Finite Group Databasebeta

← #17052740

Subgroups — #17052740

Order 4352.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic17C211yesmaximal, characteristic
2C2×SD16×Dic17C211yesmaximal, characteristic
3#14531702C211yesmaximal, characteristic, complemented
4#14531689C211yesmaximal, characteristic
5#14529512C211yesmaximal, characteristic, complemented
6#14529510C211yesmaximal, characteristic, complemented
7#14529038C211yesmaximal, characteristic, complemented
8#14529036C211yesmaximal, characteristic, complemented
9#14529031C211yesmaximal, characteristic
10#14529029C211yesmaximal, characteristic
11#14528945C211yesmaximal, characteristic, complemented
12#14528797C211yesmaximal, characteristic, complemented
13#14528792C211yesmaximal, characteristic
14C8×#63225C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C17C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19C2×D4×Dic17C2211yescharacteristic
20SD16×Dic17—24no—
21SD16×Dic17C418yescomplemented
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587488C2211yescharacteristic
26C2×#10587485C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32C2×C8×Dic17C2211yescharacteristic
33#11762707—21no—
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762705—21no—
37#11762705C2211yescharacteristic
38#11762705C2211yescharacteristic
39#11762682—21no—
40#11762681—21no—
41#11762674—21no—
42#11762674C2211yescharacteristic
43#11762674C2211yescharacteristic
44#11762667—21no—
45C4×#63225—21no—
46C4×#63225C2211yescharacteristic
47C2×#10587502—21no—
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762564C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×SD16×C17C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C4×D4×C17C2211yescharacteristic
57C4×SD16×C17—24no—
58C2×C17×#106C2211yescharacteristic
59C2×C17×#103C2211yescharacteristic
60C2×C17×#102C2211yescharacteristic
61C2×C4×C8×C17C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D4×C17C2311yescharacteristic
64C23×C4×C17—21no—
65C23×Dic17—21no—
66C23×Dic17—21no—
67C2×SD16×C17C2×C414yes—
68C2×SD16×C17D414yes—
69C2×SD16×C17Q814yes—
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C4×Q8×C17—22no—
73C4×Q8×C17—41no—
74C4×D4×C17—22no—
75C4×D4×C17—41no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84Q8×Dic17—22no—
85Q8×Dic17—22no—
86Q8×Dic17—41no—
87Q8×Dic17C2×C414yescomplemented
88D4×Dic17—22no—
89D4×Dic17—22no—
90D4×Dic17—41no—
91D4×Dic17C2×C414yes—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225D411yescharacteristic
103C2×#63225D411yescharacteristic
104C2×C4×Dic17—21no—
105C2×C4×Dic17—21no—
106C2×C4×Dic17C2311yescharacteristic
107C2×C4×Dic17C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C17×#106—22no—
111C17×#103—22no—
112C17×#102—22no—
113C4×C8×C17—22no—
114#10587506—22no—
115#10587506C2×C414yescomplemented
116#10587503—22no—
117#10587503C2×C414yes—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502D411yescharacteristic
127#10587502D411yescharacteristic
128#10587488—22no—
129#10587488C2×C414yescomplemented
130#10587485—22no—
131C8×Dic17—22no—
132C8×Dic17C2×C414yescomplemented
133#10587478—41no—
134#10587478—41no—
135C24×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17C22×C412yes—
140C2×Q8×C17C2×D412yes—
141C2×Q8×C17C2×Q812yes—
142C2×D4×C17—21no—
143C2×D4×C17—21no—
144C2×D4×C17—21no—
145C2×D4×C17C22×C412yes—
146C2×D4×C17C2×D412yes—
147C2×D4×C17C2×Q812yes—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—42no—
152C22×C4×C17—42no—
153C22×C4×C17—81no—
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2411yescharacteristic
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—24no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—42no—
163C22×Dic17—42no—
164C22×Dic17—81no—
165C22×Dic17—81no—
166C22×Dic17—81no—
167C22×Dic17C2×D411yescharacteristic
168C22×Dic17C2×D411yescharacteristic
169C22×Dic17C2×D411yescharacteristic
170C22×Dic17C2×D411yescharacteristic
171SD16×C17C4⋊C4116yescomplemented
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17C22×C412yes—
176C2×C8×C17C2×D412yes—
177C2×C8×C17C2×Q812yes—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—42no—
183C22⋊C4×C17—42no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—42no—
187C22⋊Dic17—24no—
