Finite Group Databasebeta

← #16955837

Subgroups — #16955837

Order 4352.

2298 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×#11762680C212yesmaximal
2C2×#11762660C211yesmaximal, characteristic, complemented
3C2×C4×#63227C211yesmaximal, characteristic
4C2×#11762656C212yesmaximal
5C2×#95×D17C211yesmaximal, characteristic
6#14527016C212yesmaximal, complemented
7#14527015C212yesmaximal, complemented
8#14527008C211yesmaximal, characteristic, complemented
9#14527005C211yesmaximal, characteristic
10C4×#10587502C211yesmaximal, characteristic, complemented
11C17×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C23×C4×D17C2212yes—
14C22×#63225C2212yes—
15C22×#63227C2212yes—
16C22×#63227C2211yescharacteristic
17C22×#63227C2211yescharacteristic
18C22×C4×Dic17C2212yes—
19C22×C4×Dic17C2212yes—
20C22×C4×Dic17C2211yescharacteristic
21#11762680C4116yescomplemented
22#11762660—24no—
23C4×#63227C418yescomplemented
24#11762656—28no—
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2211yescharacteristic
34#95×D17—24no—
35C22×C4⋊C4×C17C2212yes—
36C22×C42×C17C2211yescharacteristic
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2212yes—
40C24×D17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2312yes—
51C22×C4×D17—24no—
52C22×C4×D17—24no—
53C22×C4×D17—24no—
54C22×C4×D17—22no—
55C22×C4×D17—22no—
56C22×C4×D17—24no—
57C22×C4×D17—24no—
58C22×C4×D17—42no—
59C22×C4×D17—42no—
60C22×C4×D17—42no—
61C22×C4×D17C2×C414yes—
62C22×C4×D17C2×C418yes—
63C2×C4⋊C4×C17—42no—
64C2×C4⋊C4×C17—42no—
65C2×C4⋊C4×C17C2×C418yescomplemented
66C2×C42×C17—42no—
67C2×C42×C17C2×C414yescomplemented
68C2×#63227—24no—
69C2×#63227—24no—
70C2×#63227C2×C418yes—
71C2×#63227C2×C414yes—
72C2×#63227C2×C418yes—
73C2×#63227C2×C414yes—
74C2×#63227D414yes—
75C2×#63227Q814yes—
76C2×#63225—24no—
77C2×#63225—24no—
78C2×#63225C2×C418yescomplemented
79C2×C4×Dic17—22no—
80C2×C4×Dic17—24no—
81C2×C4×Dic17—22no—
82C2×C4×Dic17—24no—
83C2×C4×Dic17—24no—
84C2×C4×Dic17—24no—
85C2×C4×Dic17—42no—
86C2×C4×Dic17—42no—
87C2×C4×Dic17C2×C414yescomplemented
88C17×#95—22no—
89C17×#95—24no—
90C17×#95—22no—
91#10587502—22no—
92#10587502—24no—
93#10587502—24no—
94#10587502—42no—
95#10587502C2×C414yescomplemented
96#10587502C2×C414yescomplemented
97#10587502C2×C418yescomplemented
98#10587502C2×C418yescomplemented
99#10587502D414yes—
100C24×C17C2411yescharacteristic
101C23×D17—21no—
102C23×D17—21no—
103C23×D17—22no—
104C23×D17—21no—
105C23×D17—22no—
106C23×D17—21no—
107C23×D17C22×C412yes—
108C23×D17C22×C414yes—
109C23×D17C2×D412yes—
110C23×D17C2×Q812yes—
111C23×D17#4114yes—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
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×C17C22×C414yes—
132C22×C4×C17C22×C414yes—
133C22×C4×C17C22×C412yes—
