Finite Group Databasebeta

← #17043875

Subgroups — #17043875

Order 4352.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762870C211yesmaximal, characteristic
2#14531393C211yesmaximal, characteristic
3#14529758C211yesmaximal, characteristic, complemented
4#14529754C211yesmaximal, characteristic, complemented
5#14529750C211yesmaximal, characteristic, complemented
6#14529746C211yesmaximal, characteristic, complemented
7#14529742C211yesmaximal, characteristic, complemented
8#14529742C211yesmaximal, characteristic, complemented
9#14529737C211yesmaximal, characteristic, complemented
10#14529733C211yesmaximal, characteristic, complemented
11#14528879C211yesmaximal, characteristic
12#14528879C211yesmaximal, characteristic
13#14528874C211yesmaximal, characteristic
14#14528870C211yesmaximal, characteristic
15C2×C17×#544C211yesmaximal, characteristic
16#11762870—24no—
17C2×C4⋊#3622C2211yescharacteristic
18C2×C4⋊#3622C2211yescharacteristic
19C2×C22⋊#3622C2211yescharacteristic
20C2×C22⋊#3622C2211yescharacteristic
21C2×#10587474C2211yescharacteristic
22C2×C4×#3622C2211yescharacteristic
23#11762702—21no—
24#11762702—21no—
25#11762695—21no—
26#11762695—21no—
27#11762679C2211yescharacteristic
28#11762675C2211yescharacteristic
29#11762669C2211yescharacteristic
30#11762669C2211yescharacteristic
31C4×C4⋊Dic17C2211yescharacteristic
32C4×C4⋊Dic17C2211yescharacteristic
33#11762571C2211yescharacteristic
34#11762571C2211yescharacteristic
35#11762571C2211yescharacteristic
36#11762571C2211yescharacteristic
37#11762571C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49C17×#544—24no—
50C17×#544C418yescomplemented
51C2×C17×C4⋊Q8—21no—
52C2×C17×C4⋊Q8C2211yescharacteristic
53C2×C17×C4⋊Q8C2211yescharacteristic
54C2×C17×#125—21no—
55C2×C17×#125C2211yescharacteristic
56C2×C17×#125C2211yescharacteristic
57C2×C17×C22⋊Q8—21no—
58C2×C17×C22⋊Q8—21no—
59C2×C17×#117C2211yescharacteristic
60C2×C17×#117C2211yescharacteristic
61C22×C4⋊C4×C17C2211yescharacteristic
62C22×Q8×C17—21no—
63C22×Q8×C17—21no—
64C23×C4×C17—21no—
65C23×C4×C17C2311yescharacteristic
66C17×C4⋊Q8—28no—
67C17×C4⋊Q8—22no—
68C17×C4⋊Q8—22no—
69C17×C4⋊Q8C2×C414yes—
70C17×C4⋊Q8C2×C414yes—
71C17×#125—28no—
72C17×#125—22no—
73C17×#125—22no—
74C17×#125C2×C414yes—
75C17×#125C2×C414yes—
76C17×C22⋊Q8—28no—
77C17×C22⋊Q8—28no—
78C17×#117—22no—
79C17×#117C2×C414yes—
80C17×#117C2×C414yes—
81C17×#117D414yes—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17—22no—
84C2×C4⋊C4×C17—21no—
85C2×C4⋊C4×C17—21no—
86C2×C4⋊C4×C17—21no—
87C2×C4⋊C4×C17—21no—
88C2×C4⋊C4×C17—21no—
89C2×C4⋊C4×C17—21no—
90C2×C4⋊C4×C17C2×C414yes—
91C2×C4⋊C4×C17D411yescharacteristic
92C2×C4⋊C4×C17D411yescharacteristic
93C2×C4⋊C4×C17C2311yescharacteristic
94C2×C4⋊C4×C17C2311yescharacteristic
95C2×C4⋊C4×C17C2311yescharacteristic
96C2×C4⋊C4×C17C2311yescharacteristic
97C2×C22⋊C4×C17—21no—
98C2×C22⋊C4×C17D411yescharacteristic
99C2×C22⋊C4×C17D411yescharacteristic
