Finite Group Databasebeta

← #17051595

Subgroups — #17051595

Order 4352.

934 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×#11762819C211yesmaximal, characteristic
2#14531679C211yesmaximal, characteristic
3#14531616C211yesmaximal, characteristic
4#14529472C211yesmaximal, characteristic, complemented
5#14529458C211yesmaximal, characteristic, complemented
6#14529424C211yesmaximal, characteristic, complemented
7#14529417C211yesmaximal, characteristic, complemented
8#14529049C211yesmaximal, characteristic, complemented
9#14529035C211yesmaximal, characteristic, complemented
10#14528876C211yesmaximal, characteristic
11#14528718C211yesmaximal, characteristic
12#14528613C211yesmaximal, characteristic, complemented
13#14528603C211yesmaximal, characteristic, complemented
14#14528565C211yesmaximal, characteristic
15C17×#1809C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C2×#10587573C2211yescharacteristic
18C2×#10587573C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20#11762819C418yescomplemented
21C2×#10587487C2211yescharacteristic
22C2×#10587487C2211yescharacteristic
23C2×#10587487C2211yescharacteristic
24C2×#10587487C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32#11762714C2211yescharacteristic
33#11762714C2211yescharacteristic
34#11762713—21no—
35#11762713—21no—
36#11762705C2211yescharacteristic
37#11762705C2211yescharacteristic
38#11762697—21no—
39#11762697—21no—
40#11762676—21no—
41#11762667—21no—
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44#11762660—21no—
45C22⋊C4×Dic17—21no—
46C4×C22⋊Dic17C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C2×C17×#107C2211yescharacteristic
55C2×C17×#106C2211yescharacteristic
56C17×#380C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17—21no—
62C22×Dic34—22no—
63C22×Dic34—22no—
64C22×Dic34—21no—
65C22×Dic34—22no—
66C22×Dic34—41no—
67C22×Dic34—41no—
68C22×Dic34C2×C414yes—
69C22×Dic34C2311yescharacteristic
70C22×Dic34C2311yescharacteristic
71C2×M16×C17—41no—
72C2×M16×C17—41no—
73C2×M16×C17C2×C414yes—
74C22×C8×C17C2311yescharacteristic
75C22×C8×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17D411yescharacteristic
84C2×C22⋊Dic17D411yescharacteristic
85#10587612—22no—
86#10587612C2×C414yes—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17C2311yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102#10587573—22no—
103#10587573—22no—
104#10587573—41no—
105#10587573—41no—
106C17×#107—41no—
107C17×#106—41no—
108C17×#95—21no—
109C17×#95—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502D411yescharacteristic
119#10587502D411yescharacteristic
120C22⋊#3622—22no—
121#10587487C2×C414yes—
122#10587487C2×C414yes—
123#10587487C2×C414yes—
124#10587487C2×C414yes—
125#10587486—41no—
126C8×Dic17—41no—
127#10587481—22no—
128#10587481—22no—
129#10587481—41no—
130#10587481—41no—
131C24×C17C2×D411yescharacteristic
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17C22×C412yes—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17#4112yes—
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—24no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—42no—
150C22×Dic17—42no—
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C2×Dic34—28no—
163C2×Dic34—24no—
164C2×Dic34—21no—
165C2×Dic34—44no—
166C2×Dic34—44no—
167C2×Dic34—44no—
168C2×Dic34—42no—
169C2×Dic34—42no—
170C2×Dic34C22×C412yes—
171C2×Dic34C22×C412yes—
172C2×Dic34#4112yes—
173C2×Dic34#4112yes—
174M16×C17—44no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17C22×C412yes—
180C2×C8×C17C22×C412yes—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C22⋊C4×C17—42no—
190C22⋊C4×C17—81no—
191C22⋊Dic17—24no—
192C22⋊Dic17—22no—
193C22⋊Dic17—41no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17C22×C412yes—
197C4⋊Dic17C22×C412yes—
198#63225—42no—
199#63225—42no—
200#63225—42no—
201#63225—42no—
202C4×Dic17—24no—
203C4×Dic17—21no—
