Finite Group Databasebeta

← #16955961

Subgroups — #16955961

Order 4352.

1998 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762711C211yesmaximal, characteristic
2C2×#11762687C212yesmaximal
3#14527020C212yesmaximal, complemented
4#14527020C212yesmaximal, complemented
5#14527018C212yesmaximal, complemented
6#14527011C211yesmaximal, characteristic, complemented
7#14527005C212yesmaximal, complemented
8#14527005C212yesmaximal, complemented
9C17×#1434C211yesmaximal, characteristic, complemented
10C23×D68C2211yescharacteristic
11C23×C4×D17C2212yes—
12C22×C4⋊Dic17C2211yescharacteristic
13C22×#63227C2212yes—
14C22×#63227C2212yes—
15C22×#63227C2212yes—
16C22×#63227C2212yes—
17C22×#63227C2212yes—
18C22×C4×Dic17C2212yes—
19#11762711C418yescomplemented
20#11762687C4116yescomplemented
21C2×#10587502C2211yescharacteristic
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2212yes—
28C22×C42×C17C2211yescharacteristic
29C2×C17×#95C2211yescharacteristic
30C2×C17×#95C2212yes—
31C2×C17×#95C2211yescharacteristic
32C2×C17×#95C2212yes—
33C24×D17C2312yes—
34C23×C4×C17C2311yescharacteristic
35C23×C4×C17C2311yescharacteristic
36C23×C4×C17C2312yes—
37C23×C4×C17C2312yes—
38C23×C4×C17C2311yescharacteristic
39C23×Dic17C2312yes—
40C23×Dic17C2312yes—
41C23×Dic17C2312yes—
42C22×D68—41no—
43C22×D68—41no—
44C22×D68—42no—
45C22×D68C2×C414yes—
46C22×D68C2×C418yes—
47C22×C4×D17—42no—
48C22×C4×D17—42no—
49C22×C4×D17—42no—
50C22×C4×D17—42no—
51C22×C4×D17—42no—
52C22×C4×D17—42no—
53C22×C4×D17C2×C418yescomplemented
54C2×C42×C17—42no—
55C2×C42×C17C2×C414yescomplemented
56C2×#63227—42no—
57C2×#63227—42no—
58C2×#63227—42no—
59C2×#63227—42no—
60C2×#63227—42no—
61C2×#63227—42no—
62C2×#63227—42no—
63C2×#63227—42no—
64C2×#63227—42no—
65C2×#63227—42no—
66C2×#63227C2×C418yes—
67C2×#63227C2×C418yes—
68C2×#63227C2×C418yescomplemented
69C2×#63227C2×C418yescomplemented
70C2×#63227C2×C418yescomplemented
71C2×C4⋊Dic17—41no—
72C2×C4⋊Dic17—41no—
73C2×C4⋊Dic17C2×C414yescomplemented
74C2×C4×Dic17—42no—
75C2×C4×Dic17—42no—
76C2×C4×Dic17—42no—
77C17×#95—41no—
78C17×#95—42no—
79C17×#95—41no—
80C17×#95C2×C418yescomplemented
81#10587502—41no—
82#10587502—42no—
83#10587502—42no—
84#10587502—42no—
85#10587502—42no—
86#10587502—41no—
87#10587502C2×C418yescomplemented
88C24×C17C2411yescharacteristic
89C23×D17—22no—
90C23×D17—22no—
91C23×D17—22no—
92C23×D17—22no—
93C23×D17—22no—
94C23×D17—22no—
95C23×D17—22no—
96C23×D17—22no—
97C23×D17—22no—
98C23×D17—22no—
99C23×D17—22no—
100C23×D17—22no—
101C23×D17C22×C414yes—
102C23×D17C22×C414yes—
103C23×D17C22×C414yes—
104C22×C4×C17—21no—
105C22×C4×C17—21no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—21no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—21no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—22no—
127C22×C4×C17C22×C414yes—
128C22×C4×C17C22×C414yes—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C22×C414yes—
132C22×C4×C17C22×C412yes—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17C22×C414yes—
152C22×Dic17C22×C414yes—
153C22×Dic17C22×C414yes—
