Finite Group Databasebeta

← #16955571

Subgroups — #16955571

Order 4352.

2102 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587563C211yesmaximal, characteristic
2C2×#11762667C212yesmaximal
3C2×C4×#63225C211yesmaximal, characteristic, complemented
4C2×#11762660C211yesmaximal, characteristic, complemented
5#14527021C211yesmaximal, characteristic
6#14527021C212yesmaximal
7#14527019C211yesmaximal, characteristic, complemented
8#14527017C211yesmaximal, characteristic, complemented
9#14527015C212yesmaximal, complemented
10#14527012C212yesmaximal, complemented
11C17×#1437C211yesmaximal, characteristic
12C24×Dic17C2211yescharacteristic
13C2×#10587563—24no—
14C2×#10587563—24no—
15C2×#10587563C418yescomplemented
16C22×#63225C2212yes—
17C22×#63225C2212yes—
18C22×#63225C2211yescharacteristic
19C22×#63225C2211yescharacteristic
20C22×C22⋊Dic17C2211yescharacteristic
21C22×C22⋊Dic17C2212yes—
22C22×C4×Dic17C2211yescharacteristic
23C22×C4×Dic17C2212yes—
24C22×C4×Dic17C2211yescharacteristic
25C22×C4×Dic17C2211yescharacteristic
26#11762667C4116yescomplemented
27C4×#63225—24no—
28#11762660—24no—
29C2×#10587502C2212yes—
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C2×#10587502C2212yes—
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17C2212yes—
38C22×C42×C17C2211yescharacteristic
39C2×C17×#95C2211yescharacteristic
40C2×C17×#95C2212yes—
41C25×C17C2311yescharacteristic
42C23×C4×C17C2312yes—
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17—22no—
47C23×Dic17—22no—
48C23×Dic17—24no—
49C23×Dic17C2×C414yes—
50C23×Dic17C2×C418yes—
51C23×Dic17C2312yes—
52C23×Dic17C2312yes—
53C23×Dic17C2311yescharacteristic
54C23×Dic17C2311yescharacteristic
55C23×Dic17C2311yescharacteristic
56C23×Dic17C2311yescharacteristic
57C2×C22⋊C4×C17—22no—
58C2×C22⋊C4×C17—22no—
59C2×C22⋊C4×C17—42no—
60C2×C22⋊C4×C17—42no—
61C2×C22⋊C4×C17C2×C414yes—
62C2×C22⋊C4×C17C2×C418yes—
63C2×C42×C17—22no—
64C2×C42×C17—42no—
65C2×C22⋊Dic17—22no—
66C2×C22⋊Dic17—22no—
67C2×C22⋊Dic17—42no—
68C2×C22⋊Dic17—42no—
69C2×C22⋊Dic17C2×C414yes—
70C2×C22⋊Dic17C2×C418yes—
71C2×#63225—22no—
72C2×#63225—22no—
73C2×#63225—22no—
74C2×#63225—22no—
75C2×#63225—24no—
76C2×#63225—24no—
77C2×#63225—24no—
78C2×#63225—24no—
79C2×#63225C2×C414yes—
80C2×#63225C2×C414yes—
81C2×#63225C2×C418yes—
82C2×#63225C2×C418yes—
83C2×C4×Dic17—22no—
84C2×C4×Dic17—22no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17—24no—
87C2×C4×Dic17—42no—
88C2×C4×Dic17—42no—
89C2×C4×Dic17—42no—
90C2×C4×Dic17C2×C414yes—
91C2×C4×Dic17C2×C414yes—
92C2×C4×Dic17D414yes—
93C2×C4×Dic17D414yes—
94#10587563—232no—
95C17×#95—22no—
96C17×#95—42no—
97#10587502—22no—
98#10587502—24no—
99#10587502—42no—
100#10587502—42no—
101#10587502—42no—
102#10587502C2×C418yes—
103#10587502C2×C418yes—
104C24×C17—21no—
105C24×C17—21no—
106C24×C17—22no—
107C24×C17—22no—
108C24×C17—22no—
109C24×C17—22no—
110C24×C17—22no—
111C24×C17C22×C412yes—
112C24×C17C22×C414yes—
113C24×C17C2411yescharacteristic
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
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—22no—
