Finite Group Databasebeta

← #17056619

Subgroups — #17056619

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×#11763031C211yesmaximal, characteristic
2C2×#11762901C211yesmaximal, characteristic, complemented
3C2×#11762894C211yesmaximal, characteristic
4#14531855C211yesmaximal, characteristic
5#14531705C211yesmaximal, characteristic, complemented
6#14529595C211yesmaximal, characteristic
7#14529552C211yesmaximal, characteristic, complemented
8#14529468C211yesmaximal, characteristic
9#14529461C211yesmaximal, characteristic, complemented
10#14529137C211yesmaximal, characteristic
11#14529123C211yesmaximal, characteristic, complemented
12#14529053C211yesmaximal, characteristic, complemented
13#14529032C211yesmaximal, characteristic
14C8×C4⋊Dic17C211yesmaximal, characteristic, complemented
15C4×C17×#103C211yesmaximal, characteristic, complemented
16C22×#63231C2211yescharacteristic
17C22×#63230C2211yescharacteristic
18C2×#10587561C2211yescharacteristic
19C2×Q8×Dic17C2211yescharacteristic
20C2×Q8×Dic17C2211yescharacteristic
21C2×#10587573C2211yescharacteristic
22C2×#10587579C2211yescharacteristic
23#11763031C418yescomplemented
24#11762901—24no—
25#11762894—24no—
26C2×#10587478C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587492C2211yescharacteristic
33C2×#10587492C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36C2×C8×Dic17C2211yescharacteristic
37#11762714C2211yescharacteristic
38#11762710C2211yescharacteristic
39#11762708—21no—
40#11762708—21no—
41#11762708C2211yescharacteristic
42#11762708C2211yescharacteristic
43#11762701—21no—
44#11762700—21no—
45#11762683—21no—
46#11762675—21no—
47#11762675C2211yescharacteristic
48C4×C4⋊Dic17—21no—
49C4×C4⋊Dic17C2211yescharacteristic
50C4×#63227—21no—
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762563C2211yescharacteristic
54#11762563C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C17×#103C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C2×C4×C8×C17C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×Dic34C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C4×Q8×C17—22no—
69C4×Q8×C17C2×C414yescomplemented
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C42×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17C2311yescharacteristic
77Q8×Dic17—22no—
78Q8×Dic17—22no—
79Q8×Dic17—41no—
80Q8×Dic17—41no—
81C2×#63231—22no—
82C2×#63231—22no—
83C2×#63231C2×C414yescomplemented
84C2×#63230—22no—
85C2×#63230—22no—
86C2×#63230C2×C414yes—
87C2×#63227—21no—
88C2×#63227—21no—
89C2×#63227—21no—
90C2×#63227—21no—
91C2×#63227—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17—21no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17D411yescharacteristic
97C2×C4⋊Dic17D411yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×#63225—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17C2311yescharacteristic
105C2×C4×Dic17C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108#10587579—22no—
109#10587579—41no—
110#10587573—22no—
111#10587573—41no—
112#10587561—22no—
113#10587561C2×C414yes—
114C17×#103—22no—
115C17×#103D414yes—
116C4×C8×C17—22no—
117C17×#95—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502D411yescharacteristic
125#10587502D411yescharacteristic
126#10587492—22no—
127#10587492—41no—
128#10587487—22no—
129#10587487—41no—
130C8×Dic17—22no—
131C8×Dic17—22no—
132#10587481C2×C414yescomplemented
133#10587480C2×C414yes—
134#10587479—22no—
135#10587478—22no—
136C4⋊#3622C2×C414yescomplemented
137C23×D17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17C22×C412yes—
142C2×Q8×C17C2×D412yes—
143C2×Q8×C17#4112yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17C2×D411yescharacteristic
155C22×Dic17C2×D411yescharacteristic
156C22×Dic17C2×D411yescharacteristic
157C22×Dic17C2×D411yescharacteristic
158C2×D68—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68—21no—
163C2×D68C22×C412yes—
164C2×C4×D17—42no—
165C2×C4×D17—42no—
166C2×C4×D17—81no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34C22×C412yes—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17C2×D412yes—
179C4⋊C4×C17—24no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17C22×C412yes—
185C4⋊C4×C17C2×D412yes—
