Finite Group Databasebeta

← #17094252

Subgroups — #17094252

Order 4352.

854 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763147C211yesmaximal, characteristic, complemented
2C2×#11762849C211yesmaximal, characteristic, complemented
3#14531733C211yesmaximal, characteristic
4#14530327C211yesmaximal, characteristic, complemented
5#14530260C211yesmaximal, characteristic, complemented
6#14530220C211yesmaximal, characteristic, complemented
7#14529977C211yesmaximal, characteristic, complemented
8#14529422C211yesmaximal, characteristic, complemented
9#14529373C211yesmaximal, characteristic, complemented
10#14529218C211yesmaximal, characteristic, complemented
11#14529097C211yesmaximal, characteristic, complemented
12#14529082C211yesmaximal, characteristic
13#14528981C211yesmaximal, characteristic, complemented
14#14528977C211yesmaximal, characteristic, complemented
15C17×#2417C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587612C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19#11763147—24no—
20#11762849—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×C22⋊#3622C2211yescharacteristic
32C2×C22⋊#3622C2211yescharacteristic
33C2×C22⋊#3622C2211yescharacteristic
34#11762712—21no—
35#11762702—21no—
36#11762702—21no—
37#11762679C2211yescharacteristic
38#11762669C2211yescharacteristic
39#11762669C2211yescharacteristic
40C4⋊C4×Dic17—21no—
41C4×C4⋊Dic17C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×C22⋊Q8C2211yescharacteristic
51C2×C17×C22⋊Q8C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C22×C4⋊C4×C17C2211yescharacteristic
54C17×#398—21no—
55C17×#396—21no—
56C17×#391C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#386—21no—
59C17×#380—21no—
60C17×#380C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C22×D4×C17D411yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17C2311yescharacteristic
65C23×C4×C17C2311yescharacteristic
66C17×C22⋊Q8—22no—
67C17×C22⋊Q8—22no—
68C17×C22⋊Q8—22no—
69C17×C22⋊Q8—41no—
70C4×D4×C17—22no—
71C4×D4×C17—22no—
72C2×C4⋊C4×C17—22no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—41no—
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17—21no—
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C22⋊C4×C17D411yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C22⋊Dic17—21no—
89C2×C22⋊Dic17—21no—
90C2×C22⋊Dic17—21no—
91C2×C22⋊Dic17—21no—
92#10587612—22no—
93#10587612—22no—
94#10587612—41no—
95#10587612—41no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17—21no—
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4⋊Dic17C2311yescharacteristic
101C2×C4⋊Dic17C2311yescharacteristic
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17—21no—
105C2×#63223—22no—
106C2×#63223—22no—
107C2×#63223—22no—
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C22×#3622C2311yescharacteristic
111C22×#3622C2311yescharacteristic
112C17×#95—21no—
113C17×#95—21no—
114C17×#95—21no—
115C17×#95—21no—
116#10587506—22no—
117#10587506—22no—
118#10587506—22no—
119#10587506—41no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123C22⋊#3622—22no—
124C22⋊#3622—22no—
125C22⋊#3622—22no—
126#10587479—22no—
127#10587479—22no—
128#10587479—22no—
129#10587479—41no—
130#10587478—22no—
131#10587478D414yescomplemented
132C24×C17C2×D411yescharacteristic
133C24×C17C2×D411yescharacteristic
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×D4×C17—22no—
141C2×D4×C17—22no—
142C2×D4×C17—41no—
143C22×C4×C17—21no—
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×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—42no—
158C22×C4×C17—42no—
159C22×C4×C17—81no—
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17#4112yes—
165C22×C4×C17#4112yes—
166C22×C4×C17C2411yescharacteristic
167C22×Dic17—21no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C22×Dic17—21no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—42no—
178C4⋊C4×C17—42no—
179C4⋊C4×C17—42no—
180C4⋊C4×C17—81no—
181C4⋊C4×C17C2×D412yes—
182C22⋊C4×C17—22no—
183C22⋊C4×C17—22no—
184C22⋊C4×C17—42no—
185C22⋊C4×C17—42no—
186C22⋊C4×C17—81no—
187C42×C17—21no—
