Finite Group Databasebeta

← #16957982

Subgroups — #16957982

Order 4352.

2222 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762690C212yesmaximal, complemented
2C2×#11762682C211yesmaximal, characteristic
3C2×C4×C22⋊Dic17C211yesmaximal, characteristic
4#14527043C212yesmaximal, complemented
5#14527034C212yesmaximal, complemented
6#14527016C212yesmaximal, complemented
7#14527007C211yesmaximal, characteristic
8#14527007C211yesmaximal, characteristic
9#14527007C211yesmaximal, characteristic
10#14527007C211yesmaximal, characteristic
11C23×C4⋊C4×C17C211yesmaximal, characteristic
12C22×#63225C2212yes—
13C22×C22⋊Dic17C2211yescharacteristic
14C22×C22⋊Dic17C2211yescharacteristic
15C22×C22⋊Dic17C2211yescharacteristic
16C22×C22⋊Dic17C2211yescharacteristic
17C22×C4×Dic17C2211yescharacteristic
18C22×C4×Dic17C2211yescharacteristic
19#11762690—28no—
20#11762682C418yescomplemented
21C4×C22⋊Dic17—24no—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2211yescharacteristic
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C24×C4×C17C2211yescharacteristic
34C24×C4×C17C2211yescharacteristic
35C24×C4×C17C2211yescharacteristic
36C22×C4⋊C4×C17—28no—
37C22×C4⋊C4×C17—24no—
38C22×C4⋊C4×C17C2212yes—
39C22×C4⋊C4×C17C2212yes—
40C25×C17C2311yescharacteristic
41C23×C4×C17—22no—
42C23×C4×C17—22no—
43C23×C4×C17—24no—
44C23×C4×C17—24no—
45C23×C4×C17—22no—
46C23×C4×C17—22no—
47C23×C4×C17—22no—
48C23×C4×C17—24no—
49C23×C4×C17—24no—
50C23×C4×C17—22no—
51C23×C4×C17—22no—
52C23×C4×C17—42no—
53C23×C4×C17—42no—
54C23×C4×C17C2×C414yes—
55C23×C4×C17D414yes—
56C23×C4×C17C2312yes—
57C23×C4×C17C2311yescharacteristic
58C23×C4×C17C2312yes—
59C23×C4×C17C2311yescharacteristic
60C23×Dic17C2312yes—
61C23×Dic17C2311yescharacteristic
62C23×Dic17C2311yescharacteristic
63C23×Dic17C2312yes—
64C23×Dic17C2311yescharacteristic
65C23×Dic17C2311yescharacteristic
66C2×C4⋊C4×C17—232no—
67C2×C4⋊C4×C17—24no—
68C2×C4⋊C4×C17—24no—
69C2×C4⋊C4×C17—24no—
70C2×C4⋊C4×C17—24no—
71C2×C4⋊C4×C17—24no—
72C2×C4⋊C4×C17D418yes—
73C2×C22⋊Dic17—22no—
74C2×C22⋊Dic17—22no—
75C2×C22⋊Dic17—22no—
76C2×C22⋊Dic17—22no—
77C2×C22⋊Dic17—22no—
78C2×C22⋊Dic17—22no—
79C2×C22⋊Dic17—22no—
80C2×C22⋊Dic17—22no—
81C2×C22⋊Dic17—22no—
82C2×C22⋊Dic17C2×C414yes—
83C2×C22⋊Dic17C2×C414yes—
84C2×C22⋊Dic17Q814yes—
85C2×#63225—24no—
86C2×#63225—24no—
87C2×#63225C2×C418yescomplemented
88C2×C4×Dic17—24no—
89C2×C4×Dic17—22no—
90C2×C4×Dic17—22no—
91C2×C4×Dic17—42no—
92#10587502—24no—
93#10587502—24no—
94#10587502—24no—
95#10587502—24no—
96#10587502—22no—
97#10587502—22no—
98#10587502—24no—
99#10587502—42no—
100#10587502—42no—
101#10587502—42no—
102#10587502C2×C418yescomplemented
103C24×C17—22no—
104C24×C17—22no—
105C24×C17—22no—
106C24×C17—22no—
107C24×C17C22×C412yes—
108C24×C17C2×D412yes—
109C24×C17C2×Q812yes—
110C24×C17#4112yes—
111C24×C17#4112yes—
112C24×C17#4112yes—
113C24×C17#4112yes—
114C24×C17C2411yescharacteristic
115C22×C4×C17—216no—
116C22×C4×C17—28no—
117C22×C4×C17—28no—
118C22×C4×C17—28no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17—216no—
127C22×C4×C17—28no—
128C22×C4×C17—28no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—28no—
140C22×C4×C17—44no—
141C22×C4×C17—48no—
142C22×C4×C17—48no—
143C22×C4×C17—48no—
144C22×C4×C17—48no—
145C22×C4×C17—44no—
146C22×C4×C17—44no—
147C22×C4×C17C22×C414yes—
148C22×C4×C17C2×D414yes—
149C22×C4×C17C2×D412yes—
