Finite Group Databasebeta

← C2×#14533084 #17203844

Subgroups — C2×#14533084 #17203844

Order 4352.

3070 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D68C211yesmaximal, characteristic
2C22×#10587578C211yesmaximal, characteristic, complemented
3#14533084C2116yesmaximal, direct product factor
4C2×#11762735C212yesmaximal, complemented
5C2×#11762727C212yesmaximal
6C2×#11762707C211yesmaximal, characteristic, complemented
7C2×#11762699C212yesmaximal, complemented
8C2×#11762696C212yesmaximal, complemented
9C2×#11762666C211yesmaximal, characteristic, complemented
10C2×#11762666C211yesmaximal, characteristic, complemented
11C2×#11762666C211yesmaximal, characteristic, complemented
12C2×C17×#389C211yesmaximal, characteristic, complemented
13C23×D68C2211yescharacteristic
14C23×C4×D17C2211yescharacteristic
15C23×C4×D17C2211yescharacteristic
16C2×C4×D68—28no—
17C2×C4×D68C2218yes—
18C2×#10587578—28no—
19C2×#10587578C2218yescomplemented
20C22×C4⋊Dic17C2211yescharacteristic
21C22×C4⋊Dic17C2211yescharacteristic
22C22×C4⋊Dic17C2211yescharacteristic
23C22×#63225C2212yes—
24C22×#63227C2212yes—
25C22×#63227C2212yes—
26C22×#63227C2211yescharacteristic
27C22×#63227C2211yescharacteristic
28C22×#63227C2212yes—
29C22×#63227C2212yes—
30C22×C4×Dic17C2212yes—
31#11762735C22116yescomplemented
32#11762727C22116yes—
33#11762707C2218yescomplemented
34#11762699C22116yescomplemented
35#11762696C22116yescomplemented
36#11762666C2218yescomplemented
37#11762666C2218yescomplemented
38#11762666C2218yescomplemented
39C2×#10587502C2212yes—
40C2×#10587502C2212yes—
41C2×#10587502C2212yes—
42C2×#10587502C2212yes—
43C22×C4⋊C4×C17C2211yescharacteristic
44C22×C4⋊C4×C17C2211yescharacteristic
45C22×C4⋊C4×C17C2212yes—
46C22×C42×C17C2211yescharacteristic
47C17×#389C2218yescomplemented
48C2×C17×#95C2212yes—
49C24×D17C2311yescharacteristic
50C24×D17C2311yescharacteristic
51C23×C4×C17C2311yescharacteristic
52C23×C4×C17C2312yes—
53C23×C4×C17C2312yes—
54C23×C4×C17C2311yescharacteristic
55C23×C4×C17C2311yescharacteristic
56C23×Dic17C2312yes—
57C23×Dic17C2312yes—
58C23×Dic17C2311yescharacteristic
59C23×Dic17C2311yescharacteristic
60C22×D68—24no—
61C22×D68—24no—
62C22×D68—24no—
63C22×D68C2314yes—
64C22×C4×D17—24no—
65C22×C4×D17—24no—
66C22×C4×D17—24no—
67C22×C4×D17—24no—
68C22×C4×D17—24no—
69C22×C4×D17D418yes—
70C22×C4×D17C2314yes—
71C22×C4×D17C2314yes—
72C2×C4⋊C4×C17—24no—
73C2×C4⋊C4×C17—24no—
74C2×C4⋊C4×C17—44no—
75C2×C4⋊C4×C17C2314yes—
76C2×C4⋊C4×C17C2314yes—
77C2×C4⋊C4×C17C2318yes—
78C2×C42×C17—24no—
79C2×C42×C17C2314yes—
80C2×#63227—28no—
81C2×#63227—24no—
82C2×#63227—28no—
83C2×#63227—24no—
84C2×#63227—44no—
85C2×#63227—44no—
86C2×#63227C2314yes—
87C2×#63227C2318yes—
88C2×#63227C2318yes—
89C2×#63227C2314yes—
90C2×#63227C2318yes—
91C2×#63227C2318yes—