100C2×C42×C17—21no—
101C2×C42×C17C2311yescharacteristic
102C2×C42×C17C2311yescharacteristic
103C2×C22⋊Dic17—21no—
104C2×C22⋊Dic17—21no—
105C2×C4⋊Dic17—21no—
106C2×C4⋊Dic17—21no—
107C2×C4⋊Dic17C2311yescharacteristic
108C2×C4⋊Dic17C2311yescharacteristic
109C2×C4⋊Dic17C2311yescharacteristic
110C2×C4⋊Dic17C2311yescharacteristic
111C2×C4×Dic17—21no—
112C2×C4×Dic17—21no—
113C22×#3622C2311yescharacteristic
114C22×#3622C2311yescharacteristic
115C22×#3622C2311yescharacteristic
116C22×#3622C2311yescharacteristic
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123C22⋊#3622—22no—
124C22⋊#3622—22no—
125C4⋊#3622—22no—
126C4⋊#3622—22no—
127#10587474—22no—
128C4×#3622—22no—
129C24×C17C2×D411yescharacteristic
130C2×Q8×C17—24no—
131C2×Q8×C17—24no—
132C2×Q8×C17—81no—
133C2×Q8×C17—81no—
134C2×Q8×C17—81no—
135C2×Q8×C17—81no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—24no—
145C22×C4×C17—81no—
146C22×C4×C17—81no—
147C22×C4×C17C22×C412yes—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D412yes—
154C22×C4×C17C2411yescharacteristic
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C4⋊C4×C17—24no—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—24no—
162C4⋊C4×C17—24no—
163C4⋊C4×C17—24no—
164C4⋊C4×C17—24no—
165C4⋊C4×C17—24no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17C22⋊C414yes—
171C4⋊C4×C17C22⋊C414yes—
172C4⋊C4×C17C22×C412yes—
173C4⋊C4×C17C22×C412yes—
174C4⋊C4×C17C22×C412yes—
175C4⋊C4×C17C22×C412yes—
176C22⋊C4×C17—24no—
177C22⋊C4×C17C22⋊C414yes—
178C22⋊C4×C17C22⋊C414yes—
179C42×C17—24no—
180C42×C17C22×C412yes—
181C42×C17C22×C412yes—
182C42×C17C2×D412yes—
183C42×C17C2×D412yes—
184C22⋊Dic17—81no—
185C22⋊Dic17—81no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—81no—
195C4⋊Dic17—81no—
196C4×Dic17—81no—
197C4×Dic17—81no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210#21843—171nomaximal, Sylow
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17C2×C22⋊C412yes—
218C23×C17C22×D411yescharacteristic
219Q8×C17—84no—
220Q8×C17—84no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—84no—
248C2×C4×C17C2×C22⋊C412yes—
249C2×C4×C17C2×C22⋊C412yes—
250C2×C4×C17C2×C22⋊C412yes—
251C2×C4×C17C2×C22⋊C412yes—
252C2×C4×C17C2×C22⋊C412yes—
253C2×C4×C17C23×C411yescharacteristic
254C2×C4×C17C22×D411yescharacteristic
255C2×C4×C172−1+411yescharacteristic
256C2×C4×C172−1+411yescharacteristic
257C2×C4×C172−1+411yescharacteristic
258C2×C4×C172−1+411yescharacteristic
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271#3622—41no—
272#3622—41no—
273#3622—41no—
274#3622—41no—
275C2×#544D1711yescharacteristic
276C2×#433—171no—
277#2314—171no—
278#1993—171no—
279#1992—171no—
280#1991—171no—
281#1990—171no—
282#1989—171no—
283#1989—171no—
284#1988—171no—
285#1987—171no—
286#1825—171no—
287#1825—171no—
288#1824—171no—
289#1823—171no—
290C22×C17—22no—
291C22×C17—22no—
292C22×C17—22no—
