Finite Group Databasebeta

← #17037352

Subgroups — #17037352

Order 4352.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762921C211yesmaximal, characteristic
2#14531246C211yesmaximal, characteristic, complemented
3C4×D4×Dic17C211yesmaximal, characteristic, complemented
4#14529414C211yesmaximal, characteristic, complemented
5#14529401C211yesmaximal, characteristic, complemented
6#14529370C211yesmaximal, characteristic, complemented
7#14529361C211yesmaximal, characteristic, complemented
8#14529134C211yesmaximal, characteristic
9#14529045C211yesmaximal, characteristic, complemented
10#14529029C211yesmaximal, characteristic, complemented
11#14528680C211yesmaximal, characteristic, complemented
12#14528611C211yesmaximal, characteristic, complemented
13#14528611C211yesmaximal, characteristic, complemented
14#14528531C211yesmaximal, characteristic, complemented
15C17×#1879C211yesmaximal, characteristic
16C2×D4×Dic17C2211yescharacteristic
17C2×D4×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19C2×#10587563—21no—
20C22×C4×Dic17—21no—
21#11762921C418yescomplemented
22C2×#10587478C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31#11762712C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762705C2211yescharacteristic
34#11762703—21no—
35#11762697—21no—
36C4⋊C4×Dic17C2211yescharacteristic
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40C22⋊C4×Dic17—21no—
41C22⋊C4×Dic17—21no—
42C22⋊C4×Dic17—21no—
43C4×C22⋊Dic17—21no—
44C42×Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762564C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×D8×C17C2211yescharacteristic
54C2×C4×D4×C17C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C17×#390C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×Dic17—21no—
64C23×Dic17—21no—
65C23×Dic17—21no—
66C2×D8×C17—22no—
67C2×D8×C17—22no—
68C2×D8×C17C2×C414yes—
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C4×D4×C17—22no—
72C4×D4×C17—22no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C42×C17C2311yescharacteristic
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84D4×Dic17—22no—
85D4×Dic17—22no—
86D4×Dic17—22no—
87D4×Dic17—22no—
88D4×Dic17C2×C414yes—
89D4×Dic17C2×C414yescomplemented
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—44no—
101C2×C4×Dic17D411yescharacteristic
102C2×C4×Dic17D411yescharacteristic
103C2×C4×Dic17D411yescharacteristic
104C2×C4×Dic17D411yescharacteristic
105C2×C4×Dic17C2311yescharacteristic
106C2×C4×Dic17C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109#10587563—44no—
110C17×#102—22no—
111C17×#102—22no—
112#10587503—22no—
113#10587503—22no—
114#10587503C2×C414yes—
115#10587503C2×C414yescomplemented
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587488C2×C414yescomplemented
123#10587486—22no—
124#10587486C2×C414yescomplemented
125#10587479—22no—
126#10587478—22no—
127C24×C17—21no—
128C24×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17C22×C412yes—
138C2×D4×C17C22×C412yes—
139C2×D4×C17#4112yes—
140C2×D4×C17#4112yes—
141C22×C4×C17—24no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2411yescharacteristic
149C22×Dic17—24no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—24no—
154C22×Dic17—24no—
155C22×Dic17—21no—
156C22×Dic17—42no—
157C22×Dic17—42no—
158C22×Dic17—42no—
159C22×Dic17—42no—
160C22×Dic17—42no—
161C22×Dic17—42no—
162C22×Dic17C2×D411yescharacteristic
163C22×Dic17C2×D411yescharacteristic
164C22×Dic17C2×D411yescharacteristic
165C22×Dic17C2×D411yescharacteristic
166D8×C17—28no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17C22×C412yes—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17#4112yes—
177C22⋊C4×C17—24no—
178C22⋊C4×C17—42no—
179C22⋊C4×C17—42no—
180C42×C17—21no—
181C42×C17—21no—
182C22⋊Dic17—24no—
183C22⋊Dic17—24no—
184C22⋊Dic17—24no—
185C22⋊Dic17—42no—
186C22⋊Dic17—42no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17C22×C412yes—
193C4⋊Dic17C22×C412yes—
194C4⋊Dic17#4112yes—
195#63225—42no—
196#63225—42no—
197C4×Dic17—22no—
198C4×Dic17—21no—
199C4×Dic17—21no—
200C4×Dic17—22no—
201C4×Dic17—41no—
202C4×Dic17—41no—
203C4×Dic17—42no—
204C4×Dic17—42no—
205C4×Dic17—42no—
206C4×Dic17—42no—
207C4×Dic17C22×C412yes—