92C2×C4⋊Dic17—24no—
93C2×C4⋊Dic17—24no—
94C2×C4⋊Dic17—24no—
95C2×C4⋊Dic17C2314yes—
96C2×C4⋊Dic17C2314yes—
97C2×C4⋊Dic17C2314yes—
98C2×#63225—28no—
99C2×#63225C2318yes—
100C2×C4×Dic17—28no—
101C2×C4×Dic17C2318yes—
102#10587578—232no—
103C4×D68—232no—
104C17×#95C2318yes—
105#10587502C2318yes—
106#10587502C2318yes—
107#10587502C2318yes—
108#10587502C2318yes—
109C24×C17C2411yescharacteristic
110C23×D17—22no—
111C23×D17—22no—
112C23×D17—22no—
113C23×D17—22no—
114C23×D17—22no—
115C23×D17—22no—
116C23×D17—22no—
117C23×D17—22no—
118C23×D17—22no—
119C23×D17—22no—
120C23×D17—22no—
121C23×D17—22no—
122C23×D17C2×D414yes—
123C23×D17#4114yes—
124C23×D17C2412yes—
125C23×D17C2412yes—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—24no—
129C22×C4×C17—22no—
130C22×C4×C17—24no—
131C22×C4×C17—24no—
132C22×C4×C17—24no—
133C22×C4×C17—24no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—24no—
137C22×C4×C17C2×D414yes—
138C22×C4×C17#4114yes—
139C22×C4×C17#4114yes—
140C22×C4×C17#4114yes—
141C22×C4×C17C2414yes—
142C22×C4×C17C2412yes—
143C22×C4×C17C2414yes—
144C22×C4×C17C2412yes—
145C22×C4×C17C2412yes—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—24no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—24no—
153C22×Dic17—24no—
154C22×Dic17—24no—
155C22×Dic17—24no—
156C22×Dic17—24no—
157C22×Dic17C2×D414yes—
158C22×Dic17C2412yes—
159C22×Dic17C2414yes—
160C22×Dic17C2414yes—
161C22×Dic17C2412yes—
162C2×D68—232no—
163C2×D68—28no—
164C2×D68—28no—
165C2×D68—28no—
166C2×C4×D17—232no—
167C2×C4×D17—28no—
168C2×C4×D17—28no—
169C2×C4×D17—28no—
170C2×C4×D17—28no—
171C2×C4×D17—28no—
172C2×C4×D17—416no—
173C2×C4×D17C2×D4116yes—
174C4⋊C4×C17—28no—
175C4⋊C4×C17—28no—
176C4⋊C4×C17—48no—
177C42×C17—28no—
178#63227—28no—
179#63227—216no—
180#63227—216no—
181#63227—28no—
182#63227—48no—
183#63227—48no—
184C4⋊Dic17—28no—
185C4⋊Dic17—28no—
186C4⋊Dic17—28no—
187#63225—216no—
188C4×Dic17—216no—
189C2×#2485—171nomaximal, Sylow
190C23×C17C22×D412yes—
191C23×C17C2×#4112yes—
192C23×C17C2×#4112yes—
193C23×C17C2×#4112yes—
194C23×C17C2×#4112yes—
195C23×C17H(5,𝔽2)12yes—
196C23×C172−1+412yes—
197C23×C17C2511yescharacteristic
198C22×D17—28no—
199C22×D17—28no—
200C22×D17—28no—
201C22×D17—28no—
202C22×D17—28no—
203C22×D17—28no—
204C22×D17—22no—
205C22×D17—22no—
206C22×D17—22no—
207C22×D17—22no—
208C22×D17—22no—
209C22×D17—22no—
210C22×D17—22no—
211C22×D17—22no—
212C22×D17—22no—
213C22×D17—22no—
214C22×D17—22no—
215C22×D17—22no—
216C22×D17—44no—
217C22×D17—44no—
218C22×D17—44no—
219C22×D17—44no—
220C22×D17—44no—
221C22×D17—44no—
222C22×D17—44no—
223C22×D17—44no—
224C22×D17C22×D414yes—
225C22×D17C2×#4114yes—
226C2×C4×C17—28no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—22no—
234C2×C4×C17—24no—
