Finite Group Databasebeta

← #16955566

Subgroups — #16955566

Order 4352.

2318 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D17C211yesmaximal, characteristic
2C2×#11762671C212yesmaximal
3C2×C4×#63227C211yesmaximal, characteristic
4C2×#11762656C211yesmaximal, characteristic
5#14527019C211yesmaximal, characteristic, complemented
6#14527018C212yesmaximal, complemented
7#14527017C211yesmaximal, characteristic, complemented
8#14527016C211yesmaximal, characteristic, complemented
9#14527015C212yesmaximal, complemented
10#14527005C212yesmaximal
11C17×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C23×C4×D17C2211yescharacteristic
14C23×C4×D17C2211yescharacteristic
15C22×C4⋊Dic17C2211yescharacteristic
16C22×#63225C2211yescharacteristic
17C22×#63225C2211yescharacteristic
18C2×C4⋊C4×D17—24no—
19C2×C4⋊C4×D17—24no—
20C2×C4⋊C4×D17C418yescomplemented
21C22×#63227C2212yes—
22C22×#63227C2212yes—
23C22×C4×Dic17C2212yes—
24C22×C4×Dic17C2211yescharacteristic
25#11762671C4116yescomplemented
26C4×#63227—24no—
27#11762656—24no—
28C2×#10587502C2212yes—
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C2×#10587502C2212yes—
36C22×C4⋊C4×C17C2211yescharacteristic
37C22×C42×C17C2211yescharacteristic
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2212yes—
40C2×C17×#95C2212yes—
41C24×D17C2311yescharacteristic
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2312yes—
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C23×Dic17C2311yescharacteristic
52C22×C4×D17—22no—
53C22×C4×D17—22no—
54C22×C4×D17—24no—
55C22×C4×D17—22no—
56C22×C4×D17—22no—
57C22×C4×D17—42no—
58C22×C4×D17—42no—
59C22×C4×D17—42no—
60C22×C4×D17—42no—
61C22×C4×D17C2×C414yes—
62C22×C4×D17C2×C418yes—
63C22×C4×D17C2×C414yes—
64C22×C4×D17C2×C414yes—
65C22×C4×D17D414yes—
66C22×C4×D17D414yes—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17—22no—
69C2×C4⋊C4×C17C2×C414yes—
70C2×C42×C17—22no—
71C2×C42×C17—42no—
72C2×#63227—24no—
73C2×#63227—24no—
74C2×#63227—24no—
75C2×#63227—24no—
76C2×#63227C2×C418yes—
77C2×#63227C2×C418yes—
78C2×C4⋊Dic17—22no—
79C2×C4⋊Dic17—22no—
80C2×C4⋊Dic17C2×C414yes—
81C2×#63225—22no—
82C2×#63225—22no—
83C2×#63225—22no—
84C2×#63225—22no—
85C2×#63225C2×C414yes—
86C2×#63225C2×C414yes—
87C2×C4×Dic17—22no—
88C2×C4×Dic17—24no—
89C2×C4×Dic17—42no—
90C2×C4×Dic17—42no—
91C2×C4×Dic17—42no—
92C4⋊C4×D17—232no—
93C17×#95—22no—
94C17×#95—42no—
95C17×#95C2×C418yescomplemented
96#10587502—22no—
97#10587502—24no—
98#10587502—42no—
99#10587502—42no—
100#10587502—42no—
101#10587502C2×C418yescomplemented
102#10587502C2×C418yescomplemented
103#10587502C2×C418yescomplemented
104C24×C17C2411yescharacteristic
105C23×D17—22no—
106C23×D17—22no—
107C23×D17—22no—
108C23×D17—22no—
109C23×D17C22×C412yes—
110C23×D17C22×C414yes—
111C23×D17C2×D412yes—
112C23×D17C2×D412yes—
113C23×D17#4114yes—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
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×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17C22×C412yes—
136C22×C4×C17C22×C414yes—
137C22×C4×C17C22×C414yes—
138C22×C4×C17C22×C414yes—
139C22×C4×C17C22×C412yes—
140C22×C4×C17C22×C412yes—
141C22×C4×C17C2×D412yes—
