Finite Group Databasebeta

← #17051189

Subgroups — #17051189

Order 4352.

1054 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763532C211yesmaximal, characteristic
2C2×#11762940C211yesmaximal, characteristic, complemented
3#14529716C211yesmaximal, characteristic
4#14529716C211yesmaximal, characteristic
5#14529707C211yesmaximal, characteristic, complemented
6#14529603C211yesmaximal, characteristic, complemented
7#14529598C211yesmaximal, characteristic
8#14529593C211yesmaximal, characteristic
9#14528796C211yesmaximal, characteristic, complemented
10#14528791C211yesmaximal, characteristic
11#14528786C211yesmaximal, characteristic, complemented
12#14528617C211yesmaximal, characteristic, complemented
13#14528612C211yesmaximal, characteristic
14C4×#10587481C211yesmaximal, characteristic, complemented
15C17×#2355C211yesmaximal, characteristic, complemented
16#11763532—24no—
17#11763532C418yescomplemented
18C2×#10587555C2211yescharacteristic
19C2×#10587584—21no—
20C2×#10587584—21no—
21C2×#10587561C2211yescharacteristic
22C2×#10587561C2211yescharacteristic
23C2×#10587632—21no—
24C2×C4×Dic34C2211yescharacteristic
25C2×C4×Dic34C2211yescharacteristic
26C2×C4×D68C2211yescharacteristic
27C2×#10587578—21no—
28#11762940—24no—
29C2×C4⋊#3622C2211yescharacteristic
30C2×C4⋊#3622C2211yescharacteristic
31C2×C4⋊#3622C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×#10587481C2211yescharacteristic
35C2×#10587481C2211yescharacteristic
36C2×#10587481C2211yescharacteristic
37C2×#10587481C2211yescharacteristic
38C2×#10587481C2211yescharacteristic
39C2×#10587481C2211yescharacteristic
40C2×#10587480C2211yescharacteristic
41C2×#10587480C2211yescharacteristic
42C2×#10587480C2211yescharacteristic
43C2×#10587480C2211yescharacteristic
44C2×#10587474C2211yescharacteristic
45C2×#10587474C2211yescharacteristic
46C2×C4×#3622C2211yescharacteristic
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×Dic34C2311yescharacteristic
66C22×Dic34C2311yescharacteristic
67C4×Q8×C17C2×C414yes—
68C4×Q8×C17D414yes—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17D411yescharacteristic
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C42×C17—21no—
79C2×C42×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84#10587632—28no—
85C2×#63227—21no—
86C2×#63227—21no—
87C2×#63227—21no—
88C2×#63227—21no—
89C2×#63227—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×#63225—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587584—28no—
102#10587584—28no—
103#10587578—28no—
104#10587561—22no—
105#10587561—22no—
106#10587561C2×C414yes—
107#10587561C2×C414yes—
108C4×D68—22no—
109C4×D68C2×C414yes—
110#10587555—22no—
111#10587555C2×C414yes—
112C4×Dic34—22no—
113C4×Dic34—22no—
114C4×Dic34C2×C414yes—
115C4×Dic34C2×C414yes—
116C17×#95—21no—
117C17×#95—21no—
118C17×#95—21no—
119C17×#95—21no—
120C17×#95—21no—
121C17×#95—21no—
122C17×#95—21no—
123#10587481—22no—
124#10587481—22no—
125#10587481—22no—
126#10587481—22no—
127#10587481—41no—
128#10587481—41no—
129#10587481—41no—
130#10587481—41no—
131#10587480—22no—
132#10587480—22no—
133#10587480—41no—
134#10587480—41no—
135C4⋊#3622—22no—
136C4⋊#3622—22no—
137C4⋊#3622—22no—
138#10587474—22no—
139#10587474—22no—