208C4×Dic17#4112yes—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622C22×C412yes—
215#21122—171nomaximal, Sylow
216C23×C17—22no—
217C23×C17—22no—
218C23×C17—22no—
219C23×C17—22no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—41no—
229C23×C17—41no—
230C23×C17C22×D411yescharacteristic
231D4×C17—22no—
232D4×C17—22no—
233D4×C17—22no—
234D4×C17—22no—
235D4×C17—22no—
236D4×C17—41no—
237D4×C17—41no—
238D4×C17#11714yescomplemented
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—22no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—48no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17AGL(1,ℤ/8)12yes—
252C2×C4×C17C23×C411yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—21no—
267C2×Dic17—22no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—48no—
271C2×Dic17—48no—
272C2×Dic17—48no—
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—41no—
282C2×Dic17C4×D412yes—
283C2×Dic17C4×D412yes—
284C2×Dic17AGL(1,ℤ/8)12yes—
285C8×C17—22no—
286C8×C17—41no—
287#3622—22no—
288#3622—41no—
289C2×#442—171no—
290#2293—171no—
291C42×D4—171no—
292#1926—171no—
293#1925—171no—
294#1920—171no—
295#1919—171no—
296#1879—171no—
297#1879D1711yescharacteristic
298#1864—171no—
299#1862—171no—
300#1776—171no—
301#1762—171no—
302#1762—171no—
303#1745—171no—
304C22×C17—24no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—42no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—42no—
311C22×C17—42no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17—81no—
315C22×C17—81no—
316C22×C17—81no—
317C22×C17—81no—
318C22×C17C2×C4×D411yescharacteristic
319C22×C17#52511yescharacteristic
320C22×C17#52511yescharacteristic
321C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
322C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
323C22×C17#57511yescharacteristic
324C22×C17#57511yescharacteristic
325C4×C17—22no—
326C4×C17—42no—
327C4×C17—41no—
328C4×C17—81no—
329C4×C17C2×#11711yescharacteristic
330C4×C17C4×#4111yescharacteristic
331C4×C17#52911yescharacteristic
332C4×C17#52911yescharacteristic
333Dic17—22no—
334Dic17—22no—
335Dic17—22no—
336Dic17—42no—
337Dic17—41no—
338Dic17—42no—
339Dic17—42no—
340Dic17—81no—
341C22×D8C2×D1711yescharacteristic
342C2×C4×D4—171no—
343C2×C4×D4—171no—
344C2×C4×D4—171no—
345C2×C4×D4C2×D1711yescharacteristic
346C2×#117—341no—
347C22×C42—341no—
348#442—178no—
349C2×#107—171no—
350C2×#106—171no—
351C2×#106—171no—
352C2×#102—171no—
353C2×#102—171no—
354C2×#102—171no—
355C2×#102—171no—
356C2×#102C2×D1711yescharacteristic
357C2×#102C2×D1711yescharacteristic
358C2×#97—171no—
359C2×#97—171no—
360#390—171no—
361#390C2×D1711yescharacteristic
362#389—171no—
363#389—171no—
364#388—341no—
365#388—341no—
366C4×C4⋊C4—171no—
367C4×C4⋊C4—171no—
368C4×C4⋊C4—171no—
369C4×C4⋊C4—171no—
370C4×C22⋊C4—341no—
371C4×C22⋊C4—341no—
372C4×C22⋊C4—341no—
373C4×C22⋊C4—341no—
374C43—171no—
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#340—171no—
380#340C2×D1711yescharacteristic
381#336—171no—
382#336—171no—
383#336—171no—
384#336C2×D1711yescharacteristic
385C2×C17—44no—
386C2×C17—82no—
387C2×C17#229011yescharacteristic
388C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
389C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
390C2×C17#310911yescharacteristic
391C2×C17#311111yescharacteristic
392C2×C17#311411yescharacteristic
393C2×C17#311511yescharacteristic
394C22×D4C22×D1711yescharacteristic
395C22×D4C22×D1711yescharacteristic
396C23×C4—21no—
397C23×C4—341no—
398C23×C4—341no—
399C23×C4—341no—
400C2×D8—22no—
401C2×D8—22no—
402C2×D8C4×D1714yes—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8C22×D1711yescharacteristic
406C22×C8C22×D1711yescharacteristic
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—174no—
410C4×D4—174no—
411C4×D4—342no—
412C4×D4—342no—
413C4×D4—342no—
414C4×D4—342no—
415#117—684no—
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⋊C4C22×D1711yescharacteristic
423C2×C4⋊C4C22×D1711yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—341no—
428C2×C22⋊C4—341no—
429C2×C22⋊C4—341no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C42—171no—
433C2×C42—171no—
434C2×C42—171no—
435C2×C42—171no—
436C2×C42—171no—
437C2×C42—171no—
438C2×C42—341no—
