Finite Group Databasebeta

← #17040864

Subgroups — #17040864

Order 4352.

954 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763048C211yesmaximal, characteristic
2C2×C4×#63231C211yesmaximal, characteristic, complemented
3C2×#11762765C211yesmaximal, characteristic
4#14531466C211yesmaximal, characteristic
5#14529644C211yesmaximal, characteristic, complemented
6#14529514C211yesmaximal, characteristic, complemented
7#14529477C211yesmaximal, characteristic, complemented
8#14529473C211yesmaximal, characteristic, complemented
9#14529218C211yesmaximal, characteristic
10#14529215C211yesmaximal, characteristic
11#14528811C211yesmaximal, characteristic
12C4×#10587478C211yesmaximal, characteristic, complemented
13#14528617C211yesmaximal, characteristic, complemented
14#14528614C211yesmaximal, characteristic, complemented
15C17×#2323C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C22×#63223C2211yescharacteristic
18C23×#3622C2211yescharacteristic
19C2×#10587610C2211yescharacteristic
20C2×C4×Dic34C2211yescharacteristic
21#11763048C418yescomplemented
22C4×#63231—24no—
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32#11762765—24no—
33C2×#10587474C2211yescharacteristic
34C2×#10587474C2211yescharacteristic
35C2×C4×#3622C2211yescharacteristic
36C2×C4×#3622C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762704—21no—
40#11762703—21no—
41#11762697—21no—
42#11762690—21no—
43#11762669C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51C2×C17×C22⋊Q8C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C2×C17×#117C2211yescharacteristic
54C17×#391C2211yescharacteristic
55C17×#389—21no—
56C17×#389C2211yescharacteristic
57C17×#388—21no—
58C17×#388—21no—
59C17×#387—21no—
60C17×#380C2211yescharacteristic
61C4×C4⋊C4×C17C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C23×C4×C17C2311yescharacteristic
64C22×Dic34C2311yescharacteristic
65C17×C22⋊Q8—22no—
66C17×C22⋊Q8C2×C414yes—
67C4×Q8×C17—22no—
68C4×Q8×C17—22no—
69C17×#117—22no—
70C17×#117—22no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17D411yescharacteristic
82C2×C22⋊C4×C17D411yescharacteristic
83C2×C42×C17—21no—
84C2×C42×C17—21no—
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C22⋊Dic17—21no—
88C2×C22⋊Dic17—21no—
89C2×#63231—22no—
90C2×#63231—22no—
91C2×#63231C2×C414yes—
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95#10587610—22no—
96#10587610C2×C414yes—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×#63223—22no—
103C2×#63223—22no—
104C2×#63223—22no—
105C22×#3622—22no—
106C22×#3622—22no—
107C22×#3622C2×C414yes—
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C22×#3622C2311yescharacteristic
111C22×#3622C2311yescharacteristic
112C4×Dic34—22no—
113C4×Dic34—22no—
114C17×#95—21no—
115C17×#95—21no—
116C17×#95—21no—
117C17×#95—21no—
118C17×#95—21no—
119#10587506—22no—
120#10587506—22no—
121#10587506C2×C414yes—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587481—22no—
128#10587481—22no—
129#10587481C2×C414yes—
130#10587478—22no—
131#10587478—22no—
132#10587478C2×C414yes—
133#10587474—22no—
134#10587474—22no—
135C4×#3622—22no—
136C4×#3622D414yes—
137C24×C17C2×D411yescharacteristic
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17C22×C412yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17C22×C412yes—
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17#4112yes—
159C22×C4×C17C2411yescharacteristic
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34C22×C412yes—
170C4⋊C4×C17—24no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—22no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—42no—
179C4⋊C4×C17—81no—
180C4⋊C4×C17C22×C412yes—
181C22⋊C4×C17—24no—
182C22⋊C4×C17—22no—
183C22⋊C4×C17—22no—
184C22⋊C4×C17—81no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—42no—
188C42×C17—81no—
189C42×C17C2×D412yes—
190C42×C17#4112yes—
191C22⋊Dic17—24no—
192C22⋊Dic17—81no—
