Finite Group Databasebeta

← C4×#11762744 #16956151

Subgroups — C4×#11762744 #16956151

Order 4352.

1966 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762744C211yesmaximal, characteristic
2C2×C4×#63225C212yesmaximal, complemented
3C2×C4×#63227C212yesmaximal
4C2×C4×#63227C211yesmaximal, characteristic
5#14527052C211yesmaximal, characteristic, complemented
6#14527050C212yesmaximal, complemented
7#14527047C211yesmaximal, characteristic, complemented
8#14527028C212yesmaximal
9#14527026C211yesmaximal, characteristic
10C4×#10587502C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C17C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C22×#63225C2212yes—
14C22×#63225C2212yes—
15C22×#63227C2212yes—
16C22×#63227C2211yescharacteristic
17C22×#63227C2211yescharacteristic
18C22×#63227C2212yes—
19C22×C4×Dic17C2211yescharacteristic
20C22×C4×Dic17C2212yes—
21#11762744C418yesdirect product factor
22C4×#63225—28no—
23C4×#63227—24no—
24C4×#63227—28no—
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2211yescharacteristic
34C22×C4⋊C4×C17C2211yescharacteristic
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C42×C17C2212yes—
37C22×C42×C17C2211yescharacteristic
38C4×C4⋊C4×C17—24no—
39C2×C17×#95C2212yes—
40C24×D17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2312yes—
51C22×C4×D17—24no—
52C22×C4×D17—22no—
53C22×C4×D17—24no—
54C22×C4×D17—22no—
55C22×C4×D17Q814yes—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17—24no—
59C2×C4⋊C4×C17—22no—
60C2×C4⋊C4×C17C2×C414yescomplemented
61C2×C42×C17—24no—
62C2×C42×C17—24no—
63C2×C42×C17—24no—
64C2×C42×C17—24no—
65C2×C42×C17D414yes—
66C2×#63227—24no—
67C2×#63227—22no—
68C2×#63227—22no—
69C2×#63227—22no—
70C2×#63227—24no—
71C2×#63227—22no—
72C2×#63227—24no—
73C2×#63227—24no—
74C2×#63227—22no—
75C2×#63227—24no—
76C2×#63227C2×C414yes—
77C2×#63227C2×C418yes—
78C2×#63225—24no—
79C2×#63225—24no—
80C2×#63225—24no—
81C2×#63225—24no—
82C2×#63225—24no—
83C2×#63225C2×C418yescomplemented
84C2×C4×Dic17—22no—
85C2×C4×Dic17—24no—
86C2×C4×Dic17—24no—
87C2×C4×Dic17—24no—
88C2×C4×Dic17D418yes—
89C17×#95—24no—
90#10587502—22no—
91#10587502—22no—
92#10587502—22no—
93#10587502—24no—
94#10587502—24no—
95#10587502—24no—
96#10587502—24no—
97#10587502—24no—
98#10587502C2×C414yescomplemented
99C24×C17C2411yescharacteristic
100C23×D17—22no—
101C23×D17—21no—
102C23×D17—22no—
103C23×D17—21no—
104C23×D17—21no—
105C23×D17—22no—
106C23×D17—22no—
107C23×D17—21no—
108C23×D17C22×C412yes—
109C23×D17C2×Q812yes—
110C23×D17#4112yes—
111C22×C4×C17—22no—
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—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17C22×C414yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C2×D414yes—
132C22×C4×C17C2×D412yes—
133C22×C4×C17C2×D412yes—
134C22×C4×C17C2×D412yes—
135C22×C4×C17C2×Q812yes—
136C22×C4×C17#4114yes—
137C22×C4×C17#4112yes—
138C22×C4×C17#4114yes—
139C22×C4×C17#4112yes—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—21no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—21no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—21no—
162C22×Dic17C22×C412yes—
163C22×Dic17C22×C414yes—
164C22×Dic17C2×D414yes—
165C22×Dic17C2×D414yes—
166C22×Dic17C2×Q812yes—
167C22×Dic17#4112yes—
168C22×Dic17#4114yes—
169C2×C4×D17—216no—
