Finite Group Databasebeta

← #17064263

Subgroups — #17064263

Order 4352.

858 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D17×#389C211yesmaximal, characteristic
2#14531190C211yesmaximal, characteristic
3#14530353C211yesmaximal, characteristic, complemented
4#14530352C211yesmaximal, characteristic, complemented
5#14530351C211yesmaximal, characteristic, complemented
6#14530350C211yesmaximal, characteristic, complemented
7#14530349C211yesmaximal, characteristic, complemented
8#14530345C211yesmaximal, characteristic, complemented
9#14530344C211yesmaximal, characteristic, complemented
10#14528502C211yesmaximal, characteristic
11#14528502C211yesmaximal, characteristic
12#14528502C211yesmaximal, characteristic
13#14528502C211yesmaximal, characteristic
14#14528476C211yesmaximal, characteristic
15C17×#2101C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C22×#63218C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19C2×#10587578C2211yescharacteristic
20C2×C4⋊C4×D17—21no—
21C2×C4⋊C4×D17C2211yescharacteristic
22C2×C4⋊C4×D17C2211yescharacteristic
23C2×C42×D17C2211yescharacteristic
24#11762708C2211yescharacteristic
25#11762707C2211yescharacteristic
26#11762706—21no—
27#11762705C2211yescharacteristic
28#11762704—21no—
29#11762678C2211yescharacteristic
30#11762678C2211yescharacteristic
31#11762675C2211yescharacteristic
32#11762668—21no—
33#11762668—21no—
34#11762660—21no—
35#11762660—21no—
36#95×D17—21no—
37#11762572C2211yescharacteristic
38#11762572C2211yescharacteristic
39#11762572C2211yescharacteristic
40#11762572C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762564C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#382C2211yescharacteristic
54C17×#340C2211yescharacteristic
55C17×#340C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C22×C4×D17—21no—
60C22×C4×D17—21no—
61C22×C4×D17D411yescharacteristic
62C22×C4×D17D411yescharacteristic
63C22×C4×D17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×#63227—21no—
73C2×#63227—21no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17C2311yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—21no—
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C2×#63218—22no—
95C2×#63218—22no—
96C2×#63218—22no—
97C2×#63218—22no—
98C2×#63218—22no—
99C2×#63218—22no—
100#10587578—22no—
101#10587578—22no—
102#10587578—22no—
103#10587578—22no—
104C4⋊C4×D17—22no—
105C4⋊C4×D17—22no—
106C4⋊C4×D17—22no—
107C4⋊C4×D17—44no—
108C4⋊C4×D17Q814yes—
109C42×D17—22no—
110C42×D17—22no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119C23×D17C2×D411yescharacteristic
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2411yescharacteristic
127C22×Dic17—21no—
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17C2×D411yescharacteristic
132C22×Dic17C2×D411yescharacteristic
133C22×Dic17C2×D411yescharacteristic
134C2×C4×D17—24no—
135C2×C4×D17—22no—
136C2×C4×D17—22no—
137C2×C4×D17—22no—
138C2×C4×D17—22no—
139C2×C4×D17—41no—
140C2×C4×D17—41no—
141C2×C4×D17—42no—
142C2×C4×D17—42no—
143C2×C4×D17—42no—
144C2×C4×D17—42no—
145C2×C4×D17—42no—
146C2×C4×D17C2×Q812yes—
147C2×C4×D17#4112yes—
148C2×C4×D17#4112yes—
149C2×C4×D17#4112yes—
150C2×C4×D17#4112yes—
151C2×C4×D17#4112yes—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—42no—
166C4⋊C4×C17C2×Q812yes—
167C42×C17—21no—
168C42×C17—21no—
169#63227—42no—
170#63227—42no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—42no—
179C4⋊Dic17C2×Q812yes—
180#63225—24no—
181#63225—42no—
182#63225—42no—
183#63225—42no—
184#63225—42no—
185#63225—42no—
186#63225—42no—
187#63225—42no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—22no—
191C4×Dic17—22no—