188C22⋊Dic17—42no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—24no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—42no—
197C4⋊Dic17C22×C412yes—
198C4⋊Dic17C22×C412yes—
199#63225—22no—
200#63225—22no—
201#63225—42no—
202#63225—42no—
203#63225—81no—
204#63225—81no—
205C4×Dic17—21no—
206C4×Dic17—21no—
207C4×Dic17—24no—
208C4×Dic17—42no—
209C4×Dic17C22×C412yes—
210C4×Dic17#4112yes—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622C22×C412yes—
217#22683—171nomaximal, Sylow
218C23×C17—22no—
219C23×C17—22no—
220C23×C17—22no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17C22×D411yescharacteristic
226Q8×C17—22no—
227Q8×C17—22no—
228Q8×C17—22no—
229Q8×C17C2×C4⋊C414yes—
230D4×C17—22no—
231D4×C17—22no—
232D4×C17—22no—
233D4×C17C2×C4⋊C414yes—
234C2×C4×C17—22no—
235C2×C4×C17—21no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
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—84no—
249C2×C4×C17#13512yes—
250C2×C4×C17C23×C411yescharacteristic
251C2×C4×C17C22×D411yescharacteristic
252C2×C4×C17C22×Q811yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17H(5,𝔽2)11yescharacteristic
255C2×C4×C172−1+411yescharacteristic
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—84no—
278C2×Dic17—84no—
279C2×Dic17C4×D412yes—
280C2×Dic17C4×D412yes—
281C2×Dic17#13512yes—
282C8×C17—41no—
283C8×C17C2×C4⋊C414yes—
284#3622—22no—
285#3622—41no—
286C2×C4×SD16—171no—
287C2×C4×SD16—171no—
288C2×C4×SD16D1711yescharacteristic
289#2351—171no—
290#2349—171no—
291#1941—171no—
292#1941—171no—
293#1863—171no—
294#1863—171no—
295#1862—171no—
296#1862—171no—
297#1846—171no—
298#1805—171no—
299#1804—171no—
300C8×C4⋊C4—171no—
301C22×C17—24no—
302C22×C17—42no—
303C22×C17—42no—
304C22×C17—42no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—42no—
308C22×C17C2×C4×D411yescharacteristic
309C22×C17D4211yescharacteristic
310C22×C17D4×Q811yescharacteristic
311C22×C17C2×#13511yescharacteristic
312C22×C17C2×#13511yescharacteristic
313C22×C17#57711yescharacteristic
314C22×C17#57711yescharacteristic
315C4×C17—24no—
316C4×C17—22no—
317C4×C17—41no—
318C4×C17—81no—
319C4×C17—81no—
320C4×C17C22×C4⋊C411yescharacteristic
321C4×C17#52011yescharacteristic
322C4×C17#54711yescharacteristic
323C4×C17#55411yescharacteristic
324Dic17—22no—
325Dic17—22no—
326Dic17—22no—
327Dic17—42no—
328Dic17—42no—
329Dic17—41no—
330Dic17—81no—
331Dic17—81no—
332C22×SD16C2×D1711yescharacteristic
333C2×C4×Q8—171no—
334C2×C4×Q8—171no—
335C2×C4×Q8C2×D1711yescharacteristic
336C2×C4×D4—171no—
337C2×C4×D4—171no—
338C2×C4×D4C2×D1711yescharacteristic
339C4×SD16—24no—
340C4×SD16—178no—
341C4×SD16—344no—
342C2×#107—171no—
343C2×#107—171no—
344C2×#106—171no—
345C2×#106—171no—
346C2×#106C2×D1711yescharacteristic
347C2×C4⋊C8—171no—
348C2×#103—171no—
349C2×#103—171no—
350C2×#103C2×D1711yescharacteristic
351C2×#102—171no—
352C2×#102—171no—
353C2×#102C2×D1711yescharacteristic
354C2×C4×C8—171no—
355C2×C4×C8—171no—
356C2×C4×C8C2×D1711yescharacteristic
357#389—171no—
358#389—171no—
359#389—171no—
360#389—171no—
361#389—341no—
362#389—341no—
363#385—341no—
364#385—341no—
365#382—171no—
366#382—171no—
367#382—341no—
368#380—341no—
369C4×C4⋊C4—171no—
370C4×C4⋊C4—341no—
371C2×#95—341no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#336—171no—
377#336—171no—
378C2×C17—44no—
379C2×C17D4×C4⋊C411yescharacteristic
380C2×C17C4×#13511yescharacteristic
381C2×C17#294911yescharacteristic
382C2×C17#328211yescharacteristic
383C2×C17#328211yescharacteristic
384C2×C17#338111yescharacteristic
385C2×C17#338111yescharacteristic
386C22×Q8C22×D1711yescharacteristic
387C22×D4C22×D1711yescharacteristic
388C23×C4—21no—
389C23×C4—341no—
390C23×C4—341no—
391C2×SD16Dic3414yes—
392C2×SD16C4×D1714yes—
393C2×SD16C22⋊D1714yes—
394C22×C8—171no—
395C22×C8—171no—
396C22×C8C22×D1711yescharacteristic
397C22×C8C22×D1711yescharacteristic
398C4×Q8—22no—
399C4×Q8—41no—
400C4×Q8—174no—
401C4×Q8—342no—
402C4×Q8—342no—
403C4×Q8—681no—
404C4×D4—22no—
405C4×D4—41no—
406C4×D4—174no—
407C4×D4—342no—
408C4×D4—342no—
409C4×D4—681no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4C22×D1711yescharacteristic
424C2×C4⋊C4C22×D1711yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—341no—
