Finite Group Databasebeta

← #16956027

Subgroups — #16956027

Order 4352.

1814 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#63225C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Dic17C211yesmaximal, characteristic
3C2×C4×C22⋊Dic17C211yesmaximal, characteristic
4C2×#11762655C211yesmaximal, characteristic, complemented
5#14527044C212yesmaximal, complemented
6#14527042C212yesmaximal, complemented
7#14527031C212yesmaximal
8#14527027C212yesmaximal
9C4×#10587502C211yesmaximal, characteristic, complemented
10C4×#10587502C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C17C211yesmaximal, characteristic
12C22×#63225C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2211yescharacteristic
15C22×C22⋊Dic17C2211yescharacteristic
16C22×C4×Dic17C2211yescharacteristic
17C22×C4×Dic17C2211yescharacteristic
18C22×C4×Dic17C2211yescharacteristic
19C22×C4×Dic17C2211yescharacteristic
20C4×#63225—24no—
21C4×C22⋊Dic17—24no—
22C4×C22⋊Dic17C418yescomplemented
23#11762655—24no—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2211yescharacteristic
34C24×C4×C17C2211yescharacteristic
35C22×C22⋊C4×C17C2212yes—
36C22×C42×C17C2211yescharacteristic
37C22×C42×C17C2211yescharacteristic
38C4×C22⋊C4×C17—24no—
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C23×C4×C17—22no—
42C23×C4×C17—22no—
43C23×C4×C17—22no—
44C23×C4×C17—22no—
45C23×C4×C17—22no—
46C23×C4×C17—22no—
47C23×C4×C17C2×C414yes—
48C23×C4×C17C2311yescharacteristic
49C23×C4×C17C2312yes—
50C23×C4×C17C2312yes—
51C23×C4×C17C2311yescharacteristic
52C23×Dic17C2311yescharacteristic
53C23×Dic17C2311yescharacteristic
54C23×Dic17C2311yescharacteristic
55C23×Dic17C2312yes—
56C23×Dic17C2312yes—
57C23×Dic17C2311yescharacteristic
58C2×C22⋊C4×C17—24no—
59C2×C22⋊C4×C17—24no—
60C2×C22⋊C4×C17—24no—
61C2×C42×C17—22no—
62C2×C42×C17—22no—
63C2×C42×C17—22no—
64C2×C42×C17—22no—
65C2×C42×C17—22no—
66C2×C42×C17D414yes—
67C2×C22⋊Dic17—24no—
68C2×C22⋊Dic17—24no—
69C2×C22⋊Dic17—24no—
70C2×C22⋊Dic17—24no—
71C2×C22⋊Dic17—24no—
72C2×C22⋊Dic17C2×C414yes—
73C2×C22⋊Dic17C2×C414yes—
74C2×#63225—24no—
75C2×#63225—24no—
76C2×#63225—24no—
77C2×C4×Dic17—22no—
78C2×C4×Dic17—22no—
79C2×C4×Dic17—22no—
80C2×C4×Dic17—22no—
81C2×C4×Dic17—22no—
82C2×C4×Dic17—22no—
83C2×C4×Dic17—22no—
84C2×C4×Dic17—22no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17C2×C414yes—
87C2×C4×Dic17C2×C414yes—
88C2×C4×Dic17D414yes—
89C17×#95—24no—
90#10587502—24no—
91#10587502—24no—
92#10587502—24no—
93#10587502—24no—
94#10587502—24no—
95#10587502—24no—
96#10587502—24no—
97#10587502—24no—
98#10587502C2×C414yes—
99#10587502C2×C414yes—
100C24×C17—21no—
101C24×C17—21no—
102C24×C17—21no—
103C24×C17—21no—
104C24×C17—21no—
105C24×C17—21no—
106C24×C17—22no—
107C24×C17—22no—
108C24×C17—22no—
109C24×C17C22×C412yes—
110C24×C17#4114yes—
111C24×C17C2411yescharacteristic
112C22×C4×C17—21no—
113C22×C4×C17—21no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—28no—
131C22×C4×C17—28no—
132C22×C4×C17—28no—
133C22×C4×C17—44no—
134C22×C4×C17—44no—
135C22×C4×C17—44no—
136C22×C4×C17—44no—
137C22×C4×C17C22×C412yes—
138C22×C4×C17C22×C412yes—
139C22×C4×C17C2×D412yes—
140C22×C4×C17C2×D414yes—