186C42×C17—24no—
187C42×C17—81no—
188C42×C17C22×C412yes—
189C42×C17#4112yes—
190#63231—28no—
191#63230—28no—
192#63227—24no—
193#63227—24no—
194#63227—42no—
195#63227—42no—
196#63227—81no—
197C4⋊Dic17—22no—
198C4⋊Dic17—22no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—42no—
204C4⋊Dic17—42no—
205C4⋊Dic17—42no—
206C4⋊Dic17—42no—
207#63225—42no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—21no—
212C4×Dic17—42no—
213C4×Dic17—42no—
214C4×Dic17—42no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622C22×C412yes—
220C2×#3622C22×C412yes—
221#23000—171nomaximal, Sylow
222C23×C17C22×D411yescharacteristic
223C22×D17—22no—
224C22×D17—41no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230Q8×C17—22no—
231Q8×C17—22no—
232Q8×C17—22no—
233Q8×C17C4×D414yes—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17C4×D412yes—
251C2×C4×C17C4×D412yes—
252C2×C4×C17C23×C411yescharacteristic
253C2×C4×C17C22×D411yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17H(5,𝔽2)11yescharacteristic
257C2×C4×C172−1+411yescharacteristic
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—22no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17#13512yes—
278C2×Dic17#13512yes—
279D68—22no—
280D68—22no—
281D68—41no—
282D68—41no—
283C4×D17—84no—
284Dic34—22no—
285Dic34—22no—
286Dic34—41no—
287Dic34—41no—
288C8×C17—22no—
289C8×C17—41no—
290#3622—22no—
291#3622—22no—
292C2×#464—171no—
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×#103D1711yescharacteristic, complemented
307C22×C17C2×C4×D411yescharacteristic
308C22×C17D4211yescharacteristic
309C22×C17#54811yescharacteristic
310C22×C17C2×#13511yescharacteristic
311C22×C17C2×#13511yescharacteristic
312C22×C17#57711yescharacteristic
313C22×C17#57811yescharacteristic
314C2×D17—42no—
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—81no—
318C2×D17—81no—
319C2×D17—81no—
320C2×D17—81no—
321C4×C17—24no—
322C4×C17—22no—
323C4×C17—22no—
324C4×C17—42no—
325C4×C17—42no—
326C4×C17—81no—
327C4×C17C2×C4×D411yescharacteristic
328C4×C17#52011yescharacteristic
329C4×C17#54711yescharacteristic
330C4×C17#55311yescharacteristic
331Dic17—22no—
332Dic17—22no—
333Dic17—42no—
334Dic17—41no—
335Dic17—41no—
336Dic17—81no—
337Dic17—81no—
338C22×Q16—171no—
339C22×SD16—171no—
340C2×#124—171no—
341C2×C4×Q8—171no—
342C2×C4×Q8—171no—
343C2×C4×Q8—171no—
344C2×C4×Q8C2×D1711yescharacteristic
345C2×C4×D4—171no—
346#464—178no—
347C4×Q16—344no—
348C4×SD16—344no—
349C2×#107—171no—
350C2×#106—171no—
351C2×C4⋊C8—171no—
352C2×#103—171no—
353C2×#103—171no—
354C2×#103—171no—
355C2×#103C2×D1711yescharacteristic
356C2×#103C2×D1711yescharacteristic
357C2×#102—171no—
358C2×#102—171no—
359C2×#102—171no—
360C2×C4×C8—171no—
361C2×C4×C8—171no—
362C2×C4×C8C2×D1711yescharacteristic
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—341no—
376C4×C4⋊C4C2×D1711yescharacteristic
377C4×C22⋊C4—341no—
378#340—171no—
379#340—171no—
380#340C2×D1711yescharacteristic
381#340C2×D1711yescharacteristic
382#336—171no—
383#336—171no—
384C2×C17#236111yescharacteristic
385C2×C17C4×#13511yescharacteristic
386C2×C17#294711yescharacteristic
387C2×C17#328211yescharacteristic
388C2×C17#328511yescharacteristic
389C2×C17#336411yescharacteristic
390C2×C17#336511yescharacteristic
391D17—82no—
392C22×Q8—171no—
393C22×Q8C22×D1711yescharacteristic
394C22×D4—171no—
395C23×C4—341no—
396C2×Q16—174no—
397C2×Q16—342no—
398C2×Q16—342no—
399C2×SD16—174no—
400C2×SD16—342no—
401C2×SD16—342no—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8C22×D1711yescharacteristic
405C22×C8C22×D1711yescharacteristic
406#124—174no—
407#124—342no—
408C4×Q8—22no—
409C4×Q8—342no—
410C4×Q8—342no—
411C4×Q8—342no—
412C4×Q8—681no—
413C4×Q8—681no—
414C4×Q8—681no—
415C4×Q8C4×D1714yescomplemented
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—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C4⋊C4C22×D1711yescharacteristic
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C22⋊C4—341no—
435C2×C22⋊C4—341no—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—171no—
439C2×C42—171no—
440C2×C42—341no—
441C2×C42—341no—
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444#107—342no—
445#106—342no—
446C4⋊C8—174no—
447#103—22no—
