Finite Group Databasebeta

← #16956313

Subgroups — #16956313

Order 4352.

2222 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D17C211yesmaximal, characteristic
2C2×#11762674C211yesmaximal, characteristic, complemented
3C2×#11762666C211yesmaximal, characteristic
4C2×C4×#63227C211yesmaximal, characteristic
5#14527042C214yesmaximal, complemented
6#14527014C211yesmaximal, characteristic, complemented
7#14527013C211yesmaximal, characteristic, complemented
8#14527005C214yesmaximal
9C2×C17×#382C211yesmaximal, characteristic, complemented
10C23×C4×D17C2212yes—
11C23×C4×D17C2211yescharacteristic
12C22×C4⋊Dic17C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14C22×#63225C2212yes—
15C2×C4⋊C4×D17—24no—
16C2×C4⋊C4×D17—24no—
17C2×C4⋊C4×D17—24no—
18C22×#63227C2212yes—
19C22×#63227C2212yes—
20C22×C4×Dic17C2211yescharacteristic
21#11762674—24no—
22#11762666—24no—
23C4×#63227C418yescomplemented
24C2×#10587502C2214yes—
25C2×#10587502C2212yes—
26C2×#10587502C2214yes—
27C2×#10587502C2214yes—
28C2×#10587502C2212yes—
29C22×C4⋊C4×C17C2211yescharacteristic
30C22×C4⋊C4×C17C2211yescharacteristic
31C22×C42×C17C2211yescharacteristic
32C17×#382—24no—
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×D17—24no—
48C22×C4×D17—24no—
49C22×C4×D17—24no—
50C22×C4×D17—28no—
51C22×C4×D17—44no—
52C22×C4×D17C2×C414yes—
53C22×C4×D17D414yes—
54C22×C4×D17Q814yes—
55C2×C4⋊C4×C17—22no—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17—22no—
59C2×C4⋊C4×C17—22no—
60C2×C4⋊C4×C17D414yes—
61C2×C42×C17—22no—
62C2×C42×C17—22no—
63C2×C42×C17C2×C414yescomplemented
64C2×#63227—24no—
65C2×#63227—24no—
66C2×#63227—24no—
67C2×#63227—28no—
68C2×#63227C2×C418yes—
69C2×C4⋊Dic17—22no—
70C2×C4⋊Dic17—22no—
71C2×C4⋊Dic17—22no—
72C2×C4⋊Dic17—22no—
73C2×C4⋊Dic17—22no—
74C2×C4⋊Dic17D414yes—
75C2×#63225—24no—
76C2×#63225—24no—
77C2×#63225—24no—
78C2×C4×Dic17—22no—
79C2×C4×Dic17—22no—
80C2×C4×Dic17C2×C414yescomplemented
81C4⋊C4×D17—232no—
82C17×#95—28no—
83#10587502—24no—
84#10587502—28no—
85#10587502—44no—
86#10587502—44no—
87#10587502C2×C418yescomplemented
88C24×C17C2411yescharacteristic
89C23×D17—21no—
90C23×D17—21no—
91C23×D17—21no—
92C23×D17—21no—
93C23×D17—22no—
94C23×D17—22no—
95C23×D17C22×C412yes—
96C23×D17C2×D412yes—
97C23×D17C2×Q812yes—
98C23×D17#4114yes—
99C23×D17#4114yes—
100C22×C4×C17—22no—
101C22×C4×C17—22no—
102C22×C4×C17—22no—
103C22×C4×C17—24no—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—24no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—24no—
114C22×C4×C17—24no—
115C22×C4×C17C22×C414yes—
116C22×C4×C17C22×C412yes—
117C22×C4×C17C2×D412yes—
118C22×C4×C17C2×D412yes—
119C22×C4×C17C2×D414yes—
120C22×C4×C17C2×D412yes—
121C22×C4×C17C2×Q812yes—
122C22×C4×C17#4112yes—
123C22×C4×C17#4112yes—
124C22×Dic17—21no—
125C22×Dic17—21no—
126C22×Dic17—21no—
127C22×Dic17—21no—
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—22no—
131C22×Dic17—24no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—24no—
136C22×Dic17—22no—
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×D17—216no—
