Finite Group Databasebeta

← #17005259

Subgroups — #17005259

Order 4352.

602 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538821C211yesmaximal, characteristic
2Q8×#63218C211yesmaximal, characteristic
3#14537732C211yesmaximal, characteristic
4#14527680C211yesmaximal, characteristic, complemented
5#14527679C211yesmaximal, characteristic, complemented
6#14527636C211yesmaximal, characteristic, complemented
7#14527631C211yesmaximal, characteristic, complemented
8#14527631C211yesmaximal, characteristic, complemented
9Q16×#3622C211yesmaximal, characteristic, complemented
10C8×#63231C211yesmaximal, characteristic, complemented
11#14527557C211yesmaximal, characteristic
12#14527237C211yesmaximal, characteristic
13D17×Q8⋊C8C211yesmaximal, characteristic
14#14527063C211yesmaximal, characteristic
15C17×#1584C211yesmaximal, characteristic, complemented
16#11763363C2211yescharacteristic
17C4×Q8×D17C2211yescharacteristic
18#11763146C411yescharacteristic
19#11763143C411yescharacteristic
20#11763103C411yescharacteristic
21#11763102C411yescharacteristic
22#11763018C411yescharacteristic
23#11763017C411yescharacteristic
24#11762966C411yescharacteristic
25#11762963C411yescharacteristic
26#11762939—21no—
27#11762939C2211yescharacteristic
28#11762937—21no—
29#11762936—21no—
30#11762936—21no—
31#11762936C2211yescharacteristic
32#11762936C2211yescharacteristic
33Q8×#3622C2211yescharacteristic
34Q8×#3622C2211yescharacteristic
35C8×Dic34C2211yescharacteristic
36C4×Dic68C2211yescharacteristic
37C4×#63231C2211yescharacteristic
38C4×#63231C2211yescharacteristic
39#11762860C2211yescharacteristic
40#11762834—21no—
41#11762831—21no—
42#11762831C2211yescharacteristic
43C4⋊C8×D17C2211yescharacteristic
44#11762766—21no—
45C4×#63218—21no—
46C4×#63218C2211yescharacteristic
47C4×C8×D17C2211yescharacteristic
48#11762562C2211yescharacteristic
49#11762561C2211yescharacteristic
50#11762561C2211yescharacteristic
51#11762541C2211yescharacteristic
52#11762541C2211yescharacteristic
53Q8⋊#3622C2211yescharacteristic
54Q8⋊#3622C2211yescharacteristic
55#11762539C2211yescharacteristic
56#11762539C2211yescharacteristic
57#11762531C2211yescharacteristic
58C8×#3622C2211yescharacteristic
59C8×#3622C2211yescharacteristic
60C17×#446C2211yescharacteristic
61C8×Q8×C17C2211yescharacteristic
62C4×Q16×C17C2211yescharacteristic
63C17×#335C2211yescharacteristic
64C17×Q8⋊C8C2211yescharacteristic
65C17×Q8⋊C8C2211yescharacteristic
66C17×#322C2211yescharacteristic
67C2×Q8×D17C2×C411yescharacteristic
68C2×Q16×C17C2×C411yescharacteristic
69C4×Q8×C17C2311yescharacteristic
70C4×Q8×C17C2311yescharacteristic
71#10587632C2×C411yescharacteristic
72Q8×Dic17C2×C411yescharacteristic
73#10587630C2×C411yescharacteristic
74C2×#63231C2×C411yescharacteristic
75C2×#63231C2×C411yescharacteristic
76C2×Dic68C2×C411yescharacteristic
77C2×#63218—21no—
78C2×#63218D411yescharacteristic
79C2×#63218D411yescharacteristic
80C2×C8×D17—21no—
81C2×C8×D17—21no—
82#10587583C2×C411yescharacteristic
83#10587582—21no—
84C4⋊C4×D17—21no—
85C4⋊C4×D17C2×C411yescharacteristic
86#10587575—21no—
87#10587574C2×C411yescharacteristic
88#10587573—21no—
89#10587573C2×C411yescharacteristic
90#10587558—21no—
91C42×D17—21no—
92C42×D17C2311yescharacteristic
93C4×Dic34—21no—
94C4×Dic34—21no—
95C4×Dic34C2311yescharacteristic
96C4×Dic34C2311yescharacteristic