141C22×C4×C17C2×D412yes—
142C22×C4×C17#4114yes—
143C22×C4×C17#4112yes—
144C22×C4×C17#4112yes—
145C22×C4×C17#4112yes—
146C22×C4×C17#4112yes—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—21no—
162C22×Dic17—22no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17C22×C412yes—
175C22×Dic17C22×C412yes—
176C22×Dic17C22×C412yes—
177C22×Dic17C22×C412yes—
178C22×Dic17C2×D414yes—
179C22×Dic17#4114yes—
180C22×Dic17#4114yes—
181C22⋊C4×C17—216no—
182C42×C17—28no—
183C42×C17—44no—
184C22⋊Dic17—216no—
185C22⋊Dic17—28no—
186C22⋊Dic17—28no—
187#63225—216no—
188C4×Dic17—28no—
189C4×Dic17—28no—
190C4×Dic17—28no—
191C4×Dic17—44no—
192#13842—171nomaximal, Sylow
193C23×C17—22no—
194C23×C17—22no—
195C23×C17—22no—
196C23×C17—22no—
197C23×C17—22no—
198C23×C17—22no—
199C23×C17—24no—
200C23×C17—24no—
201C23×C17—24no—
202C23×C17—24no—
203C23×C17—24no—
204C23×C17—24no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—42no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17—42no—
211C23×C17—42no—
212C23×C17—42no—
213C23×C17—42no—
214C23×C17—42no—
215C23×C17#11714yes—
216C23×C17C23×C411yescharacteristic
217C23×C17C22×D411yescharacteristic
218C23×C17C22×D411yescharacteristic
219C23×C17C2×#4112yes—
220C23×C17C2×#4111yescharacteristic
221C23×C17C2×#4112yes—
222C23×C17C2×#4111yescharacteristic
223C23×C17C2×#4111yescharacteristic
224C23×C17C2×#4111yescharacteristic
225C23×C17C2×#4112yes—
226C23×C17C2×#4112yes—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—24no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—416no—
249C2×C4×C17#11714yes—
250C2×C4×C17C4×D414yes—
251C2×C4×C17C4×D414yes—
252C2×C4×C17#12114yes—
253C2×C4×C17#12314yes—
254C2×C4×C17#12414yes—
255C2×C4×C17#12614yes—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—22no—
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—24no—
270C2×Dic17—24no—
271C2×Dic17—24no—
272C2×Dic17—24no—
273C2×Dic17—24no—
274C2×Dic17—24no—
275C2×Dic17—42no—
276C2×Dic17—42no—
277C2×Dic17—42no—
278C2×Dic17—42no—
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#11714yes—
290C2×C4×C4⋊C4—171no—
291C2×C4×C22⋊C4—171no—
292C2×C4×C22⋊C4—171no—
293C2×C4×C22⋊C4D1711yescharacteristic
294C2×#376—171no—
295#1443—172no—
296#1442—172no—
297#1440—172no—
298#1439—172no—
299C4×#95—171no—
300C4×#95—171no—
301C22×C17—24no—
302C22×C17—24no—
303C22×C17—24no—
304C22×C17—44no—
305C22×C17—44no—
306C22×C17—44no—
307C22×C17—44no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—42no—
311C22×C17—42no—
312C22×C17C2×#11711yescharacteristic
313C22×C17C2×#11711yescharacteristic
314C22×C17C2×#11711yescharacteristic
315C22×C17C2×C4×D411yescharacteristic
316C22×C17C2×C4×D411yescharacteristic
317C22×C17C4×#4111yescharacteristic
318C22×C17C4×#4111yescharacteristic
319C22×C17C2×#12111yescharacteristic
320C22×C17C2×#12311yescharacteristic
321C22×C17#52512yes—
322C22×C17#52512yes—
323C22×C17#52512yes—
324C22×C17C2×#12411yescharacteristic
325C22×C17C2×#12611yescharacteristic
326C22×C17#52912yes—
327C22×C17#52912yes—
328C22×C17#52912yes—
329C22×C17#52912yes—
330C22×C17#52912yes—
331C22×C17#52912yes—
332C22×C17#52912yes—
333C22×C17#52912yes—
334C22×C17#53212yes—
335C4×C17—44no—