140C4×#3622—22no—
141C23×D17—21no—
142C2×Q8×C17—41no—
143C2×Q8×C17—41no—
144C2×Q8×C17C22⋊C414yes—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2411yescharacteristic
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68—21no—
163C2×D68—21no—
164C2×D68C22×C412yes—
165C2×C4×D17—24no—
166C2×C4×D17—42no—
167C2×C4×D17—42no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34—21no—
176C2×Dic34—21no—
177C2×Dic34—21no—
178C2×Dic34C22×C412yes—
179C2×Dic34C22×C412yes—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—42no—
188C4⋊C4×C17—81no—
189C4⋊C4×C17C22⋊C414yes—
190C4⋊C4×C17C22⋊C414yes—
191C4⋊C4×C17C22⋊C414yes—
192C4⋊C4×C17#4112yes—
193C42×C17—42no—
194C42×C17—81no—
195C42×C17—81no—
196C42×C17C22×C412yes—
197C42×C17C22×C412yes—
198C42×C17C22×C412yes—
199C42×C17C2×D412yes—
200C42×C17C2×D412yes—
201C42×C17C2×D412yes—
202#63227—24no—
203#63227—24no—
204#63227—24no—
205#63227—24no—
206#63227—24no—
207C4⋊Dic17—24no—
208C4⋊Dic17—24no—
209C4⋊Dic17—21no—
210C4⋊Dic17C22×C412yes—
211#63225—24no—
212#63225—24no—
213C4×Dic17—24no—
214C4×Dic17—24no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622#4112yes—
227#22548—171nomaximal, Sylow
228C23×C17C22×D411yescharacteristic
229C22×D17—22no—
230C22×D17—41no—
231C22×D17—41no—
232C22×D17—41no—
233C22×D17—41no—
234C22×D17—41no—
235C22×D17—41no—
236Q8×C17—44no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
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—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17C2×C22⋊C412yes—
262C2×C4×C17C2×C22⋊C412yes—
263C2×C4×C17C2×C22⋊C412yes—
264C2×C4×C17C2×C22⋊C412yes—
265C2×C4×C17C2×C22⋊C412yes—
266C2×C4×C17C2×C22⋊C412yes—
267C2×C4×C17C23×C411yescharacteristic
268C2×C4×C17C22×D411yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×C4×C17C2×#4111yescharacteristic
271C2×C4×C17H(5,𝔽2)11yescharacteristic
272C2×C4×C172−1+411yescharacteristic
273C2×Dic17—22no—
274C2×Dic17—22no—
275C2×Dic17—22no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282D68—22no—
283D68—22no—
284D68—41no—
285D68—41no—
286C4×D17—48no—
287Dic34—22no—
288Dic34—22no—
289Dic34—22no—
290Dic34—22no—
291Dic34—41no—
292Dic34—41no—
293Dic34—41no—
294Dic34—41no—
295#3622—22no—
296#3622—41no—
297#3622—41no—
298#3622—41no—
299C2×#556—171no—
300C2×#446—171no—
301#2355D1711yescharacteristic, complemented
302#1983—171no—
303#1983—171no—
304#1981—171no—
305#1960—171no—
306#1959—171no—
307#1958—171no—
308#1805—171no—
309#1804—171no—
310#1803—171no—
311#1763—171no—
312#1762—171no—
313C4×#103—171no—
314C22×C17C22×C22⋊C411yescharacteristic
315C22×C17#54811yescharacteristic
316C22×C17#55011yescharacteristic
317C22×C17#57511yescharacteristic
318C22×C17#57511yescharacteristic
319C22×C17#57711yescharacteristic
320C22×C17#57811yescharacteristic
321C2×D17—42no—
322C2×D17—42no—
323C2×D17—42no—
324C2×D17—81no—
325C2×D17—81no—
326C2×D17—81no—
327C2×D17—81no—
328C4×C17—22no—
329C4×C17—22no—
330C4×C17—22no—
331C4×C17—22no—
332C4×C17—22no—
333C4×C17—22no—
334C4×C17—42no—
335C4×C17—81no—
336C4×C17—81no—
337C4×C17—81no—
338C4×C17C4×#4111yescharacteristic
339C4×C17#52011yescharacteristic
340C4×C17#54811yescharacteristic
341C4×C17#55011yescharacteristic
342Dic17—42no—
