Finite Group Databasebeta

← #17052668

Subgroups — #17052668

Order 4352.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762910C211yesmaximal, characteristic
2C2×SD16×Dic17C211yesmaximal, characteristic
3#14531702C211yesmaximal, characteristic, complemented
4D4×#63225C211yesmaximal, characteristic
5#14529522C211yesmaximal, characteristic, complemented
6#14529515C211yesmaximal, characteristic, complemented
7#14529050C211yesmaximal, characteristic, complemented
8#14529045C211yesmaximal, characteristic
9#14529038C211yesmaximal, characteristic, complemented
10#14529029C211yesmaximal, characteristic
11#14528951C211yesmaximal, characteristic, complemented
12#14528792C211yesmaximal, characteristic
13#14528787C211yesmaximal, characteristic, complemented
14#14528660C211yesmaximal, characteristic, complemented
15C2×C17×#440C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19C2×D4×Dic17C2211yescharacteristic
20C22×#63225—21no—
21#11762910C418yescomplemented
22SD16×Dic17—24no—
23C2×#10587489C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587488C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587485C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587486C2211yescharacteristic
33C2×C8×Dic17C2211yescharacteristic
34#11762707—21no—
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762705—21no—
38#11762705C2211yescharacteristic
39#11762705C2211yescharacteristic
40#11762682—21no—
41#11762681—21no—
42#11762678C2211yescharacteristic
43#11762674—21no—
44#11762674C2211yescharacteristic
45#11762667—21no—
46C4×#63225—21no—
47C4×#63225C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762564C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×SD16×C17C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C4×D4×C17C2211yescharacteristic
57C17×#440—24no—
58C2×C17×#107C2211yescharacteristic
59C2×C17×#103C2211yescharacteristic
60C2×C17×#102C2211yescharacteristic
61C2×C17×#97C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D4×C17C2311yescharacteristic
64C23×C4×C17—21no—
65C23×Dic17—21no—
66C23×Dic17—21no—
67C2×SD16×C17C2×C414yes—
68C2×SD16×C17D414yes—
69C2×SD16×C17Q814yes—
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C4×Q8×C17—22no—
73C4×Q8×C17—41no—
74C4×D4×C17—22no—
75C4×D4×C17—41no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84Q8×Dic17—22no—
85Q8×Dic17—22no—
86Q8×Dic17—41no—
87Q8×Dic17C2×C414yescomplemented
88D4×Dic17—22no—
89D4×Dic17—22no—
90D4×Dic17—41no—
91D4×Dic17C2×C414yes—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×#63225—21no—
104C2×#63225—44no—
105C2×#63225D411yescharacteristic
106C2×#63225D411yescharacteristic
107C2×C4×Dic17—21no—
108C2×C4×Dic17—21no—
109C2×C4×Dic17C2311yescharacteristic
110C2×C4×Dic17C2311yescharacteristic
111C22×#3622C2311yescharacteristic
112C22×#3622C2311yescharacteristic
113C17×#107—22no—
114C17×#103—22no—
115C17×#102—22no—
116C17×#97—22no—
117#10587506—22no—
118#10587506C2×C414yescomplemented
119#10587503—22no—
120#10587503C2×C414yes—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502D411yescharacteristic
128#10587502D411yescharacteristic
129#10587489C2×C414yescomplemented
130#10587488—22no—
131#10587486C2×C414yescomplemented
132#10587485—22no—
133C8×Dic17—22no—
134#10587479—41no—
135#10587478—41no—
136C24×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17C22×C412yes—
141C2×Q8×C17C2×D412yes—
142C2×Q8×C17C2×Q812yes—
143C2×D4×C17—21no—
144C2×D4×C17—21no—
145C2×D4×C17—21no—
146C2×D4×C17C22×C412yes—
147C2×D4×C17C2×D412yes—
148C2×D4×C17C2×Q812yes—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—42no—
153C22×C4×C17—42no—
154C22×C4×C17—81no—
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2411yescharacteristic
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—24no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17—42no—
164C22×Dic17—42no—
165C22×Dic17—42no—
166C22×Dic17—42no—
167C22×Dic17—81no—
168C22×Dic17C2×D411yescharacteristic
169C22×Dic17C2×D411yescharacteristic
170C22×Dic17C2×D411yescharacteristic
171C22×Dic17C2×D411yescharacteristic
172SD16×C17C4⋊C4116yescomplemented
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17C22×C412yes—
177C2×C8×C17C2×D412yes—
178C2×C8×C17C2×Q812yes—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—42no—
184C22⋊C4×C17—42no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—42no—
188C22⋊Dic17—24no—