336C4×C17—44no—
337C4×C17—42no—
338C4×C17#38818yes—
339Dic17—24no—
340Dic17—44no—
341Dic17—44no—
342Dic17—42no—
343Dic17—42no—
344Dic17—42no—
345C24×C4C2×D1711yescharacteristic
346C22×C4⋊C4—172no—
347C22×C22⋊C4—171no—
348C22×C22⋊C4—171no—
349C22×C22⋊C4—172no—
350C22×C22⋊C4C2×D1712yes—
351C22×C42—171no—
352C22×C42—171no—
353C22×C42—171no—
354C22×C42—171no—
355C22×C42C2×D1711yescharacteristic
356C22×C42C2×D1711yescharacteristic
357C4×C4⋊C4—344no—
358C4×C22⋊C4—24no—
359C4×C22⋊C4—178no—
360C4×C22⋊C4—344no—
361#376—344no—
362C2×#95—171no—
363C2×#95—171no—
364C2×#95—172no—
365C2×#95—172no—
366C2×#95—172no—
367C2×#95—172no—
368C2×#95—172no—
369C2×#95—172no—
370C2×#95—172no—
371C2×#95—172no—
372C2×#95C2×D1712yes—
373C2×C17—44no—
374C2×C17#228311yescharacteristic
375C2×C17C2×#38811yescharacteristic
376C2×C17#229011yescharacteristic
377C2×C17#229011yescharacteristic
378C2×C17#229011yescharacteristic
379C2×C17#229311yescharacteristic
380C2×C17#229811yescharacteristic
381C2×C17#239212yes—
382C2×C17#239512yes—
383C2×C17#239812yes—
384C2×C17#240112yes—
385C25C22×D1711yescharacteristic
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—171no—
393C23×C4—172no—
394C23×C4—171no—
395C23×C4—171no—
396C23×C4—171no—
397C23×C4—172no—
398C23×C4C4×D1714yes—
399C23×C4C22×D1711yescharacteristic
400C23×C4C22×D1712yes—
401C23×C4C22×D1712yes—
402C23×C4C22×D1711yescharacteristic
403C2×C4⋊C4—344no—
404C2×C4⋊C4—344no—
405C2×C4⋊C4—344no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—174no—
410C2×C22⋊C4—174no—
411C2×C22⋊C4—344no—
412C2×C22⋊C4—344no—
413C2×C22⋊C4—344no—
414C2×C22⋊C4—344no—
415C2×C22⋊C4—344no—
416C2×C42—22no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—22no—
420C2×C42—22no—
421C2×C42—174no—
422C2×C42—174no—
423C2×C42—174no—
424C2×C42—342no—
425C2×C42—342no—
426C2×C42—342no—
427C2×C42—342no—
428C2×C42—342no—
429C2×C42—342no—
430C2×C42—342no—
431C2×C42—342no—
432C2×C42—342no—
433C2×C42C22⋊D1714yes—
434#95—24no—
435#95—174no—
436#95—174no—
437#95—344no—
438#95—344no—
439#95—344no—
440#95—344no—
441#95—344no—
442#95—344no—
443#95—344no—
444#95—344no—
445C17#1384211yescharacteristic, complemented, Sylow
446C24—22no—
447C24—21no—
448C24—21no—
449C24—21no—
450C24—21no—
451C24—21no—
452C24—22no—
453C24—22no—
454C24—21no—
455C24C2×C4×D1712yes—
456C24#6325214yes—
457C24C23×D1711yescharacteristic
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—172no—
484C22×C4—174no—
485C22×C4—172no—
486C22×C4—172no—
487C22×C4—172no—
488C22×C4—174no—
489C22×C4—174no—
490C22×C4—341no—
491C22×C4—342no—
492C22×C4—341no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—341no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—342no—
502C22×C4—342no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—342no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—342no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—341no—
516C22×C4—342no—
517C22×C4C2×C4×D1712yes—
518C22×C4C2×C4×D1712yes—
519C22×C4#6325214yes—
520C22×C4#6325212yes—
521C22×C4D4×D1712yes—
522C22×C4#6325412yes—
523C22×C4#6325412yes—
524C22×C4#6325412yes—
525C22×C4C2×C22⋊D1714yes—
526C22×C4C2×C22⋊D1712yes—
527C4⋊C4—3416no—
