Finite Group Databasebeta

← #16960965

Subgroups — #16960965

Order 4352.

1978 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587621C211yesmaximal, characteristic
2C2×#11762667C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×Dic17C211yesmaximal, characteristic, complemented
4#14527035C212yesmaximal, complemented
5#14527031C211yesmaximal, characteristic, complemented
6#14527031C211yesmaximal, characteristic, complemented
7#14527027C212yesmaximal, complemented
8#14527025C211yesmaximal, characteristic
9#14527021C212yesmaximal, complemented
10#14527015C212yesmaximal, complemented
11C17×#1438C211yesmaximal, characteristic
12C24×Dic17C2211yescharacteristic
13C2×#10587621—24no—
14C2×#10587621—24no—
15C2×#10587621C418yescomplemented
16C22×#63225C2211yescharacteristic
17C22×#63225C2211yescharacteristic
18C22×#63225C2212yes—
19C22×C22⋊Dic17C2211yescharacteristic
20C22×C22⋊Dic17C2211yescharacteristic
21C22×C22⋊Dic17C2211yescharacteristic
22C22×C22⋊Dic17C2211yescharacteristic
23C22×C22⋊Dic17C2212yes—
24C22×C22⋊Dic17C2211yescharacteristic
25C22×C4×Dic17C2212yes—
26#11762667—24no—
27C22⋊C4×Dic17—24no—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C23×D4×C17C2211yescharacteristic
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17C2212yes—
38C22×C22⋊C4×C17C2211yescharacteristic
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C25×C17C2311yescharacteristic
42C22×D4×C17—22no—
43C22×D4×C17—22no—
44C22×D4×C17—42no—
45C22×D4×C17—42no—
46C22×D4×C17C2×C414yes—
47C23×C4×C17C2312yes—
48C23×C4×C17C2311yescharacteristic
49C23×C4×C17C2312yes—
50C23×Dic17—42no—
51C23×Dic17—42no—
52C23×Dic17C2×C414yes—
53C23×Dic17D414yes—
54C23×Dic17D414yes—
55C23×Dic17C2312yes—
56C23×Dic17C2311yescharacteristic
57C23×Dic17C2311yescharacteristic
58C23×Dic17C2312yes—
59C23×Dic17C2311yescharacteristic
60C23×Dic17C2311yescharacteristic
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17—24no—
63C2×C22⋊C4×C17—22no—
64C2×C22⋊C4×C17—24no—
65C2×C22⋊C4×C17—42no—
66C2×C22⋊C4×C17—42no—
67C2×C22⋊C4×C17—42no—
68C2×C22⋊Dic17—22no—
69C2×C22⋊Dic17—22no—
70C2×C22⋊Dic17—22no—
71C2×C22⋊Dic17—22no—
72C2×C22⋊Dic17—22no—
73C2×C22⋊Dic17—22no—
74C2×C22⋊Dic17—22no—
75C2×C22⋊Dic17—24no—
76C2×C22⋊Dic17—22no—
77C2×C22⋊Dic17—24no—
78C2×C22⋊Dic17—42no—
79C2×C22⋊Dic17—42no—
80C2×C22⋊Dic17—42no—
81C2×C22⋊Dic17C2×C414yes—
82C2×C22⋊Dic17C2×C414yes—
83C2×C22⋊Dic17C2×C414yes—
84#10587621—232no—
85C2×#63225—22no—
86C2×#63225—22no—
87C2×#63225—22no—
88C2×#63225—22no—
89C2×#63225—24no—
90C2×#63225—42no—
91C2×#63225—42no—
92C2×#63225C2×C414yes—
93C2×#63225C2×C414yes—
94C2×C4×Dic17—24no—
95C2×C4×Dic17—42no—
96C2×C4×Dic17—42no—
97C17×#95—42no—
98#10587502—24no—
99#10587502—24no—
100#10587502—24no—
101#10587502—24no—