188C42×C17—21no—
189C22⋊Dic17—42no—
190C22⋊Dic17—42no—
191C22⋊Dic17—81no—
192C22⋊Dic17—81no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—42no—
201C4⋊Dic17—81no—
202C4⋊Dic17C2×D412yes—
203C4×Dic17—42no—
204C4×Dic17—42no—
205C4×Dic17—42no—
206#63223—28no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622C2×D412yes—
218C2×#3622#4112yes—
219#26155—171nomaximal, Sylow
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—21no—
224C23×C17—21no—
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17#12312yes—
234C23×C17C22×D411yescharacteristic
235Q8×C17—41no—
236Q8×C17—41no—
237Q8×C17—41no—
238Q8×C17—41no—
239D4×C17—44no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—22no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—41no—
268C2×C4×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17—84no—
271C2×C4×C17#12312yes—
272C2×C4×C17#13712yes—
273C2×C4×C17C22×D411yescharacteristic
274C2×C4×C17C2×#4111yescharacteristic
275C2×C4×C17C2×#4111yescharacteristic
276C2×C4×C17C2×#4111yescharacteristic
277C2×C4×C17H(5,𝔽2)11yescharacteristic
278C2×C4×C172−1+411yescharacteristic
279C2×Dic17—22no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290C2×Dic17—41no—
291#3622—22no—
292#3622—22no—
293#3622—41no—
294#3622—41no—
295C2×#485—171no—
296C2×#428—171no—
297#2417D1711yescharacteristic
298#2354—171no—
299#2096—171no—
300#2084—171no—
301#2077—171no—
302#2035—171no—
303#1928—171no—
304#1921—171no—
305#1894—171no—
306#1873—171no—
307#1870—171no—
308#1854—171no—
309#1853—171no—
310C22×C17—22no—
311C22×C17—22no—
312C22×C17—22no—
313C22×C17—41no—
314C22×C17—41no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17—41no—
319C22×C17—41no—
320C22×C17—41no—
321C22×C17—41no—
322C22×C17—41no—
323C22×C17—41no—
324C22×C17C2×#12311yescharacteristic
325C22×C17D4211yescharacteristic
326C22×C17#54711yescharacteristic
327C22×C17C2×#13711yescharacteristic
328C22×C17#57511yescharacteristic
329C22×C17#57711yescharacteristic
330C22×C17#57711yescharacteristic
331C4×C17—22no—
332C4×C17—22no—
333C4×C17—41no—
334C4×C17—41no—
335C4×C17—42no—
336C4×C17—81no—
337C4×C17—81no—
338C4×C17—81no—
339C4×C17#52511yescharacteristic
340C4×C17#53911yescharacteristic
341C4×C17#54811yescharacteristic
342C4×C17#55311yescharacteristic
343Dic17—42no—
344Dic17—81no—
345Dic17—81no—
346Dic17—81no—
347C22×M16—171no—
348C2×C22⋊Q8C2×D1711yescharacteristic
349C2×C22⋊Q8C2×D1711yescharacteristic
350C2×C4×D4C2×D1711yescharacteristic
351C2×#117—171no—
352C2×#117—171no—
353C22×C4⋊C4C2×D1711yescharacteristic
354#485—344no—
355#428—344no—
356C2×#107—171no—
357C2×#107—171no—
358C2×#106—171no—
359C2×#106—171no—
360C2×#106—171no—
361C2×#106—171no—
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×C22⋊C8—171no—
367C2×C22⋊C8—171no—
368C2×C22⋊C8—171no—
369#398—21no—
370#396—21no—
371#391C2×D1711yescharacteristic
372#390—341no—
373#389C2×D1711yescharacteristic
374#388—341no—
375#388—341no—
376#386—21no—
377#384—171no—
378#380—21no—
379#380—171no—
380#380—171no—
381#380C2×D1711yescharacteristic
382C4×C4⋊C4—171no—
383C4×C4⋊C4—341no—
384#340—171no—
385#340—171no—
386#340—171no—
387#340—171no—
388#340—171no—
389#340—171no—
390#336—171no—
391#336—171no—
392C2×C17—42no—
393C2×C17—81no—
394C2×C17#244311yescharacteristic
395C2×C17#309511yescharacteristic
396C2×C17#309911yescharacteristic
397C2×C17#326711yescharacteristic
398C2×C17#327811yescharacteristic
399C2×C17#334811yescharacteristic
400C2×C17#335611yescharacteristic
401C22×Q8C22×D1711yescharacteristic
402C22×D4C22⋊D1711yescharacteristic
403C23×C4—21no—
404C23×C4C22×D1711yescharacteristic
405C23×C4C22×D1711yescharacteristic
406C2×M16—342no—
407C2×M16—342no—
408C2×M16—342no—
409C22×C8—171no—
410C22×C8—171no—
411C22×C8—171no—
412C22×C8—171no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8—41no—
417C4×D4—22no—
418C4×D4—22no—
419#117—342no—
420#117—342no—
421#117—681no—
422#117—681no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—41no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—341no—
434C2×C4⋊C4—341no—
435C2×C4⋊C4C22⋊D1711yescharacteristic