193#63231—28no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17C22×C412yes—
200#63225—24no—
201#63225—42no—
202#63225—81no—
203C4×Dic17—42no—
204C4×Dic17—81no—
205#63223—28no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—28no—
213C2×#3622C22×C412yes—
214C2×#3622C22×C412yes—
215C2×#3622C2×D412yes—
216C2×#3622C2×D412yes—
217C2×#3622#4112yes—
218C2×#3622#4112yes—
219#21525—171nomaximal, Sylow
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—21no—
224C23×C17—21no—
225C23×C17—21no—
226C23×C17C4×D412yes—
227C23×C17C22×D411yescharacteristic
228Q8×C17—22no—
229Q8×C17—22no—
230Q8×C17—41no—
231Q8×C17—41no—
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—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17C4×D412yes—
261C2×C4×C17#12112yes—
262C2×C4×C17#12112yes—
263C2×C4×C17C23×C411yescharacteristic
264C2×C4×C17C22×D411yescharacteristic
265C2×C4×C17C2×#4111yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17C2×#4111yescharacteristic
268C2×C4×C17C2×#4111yescharacteristic
269C2×Dic17—22no—
270C2×Dic17—22no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281Dic34—22no—
282Dic34—22no—
283Dic34—41no—
284Dic34—41no—
285#3622—22no—
286#3622—22no—
287#3622—22no—
288#3622C4×D414yes—
289C2×#467—171no—
290C2×C4×Q16—171no—
291C2×#405—171no—
292#2323—171no—
293#2323D1711yescharacteristic
294#1969—171no—
295#1942—171no—
296#1935—171no—
297#1935—171no—
298#1894—171no—
299#1894—171no—
300#1808—171no—
301C4×#107—171no—
302#1763—171no—
303#1763—171no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17C2×C4×D411yescharacteristic
312C22×C17C2×#12111yescharacteristic
313C22×C17#53211yescharacteristic
314C22×C17#57611yescharacteristic
315C22×C17#57811yescharacteristic
316C22×C17#57811yescharacteristic
317C22×C17#57811yescharacteristic
318C4×C17—22no—
319C4×C17—22no—
320C4×C17—22no—
321C4×C17—42no—
322C4×C17—41no—
323C4×C17—41no—
324C4×C17—42no—
325C4×C17—81no—
326C4×C17—81no—
327C4×C17C2×C4×D411yescharacteristic
328C4×C17C4×#4111yescharacteristic
329C4×C17#52511yescharacteristic
330C4×C17#52911yescharacteristic
331Dic17—42no—
332Dic17—42no—
333Dic17—81no—
334Dic17—81no—
335C22×Q16—171no—
336C22×M16—171no—
337C23×C8—171no—
338C2×C22⋊Q8—171no—
339C2×C22⋊Q8C2×D1711yescharacteristic
340C2×C4×Q8—171no—
341C2×C4×Q8C2×D1711yescharacteristic
342C2×#117C2×D1711yescharacteristic
343#467—178no—
344C4×Q16—344no—
345C2×#107—171no—
346C2×#107—171no—
347C2×#107—171no—
348C2×#103—171no—
349C2×#103—171no—
350C2×#103—171no—
351C2×#103—171no—
352C2×#103—171no—
353C2×#103—171no—
354#405—344no—
355C2×#97—171no—
356C2×#97—171no—
357C2×C4×C8—171no—
358C2×C4×C8—171no—
359#391—171no—
360#391C2×D1711yescharacteristic
361#389—21no—
362#389—171no—
363#389—341no—
364#389C2×D1711yescharacteristic
365#388—21no—
366#388—21no—
367#388—341no—
368#388—341no—
369#387—21no—
370#387—341no—
371#380—171no—
372#380C2×D1711yescharacteristic
373C4×C4⋊C4—171no—
374C4×C4⋊C4C2×D1711yescharacteristic
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#340—171no—
380#340—171no—
381C2×C17—42no—
382C2×C17C4×#12111yescharacteristic
383C2×C17#295011yescharacteristic
384C2×C17#295111yescharacteristic
385C2×C17#305911yescharacteristic
386C2×C17#306111yescharacteristic
387C2×C17#317511yescharacteristic
388C2×C17#317711yescharacteristic
389C22×Q8—171no—
390C22×Q8C22×D1711yescharacteristic
391C23×C4C22×D1711yescharacteristic
392C2×Q16—174no—
393C2×Q16—342no—
394C2×Q16—342no—
395C2×M16—342no—
396C2×M16—342no—
397C2×M16—342no—
398C22×C8—171no—
399C22×C8—174no—
400C22×C8—171no—
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—342no—
404C22×C8—342no—
405C22⋊Q8—22no—
406C22⋊Q8—174no—
407C22⋊Q8—342no—
408C22⋊Q8C4×D1714yes—
409C4×Q8—22no—
410C4×Q8—22no—
411C4×Q8—342no—
412C4×Q8—342no—
413#117—22no—
414#117—22no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4C22⋊D1711yescharacteristic
425C2×C4⋊C4C22⋊D1711yescharacteristic
426C2×C4⋊C4C22×D1711yescharacteristic
