Finite Group Databasebeta

← #16956302

Subgroups — #16956302

Order 4352.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587578C211yesmaximal, characteristic
2C2×#11762679C211yesmaximal, characteristic, complemented
3C2×#11762674C211yesmaximal, characteristic, complemented
4C2×#11762666C211yesmaximal, characteristic
5C2×C4×#63227C211yesmaximal, characteristic
6#14527030C214yesmaximal
7#14527018C214yesmaximal, complemented
8#14527013C211yesmaximal, characteristic, complemented
9C17×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D17C2211yescharacteristic
11C2×#10587578—24no—
12C2×#10587578—24no—
13C2×#10587578—24no—
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C4⋊Dic17C2214yes—
16C22×C4⋊Dic17C2211yescharacteristic
17C22×#63227C2212yes—
18C22×#63227C2212yes—
19C22×#63227C2212yes—
20C22×C4×Dic17C2212yes—
21C22×C4×Dic17C2211yescharacteristic
22#11762679—24no—
23#11762674—24no—
24#11762666—24no—
25C4×#63227C418yescomplemented
26C2×#10587502C2212yes—
27C2×#10587502C2214yes—
28C2×#10587502C2214yes—
29C2×#10587502C2212yes—
30C22×C4⋊C4×C17C2211yescharacteristic
31C22×C4⋊C4×C17C2211yescharacteristic
32C22×C42×C17C2211yescharacteristic
33C2×C17×#95C2214yes—
34C24×D17C2311yescharacteristic
35C23×C4×C17C2312yes—
36C23×C4×C17C2312yes—
37C23×C4×C17C2312yes—
38C23×C4×C17C2311yescharacteristic
39C23×Dic17C2312yes—
40C23×Dic17C2312yes—
41C23×Dic17C2312yes—
42C23×Dic17C2311yescharacteristic
43C22×C4×D17—22no—
44C22×C4×D17—22no—
45C22×C4×D17—22no—
46C22×C4×D17—22no—
47C22×C4×D17C2×C414yes—
48C22×C4×D17D414yes—
49C22×C4×D17Q814yes—
50C2×C4⋊C4×C17—22no—
51C2×C4⋊C4×C17—22no—
52C2×C4⋊C4×C17—22no—
53C2×C4⋊C4×C17—22no—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17D414yes—
56C2×C42×C17—22no—
57C2×C42×C17—22no—
58C2×C42×C17C2×C414yes—
59C2×#63227—24no—
60C2×#63227—24no—
61C2×#63227—24no—
62C2×#63227—24no—
63C2×#63227—24no—
64C2×#63227—24no—
65C2×#63227—44no—
66C2×#63227C2×C418yes—
67C2×C4⋊Dic17—22no—
68C2×C4⋊Dic17—22no—
69C2×C4⋊Dic17—22no—
70C2×C4⋊Dic17—22no—
71C2×C4⋊Dic17—22no—
72C2×C4⋊Dic17—28no—
73C2×C4⋊Dic17—28no—
74C2×C4⋊Dic17—44no—
75C2×C4⋊Dic17D414yes—
76C2×C4×Dic17—24no—
77C2×C4×Dic17—22no—
78C2×C4×Dic17—22no—
79C2×C4×Dic17—24no—
80C2×C4×Dic17—24no—
81C2×C4×Dic17C2×C414yes—
82#10587578—232no—
83C17×#95—28no—
84#10587502—28no—
85#10587502—24no—
86#10587502—44no—
87#10587502C2×C418yes—
88C24×C17C2411yescharacteristic
89C23×D17—21no—
90C23×D17—21no—
91C23×D17—21no—
92C23×D17—21no—
93C23×D17—22no—
94C23×D17—22no—
95C23×D17—22no—
96C23×D17—22no—
97C23×D17C22×C412yes—
98C23×D17C2×D412yes—
99C23×D17C2×Q812yes—
100C22×C4×C17—22no—
101C22×C4×C17—22no—
102C22×C4×C17—22no—
103C22×C4×C17—22no—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—24no—
107C22×C4×C17—24no—
108C22×C4×C17—22no—
109C22×C4×C17—24no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—24no—
113C22×C4×C17C22×C414yes—
114C22×C4×C17C22×C412yes—
115C22×C4×C17C2×D414yes—
116C22×C4×C17C2×D412yes—
117C22×C4×C17C2×D412yes—
118C22×C4×C17C2×D412yes—
119C22×C4×C17C2×Q812yes—
120C22×C4×C17#4112yes—
121C22×C4×C17#4114yes—
122C22×C4×C17#4112yes—
123C22×C4×C17#4114yes—
124C22×Dic17—21no—
125C22×Dic17—21no—
126C22×Dic17—21no—
127C22×Dic17—21no—