155C2×C4×D17—416no—
156C2×C4×D17—416no—
157C2×C4×D17—48no—
158C2×C4×D17—48no—
159C2×C4×D17C4⋊C4116yes—
160C4⋊C4×C17—28no—
161C4⋊C4×C17—44no—
162C42×C17—28no—
163#63227—216no—
164#63227—48no—
165C4⋊Dic17—28no—
166C4⋊Dic17—44no—
167#63225—216no—
168C4×Dic17—28no—
169#13881—171nomaximal, Sylow
170C23×C17C23×C411yescharacteristic
171C23×C17C22×D411yescharacteristic
172C23×C17C22×D411yescharacteristic
173C23×C17C22×D411yescharacteristic
174C23×C17C22×Q811yescharacteristic
175C23×C17C2×#4111yescharacteristic
176C23×C17C2×#4111yescharacteristic
177C23×C17C2×#4112yes—
178C23×C17C2×#4112yes—
179C23×C17H(5,𝔽2)12yes—
180C23×C172−1+412yes—
181C22×D17—22no—
182C22×D17—22no—
183C22×D17—24no—
184C22×D17—22no—
185C22×D17—22no—
186C22×D17—22no—
187C22×D17—22no—
188C22×D17—22no—
189C22×D17—22no—
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×D17C2×C4⋊C414yes—
200C22×D17#11718yes—
201C22×D17#12414yes—
202C22×D17#12414yes—
203C22×D17#12518yes—
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—42no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—44no—
220C2×C4×C17—44no—
221C2×C4×C17C2×C4⋊C414yes—
222C2×C4×C17C4×D414yes—
223C2×C4×C17C4×D414yes—
224C2×C4×C17#12014yes—
225C2×C4×C17C22⋊Q814yes—
226C2×C4×C17C22⋊Q814yes—
227C2×C4×C17#12314yes—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—22no—
231C2×Dic17—22no—
232C2×Dic17—22no—
233C2×Dic17—22no—
234C2×Dic17—24no—
235C2×Dic17—24no—
236C2×Dic17—24no—
237C2×Dic17—24no—
238C2×Dic17—28no—
239C2×Dic17—42no—
240C2×Dic17—42no—
241C2×Dic17—44no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—44no—
250C2×Dic17—44no—
251C2×Dic17C2×C4⋊C414yes—
252C4×D17—232no—
253C4×D17—432no—
254C23×C4⋊C4—171no—
255C2×#382—171no—
256C2×#382D1711yescharacteristic, complemented
257C2×#380—171no—
258C2×C4×C22⋊C4—171no—
259#1442—174no—
260C4⋊#95—171no—
261C4⋊#95—171no—
262C22⋊#95—174no—
263C22×C17C22×C4⋊C411yescharacteristic
264C22×C17C2×#11712yes—
265C22×C17C2×C4×D411yescharacteristic
266C22×C17C2×C4×D411yescharacteristic
267C22×C17#52012yes—
268C22×C17C2×#12011yescharacteristic
269C22×C17C2×C22⋊Q811yescharacteristic
270C22×C17C2×C22⋊Q811yescharacteristic
271C22×C17C2×#12311yescharacteristic
272C22×C17C2×#12411yescharacteristic
273C22×C17C2×#12411yescharacteristic
274C22×C17C2×#12512yes—
275C22×C17#53511yescharacteristic
276C22×C17#53611yescharacteristic
277C22×C17#54412yes—
278C22×C17#54612yes—
279C22×C17#54612yes—
280C22×C17#54812yes—
281C22×C17#54812yes—
282C22×C17#55212yes—
283C22×C17#55212yes—
284C22×C17#55312yes—
285C22×C17#55312yes—
286C2×D17—24no—
287C2×D17—28no—
288C2×D17—28no—
289C2×D17—24no—
290C2×D17—24no—
291C2×D17—28no—
292C2×D17—28no—
293C2×D17—24no—
294C2×D17—24no—
295C2×D17—24no—
296C2×D17#38218yescomplemented
297C4×C17—44no—
298C4×C17—44no—
299C4×C17—82no—
300C4×C17#38518yes—
301Dic17—24no—
302Dic17—44no—
303Dic17—44no—
304Dic17—82no—
305C24×C4—171no—
306C24×C4—172no—
307C22×C4⋊C4—171no—
308C22×C4⋊C4—172no—
309C22×C4⋊C4—171no—
310C22×C4⋊C4—344no—
311C22×C4⋊C4—344no—
312C22×C4⋊C4—344no—
313C22×C4⋊C4C2×D1711yescharacteristic