235C2×C4×C17—24no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—48no—
239C2×C4×C17—44no—
240C2×C4×C17—48no—
241C2×C4×C17#123116yes—
242C2×C4×C17C22×D414yes—
243C2×C4×C17C2×#4114yes—
244C2×C4×C17C2×#4114yes—
245C2×C4×C17C2×#4114yes—
246C2×Dic17—28no—
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—24no—
251C2×Dic17—22no—
252C2×Dic17—22no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—24no—
258C2×Dic17—44no—
259C2×Dic17—48no—
260C2×Dic17—48no—
261C2×Dic17C22×D414yes—
262D68—232no—
263C4×D17—232no—
264C4×D17—416no—
265C22×C4×D4—171no—
266C22×#117—171no—
267#2485—1716no—
268C2×#397—172no—
269C2×#394—172no—
270C2×#389—171no—
271C2×#389D1711yescharacteristic, complemented
272C2×#388—172no—
273C2×#387—172no—
274C2×#380—171no—
275C2×#380—171no—
276C2×#380—171no—
277C22×C17C2×#12314yes—
278C22×C17#52514yes—
279C22×C17#53714yes—
280C22×C17#55214yes—
281C22×C17#55314yes—
282C22×C17#55514yes—
283C22×C17#55614yes—
284C22×C17C23×D411yescharacteristic
285C22×C17C22×#4111yescharacteristic
286C22×C17C22×#4111yescharacteristic
287C22×C17C22×#4111yescharacteristic
288C22×C17C22×#4111yescharacteristic
289C22×C17C2×H(5,𝔽2)11yescharacteristic
290C22×C17C2×2−1+411yescharacteristic
291C2×D17—24no—
292C2×D17—24no—
293C2×D17—24no—
294C2×D17—24no—
295C2×D17—24no—
296C2×D17—24no—
297C2×D17—42no—
298C2×D17—42no—
299C2×D17—42no—
300C2×D17—42no—
301C2×D17—42no—
302C2×D17—42no—
303C2×D17—42no—
304C2×D17—42no—
305C2×D17—416no—
306C2×D17—416no—
307C4×C17—24no—
308C4×C17—42no—
309C4×C17—44no—
310C4×C17—44no—
311C4×C17C2×#12318yes—
312Dic17—24no—
313Dic17—42no—
314Dic17—44no—
315Dic17—44no—
316C23×D4—171no—
317C24×C4—171no—
318C24×C4—171no—
319C2×C4×D4—178no—
320C2×C4×D4—348no—
321C2×#117—178no—
322C2×#117—348no—
323C22×C4⋊C4—172no—
324C22×C4⋊C4—171no—
325C22×C4⋊C4—171no—
326C22×C4⋊C4—171no—
327C22×C4⋊C4C2×D1711yescharacteristic
328C22×C4⋊C4C2×D1711yescharacteristic
329C22×C4⋊C4C2×D1712yes—
330C22×C22⋊C4—172no—
331C22×C22⋊C4—172no—
332C22×C22⋊C4—171no—
333C22×C22⋊C4—171no—
334C22×C22⋊C4—172no—
335C22×C22⋊C4—172no—
336C22×C42—172no—
337C22×C42C2×D1711yescharacteristic
338#397—1716no—
339#394—1716no—
340#389—178no—
341#389C2×D1718yescomplemented
342#388—1716no—
343#387—1716no—
344#380—178no—
345#380—178no—
346#380—178no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—172no—
351C2×#95C2×D1712yes—
352C2×C17#248518yescomplemented
353C2×C17C22×#12311yescharacteristic
354C2×C17C2×#52511yescharacteristic
355C2×C17C2×#53711yescharacteristic
356C2×C17C2×#55211yescharacteristic
357C2×C17C2×#55311yescharacteristic
358C2×C17C2×#55511yescharacteristic
359C2×C17C2×#55611yescharacteristic
360D17—44no—
361D17—44no—
362C25—171no—
363C25—171no—
364C22×D4—174no—
365C22×D4—344no—
366C22×D4—344no—
367C22×D4—344no—
368C23×C4—171no—
369C23×C4—178no—
370C23×C4—174no—