142C22×C4×C17C2×D412yes—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—22no—
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×Dic17—22no—
164C22×Dic17C22×C412yes—
165C22×Dic17C22×C414yes—
166C22×Dic17C22×C414yes—
167C22×Dic17C22×C414yes—
168C22×Dic17C22×C412yes—
169C22×Dic17C22×C412yes—
170C22×Dic17C2×D412yes—
171C22×Dic17C2×D412yes—
172C2×C4×D17—28no—
173C2×C4×D17—28no—
174C2×C4×D17—216no—
175C2×C4×D17—216no—
176C2×C4×D17—28no—
177C2×C4×D17—48no—
178C2×C4×D17—48no—
179C2×C4×D17—44no—
180C2×C4×D17—44no—
181C2×C4×D17—48no—
182C2×C4×D17—48no—
183C2×C4×D17—44no—
184C2×C4×D17—44no—
185C2×C4×D17C42116yescomplemented
186C2×C4×D17C22⋊C4116yes—
187C4⋊C4×C17—28no—
188C42×C17—44no—
189#63227—216no—
190#63227—216no—
191C4⋊Dic17—28no—
192#63225—28no—
193#63225—28no—
194C4×Dic17—48no—
195C4×Dic17—44no—
196#13776—171nomaximal, Sylow
197C23×C17C23×C411yescharacteristic
198C23×C17C23×C412yes—
199C23×C17C22×D411yescharacteristic
200C23×C17C22×D411yescharacteristic
201C23×C17C2×#4112yes—
202C23×C17H(5,𝔽2)12yes—
203C23×C17H(5,𝔽2)12yes—
204C23×C172−1+412yes—
205C23×C172−1+412yes—
206C22×D17—24no—
207C22×D17—24no—
208C22×D17—24no—
209C22×D17—24no—
210C22×D17—24no—
211C22×D17—24no—
212C22×D17—24no—
213C22×D17—22no—
214C22×D17—22no—
215C22×D17—22no—
216C22×D17—24no—
217C22×D17—22no—
218C22×D17—24no—
219C22×D17—24no—
220C22×D17—24no—
221C22×D17—24no—
222C22×D17C2×C4214yes—
223C22×D17C2×C22⋊C414yes—
224C22×D17#11714yes—
225C22×D17C4×D418yes—
226C22×D17C4×D418yes—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—22no—
233C2×C4×C17—24no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
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—42no—
244C2×C4×C17—42no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
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×C22⋊C414yes—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17—24no—
264C2×Dic17—22no—
265C2×Dic17—24no—
266C2×Dic17—24no—
267C2×Dic17—24no—
268C2×Dic17—24no—
269C2×Dic17—24no—
270C2×Dic17—24no—
271C2×Dic17—24no—
272C2×Dic17—42no—
273C2×Dic17—41no—
274C2×Dic17—41no—
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×Dic17C2×C4214yes—
284C2×Dic17C2×C22⋊C414yes—
285C4×D17—232no—
286C4×D17—416no—
287C4×D17—416no—
288C23×C4⋊C4—171no—
289C2×#381—172no—
290C2×C4×C22⋊C4—171no—
291C2×C4×C22⋊C4—171no—
292#1436—171no—
293#1436—172no—
294#1436—172no—
295#1436—171no—
296#1436—171no—
297#1436D1711yescharacteristic, complemented
298C22⋊#95—172no—
299C22×C17C22×C4211yescharacteristic
300C22×C17C22×C22⋊C411yescharacteristic
301C22×C17C2×#11711yescharacteristic
302C22×C17C2×C4×D412yes—
303C22×C17C2×C4×D412yes—
304C22×C17#51812yes—
305C22×C17#51912yes—
306C22×C17#52012yes—
307C22×C17#52012yes—
308C22×C17#52012yes—
309C22×C17#52012yes—
310C22×C17#53412yes—
311C22×C17#53411yescharacteristic
312C22×C17#53411yescharacteristic
313C22×C17#53611yescharacteristic
314C22×C17#53612yes—
315C22×C17#53611yescharacteristic
316C22×C17#53912yes—
317C22×C17#53912yes—
318C22×C17#53912yes—
319C22×C17#53912yes—
320C2×D17—24no—
321C2×D17—24no—
322C2×D17—28no—
323C2×D17—28no—
324C2×D17—28no—
325C2×D17—28no—
326C2×D17—28no—
327C2×D17—28no—
328C2×D17C4×C22⋊C418yes—
329C4×C17—24no—
330C4×C17—24no—