314C22×C4⋊C4C2×D1711yescharacteristic
315C22×C22⋊C4—172no—
316C22×C22⋊C4—172no—
317C22×C42—171no—
318C22×C42C2×D1711yescharacteristic
319#382—24no—
320#382—344no—
321#380—344no—
322C4×C22⋊C4—178no—
323C2×#95—172no—
324C2×#95—174no—
325C2×#95—172no—
326C2×#95—174no—
327C2×#95—174no—
328C2×#95C2×D1714yes—
329C2×C17C2×#38211yescharacteristic
330C2×C17C2×#38511yescharacteristic
331C2×C17#231711yescharacteristic
332C2×C17#235212yes—
333C2×C17#235212yes—
334C2×C17#243511yescharacteristic
335C2×C17#243611yescharacteristic
336C2×C17#244212yes—
337C2×C17#244212yes—
338C2×C17#250311yescharacteristic
339C2×C17#250411yescharacteristic
340D17—28no—
341C25—171no—
342C23×C4—171no—
343C23×C4—174no—
344C23×C4—174no—
345C23×C4—174no—
346C23×C4—172no—
347C23×C4—172no—
348C23×C4—172no—
349C23×C4—344no—
350C23×C4—344no—
351C23×C4—348no—
352C23×C4—342no—
353C23×C4—342no—
354C23×C4—342no—
355C23×C4—342no—
356C23×C4—344no—
357C23×C4—684no—
358C23×C4C22×D1712yes—
359C23×C4C22×D1712yes—
360C23×C4C22×D1712yes—
361C23×C4C22×D1711yescharacteristic
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—174no—
368C2×C4⋊C4—3432no—
369C2×C4⋊C4—344no—
370C2×C4⋊C4—344no—
371C2×C4⋊C4—342no—
372C2×C4⋊C4—344no—
373C2×C4⋊C4—342no—
374C2×C4⋊C4—342no—
375C2×C4⋊C4—342no—
376C2×C4⋊C4—342no—
377C2×C4⋊C4C22⋊D1714yes—
378C2×C22⋊C4—178no—
379C2×C22⋊C4—344no—
380C2×C22⋊C4—344no—
381C2×C22⋊C4—344no—
382C2×C22⋊C4—348no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—174no—
386C2×C42—342no—
387C2×C42—342no—
388C2×C42C4×D1714yescomplemented
389#95—28no—
390#95—178no—
391#95—344no—
392#95—348no—
393#95—684no—
394#95—684no—
395C17#1388111yescharacteristic, complemented, Sylow
396C24—172no—
397C24—172no—
398C24—172no—
399C24—174no—
400C24—174no—
401C24—341no—
402C24—341no—
403C24—341no—
404C24—342no—
405C24—342no—
406C24—341no—
407C24C23×D1711yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—172no—
424C22×C4—172no—
425C22×C4—172no—
426C22×C4—1716no—
427C22×C4—174no—
428C22×C4—174no—
429C22×C4—3416no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4—348no—
435C22×C4—348no—
436C22×C4—348no—
437C22×C4—348no—
438C22×C4—348no—
439C22×C4—348no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—342no—
444C22×C4—342no—
445C22×C4—344no—
446C22×C4—342no—
447C22×C4—344no—
448C22×C4—342no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4—344no—
454C22×C4—344no—
455C22×C4—6816no—
456C22×C4—6816no—
457C22×C4—688no—
458C22×C4—688no—
459C22×C4C2×C4×D1714yes—
460C22×C4C2×C4×D1712yes—
461C22×C4#6325212yes—
462C22×C4D4×D1712yes—
463C22×C4D4×D1712yes—
464C22×C4Q8×D1712yes—
465C22×C4#6325612yes—
466C22×C4C2×C22⋊D1714yes—
467C22×C4C2×C22⋊D1712yes—
468C4⋊C4—28no—
469C4⋊C4—44no—
470C4⋊C4—3416no—
471C4⋊C4—348no—
472C4⋊C4—684no—
473C22⋊C4—3416no—
474C22⋊C4—688no—
475C42—28no—
476C42—348no—
477C23—174no—
478C23—178no—
479C23—174no—
480C23—174no—
481C23—178no—
482C23—344no—
483C23—344no—
484C23—344no—
485C23—344no—
486C23—344no—
487C23—344no—
488C23—342no—
489C23—342no—
490C23—344no—
491C23—342no—
492C23—342no—
493C23—342no—