170C2×C4×D17—28no—
171C2×C4×D17—48no—
172C2×C4×D17—44no—
173C2×C4×D17—44no—
174C4⋊C4×C17—28no—
175C4⋊C4×C17—28no—
176C42×C17—28no—
177C42×C17—48no—
178#63227—216no—
179#63227—28no—
180#63227—28no—
181#63227—216no—
182#63225—216no—
183#63225—216no—
184C4×Dic17—216no—
185C4×Dic17—44no—
186C4×#399—171nomaximal, Sylow
187C23×C17C23×C411yescharacteristic
188C23×C17C22×D412yes—
189C23×C17C22×D411yescharacteristic
190C23×C17C22×Q811yescharacteristic
191C23×C17C2×#4111yescharacteristic
192C23×C17C2×#4112yes—
193C23×C17C2×#4112yes—
194C23×C17C2×#4111yescharacteristic
195C23×C17C2×#4112yes—
196C23×C17C2×#4111yescharacteristic
197C23×C17C2×#4111yescharacteristic
198C22×D17—24no—
199C22×D17—22no—
200C22×D17—24no—
201C22×D17—22no—
202C22×D17—22no—
203C22×D17—24no—
204C22×D17—22no—
205C22×D17—24no—
206C22×D17—24no—
207C22×D17—22no—
208C22×D17—22no—
209C22×D17—24no—
210C22×D17—41no—
211C22×D17—42no—
212C22×D17—42no—
213C22×D17—41no—
214C22×D17—42no—
215C22×D17—42no—
216C22×D17—42no—
217C22×D17—41no—
218C22×D17—41no—
219C22×D17—42no—
220C22×D17C4×Q814yes—
221C2×C4×C17—24no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—24no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—22no—
229C2×C4×C17—24no—
230C2×C4×C17—22no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—22no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
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×C17C4×D414yes—
245C2×C4×C17C22⋊Q818yes—
246C2×C4×C17C22⋊Q814yes—
247C2×C4×C17#12318yes—
248C2×C4×C17#12314yes—
249C2×C4×C17#12714yes—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—24no—
263C2×Dic17—22no—
264C2×Dic17—24no—
265C2×Dic17—41no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—41no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—42no—
278C2×Dic17C4×D418yes—
279C4×D17—416no—
280C2×#399—171no—
281C2×C4×C4⋊C4—172no—
282C2×C4×C4⋊C4D1711yescharacteristic, complemented
283C2×C4×C22⋊C4—172no—
284C2×C4×C22⋊C4—171no—
285#1445—172no—
286#1445—171no—
287#1444—171no—
288#1439—172no—
289#1439—171no—
290C4×#95—171no—
291C22×C17C2×C4×D412yes—
292C22×C17C2×C4×D411yescharacteristic
293C22×C17C2×C4×Q811yescharacteristic
294C22×C17C4×#4112yes—
295C22×C17C4×#4111yescharacteristic
296C22×C17C2×C22⋊Q811yescharacteristic
297C22×C17C2×C22⋊Q812yes—
298C22×C17C2×#12311yescharacteristic
299C22×C17C2×#12312yes—
300C22×C17#52511yescharacteristic
301C22×C17#52512yes—
302C22×C17#52512yes—
303C22×C17#52511yescharacteristic
304C22×C17#52912yes—
305C22×C17#52912yes—
306C22×C17#52912yes—
307C22×C17#52911yescharacteristic
308C22×C17#52912yes—
309C22×C17C2×#12711yescharacteristic
310C22×C17#53211yescharacteristic
311C22×C17#53212yes—
312C22×C17#53312yes—
313C22×C17#53311yescharacteristic
314C2×D17—24no—
315C2×D17—28no—
316C2×D17—42no—
317C2×D17—42no—
318C2×D17—42no—
319C2×D17—44no—
320C2×D17—42no—
321C2×D17—44no—
322C2×D17—42no—
323C2×D17—42no—
324C2×D17—44no—
325C4×C17—24no—
326C4×C17—44no—
327C4×C17—42no—
328C4×C17—44no—
329C4×C17#39918yescomplemented
330Dic17—28no—
331Dic17—42no—
332Dic17—42no—
333Dic17—44no—
334Dic17—42no—
335C24×C4—171no—
336C22×C4⋊C4—172no—
337C22×C4⋊C4—172no—
338C22×C4⋊C4C2×D1711yescharacteristic
339C22×C4⋊C4C2×D1711yescharacteristic
340C22×C22⋊C4—172no—
341C22×C22⋊C4—171no—
342C22×C22⋊C4—172no—
