Finite Group Databasebeta

← #17086165

Subgroups — #17086165

Order 4352.

838 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763087C211yesmaximal, characteristic
2C2×C8×Dic34C211yesmaximal, characteristic, complemented
3#14533771C211yesmaximal, characteristic, complemented
4#14533704C211yesmaximal, characteristic
5#14530275C211yesmaximal, characteristic, complemented
6#14530274C211yesmaximal, characteristic, complemented
7#14530172C211yesmaximal, characteristic
8#14530170C211yesmaximal, characteristic
9#14529743C211yesmaximal, characteristic, complemented
10#14529541C211yesmaximal, characteristic, complemented
11#14529517C211yesmaximal, characteristic, complemented
12#14529516C211yesmaximal, characteristic, complemented
13#14529469C211yesmaximal, characteristic
14#102×Dic17C211yesmaximal, characteristic
15C17×#1934C211yesmaximal, characteristic
16C2×#10587574—21no—
17C2×#10587574C2211yescharacteristic
18C2×#10587564—21no—
19C2×C4×Dic34C2211yescharacteristic
20C2×D4×Dic17C2211yescharacteristic
21#11763087—24no—
22C8×Dic34—24no—
23C2×#10587489C2211yescharacteristic
24C2×#10587489C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587485C2211yescharacteristic
29C2×#10587485C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×C8×Dic17C2211yescharacteristic
33C2×C8×Dic17C2211yescharacteristic
34#11762743—21no—
35#11762742—21no—
36#11762732—21no—
37#11762732—21no—
38#11762713—21no—
39#11762712C2211yescharacteristic
40#11762708C2211yescharacteristic
41#11762708C2211yescharacteristic
42#11762707C2211yescharacteristic
43#11762707C2211yescharacteristic
44#11762705C2211yescharacteristic
45#11762678C2211yescharacteristic
46#11762667—21no—
47C4⋊C4×Dic17C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C4×C8×C17C2211yescharacteristic
57C17×#390C2211yescharacteristic
58C17×#389C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C22×Dic34—21no—
64C22×Dic34D411yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C42×C17C2311yescharacteristic
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77D4×Dic17—22no—
78D4×Dic17—22no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17D411yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225D411yescharacteristic
93C2×#63225D411yescharacteristic
94C2×C4×Dic17—21no—
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99#10587574—22no—
100#10587574—22no—
101#10587574—44no—
102#10587564—44no—
103C4×Dic34—22no—
104C4×Dic34—22no—
105C17×#102—22no—
106C17×#102Q814yescomplemented
107C4×C8×C17—22no—
108C17×#95—21no—
109#10587503—22no—
110#10587503—22no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587489—22no—
118#10587489—22no—
119#10587485—22no—
120#10587485D414yes—
121C8×Dic17—22no—
122C8×Dic17—22no—
123#10587478—22no—
124#10587478—22no—
125C4⋊#3622—22no—
126C24×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17C2×Q812yes—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—42no—
144C22×Dic17—42no—
145C22×Dic17—42no—
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C2×Dic34—22no—
151C2×Dic34—22no—
152C2×Dic34—42no—
153C2×Dic34—42no—
154C2×Dic34—42no—
155C2×Dic34D814yes—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17C2×D412yes—
161C2×C8×C17C2×Q812yes—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—42no—
168C4⋊C4×C17C2×Q812yes—
169C22⋊C4×C17—42no—
170C22⋊C4×C17—81no—
171C42×C17—21no—
172C42×C17—21no—
173C22⋊Dic17—42no—
174C22⋊Dic17—42no—
175C22⋊Dic17—81no—
176C4⋊Dic17—21no—
177C4⋊Dic17—22no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—42no—
185C4⋊Dic17—42no—
186C4⋊Dic17#4112yes—
187#63225—22no—
188#63225—22no—
189#63225—42no—
190#63225—42no—
191#63225—42no—
192#63225—42no—