131C22×C4×C17—22no—
132C22×C4×C17C22×C414yes—
133C22×C4×C17C22×C414yes—
134C22×C4×C17C22×C412yes—
135C22×C4×C17C22×C412yes—
136C22×C4×C17C2×D412yes—
137C22×C4×C17C2×D412yes—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—28no—
144C22×Dic17—28no—
145C22×Dic17—216no—
146C22×Dic17—216no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—22no—
152C22×Dic17—22no—
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—22no—
162C22×Dic17—22no—
163C22×Dic17C42116yescomplemented
164C22×Dic17C22×C414yes—
165C22×Dic17C22×C414yes—
166C22×Dic17C22×C412yes—
167C22×Dic17C22×C414yes—
168C22×Dic17C22×C412yes—
169C22×Dic17C22×C412yes—
170C22×Dic17C22×C412yes—
171C22×Dic17C22×C414yes—
172C22×Dic17C2×D412yes—
173C22×Dic17C2×D412yes—
174C22×Dic17C2×D412yes—
175C22×Dic17C2×D412yes—
176C22×Dic17#4114yes—
177C22⋊C4×C17—28no—
178C22⋊C4×C17—48no—
179C42×C17—44no—
180C22⋊Dic17—28no—
181C22⋊Dic17—48no—
182#63225—28no—
183#63225—28no—
184#63225—216no—
185#63225—216no—
186C4×Dic17—28no—
187C4×Dic17—48no—
188C4×Dic17—44no—
189C4×Dic17C22⋊C4116yescomplemented
190#13777—171nomaximal, Sylow
191C23×C17—22no—
192C23×C17—22no—
193C23×C17—24no—
194C23×C17—24no—
195C23×C17—24no—
196C23×C17—24no—
197C23×C17—24no—
198C23×C17—24no—
199C23×C17—24no—
200C23×C17—24no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17—42no—
209C23×C17—42no—
210C23×C17—42no—
211C23×C17C2×C4214yes—
212C23×C17C23×C411yescharacteristic
213C23×C17C23×C412yes—
214C23×C17C22×D411yescharacteristic
215C23×C17C22×D411yescharacteristic
216C23×C17C2×#4112yes—
217C23×C17H(5,𝔽2)12yes—
218C23×C17H(5,𝔽2)12yes—
219C23×C172−1+412yes—
220C23×C172−1+412yes—
221C2×C4×C17—24no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—24no—
225C2×C4×C17—22no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—42no—
234C2×C4×C17—41no—
235C2×C4×C17—42no—
236C2×C4×C17—41no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17C2×C22⋊C414yes—
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—24no—
253C2×Dic17—232no—
254C2×Dic17—22no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—24no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—41no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—41no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17C2×C4214yes—
274C2×Dic17C2×C4214yes—
275C2×Dic17C2×C22⋊C414yes—
276C2×Dic17C2×C22⋊C414yes—
277C2×Dic17#11714yes—
278C2×Dic17C4×D418yes—
279C2×Dic17C4×D418yes—
280C22×#117—171no—
281C2×#380—172no—
282C2×C4×C4⋊C4—171no—
283C2×C4×C4⋊C4—171no—
284#1437—172no—
285#1437—171no—
286#1437D1711yescharacteristic
287#1436—171no—
288#1436—172no—
289#1436—171no—
290C4⋊#95—172no—
291C22×C17—24no—
292C22×C17—28no—
293C22×C17—42no—
294C22×C17—42no—
295C22×C17—44no—
296C22×C17—44no—
297C22×C17—44no—
298C22×C17—44no—
299C22×C17—44no—
300C22×C17C22×C4211yescharacteristic
301C22×C17C22×C22⋊C411yescharacteristic
302C22×C17C2×#11711yescharacteristic
303C22×C17C2×C4×D412yes—
304C22×C17C2×C4×D412yes—
305C22×C17#51812yes—
306C22×C17#51912yes—
307C22×C17#52012yes—
308C22×C17#52012yes—
309C22×C17#52012yes—
310C22×C17#52012yes—