494C23—342no—
495C23—342no—
496C23—342no—
497C23—344no—
498C23—344no—
499C23—344no—
500C23C22×C4×D1711yescharacteristic
501C23C2×#6325211yescharacteristic
502C23C2×D4×D1711yescharacteristic
503C23C2×D4×D1711yescharacteristic
504C23C2×Q8×D1711yescharacteristic
505C23C2×#6325611yescharacteristic
506C23#41×D1712yes—
507C23#41×D1712yes—
508C23#1058769612yes—
509C23#1058769712yes—
510C23C22×C22⋊D1711yescharacteristic
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—174no—
529C2×C4—342no—
530C2×C4—342no—
531C2×C4—342no—
532C2×C4—342no—
533C2×C4—342no—
534C2×C4—342no—
535C2×C4—3432no—
536C2×C4—344no—
537C2×C4—344no—
538C2×C4—344no—
539C2×C4—344no—
540C2×C4—348no—
541C2×C4—6832no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4—682no—
545C2×C4—682no—
546C2×C4—682no—
547C2×C4—684no—
548C2×C4—682no—
549C2×C4—684no—
550C2×C4—684no—
551C2×C4—682no—
552C2×C4—682no—
553C2×C4—682no—
554C2×C4C4⋊C4×D1714yes—
555C2×C4#1058757914yes—
556C2×C4#1058758014yes—
557C2×C4#1058758214yes—
558C2×C4C4×C22⋊D1714yes—
559C2×C4#1058762314yes—
560C2×C4#1058763214yes—
561C22—178no—
562C22—344no—
563C22—344no—
564C22—344no—
565C22—344no—
566C22—348no—
567C22—348no—
568C22—344no—
569C22—348no—
570C22—344no—
571C22—348no—
572C22C2×C4⋊C4×D1711yescharacteristic
573C22#117×D1712yes—
574C22C2×C4×C22⋊D1711yescharacteristic
575C22C2×#1058757911yescharacteristic
576C22#1176328912yes—
577C22C2×#1058762311yescharacteristic
578C22C2×#1058763211yescharacteristic
579C22C2×#1058758211yescharacteristic
580C22C2×#1058758011yescharacteristic
581C22#124×D1711yescharacteristic
582C22#124×D1711yescharacteristic
583C22#125×D1712yes—
584C22#1176337711yescharacteristic
585C22#1176337911yescharacteristic
586C22#1176343912yes—
587C22#1176344712yes—
588C22#1176345212yes—
589C22#1176346612yes—
590C22#1176346712yes—
591C22#1176349712yes—
592C22#1176349812yes—
593C22#1176350312yes—
594C22#1176351212yes—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—344no—
599C4—684no—
600C4—684no—
601C4—1362no—
602C4#1176268018yes—
603C2—348no—
604C2D17×#38211yescharacteristic
605C2C2×#1176268011yescharacteristic
606C2#1453140911yescharacteristic
607C2#1453171012yes—
608C2#1453171412yes—
609C2#1453243211yescharacteristic
610C2#1453243911yescharacteristic
611C2#1453250412yes—
612C2#1453251112yes—
613C2#1453326611yescharacteristic
614C2#1453327311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#382C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC2×#382D17
hypocenterC17#13881
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#11762674C2semidirect
5. #14527042C2semidirect
6. #14527014C2semidirect
7. #14527013C2semidirect
9. C2×C17×#382C2semidirect
23. C4×#63227C4semidirect
63. C2×C42×C17C2×C4semidirect
80. C2×C4×Dic17C2×C4semidirect
87. #10587502C2×C4semidirect
256. C2×#382D17semidirect
296. C2×D17#382semidirect
388. C2×C42C4×D17semidirect
395. C17#13881semidirect

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×#117626742
C2×#117626662
C2×C4×#632272
#145270422
#145270142
#145270132
#145270052
C2×C17×#3822
#1388117