439C2×C42—341no—
440C2×C42—341no—
441C2×C42—341no—
442C2×C42—684no—
443C2×C42C22×D1711yescharacteristic
444#107—342no—
445#106—174no—
446#106—342no—
447#102—22no—
448#102—22no—
449#102—174no—
450#102—174no—
451#102—342no—
452#102—342no—
453#97—174no—
454#97—342no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95—341no—
460#95—341no—
461C17#2112211yescharacteristic, complemented, Sylow
462C24—21no—
463C24—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D1712yes—
473C2×D4C2×C4×D1712yes—
474C2×D4#6325412yes—
475C2×D4#6325412yes—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4—171no—
484C22×C4—171no—
485C22×C4—344no—
486C22×C4—344no—
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—682no—
497C22×C4—682no—
498C22×C4D4×D1711yescharacteristic
499C22×C4D4×D1711yescharacteristic
500C22×C4C23×D1711yescharacteristic
501D8—28no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—172no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8C2×C4×D1712yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—172no—
518C4⋊C4—172no—
519C4⋊C4—172no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C4⋊C4#6325412yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—344no—
532C22⋊C4—344no—
533C22⋊C4—344no—
534C22⋊C4—682no—
535C22⋊C4—682no—
536C42—21no—
537C42—21no—
538C42—172no—
539C42—172no—
540C42—342no—
541C42—341no—
542C42—341no—
543C42—342no—
544C42—681no—
545C42—681no—
546C42—682no—
547C42—682no—
548C42—682no—
549C42—682no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C2×D4×D1711yescharacteristic
565D4—22no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572D4#1058756314yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—48no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—172no—
586C2×C4—172no—
587C2×C4—172no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—342no—
595C2×C4—342no—
596C2×C4—341no—
597C2×C4—342no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—688no—
601C2×C4—688no—
602C2×C4—688no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4#1058759712yes—
613C2×C4C22×C4×D1711yescharacteristic
614C2×C4C2×#6325411yescharacteristic
615C2×C4C2×#6325411yescharacteristic
616C2×C4C2×#6325411yescharacteristic
617C2×C4#41×D1711yescharacteristic
618C2×C4#41×D1711yescharacteristic
619C8—22no—
620C8—41no—
621C8—342no—
622C8—681no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C4×D4×D1711yescharacteristic
638C22#1176331511yescharacteristic
639C22#1176331911yescharacteristic
640C22AGL(1,ℤ/8)×D1711yescharacteristic
641C22C2×#1058759711yescharacteristic
642C22#1176360511yescharacteristic
643C22#1176360611yescharacteristic
644C4—22no—
645C4—42no—
646C4—41no—
647C4—81no—
648C4—342no—
649C4—342no—
650C4—342no—
651C4—682no—
652C4—682no—
653C4—682no—
654C4—681no—
655C4—1361no—
656C4C2×#1058756311yescharacteristic
657C4C4×#6325411yescharacteristic
658C4#1176333611yescharacteristic
659C4#1176334011yescharacteristic
660C2—44no—
661C2—82no—
662C2#1453122411yescharacteristic
663C2C4×#1058759711yescharacteristic
664C2#1453759511yescharacteristic
665C2#1453887411yescharacteristic
666C2#1453889011yescharacteristic
667C2#1453891711yescharacteristic
668C2#1453892811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1879C2
socleC23×C17C22×D4
prime radicalC22×D8×C17C22
cosocleC22×C4×C17C24
hypercenter#1879D17
hypocenterC17#21122
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
2. #14531246C2semidirect
3. C4×D4×Dic17C2semidirect
4. #14529414C2semidirect
5. #14529401C2semidirect
6. #14529370C2semidirect
7. #14529361C2semidirect
9. #14529045C2semidirect
10. #14529029C2semidirect
11. #14528680C2semidirect
12. #14528611C2semidirect
13. #14528611C2semidirect
14. #14528531C2semidirect
21. #11762921C4semidirect
89. D4×Dic17C2×C4semidirect
115. #10587503C2×C4semidirect
122. #10587488C2×C4semidirect
124. #10587486C2×C4semidirect
238. D4×C17#117semidirect
461. C17#21122semidirect
572. D4#10587563semidirect

Maximal subgroups

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

subgroupindex
C2×#117629212
#145312462
C4×D4×Dic172
#145294142
#145294012
#145293702
#145293612
#145291342
#145290452
#145290292
#145286802
#145286112
#145285312
C17×#18792
#2112217