102#10587502—42no—
103#10587502—42no—
104#10587502—42no—
105C24×C17—22no—
106C24×C17—22no—
107C24×C17—22no—
108C24×C17—22no—
109C24×C17—22no—
110C24×C17—22no—
111C24×C17—21no—
112C24×C17—21no—
113C24×C17—22no—
114C24×C17—22no—
115C24×C17—22no—
116C24×C17—22no—
117C24×C17—22no—
118C24×C17—22no—
119C24×C17C22×C412yes—
120C24×C17C22×C412yes—
121C24×C17C2×D412yes—
122C24×C17C2×D412yes—
123C24×C17C2411yescharacteristic
124C2×D4×C17—28no—
125C2×D4×C17—44no—
126C2×D4×C17—44no—
127C2×D4×C17—48no—
128C2×D4×C17—48no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—22no—
143C22×C4×C17—22no—
144C22×C4×C17—22no—
145C22×C4×C17—22no—
146C22×C4×C17C22×C412yes—
147C22×C4×C17#4114yes—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
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—44no—
174C22×Dic17—44no—
175C22×Dic17—48no—
176C22×Dic17—48no—
177C22×Dic17C22⋊C4116yescomplemented
178C22×Dic17C22×C412yes—
179C22×Dic17C22×C412yes—
180C22×Dic17C22×C412yes—
181C22×Dic17C22×C412yes—
182C22×Dic17C2×D412yes—
183C22×Dic17C2×D412yes—
184C22×Dic17C2×D412yes—
185C22×Dic17C2×D412yes—
186C22×Dic17#4114yes—
187C22⋊C4×C17—28no—
188C22⋊C4×C17—48no—
189C22⋊C4×C17—44no—
190C22⋊Dic17—28no—
191C22⋊Dic17—28no—
192C22⋊Dic17—28no—
193C22⋊Dic17—28no—
194C22⋊Dic17—48no—
195C22⋊Dic17—44no—
196#63225—28no—
197#63225—28no—
198#63225—48no—
199C4×Dic17—48no—
200#14287—171nomaximal, Sylow
201C23×C17—24no—
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—24no—
205C23×C17—22no—
206C23×C17—22no—
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—22no—
211C23×C17—22no—
212C23×C17—24no—
213C23×C17—22no—
214C23×C17—24no—
215C23×C17—22no—
216C23×C17—22no—
217C23×C17—24no—
218C23×C17—42no—
219C23×C17—42no—
220C23×C17—42no—
221C23×C17—42no—
222C23×C17—42no—
223C23×C17—42no—
224C23×C17—42no—
225C23×C17—42no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—42no—
229C23×C17—42no—
230C23×C17—42no—
231C23×C17—41no—
232C23×C17—41no—
233C23×C17—42no—
234C23×C17—42no—
235C23×C17—42no—
236C23×C17—42no—
237C23×C17—42no—
238C23×C17—42no—
239C23×C17—42no—
240C23×C17—42no—
241C23×C17—42no—
242C23×C17C2×C22⋊C414yes—
243C23×C17C23×C411yescharacteristic
244C23×C17C22×D411yescharacteristic
245C23×C17C22×D411yescharacteristic
246C23×C17C2×#4112yes—
247C23×C17C2×#4112yes—
248C23×C17H(5,𝔽2)12yes—
249C23×C17H(5,𝔽2)12yes—
250C23×C17H(5,𝔽2)12yes—
251C23×C172−1+412yes—
252D4×C17—416no—
253C2×C4×C17—22no—
254C2×C4×C17—22no—
255C2×C4×C17—22no—
256C2×C4×C17—24no—
257C2×C4×C17—24no—
258C2×C4×C17—24no—
259C2×C4×C17—42no—
260C2×C4×C17—42no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
263C2×C4×C17—42no—
264C2×C4×C17—42no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17—42no—
268C2×C4×C17—42no—