150C22×C4×C17C2×D414yes—
151C22×C4×C17C2×D414yes—
152C22×C4×C17C2×D412yes—
153C22×C4×C17#4114yes—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17—22no—
175C22×Dic17—22no—
176C22×Dic17—22no—
177C22×Dic17—22no—
178C22×Dic17—22no—
179C22×Dic17—22no—
180C22×Dic17—22no—
181C22×Dic17—22no—
182C22×Dic17—22no—
183C22×Dic17C22×C412yes—
184C22×Dic17C22×C414yes—
185C22×Dic17C22×C412yes—
186C22×Dic17C2×Q812yes—
187C22×Dic17C2×Q812yes—
188C4⋊C4×C17—216no—
189C4⋊C4×C17—216no—
190C22⋊Dic17—28no—
191C22⋊Dic17—28no—
192C22⋊Dic17—28no—
193C22⋊Dic17—44no—
194#63225—216no—
195C4×Dic17—44no—
196C4×Dic17—44no—
197#14039—171nomaximal, Sylow
198C23×C17—24no—
199C23×C17—24no—
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—24no—
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—24no—
206C23×C17—24no—
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—24no—
211C23×C17—24no—
212C23×C17—24no—
213C23×C17—24no—
214C23×C17#11714yes—
215C23×C17C4×D414yes—
216C23×C17C4×Q814yes—
217C23×C17C22⋊Q814yes—
218C23×C17#12314yes—
219C23×C17#12514yes—
220C23×C17#12614yes—
221C23×C17C23×C411yescharacteristic
222C23×C17C22×D411yescharacteristic
223C23×C17C22×D412yes—
224C23×C17C22×Q811yescharacteristic
225C23×C17C2×#4111yescharacteristic
226C23×C17C2×#4112yes—
227C23×C17C2×#4111yescharacteristic
228C23×C17C2×#4111yescharacteristic
229C23×C17C2×#4111yescharacteristic
230C23×C17H(5,𝔽2)12yes—
231C23×C172−1+412yes—
232C2×C4×C17—232no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—24no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—22no—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—22no—
244C2×C4×C17—24no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—24no—
248C2×C4×C17—24no—
249C2×C4×C17—416no—
250C2×C4×C17—416no—
251C2×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17C4×D418yes—
258C2×C4×C17#12014yes—
259C2×Dic17—24no—
260C2×Dic17—22no—
261C2×Dic17—22no—
262C2×Dic17—24no—
263C2×Dic17—24no—
264C2×Dic17—24no—
265C2×Dic17—24no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—24no—
269C2×Dic17—24no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—42no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17—42no—
282C2×Dic17—42no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17—42no—
289C2×Dic17—42no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17—42no—
293C23×C4⋊C4D1711yescharacteristic
294C2×#387—172no—
295C2×#385—171no—
296C2×C4×C22⋊C4—171no—
297#1442—172no—
298#1441—172no—
299#1436—172no—
300C22⋊#95—171no—
301C22⋊#95—171no—
302C22⋊#95—171no—
303C22⋊#95—171no—
304C22×C17—24no—
305C22×C17—24no—
306C22×C17—28no—
307C22×C17—28no—
308C22×C17—24no—
309C22×C17—24no—
310C22×C17—28no—
311C22×C17—28no—
312C22×C17—24no—
313C22×C17—24no—
314C22×C17#38718yescomplemented
315C22×C17C2×#11711yescharacteristic
316C22×C17C2×C4×D411yescharacteristic
317C22×C17C2×C4×D412yes—
318C22×C17C2×C4×Q811yescharacteristic
319C22×C17#52012yes—
320C22×C17C2×#12011yescharacteristic
321C22×C17C2×C22⋊Q811yescharacteristic
322C22×C17C2×#12311yescharacteristic
323C22×C17#52511yescharacteristic
324C22×C17#52512yes—
325C22×C17C2×#12511yescharacteristic
326C22×C17C2×#12611yescharacteristic
327C22×C17#53411yescharacteristic
328C22×C17#53611yescharacteristic
329C22×C17#53912yes—
330C22×C17#54612yes—
331C22×C17#54612yes—
332C22×C17#54812yes—
333C22×C17D4×Q812yes—
334C22×C17#55212yes—
335C22×C17#55212yes—