427C2×C4⋊C4C22×D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4C22⋊D1711yescharacteristic
434C2×C22⋊C4C22⋊D1711yescharacteristic
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—341no—
438C2×C42—341no—
439C2×C42C22×D1711yescharacteristic
440C2×C42C22×D1711yescharacteristic
441#107—174no—
442#107—342no—
443#107—342no—
444#103—174no—
445#103—174no—
446#103—342no—
447#103—342no—
448#103—342no—
449#103—342no—
450#97—342no—
451#97—342no—
452C4×C8—174no—
453C4×C8—342no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—341no—
460#95—341no—
461#95—341no—
462#95—341no—
463#95—341no—
464C17#2152511yescharacteristic, complemented, Sylow
465C24C2×C22⋊D1711yescharacteristic
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—172no—
472C2×Q8—341no—
473C2×Q8—341no—
474C2×Q8—341no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×Q8C2×C4×D1712yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4C2×C4×D1712yes—
491C22×C4#6325212yes—
492C22×C4C2×C22⋊D1711yescharacteristic
493C22×C4C2×C22⋊D1711yescharacteristic
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C2×C22⋊D1711yescharacteristic
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C23×D1711yescharacteristic
498Q16—348no—
499M16—348no—
500C2×C8—172no—
501C2×C8—172no—
502C2×C8—172no—
503C2×C8—172no—
504C2×C8—172no—
505C2×C8—172no—
506C2×C8—348no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—172no—
524C4⋊C4—341no—
525C4⋊C4—344no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—682no—
531C4⋊C4—1361no—
532C4⋊C4C2×C4×D1712yes—
533C22⋊C4—24no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—81no—
537C22⋊C4—344no—
538C22⋊C4—1361no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—81no—
543C42—682no—
544C42—1361no—
545C42D4×D1712yes—
546C42#6325412yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23C4×C22⋊D1712yes—
554C23C22×C22⋊D1711yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—342no—
560Q8—342no—
561Q8—681no—
562Q8—681no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—342no—
592C2×C4—342no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4C4×C22⋊D1712yes—
604C2×C4#1058762512yes—
605C2×C4#1058762512yes—
606C2×C4C22×C4×D1711yescharacteristic
607C2×C4C2×#6325211yescharacteristic
608C2×C4C2×D4×D1711yescharacteristic
609C2×C4C2×#6325411yescharacteristic
610C2×C4#41×D1711yescharacteristic
611C2×C4#41×D1711yescharacteristic
612C8—174no—
613C8—342no—
614C8—342no—
615C8—342no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D1711yescharacteristic
624C22C2×#1058762511yescharacteristic
625C22#1176335511yescharacteristic
626C22#1176361211yescharacteristic
627C22#1176362411yescharacteristic
628C22#1176362411yescharacteristic
629C22#1176362411yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—682no—
640C4—682no—
641C4—1361no—
642C4—1361no—
643C4C4×D4×D1711yescharacteristic
644C4C4×#6325411yescharacteristic
645C4#1176331811yescharacteristic
646C4#1176333911yescharacteristic
647C2—42no—
648C2C4×#1058762511yescharacteristic
649C2#1453765011yescharacteristic
650C2#1453765711yescharacteristic
651C2#1453843611yescharacteristic
652C2#1453845111yescharacteristic
653C2#1453941711yescharacteristic
654C2#1453943711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2323C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#2323D17
hypocenterC17#21525
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×C4×#63231C2semidirect
5. #14529644C2semidirect
6. #14529514C2semidirect
7. #14529477C2semidirect
8. #14529473C2semidirect
12. C4×#10587478C2semidirect
13. #14528617C2semidirect
14. #14528614C2semidirect
21. #11763048C4semidirect
464. C17#21525semidirect

Maximal subgroups

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

subgroupindex
C2×#117630482
C2×C4×#632312
C2×#117627652
#145314662
#145296442
#145295142
#145294772
#145294732
#145292182
#145292152
#145288112
C4×#105874782
#145286172
#145286142
C17×#23232
#2152517