269C2×C4×C17—42no—
270C2×C4×C17—42no—
271C2×C4×C17—42no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—22no—
275C2×Dic17—22no—
276C2×Dic17—24no—
277C2×Dic17—24no—
278C2×Dic17—24no—
279C2×Dic17—24no—
280C2×Dic17—22no—
281C2×Dic17—22no—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—24no—
285C2×Dic17—24no—
286C2×Dic17—24no—
287C2×Dic17—416no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17—41no—
293C2×Dic17—41no—
294C2×Dic17—42no—
295C2×Dic17—42no—
296C2×Dic17—42no—
297C2×Dic17—42no—
298C2×Dic17—42no—
299C2×Dic17—42no—
300C2×Dic17—42no—
301C2×Dic17—42no—
302C2×Dic17—42no—
303C2×Dic17—42no—
304C2×Dic17—42no—
305C2×Dic17C2×C22⋊C414yes—
306C2×Dic17C2×C22⋊C414yes—
307C22×#123—171no—
308C2×#380—171no—
309C2×C4×C22⋊C4—171no—
310#1441—172no—
311#1440—171no—
312#1440—171no—
313#1439—172no—
314#1438—171no—
315#1438D1711yescharacteristic
316#1437—172no—
317#1436—172no—
318C22×C17—24no—
319C22×C17—24no—
320C22×C17—24no—
321C22×C17—44no—
322C22×C17—44no—
323C22×C17—44no—
324C22×C17—44no—
325C22×C17—44no—
326C22×C17—44no—
327C22×C17—42no—
328C22×C17—42no—
329C22×C17—44no—
330C22×C17—44no—
331C22×C17—42no—
332C22×C17—82no—
333C22×C17—82no—
334C22×C17—82no—
335C22×C17—82no—
336C22×C17C22×C22⋊C411yescharacteristic
337C22×C17C4×#4112yes—
338C22×C17#51811yescharacteristic
339C22×C17#51811yescharacteristic
340C22×C17#52012yes—
341C22×C17#52511yescharacteristic
342C22×C17#52511yescharacteristic
343C22×C17#53212yes—
344C22×C17#53412yes—
345C22×C17#53412yes—
346C22×C17#53511yescharacteristic
347C22×C17#53611yescharacteristic
348C22×C17#53712yes—
349C22×C17#53812yes—
350C22×C17#53812yes—
351C22×C17#53912yes—
352C22×C17#53912yes—
353C22×C17#54012yes—
354C22×C17#54012yes—
355C22×C17#54212yes—
356C22×C17#54212yes—
357C4×C17—42no—
358C4×C17—44no—
359C4×C17—82no—
360Dic17—24no—
361Dic17—24no—
362Dic17—42no—
363Dic17—44no—
364Dic17—42no—
365Dic17—82no—
366C23×D4C2×D1711yescharacteristic
367C24×C4—171no—
368C2×#123—178no—
369C2×#123—344no—
370C2×#123—344no—
371C22×C4⋊C4—171no—
372C22×C4⋊C4—171no—
373C22×C4⋊C4—172no—
374C22×C22⋊C4—171no—
375C22×C22⋊C4—171no—
376C22×C22⋊C4—171no—
377C22×C22⋊C4—171no—
378C22×C22⋊C4—172no—
379C22×C22⋊C4—171no—
380C22×C22⋊C4C2×D1711yescharacteristic
381C22×C22⋊C4C2×D1712yes—
382C22×C22⋊C4C2×D1711yescharacteristic
383C22×C42—172no—
384#380—344no—
385C4×C22⋊C4—344no—
386C2×#95—172no—
387C2×#95—172no—
388C2×#95—172no—
389C2×#95—172no—
390C2×#95—172no—
391C2×#95—172no—
392C2×#95—172no—
393C2×#95C2×D1712yes—
394C2×C17—44no—
395C2×C17—82no—
396C2×C17#229711yescharacteristic
397C2×C17#230811yescharacteristic
398C2×C17#231111yescharacteristic
399C2×C17#233512yes—
400C2×C17#233512yes—
401C2×C17#263012yes—
402C2×C17#263012yes—
403C2×C17#266112yes—