448#103—174no—
449#103—342no—
450#103—681no—
451#103D6814yes—
452#102—174no—
453#102—342no—
454#102—681no—
455C4×C8—22no—
456C4×C8—342no—
457C4×C8—342no—
458#95—21no—
459#95—171no—
460#95—171no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467C17#2300011yescharacteristic, complemented, Sylow
468C24—341no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—172no—
473C2×Q8—341no—
474C2×Q8—341no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×Q8—341no—
478C2×Q8C2×C4×D1712yes—
479C2×Q8C2×D6812yes—
480C2×Q8#6325212yes—
481C2×D4—172no—
482C2×D4—341no—
483C2×D4—341no—
484C2×D4—341no—
485C2×D4—341no—
486C2×D4—341no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—171no—
492C22×C4—171no—
493C22×C4—171no—
494C22×C4—171no—
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—348no—
505SD16—348no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—172no—
512C2×C8—172no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8C2×D6812yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—342no—
524C4⋊C4—342no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—682no—
530C4⋊C4—682no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—682no—
534C4⋊C4C2×C4×D1712yes—
535C4⋊C4C2×D6812yes—
536C22⋊C4—344no—
537C22⋊C4—344no—
538C22⋊C4—682no—
539C22⋊C4—682no—
540C22⋊C4—1361no—
541C42—24no—
542C42—81no—
543C42—341no—
544C42—341no—
545C42—341no—
546C42—341no—
547C42—682no—
548C42—682no—
549C42—682no—
550C42C2×C4×D1712yes—
551C42#6325412yes—
552C23—342no—
553C23—681no—
554C23—681no—
555C23—681no—
556C23—681no—
557C23—681no—
558C23—681no—
559C23C2×D4×D1711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—342no—
564Q8—342no—
565Q8—681no—
566Q8—681no—
567Q8C4×D6814yes—
568D4—342no—
569D4—342no—
570D4—681no—
571D4—681no—
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—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—172no—
589C2×C4—172no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—342no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—341no—
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×C4#1058756712yes—
611C2×C4#1058756712yes—
612C2×C4C22×C4×D1711yescharacteristic
613C2×C4C22×D6811yescharacteristic
614C2×C4C2×#6325211yescharacteristic
615C2×C4C2×#6325411yescharacteristic
616C2×C4#1058769611yescharacteristic
617C2×C4#1058769711yescharacteristic
618C8—22no—
619C8—41no—
620C8—342no—
621C8—342no—
622C22—682no—
623C22—682no—
624C22—682no—
625C22—1361no—
626C22—1361no—
627C22—1361no—
628C22—1361no—
629C22C2×#1058756711yescharacteristic
630C22D4×D6811yescharacteristic
631C22#1176346811yescharacteristic
632C22#135×D1711yescharacteristic
633C22#135×D1711yescharacteristic
634C22#1176361511yescharacteristic
635C22#1176362211yescharacteristic
636C4—22no—
637C4—22no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—342no—
643C4—342no—
644C4—681no—
645C4—682no—
646C4—681no—
647C4—1361no—
648C4—1361no—
649C4C2×C4×D6811yescharacteristic
650C4#1176328911yescharacteristic
651C4#1176346311yescharacteristic
652C4#1176350511yescharacteristic
653C2—1362no—
654C2#1453179811yescharacteristic
655C2#1453755211yescharacteristic
656C2#1453762111yescharacteristic
657C2#1454049311yescharacteristic
658C2#1454052611yescharacteristic
659C2#1454147011yescharacteristic
660C2#1454148111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC4×C17×#103C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenterC4×#103D17
hypocenterC17#23000
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#11762901C2semidirect
5. #14531705C2semidirect
7. #14529552C2semidirect
9. #14529461C2semidirect
11. #14529123C2semidirect
12. #14529053C2semidirect
14. C8×C4⋊Dic17C2semidirect
15. C4×C17×#103C2semidirect
23. #11763031C4semidirect
69. C4×Q8×C17C2×C4semidirect
83. C2×#63231C2×C4semidirect
132. #10587481C2×C4semidirect
136. C4⋊#3622C2×C4semidirect
306. C4×#103D17semidirect
415. C4×Q8C4×D17semidirect
467. C17#23000semidirect

Maximal subgroups

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

subgroupindex
C2×#117630312
C2×#117629012
C2×#117628942
#145318552
#145317052
#145295952
#145295522
#145294682
#145294612
#145291372
#145291232
#145290532
#145290322
C8×C4⋊Dic172
C4×C17×#1032
#2300017