Finite Group Databasebeta

← #17050989

Subgroups — #17050989

Order 4352.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763519C211yesmaximal, characteristic
2C2×#11762940C211yesmaximal, characteristic, complemented
3#14529756C211yesmaximal, characteristic, complemented
4#14529752C211yesmaximal, characteristic, complemented
5#14529748C211yesmaximal, characteristic, complemented
6#14529741C211yesmaximal, characteristic, complemented
7#14529741C211yesmaximal, characteristic, complemented
8#14529736C211yesmaximal, characteristic, complemented
9#14528791C211yesmaximal, characteristic
10#14528791C211yesmaximal, characteristic
11#14528781C211yesmaximal, characteristic
12#14528612C211yesmaximal, characteristic
13#14528612C211yesmaximal, characteristic
14C4×#10587480C211yesmaximal, characteristic
15C17×#2355C211yesmaximal, characteristic, complemented
16#11763519—24no—
17#11763519C418yescomplemented
18C2×#10587555C2211yescharacteristic
19C2×#10587556C2211yescharacteristic
20C2×#10587556C2211yescharacteristic
21C2×#10587582—21no—
22C2×#10587582—21no—
23C2×#10587632—21no—
24C2×C4×D68C2211yescharacteristic
25C2×C4×D68C2211yescharacteristic
26C2×C4×D68C2211yescharacteristic
27C2×C4⋊C4×D17—21no—
28#11762940—24no—
29C2×C4⋊#3622C2211yescharacteristic
30C2×C4⋊#3622C2211yescharacteristic
31C2×C4⋊#3622C2211yescharacteristic
32C2×#10587480C2211yescharacteristic
33C2×#10587480C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×#10587480C2211yescharacteristic
36C2×#10587474C2211yescharacteristic
37C2×#10587474C2211yescharacteristic
38C2×C4×#3622C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C17×#391—21no—
53C17×#387—21no—
54C17×#387—21no—
55C17×#384C2211yescharacteristic
56C17×#382C2211yescharacteristic
57C17×#382C2211yescharacteristic
58C4×C4⋊C4×C17—21no—
59C4×C4⋊C4×C17C2211yescharacteristic
60C4×C4⋊C4×C17C2211yescharacteristic
61C4×C4⋊C4×C17C2211yescharacteristic
62C22×Q8×C17D411yescharacteristic
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×C4×D17—21no—
66C22×C4×D17—21no—
67C22×Dic34—21no—
68C4×Q8×C17C2×C414yes—
69C4×Q8×C17D414yes—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C42×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17—21no—
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85#10587632—28no—
86C2×#63227—21no—
87C2×#63227—21no—
88C2×#63227—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587582—28no—
102#10587582—28no—
103C4⋊C4×D17—28no—
104C4×D68—22no—
105C4×D68—22no—
106C4×D68—22no—
107C4×D68C2×C414yes—
108C4×D68C2×C414yes—
109C4×D68C2×C414yes—
110#10587556—22no—
111#10587556—22no—
112#10587556C2×C414yes—
113#10587556C2×C414yes—
114#10587555—22no—
115#10587555C2×C414yes—
116C17×#95—21no—
117C17×#95—21no—
118C17×#95—21no—
119C17×#95—21no—
120C17×#95—21no—
121C17×#95—21no—
122C17×#95—21no—
123#10587480—22no—
124#10587480—22no—
125#10587480—22no—
126#10587480Q814yescomplemented
127C4⋊#3622—22no—
128C4⋊#3622—22no—
129C4⋊#3622—22no—
130#10587474—22no—
131#10587474—22no—
132C4×#3622—22no—
133C23×D17—21no—
134C2×Q8×C17—41no—
135C2×Q8×C17—41no—