97C17×#107C2×C411yescharacteristic
98C17×C4⋊C8—21no—
99C17×C4⋊C8—21no—
100C17×C4⋊C8C2311yescharacteristic
101C17×C4⋊C8C2311yescharacteristic
102C17×#103C2×C411yescharacteristic
103C17×#103C2×C411yescharacteristic
104C17×#97—21no—
105C4×C8×C17—21no—
106C4×C8×C17C2311yescharacteristic
107C4×C8×C17C2311yescharacteristic
108C4×C8×C17C2311yescharacteristic
109#10587506C2×C411yescharacteristic
110#10587506C2×C411yescharacteristic
111#10587491C2×C411yescharacteristic
112#10587491C2×C411yescharacteristic
113#10587489C2×C411yescharacteristic
114#10587487C2×C411yescharacteristic
115#10587487C2×C411yescharacteristic
116#10587486—21no—
117#10587486—21no—
118#10587486—21no—
119#10587486D411yescharacteristic
120#10587486D411yescharacteristic
121#10587485—21no—
122#10587485—21no—
123#10587485—21no—
124#10587485—21no—
125#10587485—21no—
126#10587485—21no—
127#10587485C2×C411yescharacteristic
128#10587485C2×C411yescharacteristic
129C8×Dic17—21no—
130C8×Dic17—21no—
131#10587481C2×C411yescharacteristic
132#10587481C2×C411yescharacteristic
133#10587478C2×C411yescharacteristic
134#10587478C2×C411yescharacteristic
135C4⋊#3622—21no—
136C4⋊#3622—21no—
137C4⋊#3622C2311yescharacteristic
138C4⋊#3622C2311yescharacteristic
139C4×#3622—21no—
140C4×#3622—21no—
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C2×Q8×C17C22×C411yescharacteristic
145C2×Q8×C17C22×C411yescharacteristic
146Q8×D17—22no—
147Q8×D17—41no—
148C2×C4×D17—21no—
149C2×C4×D17—21no—
150C2×C4×D17C22×C411yescharacteristic
151C2×C4×D17C2×D411yescharacteristic
152C2×C4×D17#4111yescharacteristic
153C2×Dic34—21no—
154C2×Dic34—21no—
155C2×Dic34C22×C411yescharacteristic
156C2×Dic34C22×C411yescharacteristic
157Q16×C17—22no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17C22×C411yescharacteristic
162C2×C8×C17C22×C411yescharacteristic
163C2×C8×C17C2×D411yescharacteristic
164C2×C8×C17C2×D411yescharacteristic
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17C22×C411yescharacteristic
168C4⋊C4×C17C22×C411yescharacteristic
169C42×C17—21no—
170C42×C17—21no—
171C42×C17C2411yescharacteristic
172#63231—22no—
173#63231—41no—
174#63227—21no—
175#63227—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17C22×C411yescharacteristic
179C4⋊Dic17C22×C411yescharacteristic
180#63225—21no—
181#63225—21no—
182#63225—21no—
183#63225—21no—
184#63225—21no—
185#63225—21no—
186#63225—21no—
187#63225—21no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17C22×C411yescharacteristic
193C4×Dic17C2×D411yescharacteristic
194C4×Dic17#4111yescharacteristic
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622C22×C411yescharacteristic
199C2×#3622C22×C411yescharacteristic
200C2×#3622C2×D411yescharacteristic
201C2×#3622C2×D411yescharacteristic
202Dic68—41no—
203#63218—22no—
204#63218—42no—
205#63218Q1614yescomplemented
206C8×D17—42no—
207C8×D17—42no—
208#17998—171nomaximal, Sylow
209C22×D17C4×D411yescharacteristic
210Q8×C17—21no—
211Q8×C17—21no—
212Q8×C17—21no—
213Q8×C17#13112yes—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17C23×C411yescharacteristic
219C2×C4×C17C22×D411yescharacteristic
220C2×C4×C17C2×#4111yescharacteristic
221C2×Dic17—21no—
222C2×Dic17—21no—
223C2×Dic17—21no—
224C2×Dic17—21no—
225C2×Dic17C4×D411yescharacteristic
226C2×Dic17C4×D411yescharacteristic
227C2×Dic17C4×D411yescharacteristic
228C4×D17—21no—
229C4×D17—21no—
230C4×D17—41no—