528C22⋊C4—216no—
529C22⋊C4—348no—
530C22⋊C4—3416no—
531C22⋊C4—348no—
532C42—28no—
533C42—44no—
534C42—348no—
535C42—348no—
536C42—348no—
537C42—684no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23#1058755814yes—
561C23C22×C4×D1711yescharacteristic
562C23C2×#6325212yes—
563C23C2×#6325211yescharacteristic
564C23C2×#6325212yes—
565C23C2×D4×D1711yescharacteristic
566C23C2×#6325411yescharacteristic
567C23C2×#6325411yescharacteristic
568C23C2×#6325411yescharacteristic
569C23#41×D1712yes—
570C23#41×D1712yes—
571C23C22×C22⋊D1711yescharacteristic
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—416no—
594C2×C4—174no—
595C2×C4—342no—
596C2×C4—342no—
597C2×C4—344no—
598C2×C4—344no—
599C2×C4—342no—
600C2×C4—342no—
601C2×C4—342no—
602C2×C4—342no—
603C2×C4—342no—
604C2×C4—342no—
605C2×C4—342no—
606C2×C4—342no—
607C2×C4—342no—
608C2×C4—344no—
609C2×C4—344no—
610C2×C4—344no—
611C2×C4—344no—
612C2×C4—344no—
613C2×C4—344no—
614C2×C4—682no—
615C2×C4—682no—
616C2×C4—682no—
617C2×C4—682no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—682no—
622C2×C4—682no—
623C2×C4—682no—
624C2×C4—682no—
625C2×C4—682no—
626C2×C4—682no—
627C2×C4—682no—
628C2×C4#1058756314yes—
629C2×C4#1058756514yes—
630C2×C4#1058756714yes—
631C2×C4#1058756914yes—
632C2×C4C4×C22⋊D1714yes—
633C2×C4#1058762214yes—
634C2×C4#1058762514yes—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22#117×D1711yescharacteristic
647C22C2×#1058756311yescharacteristic
648C22C2×#1058755811yescharacteristic
649C22C2×#1058756711yescharacteristic
650C22C2×C4×C22⋊D1711yescharacteristic
651C22#1176327711yescharacteristic
652C22C4×#6325211yescharacteristic
653C22C2×#1058762511yescharacteristic
654C22C2×#1058756911yescharacteristic
655C22#1176331612yes—
656C22#1176331812yes—
657C22#1176332012yes—
658C22C2×#1058762211yescharacteristic
659C22C2×#1058756511yescharacteristic
660C22#1176333712yes—
661C22#1176333912yes—
662C22#1176333912yes—
663C22#1176333912yes—
664C22#1176333912yes—
665C22#1176333912yes—
666C22#1176334112yes—
667C22#1176334312yes—
668C22#1176335612yes—
669C4—42no—
670C4—44no—
671C4—44no—
672C4—344no—
673C4—682no—
674C4—682no—
675C4—682no—
676C4—684no—
677C4—684no—
678C4#1176270318yes—
679C2—44no—
680C2#1453117811yescharacteristic
681C2C2×#1176270311yescharacteristic
682C2#1453122111yescharacteristic
683C2#1453122311yescharacteristic
684C2#1453122711yescharacteristic
685C2#1453124711yescharacteristic
686C2#1453128011yescharacteristic
687C2#1453202612yes—
688C2#1453204512yes—
689C2#1453206812yes—
690C2#1453208412yes—
subgroupG/H
centerC23×C4C22×D17
commutatorC23×C17C23×C4
FrattiniC24C23×D17
FittingC2×C4×C22⋊C4×C17C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×C4×C22⋊C4D17
hypocenterC17#13842
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×#63225C2semidirect
4. C2×#11762655C2semidirect
5. #14527044C2semidirect
6. #14527042C2semidirect
9. C4×#10587502C2semidirect
10. C4×#10587502C2semidirect
22. C4×C22⋊Dic17C4semidirect
445. C17#13842semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#632252
C2×C4×C22⋊Dic172
C2×#117626552
#145270442
#145270422
#145270312
#145270272
C4×#105875022
C2×C4×C22⋊C4×C172
#1384217