Finite Group Databasebeta

← #17004416

Subgroups — #17004416

Order 4352.

638 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537895C211yesmaximal, characteristic, complemented
2#14537860C211yesmaximal, characteristic, complemented
3#14527663C211yesmaximal, characteristic, complemented
4#14527649C211yesmaximal, characteristic, complemented
5#14527628C211yesmaximal, characteristic, complemented
6#14527614C211yesmaximal, characteristic, complemented
7#14527608C211yesmaximal, characteristic, complemented
8SD16⋊#3622C211yesmaximal, characteristic, complemented
9SD16×#3622C211yesmaximal, characteristic, complemented
10D8×#3622C211yesmaximal, characteristic, complemented
11#14527569C211yesmaximal, characteristic, complemented
12#14527218C211yesmaximal, characteristic, complemented
13#14527183C211yesmaximal, characteristic, complemented
14#14527084C211yesmaximal, characteristic, complemented
15C17×#3106C211yesmaximal, characteristic
16Q8×#3622C2211yescharacteristic
17Q8×#3622C2211yescharacteristic
18#11762887C2211yescharacteristic
19#11762887C2211yescharacteristic
20#11762881—21no—
21#11762881—21no—
22D4×#3622—21no—
23D4×#3622—21no—
24D4×#3622C2211yescharacteristic
25D4×#3622C2211yescharacteristic
26D4×#3622C2211yescharacteristic
27D4×#3622C2211yescharacteristic
28#11762866C2211yescharacteristic
29#11762862—21no—
30#11762862C2211yescharacteristic
31#11762862C2211yescharacteristic
32C2×C4⋊#3622—21no—
33C2×C4⋊#3622C2211yescharacteristic
34#11762776—21no—
35#11762776—21no—
36#11762562C2211yescharacteristic
37#11762562C2211yescharacteristic
38#11762560C2211yescharacteristic
39#11762560C2211yescharacteristic
40Q8⋊#3622C2211yescharacteristic
41Q8⋊#3622C2211yescharacteristic
42D4⋊#3622C2211yescharacteristic
43D4⋊#3622C2211yescharacteristic
44D4⋊#3622C2211yescharacteristic
45D4⋊#3622C2211yescharacteristic
46D4⋊#3622C2211yescharacteristic
47D4⋊#3622C2211yescharacteristic
48#11762531C2211yescharacteristic
49#11762531C2211yescharacteristic
50#11762530C2211yescharacteristic
51C8×#3622C2211yescharacteristic
52C17×#532C2211yescharacteristic
53C4×#41×C17C2211yescharacteristic
54C17×#482C411yescharacteristic
55C17×#482C411yescharacteristic
56C17×#477C411yescharacteristic
57C17×#476C411yescharacteristic
58C17×#460C411yescharacteristic
59C17×#459C411yescharacteristic
60C17×#449C411yescharacteristic
61C17×#449C411yescharacteristic
62C4×SD16×C17C2211yescharacteristic
63C4×SD16×C17C2211yescharacteristic
64C4×D8×C17C2211yescharacteristic
65C4×D8×C17C2211yescharacteristic
66C17×#431C2211yescharacteristic
67C2×#41×C17—21no—
68C2×#41×C17C2×C411yescharacteristic
69C2×SD16×C17C2×C411yescharacteristic
70C2×SD16×C17C2×C411yescharacteristic
71C2×D8×C17C2×C411yescharacteristic
72C2×D8×C17C2×C411yescharacteristic
73C17×C4⋊Q8C2×C411yescharacteristic
74C17×#127C2×C411yescharacteristic
75C17×#124—21no—
76C17×#121—21no—
77C17×#121C2×C411yescharacteristic
78C17×#121C2×C411yescharacteristic
79C4×Q8×C17C2×C411yescharacteristic
80C4×Q8×C17C2311yescharacteristic
81C4×D4×C17—21no—
82C4×D4×C17—21no—
83C4×D4×C17—21no—
84C4×D4×C17C2×C411yescharacteristic
85C4×D4×C17C2311yescharacteristic
86C4×D4×C17C2311yescharacteristic
87C4×D4×C17C2311yescharacteristic
88C17×#117—21no—
89C17×#117C2×C411yescharacteristic
90C2×C42×C17—21no—
91C2×C42×C17C2311yescharacteristic
92C2×#63223—21no—
93C2×#63223—21no—
94C22×#3622—21no—
95C22×#3622—21no—
96C22×#3622—21no—
97C22×#3622—21no—
98C22×#3622—21no—
99C17×#107C2×C411yescharacteristic
100C17×#107C2×C411yescharacteristic
101C17×#106C2×C411yescharacteristic
102C17×#106C2×C411yescharacteristic