427C2×C22⋊C4—341no—
428C2×C42—21no—
429C2×C42—171no—
430C2×C42—171no—
431C2×C42—341no—
432C2×C42—341no—
433C2×C42C22×D1711yescharacteristic
434#107—681no—
435#107—681no—
436#106—22no—
437#106—174no—
438#106—342no—
439C4⋊C8—342no—
440#103—22no—
441#103—174no—
442#103—342no—
443#102—22no—
444#102—174no—
445#102—342no—
446C4×C8—22no—
447C4×C8—174no—
448C4×C8—342no—
449#95—171no—
450#95—171no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459C17#2268311yescharacteristic, complemented, Sylow
460C24—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×Dic3412yes—
465C2×Q8C2×C4×D1712yes—
466C2×Q8C2×C22⋊D1712yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×Dic3412yes—
471C2×D4C2×C4×D1712yes—
472C2×D4C2×C22⋊D1712yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—81no—
479C22×C4—171no—
480C22×C4—171no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4—344no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—682no—
489C22×C4—682no—
490C22×C4—1361no—
491C22×C4—1361no—
492C22×C4—1361no—
493C22×C4D4×D1711yescharacteristic
494C22×C4D4×D1711yescharacteristic
495C22×C4C23×D1711yescharacteristic
496SD16#63225116yescomplemented
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—172no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8C2×Dic3412yes—
507C2×C8C2×C4×D1712yes—
508C2×C8C2×C22⋊D1712yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—172no—
515C4⋊C4—172no—
516C4⋊C4—342no—
517C4⋊C4—344no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—342no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C4⋊C4—682no—
528C4⋊C4—1361no—
529C4⋊C4—1361no—
530C22⋊C4—42no—
531C22⋊C4—344no—
532C22⋊C4—682no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—172no—
537C42—172no—
538C42—344no—
539C42—341no—
540C42—341no—
541C42—682no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D1711yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8C2×#6322514yes—
554D4—22no—
555D4—22no—
556D4—22no—
557D4C2×#6322514yes—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—172no—
574C2×C4—172no—
575C2×C4—172no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—342no—
583C2×C4—342no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—1364no—
598C2×C4—1364no—
599C2×C4#1058760212yes—
600C2×C4C22×Dic3411yescharacteristic
601C2×C4C22×C4×D1711yescharacteristic
602C2×C4#1058769111yescharacteristic
603C2×C4#41×D1711yescharacteristic
604C2×C4#1058769711yescharacteristic
605C2×C4C22×C22⋊D1711yescharacteristic
606C8—41no—
607C8—342no—
608C8—681no—
609C8C2×#6322514yes—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C4×D4×D1711yescharacteristic
618C22D4×C22⋊D1711yescharacteristic
619C22D4×Dic3411yescharacteristic
620C22#135×D1711yescharacteristic
621C22C2×#1058760211yescharacteristic
622C22#1176361511yescharacteristic
623C22#1176361611yescharacteristic
624C4—24no—
625C4—22no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—342no—
630C4—342no—
631C4—342no—
632C4—682no—
633C4—682no—
634C4—681no—
635C4—1361no—
636C4—1361no—
637C4C22×#6322511yescharacteristic
638C4#1176328811yescharacteristic
639C4#1176345811yescharacteristic
640C4#1176351811yescharacteristic
641C2—44no—
642C2D4×#6322511yescharacteristic
643C2C4×#1058760211yescharacteristic
644C2#1453763811yescharacteristic
645C2#1454048511yescharacteristic
646C2#1454049111yescharacteristic
647C2#1454165711yescharacteristic
648C2#1454166011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C4×SD16×C17C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×C4×SD16D17
hypocenterC17#22683
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. #14531702C2semidirect
5. #14529512C2semidirect
6. #14529510C2semidirect
7. #14529038C2semidirect
8. #14529036C2semidirect
11. #14528945C2semidirect
12. #14528797C2semidirect
14. C8×#63225C2semidirect
21. SD16×Dic17C4semidirect
87. Q8×Dic17C2×C4semidirect
115. #10587506C2×C4semidirect
129. #10587488C2×C4semidirect
132. C8×Dic17C2×C4semidirect
171. SD16×C17C4⋊C4semidirect
459. C17#22683semidirect
496. SD16#63225semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic172
#145317022
#145316892
#145295122
#145295102
#145290382
#145290362
#145290312
#145290292
#145289452
#145287972
#145287922
C8×#632252
C2×C4×SD16×C172
#2268317