Finite Group Databasebeta

← #17037805

Subgroups — #17037805

Order 4352.

1022 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762850C211yesmaximal, characteristic
2C2×#11762848C211yesmaximal, characteristic
3C2×#11762831C211yesmaximal, characteristic
4D17×#382C211yesmaximal, characteristic
5C4×#10587578C211yesmaximal, characteristic
6#14529555C211yesmaximal, characteristic, complemented
7#14529554C211yesmaximal, characteristic, complemented
8#14529553C211yesmaximal, characteristic, complemented
9#14529552C211yesmaximal, characteristic, complemented
10#14529551C211yesmaximal, characteristic, complemented
11#14529550C211yesmaximal, characteristic, complemented
12#14529549C211yesmaximal, characteristic, complemented
13#14528502C211yesmaximal, characteristic
14#14528502C211yesmaximal, characteristic
15C17×#1949C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C22×#63218C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19C2×#10587578C2211yescharacteristic
20C2×C4⋊C4×D17C2211yescharacteristic
21C2×C4⋊C4×D17C2211yescharacteristic
22C2×C42×D17C2211yescharacteristic
23#11762850—24no—
24#11762848—24no—
25C2×#10587489C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587478C2211yescharacteristic
28C2×#10587488C2211yescharacteristic
29C2×#10587479C2211yescharacteristic
30C2×#10587479C2211yescharacteristic
31#11762831C418yescomplemented
32C2×C4⋊#3622C2211yescharacteristic
33C2×#10587485C2211yescharacteristic
34C2×#10587485C2211yescharacteristic
35#11762675C2211yescharacteristic
36#11762674C2211yescharacteristic
37#11762674C2211yescharacteristic
38#11762673—21no—
39#11762673—21no—
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43C4×#63227—21no—
44#11762656—21no—
45#11762654—21no—
46#95×D17—21no—
47#95×D17—21no—
48C42×Dic17—21no—
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C17×#107C2211yescharacteristic
56C2×C17×#106C2211yescharacteristic
57C2×C17×C4⋊C8C2211yescharacteristic
58C17×#382C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×C4×D17D411yescharacteristic
65C22×C4×D17D411yescharacteristic
66C22×C4×D17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C42×C17—21no—
74C2×C42×C17C2311yescharacteristic
75C2×#63227—21no—
76C2×#63227—21no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×#63225—21no—
82C2×#63225—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17D411yescharacteristic
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C2×#63218—22no—
94C2×#63218—22no—
95C2×#63218—22no—
96C2×#63218—22no—
97C2×#63218C2×C414yes—
98C2×#63218C2×C414yes—
99#10587578—22no—
100#10587578—22no—
101#10587578D414yes—
102#10587578Q814yes—
103C4⋊C4×D17—22no—
104C4⋊C4×D17—22no—
105C4⋊C4×D17—22no—
106C4⋊C4×D17—22no—
107C42×D17C2×C414yes—
108C42×D17Q814yes—
109C17×#107—22no—
110C17×#106—22no—
111C17×C4⋊C8C2×C414yescomplemented
112C17×#95—21no—
113C17×#95—21no—
114C17×#95—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587489—22no—
125#10587488—22no—
126#10587485C2×C414yescomplemented
127#10587485C2×C414yescomplemented
128#10587479—22no—
129#10587479—22no—
130#10587478—22no—
131#10587478—22no—
132C4⋊#3622C2×C414yescomplemented
133C23×D17C2×D411yescharacteristic