134C22×C4×C17C22×C414yes—
135C22×C4×C17C22×C412yes—
136C22×C4×C17C22×C412yes—
137C22×C4×C17C2×D412yes—
138C22×C4×C17C2×D412yes—
139C22×C4×C17C2×Q812yes—
140C22×C4×C17#4112yes—
141C22×Dic17—22no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—22no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17C22×C412yes—
163C22×Dic17C22×C414yes—
164C22×Dic17C22×C412yes—
165C22×Dic17C22×C414yes—
166C22×Dic17C22×C412yes—
167C22×Dic17C22×C414yes—
168C22×Dic17C2×D412yes—
169C22×Dic17C2×D414yes—
170C22×Dic17C2×Q812yes—
171C22×Dic17#4114yes—
172C2×C4×D17—28no—
173C2×C4×D17—216no—
174C2×C4×D17—28no—
175C2×C4×D17—216no—
176C2×C4×D17—48no—
177C2×C4×D17—44no—
178C2×C4×D17—48no—
179C2×C4×D17—48no—
180C2×C4×D17—48no—
181C2×C4×D17—48no—
182C2×C4×D17—48no—
183C2×C4×D17—48no—
184C4⋊C4×C17—48no—
185C42×C17—44no—
186#63227—216no—
187#63227C42116yescomplemented
188#63227C4⋊C4116yescomplemented
189#63225—216no—
190C4×Dic17—28no—
191C4×Dic17—48no—
192C4×Dic17—48no—
193#13814—171nomaximal, Sylow
194C23×C17C23×C411yescharacteristic
195C23×C17C23×C412yes—
196C23×C17C22×D411yescharacteristic
197C23×C17C22×D411yescharacteristic
198C23×C17C22×Q811yescharacteristic
199C23×C17C2×#4111yescharacteristic
200C23×C17C2×#4112yes—
201C23×C17C2×#4112yes—
202C23×C17H(5,𝔽2)11yescharacteristic
203C23×C17H(5,𝔽2)12yes—
204C23×C172−1+411yescharacteristic
205C22×D17—24no—
206C22×D17—22no—
207C22×D17—22no—
208C22×D17—24no—
209C22×D17—22no—
210C22×D17—24no—
211C22×D17—22no—
212C22×D17—24no—
213C22×D17—22no—
214C22×D17—24no—
215C22×D17—24no—
216C22×D17—22no—
217C22×D17—22no—
218C22×D17—24no—
219C22×D17—24no—
220C22×D17—24no—
221C22×D17—24no—
222C22×D17—22no—
223C22×D17C2×C4214yes—
224C22×D17C2×C4⋊C414yes—
225C22×D17#11714yes—
226C22×D17C4×D418yes—
227C22×D17C4×Q818yes—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—24no—
234C2×C4×C17—22no—
235C2×C4×C17—24no—
236C2×C4×C17—24no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—24no—
244C2×C4×C17—24no—
245C2×C4×C17—24no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17C2×C4214yes—
255C2×C4×C17C2×C4⋊C414yes—
256C2×C4×C17C4×D414yes—
257C2×Dic17—24no—
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—24no—
261C2×Dic17—22no—
262C2×Dic17—24no—
263C2×Dic17—24no—
264C2×Dic17—22no—
265C2×Dic17—24no—
266C2×Dic17—22no—
267C2×Dic17—24no—
268C2×Dic17—24no—
269C2×Dic17—22no—
270C2×Dic17—22no—
271C2×Dic17—24no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—22no—
275C2×Dic17—24no—
276C2×Dic17—24no—
277C2×Dic17—22no—
278C2×Dic17—24no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17—42no—
282C2×Dic17—42no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17C2×C4214yes—
286C2×Dic17C2×C4⋊C414yes—
287C2×Dic17C4×D418yes—
288C4×D17—432no—