136C2×Q8×C17C22⋊C414yes—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2411yescharacteristic
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C2×D68—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68C22×C412yes—
155C2×D68C2×Q812yes—
156C2×D68#4112yes—
157C2×C4×D17—24no—
158C2×C4×D17—24no—
159C2×C4×D17—24no—
160C2×C4×D17—42no—
161C2×C4×D17—42no—
162C2×C4×D17—81no—
163C2×C4×D17—81no—
164C2×C4×D17—81no—
165C2×C4×D17—81no—
166C2×Dic34—24no—
167C2×Dic34—81no—
168C2×Dic34—81no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17—81no—
178C4⋊C4×C17C22⋊C414yes—
179C4⋊C4×C17C22⋊C414yes—
180C4⋊C4×C17C22⋊C414yes—
181C4⋊C4×C17C2×Q812yes—
182C42×C17—42no—
183C42×C17—81no—
184C42×C17—81no—
185C42×C17C22×C412yes—
186C42×C17C22×C412yes—
187C42×C17C22×C412yes—
188C42×C17C2×D412yes—
189C42×C17C2×D412yes—
190C42×C17C2×D412yes—
191#63227—24no—
192#63227—24no—
193#63227—24no—
194C4⋊Dic17—24no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17C22×C412yes—
199C4⋊Dic17C22×C412yes—
200C4⋊Dic17C22×C412yes—
201#63225—24no—
202#63225—24no—
203#63225—24no—
204#63225—24no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622C2×Q812yes—
217#22526—171nomaximal, Sylow
218C23×C17C22×D411yescharacteristic
219C22×D17—22no—
220C22×D17—22no—
221C22×D17—22no—
222C22×D17—41no—
223C22×D17—41no—
224C22×D17—41no—
225C22×D17—41no—
226Q8×C17—44no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—22no—
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—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17C2×C22⋊C412yes—
252C2×C4×C17C2×C22⋊C412yes—
253C2×C4×C17C2×C22⋊C412yes—
254C2×C4×C17C2×C22⋊C412yes—
255C2×C4×C17C2×C22⋊C412yes—
256C2×C4×C17C2×C22⋊C412yes—
257C2×C4×C17C23×C411yescharacteristic
258C2×C4×C17C22×D411yescharacteristic
259C2×C4×C17C22×Q811yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C172−1+411yescharacteristic
262C2×C4×C172−1+411yescharacteristic
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272D68—22no—
273D68—22no—
274D68—22no—
275D68C4×Q814yes—
276C4×D17—48no—
277C4×D17—84no—
278C4×D17—84no—
279Dic34—84no—
280#3622—22no—
281#3622—41no—
282#3622—41no—
283#3622—41no—
284C2×#554—171no—
285C2×#446—171no—
286#2355D1711yescharacteristic, complemented
287#1993—171no—
288#1992—171no—
289#1991—171no—
290#1989—171no—
291#1989—171no—
292#1988—171no—
293#1804—171no—
294#1804—171no—
295#1802—171no—
296#1762—171no—
297#1762—171no—
298C4×#102—171no—
299C22×C17C22×C22⋊C411yescharacteristic
300C22×C17#54811yescharacteristic
301C22×C17D4×Q811yescharacteristic
302C22×C17#57511yescharacteristic
303C22×C17#57511yescharacteristic
304C22×C17#57611yescharacteristic
305C22×C17#57711yescharacteristic
306C2×D17—24no—
307C2×D17—42no—
308C2×D17—42no—
309C2×D17—42no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—42no—
313C4×C17—22no—
314C4×C17—22no—
315C4×C17—22no—
316C4×C17—22no—
317C4×C17—22no—
318C4×C17—22no—
319C4×C17—42no—
320C4×C17—81no—
321C4×C17—81no—
322C4×C17—81no—
323C4×C17C2×C4×Q811yescharacteristic
324C4×C17#51911yescharacteristic
325C4×C17#54811yescharacteristic