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17C2×D411yescharacteristic
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C2×C4×D17—41no—
149C2×C4×D17—41no—
150C2×C4×D17—41no—
151C2×C4×D17—41no—
152C2×C4×D17—42no—
153C2×C4×D17—42no—
154C2×C4×D17—42no—
155C2×C4×D17—42no—
156C2×C4×D17—42no—
157C2×C4×D17—42no—
158C2×C4×D17C4⋊C414yes—
159C2×C4×D17C4⋊C414yes—
160C2×C4×D17C22×C412yes—
161C2×C4×D17C2×D412yes—
162C2×C4×D17C2×Q812yes—
163C2×C4×D17C2×Q812yes—
164C2×C4×D17#4112yes—
165C2×C4×D17#4112yes—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17C22×C412yes—
171C2×C8×C17C22×C412yes—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17C2×D412yes—
179C4⋊C4×C17C2×Q812yes—
180C42×C17—42no—
181C42×C17C22×C412yes—
182C42×C17C2×Q812yes—
183#63227—24no—
184#63227—24no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17C2×D412yes—
192C4⋊Dic17C2×Q812yes—
193#63225—42no—
194#63225—42no—
195C4×Dic17—24no—
196C4×Dic17—24no—
197C4×Dic17—42no—
198C4×Dic17—42no—
199C4×Dic17—42no—
200C4×Dic17C4⋊C414yes—
201C4×Dic17C4⋊C414yes—
202C4×Dic17C22×C412yes—
203C4×Dic17C2×Q812yes—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622C22×C412yes—
209C2×#3622C22×C412yes—
210#63218—28no—
211#63218—28no—
212#21172—171nomaximal, Sylow
213C23×C17C22×D411yescharacteristic
214C22×D17—21no—
215C22×D17—21no—
216C22×D17—21no—
217C22×D17—21no—
218C22×D17C2×C4⋊C412yes—
219C22×D17#12112yes—
220C22×D17C22⋊Q812yes—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
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×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17C2×C4⋊C412yes—
238C2×C4×C17C2×C4⋊C412yes—
239C2×C4×C17C23×C411yescharacteristic
240C2×C4×C17C22×D411yescharacteristic
241C2×C4×C17C22×Q811yescharacteristic
242C2×C4×C17C22×Q811yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—22no—
250C2×Dic17—22no—
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—41no—
260C2×Dic17—41no—
261C2×Dic17C2×C4⋊C412yes—
262C2×Dic17C2×C4⋊C412yes—
263C2×Dic17C2×C4⋊C412yes—
264C2×Dic17#12112yes—
265C2×Dic17C22⋊Q812yes—
266C4×D17—44no—
267C4×D17—44no—
268C4×D17—84no—
269C4×D17—84no—
270C4×D17C4×D414yes—
271C4×D17C4×Q814yes—
272C4×D17#12514yes—
273C4×D17C4⋊Q814yes—
274C8×C17—22no—
275C8×C17—22no—
276#3622—22no—
277#3622—22no—
278C2×#428—171no—
279C2×#428—171no—
280C2×#423—171no—
281C2×#382—171no—
282C4×#117—171no—
283#1949—171no—
284#1949—171no—
285#1949—171no—
286#1949—171no—
287#1949—171no—
288#1949—171no—
289#1949—171no—
290#1949D1711yescharacteristic, complemented
291#1736—171no—
292#1736—171no—
293C22×C17C22×C4⋊C411yescharacteristic
294C22×C17C2×#12111yescharacteristic
295C22×C17C2×C22⋊Q811yescharacteristic
296C22×C17#57511yescharacteristic
297C22×C17#57511yescharacteristic
298C22×C17#57511yescharacteristic
299C22×C17#57511yescharacteristic
300C2×D17—22no—
301C2×D17—22no—
302C2×D17—22no—
303C2×D17—22no—
304C2×D17—22no—
305C2×D17—22no—
306C2×D17#38914yes—
307C4×C17—22no—
308C4×C17—22no—
309C4×C17—42no—
310C4×C17—42no—
311C4×C17—81no—
312C4×C17—81no—
313C4×C17C2×C4×D411yescharacteristic
314C4×C17C2×C4×Q811yescharacteristic
315C4×C17C2×#12511yescharacteristic
316C4×C17C2×C4⋊Q811yescharacteristic
317Dic17—22no—
318Dic17—22no—
