Finite Group Databasebeta

← #17035302

Subgroups — #17035302

Order 4352.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763277C211yesmaximal, characteristic
2C2×#11762852C211yesmaximal, characteristic, complemented
3C2×#11762811C211yesmaximal, characteristic
4C2×#11762808C211yesmaximal, characteristic
5C2×#11762708C211yesmaximal, characteristic, complemented
6#14529053C212yesmaximal, complemented
7#14529046C212yesmaximal, complemented
8#14529039C212yesmaximal, complemented
9#14529032C212yesmaximal, complemented
10#14528477C211yesmaximal, characteristic, complemented
11C17×#1736C211yesmaximal, characteristic, complemented
12C22×#63252C2211yescharacteristic
13C22×#63223C2211yescharacteristic
14#11763277—24no—
15#11763277—24no—
16C2×#10587573C2211yescharacteristic
17C2×#10587573C2211yescharacteristic
18C2×#10587579C2211yescharacteristic
19C2×#10587579C2211yescharacteristic
20C2×#10587567—21no—
21C2×#10587567—21no—
22C2×#10587558—21no—
23C22×C4⋊Dic17—21no—
24C22×C4⋊Dic17C2211yescharacteristic
25C22×C4×Dic17C2211yescharacteristic
26C2×C42×D17—21no—
27#11762852—24no—
28C2×#10587478C2212yes—
29C2×#10587479C2212yes—
30#11762811—24no—
31#11762808C418yescomplemented
32C2×#10587487C2212yes—
33C2×#10587481C2212yes—
34C2×#10587492C2212yes—
35C2×#10587480C2212yes—
36#11762708—21no—
37#11762708—21no—
38#11762708C2212yes—
39#11762708C2212yes—
40C2×#10587502—21no—
41#11762563C2212yes—
42#11762563C2212yes—
43C22×M16×C17C2211yescharacteristic
44C2×C17×#117C2211yescharacteristic
45C22×C4⋊C4×C17C2211yescharacteristic
46C17×#340C2212yes—
47C17×#340C2212yes—
48C23×C4×C17—21no—
49C23×C4×C17C2311yescharacteristic
50C22×C22⋊D17—21no—
51C23×Dic17—21no—
52C23×Dic17D411yescharacteristic
53C23×Dic17D411yescharacteristic
54C2×#63252—22no—
55C2×#63252—22no—
56C2×#63252—22no—
57C2×#63252C2×C414yes—
58C2×#63252D414yes—
59C2×#63252Q814yes—
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×Dic34C2311yescharacteristic
65C2×M16×C17—22no—
66C2×M16×C17—22no—
67C2×M16×C17—22no—
68C22×C8×C17C2312yes—
69C17×#117—22no—
70C17×#117D414yes—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—22no—
73C2×C4⋊C4×C17C2×C414yescomplemented
74C2×C4⋊C4×C17C2312yes—
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C42×C17—21no—
79C2×#63227—21no—
80C2×#63227—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—22no—
83C2×C4⋊Dic17—22no—
84C2×C4⋊Dic17—22no—
85C2×C4⋊Dic17—22no—
86C2×C4⋊Dic17—44no—
87C2×C4⋊Dic17C2312yes—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×C4×Dic17—22no—
91C2×C4×Dic17—22no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17D411yescharacteristic
95C2×C4×Dic17D411yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C2×C4×Dic17C2311yescharacteristic
98C2×#63223—22no—
99C2×#63223—22no—
100C2×#63223C2×C414yescomplemented
101C22×#3622C2312yes—
102#10587579—22no—
103#10587579—22no—
104#10587579—22no—
105#10587579—22no—
106#10587573—22no—
107#10587573—22no—
108#10587573—22no—
109#10587573—22no—
110#10587567—28no—
111#10587567—28no—
112#10587558—28no—
113C42×D17—28no—
114#10587502—22no—
115#10587502—22no—
116#10587492—24no—
117#10587487—24no—
118#10587481C2×C418yescomplemented
119#10587480C2×C418yescomplemented
120#10587479—24no—
121#10587478—24no—
122C24×C17C2×D411yescharacteristic
123C23×D17—21no—
124C22×C4×C17—24no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—22no—
128C22×C4×C17—42no—
129C22×C4×C17—42no—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C2×D412yes—
132C22×C4×C17C2×D412yes—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×Q812yes—
135C22×C4×C17#4112yes—
136C22×C4×C17#4112yes—
137C22×C4×C17C2411yescharacteristic