103C17×C4⋊C8C2×C411yescharacteristic
104C17×C4⋊C8C2×C411yescharacteristic
105C17×#103C2×C411yescharacteristic
106C17×#103C2×C411yescharacteristic
107C17×#102C2×C411yescharacteristic
108C17×#102C2×C411yescharacteristic
109C17×#102C2×C411yescharacteristic
110C17×#102C2×C411yescharacteristic
111C17×#102C2×C411yescharacteristic
112C17×#102C2×C411yescharacteristic
113C17×C22⋊C8C2×C411yescharacteristic
114C17×C22⋊C8C2×C411yescharacteristic
115C4×C8×C17C2311yescharacteristic
116C4×C8×C17C2311yescharacteristic
117C22⋊#3622—21no—
118C22⋊#3622—21no—
119C22⋊#3622—21no—
120C22⋊#3622—21no—
121C22⋊#3622—21no—
122C22⋊#3622—21no—
123C22⋊#3622—21no—
124C22⋊#3622—21no—
125C22⋊#3622D411yescharacteristic
126C22⋊#3622D411yescharacteristic
127C4⋊#3622—21no—
128C4⋊#3622—21no—
129C4⋊#3622—21no—
130C4⋊#3622—21no—
131C4⋊#3622D411yescharacteristic
132C4⋊#3622D411yescharacteristic
133C4⋊#3622C2311yescharacteristic
134C4⋊#3622C2311yescharacteristic
135C4⋊#3622C2311yescharacteristic
136C4⋊#3622C2311yescharacteristic
137#10587474—21no—
138C4×#3622—21no—
139C4×#3622—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144#41×C17—22no—
145#41×C17—22no—
146#41×C17—81no—
147#41×C17—81no—
148C2×Q8×C17—21no—
149C2×Q8×C17C22×C411yescharacteristic
150C2×D4×C17—21no—
151C2×D4×C17—21no—
152C2×D4×C17—21no—
153C2×D4×C17—21no—
154C2×D4×C17—21no—
155C2×D4×C17C22×C411yescharacteristic
156C2×D4×C17C22×C411yescharacteristic
157C2×D4×C17C22×C411yescharacteristic
158C22×C4×C17—21no—
159C22×C4×C17—21no—
160C22×C4×C17—21no—
161C22×C4×C17—21no—
162C22×C4×C17—21no—
163C22×C4×C17C22×C411yescharacteristic
164C22×C4×C17C2×D411yescharacteristic
165C22×C4×C17#4111yescharacteristic
166SD16×C17—22no—
167SD16×C17—22no—
168D8×C17—22no—
169D8×C17—22no—
170C2×C8×C17C22×C411yescharacteristic
171C2×C8×C17C22×C411yescharacteristic
172C2×C8×C17C22×C411yescharacteristic
173C2×C8×C17C22×C411yescharacteristic
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17C22×C411yescharacteristic
178C4⋊C4×C17C22×C411yescharacteristic
179C4⋊C4×C17C22×C411yescharacteristic
180C4⋊C4×C17C22×C411yescharacteristic
181C22⋊C4×C17—21no—
182C22⋊C4×C17—21no—
183C22⋊C4×C17—21no—
184C22⋊C4×C17—21no—
185C22⋊C4×C17—21no—
186C22⋊C4×C17—21no—
187C22⋊C4×C17—21no—
188C42×C17—21no—
189C42×C17—21no—
190C42×C17—21no—
191C42×C17C22×C411yescharacteristic
192C42×C17C2×D411yescharacteristic
193C42×C17#4111yescharacteristic
194C42×C17C2411yescharacteristic
195#63223—81no—
196#63223—81no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—81no—
204C2×#3622—81no—
205C2×#3622—81no—
206C2×#3622—81no—
207C2×#3622—81no—
208C2×#3622C2×D411yescharacteristic
209C2×#3622C2×D411yescharacteristic
210C2×#3622C2×D411yescharacteristic
211C2×#3622C2×D411yescharacteristic
212#17920—171nomaximal, Sylow
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17C4×D411yescharacteristic
217Q8×C17—21no—
218Q8×C17—81no—
219Q8×C17#13112yes—
220D4×C17—24no—
221D4×C17—21no—
222D4×C17—21no—
223D4×C17—21no—
224D4×C17—21no—
225D4×C17—21no—
226D4×C17—81no—
227D4×C17—81no—
228D4×C17—81no—
229D4×C17—81no—
230D4×C17#13112yes—
231C2×C4×C17—24no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
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×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—81no—
249C2×C4×C17C4×D411yescharacteristic
250C2×C4×C17C4×D411yescharacteristic
251C2×C4×C17C4×D411yescharacteristic
252C2×C4×C17C23×C411yescharacteristic
253C2×C4×C17C22×D411yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C8×C17—21no—
256C8×C17—21no—
257C8×C17—21no—
258C8×C17—21no—
259#3622—21no—
260#3622—21no—
261#3622—21no—
262#3622—21no—
