Finite Group Databasebeta

← #17097191

Subgroups — #17097191

Order 4352.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763114C211yesmaximal, characteristic, complemented
2C2×#11762851C211yesmaximal, characteristic
3#14532498C211yesmaximal, characteristic
4#14531793C211yesmaximal, characteristic
5#14530320C211yesmaximal, characteristic, complemented
6#14530274C211yesmaximal, characteristic, complemented
7#14530163C211yesmaximal, characteristic, complemented
8#14530121C211yesmaximal, characteristic, complemented
9#14529541C211yesmaximal, characteristic, complemented
10#14529361C211yesmaximal, characteristic, complemented
11#14529097C211yesmaximal, characteristic
12#14529040C211yesmaximal, characteristic, complemented
13#14528971C211yesmaximal, characteristic
14#14528967C211yesmaximal, characteristic
15C17×#1894C211yesmaximal, characteristic
16C2×#10587576C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19C22×C4⋊Dic17C2211yescharacteristic
20#11763114—24no—
21#11762851—24no—
22C2×#10587489C2211yescharacteristic
23C2×#10587489C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587488C2211yescharacteristic
27C2×#10587488C2211yescharacteristic
28C2×#10587485C2211yescharacteristic
29C2×#10587485C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587506C2211yescharacteristic
32C2×C22⋊#3622C2211yescharacteristic
33#11762745—21no—
34#11762732—21no—
35#11762716C2211yescharacteristic
36#11762708—21no—
37#11762708C2211yescharacteristic
38#11762708C2211yescharacteristic
39#11762706—21no—
40#11762705C2211yescharacteristic
41#11762696—21no—
42#11762695—21no—
43#11762684—21no—
44#11762678C2211yescharacteristic
45C4⋊C4×Dic17C2211yescharacteristic
46C22⋊C4×Dic17—21no—
47C4×C22⋊Dic17C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×M16×C17C2211yescharacteristic
55C2×C17×C22⋊Q8C2211yescharacteristic
56C2×C17×#103C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C17×#380C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C23×C4×C17C2311yescharacteristic
63C23×Dic17—21no—
64C2×M16×C17—22no—
65C2×M16×C17—22no—
66C2×M16×C17—22no—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C17×C22⋊Q8—22no—
70C17×C22⋊Q8Q814yes—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17D411yescharacteristic
79C2×C22⋊Dic17D411yescharacteristic
80Q8×Dic17—22no—
81Q8×Dic17—41no—
82#10587612—22no—
83#10587612—41no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—22no—
90C2×C4⋊Dic17—41no—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17C2311yescharacteristic
103C2×C4×Dic17C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106#10587576—22no—
107#10587576—41no—
108C17×#103—22no—
109C17×#103—22no—
110C17×#95—21no—
111#10587506—22no—
112#10587506—41no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502D411yescharacteristic
120#10587502D411yescharacteristic
121C22⋊#3622—22no—
122#10587489—22no—
123#10587489—22no—
124#10587488—22no—
125#10587488D414yes—
126#10587485—22no—
127#10587485—22no—
128#10587478—22no—
129#10587478—41no—
130C24×C17C2×D411yescharacteristic
131C2×Q8×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17C2×Q812yes—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×Q812yes—
147C22×C4×C17#4112yes—
148C22×C4×C17C2411yescharacteristic
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—42no—
154C22×Dic17—42no—
155C22×Dic17—81no—
156C22×Dic17C2×D411yescharacteristic
157C22×Dic17C2×D411yescharacteristic
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160M16×C17—28no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17C2×D412yes—
166C2×C8×C17#4112yes—
167C4⋊C4×C17—24no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17C2×Q812yes—
174C22⋊C4×C17—24no—
175C22⋊C4×C17—81no—
176C22⋊Dic17—22no—
177C22⋊Dic17—41no—