436C2×C4⋊C4C22×D1711yescharacteristic
437C2×C4⋊C4C22×D1711yescharacteristic
438C2×C4⋊C4C22×D1711yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—341no—
443C2×C22⋊C4—341no—
444C2×C22⋊C4—341no—
445C2×C22⋊C4—341no—
446C2×C22⋊C4C22⋊D1711yescharacteristic
447C2×C22⋊C4C22⋊D1711yescharacteristic
448C2×C42—341no—
449C2×C42—341no—
450C2×C42—341no—
451C2×C42C22×D1711yescharacteristic
452#107—174no—
453#107—342no—
454#106—342no—
455#106—342no—
456#106—342no—
457#106—681no—
458#103—342no—
459#103—342no—
460#103—342no—
461#103—681no—
462C22⋊C8—342no—
463C22⋊C8—342no—
464C22⋊C8—342no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—341no—
470#95—341no—
471#95—341no—
472C17#2615511yescharacteristic, complemented, Sylow
473C24C2×C22⋊D1711yescharacteristic
474C24C2×C22⋊D1711yescharacteristic
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—41no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—81no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4#6325412yes—
506C22×C4#6325412yes—
507C22×C4C2×C22⋊D1711yescharacteristic
508C22×C4C2×C22⋊D1711yescharacteristic
509C22×C4C2×C22⋊D1711yescharacteristic
510C22×C4C2×C22⋊D1711yescharacteristic
511C22×C4C23×D1711yescharacteristic
512M16—348no—
513C2×C8—172no—
514C2×C8—172no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—81no—
535C4⋊C4—172no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—341no—
539C4⋊C4—341no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—682no—
544C4⋊C4—1361no—
545C4⋊C4D4×D1712yes—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—682no—
552C22⋊C4—682no—
553C22⋊C4—1361no—
554C22⋊C4—1361no—
555C42—21no—
556C42—21no—
557C42—682no—
558C42—682no—
559C42—682no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1058762112yes—
574C23C22×C22⋊D1711yescharacteristic
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—41no—
579D4—44no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—84no—
611C2×C4—342no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4#1058762112yes—
624C2×C4#1058762812yes—
625C2×C4C2×D4×D1711yescharacteristic
626C2×C4C2×#6325411yescharacteristic
627C2×C4C2×#6325411yescharacteristic
628C2×C4#1058769111yescharacteristic
629C2×C4#41×D1711yescharacteristic
630C2×C4#1058769711yescharacteristic
631C8—342no—
632C8—342no—
633C8—681no—
634C8—681no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1058762111yescharacteristic
650C22D4×C22⋊D1711yescharacteristic
651C22#1176345811yescharacteristic
652C22C2×#1058762811yescharacteristic
653C22#1176360711yescharacteristic
654C22#1176361711yescharacteristic
655C22#1176362011yescharacteristic
656C4—22no—
657C4—22no—
658C4—41no—
659C4—42no—
660C4—41no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—682no—
665C4—1361no—
666C4—1361no—
667C4—1361no—
668C4#1176331911yescharacteristic
669C4#1176340411yescharacteristic
670C4#1176346511yescharacteristic
671C4#1176350211yescharacteristic
672C2—42no—
673C2—81no—
674C2#1453251511yescharacteristic
675C2#1453872811yescharacteristic
676C2#1453876911yescharacteristic
677C2#1454030511yescharacteristic
678C2#1454044611yescharacteristic
679C2#1454129311yescharacteristic
680C2#1454139911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2417C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenter#2417D17
hypocenterC17#26155
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2417

Complements

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

NH ≅ G/Ntype
1. C2×#11763147C2semidirect
2. C2×#11762849C2semidirect
4. #14530327C2semidirect
5. #14530260C2semidirect
6. #14530220C2semidirect
7. #14529977C2semidirect
8. #14529422C2semidirect
9. #14529373C2semidirect
10. #14529218C2semidirect
11. #14529097C2semidirect
13. #14528981C2semidirect
14. #14528977C2semidirect
131. #10587478D4semidirect
472. C17#26155semidirect

Maximal subgroups

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

subgroupindex
C2×#117631472
C2×#117628492
#145317332
#145303272
#145302602
#145302202
#145299772
#145294222
#145293732
#145292182
#145290972
#145290822
#145289812
#145289772
C17×#24172
#2615517