Finite Group Databasebeta

← #16955652

Subgroups — #16955652

Order 4352.

1710 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762701C212yesmaximal
2C2×C4×#63227C211yesmaximal, characteristic
3#14527020C212yesmaximal
4#14527018C212yesmaximal, complemented
5#14527014C211yesmaximal, characteristic, complemented
6#14527010C212yesmaximal, complemented
7#14527005C211yesmaximal, characteristic
8#14527005C211yesmaximal, characteristic
9#95×Dic17C211yesmaximal, characteristic, complemented
10#95×Dic17C211yesmaximal, characteristic, complemented
11C17×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C22×C4⋊Dic17C2212yes—
14C22×#63227C2211yescharacteristic
15C22×#63227C2212yes—
16C22×#63227C2212yes—
17C22×#63227C2211yescharacteristic
18C22×C4×Dic17C2212yes—
19C22×C4×Dic17C2211yescharacteristic
20C22×C4×Dic17C2212yes—
21#11762701C4116yescomplemented
22C4×#63227C418yescomplemented
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2212yes—
32C22×C42×C17C2211yescharacteristic
33C2×C17×#95C2211yescharacteristic
34C2×C17×#95C2211yescharacteristic
35C2×C17×#95C2212yes—
36C2×C17×#95C2212yes—
37C24×D17C2311yescharacteristic
38C23×C4×C17C2311yescharacteristic
39C23×C4×C17C2312yes—
40C23×C4×C17C2312yes—
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×Dic17C2312yes—
44C23×Dic17C2312yes—
45C23×Dic17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2311yescharacteristic
48C22×C4×D17—22no—
49C22×C4×D17—22no—
50C22×C4×D17—42no—
51C22×C4×D17—42no—
52C22×C4×D17C2×C414yes—
53C2×C42×C17—42no—
54C2×C42×C17C2×C414yes—
55C2×#63227—24no—
56C2×#63227—42no—
57C2×#63227—42no—
58C2×#63227—42no—
59C2×#63227—42no—
60C2×#63227C2×C414yes—
61C2×#63227C2×C418yes—
62C2×#63227C2×C418yes—
63C2×#63227C2×C418yes—
64C2×#63227C2×C414yes—
65C2×C4⋊Dic17—42no—
66C2×C4⋊Dic17—42no—
67C2×C4⋊Dic17C2×C418yescomplemented
68C2×C4×Dic17—24no—
69C2×C4×Dic17—42no—
70C2×C4×Dic17—42no—
71C2×C4×Dic17—42no—
72C2×C4×Dic17—42no—
73C2×C4×Dic17—42no—
74C2×C4×Dic17C2×C414yes—
75C2×C4×Dic17C2×C418yescomplemented
76C17×#95—22no—
77C17×#95—22no—
78C17×#95—42no—
79C17×#95C2×C418yescomplemented
80#10587502—22no—
81#10587502—22no—
82#10587502—24no—
83#10587502—42no—
84#10587502—42no—
85#10587502—42no—
86#10587502C2×C414yes—
87#10587502C2×C414yes—
88#10587502C2×C418yescomplemented
89C24×C17C2411yescharacteristic
90C23×D17—21no—
91C23×D17—21no—
92C23×D17—22no—
93C23×D17—22no—
94C23×D17—22no—
95C23×D17—21no—
96C23×D17—21no—
97C23×D17—21no—
98C23×D17—21no—
99C23×D17C22×C412yes—
100C23×D17C22×C414yes—
101C22×C4×C17—22no—
102C22×C4×C17—22no—
103C22×C4×C17—21no—
104C22×C4×C17—22no—
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—21no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17C22×C412yes—
123C22×C4×C17C22×C414yes—
124C22×C4×C17C22×C414yes—
125C22×C4×C17C22×C412yes—
126C22×C4×C17C22×C414yes—
127C22×C4×C17C22×C412yes—
128C22×C4×C17#4112yes—
129C22×C4×C17#4112yes—
130C22×Dic17—22no—
131C22×Dic17—21no—
132C22×Dic17—22no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