154C2×D68—44no—
155C2×D68—48no—
156C2×D68—48no—
157C2×D68—44no—
158C2×D68C42116yescomplemented
159C2×C4×D17—48no—
160C2×C4×D17—48no—
161C2×C4×D17—48no—
162C2×C4×D17—48no—
163C2×C4×D17—48no—
164C2×C4×D17—48no—
165C2×C4×D17—48no—
166C42×C17—44no—
167#63227—48no—
168#63227—48no—
169#63227—48no—
170#63227—48no—
171#63227—48no—
172C4⋊Dic17—44no—
173C4×Dic17—48no—
174#13832—171nomaximal, Sylow
175C23×C17C23×C412yes—
176C23×C17C23×C411yescharacteristic
177C23×C17H(5,𝔽2)12yes—
178C23×C17H(5,𝔽2)11yescharacteristic
179C23×C17H(5,𝔽2)11yescharacteristic
180C23×C17H(5,𝔽2)11yescharacteristic
181C23×C17H(5,𝔽2)12yes—
182C23×C17H(5,𝔽2)12yes—
183C23×C17H(5,𝔽2)11yescharacteristic
184C23×C172−1+411yescharacteristic
185C23×C172−1+411yescharacteristic
186C22×D17—24no—
187C22×D17—24no—
188C22×D17—24no—
189C22×D17—24no—
190C22×D17—24no—
191C22×D17—24no—
192C22×D17—24no—
193C22×D17—24no—
194C22×D17—24no—
195C22×D17—24no—
196C22×D17—24no—
197C22×D17—24no—
198C22×D17—24no—
199C22×D17—24no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—24no—
203C22×D17—24no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—42no—
213C22×D17—42no—
214C22×D17—42no—
215C22×D17—42no—
216C22×D17—42no—
217C22×D17—42no—
218C22×D17—42no—
219C22×D17—42no—
220C22×D17C2×C4218yes—
221C2×C4×C17—22no—
222C2×C4×C17—24no—
223C2×C4×C17—24no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—24no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
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×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17C2×C4214yes—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—24no—
258C2×Dic17—24no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271D68—416no—
272C4×D17—432no—
273C2×#390—171no—
274C2×#386—172no—
275#1437—172no—
276#1437—172no—
277#1436—172no—
278#1434—171no—
279#1434D1711yescharacteristic, complemented
280C22⋊#95—172no—
281C22⋊#95—172no—
282C22×C17C22×C4211yescharacteristic
283C22×C17#51812yes—
284C22×C17#51812yes—
285C22×C17#51812yes—
286C22×C17#51811yescharacteristic
287C22×C17#51812yes—
288C22×C17#51811yescharacteristic
289C22×C17#51811yescharacteristic
290C22×C17#51811yescharacteristic
291C22×C17#52012yes—
292C22×C17#52011yescharacteristic
293C22×C17#52011yescharacteristic
294C22×C17#52012yes—
295C22×C17#56012yes—
296C22×C17#56012yes—
297C22×C17#56012yes—
298C22×C17#56012yes—
299C22×C17H(3,𝔽4)11yescharacteristic
300C22×C17#56211yescharacteristic
301C22×C17#56212yes—
302C22×C17#56212yes—
303C22×C17#56211yescharacteristic
304C22×C17#56311yescharacteristic
305C2×D17—28no—
306C2×D17—28no—
307C2×D17—28no—
308C2×D17—44no—
309C2×D17—44no—
310C2×D17—44no—
311C2×D17—44no—
312C2×D17—44no—
313C2×D17—44no—
314C2×D17—44no—
315C2×D17—44no—
316C2×D17—44no—
317C2×D17—44no—
318C2×D17—44no—
319C2×D17—44no—
320C4×C17—24no—
321C4×C17—44no—
322C4×C17—44no—
323C4×C17—42no—
324C4×C17—42no—
325Dic17—44no—
326Dic17—44no—
327Dic17—44no—
328C23×D4—171no—
329C24×C4—172no—
330C22×C4⋊C4—171no—
331C22×C22⋊C4—172no—
332C22×C22⋊C4—172no—
333C22×C22⋊C4—172no—
334C22×C22⋊C4—172no—