138C2×C22⋊D17—24no—
139C2×C22⋊D17—24no—
140C2×C22⋊D17—24no—
141C22×Dic17—21no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—41no—
148C22×Dic17—41no—
149C22×Dic17—42no—
150C22×Dic17—42no—
151C22×Dic17—42no—
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C22×Dic17C2×D411yescharacteristic
156#63252—28no—
157#63252—28no—
158#63252—28no—
159#63252C4⋊C4116yescomplemented
160C2×D68—21no—
161C2×D68—21no—
162C2×D68—21no—
163C2×D68C22×C412yes—
164C2×D68C2×D412yes—
165C2×D68C2×Q812yes—
166C2×C4×D17—24no—
167C2×C4×D17—24no—
168C2×C4×D17—24no—
169C2×C4×D17—24no—
170C2×C4×D17—24no—
171C2×C4×D17—42no—
172C2×C4×D17—42no—
173C2×C4×D17—42no—
174C2×C4×D17—42no—
175C2×Dic34—21no—
176C2×Dic34—21no—
177C2×Dic34—21no—
178C2×Dic34C22×C412yes—
179C2×Dic34C2×D412yes—
180C2×Dic34C2×Q812yes—
181M16×C17—44no—
182C2×C8×C17—22no—
183C2×C8×C17—22no—
184C2×C8×C17—22no—
185C4⋊C4×C17—24no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—22no—
189C4⋊C4×C17C22×C414yes—
190C4⋊C4×C17C2×D412yes—
191C4⋊C4×C17C2×D412yes—
192C22⋊C4×C17—24no—
193C42×C17—24no—
194#63227—24no—
195#63227—24no—
196C4⋊Dic17—22no—
197C4⋊Dic17—22no—
198C4⋊Dic17—44no—
199C4⋊Dic17—42no—
200C4⋊Dic17—44no—
201#63225—24no—
202#63225—24no—
203C4×Dic17—24no—
204C4×Dic17—24no—
205C4×Dic17—22no—
206C4×Dic17—22no—
207C4×Dic17—21no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211#63223—28no—
212C2×#3622—22no—
213C2×#3622—22no—
214C2×#3622C22×C414yes—
215#20875—171nomaximal, Sylow
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17C4×D412yes—
221C23×C17#12012yes—
222C23×C17C22⋊Q812yes—
223C23×C17C22×D411yescharacteristic
224C22×D17—22no—
225C22×D17—22no—
226C22×D17—22no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C22×D17—41no—
231C2×C4×C17—24no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—48no—
235C2×C4×C17—41no—
236C2×C4×C17—42no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—42no—
241C2×C4×C17C2×C4⋊C414yes—
242C2×C4×C17C4×D414yes—
243C2×C4×C17C4×D412yes—
244C2×C4×C17#12012yes—
245C2×C4×C17C22⋊Q812yes—
246C2×C4×C17C22⋊Q814yes—
247C2×C4×C17#12314yes—
248C2×C4×C17C23×C411yescharacteristic
249C2×C4×C17C22×D411yescharacteristic
250C2×C4×C17C22×D411yescharacteristic
251C2×C4×C17C22×Q811yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C22⋊D17—216no—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—22no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—44no—
269C2×Dic17—44no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—84no—
274C2×Dic17AGL(1,ℤ/8)12yes—
275C2×Dic17#13712yes—
276D68—22no—
277D68—22no—
278D68—22no—
279D68C2×C4⋊C414yescomplemented
280C4×D17—216no—
281C4×D17—48no—
282C4×D17—48no—
283Dic34—22no—
284Dic34—22no—
285Dic34—22no—
286Dic34C2×C4⋊C414yescomplemented
287C8×C17—42no—
288#3622—24no—
289C2×C4×#41—171no—
290C2×#428—171no—
291C2×#417—171no—
292C2×#417—171no—
293C2×#389—171no—
294#1865—172no—
295#1864—172no—
296#1863—172no—
297#1862—172no—
298#1736D1711yescharacteristic, complemented
299#1728—171no—
300C22×C17—22no—
301C22×C17—22no—
302C22×C17—22no—
303C22×C17—22no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17#38514yes—
307C22×C17C2×C4×D411yescharacteristic
308C22×C17C2×#12011yescharacteristic
309C22×C17C2×C22⋊Q811yescharacteristic
310C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C17C2×#13711yescharacteristic
312C22×C17#57511yescharacteristic
313C22×C17#57511yescharacteristic
314C2×D17—24no—
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—42no—
318C2×D17—42no—
319C2×D17—42no—
320C2×D17—42no—
321C4×C17—44no—
322C4×C17—42no—
323C4×C17—42no—
324C4×C17#38514yes—
325C4×C17C22×C4⋊C411yescharacteristic
326C4×C17C2×C4×D411yescharacteristic