319Dic17—42no—
320Dic17—42no—
321Dic17—81no—
322Dic17—81no—
323Dic17#38914yes—
324C22×M16—171no—
325C22×M16—171no—
326C2×#117—171no—
327C2×#117—171no—
328C22×C4⋊C4—171no—
329C22×C4⋊C4—171no—
330C22×C42—171no—
331#428—344no—
332#428—344no—
333C2×#107—171no—
334C2×#107—171no—
335C2×#107—171no—
336C2×#107C2×D1711yescharacteristic
337C2×#106—171no—
338C2×#106—171no—
339C2×#106—171no—
340C2×#106C2×D1711yescharacteristic
341#423—178no—
342C2×C4⋊C8—171no—
343C2×C4⋊C8—171no—
344C2×C4⋊C8—171no—
345C2×C4⋊C8C2×D1711yescharacteristic
346#382—171no—
347#382—171no—
348#382—171no—
349#382—341no—
350#382—341no—
351#382C2×D1711yescharacteristic
352C4×C4⋊C4—171no—
353C4×C4⋊C4—171no—
354C4×C4⋊C4—171no—
355C4×C4⋊C4C2×D1711yescharacteristic
356C4×C22⋊C4—341no—
357C4×C22⋊C4—341no—
358#376—341no—
359C2×#95—341no—
360C2×#95—341no—
361C43—341no—
362#340—171no—
363#340—171no—
364#340—171no—
365#340—171no—
366#340—171no—
367#340—171no—
368#340C2×D1711yescharacteristic
369#340C2×D1711yescharacteristic
370C2×C17C2×#38911yescharacteristic
371C2×C17#291111yescharacteristic
372C2×C17#291111yescharacteristic
373C2×C17#304011yescharacteristic
374C2×C17#304011yescharacteristic
375C2×C17#308011yescharacteristic
376C2×C17#308011yescharacteristic
377D17—24no—
378C23×C4—171no—
379C23×C4—171no—
380C23×C4—171no—
381C23×C4—341no—
382C23×C4—341no—
383C2×M16—174no—
384C2×M16—174no—
385C2×M16—342no—
386C2×M16—342no—
387C2×M16—342no—
388C2×M16—342no—
389C22×C8—171no—
390C22×C8—171no—
391C22×C8C22×D1711yescharacteristic
392C22×C8C22×D1711yescharacteristic
393#117—174no—
394#117—174no—
395#117—342no—
396#117—342no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—342no—
402C2×C4⋊C4—342no—
403C2×C4⋊C4—342no—
404C2×C4⋊C4—342no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4C22×D1711yescharacteristic
408C2×C4⋊C4C22×D1711yescharacteristic
409C2×C4⋊C4C22×D1711yescharacteristic
410C2×C4⋊C4C22×D1711yescharacteristic
411C2×C22⋊C4—341no—
412C2×C22⋊C4—341no—
413C2×C42—21no—
414C2×C42—171no—
415C2×C42—171no—
416C2×C42—171no—
417C2×C42—174no—
418C2×C42—174no—
419C2×C42—341no—
420C2×C42—341no—
421C2×C42—341no—
422C2×C42—341no—
423C2×C42—341no—
424C2×C42C22×D1711yescharacteristic
425#107—22no—
426#107—342no—
427#107—342no—
428#107—342no—
429#106—22no—
430#106—342no—
431#106—342no—
432#106—342no—
433C4⋊C8—174no—
434C4⋊C8—174no—
435C4⋊C8—174no—
436C4⋊C8C4×D1714yescomplemented
437#95—21no—
438#95—21no—
439#95—21no—
440#95—341no—
441#95—341no—
442#95—341no—
443#95—341no—
444#95—341no—
445#95—341no—
446#95—341no—
447#95—341no—
448#95—341no—
449C17#2117211yescharacteristic, complemented, Sylow
450C24—171no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—171no—
456C22×C4—172no—
457C22×C4—172no—
458C22×C4—172no—
459C22×C4—172no—
460C22×C4—172no—
461C22×C4—172no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—174no—
465C22×C4—174no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—681no—
471C22×C4—681no—
472C22×C4—681no—
473C22×C4—681no—
474C22×C4—682no—
475C22×C4—682no—
476C22×C4—682no—
477C22×C4—682no—
478C22×C4—682no—
479C22×C4—682no—
480C22×C4D4×D1711yescharacteristic
481C22×C4D4×D1711yescharacteristic
482C22×C4C23×D1711yescharacteristic