335C22×C22⋊C4—172no—
336C22×C42—172no—
337C22×C42C2×D1711yescharacteristic
338#390—178no—
339#386—1716no—
340C2×#95—172no—
341C2×#95—171no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—171no—
347C2×#95C2×D1711yescharacteristic
348C2×#95C2×D1712yes—
349C2×#95C2×D1711yescharacteristic
350C2×#95C2×D1712yes—
351C2×C17#227311yescharacteristic
352C2×C17#227511yescharacteristic
353C2×C17#227511yescharacteristic
354C2×C17#237512yes—
355C2×C17#237612yes—
356C2×C17#237712yes—
357C2×C17#237711yescharacteristic
358C2×C17#237712yes—
359C2×C17#237711yescharacteristic
360C2×C17#237911yescharacteristic
361C2×C17#238011yescharacteristic
362D17—48no—
363C25—172no—
364C22×D4—178no—
365C22×D4—174no—
366C22×D4—682no—
367C22×D4—681no—
368C22×D4—681no—
369C23×C4—172no—
370C23×C4—172no—
371C23×C4—172no—
372C23×C4—178no—
373C23×C4—682no—
374C23×C4—682no—
375C23×C4—682no—
376C23×C4—682no—
377C23×C4—682no—
378C23×C4—682no—
379C23×C4C22×D1711yescharacteristic
380C23×C4C22×D1711yescharacteristic
381C23×C4C22×D1712yes—
382C23×C4C22×D1712yes—
383C23×C4C22×D1711yescharacteristic
384C2×C4⋊C4—174no—
385C2×C4⋊C4—681no—
386C2×C4⋊C4—681no—
387C2×C22⋊C4—178no—
388C2×C22⋊C4—178no—
389C2×C22⋊C4—178no—
390C2×C22⋊C4—178no—
391C2×C22⋊C4—178no—
392C2×C22⋊C4—682no—
393C2×C22⋊C4—682no—
394C2×C22⋊C4—682no—
395C2×C22⋊C4—682no—
396C2×C22⋊C4—682no—
397C2×C22⋊C4—682no—
398C2×C22⋊C4—682no—
399C2×C22⋊C4—682no—
400C2×C22⋊C4—682no—
401C2×C22⋊C4—682no—
402C2×C42—42no—
403C2×C42—682no—
404C2×C42—682no—
405C2×C42—682no—
406C2×C42C4×D1714yescomplemented
407#95—41no—
408#95—41no—
409#95—42no—
410#95—178no—
411#95—682no—
412#95—681no—
413#95—681no—
414#95—682no—
415#95—682no—
416#95—682no—
417#95C4×D1718yescomplemented
418C17#1383211yescharacteristic, complemented, Sylow
419C24—174no—
420C24—174no—
421C24—174no—
422C24—342no—
423C24—342no—
424C24—342no—
425C24—342no—
426C24—342no—
427C24—342no—
428C24—342no—
429C24—342no—
430C24—342no—
431C24—342no—
432C24—342no—
433C24—342no—
434C24C23×D1711yescharacteristic
435C2×D4—1716no—
436C2×D4—688no—
437C2×D4—688no—
438C2×D4—684no—
439C2×D4—684no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
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—21no—
463C22×C4—174no—
464C22×C4—174no—
465C22×C4—174no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—688no—
485C22×C4—688no—
486C22×C4—688no—
487C22×C4—688no—
488C22×C4—688no—
489C22×C4—688no—
490C22×C4—688no—
491C22×C4C2×C4×D1714yes—
492C22×C4C2×C4×D1714yes—
493C22×C4C2×C4×D1712yes—
494C22×C4C2×C4×D1712yes—
495C22×C4C2×C4×D1714yes—
496C22×C4C2×C4×D1712yes—
497C4⋊C4—684no—
498C22⋊C4—688no—
499C22⋊C4—688no—
500C22⋊C4—688no—
501C22⋊C4—688no—
502C22⋊C4—688no—
503C42—44no—
504C42—688no—
505C23—178no—
506C23—344no—
507C23—344no—
508C23—344no—
509C23—344no—
510C23—344no—
511C23—344no—
512C23—344no—
513C23—344no—
514C23—344no—
515C23—344no—
516C23—344no—
517C23—344no—
518C23—344no—
519C23—344no—
520C23—344no—
521C23—344no—
522C23—344no—
523C23—344no—
524C23—682no—
525C23—682no—
526C23—682no—
527C23—682no—
528C23—682no—
529C23—682no—
530C23—682no—