192C4×Dic17—42no—
193C4×Dic17—42no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200#63218—44no—
201#63218—44no—
202#23747—171nomaximal, Sylow
203C23×C17C22×D411yescharacteristic
204C22×D17—21no—
205C22×D17—21no—
206C22×D17—21no—
207C22×D17—21no—
208C22×D17C22⋊Q812yes—
209C22×D17#12312yes—
210C22×D17#12412yes—
211C2×C4×C17—22no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17C22×Q811yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—22no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17C22⋊Q812yes—
252C2×Dic17#12312yes—
253C2×Dic17#12412yes—
254C2×Dic17AGL(1,ℤ/8)12yes—
255C2×Dic17#13712yes—
256C4×D17—44no—
257C4×D17—44no—
258C4×D17—48no—
259C4×D17—84no—
260C4×D17#12514yes—
261C4×D17#12514yes—
262C4×D17#12614yes—
263C4×D17#12614yes—
264C8×C17—41no—
265C8×C17—41no—
266#3622—41no—
267#3622—41no—
268C2×#389—171no—
269#2285—171no—
270#2101—171no—
271#2101—171no—
272#2101—171no—
273#2101D1711yescharacteristic, complemented
274#2100—171no—
275#2100—171no—
276#2099—171no—
277#2099—171no—
278#1736—171no—
279#1736—171no—
280#1736—171no—
281#1736—171no—
282#1728—171no—
283C22×C17C2×C22⋊Q811yescharacteristic
284C22×C17C2×#12311yescharacteristic
285C22×C17C2×#12411yescharacteristic
286C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C17C2×#13711yescharacteristic
288C22×C17#57511yescharacteristic
289C22×C17#57511yescharacteristic
290C2×D17—22no—
291C2×D17—22no—
292C2×D17—22no—
293C2×D17—22no—
294C2×D17—22no—
295C2×D17—22no—
296C2×D17#40014yes—
297C4×C17—42no—
298C4×C17—41no—
299C4×C17—41no—
300C4×C17—81no—
301C4×C17—81no—
302C4×C17—81no—
303C4×C17C2×#12511yescharacteristic
304C4×C17C2×#12511yescharacteristic
305C4×C17C2×#12611yescharacteristic
306C4×C17C2×#12611yescharacteristic
307Dic17—22no—
308Dic17—22no—
309Dic17—42no—
310Dic17—81no—
311Dic17—81no—
312Dic17—81no—
313Dic17#40014yes—
314C22×M16—171no—
315C22×M16—171no—
316C2×#117—171no—
317C2×#117—171no—
318C22×C4⋊C4—171no—
319C22×C4⋊C4—171no—
320C22×C4⋊C4—341no—
321C22×C42—171no—
322#389—171no—
323#389—171no—
324#389—171no—
325#389—341no—
326#389—341no—
327#389C2×D1711yescharacteristic
328#384—171no—
329#384—171no—
330#382—171no—
331#382C2×D1711yescharacteristic
332#380—341no—
333#380—341no—
334C4×C4⋊C4—341no—
335C4×C4⋊C4—341no—
336C2×#95—341no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340C2×D1711yescharacteristic
350#340C2×D1711yescharacteristic
351#340C2×D1711yescharacteristic
352#340C2×D1711yescharacteristic
353#336—171no—
354#336—171no—
355#336—171no—
356#336C2×D1711yescharacteristic
357C2×C17C2×#40011yescharacteristic
358C2×C17#307711yescharacteristic
359C2×C17#307811yescharacteristic
360C2×C17#309411yescharacteristic
361C2×C17#309511yescharacteristic
362C2×C17#314111yescharacteristic
363C2×C17#314211yescharacteristic
364D17—24no—
365C23×C4—171no—
366C23×C4—171no—
367C23×C4—171no—
368C23×C4—341no—
369C23×C4—341no—
370C2×M16—342no—
371C2×M16—342no—
372C2×M16—342no—
373C2×M16—342no—
374C2×M16—342no—
375C2×M16—342no—
376C22×C8—171no—
377C22×C8—171no—
378C22×C8C22×D1711yescharacteristic
379C22×C8C22×D1711yescharacteristic
380#117—342no—
381#117—342no—
382#117—342no—
383#117—342no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—174no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—342no—
391C2×C4⋊C4—342no—
392C2×C4⋊C4—342no—
393C2×C4⋊C4—341no—
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⋊C4—684no—
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C4⋊C4C22×D1711yescharacteristic
405C2×C4⋊C4C22×D1711yescharacteristic
406C2×C4⋊C4C22×D1711yescharacteristic
407C2×C22⋊C4—341no—
408C2×C22⋊C4—341no—
409C2×C42—171no—
410C2×C42—171no—
411C2×C42—171no—