336C22×C17#55312yes—
337C22×C17#55412yes—
338C4×C17—24no—
339C4×C17—28no—
340C4×C17—42no—
341C4×C17—44no—
342Dic17—42no—
343Dic17—42no—
344Dic17—44no—
345Dic17—42no—
346Dic17—42no—
347Dic17—82no—
348C24×C4C2×D1711yescharacteristic
349C24×C4C2×D1711yescharacteristic
350C24×C4C2×D1711yescharacteristic
351C22×C4⋊C4—24no—
352C22×C4⋊C4—28no—
353C22×C4⋊C4—172no—
354C22×C4⋊C4C2×D1712yes—
355C22×C4⋊C4C2×D1712yes—
356C22×C22⋊C4—171no—
357C22×C22⋊C4—171no—
358C22×C22⋊C4—171no—
359C22×C22⋊C4—171no—
360C22×C42—171no—
361C22×C42—171no—
362#387—348no—
363#385—178no—
364C4×C22⋊C4—344no—
365C2×#95—172no—
366C2×#95—172no—
367C2×#95—172no—
368C2×#95—172no—
369C2×#95—172no—
370C2×#95—172no—
371C2×#95—171no—
372C2×#95—171no—
373C2×#95—172no—
374C2×#95—172no—
375C2×#95—172no—
376C2×C17—28no—
377C2×C17C2×#38711yescharacteristic
378C2×C17C4×#12011yescharacteristic
379C2×C17#232211yescharacteristic
380C2×C17#235212yes—
381C2×C17#236012yes—
382C2×C17#240511yescharacteristic
383C2×C17#240911yescharacteristic
384C2×C17#247312yes—
385C2×C17#248112yes—
386C2×C17#248411yescharacteristic
387C2×C17#257911yescharacteristic
388C25C22×D1711yescharacteristic
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—24no—
392C23×C4—24no—
393C23×C4—22no—
394C23×C4—22no—
395C23×C4—22no—
396C23×C4—24no—
397C23×C4—24no—
398C23×C4—22no—
399C23×C4—22no—
400C23×C4—42no—
401C23×C4—42no—
402C23×C4—171no—
403C23×C4—172no—
404C23×C4—171no—
405C23×C4—171no—
406C23×C4—171no—
407C23×C4—172no—
408C23×C4C4×D1714yes—
409C23×C4C22⋊D1714yes—
410C23×C4C22×D1711yescharacteristic
411C23×C4C22×D1712yes—
412C23×C4C22×D1712yes—
413C23×C4C22×D1711yescharacteristic
414C2×C4⋊C4—232no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—24no—
418C2×C4⋊C4—24no—
419C2×C4⋊C4—24no—
420C2×C4⋊C4—178no—
421C2×C4⋊C4—344no—
422C2×C4⋊C4—344no—
423C2×C4⋊C4C22⋊D1718yes—
424C2×C22⋊C4—174no—
425C2×C22⋊C4—174no—
426C2×C22⋊C4—174no—
427C2×C22⋊C4—342no—
428C2×C22⋊C4—342no—
429C2×C22⋊C4—342no—
430C2×C22⋊C4—342no—
431C2×C22⋊C4—342no—
432C2×C22⋊C4—342no—
433C2×C22⋊C4—342no—
434C2×C22⋊C4—342no—
435C2×C22⋊C4—342no—
436C2×C42—342no—
437C2×C42—342no—
438C2×C42—344no—
439C2×C42—682no—
440#95—178no—
441#95—344no—
442#95—342no—
443#95—342no—
444#95—344no—
445#95—344no—
446#95—344no—
447#95—344no—
448#95—682no—
449#95—682no—
450#95—682no—
451C17#1403911yescharacteristic, complemented, Sylow
452C24—22no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24C2×C4×D1712yes—
457C24#6325212yes—
458C24#6325412yes—
459C24#6325412yes—
460C24Q8×D1712yes—
461C24#6325612yes—
462C24C2×C22⋊D1712yes—
463C24C23×D1711yescharacteristic
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—216no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—28no—
475C22×C4—216no—
476C22×C4—28no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—28no—
488C22×C4—28no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—172no—
497C22×C4—174no—
498C22×C4—172no—
499C22×C4—172no—
500C22×C4—172no—
501C22×C4—342no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—342no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—341no—
512C22×C4—341no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—341no—
519C22×C4—341no—
520C22×C4—341no—
521C22×C4—341no—
522C22×C4—342no—
523C22×C4—342no—
524C22×C4—342no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4C2×C4×D1714yes—
531C22×C4#6325214yes—
532C22×C4C2×C22⋊D1712yes—