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—24no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—24no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—24no—
142C22×Dic17—24no—
143C22×Dic17C22×C414yes—
144C22×Dic17C22×C412yes—
145C22×Dic17C2×D414yes—
146C22×Dic17C2×D412yes—
147C22×Dic17C2×Q812yes—
148C2×C4×D17—28no—
149C2×C4×D17—28no—
150C2×C4×D17—28no—
151C2×C4×D17—28no—
152C2×C4×D17—28no—
153C2×C4×D17—28no—
154C2×C4×D17C4⋊C4116yescomplemented
155C4⋊C4×C17—28no—
156C4⋊C4×C17—44no—
157C42×C17—28no—
158#63227—216no—
159#63227—216no—
160#63227—48no—
161C4⋊Dic17—28no—
162C4⋊Dic17—416no—
163C4⋊Dic17—44no—
164C4×Dic17—216no—
165C4×Dic17—28no—
166#13879—171nomaximal, Sylow
167C23×C17C23×C411yescharacteristic
168C23×C17C22×D411yescharacteristic
169C23×C17C22×D411yescharacteristic
170C23×C17C22×D411yescharacteristic
171C23×C17C22×Q811yescharacteristic
172C23×C17C2×#4111yescharacteristic
173C23×C17C2×#4112yes—
174C23×C17C2×#4111yescharacteristic
175C23×C17C2×#4112yes—
176C23×C17H(5,𝔽2)12yes—
177C23×C172−1+412yes—
178C22×D17—24no—
179C22×D17—22no—
180C22×D17—22no—
181C22×D17—24no—
182C22×D17—22no—
183C22×D17—22no—
184C22×D17—22no—
185C22×D17—22no—
186C22×D17—22no—
187C22×D17—22no—
188C22×D17—24no—
189C22×D17—22no—
190C22×D17—22no—
191C22×D17—24no—
192C22×D17—42no—
193C22×D17—42no—
194C22×D17—42no—
195C22×D17—42no—
196C22×D17—42no—
197C22×D17—42no—
198C22×D17—42no—
199C22×D17—42no—
200C22×D17C2×C4⋊C414yes—
201C2×C4×C17—24no—
202C2×C4×C17—24no—
203C2×C4×C17—24no—
204C2×C4×C17—24no—
205C2×C4×C17—24no—
206C2×C4×C17—24no—
207C2×C4×C17—24no—
208C2×C4×C17—28no—
209C2×C4×C17—42no—
210C2×C4×C17—42no—
211C2×C4×C17—44no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—44no—
216C2×C4×C17—44no—
217C2×C4×C17C2×C4⋊C414yes—
218C2×C4×C17#11718yescomplemented
219C2×C4×C17C4×D414yes—
220C2×C4×C17C4×D414yes—
221C2×C4×C17#12014yes—
222C2×C4×C17C22⋊Q814yes—
223C2×C4×C17C22⋊Q814yes—
224C2×C4×C17#12314yes—
225C2×Dic17—22no—
226C2×Dic17—22no—
227C2×Dic17—22no—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—22no—
231C2×Dic17—24no—
232C2×Dic17—24no—
233C2×Dic17—24no—
234C2×Dic17—24no—
235C2×Dic17—28no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240C2×Dic17—44no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—44no—
247C2×Dic17—44no—
248C2×Dic17C2×C4⋊C414yes—
249C4×D17—232no—
250C22×#117—171no—
251C2×#384—171no—
252C2×#382—171no—
253C2×#380—171no—
254C2×C4×C22⋊C4—171no—
255#1439—174no—
256#1436—174no—
257C4⋊#95—171no—
258C4⋊#95D1711yescharacteristic, complemented
259C22×C17C22×C4⋊C411yescharacteristic
260C22×C17C2×#11712yes—
261C22×C17C2×C4×D411yescharacteristic
262C22×C17C2×C4×D411yescharacteristic
263C22×C17#52012yes—
264C22×C17C2×#12011yescharacteristic
265C22×C17C2×C22⋊Q811yescharacteristic
266C22×C17C2×C22⋊Q811yescharacteristic
267C22×C17C2×#12311yescharacteristic
268C22×C17#53211yescharacteristic
269C22×C17#53211yescharacteristic
270C22×C17#53312yes—
271C22×C17#53511yescharacteristic
272C22×C17#53611yescharacteristic
273C22×C17#54412yes—
274C22×C17#54612yes—
275C22×C17#54712yes—
276C22×C17#54812yes—
277C22×C17#55012yes—
278C22×C17#55212yes—
279C22×C17#55312yes—
280C22×C17#55512yes—
281C22×C17#55612yes—
282C2×D17—24no—
283C2×D17—24no—
284C2×D17—24no—
285C2×D17—44no—
286C2×D17—44no—
