D4—171no—
346#464Dic1718yescomplemented
347C4×Q16—344no—
348C4×SD16—344no—
349C2×#107—171no—
350C2×#106—171no—
351C2×C4⋊C8C2×D1711yescharacteristic
352C2×#103—171no—
353C2×#103—171no—
354C2×#103—171no—
355C2×#103—171no—
356C2×#103C2×D1711yescharacteristic
357C2×#102—171no—
358C2×#102—171no—
359C2×#102C2×D1711yescharacteristic
360C2×C4×C8—171no—
361C2×C4×C8—171no—
362C2×C4×C8—171no—
363#391—171no—
364#390—171no—
365#389—171no—
366#389—171no—
367#389—341no—
368#389—341no—
369#388—341no—
370#388—341no—
371#385—341no—
372#382—171no—
373#382—341no—
374C4×C4⋊C4—171no—
375C4×C4⋊C4—171no—
376C4×C4⋊C4—341no—
377C4×C22⋊C4—341no—
378#340—171no—
379#340—171no—
380#340—171no—
381#340—171no—
382#336—171no—
383#336—171no—
384C2×C17—82no—
385C2×C17#236111yescharacteristic
386C2×C17C4×#13511yescharacteristic
387C2×C17#294711yescharacteristic
388C2×C17#328211yescharacteristic
389C2×C17#328511yescharacteristic
390C2×C17#336411yescharacteristic
391C2×C17#336511yescharacteristic
392C22×Q8C22×D1711yescharacteristic
393C22×Q8C22×D1711yescharacteristic
394C22×D4C22×D1711yescharacteristic
395C23×C4—341no—
396C2×Q16—22no—
397C2×Q16—22no—
398C2×Q16C2×Dic1714yescomplemented
399C2×SD16—22no—
400C2×SD16—22no—
401C2×SD16C2×Dic1714yes—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8C22×D1711yescharacteristic
405C22×C8C22×D1711yescharacteristic
406#124—22no—
407#124C2×Dic1714yes—
408C4×Q8—22no—
409C4×Q8—342no—
410C4×Q8—342no—
411C4×Q8—342no—
412C4×Q8—681no—
413C4×Q8—681no—
414C4×Q8—681no—
415C4×Q8C2×Dic1714yescomplemented
416C4×D4—342no—
417C4×D4—681no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—341no—
434C2×C22⋊C4—341no—
435C2×C22⋊C4—341no—
436C2×C42—21no—
437C2×C42—171no—
438C2×C42—171no—
439C2×C42—341no—
440C2×C42—341no—
441C2×C42—341no—
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444#107—342no—
445#106—342no—
446C4⋊C8C2×Dic1714yescomplemented
447#103—174no—
448#103—342no—
449#103—342no—
450#103—681no—
451#103C2×Dic1714yescomplemented
452#102—342no—
453#102—681no—
454#102C2×Dic1714yes—
455C4×C8—342no—
456C4×C8—342no—
457C4×C8—342no—
458#95—171no—
459#95—171no—
460#95—341no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467C17#2300011yescharacteristic, complemented, Sylow
468C24—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8D4×D1712yes—
478C2×Q8#6325412yes—
479C2×Q8C22×Dic1712yes—
480C2×Q8C22×Dic1712yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C22×Dic1712yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—171no—
491C22×C4—171no—
492C22×C4—171no—
493C22×C4—171no—
494C22×C4—341no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—682no—
499C22×C4—682no—
500C22×C4—1361no—
501C22×C4D4×D1711yescharacteristic
502C22×C4D4×D1711yescharacteristic
503C22×C4C23×D1711yescharacteristic
504Q16—28no—
505SD16—28no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—172no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8C22×Dic1712yes—
517C2×C8C22×Dic1712yes—
518C4⋊C4—21no—
519C4⋊C4—24no—
520C4⋊C4—172no—
521C4⋊C4—342no—
522C4⋊C4—342no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—682no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—682no—
534C4⋊C4—682no—
535C4⋊C4C22×Dic1712yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—682no—
539C22⋊C4—682no—
540C22⋊C4—1361no—
541C42—24no—
542C42—341no—
543C42—341no—
544C42—341no—
545C42—341no—
546C42—682no—
547C42—682no—
548C42—682no—
549C42—1361no—
550C42#6325412yes—
551C42C22×Dic1712yes—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D1711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8D4×Dic1714yes—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—172no—
586C2×C4—172no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—342no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—1364no—
610C2×C4D4×Dic1712yes—
611C2×C4D4×Dic1712yes—
612C2×C4C2×D4×D1711yescharacteristic
613C2×C4C2×#6325411yescharacteristic
614C2×C4C2×#6325411yescharacteristic
615C2×C4#1058769611yescharacteristic
616C2×C4#1058769711yescharacteristic
617C2×C4C23×Dic1711yescharacteristic
618C8—22no—
619C8—22no—
620C8—342no—
621C8—681no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×D4×Dic1711yescharacteristic
630C22#1176344311yescharacteristic
631C22#1176346511yescharacteristic
632C22#135×D1711yescharacteristic
633C22#135×D1711yescharacteristic
634C22#1176361811yescharacteristic
635C22#1176362511yescharacteristic
636C4—22no—
637C4—22no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—342no—
642C4—342no—
643C4—681no—
644C4—682no—
645C4—681no—
646C4—1361no—
647C4—1361no—
648C4—1361no—
649C4C2×D4×Dic1711yescharacteristic
650C4#1176329211yescharacteristic
651C4#1176345911yescharacteristic
652C4#1176351111yescharacteristic
653C2—82no—
654C2#1453180411yescharacteristic
655C2#135×Dic1711yescharacteristic
656C2#1453762811yescharacteristic
657C2#1454048611yescharacteristic
658C2#1454052011yescharacteristic
659C2#1454146411yescharacteristic
660C2#1454147511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#464C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×#464D17
hypocenterC17#23000
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
1. C2×Q16×Dic17C2semidirect
4. #14531705C2semidirect
6. #14529554C2semidirect
8. #14529462C2semidirect
10. #14529127C2semidirect
11. #14529054C2semidirect
13. #14528649C2semidirect
14. C4×#10587506C2semidirect
58. C17×#464C4semidirect
68. C2×Q16×C17C2×C4semidirect
77. C4×Q8×C17C2×C4semidirect
114. C17×C4⋊C8C2×C4semidirect
115. C17×#103C2×C4semidirect
346. #464Dic17semidirect
398. C2×Q16C2×Dic17semidirect
415. C4×Q8C2×Dic17semidirect
446. C4⋊C8C2×Dic17semidirect
451. #103C2×Dic17semidirect
467. C17#23000semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×Dic172
C2×SD16×Dic172
#145318592
#145317052
#145295962
#145295542
#145294692
#145294622
#145291412
#145291272
#145290542
#145290332
#145286492
C4×#105875062
C2×C17×#4642
#2300017