326C4×C17D4×Q811yescharacteristic
327Dic17—42no—
328Dic17—42no—
329Dic17—42no—
330#554—178no—
331#554—344no—
332C2×C4⋊Q8—171no—
333C2×#125—171no—
334C2×#125—171no—
335C2×C22⋊Q8—341no—
336C2×C22⋊Q8—341no—
337C2×C22⋊Q8—341no—
338C2×C4×Q8C2×D1711yescharacteristic
339C2×C4×D4—171no—
340C2×C4×D4—171no—
341C2×C4×D4—171no—
342C22×C4⋊C4—341no—
343#446—344no—
344C2×C4⋊C8—171no—
345C2×C4⋊C8—171no—
346C2×C4⋊C8—171no—
347C2×#102—171no—
348C2×#102—171no—
349C2×#102—171no—
350C2×#102—171no—
351C2×#97—171no—
352C2×#97—171no—
353C2×C4×C8—171no—
354#391—21no—
355#387—21no—
356#387—21no—
357#384C2×D1711yescharacteristic
358#382C2×D1711yescharacteristic
359#382C2×D1711yescharacteristic
360C4×C4⋊C4—21no—
361C4×C4⋊C4C2×D1711yescharacteristic
362C4×C4⋊C4C2×D1711yescharacteristic
363C4×C4⋊C4C2×D1711yescharacteristic
364#340—171no—
365#340—171no—
366#340—171no—
367#340—171no—
368#340—171no—
369#340—171no—
370#340—171no—
371#340—171no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376C2×C17Q8×C22⋊C411yescharacteristic
377C2×C17#289611yescharacteristic
378C2×C17#289711yescharacteristic
379C2×C17#337011yescharacteristic
380C2×C17#337611yescharacteristic
381C2×C17#338811yescharacteristic
382C2×C17#338911yescharacteristic
383D17—44no—
384C22×Q8—341no—
385C22×Q8C22⋊D1711yescharacteristic
386C22×D4—171no—
387C23×C4—341no—
388C23×C4—341no—
389C23×C4—341no—
390C22×C8—171no—
391C22×C8—171no—
392C22×C8—171no—
393C22×C8—171no—
394C4⋊Q8—174no—
395C4⋊Q8—342no—
396#125—174no—
397#125—174no—
398#125—342no—
399#125—342no—
400C22⋊Q8—348no—
401C22⋊Q8—348no—
402C22⋊Q8—348no—
403C4×Q8C4×D1714yes—
404C4×Q8D6814yes—
405C4×D4—174no—
406C4×D4—174no—
407C4×D4—174no—
408C4×D4—342no—
409C4×D4—342no—
410C4×D4—342no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—348no—
422C2×C4⋊C4C22⋊D1711yescharacteristic
423C2×C4⋊C4C22⋊D1711yescharacteristic
424C2×C4⋊C4C22⋊D1711yescharacteristic
425C2×C4⋊C4C22×D1711yescharacteristic
426C2×C4⋊C4C22×D1711yescharacteristic
427C2×C4⋊C4C22×D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C22⋊C4—341no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C42—21no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42C22×D1711yescharacteristic
436C2×C42C22×D1711yescharacteristic
437C2×C42C22×D1711yescharacteristic
438C4⋊C8—342no—
439C4⋊C8—342no—
440C4⋊C8—342no—
441#102—174no—
442#102—342no—
443#102—342no—
444#102—342no—
445#97—342no—
446#97—342no—
447C4×C8—342no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455C17#2252611yescharacteristic, complemented, Sylow
456C24—341no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—344no—
460C2×Q8—1361no—
461C2×Q8—1361no—
462C2×Q8#6322714yes—
463C2×D4—172no—
464C2×D4—172no—
465C2×D4—172no—
466C2×D4—341no—
467C2×D4—341no—
468C2×D4—341no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—341no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4—344no—
477C22×C4—344no—
478C22×C4—344no—
479C22×C4—682no—
480C22×C4—682no—
481C22×C4—1361no—
482C22×C4—1361no—
483C22×C4—1361no—
484C22×C4—1361no—
485C22×C4C2×C22⋊D1711yescharacteristic