287C2×D17—42no—
288C2×D17—42no—
289C2×D17—44no—
290C2×D17—44no—
291C2×D17—42no—
292C2×D17—42no—
293C4×C17—28no—
294C4×C17—44no—
295C4×C17—82no—
296C4×C17#38518yes—
297Dic17—24no—
298Dic17—44no—
299Dic17—44no—
300Dic17—82no—
301C24×C4—171no—
302C2×#117—344no—
303C2×#117—344no—
304C2×#117—344no—
305C22×C4⋊C4—171no—
306C22×C4⋊C4—171no—
307C22×C4⋊C4—174no—
308C22×C4⋊C4C2×D1711yescharacteristic
309C22×C4⋊C4C2×D1711yescharacteristic
310C22×C22⋊C4—172no—
311C22×C22⋊C4—172no—
312C22×C22⋊C4—172no—
313C22×C42—171no—
314C22×C42—172no—
315C22×C42C2×D1711yescharacteristic
316#384—344no—
317#382—344no—
318#380—344no—
319C4×C22⋊C4—178no—
320C2×#95—172no—
321C2×#95—172no—
322C2×#95—174no—
323C2×#95—174no—
324C2×#95C2×D1714yes—
325C2×C17#228511yescharacteristic
326C2×C17C2×#38511yescharacteristic
327C2×C17#231711yescharacteristic
328C2×C17#235212yes—
329C2×C17#235412yes—
330C2×C17#243111yescharacteristic
331C2×C17#243211yescharacteristic
332C2×C17#243912yes—
333C2×C17#244112yes—
334C2×C17#249911yescharacteristic
335C2×C17#250011yescharacteristic
336D17—44no—
337C25—171no—
338C23×C4—171no—
339C23×C4—174no—
340C23×C4—174no—
341C23×C4—174no—
342C23×C4—172no—
343C23×C4—172no—
344C23×C4—172no—
345C23×C4—342no—
346C23×C4—342no—
347C23×C4—342no—
348C23×C4—342no—
349C23×C4C22×D1712yes—
350C23×C4C22×D1712yes—
351C23×C4C22×D1712yes—
352C23×C4C22×D1711yescharacteristic
353#117—3432no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—174no—
360C2×C4⋊C4—342no—
361C2×C4⋊C4—342no—
362C2×C4⋊C4—348no—
363C2×C4⋊C4—342no—
364C2×C4⋊C4—342no—
365C2×C4⋊C4—342no—
366C2×C4⋊C4—348no—
367C2×C4⋊C4—684no—
368C2×C4⋊C4D6814yes—
369C2×C22⋊C4—178no—
370C2×C22⋊C4—344no—
371C2×C22⋊C4—344no—
372C2×C22⋊C4—344no—
373C2×C22⋊C4—344no—
374C2×C22⋊C4—344no—
375C2×C22⋊C4—344no—
376C2×C22⋊C4—684no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—174no—
380C2×C42—344no—
381C2×C42—344no—
382C2×C42—344no—
383C2×C42—342no—
384C2×C42—342no—
385C2×C42C4×D1714yes—
386#95—28no—
387#95—178no—
388#95—344no—
389#95—348no—
390#95—684no—
391C17#1387911yescharacteristic, complemented, Sylow
392C24—172no—
393C24—172no—
394C24—172no—
395C24—341no—
396C24—341no—
397C24—341no—
398C24—342no—
399C24—341no—
400C24—342no—
401C24—342no—
402C24—342no—
403C24C23×D1711yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—172no—
418C22×C4—172no—
419C22×C4—172no—
420C22×C4—1716no—
421C22×C4—174no—
422C22×C4—174no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—348no—
428C22×C4—348no—
429C22×C4—348no—
430C22×C4—348no—
431C22×C4—348no—
432C22×C4—348no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—344no—
440C22×C4—342no—
441C22×C4—344no—
442C22×C4—342no—
443C22×C4—342no—
444C22×C4—342no—
445C22×C4—342no—
446C22×C4—344no—
447C22×C4—344no—
448C22×C4C2×C4×D1714yes—
449C22×C4C2×C4×D1712yes—
450C22×C4C2×D6814yes—
451C22×C4C2×D6812yes—
452C22×C4#6325212yes—
453C22×C4D4×D1712yes—
454C22×C4D4×D1712yes—
455C22×C4#6325414yes—
456C22×C4#6325412yes—
457C22×C4Q8×D1712yes—
458C22×C4#6325614yes—
459C4⋊C4—28no—
460C4⋊C4—44no—
461C4⋊C4—348no—
462C4⋊C4—6816no—
463C4⋊C4—684no—
464C22⋊C4—3416no—
465C22⋊C4—3416no—
466C22⋊C4—688no—
467C42—28no—
468C42—3416no—
469C42—348no—
470C23—174no—
471C23—344no—
472C23—342no—
473C23—342no—