327C4×C17C2×C22⋊Q811yescharacteristic
328C4×C17C2×#12311yescharacteristic
329Dic17—22no—
330Dic17—22no—
331Dic17—24no—
332Dic17—41no—
333Dic17—41no—
334Dic17—82no—
335C22×#41—171no—
336C22×M16—171no—
337C22×M16C2×D1711yescharacteristic
338C4×#41—344no—
339C4×#41—344no—
340C2×C4×Q8—171no—
341C2×C4×Q8—171no—
342C2×C4×D4—171no—
343C2×C4×D4—171no—
344C2×C4×D4—341no—
345C2×C4×D4—341no—
346C2×#117—341no—
347C2×#117C2×D1711yescharacteristic
348C22×C4⋊C4—171no—
349C22×C4⋊C4—341no—
350C22×C4⋊C4C2×D1711yescharacteristic
351C22×C42—171no—
352C22×C42—341no—
353#428—344no—
354C2×#107—172no—
355C2×#106—172no—
356#417—178no—
357#417—344no—
358C2×#103—172no—
359C2×#103—172no—
360C2×#102—172no—
361C2×#102—172no—
362#389—172no—
363#389—172no—
364#389—341no—
365#389—341no—
366C2×#95—341no—
367#340C2×D1712yes—
368#340C2×D1712yes—
369#336—172no—
370#336—172no—
371C2×C17—24no—
372C2×C17C2×#38511yescharacteristic
373C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
374C2×C17C4×#13711yescharacteristic
375C2×C17#300611yescharacteristic
376C2×C17#300711yescharacteristic
377C2×C17#307911yescharacteristic
378C2×C17#308011yescharacteristic
379D17—44no—
380C2×#41—174no—
381C2×#41—174no—
382C2×#41—174no—
383C2×#41—342no—
384C2×#41—342no—
385C2×#41—342no—
386C22×Q8—171no—
387C22×D4—171no—
388C22×D4—341no—
389C23×C4—21no—
390C23×C4—171no—
391C23×C4—171no—
392C23×C4—341no—
393C23×C4—341no—
394C23×C4—341no—
395C23×C4—341no—
396C23×C4C22×D1711yescharacteristic
397C2×M16—22no—
398C2×M16—22no—
399C2×M16—22no—
400C2×M16—174no—
401C2×M16—342no—
402C2×M16—342no—
403C22×C8—172no—
404C22×C8C22×D1712yes—
405C4×Q8—342no—
406C4×Q8—342no—
407C4×Q8—342no—
408C4×Q8—342no—
409C4×D4—348no—
410C4×D4—348no—
411C4×D4—342no—
412C4×D4—342no—
413C4×D4—342no—
414C4×D4—342no—
415#117—22no—
416#117—348no—
417#117D6814yes—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—172no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—342no—
424C2×C4⋊C4—342no—
425C2×C4⋊C4—342no—
426C2×C4⋊C4—342no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—684no—
429C2×C4⋊C4C4×D1714yescomplemented
430C2×C4⋊C4C22×D1712yes—
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—341no—
435C2×C22⋊C4—341no—
436C2×C42—21no—
437C2×C42—171no—
438C2×C42—171no—
439C2×C42—171no—
440C2×C42—171no—
441C2×C42—348no—
442C2×C42—341no—
443C2×C42—341no—
444C2×C42—342no—
445C2×C42—342no—
446#107—344no—
447#106—344no—
448#103—178no—
449#103—344no—
450#102—178no—
451#102—344no—
452#95—342no—
453#95—342no—
454C17#2087511yescharacteristic, complemented, Sylow
455C24—341no—
456C24D4×D1711yescharacteristic
457#41—1716no—
458#41—348no—
459#41—348no—
460#41—348no—
461C2×Q8—172no—
462C2×Q8—172no—
463C2×Q8—172no—
464C2×Q8—341no—
465C2×Q8—341no—
466C2×Q8—341no—
467C2×D4—172no—
468C2×D4—172no—
469C2×D4—172no—
470C2×D4—344no—
471C2×D4—344no—
472C2×D4—344no—
473C2×D4—341no—
474C2×D4—341no—
475C2×D4—341no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—24no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—171no—
483C22×C4—171no—
484C22×C4—171no—
485C22×C4—171no—
486C22×C4—344no—
487C22×C4—344no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—341no—
491C22×C4—344no—
492C22×C4—344no—
493C22×C4—344no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—682no—
498C22×C4—682no—
499C22×C4—682no—
500C22×C4—682no—
501C22×C4—681no—
502C22×C4—681no—
503C22×C4—682no—
504C22×C4—682no—
505C22×C4—682no—
506C22×C4C2×C4×D1712yes—
507C22×C4C2×D6812yes—
508C22×C4#6325212yes—
509C22×C4D4×D1712yes—
510C22×C4D4×D1711yescharacteristic
511C22×C4#6325412yes—
512C22×C4Q8×D1712yes—
513C22×C4C23×D1711yescharacteristic
514M16—44no—