533C22×C4C2×C22⋊D1714yes—
534C22×C4C2×C22⋊D1714yes—
535C22×C4C2×C22⋊D1714yes—
536C22×C4C2×C22⋊D1712yes—
537C4⋊C4—216no—
538C4⋊C4—216no—
539C4⋊C4—3416no—
540C22⋊C4—348no—
541C22⋊C4—348no—
542C22⋊C4—348no—
543C22⋊C4—684no—
544C42—684no—
545C42—684no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23#1058757314yes—
563C23#1058757514yes—
564C23#1058757814yes—
565C23#1058758414yes—
566C23C4×C22⋊D1714yes—
567C23#1058762114yes—
568C23#1058763214yes—
569C23C22×C4×D1711yescharacteristic
570C23C2×#6325212yes—
571C23C2×#6325211yescharacteristic
572C23C2×#6325411yescharacteristic
573C23C2×#6325411yescharacteristic
574C23#1058769112yes—
575C23C2×Q8×D1711yescharacteristic
576C23C2×#6325611yescharacteristic
577C23#1058769412yes—
578C23C22×C22⋊D1712yes—
579C23C22×C22⋊D1711yescharacteristic
580C2×C4—232no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—344no—
606C2×C4—344no—
607C2×C4—344no—
608C2×C4—342no—
609C2×C4—342no—
610C2×C4—344no—
611C2×C4—344no—
612C2×C4—344no—
613C2×C4—344no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—682no—
617C2×C4—682no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—682no—
624C2×C4—681no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—682no—
631C2×C4—682no—
632C2×C4—682no—
633C2×C4—682no—
634C2×C4—682no—
635C2×C4—682no—
636C2×C4—682no—
637C2×C4—682no—
638C2×C4—682no—
639C2×C4C4×C22⋊D1718yes—
640C2×C4#1058763914yes—
641C22—24no—
642C22—24no—
643C22—28no—
644C22—24no—
645C22—28no—
646C22—28no—
647C22—28no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22#1176269018yescomplemented
652C22C2×#1058757811yescharacteristic
653C22C2×C4×C22⋊D1711yescharacteristic
654C22C2×C4×C22⋊D1712yes—
655C22C2×#1058757311yescharacteristic
656C22#1176329012yes—
657C22C2×#1058763911yescharacteristic
658C22C2×#1058763211yescharacteristic
659C22C2×#1058762111yescharacteristic
660C22#1176332012yes—
661C22#1176332011yescharacteristic
662C22C2×#1058757511yescharacteristic
663C22C2×#1058758411yescharacteristic
664C22#1176336811yescharacteristic
665C22#1176338311yescharacteristic
666C22#1176340612yes—
667C22#1176345312yes—
668C22#1176345312yes—
669C22#1176347012yes—
670C22Q8×C22⋊D1712yes—
671C22#1176349412yes—
672C22#1176349412yes—
673C22#1176350712yes—
674C22#1176351412yes—
675C4—24no—
676C4—28no—
677C4—42no—
678C4—44no—
679C4—682no—
680C4—682no—
681C4—684no—
682C4—682no—
683C4—682no—
684C4—1362no—
685C2—28no—
686C2C2×#1176269011yescharacteristic
687C2C4×#1058763911yescharacteristic
688C2#1453145311yescharacteristic
689C2#1453170912yes—
690C2#1453178912yes—
691C2#1453210711yescharacteristic
692C2#1453214311yescharacteristic
693C2#1453291912yes—
694C2#1453303912yes—
695C2#1453307111yescharacteristic
696C2#1453413111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC23×C4⋊C4×C17C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC23×C4⋊C4D17
hypocenterC17#14039
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11762690C2semidirect
4. #14527043C2semidirect
5. #14527034C2semidirect
6. #14527016C2semidirect
20. #11762682C4semidirect
87. C2×#63225C2×C4semidirect
102. #10587502C2×C4semidirect
314. C22×C17#387semidirect
451. C17#14039semidirect
651. C22#11762690semidirect

Maximal subgroups

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

subgroupindex
C2×#117626902
C2×#117626822
C2×C4×C22⋊Dic172
#145270432
#145270342
#145270162
#145270072
C23×C4⋊C4×C172
#1403917