263#3622—41no—
264#3622—41no—
265#3622—41no—
266#3622—41no—
267#3622—41no—
268#3622—41no—
269#3106D1711yescharacteristic
270#2988—171no—
271#2981—171no—
272#1588—171no—
273#1586—171no—
274#1583—171no—
275#1581—171no—
276#1580—171no—
277SD16⋊C8—171no—
278C8×SD16—171no—
279C8×D8—171no—
280#1573—171no—
281#1488—171no—
282#1480—171no—
283#1454—171no—
284C22×C17—21no—
285C22×C17—22no—
286C22×C17—21no—
287C22×C17—21no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17C2×C4×D411yescharacteristic
297C4×C17—21no—
298C4×C17—22no—
299C4×C17—21no—
300C4×C17—21no—
301C4×C17—21no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—41no—
306C4×C17C2×#13111yescharacteristic
307C4×C17C2×#13111yescharacteristic
308C4×C17#57611yescharacteristic
309C4×C17#57711yescharacteristic
310#532C2×D1711yescharacteristic
311C4×#41C2×D1711yescharacteristic
312#482Dic1711yescharacteristic
313#482Dic1711yescharacteristic
314#477Dic1711yescharacteristic
315#476Dic1711yescharacteristic
316#460Dic1711yescharacteristic
317#459Dic1711yescharacteristic
318#449Dic1711yescharacteristic
319#449Dic1711yescharacteristic
320C8×Q8—171no—
321C8×Q8—171no—
322C4×SD16C2×D1711yescharacteristic
323C4×SD16C2×D1711yescharacteristic
324C4×D8C2×D1711yescharacteristic
325C4×D8C2×D1711yescharacteristic
326#436—171no—
327#436—171no—
328#435—341no—
329#435—341no—
330C8×D4—171no—
331C8×D4—171no—
332C8×D4—171no—
333C8×D4—171no—
334C8×D4—341no—
335C8×D4—341no—
336#432—171no—
337#431—171no—
338#431—171no—
339#431—341no—
340#431C2×D1711yescharacteristic
341C2×C4⋊C8—171no—
342C2×C4⋊C8—341no—
343#408—341no—
344#408—341no—
345#335—171no—
346#335—171no—
347#334—171no—
348#334—171no—
349Q8⋊C8—171no—
350Q8⋊C8—171no—
351D4⋊C8—171no—
352D4⋊C8—171no—
353D4⋊C8—171no—
354D4⋊C8—171no—
355D4⋊C8—171no—
356D4⋊C8—171no—
357#322—171no—
358#322—171no—
359#322—171no—
360C82—171no—
361C2×C17—41no—
362C2×C17—42no—
363C2×C17—81no—
364C2×C17—81no—
365C2×C17#292911yescharacteristic
366C2×C17#294611yescharacteristic
367C2×C17#295711yescharacteristic
368C2×#41—21no—
369C2×#41C2×Dic1711yescharacteristic
370C2×SD16C2×Dic1711yescharacteristic
371C2×SD16C2×Dic1711yescharacteristic
372C2×D8C2×Dic1711yescharacteristic
373C2×D8C2×Dic1711yescharacteristic
374C2×M16—341no—
375C2×M16—341no—
376C22×C8—341no—
377C22×C8—341no—
378C22×C8—341no—
379C22×C8—341no—
380C22×C8—341no—
381C4⋊Q8C2×Dic1711yescharacteristic
382#127C2×Dic1711yescharacteristic
383#124—21no—
384#121—21no—
385#121C2×Dic1711yescharacteristic
386#121C2×Dic1711yescharacteristic
387C4×Q8C2×Dic1711yescharacteristic
388C4×Q8C22×D1711yescharacteristic
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4C2×Dic1711yescharacteristic
393C4×D4C22×D1711yescharacteristic
394C4×D4C22×D1711yescharacteristic
395C4×D4C22×D1711yescharacteristic
396#117—21no—
397#117C2×Dic1711yescharacteristic
398C2×C42—21no—
399C2×C42C22×D1711yescharacteristic
400#107C2×Dic1711yescharacteristic
401#107C2×Dic1711yescharacteristic
402#106C2×Dic1711yescharacteristic
403#106C2×Dic1711yescharacteristic
404C4⋊C8—171no—
405C4⋊C8—171no—
406C4⋊C8—171no—
407C4⋊C8—171no—
408C4⋊C8—171no—
409C4⋊C8—171no—
410C4⋊C8—341no—
411C4⋊C8—341no—
412C4⋊C8—341no—
413C4⋊C8—341no—
414C4⋊C8C2×Dic1711yescharacteristic
415C4⋊C8C2×Dic1711yescharacteristic
416#103C2×Dic1711yescharacteristic
417#103C2×Dic1711yescharacteristic
418#102C2×Dic1711yescharacteristic
419#102C2×Dic1711yescharacteristic
420#102C2×Dic1711yescharacteristic
421#102C2×Dic1711yescharacteristic
422#102C2×Dic1711yescharacteristic
423#102C2×Dic1711yescharacteristic
424C22⋊C8—171no—
425C22⋊C8—171no—
426C22⋊C8—341no—