178C22⋊Dic17—42no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—42no—
186C4⋊Dic17—42no—
187C4⋊Dic17—42no—
188C4⋊Dic17—81no—
189C4⋊Dic17—81no—
190C4⋊Dic17C2×D412yes—
191C4⋊Dic17C2×D412yes—
192#63225—42no—
193#63225—42no—
194#63225—42no—
195#63225—81no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C4×Dic17—21no—
199C4×Dic17—21no—
200C4×Dic17—42no—
201C4×Dic17—42no—
202C4×Dic17—81no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209#26362—171nomaximal, Sylow
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17C22⋊Q812yes—
217C23×C17C22×D411yescharacteristic
218Q8×C17—22no—
219Q8×C17—22no—
220Q8×C17—41no—
221Q8×C17—41no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
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×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17C22⋊Q812yes—
241C2×C4×C17C22×D411yescharacteristic
242C2×C4×C17C22×Q811yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17H(5,𝔽2)11yescharacteristic
246C2×C4×C172−1+411yescharacteristic
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—84no—
271C2×Dic17#13712yes—
272C8×C17—22no—
273C8×C17—22no—
274#3622—41no—
275#3622—41no—
276C2×#479—171no—
277C2×#428—171no—
278#2441—171no—
279#2360—171no—
280#2095—171no—
281#2086—171no—
282#2066—171no—
283#2059—171no—
284#1947—171no—
285#1919—171no—
286#1894D1711yescharacteristic
287#1873—171no—
288#1863—171no—
289#1852—171no—
290#1851—171no—
291C22×C17—22no—
292C22×C17—22no—
293C22×C17—22no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17—41no—
298C22×C17C2×C22⋊Q811yescharacteristic
299C22×C17D4211yescharacteristic
300C22×C17#54711yescharacteristic
301C22×C17C2×#13711yescharacteristic
302C22×C17#57511yescharacteristic
303C22×C17#57611yescharacteristic
304C22×C17#57811yescharacteristic
305C4×C17—22no—
306C4×C17—42no—
307C4×C17—42no—
308C4×C17—81no—
309C4×C17—81no—
310C4×C17#52511yescharacteristic
311C4×C17#53911yescharacteristic
312C4×C17D4×Q811yescharacteristic
313C4×C17#55411yescharacteristic
314Dic17—22no—
315Dic17—41no—
316Dic17—41no—
317Dic17—42no—
318Dic17—41no—
319Dic17—42no—
320Dic17—81no—
321Dic17—81no—
322C22×M16C2×D1711yescharacteristic
323C2×#125—171no—
324C2×C22⋊Q8C2×D1711yescharacteristic
325C2×C4×Q8—171no—
326C2×#117—171no—
327C22×C4⋊C4—171no—
328#479—344no—
329#428—344no—
330C2×#107—171no—
331C2×#107—171no—
332C2×#107—171no—
333C2×#107—171no—
334C2×#106—171no—
335C2×#106—171no—
336C2×C4⋊C8—171no—
337C2×C4⋊C8—171no—
338C2×#103—171no—
339C2×#103—171no—
340C2×#103C2×D1711yescharacteristic
341C2×#103C2×D1711yescharacteristic
342C2×C22⋊C8—171no—
343#399—341no—
344#396—341no—
345#391—171no—
346#389—171no—
347#389—171no—
348#389—171no—
349#389—341no—
350#389—341no—
351#387—341no—
352#387—341no—
353#385—341no—
354#384—171no—
355#380C2×D1711yescharacteristic
356C4×C4⋊C4—171no—
357C4×C22⋊C4—171no—
358C4×C22⋊C4—341no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340C2×D1711yescharacteristic
364#340C2×D1711yescharacteristic
365#336—171no—
366#336—171no—
367C2×C17—42no—
368C2×C17#244911yescharacteristic
369C2×C17#307811yescharacteristic
370C2×C17#308311yescharacteristic
371C2×C17#326811yescharacteristic
372C2×C17#327611yescharacteristic
373C2×C17#334611yescharacteristic
374C2×C17#335211yescharacteristic
375C22×Q8C22×D1711yescharacteristic
376C23×C4—341no—
377C23×C4C22×D1711yescharacteristic
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—22no—
381C22×C8—171no—
382C22×C8—171no—
383C22×C8C22×D1711yescharacteristic
384C22×C8C22×D1711yescharacteristic
385#125—342no—
386#125—681no—
387C22⋊Q8—22no—
388C22⋊Q8Dic3414yes—
389C4×Q8—342no—
390C4×Q8—681no—
391#117—342no—
392#117—681no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—342no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4—681no—