412C2×C42—342no—
413C2×C42—342no—
414C2×C42—341no—
415C2×C42—341no—
416C2×C42C22×D1711yescharacteristic
417#95—21no—
418#95—21no—
419#95—21no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425C17#2374711yescharacteristic, complemented, Sylow
426C24—171no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—171no—
432C22×C4—172no—
433C22×C4—172no—
434C22×C4—172no—
435C22×C4—172no—
436C22×C4—172no—
437C22×C4—172no—
438C22×C4—171no—
439C22×C4—171no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—342no—
444C22×C4—344no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—681no—
450C22×C4—681no—
451C22×C4—682no—
452C22×C4—682no—
453C22×C4—682no—
454C22×C4—682no—
455C22×C4—682no—
456C22×C4D4×D1711yescharacteristic
457C22×C4D4×D1711yescharacteristic
458C22×C4C23×D1711yescharacteristic
459M16—684no—
460M16—684no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—341no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—172no—
482C4⋊C4—344no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—682no—
491C4⋊C4—682no—
492C4⋊C4—682no—
493C4⋊C4—682no—
494C4⋊C4—682no—
495C4⋊C4—682no—
496C4⋊C4—682no—
497C4⋊C4—682no—
498C4⋊C4Q8×D1712yes—
499C22⋊C4—682no—
500C22⋊C4—682no—
501C42—21no—
502C42—21no—
503C42—342no—
504C42—342no—
505C42—341no—
506C42—341no—
507C42—682no—
508C42—682no—
509C23—172no—
510C23—172no—
511C23—172no—
512C23—341no—
513C23—341no—
514C23—341no—
515C23—341no—
516C23C2×D4×D1711yescharacteristic
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—172no—
536C2×C4—172no—
537C2×C4—172no—
538C2×C4—174no—
539C2×C4—174no—
540C2×C4—174no—
541C2×C4—174no—
542C2×C4—172no—
543C2×C4—172no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—342no—
549C2×C4—684no—
550C2×C4—684no—
551C2×C4—688no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—1364no—
564C2×C4C2×Q8×D1711yescharacteristic
565C2×C4#41×D1711yescharacteristic
566C2×C4#41×D1711yescharacteristic
567C2×C4#41×D1711yescharacteristic
568C2×C4#41×D1711yescharacteristic
569C2×C4#41×D1711yescharacteristic
570C8—41no—
571C8—41no—
572C8—681no—
573C8—681no—
574C22—174no—
575C22—342no—
576C22—342no—
577C22—342no—
578C22—342no—
579C22—342no—
580C22—342no—
581C22C22⋊Q8×D1711yescharacteristic
582C22#123×D1711yescharacteristic
583C22#124×D1711yescharacteristic
584C22AGL(1,ℤ/8)×D1711yescharacteristic
585C22#137×D1711yescharacteristic
586C22#1176360611yescharacteristic
587C22#1176360611yescharacteristic
588C4—42no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—174no—
595C4—342no—
596C4—342no—
597C4—682no—
598C4—1361no—
599C4—1361no—
600C4—1361no—
601C4#125×D1711yescharacteristic
602C4#125×D1711yescharacteristic
603C4#126×D1711yescharacteristic
604C4#126×D1711yescharacteristic
605C2—344no—
606C2D17×#40011yescharacteristic
607C2#1453857911yescharacteristic
608C2#1453858711yescharacteristic
609C2#1453872311yescharacteristic
610C2#1453873111yescharacteristic
611C2#1453917611yescharacteristic
612C2#1453918111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2101C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
hypercenter#2101D17
hypocenterC17#23747
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
3. #14530353C2semidirect
4. #14530352C2semidirect
5. #14530351C2semidirect
6. #14530350C2semidirect
7. #14530349C2semidirect
8. #14530345C2semidirect
9. #14530344C2semidirect
15. C17×#2101C2semidirect
273. #2101D17semidirect
425. C17#23747semidirect

Maximal subgroups

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

subgroupindex
D17×#3892
#145311902
#145303532
#145303522
#145303512
#145303502
#145303492
#145303452
#145303442
#145285022
#145284762
C17×#21012
#2374717