231C4×D17—42no—
232C4×D17—81no—
233C4×D17C2×Q1612yes—
234C4×D17#13512yes—
235Dic34—21no—
236Dic34—21no—
237Dic34—21no—
238Dic34—21no—
239Dic34—42no—
240Dic34—81no—
241C8×C17—21no—
242C8×C17—21no—
243C8×C17—21no—
244C8×C17—41no—
245C8×C17—41no—
246C8×C17—41no—
247C8×C17C2×Q1612yes—
248#3622—22no—
249#3622—21no—
250#3622—21no—
251#3622—21no—
252#3622—41no—
253#3622—41no—
254#3622C2×Q1612yes—
255#3104—171no—
256Q8×M16—171no—
257#2962—171no—
258#1591—171no—
259#1591—171no—
260#1584—171no—
261#1584—171no—
262#1584—171no—
263#1584D1711yescharacteristic, complemented
264C8×Q16—171no—
265C8×Q16—171no—
266#1571—171no—
267#1493—171no—
268C2×Q8⋊C8—171no—
269C8×M16—171no—
270C22×C17C2×C4×D411yescharacteristic
271C2×D17—21no—
272C2×D17—21no—
273C2×D17C4×Q1612yes—
274C4×C17—22no—
275C4×C17—21no—
276C4×C17—41no—
277C4×C17—41no—
278C4×C17—41no—
279C4×C17C2×#13111yescharacteristic
280C4×C17#56811yescharacteristic
281C4×C17C22×Q1611yescharacteristic
282C4×C17C2×#13511yescharacteristic
283Dic17—21no—
284Dic17—21no—
285Dic17—41no—
286Dic17—41no—
287Dic17—41no—
288Dic17—41no—
289Dic17C4×Q1612yes—
290#533—171no—
291C2×C4×Q8—171no—
292#484—171no—
293#484—171no—
294#477—171no—
295#477—171no—
296#462—171no—
297#462—171no—
298#451—171no—
299#451—171no—
300#446—171no—
301#446—171no—
302#446—171no—
303#446—341no—
304#446—341no—
305#446—341no—
306#446—341no—
307#446C2×D1711yescharacteristic
308C8×Q8—171no—
309C8×Q8—171no—
310C8×Q8—171no—
311C8×Q8C2×D1711yescharacteristic
312C4×Q16—171no—
313C4×Q16—171no—
314C4×Q16—171no—
315C4×Q16C2×D1711yescharacteristic
316#431—171no—
317#424—341no—
318#423—171no—
319#423—341no—
320C2×C4⋊C8—171no—
321#405—341no—
322C4×M16—171no—
323C4×M16—341no—
324C2×C4×C8—171no—
325#335—171no—
326#335—171no—
327#335—171no—
328#335C2×D1711yescharacteristic
329Q8⋊C8—171no—
330Q8⋊C8—171no—
331Q8⋊C8—171no—
332Q8⋊C8—171no—
333Q8⋊C8—171no—
334Q8⋊C8—171no—
335Q8⋊C8C2×D1711yescharacteristic
336Q8⋊C8C2×D1711yescharacteristic
337#322—171no—
338#322C2×D1711yescharacteristic
339C82—171no—
340C82—171no—
341C2×C17#293311yescharacteristic
342C2×C17C2×C4×Q1611yescharacteristic
343C2×C17#295611yescharacteristic
344D17—22no—
345C22×Q8—171no—
346C2×Q16—171no—
347C2×Q16—171no—
348C2×Q16—171no—
349C2×Q16C4×D1711yescharacteristic
350C2×M16—171no—
351C2×M16—171no—
352C2×M16—341no—
353C22×C8—341no—
354C22×C8—341no—
355C4⋊Q8—171no—
356C4⋊Q8—171no—
357#125—341no—
358C22⋊Q8—171no—
359C22⋊Q8—171no—
360C22⋊Q8—341no—
361C4×Q8—171no—
362C4×Q8—171no—
363C4×Q8—171no—
364C4×Q8—171no—
365C4×Q8—341no—
366C4×Q8—341no—
367C4×Q8—341no—
368C4×Q8C22×D1711yescharacteristic
369C4×Q8C22×D1711yescharacteristic
370#117—341no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—341no—
373C2×C42—171no—
374C2×C42—341no—
375#107—171no—
376#107—171no—
377#107—171no—
378#107C4×D1711yescharacteristic
379C4⋊C8—21no—
380C4⋊C8—21no—
381C4⋊C8—171no—
382C4⋊C8—171no—
383C4⋊C8—171no—
384C4⋊C8—171no—
385C4⋊C8—341no—
386C4⋊C8—341no—
387C4⋊C8—341no—
388C4⋊C8—341no—
389C4⋊C8—341no—
390C4⋊C8—341no—
391C4⋊C8—341no—
392C4⋊C8—341no—
393C4⋊C8C22×D1711yescharacteristic
394C4⋊C8C22×D1711yescharacteristic
395#103—171no—
396#103—171no—
397#103—171no—
398#103—171no—
399#103—171no—
400#103—171no—
401#103C4×D1711yescharacteristic
402#103C4×D1711yescharacteristic
403C22⋊C8—171no—
404C22⋊C8—171no—
405#97—21no—
406#97—171no—
407#97—171no—