343Dic17—42no—
344Dic17—42no—
345#556—178no—
346#556—344no—
347C2×C4⋊Q8—171no—
348C2×#126—341no—
349C2×#126—341no—
350C2×#124—171no—
351C2×#124—171no—
352C2×C22⋊Q8—341no—
353C2×C4×Q8—171no—
354C2×C4×Q8—171no—
355C2×C4×Q8C2×D1711yescharacteristic
356C2×C4×D4—171no—
357C2×#117—341no—
358#446—344no—
359C2×C4⋊C8—171no—
360C2×C4⋊C8—171no—
361C2×C4⋊C8—171no—
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#103—171no—
369C2×#103—171no—
370C2×#102—171no—
371C2×#102—171no—
372C2×#102—171no—
373C2×#102—171no—
374C2×#97—171no—
375C2×#97—171no—
376C2×C4×C8—171no—
377#391—21no—
378#387—21no—
379#387—21no—
380#384C2×D1711yescharacteristic
381#382C2×D1711yescharacteristic
382#382C2×D1711yescharacteristic
383C4×C4⋊C4—21no—
384C4×C4⋊C4C2×D1711yescharacteristic
385C4×C4⋊C4C2×D1711yescharacteristic
386C4×C4⋊C4C2×D1711yescharacteristic
387#340—171no—
388#340—171no—
389#340—171no—
390#340—171no—
391C2×C17#234111yescharacteristic
392C2×C17#289611yescharacteristic
393C2×C17#289811yescharacteristic
394C2×C17#337611yescharacteristic
395C2×C17#337711yescharacteristic
396C2×C17#340111yescharacteristic
397C2×C17#340211yescharacteristic
398D17—82no—
399C22×Q8—171no—
400C22×Q8—171no—
401C22×Q8C22⋊D1711yescharacteristic
402C22×D4—171no—
403C23×C4—341no—
404C22×C8—171no—
405C22×C8—171no—
406C22×C8—171no—
407C22×C8—171no—
408C4⋊Q8—174no—
409C4⋊Q8—342no—
410#126—348no—
411#126—348no—
412#124—174no—
413#124—174no—
414#124—342no—
415#124—342no—
416C22⋊Q8—348no—
417C4×Q8—174no—
418C4×Q8—174no—
419C4×Q8—342no—
420C4×Q8—342no—
421C4×Q8C4×D1714yes—
422C4×Q8D6814yes—
423C4×D4—174no—
424C4×D4—342no—
425#117—348no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—171no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—341no—
431C2×C4⋊C4—341no—
432C2×C4⋊C4—341no—
433C2×C4⋊C4C22⋊D1711yescharacteristic
434C2×C4⋊C4C22⋊D1711yescharacteristic
435C2×C4⋊C4C22⋊D1711yescharacteristic
436C2×C4⋊C4C22×D1711yescharacteristic
437C2×C4⋊C4C22×D1711yescharacteristic
438C2×C4⋊C4C22×D1711yescharacteristic
439C2×C4⋊C4C22×D1711yescharacteristic
440C2×C22⋊C4—341no—
441C2×C22⋊C4—341no—
442C2×C22⋊C4—341no—
443C2×C22⋊C4—341no—
444C2×C22⋊C4—341no—
445C2×C42—21no—
446C2×C42—21no—
447C2×C42—21no—
448C2×C42—341no—
449C2×C42—341no—
450C2×C42C22×D1711yescharacteristic
451C2×C42C22×D1711yescharacteristic
452C2×C42C22×D1711yescharacteristic
453C4⋊C8—342no—
454C4⋊C8—342no—
455C4⋊C8—342no—
456#103—342no—
457#103—342no—
458#103—342no—
459#103—342no—
460#103—681no—
461#103—681no—
462#103—681no—
463#103—681no—
464#102—342no—
465#102—342no—
466#102—681no—
467#102—681no—
468#97—342no—
469#97—342no—
470C4×C8—342no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95—21no—
477#95—21no—
478C17#2254811yescharacteristic, complemented, Sylow
479C24—341no—
480C2×Q8—41no—
481C2×Q8—41no—
482C2×Q8—172no—
483C2×Q8—172no—
484C2×Q8—341no—
485C2×Q8—341no—
486C2×Q8—341no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×Q8—341no—
494C2×Q8#6322714yes—
495C2×D4—172no—
496C2×D4—341no—
497C2×D4—341no—
498C2×D4—341no—
499C2×D4—341no—
500C2×D4—341no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—344no—
509C22×C4—682no—
510C22×C4—682no—
511C22×C4C2×C22⋊D1711yescharacteristic