474C23—344no—
475C23—342no—
476C23—342no—
477C23—342no—
478C23—342no—
479C23—342no—
480C23—342no—
481C23—344no—
482C23—342no—
483C23—342no—
484C23—344no—
485C23—682no—
486C23—682no—
487C23—682no—
488C23—682no—
489C23—682no—
490C23—682no—
491C23—682no—
492C23—682no—
493C23C22×C4×D1711yescharacteristic
494C23C22×D6811yescharacteristic
495C23C2×#6325211yescharacteristic
496C23C2×D4×D1711yescharacteristic
497C23C2×D4×D1711yescharacteristic
498C23C2×#6325412yes—
499C23C2×#6325411yescharacteristic
500C23C2×Q8×D1711yescharacteristic
501C23C2×#6325612yes—
502C23#1058769612yes—
503C23#1058769712yes—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—174no—
521C2×C4—342no—
522C2×C4—342no—
523C2×C4—342no—
524C2×C4—342no—
525C2×C4—342no—
526C2×C4—342no—
527C2×C4—3432no—
528C2×C4—344no—
529C2×C4—344no—
530C2×C4—344no—
531C2×C4—344no—
532C2×C4—348no—
533C2×C4—682no—
534C2×C4—684no—
535C2×C4—682no—
536C2×C4—682no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—684no—
541C2×C4—682no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4—684no—
545C2×C4C4×D6814yes—
546C2×C4#1058756714yes—
547C2×C4#1058756814yes—
548C2×C4#1058756914yes—
549C2×C4C4⋊C4×D1714yes—
550C2×C4#1058757818yescomplemented
551C2×C4#1058758214yes—
552C2×C4#1058758314yes—
553C22—344no—
554C22—344no—
555C22—344no—
556C22—682no—
557C22—682no—
558C22—682no—
559C22—684no—
560C22—682no—
561C22—684no—
562C22—684no—
563C22—684no—
564C22C2×C4⋊C4×D1711yescharacteristic
565C22C2×#1058757812yes—
566C22C2×#1058756711yescharacteristic
567C22C2×C4×D6811yescharacteristic
568C22#1176328912yes—
569C22C2×#1058756811yescharacteristic
570C22C2×#1058758211yescharacteristic
571C22C2×#1058758311yescharacteristic
572C22C2×#1058756911yescharacteristic
573C22#1176335711yescharacteristic
574C22#1176336011yescharacteristic
575C22#1176336412yes—
576C22#1176337711yescharacteristic
577C22#1176337911yescharacteristic
578C22#1176343912yes—
579C22#1176345012yes—
580C22#1176346312yes—
581C22#1176346512yes—
582C22#1176348112yes—
583C22#1176350112yes—
584C22#1176351112yes—
585C22#1176352412yes—
586C22#1176353212yes—
587C4—28no—
588C4—44no—
589C4—82no—
590C4—344no—
591C4—684no—
592C4—684no—
593C4—1362no—
594C4#1176268318yes—
595C2—684no—
596C2#1453118911yescharacteristic
597C2C2×#1176268311yescharacteristic
598C2#1453140911yescharacteristic
599C2#1453171512yes—
600C2#1453173112yes—
601C2#1453240511yescharacteristic
602C2#1453241411yescharacteristic
603C2#1453247712yes—
604C2#1453249512yes—
605C2#1453323911yescharacteristic
606C2#1453324811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×C4⋊#95C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC4⋊#95D17
hypocenterC17#13879
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#11762679C2semidirect
3. C2×#11762674C2semidirect
7. #14527018C2semidirect
8. #14527013C2semidirect
9. C17×C4⋊#95C2semidirect
25. C4×#63227C4semidirect
154. C2×C4×D17C4⋊C4semidirect
218. C2×C4×C17#117semidirect
258. C4⋊#95D17semidirect
391. C17#13879semidirect
550. C2×C4#10587578semidirect

Maximal subgroups

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

subgroupindex
C22×#105875782
C2×#117626792
C2×#117626742
C2×#117626662
C2×C4×#632272
#145270302
#145270182
#145270132
C17×C4⋊#952
#1387917