331C4×C17—42no—
332C4×C17—44no—
333C4×C17—44no—
334Dic17—24no—
335Dic17—24no—
336Dic17—42no—
337Dic17—44no—
338Dic17—44no—
339C24×C4—171no—
340C24×C4—171no—
341C24×C4—171no—
342C22×C4⋊C4—171no—
343C22×C4⋊C4—171no—
344C22×C4⋊C4—171no—
345C22×C4⋊C4—178no—
346C22×C4⋊C4—344no—
347C22×C4⋊C4—344no—
348C22×C4⋊C4C2×D1711yescharacteristic
349C22×C22⋊C4—172no—
350C22×C22⋊C4—172no—
351C22×C42—172no—
352C22×C42—171no—
353C22×C42C2×D1711yescharacteristic
354#381—1716no—
355C4×C22⋊C4—344no—
356C4×C22⋊C4—344no—
357C2×#95—172no—
358C2×#95—172no—
359C2×#95—172no—
360C2×#95—172no—
361C2×#95—171no—
362C2×#95—172no—
363C2×#95—172no—
364C2×#95—172no—
365C2×#95C2×D1711yescharacteristic
366C2×#95C2×D1712yes—
367C2×#95C2×D1712yes—
368C2×C17C2×C4×C22⋊C411yescharacteristic
369C2×C17#227412yes—
370C2×C17#230811yescharacteristic
371C2×C17#231011yescharacteristic
372C2×C17#231312yes—
373C2×C17#232212yes—
374C2×C17#232212yes—
375C2×C17#232212yes—
376C2×C17#232212yes—
377D17—28no—
378C25—171no—
379C23×C4—171no—
380C23×C4—171no—
381C23×C4—174no—
382C23×C4—178no—
383C23×C4—174no—
384C23×C4—174no—
385C23×C4—174no—
386C23×C4—172no—
387C23×C4—172no—
388C23×C4—171no—
389C23×C4—174no—
390C23×C4—342no—
391C23×C4—342no—
392C23×C4—344no—
393C23×C4—342no—
394C23×C4—342no—
395C23×C4—682no—
396C23×C4—682no—
397C23×C4—682no—
398C23×C4—682no—
399C23×C4C22×D1712yes—
400C23×C4C22×D1712yes—
401C23×C4C22×D1711yescharacteristic
402C23×C4C22×D1711yescharacteristic
403C23×C4C22×D1711yescharacteristic
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—174no—
407C2×C4⋊C4—174no—
408C2×C4⋊C4—174no—
409C2×C4⋊C4—3432no—
410C2×C4⋊C4—342no—
411C2×C4⋊C4—342no—
412C2×C4⋊C4—342no—
413C2×C4⋊C4—342no—
414C2×C4⋊C4—342no—
415C2×C4⋊C4—342no—
416C2×C4⋊C4C4×D1714yes—
417C2×C22⋊C4—178no—
418C2×C22⋊C4—178no—
419C2×C22⋊C4—344no—
420C2×C22⋊C4—344no—
421C2×C22⋊C4—344no—
422C2×C22⋊C4—344no—
423C2×C42—22no—
424C2×C42—42no—
425C2×C42—344no—
426C2×C42—342no—
427C2×C42—682no—
428C2×C42—682no—
429C2×C42—682no—
430#95—22no—
431#95—42no—
432#95—178no—
433#95—178no—
434#95—178no—
435#95—344no—
436#95—342no—
437#95—682no—
438#95—682no—
439#95—682no—
440#95C4×D1718yescomplemented
441C17#1377611yescharacteristic, complemented, Sylow
442C24—174no—
443C24—172no—
444C24—172no—
445C24—172no—
446C24—174no—
447C24—342no—
448C24—342no—
449C24—342no—
450C24—342no—
451C24C23×D1711yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
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—174no—
474C22×C4—172no—
475C22×C4—172no—
476C22×C4—172no—
477C22×C4—172no—
478C22×C4—1716no—
479C22×C4—1716no—
480C22×C4—174no—
481C22×C4—174no—
482C22×C4—172no—
483C22×C4—348no—
484C22×C4—348no—
485C22×C4—3416no—
486C22×C4—3416no—
487C22×C4—348no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—342no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—342no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—688no—
510C22×C4—688no—
511C22×C4—684no—
512C22×C4—684no—
513C22×C4—688no—
514C22×C4—688no—
515C22×C4—684no—
516C22×C4—684no—
517C22×C4C2×C4×D1714yes—
518C22×C4C2×C4×D1714yes—
519C22×C4C2×C4×D1712yes—
520C22×C4C2×C4×D1714yes—
521C22×C4C2×C4×D1712yes—
522C22×C4C2×C4×D1712yes—
523C22×C4D4×D1712yes—