483M16—348no—
484M16—348no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—172no—
490C2×C8—172no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8C2×C4×D1712yes—
496C2×C8C2×C4×D1712yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—172no—
504C4⋊C4—172no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—682no—
512C4⋊C4—682no—
513C4⋊C4D4×D1712yes—
514C4⋊C4Q8×D1712yes—
515C22⋊C4—344no—
516C22⋊C4—344no—
517C42—42no—
518C42—172no—
519C42—172no—
520C42—174no—
521C42—174no—
522C42—344no—
523C42—344no—
524C42—682no—
525C42—682no—
526C42—682no—
527C42C2×C4×D1712yes—
528C42Q8×D1712yes—
529C23—172no—
530C23—172no—
531C23—172no—
532C23—341no—
533C23—341no—
534C23—341no—
535C23—341no—
536C23C2×D4×D1711yescharacteristic
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—172no—
554C2×C4—172no—
555C2×C4—172no—
556C2×C4—174no—
557C2×C4—174no—
558C2×C4—174no—
559C2×C4—174no—
560C2×C4—172no—
561C2×C4—172no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—342no—
567C2×C4—342no—
568C2×C4—684no—
569C2×C4—684no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—1364no—
581C2×C4—1364no—
582C2×C4C4⋊C4×D1712yes—
583C2×C4C4⋊C4×D1712yes—
584C2×C4C22×C4×D1711yescharacteristic
585C2×C4C2×D4×D1711yescharacteristic
586C2×C4C2×Q8×D1711yescharacteristic
587C2×C4C2×Q8×D1711yescharacteristic
588C2×C4#41×D1711yescharacteristic
589C2×C4#41×D1711yescharacteristic
590C8—22no—
591C8—22no—
592C8—342no—
593C8—342no—
594C22—174no—
595C22—342no—
596C22—342no—
597C22—342no—
598C22—342no—
599C22—342no—
600C22—342no—
601C22C2×C4⋊C4×D1711yescharacteristic
602C22#121×D1711yescharacteristic
603C22C22⋊Q8×D1711yescharacteristic
604C22#1176360511yescharacteristic
605C22#1176360611yescharacteristic
606C22#1176360611yescharacteristic
607C22#1176360811yescharacteristic
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4—174no—
615C4—342no—
616C4—342no—
617C4—682no—
618C4—682no—
619C4—1361no—
620C4—1361no—
621C4C4×D4×D1711yescharacteristic
622C4C4×Q8×D1711yescharacteristic
623C4#125×D1711yescharacteristic
624C4C4⋊Q8×D1711yescharacteristic
625C2—344no—
626C2D17×#38911yescharacteristic
627C2#1453737111yescharacteristic
628C2#1453737211yescharacteristic
629C2#1453827311yescharacteristic
630C2#1453827511yescharacteristic
631C2#1453860111yescharacteristic
632C2#1453860311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1949C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1949D17
hypocenterC17#21172
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
6. #14529555C2semidirect
7. #14529554C2semidirect
8. #14529553C2semidirect
9. #14529552C2semidirect
10. #14529551C2semidirect
11. #14529550C2semidirect
12. #14529549C2semidirect
15. C17×#1949C2semidirect
31. #11762831C4semidirect
111. C17×C4⋊C8C2×C4semidirect
126. #10587485C2×C4semidirect
127. #10587485C2×C4semidirect
132. C4⋊#3622C2×C4semidirect
290. #1949D17semidirect
436. C4⋊C8C4×D17semidirect
449. C17#21172semidirect

Maximal subgroups

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

subgroupindex
C2×#117628502
C2×#117628482
C2×#117628312
D17×#3822
C4×#105875782
#145295552
#145295542
#145295532
#145295522
#145295512
#145295502
#145295492
#145285022
C17×#19492
#2117217