289C4×D17—416no—
290C2×#385—172no—
291C2×C4×C4⋊C4—171no—
292C2×C4×C22⋊C4—171no—
293C2×C4×C22⋊C4—172no—
294C22×#95—171no—
295#1436—172no—
296#1436—172no—
297C4⋊#95D1711yescharacteristic, complemented
298#1434—171no—
299C22⋊#95—171no—
300C4×#95—171no—
301C22×C17C22×C4211yescharacteristic
302C22×C17C22×C4⋊C411yescharacteristic
303C22×C17C2×#11711yescharacteristic
304C22×C17C2×C4×D412yes—
305C22×C17C2×C4×D412yes—
306C22×C17C2×C4×D411yescharacteristic
307C22×C17C2×C4×Q812yes—
308C22×C17C4×#4111yescharacteristic
309C22×C17C4×#4112yes—
310C22×C17#51811yescharacteristic
311C22×C17#51812yes—
312C22×C17#52011yescharacteristic
313C22×C17#52012yes—
314C22×C17D4211yescharacteristic
315C22×C17#54612yes—
316C22×C17#54612yes—
317C22×C17#54711yescharacteristic
318C22×C17D4×Q811yescharacteristic
319C22×C17#55112yes—
320C22×C17#55112yes—
321C22×C17#55212yes—
322C22×C17#55412yes—
323C22×C17#55411yescharacteristic
324C2×D17—24no—
325C2×D17—24no—
326C2×D17—24no—
327C2×D17—24no—
328C2×D17—28no—
329C2×D17—28no—
330C2×D17—28no—
331C2×D17—28no—
332C2×D17—24no—
333C2×D17—24no—
334C2×D17C4×C4⋊C418yescomplemented
335C4×C17—24no—
336C4×C17—24no—
337C4×C17—24no—
338C4×C17—44no—
339C4×C17—44no—
340Dic17—24no—
341Dic17—28no—
342Dic17—24no—
343Dic17—42no—
344Dic17—44no—
345C24×C4—171no—
346C24×C4—172no—
347C22×C4⋊C4—172no—
348C22×C4⋊C4C2×D1712yes—
349C22×C22⋊C4—171no—
350C22×C22⋊C4—171no—
351C22×C22⋊C4—172no—
352C22×C42—171no—
353C22×C42—172no—
354C22×C42—172no—
355C22×C42C2×D1711yescharacteristic
356#385—1716no—
357C4×C4⋊C4—344no—
358C4×C22⋊C4—178no—
359C4×C22⋊C4—348no—
360C2×#95—172no—
361C2×#95—171no—
362C2×#95—172no—
363C2×#95—171no—
364C2×#95—171no—
365C2×#95—171no—
366C2×#95—172no—
367C2×#95—172no—
368C2×#95—172no—
369C2×#95—344no—
370C2×#95C2×D1711yescharacteristic
371C2×#95C2×D1711yescharacteristic
372C2×#95C2×D1712yes—
373C2×C17C2×C4×C4⋊C411yescharacteristic
374C2×C17C42×D411yescharacteristic
375C2×C17#227511yescharacteristic
376C2×C17#234211yescharacteristic
377C2×C17#234311yescharacteristic
378C2×C17D4×C4⋊C411yescharacteristic
379C2×C17#234911yescharacteristic
380C2×C17#235812yes—
381C2×C17#235912yes—
382C2×C17#236012yes—
383C2×C17#236812yes—
384D17—28no—
385C25—171no—
386C23×C4—171no—
387C23×C4—171no—
388C23×C4—172no—
389C23×C4—172no—
390C23×C4—171no—
391C23×C4—174no—
392C23×C4—178no—
393C23×C4—344no—
394C23×C4—344no—
395C23×C4—344no—
396C23×C4—344no—
397C23×C4—344no—
398C23×C4—342no—
399C23×C4—342no—
400C23×C4—682no—
401C23×C4—682no—
402C23×C4—682no—
403C23×C4C22×D1712yes—
404C23×C4C22×D1712yes—
405C23×C4C22×D1711yescharacteristic
406C23×C4C22×D1711yescharacteristic
407C23×C4C22×D1711yescharacteristic
408C2×C4⋊C4—42no—
409C2×C4⋊C4—42no—
410C2×C4⋊C4—178no—
411C2×C4⋊C4—344no—
412C2×C4⋊C4—344no—