293C22×C17—41no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17C22×C22⋊C411yescharacteristic
298C22×C17#53611yescharacteristic
299C22×C17#53611yescharacteristic
300C22×C17#57611yescharacteristic
301C22×C17#57711yescharacteristic
302C22×C17#57711yescharacteristic
303C22×C17#57811yescharacteristic
304C4×C17—22no—
305C4×C17—22no—
306C4×C17—22no—
307C4×C17—22no—
308C4×C17—22no—
309C4×C17—42no—
310C4×C17—42no—
311C4×C17—42no—
312C4×C17—42no—
313C4×C17#51911yescharacteristic
314C4×C17#51911yescharacteristic
315C4×C17#53611yescharacteristic
316C4×C17#53611yescharacteristic
317Dic17—81no—
318Dic17—81no—
319Dic17—81no—
320Dic17—81no—
321#544—24no—
322#544Dic1718yescomplemented
323C2×C4⋊Q8—21no—
324C2×C4⋊Q8C2×D1711yescharacteristic
325C2×C4⋊Q8C2×D1711yescharacteristic
326C2×#125—21no—
327C2×#125C2×D1711yescharacteristic
328C2×#125C2×D1711yescharacteristic
329C2×C22⋊Q8—21no—
330C2×C22⋊Q8—21no—
331C2×#117C2×D1711yescharacteristic
332C2×#117C2×D1711yescharacteristic
333C22×C4⋊C4C2×D1711yescharacteristic
334#433—344no—
335C2×C4⋊C8—171no—
336C2×C4⋊C8—171no—
337C2×C22⋊C8—171no—
338C2×C22⋊C8—171no—
339C2×#97—171no—
340C2×C4×C8—171no—
341#388—341no—
342#388—341no—
343#387—341no—
344#387—341no—
345#384—171no—
346#382—171no—
347#380—171no—
348#380—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#340—171no—
365#340—171no—
366#340—171no—
367C2×C17—42no—
368C2×C17#231011yescharacteristic
369C2×C17#289711yescharacteristic
370C2×C17#289811yescharacteristic
371C2×C17#324111yescharacteristic
372C2×C17#324211yescharacteristic
373C2×C17#324311yescharacteristic
374C2×C17#324411yescharacteristic
375C22×Q8—21no—
376C22×Q8—21no—
377C23×C4—21no—
378C23×C4C22×D1711yescharacteristic
379C22×C8—171no—
380C22×C8—171no—
381C22×C8—171no—
382C22×C8—171no—
383C4⋊Q8—28no—
384C4⋊Q8—22no—
385C4⋊Q8—22no—
386C4⋊Q8C2×Dic1714yes—
387C4⋊Q8C2×Dic1714yes—
388#125—28no—
389#125—22no—
390#125—22no—
391#125C2×Dic1714yes—
392#125C2×Dic1714yes—
393C22⋊Q8—28no—
394C22⋊Q8—28no—
395#117—22no—
396#117C2×Dic1714yes—
397#117C2×Dic1714yes—
398#117C22⋊D1714yes—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—171no—
410C2×C4⋊C4—171no—
411C2×C4⋊C4—341no—
412C2×C4⋊C4—341no—
413C2×C4⋊C4C2×Dic1714yes—
414C2×C4⋊C4C22⋊D1711yescharacteristic
415C2×C4⋊C4C22⋊D1711yescharacteristic
416C2×C4⋊C4C22×D1711yescharacteristic
417C2×C4⋊C4C22×D1711yescharacteristic
418C2×C4⋊C4C22×D1711yescharacteristic
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—341no—
422C2×C22⋊C4—341no—
423C2×C22⋊C4C22⋊D1711yescharacteristic
424C2×C22⋊C4C22⋊D1711yescharacteristic
425C2×C42—21no—
426C2×C42—341no—
427C2×C42—341no—
428C2×C42C22×D1711yescharacteristic
429C2×C42C22×D1711yescharacteristic
430C4⋊C8—342no—
431C4⋊C8—342no—
432C22⋊C8—342no—
433C22⋊C8—342no—
434#97—342no—
435C4×C8—342no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441#95—341no—
442C17#2184311yescharacteristic, complemented, Sylow
443C24C2×C22⋊D1711yescharacteristic