311C22×C17#53511yescharacteristic
312C22×C17#53511yescharacteristic
313C22×C17#53611yescharacteristic
314C22×C17#53611yescharacteristic
315C22×C17#53712yes—
316C22×C17#53712yes—
317C22×C17#54012yes—
318C22×C17#54112yes—
319C22×C17#54212yes—
320C22×C17#54212yes—
321C4×C17—24no—
322C4×C17—42no—
323C4×C17—44no—
324C4×C17—44no—
325Dic17—24no—
326Dic17—24no—
327Dic17—42no—
328Dic17—44no—
329Dic17—44no—
330Dic17C4×C22⋊C418yes—
331C24×C4—171no—
332C2×#117—178no—
333C2×#117—344no—
334C2×#117—344no—
335C22×C4⋊C4—171no—
336C22×C4⋊C4—171no—
337C22×C4⋊C4—172no—
338C22×C4⋊C4—172no—
339C22×C22⋊C4—171no—
340C22×C22⋊C4—172no—
341C22×C22⋊C4C2×D1711yescharacteristic
342C22×C22⋊C4C2×D1712yes—
343C22×C42—171no—
344C22×C42—171no—
345C22×C42—172no—
346C22×C42—171no—
347C22×C42C2×D1711yescharacteristic
348#380—1716no—
349C4×C4⋊C4—344no—
350C4×C4⋊C4—344no—
351C2×#95—172no—
352C2×#95—172no—
353C2×#95—172no—
354C2×#95—171no—
355C2×#95—172no—
356C2×#95—172no—
357C2×#95—172no—
358C2×#95C2×D1711yescharacteristic
359C2×#95C2×D1712yes—
360C2×C17—44no—
361C2×C17C2×C4×C22⋊C411yescharacteristic
362C2×C17#227412yes—
363C2×C17#231111yescharacteristic
364C2×C17#231111yescharacteristic
365C2×C17#231412yes—
366C2×C17#232312yes—
367C2×C17#232312yes—
368C2×C17#232412yes—
369C2×C17#232412yes—
370C25C22×D1711yescharacteristic
371C23×C4—171no—
372C23×C4—174no—
373C23×C4—178no—
374C23×C4—171no—
375C23×C4—171no—
376C23×C4—172no—
377C23×C4—172no—
378C23×C4—171no—
379C23×C4—342no—
380C23×C4—342no—
381C23×C4—344no—
382C23×C4C22×D1712yes—
383C23×C4C22×D1712yes—
384C23×C4C22×D1711yescharacteristic
385C23×C4C22×D1711yescharacteristic
386#117—3432no—
387C2×C4⋊C4—174no—
388C2×C4⋊C4—174no—
389C2×C4⋊C4—178no—
390C2×C4⋊C4—178no—
391C2×C4⋊C4—342no—
392C2×C4⋊C4—342no—
393C2×C4⋊C4—344no—
394C2×C4⋊C4—344no—
395C2×C4⋊C4—344no—
396C2×C4⋊C4—344no—
397C2×C4⋊C4—342no—
398C2×C4⋊C4—342no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—42no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—174no—
404C2×C22⋊C4—178no—
405C2×C22⋊C4—342no—
406C2×C22⋊C4—342no—
407C2×C22⋊C4—682no—
408C2×C22⋊C4—682no—
409C2×C22⋊C4C4×D1714yes—
410C2×C22⋊C4C4×D1718yes—
411C2×C42—22no—
412C2×C42—42no—
413C2×C42—174no—
414C2×C42—174no—
415C2×C42—174no—
416C2×C42—174no—
417C2×C42—342no—
418C2×C42—342no—
419C2×C42—344no—
420C2×C42—342no—
421C2×C42—682no—
422C2×C42—682no—
423C2×C42—682no—
424#95—22no—
425#95—42no—
426#95—178no—
427#95—178no—
428#95—344no—
429#95—342no—
430#95—682no—
431#95—682no—
432#95—682no—
433C17#1377711yescharacteristic, complemented, Sylow
434C24—21no—
435C24—21no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24C2×C4×D1712yes—
442C24C2×C4×D1714yes—
443C24C23×D1711yescharacteristic
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—172no—
463C22×C4—174no—
464C22×C4—172no—
465C22×C4—172no—
466C22×C4—174no—
467C22×C4—174no—
468C22×C4—172no—
469C22×C4—172no—
470C22×C4—172no—
471C22×C4—172no—
472C22×C4—1716no—
473C22×C4—174no—
474C22×C4—174no—
475C22×C4—172no—
476C22×C4—348no—
477C22×C4—348no—
478C22×C4—3416no—
479C22×C4—3416no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4C2×C4×D1714yes—