204C4×Dic17—21no—
205C4×Dic17—21no—
206C4×Dic17—21no—
207C4×Dic17—42no—
208C4×Dic17—42no—
209C4×Dic17—81no—
210C4×Dic17—81no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217#22581—171nomaximal, Sylow
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—21no—
224C23×C17C4×D412yes—
225C23×C17C22×D411yescharacteristic
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17C4×D412yes—
245C2×C4×C17C23×C411yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17H(5,𝔽2)11yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×Dic17—22no—
252C2×Dic17—22no—
253C2×Dic17—22no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—48no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17#13712yes—
276Dic34—22no—
277Dic34—22no—
278Dic34—22no—
279Dic34—22no—
280Dic34—22no—
281Dic34—48no—
282Dic34—41no—
283Dic34—41no—
284Dic34#11714yes—
285C8×C17—22no—
286C8×C17—22no—
287#3622—41no—
288#3622—41no—
289C2×#419—171no—
290#2347—171no—
291Q8×C22⋊C4—171no—
292#1935—171no—
293#1933—171no—
294#1928—171no—
295#1927—171no—
296#1865—171no—
297#1863—171no—
298#1825—171no—
299#1809D1711yescharacteristic
300#1787—171no—
301#1763—171no—
302C4×#103—171no—
303#1753—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×C17#54611yescharacteristic
313C22×C17#54611yescharacteristic
314C22×C17C2×#13711yescharacteristic
315C22×C17#57511yescharacteristic
316C22×C17#57711yescharacteristic
317C22×C17#57811yescharacteristic
318C4×C17—22no—
319C4×C17—81no—
320C4×C17—81no—
321C4×C17—81no—
322C4×C17—81no—
323C4×C17C2×#11711yescharacteristic
324C4×C17#51811yescharacteristic
325C4×C17#55111yescharacteristic
326C4×C17#55111yescharacteristic
327Dic17—24no—
328Dic17—22no—
329Dic17—41no—
330Dic17—42no—
331Dic17—42no—
332Dic17—42no—
333Dic17—41no—
334Dic17—41no—
335C23×Q8—171no—
336C22×M16C2×D1711yescharacteristic
337C2×C4×Q8—171no—
338C2×C4×Q8—171no—
339C2×#117—171no—
340C2×#107C2×D1711yescharacteristic
341C2×#106C2×D1711yescharacteristic
342#419—178no—
343C2×#103—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#103—171no—
347C2×#103—171no—
348C2×#103—171no—
349C2×#103—171no—
350C2×#103—171no—
351C2×C22⋊C8—171no—
352C2×#97—171no—
353C2×C4×C8—171no—
354#391—171no—
355#391—171no—
356#391—341no—
357#391—341no—
358#389—171no—
359#389—171no—
360#388—341no—
361#388—341no—
362#383—341no—
363#380—341no—
364#380C2×D1711yescharacteristic
365#380C2×D1711yescharacteristic
366C4×C4⋊C4—171no—
367C4×C4⋊C4—171no—
368C4×C4⋊C4—341no—
369C4×C22⋊C4—171no—
370C4×C22⋊C4—341no—
371#340—171no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340C2×D1711yescharacteristic
376#340C2×D1711yescharacteristic
377#336—171no—
378#336—171no—
379C2×C17—42no—
380C2×C17#234211yescharacteristic
381C2×C17C4×#13711yescharacteristic
382C2×C17#294711yescharacteristic
383C2×C17#331311yescharacteristic
384C2×C17#332011yescharacteristic
385C2×C17#332611yescharacteristic
386C2×C17#332811yescharacteristic
387C22×Q8—174no—
388C22×Q8—171no—
389C22×Q8—171no—
390C22×Q8—342no—
391C22×Q8—342no—
392C22×Q8—341no—
393C22×Q8—342no—
394C22×Q8—681no—
395C22×Q8—681no—
396C23×C4—341no—
397C23×C4C22×D1711yescharacteristic
398C2×M16—41no—
399C2×M16—41no—
400C2×M16C4×D1714yes—
401C22×C8—171no—
402C22×C8—171no—
403C22×C8C22×D1711yescharacteristic
404C22×C8C22×D1711yescharacteristic
405C4×Q8—342no—
406C4×Q8—342no—
407C4×Q8—681no—
408C4×Q8—681no—
409#117—174no—
410#117—342no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—341no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4C22×D1711yescharacteristic
418C2×C4⋊C4C22×D1711yescharacteristic
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C4⋊C4C22×D1711yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—171no—
424C2×C22⋊C4—171no—
425C2×C22⋊C4—341no—
426C2×C42—171no—
427C2×C42—171no—
428C2×C42—341no—
429C2×C42—341no—
430C2×C42—341no—
431C2×C42—341no—
432C2×C42—341no—
433#107—41no—
434#106—41no—