371C23×C4—172no—
372C23×C4—172no—
373C23×C4—171no—
374C23×C4—174no—
375C23×C4—344no—
376C23×C4—344no—
377C23×C4—344no—
378C23×C4—344no—
379C23×C4—344no—
380C23×C4C22×D1712yes—
381C23×C4C22×D1711yescharacteristic
382C23×C4C22×D1712yes—
383C23×C4C22×D1711yescharacteristic
384C23×C4C22×D1711yescharacteristic
385C4×D4—3432no—
386#117—3432no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—44no—
390C2×C4⋊C4—174no—
391C2×C4⋊C4—174no—
392C2×C4⋊C4—174no—
393C2×C4⋊C4—178no—
394C2×C4⋊C4—344no—
395C2×C4⋊C4—344no—
396C2×C4⋊C4—348no—
397C2×C4⋊C4—344no—
398C2×C4⋊C4C22×D1714yes—
399C2×C4⋊C4C22×D1714yes—
400C2×C4⋊C4C22×D1718yes—
401C2×C22⋊C4—178no—
402C2×C22⋊C4—174no—
403C2×C22⋊C4—174no—
404C2×C22⋊C4—178no—
405C2×C22⋊C4—178no—
406C2×C22⋊C4—178no—
407C2×C22⋊C4—344no—
408C2×C22⋊C4—348no—
409C2×C22⋊C4—344no—
410C2×C22⋊C4—348no—
411C2×C22⋊C4—684no—
412C2×C22⋊C4—684no—
413C2×C42—24no—
414C2×C42—178no—
415C2×C42—348no—
416C2×C42C22×D1714yes—
417#95—178no—
418#95—178no—
419#95—178no—
420#95—178no—
421#95C22×D1718yes—
422C17C2×#248511yescharacteristic, complemented, Sylow
423C24—174no—
424C24—174no—
425C24—172no—
426C24—172no—
427C24—342no—
428C24—342no—
429C24—342no—
430C24—342no—
431C24—342no—
432C24—342no—
433C24—342no—
434C24—342no—
435C24—342no—
436C24—342no—
437C24—342no—
438C24—342no—
439C24C23×D1711yescharacteristic
440C2×D4—348no—
441C2×D4—348no—
442C2×D4—3432no—
443C2×D4—348no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—174no—
456C22×C4—172no—
457C22×C4—1716no—
458C22×C4—172no—
459C22×C4—174no—
460C22×C4—174no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—348no—
464C22×C4—348no—
465C22×C4—3432no—
466C22×C4—342no—
467C22×C4—344no—
468C22×C4—342no—
469C22×C4—344no—
470C22×C4—344no—
471C22×C4—344no—
472C22×C4—344no—
473C22×C4—342no—
474C22×C4—344no—
475C22×C4—348no—
476C22×C4—348no—
477C22×C4—348no—
478C22×C4—6816no—
479C22×C4#6325214yes—
480C22×C4D4×D1714yes—
481C22×C4#6325614yes—
482C22×C4#6325614yes—
483C22×C4C23×D1714yes—
484C22×C4C23×D1712yes—
485C22×C4C23×D1714yes—
486C22×C4C23×D1712yes—
487C22×C4C23×D1712yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—48no—
491C4⋊C4—348no—
492C4⋊C4—348no—
493C4⋊C4—3416no—
494C4⋊C4—348no—
495C22⋊C4—348no—
496C22⋊C4—3416no—
497C22⋊C4—348no—
498C22⋊C4—3416no—
499C22⋊C4—688no—
500C22⋊C4—688no—
501C42—28no—
502C42—3416no—
503C23—174no—
504C23—174no—
505C23—348no—
506C23—348no—
507C23—348no—
508C23—348no—
509C23—348no—
510C23—342no—
511C23—342no—
512C23—342no—
513C23—342no—
514C23—342no—
515C23—342no—
516C23—342no—
517C23—342no—
518C23—342no—
519C23—342no—
520C23—342no—
521C23—342no—
522C23—348no—
523C23—684no—
524C23—684no—
525C23—684no—
526C23—684no—
527C23—684no—
528C23—684no—
529C23—684no—
530C23—684no—
531C23C2×#6325212yes—
532C23C2×D4×D1712yes—