512C22×C4C2×C22⋊D1711yescharacteristic
513C22×C4C2×C22⋊D1711yescharacteristic
514C22×C4C2×C22⋊D1711yescharacteristic
515C22×C4C2×C22⋊D1711yescharacteristic
516C22×C4C2×C22⋊D1711yescharacteristic
517C22×C4C23×D1711yescharacteristic
518C2×C8—172no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C2×C8—341no—
528C2×C8—341no—
529C2×C8—341no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—42no—
538C4⋊C4—81no—
539C4⋊C4—172no—
540C4⋊C4—341no—
541C4⋊C4—344no—
542C4⋊C4—344no—
543C4⋊C4—344no—
544C4⋊C4—344no—
545C4⋊C4#6322714yes—
546C4⋊C4#6322714yes—
547C4⋊C4#6322714yes—
548C4⋊C4#6325612yes—
549C22⋊C4—344no—
550C22⋊C4—344no—
551C22⋊C4—344no—
552C22⋊C4—344no—
553C22⋊C4—344no—
554C42—42no—
555C42—81no—
556C42—81no—
557C42—344no—
558C42—344no—
559C42C2×C4×D1712yes—
560C42C2×C4×D1712yes—
561C42C2×C4×D1712yes—
562C42C2×D6812yes—
563C42C2×D6812yes—
564C42C2×D6812yes—
565C23—342no—
566C23—681no—
567C23—681no—
568C23—681no—
569C23—681no—
570C23—681no—
571C23—681no—
572C23C22×C22⋊D1711yescharacteristic
573Q8—44no—
574Q8—342no—
575Q8—342no—
576Q8—342no—
577Q8—342no—
578Q8—681no—
579Q8—681no—
580Q8—681no—
581Q8—681no—
582D4—342no—
583D4—342no—
584D4—681no—
585D4—681no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—342no—
611C2×C4—342no—
612C2×C4—342no—
613C2×C4—688no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4C2×#6322712yes—
621C2×C4C2×#6322712yes—
622C2×C4C2×#6322712yes—
623C2×C4C2×#6322712yes—
624C2×C4C2×#6322712yes—
625C2×C4C2×#6322712yes—
626C2×C4C22×C4×D1711yescharacteristic
627C2×C4C22×D6811yescharacteristic
628C2×C4C2×#6325611yescharacteristic
629C2×C4#1058769411yescharacteristic
630C2×C4#41×D1711yescharacteristic
631C2×C4#1058769611yescharacteristic
632C8—342no—
633C8—681no—
634C8—681no—
635C8—681no—
636C22—682no—
637C22—682no—
638C22—682no—
639C22—1361no—
640C22—1361no—
641C22—1361no—
642C22—1361no—
643C22C22×#6322711yescharacteristic
644C22#1176347011yescharacteristic
645C22#1176347811yescharacteristic
646C22#1176360711yescharacteristic
647C22#1176360711yescharacteristic
648C22#1176362011yescharacteristic
649C22#1176362411yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—682no—
661C4—682no—
662C4—682no—
663C4C4×#6325611yescharacteristic
664C4#1176329111yescharacteristic
665C4#1176347111yescharacteristic
666C4#1176348111yescharacteristic
667C2—1362no—
668C2#1453162211yescharacteristic
669C2#1453730111yescharacteristic
670C2#1453731411yescharacteristic
671C2#1454161111yescharacteristic
672C2#1454162611yescharacteristic
673C2#1454184411yescharacteristic
674C2#1454185911yescharacteristic
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#22548
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
5. #14529707C2semidirect
6. #14529603C2semidirect
9. #14528796C2semidirect
11. #14528786C2semidirect
12. #14528617C2semidirect
14. C4×#10587481C2semidirect
15. C17×#2355C2semidirect
17. #11763532C4semidirect
301. #2355D17semidirect
478. C17#22548semidirect

Maximal subgroups

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

subgroupindex
C2×#117635322
C2×#117629402
#145297162
#145297072
#145296032
#145295982
#145295932
#145287962
#145287912
#145287862
#145286172
#145286122
C4×#105874812
C17×#23552
#2254817