Finite Group Databasebeta

← #17097356

Subgroups — #17097356

Order 4352.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763092C211yesmaximal, characteristic, complemented
2C2×SD16×Dic17C211yesmaximal, characteristic
3#14532637C211yesmaximal, characteristic
4Q8×C4⋊Dic17C211yesmaximal, characteristic, complemented
5#14530317C211yesmaximal, characteristic
6#14530262C211yesmaximal, characteristic, complemented
7#14530154C211yesmaximal, characteristic
8#14530116C211yesmaximal, characteristic, complemented
9#14529511C211yesmaximal, characteristic, complemented
10#14529340C211yesmaximal, characteristic, complemented
11#14529141C211yesmaximal, characteristic
12#14529052C211yesmaximal, characteristic, complemented
13#14529050C211yesmaximal, characteristic, complemented
14#14529047C211yesmaximal, characteristic
15C17×#1880C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×#10587564—21no—
18C2×Q8×Dic17C2211yescharacteristic
19C2×Q8×Dic17C2211yescharacteristic
20C2×D4×Dic17C2211yescharacteristic
21#11763092—24no—
22SD16×Dic17—24no—
23C2×#10587488C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587485C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×C8×Dic17C2211yescharacteristic
34#11762732—21no—
35#11762721—21no—
36#11762712C2211yescharacteristic
37#11762708—21no—
38#11762708—21no—
39#11762708C2211yescharacteristic
40#11762708C2211yescharacteristic
41#11762707C2211yescharacteristic
42#11762705C2211yescharacteristic
43#11762681—21no—
44#11762679—21no—
45#11762679C2211yescharacteristic
46C4×C4⋊Dic17—21no—
47C4×C4⋊Dic17C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762564C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×SD16×C17C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C17×#103C2211yescharacteristic
57C2×C17×#102C2211yescharacteristic
58C17×#390C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×Dic17—21no—
64C22×Dic34—21no—
65C2×SD16×C17—22no—
66C2×SD16×C17—22no—
67C2×SD16×C17Q814yescomplemented
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C4×Q8×C17—22no—
71C4×Q8×C17—22no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C42×C17—21no—
78C2×C42×C17C2311yescharacteristic
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82Q8×Dic17—22no—
83Q8×Dic17—22no—
84Q8×Dic17—41no—
85Q8×Dic17—41no—
86D4×Dic17—22no—
87D4×Dic17—41no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17D411yescharacteristic
95C2×C4⋊Dic17D411yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17—21no—
105C2×C4×Dic17C2311yescharacteristic
106C2×C4×Dic17C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109#10587574—22no—
110#10587574—41no—
111#10587564—44no—
112C17×#103—22no—
113C17×#102—22no—
114#10587506—22no—
115#10587506D414yes—
116#10587503—22no—
117#10587503—41no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502D411yescharacteristic
123#10587502D411yescharacteristic
124#10587488—22no—
125#10587488—22no—
126#10587485—22no—
127C8×Dic17—22no—
128#10587479—22no—
129#10587479—22no—
130#10587479—41no—
131C24×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17C2×D412yes—
136C2×Q8×C17C2×Q812yes—
137C2×Q8×C17#4112yes—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17—21no—
141C2×D4×C17—21no—
142C2×D4×C17—21no—
143C2×D4×C17C2×Q812yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—42no—
155C22×Dic17—42no—
156C22×Dic17—81no—
157C22×Dic17C2×D411yescharacteristic
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C2×Dic34—42no—
162C2×Dic34—42no—
163C2×Dic34—81no—
164SD16×C17—28no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17C2×Q812yes—
171C4⋊C4×C17—24no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17#4112yes—
176C22⋊C4×C17—42no—
177C22⋊C4×C17—81no—
178C42×C17—24no—
179C42×C17—21no—
180C42×C17—21no—
181C22⋊Dic17—42no—
182C22⋊Dic17—42no—
183C22⋊Dic17—42no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—41no—
192C4⋊Dic17—42no—
193C4⋊Dic17—42no—
194C4⋊Dic17—42no—
195C4⋊Dic17—81no—
196C4⋊Dic17—81no—
197C4⋊Dic17—81no—
198C4⋊Dic17SD1614yescomplemented
199C4⋊Dic17C2×D412yes—
200#63225—42no—
201#63225—42no—