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×Dic17—22no—
152C22×Dic17C22×C412yes—
153C22×Dic17C22×C414yes—
154C22×Dic17C22×C414yes—
155C22×Dic17C22×C412yes—
156C22×Dic17C22×C414yes—
157C22×Dic17C22×C412yes—
158C22×Dic17#4114yes—
159C2×C4×D17—28no—
160C2×C4×D17—44no—
161C2×C4×D17—44no—
162C2×C4×D17—48no—
163C2×C4×D17—48no—
164C42×C17—44no—
165#63227—28no—
166#63227—48no—
167#63227—48no—
168#63227C42116yescomplemented
169C4⋊Dic17—48no—
170C4×Dic17—44no—
171C4×Dic17—48no—
172C4×Dic17—48no—
173#13788—171nomaximal, Sylow
174C23×C17C23×C411yescharacteristic
175C23×C17C23×C412yes—
176C23×C17C2×#4111yescharacteristic
177C23×C17C2×#4111yescharacteristic
178C23×C17C2×#4112yes—
179C23×C17H(5,𝔽2)12yes—
180C23×C17H(5,𝔽2)12yes—
181C23×C17H(5,𝔽2)11yescharacteristic
182C23×C17H(5,𝔽2)11yescharacteristic
183C23×C172−1+411yescharacteristic
184C23×C172−1+411yescharacteristic
185C22×D17—22no—
186C22×D17—22no—
187C22×D17—24no—
188C22×D17—24no—
189C22×D17—22no—
190C22×D17—22no—
191C22×D17—24no—
192C22×D17—24no—
193C22×D17—24no—
194C22×D17—22no—
195C22×D17—22no—
196C22×D17—24no—
197C22×D17—42no—
198C22×D17—42no—
199C22×D17—42no—
200C22×D17—42no—
201C22×D17—42no—
202C22×D17—41no—
203C22×D17—41no—
204C22×D17—42no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17C2×C4214yes—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—24no—
212C2×C4×C17—24no—
213C2×C4×C17—22no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—22no—
217C2×C4×C17—24no—
218C2×C4×C17—24no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—22no—
224C2×C4×C17—24no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17C2×C4214yes—
236C2×C4×C17#11714yes—
237C2×Dic17—22no—
238C2×Dic17—22no—
239C2×Dic17—24no—
240C2×Dic17—22no—
241C2×Dic17—22no—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—22no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17C2×C4214yes—
265C2×Dic17#11714yes—
266C4×D17—416no—
267C2×#388—172no—
268C2×C4×C22⋊C4—171no—
269#1437—172no—
270#1436—172no—
271C4⋊#95—171no—
272#1434—172no—
273#1434D1711yescharacteristic, complemented
274C22⋊#95—171no—
275C22⋊#95—171no—
276C4×#95—171no—
277C4×#95—171no—
278C22×C17C22×C4211yescharacteristic
279C22×C17C2×#11711yescharacteristic
280C22×C17C2×#11711yescharacteristic
281C22×C17C4×#4112yes—
282C22×C17C4×#4112yes—
283C22×C17#51812yes—
284C22×C17#51811yescharacteristic
285C22×C17#51811yescharacteristic
286C22×C17#51812yes—
287C22×C17#52012yes—
288C22×C17#52011yescharacteristic
289C22×C17#52011yescharacteristic
290C22×C17#52012yes—
291C22×C17#53812yes—
292C22×C17#53812yes—
293C22×C17#53812yes—
294C22×C17#53912yes—
295C22×C17#54012yes—
296C22×C17#54011yescharacteristic
297C22×C17#54111yescharacteristic
298C22×C17#54212yes—
299C22×C17#54211yescharacteristic
300C22×C17#54211yescharacteristic
301C2×D17—24no—
302C2×D17—24no—
303C2×D17—24no—
304C2×D17—42no—
305C2×D17—42no—
306C2×D17—42no—
307C2×D17—42no—
308C2×D17—44no—
309C2×D17—44no—
310C2×D17—44no—
311C2×D17—44no—
312C4×C17—24no—
313C4×C17—24no—
314C4×C17—42no—
315C4×C17—44no—
316C4×C17—44no—
317Dic17—24no—