502C22×C4C2×C4×D1714yes—
503C22×C4C2×C4×D1712yes—
504C22×C4C2×C4×D1712yes—
505C22×C4D4×D1712yes—
506C22×C4D4×D1712yes—
507C4⋊C4—348no—
508C4⋊C4—3416no—
509C4⋊C4—3416no—
510C4⋊C4—348no—
511C22⋊C4—28no—
512C22⋊C4—48no—
513C22⋊C4—348no—
514C22⋊C4—688no—
515C42—44no—
516C42—1716no—
517C42—348no—
518C42—684no—
519C42—688no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C42×D1714yes—
541C23C22×C4×D1711yescharacteristic
542C23C22×C4×D1712yes—
543C23C2×D4×D1711yescharacteristic
544C23C2×D4×D1711yescharacteristic
545C23#1058769112yes—
546C23#1058769112yes—
547C23#41×D1712yes—
548C23#1058769712yes—
549C23#1058769712yes—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—174no—
574C2×C4—174no—
575C2×C4—174no—
576C2×C4—178no—
577C2×C4—178no—
578C2×C4—174no—
579C2×C4—174no—
580C2×C4—3432no—
581C2×C4—342no—
582C2×C4—342no—
583C2×C4—344no—
584C2×C4—344no—
585C2×C4—344no—
586C2×C4—342no—
587C2×C4—342no—
588C2×C4—344no—
589C2×C4—342no—
590C2×C4—344no—
591C2×C4—344no—
592C2×C4—344no—
593C2×C4—344no—
594C2×C4—344no—
595C2×C4—344no—
596C2×C4—344no—
597C2×C4—682no—
598C2×C4—682no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—682no—
602C2×C4—682no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4—682no—
606C2×C4—682no—
607C2×C4—682no—
608C2×C4C22⋊C4×D1714yes—
609C22—24no—
610C22—28no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22C2×C42×D1711yescharacteristic
619C22C2×C22⋊C4×D1711yescharacteristic
620C22#117×D1711yescharacteristic
621C22C4×D4×D1712yes—
622C22C4×D4×D1712yes—
623C22#1176328212yes—
624C22#1176328612yes—
625C22#1176328812yes—
626C22#1176328812yes—
627C22#1176328812yes—
628C22#1176328812yes—
629C22#1176337211yescharacteristic
630C22#1176337411yescharacteristic
631C22#1176337911yescharacteristic
632C22#1176338111yescharacteristic
633C22#1176338612yes—
634C22#1176338812yes—
635C22#1176340912yes—
636C22#1176341612yes—
637C22#1176342312yes—
638C22#1176342712yes—
639C4—24no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—178no—
644C4—344no—
645C4—344no—
646C4—682no—
647C4—684no—
648C4—684no—
649C2—44no—
650C2C4×C22⋊C4×D1711yescharacteristic
651C2#1453113912yes—
652C2#1453137111yescharacteristic
653C2#1453137411yescharacteristic
654C2#1453138812yes—
655C2#1453145912yes—
656C2#1453146312yes—
657C2#1453147012yes—
658C2#1453147412yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC17×#1437C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenter#1437D17
hypocenterC17#13777
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
3. C2×C4×#63225C2semidirect
4. C2×#11762660C2semidirect
7. #14527019C2semidirect
8. #14527017C2semidirect
9. #14527015C2semidirect
10. #14527012C2semidirect
15. C2×#10587563C4semidirect
26. #11762667C4semidirect
163. C22×Dic17C42semidirect
189. C4×Dic17C22⋊C4semidirect
433. C17#13777semidirect

Maximal subgroups

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

subgroupindex
C22×#105875632
C2×#117626672
C2×C4×#632252
C2×#117626602
#145270212
#145270192
#145270172
#145270152
#145270122
C17×#14372
#1377717