408#97—341no—
409#97—341no—
410#97—341no—
411C4×C8—21no—
412C4×C8—171no—
413C4×C8—171no—
414C4×C8—171no—
415C4×C8—341no—
416C4×C8—341no—
417C4×C8—341no—
418C4×C8—341no—
419C4×C8C22×D1711yescharacteristic
420C4×C8C22×D1711yescharacteristic
421C4×C8C22×D1711yescharacteristic
422C17#1799811yescharacteristic, complemented, Sylow
423C2×Q8—171no—
424C2×Q8—171no—
425C2×Q8—341no—
426C2×Q8—341no—
427C2×Q8—342no—
428C2×Q8—681no—
429C2×Q8C2×C4×D1711yescharacteristic
430C2×Q8C2×C4×D1711yescharacteristic
431C22×C4—171no—
432C22×C4—171no—
433C22×C4—171no—
434C22×C4—341no—
435C22×C4—341no—
436Q16—22no—
437Q16—342no—
438Q16—681no—
439Q16—681no—
440M16—174no—
441M16—342no—
442M16—682no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—171no—
447C2×C8—171no—
448C2×C8—171no—
449C2×C8—171no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—682no—
454C2×C8—682no—
455C2×C8C2×C4×D1711yescharacteristic
456C2×C8C2×C4×D1711yescharacteristic
457C2×C8D4×D1711yescharacteristic
458C2×C8D4×D1711yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—171no—
462C4⋊C4—171no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4C2×C4×D1711yescharacteristic
474C4⋊C4C2×C4×D1711yescharacteristic
475C22⋊C4—341no—
476C22⋊C4—341no—
477C42—21no—
478C42—21no—
479C42—171no—
480C42—171no—
481C42—171no—
482C42—341no—
483C42—341no—
484C42—341no—
485C42—341no—
486C42C23×D1711yescharacteristic
487C23—171no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—341no—
492Q8—341no—
493Q8—341no—
494Q8—341no—
495Q8—682no—
496Q8—1361no—
497Q8#1058759312yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—171no—
503C2×C4—172no—
504C2×C4—172no—
505C2×C4—171no—
506C2×C4—171no—
507C2×C4—341no—
508C2×C4—341no—
509C2×C4—341no—
510C2×C4—341no—
511C2×C4—341no—
512C2×C4—341no—
513C2×C4—681no—
514C2×C4—682no—
515C2×C4—1361no—
516C2×C4C22×C4×D1711yescharacteristic
517C2×C4C2×D4×D1711yescharacteristic
518C2×C4#41×D1711yescharacteristic
519C8—21no—
520C8—21no—
521C8—21no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—172no—
526C8—342no—
527C8—341no—
528C8—341no—
529C8—341no—
530C8—681no—
531C8—681no—
532C8Q16×D1712yes—
533C22—172no—
534C22—341no—
535C22—341no—
536C22C4×D4×D1711yescharacteristic
537C4—22no—
538C4—21no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—172no—
543C4—341no—
544C4—341no—
545C4—681no—
546C4—681no—
547C4—681no—
548C4—681no—
549C4C2×#1058759311yescharacteristic
550C4#1176357411yescharacteristic
551C4C2×Q16×D1711yescharacteristic
552C4#135×D1711yescharacteristic
553C2—342no—
554C2#1453751011yescharacteristic
555C2C4×Q16×D1711yescharacteristic
556C2#1453769211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1584C2
socleC22×C17C2×C4×D4
prime radicalC22×D17C4×D4
cosocleC42×C17C24
hypercenter#1584D17
hypocenterC17#17998
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #14527680C2semidirect
5. #14527679C2semidirect
6. #14527636C2semidirect
7. #14527631C2semidirect
8. #14527631C2semidirect
9. Q16×#3622C2semidirect
10. C8×#63231C2semidirect
15. C17×#1584C2semidirect
205. #63218Q16semidirect
263. #1584D17semidirect
422. C17#17998semidirect

Maximal subgroups

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

subgroupindex
#145388212
Q8×#632182
#145377322
#145276802
#145276792
#145276362
#145276312
Q16×#36222
C8×#632312
#145275572
#145272372
D17×Q8⋊C82
#145270632
C17×#15842
#1799817