202#63225—42no—
203C4×Dic17—21no—
204C4×Dic17—21no—
205C4×Dic17—21no—
206C4×Dic17—21no—
207C4×Dic17—42no—
208C4×Dic17—42no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622C2×D412yes—
215#26373—171nomaximal, Sylow
216C23×C17—22no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17C22×D411yescharacteristic
224Q8×C17—22no—
225Q8×C17—22no—
226Q8×C17—22no—
227Q8×C17C22⋊Q814yescomplemented
228D4×C17—22no—
229D4×C17—22no—
230D4×C17—41no—
231D4×C17—41no—
232C2×C4×C17—21no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—22no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17C2×SD1612yes—
247C2×C4×C17C22×D411yescharacteristic
248C2×C4×C17C22×Q811yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17H(5,𝔽2)11yescharacteristic
252C2×C4×C172−1+411yescharacteristic
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—22no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
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—84no—
276C2×Dic17C2×SD1612yes—
277C2×Dic17C2×SD1612yes—
278Dic34—84no—
279C8×C17—22no—
280C8×C17—41no—
281#3622—22no—
282#3622—41no—
283C2×#475—171no—
284C2×C4×SD16—171no—
285#2452—171no—
286Q8×C4⋊C4—171no—
287#2094—171no—
288#2084—171no—
289#2065—171no—
290#2058—171no—
291#1941—171no—
292#1916—171no—
293#1880—171no—
294#1880D1711yescharacteristic
295#1865—171no—
296#1865—171no—
297#1864—171no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—81no—
302C22×C17—81no—
303C22×C17—81no—
304C22×C17—81no—
305C22×C17#52511yescharacteristic
306C22×C17D4211yescharacteristic
307C22×C17D4×Q811yescharacteristic
308C22×C17C22×SD1611yescharacteristic
309C22×C17#57511yescharacteristic
310C22×C17#57711yescharacteristic
311C22×C17#57711yescharacteristic
312C4×C17—22no—
313C4×C17—24no—
314C4×C17—41no—
315C4×C17—42no—
316C4×C17—81no—
317C4×C17C2×C22⋊Q811yescharacteristic
318C4×C17#53911yescharacteristic
319C4×C17#54711yescharacteristic
320C4×C17#55411yescharacteristic
321Dic17—22no—
322Dic17—22no—
323Dic17—41no—
324Dic17—41no—
325Dic17—42no—
326Dic17—81no—
327Dic17—81no—
328Dic17—81no—
329C22×SD16C2×D1711yescharacteristic
330C2×C4⋊Q8—171no—
331C2×C22⋊Q8—341no—
332C2×C4×Q8—171no—
333C2×C4×Q8—171no—
334C2×C4×Q8C2×D1711yescharacteristic
335C2×C4×D4—171no—
336#475—344no—
337C4×SD16—344no—
338C2×#106—171no—
339C2×#106—171no—
340C2×#106—171no—
341C2×#106—171no—
342C2×#106—171no—
343C2×C4⋊C8—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#103C2×D1711yescharacteristic
347C2×#102—171no—
348C2×#102—171no—
349C2×#102C2×D1711yescharacteristic
350C2×C4×C8—171no—
351#396—341no—
352#393—341no—
353#390—171no—
354#390C2×D1711yescharacteristic
355#389—171no—
356#389—171no—
357#389—171no—
358#389—171no—
359#389—341no—
360#389—341no—
361#385—341no—
362#384—171no—
363#384—341no—
364C4×C4⋊C4—171no—
365C4×C4⋊C4—341no—
366#340—171no—
367#340—171no—
368#340—171no—
369#340C2×D1711yescharacteristic
370#336—171no—
371#336—171no—
372#336—171no—
373#336C2×D1711yescharacteristic
374C2×C17—82no—
375C2×C17#244911yescharacteristic
376C2×C17#310311yescharacteristic
377C2×C17#312511yescharacteristic
378C2×C17#327111yescharacteristic
379C2×C17D4×SD1611yescharacteristic
380C2×C17Q8×SD1611yescharacteristic
381C2×C17#338811yescharacteristic
382C22×Q8—341no—
383C22×Q8C22×D1711yescharacteristic
384C22×D4C22×D1711yescharacteristic
385C23×C4—341no—
386C2×SD16—22no—
387C2×SD16—22no—
388C2×SD16Dic3414yescomplemented
389C22×C8—171no—
390C22×C8—171no—
391C22×C8C22×D1711yescharacteristic
392C22×C8C22×D1711yescharacteristic
393C4⋊Q8—342no—
394C4⋊Q8—681no—
395C22⋊Q8—684no—
396C4×Q8—22no—
397C4×Q8—22no—
398C4×Q8—342no—
399C4×Q8—342no—
400C4×Q8—681no—
401C4×Q8—681no—
402C4×D4—342no—
403C4×D4—681no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—171no—
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⋊C4—341no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4C22×D1711yescharacteristic
421C2×C4⋊C4C22×D1711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—341no—
425C2×C22⋊C4—341no—
426C2×C22⋊C4—341no—
427C2×C42—21no—
428C2×C42—171no—
429C2×C42—171no—