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—81no—
447C2×Q8—81no—
448C2×Q8—81no—
449C2×Q8—81no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4C22×Dic1712yes—
466C22×C4C2×C22⋊D1712yes—
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C2×C22⋊D1711yescharacteristic
469C22×C4C2×C22⋊D1711yescharacteristic
470C22×C4C2×C22⋊D1711yescharacteristic
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C23×D1711yescharacteristic
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—1361no—
505C4⋊C4—1361no—
506C4⋊C4C22⋊Dic1714yes—
507C4⋊C4C22⋊Dic1714yes—
508C4⋊C4C22×Dic1712yes—
509C4⋊C4C22×Dic1712yes—
510C4⋊C4C22×Dic1712yes—
511C4⋊C4C22×Dic1712yes—
512C22⋊C4—24no—
513C22⋊C4—1361no—
514C22⋊C4—1361no—
515C22⋊C4C22⋊Dic1714yes—
516C22⋊C4C22⋊Dic1714yes—
517C42—24no—
518C42—1361no—
519C42—1361no—
520C42C22×Dic1712yes—
521C42C22×Dic1712yes—
522C42C2×C22⋊D1712yes—
523C42C2×C22⋊D1712yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×C22⋊Dic1712yes—
531C23C22×C22⋊D1711yescharacteristic
532Q8—84no—
533Q8—84no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4C2×C22⋊Dic1712yes—
574C2×C4C2×C22⋊Dic1712yes—
575C2×C4C2×C22⋊Dic1712yes—
576C2×C4C2×C22⋊Dic1712yes—
577C2×C4C2×C22⋊Dic1712yes—
578C2×C4#1058769411yescharacteristic
579C2×C4#1058769411yescharacteristic
580C2×C4#1058769411yescharacteristic
581C2×C4#1058769411yescharacteristic
582C2×C4C23×Dic1711yescharacteristic
583C2×C4C22×C22⋊D1711yescharacteristic
584C8—681no—
585C8—681no—
586C8—681no—
587C8—681no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×C22⋊Dic1711yescharacteristic
596C22#1176338311yescharacteristic
597C22#1176338311yescharacteristic
598C22#1176361211yescharacteristic
599C22#1176361711yescharacteristic
600C22#1176362011yescharacteristic
601C22#1176362411yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—1361no—
612C4—1361no—
613C4—1361no—
614C4—1361no—
615C4#1176328711yescharacteristic
616C4#1176328711yescharacteristic
617C4#1176338311yescharacteristic
618C4#1176338311yescharacteristic
619C2—42no—
620C2#1453137011yescharacteristic
621C2#1453731011yescharacteristic
622C2#1453731711yescharacteristic
623C2#1454003711yescharacteristic
624C2#1454005211yescharacteristic
625C2#1454006711yescharacteristic
626C2#1454008211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#544C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenterC2×#544D17
hypocenterC17#21843
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #14529758C2semidirect
4. #14529754C2semidirect
5. #14529750C2semidirect
6. #14529746C2semidirect
7. #14529742C2semidirect
8. #14529742C2semidirect
9. #14529737C2semidirect
10. #14529733C2semidirect
50. C17×#544C4semidirect
322. #544Dic17semidirect
442. C17#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#117628702
#145313932
#145297582
#145297542
#145297502
#145297462
#145297422
#145297372
#145297332
#145288792
#145288742
#145288702
C2×C17×#5442
#2184317