486C22×C4C2×C22⋊D1711yescharacteristic
487C22×C4C2×C22⋊D1711yescharacteristic
488C22×C4C2×C22⋊D1711yescharacteristic
489C22×C4C2×C22⋊D1711yescharacteristic
490C22×C4C2×C22⋊D1711yescharacteristic
491C22×C4C23×D1711yescharacteristic
492C2×C8—172no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—172no—
514C4⋊C4—172no—
515C4⋊C4—172no—
516C4⋊C4—341no—
517C4⋊C4—341no—
518C4⋊C4—341no—
519C4⋊C4—344no—
520C4⋊C4—344no—
521C4⋊C4—344no—
522C4⋊C4—344no—
523C4⋊C4—344no—
524C4⋊C4#6322714yes—
525C4⋊C4#6322714yes—
526C4⋊C4#6322714yes—
527C4⋊C4Q8×D1712yes—
528C22⋊C4—344no—
529C22⋊C4—344no—
530C22⋊C4—344no—
531C42—42no—
532C42—81no—
533C42—81no—
534C42C2×C4×D1712yes—
535C42C2×C4×D1712yes—
536C42C2×C4×D1712yes—
537C42C2×D6812yes—
538C42C2×D6812yes—
539C42C2×D6812yes—
540C23—342no—
541C23—342no—
542C23—342no—
543C23—681no—
544C23—681no—
545C23—681no—
546C23—681no—
547C23C22×C22⋊D1711yescharacteristic
548Q8—44no—
549Q8—1364no—
550D4—174no—
551D4—342no—
552D4—342no—
553D4—342no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—342no—
579C2×C4—342no—
580C2×C4—342no—
581C2×C4—688no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—1364no—
589C2×C4—1364no—
590C2×C4C2×#6322712yes—
591C2×C4C2×#6322712yes—
592C2×C4C2×#6322712yes—
593C2×C4C2×#6322712yes—
594C2×C4C2×#6322712yes—
595C2×C4C2×#6322712yes—
596C2×C4C22×C4×D1711yescharacteristic
597C2×C4C22×D6811yescharacteristic
598C2×C4C2×Q8×D1711yescharacteristic
599C2×C4#1058769411yescharacteristic
600C2×C4#41×D1711yescharacteristic
601C2×C4#1058769711yescharacteristic
602C8—342no—
603C8—681no—
604C8—681no—
605C8—681no—
606C22—344no—
607C22—682no—
608C22—682no—
609C22—682no—
610C22—682no—
611C22—682no—
612C22—682no—
613C22C22×#6322711yescharacteristic
614C22#1176346611yescharacteristic
615C22Q8×C22⋊D1711yescharacteristic
616C22#1176360711yescharacteristic
617C22#1176360711yescharacteristic
618C22#1176361211yescharacteristic
619C22#1176361711yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—682no—
631C4—682no—
632C4—682no—
633C4C4×Q8×D1711yescharacteristic
634C4#1176328511yescharacteristic
635C4#1176347111yescharacteristic
636C4Q8×D6811yescharacteristic
637C2—684no—
638C2Q8×#6322711yescharacteristic
639C2#1453730111yescharacteristic
640C2#1453730711yescharacteristic
641C2#1454153111yescharacteristic
642C2#1454160511yescharacteristic
643C2#1454172311yescharacteristic
644C2#1454173811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2355C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#2355D17
hypocenterC17#22526
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. C2×#11762940C2semidirect
3. #14529756C2semidirect
4. #14529752C2semidirect
5. #14529748C2semidirect
6. #14529741C2semidirect
7. #14529741C2semidirect
8. #14529736C2semidirect
15. C17×#2355C2semidirect
17. #11763519C4semidirect
126. #10587480Q8semidirect
286. #2355D17semidirect
455. C17#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#117635192
C2×#117629402
#145297562
#145297522
#145297482
#145297412
#145297362
#145287912
#145287812
#145286122
C4×#105874802
C17×#23552
#2252617