318Dic17—24no—
319Dic17—42no—
320Dic17—44no—
321Dic17—44no—
322C24×C4—171no—
323C22×C4⋊C4—172no—
324C22×C22⋊C4—171no—
325C22×C22⋊C4—171no—
326C22×C22⋊C4—172no—
327C22×C22⋊C4—172no—
328C22×C42—171no—
329C22×C42—172no—
330C22×C42—172no—
331C22×C42C2×D1711yescharacteristic
332#388—1716no—
333C4×C22⋊C4—178no—
334C2×#95—172no—
335C2×#95—171no—
336C2×#95—171no—
337C2×#95—171no—
338C2×#95—171no—
339C2×#95—172no—
340C2×#95—172no—
341C2×#95—172no—
342C2×#95—172no—
343C2×#95C2×D1711yescharacteristic
344C2×#95C2×D1711yescharacteristic
345C2×#95C2×D1712yes—
346C2×#95C2×D1712yes—
347C2×C17C4×#11711yescharacteristic
348C2×C17#227511yescharacteristic
349C2×C17#227511yescharacteristic
350C2×C17#231211yescharacteristic
351C2×C17#231411yescharacteristic
352C2×C17#231811yescharacteristic
353C2×C17#231911yescharacteristic
354C2×C17#233312yes—
355C2×C17#233412yes—
356C2×C17#233512yes—
357C2×C17#233712yes—
358D17—44no—
359C25—171no—
360C23×C4—171no—
361C23×C4—171no—
362C23×C4—172no—
363C23×C4—172no—
364C23×C4—171no—
365C23×C4—174no—
366C23×C4—342no—
367C23×C4—342no—
368C23×C4—682no—
369C23×C4—682no—
370C23×C4C22×D1711yescharacteristic
371C23×C4C22×D1712yes—
372C23×C4C22×D1712yes—
373C23×C4C22×D1711yescharacteristic
374C23×C4C22×D1711yescharacteristic
375C2×C4⋊C4—178no—
376C2×C4⋊C4—682no—
377C2×C4⋊C4—682no—
378C2×C22⋊C4—174no—
379C2×C22⋊C4—178no—
380C2×C22⋊C4—174no—
381C2×C22⋊C4—178no—
382C2×C22⋊C4—178no—
383C2×C22⋊C4—344no—
384C2×C22⋊C4—682no—
385C2×C22⋊C4—682no—
386C2×C22⋊C4—682no—
387C2×C22⋊C4—682no—
388C2×C42—42no—
389C2×C42—174no—
390C2×C42—178no—
391C2×C42—344no—
392C2×C42—682no—
393C2×C42—682no—
394C2×C42—682no—
395C2×C42—682no—
396C2×C42—682no—
397C2×C42C4×D1714yes—
398#95—22no—
399#95—22no—
400#95—42no—
401#95—174no—
402#95—174no—
403#95—178no—
404#95—344no—
405#95—342no—
406#95—342no—
407#95—682no—
408#95—682no—
409#95—682no—
410#95C4×D1718yescomplemented
411C17#1378811yescharacteristic, complemented, Sylow
412C24—174no—
413C24—172no—
414C24—341no—
415C24—342no—
416C24—341no—
417C24—342no—
418C24—341no—
419C24—341no—
420C24—342no—
421C24—341no—
422C24—341no—
423C24C23×D1711yescharacteristic
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—172no—
446C22×C4—174no—
447C22×C4—172no—
448C22×C4—174no—
449C22×C4—172no—
450C22×C4—174no—
451C22×C4—174no—
452C22×C4—342no—
453C22×C4—341no—
454C22×C4—342no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—341no—
465C22×C4—342no—
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—348no—
475C22×C4—684no—
476C22×C4—684no—
477C22×C4—688no—
478C22×C4—688no—
479C22×C4C2×C4×D1714yes—
480C22×C4C2×C4×D1712yes—
481C22×C4C2×C4×D1714yes—
482C22×C4C2×C4×D1712yes—
483C22×C4C2×C4×D1714yes—
484C22×C4C2×C4×D1712yes—
485C22×C4#6325412yes—
486C22×C4#6325612yes—
487C4⋊C4—688no—
488C22⋊C4—1716no—
489C22⋊C4—348no—
490C22⋊C4—688no—
491C22⋊C4—688no—
492C42—44no—
493C42—684no—
494C42—688no—
495C42—688no—
496C23—174no—
497C23—342no—
498C23—342no—
499C23—344no—
500C23—344no—
501C23—344no—
502C23—342no—
503C23—344no—
504C23—342no—
505C23—344no—