404C2×C17#266312yes—
405C25C22×D1711yescharacteristic
406C25C22×D1711yescharacteristic
407C22×D4—22no—
408C22×D4—22no—
409C22×D4—42no—
410C22×D4—42no—
411C22×D4C4×D1714yes—
412C23×C4—171no—
413C23×C4—171no—
414C23×C4—174no—
415C23×C4—174no—
416C23×C4—174no—
417C23×C4—171no—
418C23×C4—171no—
419C23×C4—172no—
420C23×C4—172no—
421C23×C4—682no—
422C23×C4—682no—
423C23×C4C22×D1712yes—
424C23×C4C22×D1711yescharacteristic
425C23×C4C22×D1712yes—
426#123—3432no—
427C2×C4⋊C4—174no—
428C2×C4⋊C4—174no—
429C2×C4⋊C4—342no—
430C2×C4⋊C4—342no—
431C2×C4⋊C4—342no—
432C2×C4⋊C4—342no—
433C2×C4⋊C4—344no—
434C2×C4⋊C4—682no—
435C2×C4⋊C4—682no—
436C2×C22⋊C4—22no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4—22no—
439C2×C22⋊C4—24no—
440C2×C22⋊C4—42no—
441C2×C22⋊C4—42no—
442C2×C22⋊C4—42no—
443C2×C22⋊C4—174no—
444C2×C22⋊C4—174no—
445C2×C22⋊C4—174no—
446C2×C22⋊C4—342no—
447C2×C22⋊C4—342no—
448C2×C22⋊C4—342no—
449C2×C22⋊C4—342no—
450C2×C22⋊C4—344no—
451C2×C22⋊C4—342no—
452C2×C22⋊C4—342no—
453C2×C22⋊C4—342no—
454C2×C22⋊C4—344no—
455C2×C22⋊C4—342no—
456C2×C22⋊C4—682no—
457C2×C22⋊C4—682no—
458C2×C22⋊C4—682no—
459C2×C42—344no—
460C2×C42—682no—
461C2×C42—682no—
462#95—42no—
463#95—344no—
464#95—344no—
465#95—344no—
466#95—344no—
467#95—682no—
468#95—682no—
469#95—682no—
470C17#1428711yescharacteristic, complemented, Sylow
471C24—22no—
472C24—22no—
473C24—22no—
474C24—22no—
475C24—22no—
476C24—22no—
477C24—21no—
478C24—21no—
479C24—22no—
480C24—22no—
481C24—22no—
482C24—22no—
483C24—22no—
484C24—22no—
485C24C2×C4×D1712yes—
486C24C2×C4×D1712yes—
487C24D4×D1712yes—
488C24D4×D1712yes—
489C24C23×D1711yescharacteristic
490C2×D4—28no—
491C2×D4—44no—
492C2×D4—44no—
493C2×D4—48no—
494C2×D4—48no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—172no—
513C22×C4—172no—
514C22×C4—172no—
515C22×C4—172no—
516C22×C4—172no—
517C22×C4—172no—
518C22×C4—1716no—
519C22×C4—172no—
520C22×C4—172no—
521C22×C4—174no—
522C22×C4—342no—
523C22×C4—342no—
524C22×C4—342no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—341no—
528C22×C4—341no—
529C22×C4—341no—
530C22×C4—341no—
531C22×C4—342no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—342no—
542C22×C4—342no—
543C22×C4—342no—
544C22×C4—342no—
545C22×C4—342no—
546C22×C4—342no—
547C22×C4—684no—
548C22×C4—684no—
549C22×C4—688no—
550C22×C4—688no—
551C22×C4C2×C4×D1712yes—
552C22×C4#6325414yes—
553C4⋊C4—348no—
554C4⋊C4—348no—
555C4⋊C4—688no—
556C22⋊C4—28no—
557C22⋊C4—48no—
558C22⋊C4—44no—
559C22⋊C4—348no—
560C22⋊C4—348no—
561C22⋊C4—348no—
562C22⋊C4—348no—
563C22⋊C4—688no—
564C22⋊C4—684no—
565C42—688no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—22no—
579C23—24no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—41no—