413C2×C4⋊C4C4×D1718yescomplemented
414C2×C22⋊C4—174no—
415C2×C22⋊C4—174no—
416C2×C22⋊C4—178no—
417C2×C22⋊C4—174no—
418C2×C22⋊C4—174no—
419C2×C22⋊C4—178no—
420C2×C22⋊C4—344no—
421C2×C22⋊C4—344no—
422C2×C42—42no—
423C2×C42—174no—
424C2×C42—342no—
425C2×C42—342no—
426C2×C42—344no—
427C2×C42—344no—
428C2×C42—344no—
429C2×C42—344no—
430C2×C42—682no—
431C2×C42—682no—
432C2×C42C4×D1714yescomplemented
433#95—22no—
434#95—24no—
435#95—22no—
436#95—174no—
437#95—174no—
438#95—178no—
439#95—178no—
440#95—174no—
441#95—344no—
442#95—344no—
443#95—342no—
444#95—682no—
445C17#1381411yescharacteristic, complemented, Sylow
446C24—172no—
447C24—174no—
448C24—172no—
449C24—172no—
450C24—174no—
451C24—341no—
452C24—342no—
453C24—341no—
454C24—341no—
455C24—341no—
456C24—342no—
457C24C23×D1711yescharacteristic
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—172no—
478C22×C4—172no—
479C22×C4—174no—
480C22×C4—174no—
481C22×C4—174no—
482C22×C4—172no—
483C22×C4—172no—
484C22×C4—174no—
485C22×C4—174no—
486C22×C4—172no—
487C22×C4—3416no—
488C22×C4—348no—
489C22×C4—3416no—
490C22×C4—341no—
491C22×C4—342no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—341no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—341no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—348no—
512C22×C4—684no—
513C22×C4—688no—
514C22×C4—688no—
515C22×C4—688no—
516C22×C4—688no—
517C22×C4—688no—
518C22×C4—688no—
519C22×C4—688no—
520C22×C4C2×C4×D1714yes—
521C22×C4C2×C4×D1714yes—
522C22×C4C2×C4×D1712yes—
523C22×C4C2×C4×D1714yes—
524C22×C4C2×C4×D1712yes—
525C22×C4C2×C4×D1712yes—
526C22×C4D4×D1712yes—
527C22×C4D4×D1712yes—
528C22×C4Q8×D1712yes—
529C22×C4#6325612yes—
530C4⋊C4—48no—
531C4⋊C4—3416no—
532C22⋊C4—1716no—
533C22⋊C4—1716no—
534C22⋊C4—3416no—
535C42—44no—
536C42—348no—
537C42—688no—
538C42—688no—
539C23—174no—
540C23—174no—
541C23—174no—
542C23—178no—
543C23—178no—
544C23—342no—
545C23—342no—
546C23—344no—
547C23—344no—
548C23—344no—
549C23—344no—
550C23—342no—
551C23—344no—
552C23—342no—
553C23—342no—
554C23—344no—
555C23—344no—
556C23—342no—
557C23—342no—
558C23—344no—
559C23—342no—
560C23—344no—
561C23—344no—
562C23C22×C4×D1711yescharacteristic
563C23C22×C4×D1712yes—
564C23C2×D4×D1711yescharacteristic
565C23C2×D4×D1711yescharacteristic
566C23#1058769112yes—
567C23#1058769111yescharacteristic
568C23C2×Q8×D1711yescharacteristic
569C23C2×#6325611yescharacteristic
570C23#1058769411yescharacteristic
571C23#41×D1712yes—
572C23#41×D1712yes—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—174no—
600C2×C4—174no—
601C2×C4—178no—
602C2×C4—342no—
603C2×C4—342no—
604C2×C4—344no—
605C2×C4—342no—
606C2×C4—344no—
607C2×C4—344no—
608C2×C4—342no—
609C2×C4—342no—
610C2×C4—342no—
611C2×C4—344no—
612C2×C4—344no—