506C23—342no—
507C23—344no—
508C23—342no—
509C23—682no—
510C23—682no—
511C23—682no—
512C23—682no—
513C23—682no—
514C23—681no—
515C23—681no—
516C23—682no—
517C23—681no—
518C23—681no—
519C23C22×C4×D1711yescharacteristic
520C23C22×C4×D1712yes—
521C23C2×#6325411yescharacteristic
522C23#1058769112yes—
523C23C2×#6325611yescharacteristic
524C23#41×D1712yes—
525C23#1058769611yescharacteristic
526C23#1058769612yes—
527C23#1058769611yescharacteristic
528C23#1058769711yescharacteristic
529C23#1058769711yescharacteristic
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—174no—
558C2×C4—174no—
559C2×C4—342no—
560C2×C4—344no—
561C2×C4—342no—
562C2×C4—344no—
563C2×C4—342no—
564C2×C4—342no—
565C2×C4—344no—
566C2×C4—344no—
567C2×C4—342no—
568C2×C4—344no—
569C2×C4—344no—
570C2×C4—344no—
571C2×C4—344no—
572C2×C4—342no—
573C2×C4—344no—
574C2×C4—344no—
575C2×C4—342no—
576C2×C4—6816no—
577C2×C4—682no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4C42×D1714yes—
588C2×C4#1058757814yes—
589C22—344no—
590C22—344no—
591C22—344no—
592C22—684no—
593C22—682no—
594C22—682no—
595C22—682no—
596C22—682no—
597C22—684no—
598C22—684no—
599C22—684no—
600C22C2×C42×D1711yescharacteristic
601C22#117×D1711yescharacteristic
602C22C2×#1058757811yescharacteristic
603C22C4×#6325412yes—
604C22C4×#6325612yes—
605C22#1176328112yes—
606C22#1176328211yescharacteristic
607C22#1176328311yescharacteristic
608C22#1176328312yes—
609C22#1176328812yes—
610C22#1176328912yes—
611C22#1176328911yescharacteristic
612C22#1176328911yescharacteristic
613C22#1176339612yes—
614C22#1176339812yes—
615C22#1176339912yes—
616C22#1176340012yes—
617C22#1176341112yes—
618C22#1176341511yescharacteristic
619C22#1176342211yescharacteristic
620C22#1176342712yes—
621C22#1176342811yescharacteristic
622C22#1176342811yescharacteristic
623C4—24no—
624C4—24no—
625C4—44no—
626C4—42no—
627C4—44no—
628C4—344no—
629C4—344no—
630C4—682no—
631C4—684no—
632C4—684no—
633C2—684no—
634C2C4×#1058757811yescharacteristic
635C2#1453114311yescharacteristic
636C2#1453114611yescharacteristic
637C2#1453138111yescharacteristic
638C2#1453139411yescharacteristic
639C2#1453141511yescharacteristic
640C2#1453142411yescharacteristic
641C2#1453154812yes—
642C2#1453156212yes—
643C2#1453156712yes—
644C2#1453159112yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1434D17
hypocenterC17#13788
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #14527018C2semidirect
5. #14527014C2semidirect
6. #14527010C2semidirect
9. #95×Dic17C2semidirect
10. #95×Dic17C2semidirect
11. C17×#1434C2semidirect
21. #11762701C4semidirect
22. C4×#63227C4semidirect
67. C2×C4⋊Dic17C2×C4semidirect
75. C2×C4×Dic17C2×C4semidirect
79. C17×#95C2×C4semidirect
88. #10587502C2×C4semidirect
168. #63227C42semidirect
273. #1434D17semidirect
410. #95C4×D17semidirect
411. C17#13788semidirect

Maximal subgroups

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

subgroupindex
C2×#117627012
C2×C4×#632272
#145270202
#145270182
#145270142
#145270102
#145270052
#95×Dic172
C17×#14342
#1378817