430C2×C42—341no—
431C2×C42—341no—
432C2×C42C22×D1711yescharacteristic
433#106—342no—
434#106—342no—
435#106—342no—
436#106—342no—
437#106—681no—
438C4⋊C8—342no—
439#103—22no—
440#103—174no—
441#103—342no—
442#102—22no—
443#102—342no—
444#102—681no—
445C4×C8—342no—
446#95—171no—
447#95—171no—
448#95—341no—
449#95—341no—
450#95—341no—
451#95—341no—
452C17#2637311yescharacteristic, complemented, Sylow
453C24—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—682no—
458C2×Q8—682no—
459C2×Q8—1361no—
460C2×Q8C2×Dic3412yes—
461C2×Q8D4×D1712yes—
462C2×Q8#6325412yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×Dic3412yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—171no—
473C22×C4—171no—
474C22×C4—171no—
475C22×C4—171no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—682no—
481C22×C4—682no—
482C22×C4—1361no—
483C22×C4D4×D1711yescharacteristic
484C22×C4D4×D1711yescharacteristic
485C22×C4C23×D1711yescharacteristic
486SD16—28no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—172no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8C2×Dic3412yes—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—174no—
504C4⋊C4—172no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—341no—
512C4⋊C4—681no—
513C4⋊C4—682no—
514C4⋊C4—682no—
515C4⋊C4—682no—
516C4⋊C4—682no—
517C4⋊C4—682no—
518C4⋊C4—682no—
519C4⋊C4—1361no—
520C4⋊C4—1361no—
521C4⋊C4—1361no—
522C4⋊C4#6325412yes—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—682no—
526C22⋊C4—682no—
527C22⋊C4—682no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—341no—
532C42—341no—
533C42—341no—
534C42—341no—
535C42—682no—
536C42—682no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×D4×D1711yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—1364no—
549Q8#1058756414yescomplemented
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—172no—
569C2×C4—172no—
570C2×C4—341no—
571C2×C4—341no—
572C2×C4—341no—
573C2×C4—341no—
574C2×C4—341no—
575C2×C4—341no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—342no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—1364no—
593C2×C4SD16×D1712yes—
594C2×C4C22×Dic3411yescharacteristic
595C2×C4C2×D4×D1711yescharacteristic
596C2×C4C2×#6325411yescharacteristic
597C2×C4C2×#6325411yescharacteristic
598C2×C4#1058769611yescharacteristic
599C2×C4#1058769711yescharacteristic
600C8—22no—
601C8—41no—
602C8—342no—
603C8—681no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1176331511yescharacteristic
612C22#1176344311yescharacteristic
613C22D4×Dic3411yescharacteristic
614C22C2×SD16×D1711yescharacteristic
615C22#1176360811yescharacteristic
616C22#1176361611yescharacteristic
617C22#1176361911yescharacteristic
618C4—22no—
619C4—24no—
620C4—41no—
621C4—42no—
622C4—81no—
623C4—342no—
624C4—342no—
625C4—681no—
626C4—681no—
627C4—682no—
628C4—1361no—
629C4—1361no—
630C4—1361no—
631C4C2×#1058756411yescharacteristic
632C4#1176340311yescharacteristic
633C4#1176345911yescharacteristic
634C4#1176351611yescharacteristic
635C2—82no—
636C2#1453259911yescharacteristic
637C2#1453881011yescharacteristic
638C2#1453906211yescharacteristic
639C2#1454037511yescharacteristic
640C2#1454047511yescharacteristic
641C2SD16×Dic3411yescharacteristic
642C2#1454173011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1880C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#1880D17
hypocenterC17#26373
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. C2×#11763092C2semidirect
4. Q8×C4⋊Dic17C2semidirect
6. #14530262C2semidirect
8. #14530116C2semidirect
9. #14529511C2semidirect
10. #14529340C2semidirect
12. #14529052C2semidirect
13. #14529050C2semidirect
67. C2×SD16×C17Q8semidirect
198. C4⋊Dic17SD16semidirect
227. Q8×C17C22⋊Q8semidirect
388. C2×SD16Dic34semidirect
452. C17#26373semidirect
549. Q8#10587564semidirect

Maximal subgroups

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

subgroupindex
C2×#117630922
C2×SD16×Dic172
#145326372
Q8×C4⋊Dic172
#145303172
#145302622
#145301542
#145301162
#145295112
#145293402
#145291412
#145290522
#145290502
#145290472
C17×#18802
#2637317