409C2×C4⋊C4C22×D1711yescharacteristic
410C2×C4⋊C4C22×D1711yescharacteristic
411C2×C4⋊C4C22×D1711yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—171no—
415C2×C22⋊C4—171no—
416C2×C22⋊C4—341no—
417C2×C42—171no—
418C2×C42—171no—
419C2×C42—341no—
420C2×C42—341no—
421C2×C42—341no—
422#107—342no—
423#107—342no—
424#107—342no—
425#107—681no—
426#106—174no—
427#106—342no—
428C4⋊C8—342no—
429C4⋊C8—342no—
430#103—22no—
431#103—22no—
432#103—342no—
433#103—681no—
434C22⋊C8—342no—
435#95—21no—
436#95—171no—
437#95—171no—
438#95—341no—
439#95—341no—
440#95—341no—
441#95—341no—
442#95—341no—
443#95—341no—
444C17#2636211yescharacteristic, complemented, Sylow
445C24D4×D1711yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×Dic3412yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—171no—
461C22×C4—171no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—682no—
469C22×C4—682no—
470C22×C4—1361no—
471C22×C4C2×Dic3412yes—
472C22×C4D4×D1711yescharacteristic
473C22×C4#6325412yes—
474C22×C4C23×D1711yescharacteristic
475M16—28no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8D4×D1712yes—
487C2×C8#6325412yes—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—172no—
495C4⋊C4—172no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—682no—
503C4⋊C4—682no—
504C4⋊C4—682no—
505C4⋊C4—682no—
506C4⋊C4—682no—
507C4⋊C4—682no—
508C4⋊C4—1361no—
509C4⋊C4—1361no—
510C4⋊C4—1361no—
511C4⋊C4C2×Dic3412yes—
512C22⋊C4—24no—
513C22⋊C4—81no—
514C22⋊C4—342no—
515C22⋊C4—681no—
516C22⋊C4—682no—
517C42—341no—
518C42—341no—
519C42—341no—
520C42—341no—
521C42—682no—
522C42—682no—
523C42—1361no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1058756412yes—
531C23C2×D4×D1711yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
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—41no—
554C2×C4—172no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—342no—
567C2×C4—342no—
568C2×C4—681no—
569C2×C4—681no—
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—1364no—
579C2×C4#1058756412yes—
580C2×C4C22×Dic3411yescharacteristic
581C2×C4C2×D4×D1711yescharacteristic
582C2×C4C2×#6325411yescharacteristic
583C2×C4C2×#6325411yescharacteristic
584C2×C4#1058769611yescharacteristic
585C2×C4#1058769711yescharacteristic
586C8—22no—
587C8—22no—
588C8—681no—
589C8—681no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#1058756411yescharacteristic
598C22#1176344311yescharacteristic
599C22#1176345911yescharacteristic
600C22#137×D1711yescharacteristic
601C22#1176360811yescharacteristic
602C22#1176361311yescharacteristic
603C22#1176362511yescharacteristic
604C4—22no—
605C4—42no—
606C4—42no—
607C4—81no—
608C4—81no—
609C4—342no—
610C4—681no—
611C4—681no—
612C4—682no—
613C4—681no—
614C4—682no—
615C4—1361no—
616C4—1361no—
617C4#1176331511yescharacteristic
618C4#1176340311yescharacteristic
619C4D4×Dic3411yescharacteristic
620C4#1176351611yescharacteristic
621C2—42no—
622C2#1453259911yescharacteristic
623C2#1453858611yescharacteristic
624C2#1453863911yescharacteristic
625C2#1454033111yescharacteristic
626C2#1454042711yescharacteristic
627C2#1454126411yescharacteristic
628C2#1454135411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1894C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1894D17
hypocenterC17#26362
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
1. C2×#11763114C2semidirect
5. #14530320C2semidirect
6. #14530274C2semidirect
7. #14530163C2semidirect
8. #14530121C2semidirect
9. #14529541C2semidirect
10. #14529361C2semidirect
12. #14529040C2semidirect
444. C17#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#117631142
C2×#117628512
#145324982
#145317932
#145303202
#145302742
#145301632
#145301212
#145295412
#145293612
#145290972
#145290402
#145289712
#145289672
C17×#18942
#2636217