193#63225—42no—
194#63225—81no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—42no—
198C4×Dic17C2×D412yes—
199C4×Dic17#4112yes—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622C2×D412yes—
206#25469—171nomaximal, Sylow
207C23×C17—22no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17—41no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17C22×D411yescharacteristic
215D4×C17—22no—
216D4×C17—22no—
217D4×C17—41no—
218D4×C17—41no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17C2×D812yes—
234C2×C4×C17C22×D411yescharacteristic
235C2×C4×C17C22×Q811yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C172−1+411yescharacteristic
239C2×C4×C172−1+411yescharacteristic
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—22no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—84no—
260C2×Dic17#12112yes—
261C2×Dic17#12112yes—
262C2×Dic17C2×D812yes—
263C2×Dic17C2×D812yes—
264C2×Dic17#13512yes—
265Dic34—28no—
266Dic34—84no—
267C8×C17—22no—
268C8×C17—22no—
269#3622—22no—
270#3622—41no—
271C2×#474—171no—
272C2×C8×Q8—171no—
273#2546—171no—
274#2538—171no—
275#2086—171no—
276#2086—171no—
277#2067—171no—
278#2067—171no—
279#1990—171no—
280#1947—171no—
281#1942—171no—
282#1942—171no—
283#1934—171no—
284#1934D1711yescharacteristic
285C4×#102—171no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—42no—
289C22×C17—81no—
290C22×C17—81no—
291C22×C17—81no—
292C22×C17—81no—
293C22×C17C2×#12111yescharacteristic
294C22×C17#54811yescharacteristic
295C22×C17D4×Q811yescharacteristic
296C22×C17C22×D811yescharacteristic
297C22×C17C2×#13511yescharacteristic
298C22×C17#57711yescharacteristic
299C22×C17#57811yescharacteristic
300C4×C17—22no—
301C4×C17—42no—
302C4×C17—41no—
303C4×C17—81no—
304C4×C17—81no—
305C4×C17#53311yescharacteristic
306C4×C17#54111yescharacteristic
307C4×C17#54711yescharacteristic
308C4×C17D4×Q811yescharacteristic
309Dic17—22no—
310Dic17—22no—
311Dic17—42no—
312Dic17—41no—
313Dic17—81no—
314Dic17—81no—
315Dic17—81no—
316Dic17#46614yes—
317C2×C4⋊Q8—171no—
318C2×C4⋊Q8—341no—
319C2×C22⋊Q8—341no—
320C2×C4×Q8—171no—
321C2×C4×D4—171no—
322#474—344no—
323C8×Q8—344no—
324C2×#107—171no—
325C2×#107—171no—
326C2×#107—171no—
327C2×#107—171no—
328C2×C4⋊C8—171no—
329C2×C4⋊C8—171no—
330C2×C4⋊C8—171no—
331C2×#102—171no—
332C2×#102—171no—
333C2×#102C2×D1711yescharacteristic
334C2×#102C2×D1711yescharacteristic
335C2×C4×C8—171no—
336C2×C4×C8—171no—
337C2×C4×C8C2×D1711yescharacteristic
338#398—341no—
339#398—341no—
340#396—341no—
341#396—341no—
342#391—341no—
343#390—171no—
344#390C2×D1711yescharacteristic
345#389—171no—
346#389—171no—
347#389—171no—
348#389—171no—
349#389—171no—
350#389C2×D1711yescharacteristic
351#384—171no—
352#380—341no—
353C4×C4⋊C4—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340C2×D1711yescharacteristic
361#340C2×D1711yescharacteristic
362C2×C17—82no—
363C2×C17#242911yescharacteristic
364C2×C17C2×#46611yescharacteristic
365C2×C17#306311yescharacteristic
366C2×C17#335811yescharacteristic
367C2×C17#336411yescharacteristic
368C2×C17Q8×D811yescharacteristic
369C2×C17#338111yescharacteristic
370C22×Q8—171no—
371C22×Q8—341no—
372C22×D4C22×D1711yescharacteristic
373C23×C4—341no—
374C22×C8—171no—
375C22×C8—171no—
376C22×C8C22×D1711yescharacteristic
377C22×C8C22×D1711yescharacteristic
378C4⋊Q8—342no—
379C4⋊Q8—342no—
380C4⋊Q8—684no—
381C22⋊Q8—684no—
382C4×Q8—342no—
383C4×Q8—342no—
384C4×D4—342no—
385C4×D4—342no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C4⋊C4C22×D1711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—341no—
408C2×C22⋊C4—341no—
409C2×C22⋊C4—341no—
410C2×C42—171no—
411C2×C42—171no—
412C2×C42—341no—