533C23#1058769112yes—
534C23C2×#6325612yes—
535C23C2×#6325612yes—
536C23#41×D1712yes—
537C23#1058769712yes—
538C23C24×D1711yescharacteristic
539D4—3432no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—174no—
556C2×C4—348no—
557C2×C4—342no—
558C2×C4—342no—
559C2×C4—3432no—
560C2×C4—344no—
561C2×C4—342no—
562C2×C4—344no—
563C2×C4—342no—
564C2×C4—344no—
565C2×C4—344no—
566C2×C4—344no—
567C2×C4—342no—
568C2×C4—344no—
569C2×C4—688no—
570C2×C4—684no—
571C2×C4—688no—
572C2×C4—6816no—
573C2×C4#10587580116yes—
574C2×C4C2×#6325214yes—
575C2×C4C2×D4×D1714yes—
576C2×C4C2×#6325614yes—
577C2×C4C2×#6325614yes—
578C22—344no—
579C22—344no—
580C22—344no—
581C22—344no—
582C22—344no—
583C22—344no—
584C22—682no—
585C22—682no—
586C22—682no—
587C22—6816no—
588C22—6816no—
589C22—682no—
590C22—682no—
591C22—682no—
592C22—682no—
593C22—682no—
594C22C2×#1058758014yes—
595C22#1176332214yes—
596C22#1176338814yes—
597C22#1176349714yes—
598C22#1176350914yes—
599C22#1176352514yes—
600C22#1176353214yes—
601C22C22×D4×D1711yescharacteristic
602C22C2×#41×D1711yescharacteristic
603C22C22×#6325211yescharacteristic
604C22C22×#6325611yescharacteristic
605C22C22×#6325611yescharacteristic
606C22C2×#1058769111yescharacteristic
607C22C2×#1058769711yescharacteristic
608C4—24no—
609C4—42no—
610C4—44no—
611C4—44no—
612C4—344no—
613C4—682no—
614C4—684no—
615C4—684no—
616C4C2×#1058758018yes—
617C2—684no—
618C2—684no—
619C2#1453308418yesdirect product factor
620C2C22×#1058758011yescharacteristic
621C2C2×#1176332211yescharacteristic
622C2C2×#1176338811yescharacteristic
623C2C2×#1176349711yescharacteristic
624C2C2×#1176350911yescharacteristic
625C2C2×#1176352511yescharacteristic
626C2C2×#1176353211yescharacteristic
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC2×C17×#389C2
socleC24×C17C24
prime radicalC23×D68C22
hypercenterC2×#389D17
hypocenterC17C2×#2485
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×#10587578C2semidirect
3. #14533084C2direct (N × H)
4. C2×#11762735C2semidirect
6. C2×#11762707C2semidirect
7. C2×#11762699C2semidirect
8. C2×#11762696C2semidirect
9. C2×#11762666C2semidirect
10. C2×#11762666C2semidirect
11. C2×#11762666C2semidirect
12. C2×C17×#389C2semidirect
19. C2×#10587578C22semidirect
31. #11762735C22semidirect
33. #11762707C22semidirect
34. #11762699C22semidirect
35. #11762696C22semidirect
36. #11762666C22semidirect
37. #11762666C22semidirect
38. #11762666C22semidirect
47. C17×#389C22semidirect
271. C2×#389D17semidirect
341. #389C2×D17semidirect
352. C2×C17#2485semidirect
422. C17C2×#2485semidirect
619. C2#14533084direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D682
C22×#105875782
#145330842
C2×#117627352
C2×#117627272
C2×#117627072
C2×#117626992
C2×#117626962
C2×#117626662
C2×C17×#3892
C2×#248517