427C22⋊C8—341no—
428C22⋊C8—341no—
429C22⋊C8—341no—
430C22⋊C8—341no—
431C22⋊C8—341no—
432C22⋊C8—341no—
433C22⋊C8—341no—
434C22⋊C8C2×Dic1711yescharacteristic
435C22⋊C8C2×Dic1711yescharacteristic
436#97—341no—
437C4×C8—171no—
438C4×C8—171no—
439C4×C8—171no—
440C4×C8—171no—
441C4×C8—341no—
442C4×C8—341no—
443C4×C8C22×D1711yescharacteristic
444C4×C8C22×D1711yescharacteristic
445C17#1792011yescharacteristic, complemented, Sylow
446#41—22no—
447#41—22no—
448#41—81no—
449#41—81no—
450C2×Q8—21no—
451C2×Q8C22×Dic1711yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4C22×Dic1711yescharacteristic
458C2×D4C22×Dic1711yescharacteristic
459C2×D4C22×Dic1711yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4D4×D1711yescharacteristic
466C22×C4#6325411yescharacteristic
467C22×C4C22×Dic1711yescharacteristic
468SD16—22no—
469SD16—22no—
470D8—22no—
471D8—22no—
472M16—1361no—
473M16—1361no—
474C2×C8—171no—
475C2×C8—171no—
476C2×C8—171no—
477C2×C8—171no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—1361no—
485C2×C8—1361no—
486C2×C8—1361no—
487C2×C8—1361no—
488C2×C8—1361no—
489C2×C8C22×Dic1711yescharacteristic
490C2×C8C22×Dic1711yescharacteristic
491C2×C8C22×Dic1711yescharacteristic
492C2×C8C22×Dic1711yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C22×Dic1711yescharacteristic
497C4⋊C4C22×Dic1711yescharacteristic
498C4⋊C4C22×Dic1711yescharacteristic
499C4⋊C4C22×Dic1711yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42D4×D1711yescharacteristic
511C42#6325411yescharacteristic
512C42C22×Dic1711yescharacteristic
513C42C23×D1711yescharacteristic
514C23—21no—
515C23—21no—
516C23—21no—
517C23D4×Dic1711yescharacteristic
518Q8—21no—
519Q8—81no—
520Q8#1058763412yes—
521D4—24no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—81no—
528D4—81no—
529D4—81no—
530D4—81no—
531D4#1058763412yes—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—81no—
550C2×C4D4×Dic1711yescharacteristic
551C2×C4D4×Dic1711yescharacteristic
552C2×C4D4×Dic1711yescharacteristic
553C2×C4C2×D4×D1711yescharacteristic
554C2×C4C2×#6325411yescharacteristic
555C2×C4C23×Dic1711yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—341no—
561C8—341no—
562C8—341no—
563C8—341no—
564C8—681no—
565C8—681no—
566C8—681no—
567C8—681no—
568C8—681no—
569C8—681no—
570C22—21no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22C2×D4×Dic1711yescharacteristic
583C4—21no—
584C4—22no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4C2×#1058763411yescharacteristic
593C4C2×#1058763411yescharacteristic
594C4#1176361311yescharacteristic
595C4#1176361911yescharacteristic
596C2—41no—
597C2—42no—
598C2—81no—
599C2—81no—
600C2#1453747911yescharacteristic
601C2#1453761711yescharacteristic
602C2#1453770111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
Fitting, prime radicalC17×#3106C2
socleC22×C17C2×C4×D4
cosocleC42×C17C24
hypercenter#3106D17
hypocenterC17#17920
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #14537895C2semidirect
2. #14537860C2semidirect
3. #14527663C2semidirect
4. #14527649C2semidirect
5. #14527628C2semidirect
6. #14527614C2semidirect
7. #14527608C2semidirect
8. SD16⋊#3622C2semidirect
9. SD16×#3622C2semidirect
10. D8×#3622C2semidirect
11. #14527569C2semidirect
12. #14527218C2semidirect
13. #14527183C2semidirect
14. #14527084C2semidirect
445. C17#17920semidirect

Maximal subgroups

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

subgroupindex
#145378952
#145378602
#145276632
#145276492
#145276282
#145276142
#145276082
SD16⋊#36222
SD16×#36222
D8×#36222
#145275692
#145272182
#145271832
#145270842
C17×#31062
#1792017