413C2×C42C22×D1711yescharacteristic
414#107—342no—
415#107—342no—
416#107—342no—
417#107—342no—
418C4⋊C8—174no—
419C4⋊C8—342no—
420C4⋊C8—342no—
421#102—22no—
422#102—342no—
423#102—342no—
424#102Dic3414yescomplemented
425C4×C8—22no—
426C4×C8—342no—
427C4×C8—342no—
428#95—21no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435C17#2546911yescharacteristic, complemented, Sylow
436C24—21no—
437C2×Q8—174no—
438C2×Q8—342no—
439C2×Q8—342no—
440C2×Q8—682no—
441C2×Q8—682no—
442C2×Q8—682no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×Dic3412yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—171no—
455C22×C4—171no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—682no—
461C22×C4—682no—
462C22×C4—682no—
463C22×C4D4×D1711yescharacteristic
464C22×C4D4×D1711yescharacteristic
465C22×C4C23×D1711yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—172no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8C2×Dic3412yes—
477C2×C8D4×D1712yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—172no—
485C4⋊C4—342no—
486C4⋊C4—342no—
487C4⋊C4—342no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—682no—
496C4⋊C4—682no—
497C4⋊C4—682no—
498C4⋊C4—682no—
499C4⋊C4—682no—
500C4⋊C4—682no—
501C4⋊C4—682no—
502C4⋊C4—1361no—
503C4⋊C4C2×Dic3412yes—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—682no—
507C22⋊C4—682no—
508C22⋊C4—1361no—
509C42—21no—
510C42—21no—
511C42—172no—
512C42—172no—
513C42—341no—
514C42—341no—
515C42—682no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C2×D4×D1711yescharacteristic
524Q8—348no—
525Q8—1364no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—172no—
545C2×C4—172no—
546C2×C4—172no—
547C2×C4—172no—
548C2×C4—172no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—342no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—1364no—
569C2×C4D8×D1712yes—
570C2×C4C22×Dic3411yescharacteristic
571C2×C4C2×D4×D1711yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4#41×D1711yescharacteristic
574C2×C4#1058769711yescharacteristic
575C2×C4#1058769711yescharacteristic
576C8—22no—
577C8—22no—
578C8—342no—
579C8—681no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22#121×D1711yescharacteristic
588C22#1176346711yescharacteristic
589C22D4×Dic3411yescharacteristic
590C22C2×D8×D1711yescharacteristic
591C22#135×D1711yescharacteristic
592C22#1176361511yescharacteristic
593C22#1176362311yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—174no—
600C4—342no—
601C4—342no—
602C4—682no—
603C4—681no—
604C4—1361no—
605C4—1361no—
606C4—1361no—
607C4#1176336211yescharacteristic
608C4#1176341611yescharacteristic
609C4#1176345711yescharacteristic
610C4D4×Dic3411yescharacteristic
611C2—82no—
612C2#1453238011yescharacteristic
613C2D17×#46611yescharacteristic
614C2#1453846811yescharacteristic
615C2#1454141611yescharacteristic
616C2#1454146611yescharacteristic
617C2D8×Dic3411yescharacteristic
618C2#1454166011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1934C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#1934D17
hypocenterC17#25469
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. C2×C8×Dic34C2semidirect
3. #14533771C2semidirect
5. #14530275C2semidirect
6. #14530274C2semidirect
9. #14529743C2semidirect
10. #14529541C2semidirect
11. #14529517C2semidirect
12. #14529516C2semidirect
106. C17×#102Q8semidirect
424. #102Dic34semidirect
435. C17#25469semidirect

Maximal subgroups

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

subgroupindex
C2×#117630872
C2×C8×Dic342
#145337712
#145337042
#145302752
#145302742
#145301722
#145301702
#145297432
#145295412
#145295172
#145295162
#145294692
#102×Dic172
C17×#19342
#2546917