592C23—41no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—41no—
597C23—41no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23C22⋊C4×D1714yes—
608C23C22×C4×D1711yescharacteristic
609C23C2×D4×D1711yescharacteristic
610C23C2×D4×D1711yescharacteristic
611C23C2×#6325412yes—
612C23#1058769112yes—
613C23#1058769112yes—
614C23#1058769112yes—
615C23#41×D1712yes—
616C23#1058769712yes—
617D4—416no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—174no—
638C2×C4—174no—
639C2×C4—342no—
640C2×C4—342no—
641C2×C4—342no—
642C2×C4—342no—
643C2×C4—344no—
644C2×C4—344no—
645C2×C4—344no—
646C2×C4—344no—
647C2×C4—342no—
648C2×C4—342no—
649C2×C4—344no—
650C2×C4—344no—
651C2×C4—344no—
652C2×C4—344no—
653C2×C4—344no—
654C2×C4—6816no—
655C2×C4—681no—
656C2×C4—681no—
657C2×C4—682no—
658C2×C4—682no—
659C2×C4—681no—
660C2×C4—681no—
661C2×C4—682no—
662C2×C4—682no—
663C2×C4—682no—
664C2×C4—682no—
665C2×C4—682no—
666C2×C4—682no—
667C2×C4—682no—
668C2×C4—682no—
669C2×C4—682no—
670C2×C4—682no—
671C2×C4—682no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—44no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22C2×C22⋊C4×D1711yescharacteristic
691C22C4×#6325412yes—
692C22#1176328211yescharacteristic
693C22#1176328211yescharacteristic
694C22#1176328812yes—
695C22#1176331511yescharacteristic
696C22#1176331711yescharacteristic
697C22#1176335412yes—
698C22#1176336912yes—
699C22#1176336912yes—
700C22#1176337411yescharacteristic
701C22#1176338111yescharacteristic
702C22#1176338612yes—
703C22#1176339112yes—
704C22#1176339412yes—
705C22#1176340212yes—
706C22#1176340412yes—
707C22#1176340912yes—
708C22#1176341312yes—
709C22#1176342312yes—
710C22#1176342612yes—
711C4—42no—
712C4—44no—
713C4—82no—
714C4—344no—
715C4—344no—
716C4—682no—
717C4—682no—
718C4—684no—
719C4—1362no—
720C2—44no—
721C2—82no—
722C2#1453126911yescharacteristic
723C2#1453136011yescharacteristic
724C2#1453137111yescharacteristic
725C2#1453157112yes—
726C2#1453157512yes—
727C2#1453462012yes—
728C2#1453462412yes—
729C2#1453490912yes—
730C2#1453493112yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1438C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenter#1438D17
hypocenterC17#14287
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#11762667C2semidirect
3. C2×C22⋊C4×Dic17C2semidirect
4. #14527035C2semidirect
5. #14527031C2semidirect
6. #14527031C2semidirect
7. #14527027C2semidirect
9. #14527021C2semidirect
10. #14527015C2semidirect
15. C2×#10587621C4semidirect
177. C22×Dic17C22⋊C4semidirect
470. C17#14287semidirect

Maximal subgroups

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

subgroupindex
C22×#105876212
C2×#117626672
C2×C22⋊C4×Dic172
#145270352
#145270312
#145270272
#145270252
#145270212
#145270152
C17×#14382
#1428717