524C22×C4D4×D1712yes—
525C4⋊C4—28no—
526C4⋊C4—348no—
527C4⋊C4—348no—
528C4⋊C4—348no—
529C22⋊C4—3416no—
530C22⋊C4—3416no—
531C42—44no—
532C42—684no—
533C42—688no—
534C23—174no—
535C23—174no—
536C23—174no—
537C23—178no—
538C23—178no—
539C23—344no—
540C23—344no—
541C23—344no—
542C23—344no—
543C23—344no—
544C23—344no—
545C23—344no—
546C23—342no—
547C23—342no—
548C23—342no—
549C23—344no—
550C23—342no—
551C23—344no—
552C23—344no—
553C23—344no—
554C23—344no—
555C23C22×C4×D1711yescharacteristic
556C23C22×C4×D1712yes—
557C23C2×D4×D1711yescharacteristic
558C23C2×D4×D1711yescharacteristic
559C23#1058769112yes—
560C23#1058769412yes—
561C23#41×D1712yes—
562C23#1058769612yes—
563C23#1058769712yes—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—174no—
592C2×C4—174no—
593C2×C4—3432no—
594C2×C4—342no—
595C2×C4—342no—
596C2×C4—344no—
597C2×C4—344no—
598C2×C4—344no—
599C2×C4—344no—
600C2×C4—342no—
601C2×C4—342no—
602C2×C4—342no—
603C2×C4—344no—
604C2×C4—344no—
605C2×C4—344no—
606C2×C4—344no—
607C2×C4—344no—
608C2×C4—344no—
609C2×C4—344no—
610C2×C4—6816no—
611C2×C4—6816no—
612C2×C4—682no—
613C2×C4—682no—
614C2×C4—681no—
615C2×C4—682no—
616C2×C4—681no—
617C2×C4—682no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—682no—
622C2×C4—682no—
623C2×C4C42×D1714yes—
624C2×C4C22⋊C4×D1714yes—
625C22—178no—
626C22—344no—
627C22—344no—
628C22—348no—
629C22—348no—
630C22—348no—
631C22—348no—
632C22—348no—
633C22—348no—
634C22C2×C42×D1711yescharacteristic
635C22C2×C22⋊C4×D1711yescharacteristic
636C22#117×D1711yescharacteristic
637C22C4×D4×D1712yes—
638C22C4×D4×D1712yes—
639C22#1176328112yes—
640C22#1176328512yes—
641C22#1176328812yes—
642C22#1176328912yes—
643C22#1176329012yes—
644C22#1176329112yes—
645C22#1176336712yes—
646C22#1176336911yescharacteristic
647C22#1176337011yescharacteristic
648C22#1176337911yescharacteristic
649C22#1176338212yes—
650C22#1176338411yescharacteristic
651C22#1176340012yes—
652C22#1176340212yes—
653C22#1176340512yes—
654C22#1176340712yes—
655C4—24no—
656C4—24no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—344no—
661C4—344no—
662C4—682no—
663C4—684no—
664C4—684no—
665C2—348no—
666C2C4×C22⋊C4×D1711yescharacteristic
667C2#1453113812yes—
668C2#1453135811yescharacteristic
669C2#1453136711yescharacteristic
670C2#1453138312yes—
671C2#1453144812yes—
672C2#1453145112yes—
673C2#1453145412yes—
674C2#1453145712yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC17×#1436C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1436D17
hypocenterC17#13776
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
5. #14527019C2semidirect
6. #14527018C2semidirect
7. #14527017C2semidirect
8. #14527016C2semidirect
9. #14527015C2semidirect
11. C17×#1436C2semidirect
20. C2×C4⋊C4×D17C4semidirect
25. #11762671C4semidirect
95. C17×#95C2×C4semidirect
101. #10587502C2×C4semidirect
102. #10587502C2×C4semidirect
103. #10587502C2×C4semidirect
185. C2×C4×D17C42semidirect
297. #1436D17semidirect
440. #95C4×D17semidirect
441. C17#13776semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D172
C2×#117626712
C2×C4×#632272
C2×#117626562
#145270192
#145270182
#145270172
#145270162
#145270152
#145270052
C17×#14362
#1377617