613C2×C4—342no—
614C2×C4—342no—
615C2×C4—344no—
616C2×C4—344no—
617C2×C4—344no—
618C2×C4—342no—
619C2×C4—344no—
620C2×C4—342no—
621C2×C4—344no—
622C2×C4—342no—
623C2×C4—344no—
624C2×C4—6816no—
625C2×C4—6832no—
626C2×C4—682no—
627C2×C4—682no—
628C2×C4—682no—
629C2×C4—682no—
630C2×C4—682no—
631C2×C4—682no—
632C2×C4C42×D1714yes—
633C2×C4C4⋊C4×D1714yes—
634C2×C4#1058757914yes—
635C22—178no—
636C22—348no—
637C22—344no—
638C22—344no—
639C22—344no—
640C22—344no—
641C22—344no—
642C22—344no—
643C22—348no—
644C22—348no—
645C22—348no—
646C22C2×C42×D1711yescharacteristic
647C22C2×C4⋊C4×D1711yescharacteristic
648C22#117×D1711yescharacteristic
649C22C4×D4×D1712yes—
650C22C4×D4×D1712yes—
651C22C2×#1058757911yescharacteristic
652C22C4×Q8×D1712yes—
653C22#1176327711yescharacteristic
654C22C4×#6325612yes—
655C22#1176328212yes—
656C22#1176328211yescharacteristic
657C22#1176329011yescharacteristic
658C22#1176329012yes—
659C22#1176344411yescharacteristic
660C22#1176344812yes—
661C22#1176344812yes—
662C22#1176346211yescharacteristic
663C22#1176347411yescharacteristic
664C22#1176348612yes—
665C22#1176348912yes—
666C22#1176349112yes—
667C22#1176351312yes—
668C22#1176352011yescharacteristic
669C4—24no—
670C4—24no—
671C4—24no—
672C4—44no—
673C4—44no—
674C4—344no—
675C4—344no—
676C4—348no—
677C4—682no—
678C4—684no—
679C2—348no—
680C2C4×C4⋊C4×D1711yescharacteristic
681C2C4×#1058757911yescharacteristic
682C2#1453114211yescharacteristic
683C2#1453163011yescharacteristic
684C2#1453163711yescharacteristic
685C2#1453168411yescharacteristic
686C2#1453168811yescharacteristic
687C2#1453176512yes—
688C2#1453177112yes—
689C2#1453178612yes—
690C2#1453186312yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC17×C4⋊#95C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC4⋊#95D17
hypocenterC17#13814
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#11762660C2semidirect
6. #14527016C2semidirect
7. #14527015C2semidirect
8. #14527008C2semidirect
10. C4×#10587502C2semidirect
11. C17×C4⋊#95C2semidirect
21. #11762680C4semidirect
23. C4×#63227C4semidirect
65. C2×C4⋊C4×C17C2×C4semidirect
67. C2×C42×C17C2×C4semidirect
78. C2×#63225C2×C4semidirect
87. C2×C4×Dic17C2×C4semidirect
95. #10587502C2×C4semidirect
96. #10587502C2×C4semidirect
97. #10587502C2×C4semidirect
98. #10587502C2×C4semidirect
187. #63227C42semidirect
188. #63227C4⋊C4semidirect
297. C4⋊#95D17semidirect
334. C2×D17C4×C4⋊C4semidirect
413. C2×C4⋊C4C4×D17semidirect
432. C2×C42C4×D17semidirect
445. C17#13814semidirect

Maximal subgroups

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

subgroupindex
C2×#117626802
C2×#117626602
C2×C4×#632272
C2×#117626562
C2×#95×D172
#145270162
#145270152
#145270082
#145270052
C4×#105875022
C17×C4⋊#952
#1381417