189C22⋊Dic17—42no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—24no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—42no—
198C4⋊Dic17C22×C412yes—
199C4⋊Dic17C22×C412yes—
200#63225—22no—
201#63225—41no—
202#63225—42no—
203#63225—42no—
204#63225—42no—
205#63225—42no—
206#63225—42no—
207#63225—81no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—24no—
211C4×Dic17—42no—
212C4×Dic17C22×C412yes—
213C4×Dic17#4112yes—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622C22×C412yes—
220#22677—171nomaximal, Sylow
221C23×C17—22no—
222C23×C17—22no—
223C23×C17—22no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17C22×D411yescharacteristic
229Q8×C17—22no—
230Q8×C17—22no—
231Q8×C17—22no—
232Q8×C17C2×C4⋊C414yescomplemented
233D4×C17—22no—
234D4×C17—22no—
235D4×C17—22no—
236D4×C17C2×C4⋊C414yes—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—84no—
253C2×C4×C17C23×C411yescharacteristic
254C2×C4×C17C22×D411yescharacteristic
255C2×C4×C17C22×Q811yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17H(5,𝔽2)11yescharacteristic
258C2×C4×C172−1+411yescharacteristic
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—48no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—84no—
282C2×Dic17C4×D412yes—
283C2×Dic17C4×D412yes—
284C2×Dic17AGL(1,ℤ/8)12yes—
285C8×C17—41no—
286C8×C17C2×C4⋊C414yescomplemented
287#3622—22no—
288#3622—41no—
289C2×#440—171no—
290C2×#440D1711yescharacteristic
291C2×C4×SD16—171no—
292#2351—171no—
293D4×C4⋊C4—171no—
294#1943—171no—
295#1942—171no—
296#1865—171no—
297#1864—171no—
298#1863—171no—
299#1862—171no—
300#1847—171no—
301#1804—171no—
302#1803—171no—
303#1771—171no—
304C22×C17—24no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—42no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—42no—
311C22×C17C2×C4×D411yescharacteristic
312C22×C17D4211yescharacteristic
313C22×C17D4×Q811yescharacteristic
314C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C17#57511yescharacteristic
316C22×C17#57711yescharacteristic
317C22×C17#57811yescharacteristic
318C4×C17—24no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17—81no—
322C4×C17—81no—
323C4×C17C22×C4⋊C411yescharacteristic
324C4×C17#52011yescharacteristic
325C4×C17#54711yescharacteristic
326C4×C17#55411yescharacteristic
327Dic17—22no—
328Dic17—22no—
329Dic17—22no—
330Dic17—42no—
331Dic17—42no—
332Dic17—41no—
333Dic17—81no—
334Dic17—81no—
335C22×SD16C2×D1711yescharacteristic
336C2×C4×Q8—171no—
337C2×C4×Q8—171no—
338C2×C4×Q8C2×D1711yescharacteristic
339C2×C4×D4—171no—
340C2×C4×D4—171no—
341C2×C4×D4C2×D1711yescharacteristic
342C22×C4⋊C4—341no—
343#440—24no—
344#440—178no—
345C4×SD16—344no—
346C2×#107—171no—
347C2×#107—171no—
348C2×#107C2×D1711yescharacteristic
349C2×#106—171no—
350C2×#106—171no—
351C2×C4⋊C8—171no—
352C2×#103—171no—
353C2×#103—171no—
354C2×#103C2×D1711yescharacteristic
355C2×#102—171no—
356C2×#102—171no—
357C2×#102C2×D1711yescharacteristic
358C2×#97—171no—
359C2×#97C2×D1711yescharacteristic
360C2×C4×C8—171no—
361#389—171no—
362#389—171no—
363#389—171no—
364#389—171no—
365#389—341no—
366#389—341no—
367#385—341no—
368#385—341no—
369#384—171no—
370#382—171no—
371#382—341no—
372#380—341no—
373C4×C4⋊C4—171no—
374C4×C4⋊C4—341no—
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#336—171no—
380#336—171no—
381C2×C17—44no—
382C2×C17D4×C4⋊C411yescharacteristic
383C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
384C2×C17#294711yescharacteristic
385C2×C17#326511yescharacteristic
386C2×C17#326911yescharacteristic
387C2×C17#338611yescharacteristic
388C2×C17#338811yescharacteristic
389C22×Q8C22×D1711yescharacteristic
390C22×D4C22×D1711yescharacteristic
391C23×C4—21no—
392C23×C4—341no—
393C23×C4—341no—
394C2×SD16Dic3414yes—
395C2×SD16C4×D1714yes—
396C2×SD16C22⋊D1714yes—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8C22×D1711yescharacteristic
400C22×C8C22×D1711yescharacteristic
401C4×Q8—22no—
402C4×Q8—41no—
403C4×Q8—174no—
404C4×Q8—342no—
405C4×Q8—342no—
406C4×Q8—681no—
407C4×D4—22no—
408C4×D4—41no—
409C4×D4—174no—
410C4×D4—342no—
411C4×D4—342no—
412C4×D4—681no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—684no—
429C2×C4⋊C4C22×D1711yescharacteristic
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C42—21no—
435C2×C42—171no—
436C2×C42—171no—
437C2×C42—341no—
438C2×C42—341no—
439C2×C42C22×D1711yescharacteristic