435#103—174no—
436#103—174no—
437#103—174no—
438#103—174no—
439#103—342no—
440#103—342no—
441#103—681no—
442#103—681no—
443C22⋊C8—342no—
444#97—681no—
445C4×C8—681no—
446#95—21no—
447#95—21no—
448#95—171no—
449#95—171no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458C17#2258111yescharacteristic, complemented, Sylow
459C24D4×D1711yescharacteristic
460C2×Q8—172no—
461C2×Q8—172no—
462C2×Q8—172no—
463C2×Q8—172no—
464C2×Q8—348no—
465C2×Q8—341no—
466C2×Q8—341no—
467C2×Q8—341no—
468C2×Q8—341no—
469C2×Q8—344no—
470C2×Q8—341no—
471C2×Q8—341no—
472C2×Q8—341no—
473C2×Q8—341no—
474C2×Q8—682no—
475C2×Q8—682no—
476C2×Q8—684no—
477C2×Q8—684no—
478C2×Q8—684no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—171no—
488C22×C4—171no—
489C22×C4—171no—
490C22×C4—171no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—341no—
495C22×C4—344no—
496C22×C4—682no—
497C22×C4—682no—
498C22×C4C2×C4×D1712yes—
499C22×C4D4×D1711yescharacteristic
500C22×C4#6325612yes—
501C22×C4C23×D1711yescharacteristic
502M16—44no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8C2×C4×D1712yes—
514C2×C8C2×C4×D1712yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—172no—
524C4⋊C4—172no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—682no—
528C4⋊C4—682no—
529C4⋊C4—682no—
530C4⋊C4—682no—
531C22⋊C4—42no—
532C22⋊C4—81no—
533C22⋊C4—342no—
534C22⋊C4—344no—
535C22⋊C4—681no—
536C42—341no—
537C42—341no—
538C42—341no—
539C42—341no—
540C42—344no—
541C42—682no—
542C42—682no—
543C42—1361no—
544C42—1361no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23#1058757912yes—
552C23C2×D4×D1711yescharacteristic
553Q8—174no—
554Q8—342no—
555Q8—342no—
556Q8—342no—
557Q8—342no—
558Q8—342no—
559Q8—688no—
560Q8—681no—
561Q8—681no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—172no—
581C2×C4—342no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—342no—
585C2×C4—342no—
586C2×C4—342no—
587C2×C4—342no—
588C2×C4—342no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—688no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4#1058757912yes—
606C2×C4C22×C4×D1711yescharacteristic
607C2×C4#1058769111yescharacteristic
608C2×C4#1058769111yescharacteristic
609C2×C4C2×#6325611yescharacteristic
610C2×C4#41×D1711yescharacteristic
611C2×C4#41×D1711yescharacteristic
612C8—22no—
613C8—22no—
614C8—681no—
615C8—681no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×#1058757911yescharacteristic
624C22#1176344811yescharacteristic
625C22#1176344811yescharacteristic
626C22#137×D1711yescharacteristic
627C22#1176360511yescharacteristic
628C22#1176361911yescharacteristic
629C22#1176362511yescharacteristic
630C4—22no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—344no—
636C4—342no—
637C4—681no—
638C4—681no—
639C4—682no—
640C4—682no—
641C4—682no—
642C4—681no—
643C4#117×D1711yescharacteristic
644C4#1176328211yescharacteristic
645C4#1176348611yescharacteristic
646C4#1176348611yescharacteristic
647C2—42no—
648C2#1453163011yescharacteristic
649C2#1453760211yescharacteristic
650C2#1453762511yescharacteristic
651C2#1454084511yescharacteristic
652C2#1454095011yescharacteristic
653C2#1454104311yescharacteristic
654C2#1454106211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1809C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#1809D17
hypocenterC17#22581
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #14529472C2semidirect
5. #14529458C2semidirect
6. #14529424C2semidirect
7. #14529417C2semidirect
8. #14529049C2semidirect
9. #14529035C2semidirect
12. #14528613C2semidirect
13. #14528603C2semidirect
20. #11762819C4semidirect
458. C17#22581semidirect

Maximal subgroups

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

subgroupindex
C2×#117628192
#145316792
#145316162
#145294722
#145294582
#145294242
#145294172
#145290492
#145290352
#145288762
#145287182
#145286132
#145286032
#145285652
C17×#18092
#2258117