515M16—348no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—174no—
520C2×C8—342no—
521C2×C8—342no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—344no—
527C4⋊C4—344no—
528C4⋊C4—342no—
529C4⋊C4—342no—
530C4⋊C4—684no—
531C4⋊C4—684no—
532C4⋊C4—682no—
533C4⋊C4C2×C4×D1714yes—
534C4⋊C4C2×D6812yes—
535C4⋊C4C2×D6812yes—
536C22⋊C4—24no—
537C22⋊C4—344no—
538C22⋊C4—344no—
539C42—24no—
540C42—344no—
541C42—344no—
542C42—341no—
543C42—341no—
544C42—341no—
545C42—341no—
546C42—342no—
547C42—342no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—342no—
553C23—342no—
554C23—342no—
555C23—681no—
556C23—681no—
557C23—681no—
558C23—681no—
559C23#1058756712yes—
560C23#1058756812yes—
561C23#1058758212yes—
562C23C2×D4×D1711yescharacteristic
563Q8—174no—
564Q8—342no—
565Q8—342no—
566Q8—342no—
567D4—174no—
568D4—3416no—
569D4—342no—
570D4—342no—
571D4—342no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—172no—
583C2×C4—172no—
584C2×C4—342no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—342no—
588C2×C4—342no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—3416no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—688no—
599C2×C4—688no—
600C2×C4—684no—
601C2×C4—684no—
602C2×C4—682no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4—1364no—
606C2×C4C4×D6814yes—
607C2×C4#1058756712yes—
608C2×C4#1058756812yes—
609C2×C4#1058756914yes—
610C2×C4C4⋊C4×D1714yes—
611C2×C4#1058758212yes—
612C2×C4#1058758314yes—
613C2×C4C22×C4×D1711yescharacteristic
614C2×C4C22×D6811yescharacteristic
615C2×C4C2×#6325211yescharacteristic
616C2×C4C2×D4×D1711yescharacteristic
617C2×C4C2×#6325411yescharacteristic
618C2×C4C2×Q8×D1711yescharacteristic
619C8—42no—
620C8—344no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—344no—
628C22—682no—
629C22—682no—
630C22—682no—
631C22—682no—
632C22—682no—
633C22—682no—
634C22#1176268314yes—
635C22C2×#1058756711yescharacteristic
636C22C2×#1058756811yescharacteristic
637C22C2×#1058758211yescharacteristic
638C22AGL(1,ℤ/8)×D1711yescharacteristic
639C22#137×D1711yescharacteristic
640C22#1176360511yescharacteristic
641C22#1176360811yescharacteristic
642C4—42no—
643C4—42no—
644C4—44no—
645C4—342no—
646C4—342no—
647C4—344no—
648C4—681no—
649C4—681no—
650C4—1362no—
651C4#1176268314yes—
652C4C2×C4⋊C4×D1711yescharacteristic
653C4C2×C4×D6811yescharacteristic
654C4C2×#1058758311yescharacteristic
655C4C2×#1058756911yescharacteristic
656C2—24no—
657C2—684no—
658C2C2×#1176268311yescharacteristic
659C2#1453758811yescharacteristic
660C2#1453759911yescharacteristic
661C2#1453803711yescharacteristic
662C2#1453804511yescharacteristic
663C2#1453859711yescharacteristic
664C2#1453860511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1736C2
socleC23×C17C22×D4
prime radicalC22×#63252C22
cosocleC22×C4×C17C24
hypercenter#1736D17
hypocenterC17#20875
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#11762852C2semidirect
5. C2×#11762708C2semidirect
6. #14529053C2semidirect
7. #14529046C2semidirect
8. #14529039C2semidirect
9. #14529032C2semidirect
10. #14528477C2semidirect
11. C17×#1736C2semidirect
31. #11762808C4semidirect
73. C2×C4⋊C4×C17C2×C4semidirect
100. C2×#63223C2×C4semidirect
118. #10587481C2×C4semidirect
119. #10587480C2×C4semidirect
159. #63252C4⋊C4semidirect
279. D68C2×C4⋊C4semidirect
286. Dic34C2×C4⋊C4semidirect
298. #1736D17semidirect
429. C2×C4⋊C4C4×D17semidirect
454. C17#20875semidirect

Maximal subgroups

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

subgroupindex
C2×#117632772
C2×#117628522
C2×#117628112
C2×#117628082
C2×#117627082
#145290532
#145290462
#145290392
#145290322
#145284772
C17×#17362
#2087517