531C23—682no—
532C23—682no—
533C23—682no—
534C23—682no—
535C23—682no—
536C23—682no—
537C23—682no—
538C23—682no—
539C23—682no—
540C23C22×C4×D1712yes—
541C23C22×C4×D1711yescharacteristic
542C23#1058769112yes—
543C23#1058769111yescharacteristic
544C23#1058769411yescharacteristic
545C23#1058769611yescharacteristic
546C23#1058769611yescharacteristic
547C23#1058769612yes—
548C23#1058769612yes—
549C23#1058769611yescharacteristic
550C23#1058769711yescharacteristic
551D4—6816no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—344no—
581C2×C4—344no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—344no—
585C2×C4—344no—
586C2×C4—344no—
587C2×C4—344no—
588C2×C4—344no—
589C2×C4—6832no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—682no—
599C2×C4—682no—
600C2×C4—682no—
601C2×C4—682no—
602C2×C4C42×D1714yes—
603C22—348no—
604C22—348no—
605C22—348no—
606C22—684no—
607C22—684no—
608C22—684no—
609C22—684no—
610C22—684no—
611C22—684no—
612C22—684no—
613C22—684no—
614C22—684no—
615C22—684no—
616C22—684no—
617C22—684no—
618C22C2×C42×D1711yescharacteristic
619C22#1176328112yes—
620C22#1176328112yes—
621C22#1176328112yes—
622C22#1176328211yescharacteristic
623C22#1176328311yescharacteristic
624C22#1176328311yescharacteristic
625C22#1176328312yes—
626C22#1176328311yescharacteristic
627C22#1176328912yes—
628C22#1176328911yescharacteristic
629C22#1176329011yescharacteristic
630C22#1176329112yes—
631C22#1176354712yes—
632C22#1176354912yes—
633C22#1176355012yes—
634C22#1176355012yes—
635C22#1176355111yescharacteristic
636C22#1176355311yescharacteristic
637C22#1176355512yes—
638C22#1176355612yes—
639C22#1176355711yescharacteristic
640C22#1176356311yescharacteristic
641C4—24no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—684no—
647C4—684no—
648C4—684no—
649C2—688no—
650C2#1453113611yescharacteristic
651C2#1453114411yescharacteristic
652C2#1453114611yescharacteristic
653C2#1453192412yes—
654C2#1453193112yes—
655C2#1453193712yes—
656C2#1453193811yescharacteristic
657C2#1453193912yes—
658C2#1453194311yescharacteristic
659C2#1453195511yescharacteristic
660C2#1453196011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC23×D68C22
hypercenter#1434D17
hypocenterC17#13832
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
3. #14527020C2semidirect
4. #14527020C2semidirect
5. #14527018C2semidirect
6. #14527011C2semidirect
7. #14527005C2semidirect
8. #14527005C2semidirect
9. C17×#1434C2semidirect
19. #11762711C4semidirect
20. #11762687C4semidirect
53. C22×C4×D17C2×C4semidirect
55. C2×C42×C17C2×C4semidirect
68. C2×#63227C2×C4semidirect
69. C2×#63227C2×C4semidirect
70. C2×#63227C2×C4semidirect
73. C2×C4⋊Dic17C2×C4semidirect
80. C17×#95C2×C4semidirect
87. #10587502C2×C4semidirect
158. C2×D68C42semidirect
279. #1434D17semidirect
406. C2×C42C4×D17semidirect
417. #95C4×D17semidirect
418. C17#13832semidirect

Maximal subgroups

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

subgroupindex
C2×#117627112
C2×#117626872
#145270202
#145270182
#145270112
#145270052
C17×#14342
#1383217