440#107—22no—
441#107—174no—
442#107—681no—
443#106—342no—
444#106—681no—
445C4⋊C8—342no—
446#103—22no—
447#103—174no—
448#103—342no—
449#102—22no—
450#102—174no—
451#102—342no—
452#97—22no—
453#97—174no—
454C4×C8—342no—
455#95—171no—
456#95—171no—
457#95—341no—
458#95—341no—
459#95—341no—
460#95—341no—
461#95—341no—
462#95—341no—
463C17#2267711yescharacteristic, complemented, Sylow
464C24—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C2×Dic3412yes—
469C2×Q8C2×C4×D1712yes—
470C2×Q8C2×C22⋊D1712yes—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4C2×Dic3412yes—
475C2×D4C2×C4×D1712yes—
476C2×D4C2×C22⋊D1712yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—171no—
484C22×C4—171no—
485C22×C4—171no—
486C22×C4—171no—
487C22×C4—344no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—682no—
493C22×C4—682no—
494C22×C4—682no—
495C22×C4—682no—
496C22×C4—1361no—
497C22×C4D4×D1711yescharacteristic
498C22×C4D4×D1711yescharacteristic
499C22×C4C23×D1711yescharacteristic
500SD16#63225116yescomplemented
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—172no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8C2×Dic3412yes—
511C2×C8C2×C4×D1712yes—
512C2×C8C2×C22⋊D1712yes—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—172no—
519C4⋊C4—172no—
520C4⋊C4—342no—
521C4⋊C4—344no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—681no—
529C4⋊C4—682no—
530C4⋊C4—682no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—682no—
534C4⋊C4—682no—
535C4⋊C4—1361no—
536C22⋊C4—42no—
537C22⋊C4—344no—
538C22⋊C4—682no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—172no—
543C42—172no—
544C42—344no—
545C42—341no—
546C42—341no—
547C42—682no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×D4×D1711yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—22no—
559Q8C2×#6322514yescomplemented
560D4—22no—
561D4—22no—
562D4—22no—
563D4C2×#6322514yes—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—172no—
581C2×C4—172no—
582C2×C4—172no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—342no—
590C2×C4—342no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—688no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—1364no—
606C2×C4C22×Dic3411yescharacteristic
607C2×C4C22×C4×D1711yescharacteristic
608C2×C4#1058769111yescharacteristic
609C2×C4#41×D1711yescharacteristic
610C2×C4#1058769711yescharacteristic
611C2×C4C22×C22⋊D1711yescharacteristic
612C8—41no—
613C8—342no—
614C8—681no—
615C8C2×#6322514yescomplemented
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C4×D4×D1711yescharacteristic
624C22D4×C22⋊D1711yescharacteristic
625C22D4×Dic3411yescharacteristic
626C22AGL(1,ℤ/8)×D1711yescharacteristic
627C22#1176360811yescharacteristic
628C22#1176361911yescharacteristic
629C22#1176362511yescharacteristic
630C4—24no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—342no—
636C4—342no—
637C4—342no—
638C4—681no—
639C4—682no—
640C4—682no—
641C4—1361no—
642C4—1361no—
643C4C22×#6322511yescharacteristic
644C4#1176328811yescharacteristic
645C4#1176345811yescharacteristic
646C4#1176351811yescharacteristic
647C2—44no—
648C2D4×#6322511yescharacteristic
649C2#1453758911yescharacteristic
650C2#1453762411yescharacteristic
651C2#1454027311yescharacteristic
652C2#1454033311yescharacteristic
653C2#1454170611yescharacteristic
654C2#1454173011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#440C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×#440D17
hypocenterC17#22677
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. #14531702C2semidirect
5. #14529522C2semidirect
6. #14529515C2semidirect
7. #14529050C2semidirect
9. #14529038C2semidirect
11. #14528951C2semidirect
13. #14528787C2semidirect
14. #14528660C2semidirect
21. #11762910C4semidirect
87. Q8×Dic17C2×C4semidirect
118. #10587506C2×C4semidirect
129. #10587489C2×C4semidirect
131. #10587486C2×C4semidirect
172. SD16×C17C4⋊C4semidirect
232. Q8×C17C2×C4⋊C4semidirect
286. C8×C17C2×C4⋊C4semidirect
463. C17#22677semidirect
500. SD16#63225semidirect
559. Q8C2×#63225semidirect
615. C8C2×#63225semidirect

Maximal subgroups

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

subgroupindex
C2×#117629102
C2×SD16×Dic172
#145317022
D4×#632252
#145295222
#145295152
#145290502
#145290452
#145290382
#145290292
#145289512
#145287922
#145287872
#145286602
C2×C17×#4402
#2267717