343C22×C22⋊C4—171no—
344C22×C42—172no—
345C22×C42—171no—
346C22×C42C2×D1712yes—
347C22×C42C2×D1711yescharacteristic
348#399—178no—
349C4×C4⋊C4—24no—
350C4×C4⋊C4—348no—
351C4×C22⋊C4—348no—
352C4×C22⋊C4—344no—
353C2×#95—171no—
354C2×#95—172no—
355C2×#95—171no—
356C2×#95—172no—
357C2×#95—172no—
358C2×#95—172no—
359C2×#95—171no—
360C2×#95—172no—
361C2×#95—171no—
362C2×#95C2×D1712yes—
363C2×C17C4×#12311yescharacteristic
364C2×C17C4×#12312yes—
365C2×C17C4×C22⋊Q811yescharacteristic
366C2×C17C4×C22⋊Q812yes—
367C2×C17C4×#12711yescharacteristic
368C2×C17C2×#39911yescharacteristic
369C2×C17#239511yescharacteristic
370C2×C17#239512yes—
371C2×C17#239611yescharacteristic
372C2×C17#239612yes—
373C2×C17#239711yescharacteristic
374D17—44no—
375C25—171no—
376C23×C4—172no—
377C23×C4—171no—
378C23×C4—171no—
379C23×C4—172no—
380C23×C4—171no—
381C23×C4—174no—
382C23×C4—344no—
383C23×C4—344no—
384C23×C4—342no—
385C23×C4—342no—
386C23×C4C22×D1712yes—
387C23×C4C22×D1712yes—
388C23×C4C22×D1711yescharacteristic
389C23×C4C22×D1711yescharacteristic
390C23×C4C22×D1711yescharacteristic
391C2×C4⋊C4—24no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—178no—
396C2×C4⋊C4—344no—
397C2×C4⋊C4—344no—
398C2×C4⋊C4—344no—
399C2×C4⋊C4—344no—
400C2×C4⋊C4—344no—
401C2×C4⋊C4C4×D1714yescomplemented
402C2×C22⋊C4—178no—
403C2×C22⋊C4—174no—
404C2×C22⋊C4—344no—
405C2×C22⋊C4—344no—
406C2×C22⋊C4—344no—
407C2×C22⋊C4—342no—
408C2×C22⋊C4—342no—
409C2×C22⋊C4—344no—
410C2×C22⋊C4—344no—
411C2×C22⋊C4—342no—
412C2×C22⋊C4—342no—
413C2×C22⋊C4—342no—
414C2×C42—24no—
415C2×C42—24no—
416C2×C42—24no—
417C2×C42—24no—
418C2×C42—178no—
419C2×C42—344no—
420C2×C42—344no—
421C2×C42—344no—
422C2×C42—342no—
423C2×C42C22⋊D1714yes—
424#95—24no—
425#95—174no—
426#95—344no—
427#95—344no—
428#95—344no—
429#95—344no—
430#95—344no—
431#95—342no—
432#95—342no—
433#95—342no—
434C17C4×#39911yescharacteristic, complemented, Sylow
435C24—172no—
436C24—172no—
437C24—172no—
438C24—341no—
439C24—342no—
440C24—342no—
441C24—342no—
442C24—341no—
443C24—342no—
444C24—341no—
445C24—341no—
446C24C23×D1711yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—174no—
466C22×C4—174no—
467C22×C4—174no—
468C22×C4—172no—
469C22×C4—174no—
470C22×C4—172no—
471C22×C4—172no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—341no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—341no—
487C22×C4—342no—
488C22×C4—341no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—3416no—
495C22×C4—348no—
496C22×C4—688no—
497C22×C4—684no—
498C22×C4—684no—
499C22×C4C2×C4×D1714yes—
500C22×C4C2×C4×D1712yes—
501C22×C4#6325214yes—
502C22×C4#6325212yes—
503C22×C4#6325214yes—
504C22×C4D4×D1714yes—
505C22×C4Q8×D1712yes—
506C22×C4#6325612yes—
507C22×C4C2×C22⋊D1712yes—
508C22×C4C2×C22⋊D1712yes—
509C22×C4C2×C22⋊D1712yes—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—3416no—
513C4⋊C4—3416no—
514C22⋊C4—3416no—
515C22⋊C4—348no—
516C22⋊C4—3416no—
517C22⋊C4—348no—
518C42—28no—
519C42—48no—
520C42—3416no—
521C42—684no—
522C23—174no—
523C23—344no—
524C23—342no—
525C23—344no—
526C23—342no—
527C23—342no—
528C23—344no—
529C23—342no—
530C23—344no—
531C23—344no—
532C23—342no—
533C23—342no—
534C23—344no—
535C23—681no—
536C23—682no—
537C23—682no—
538C23—681no—
539C23—682no—
540C23—682no—
541C23—682no—
542C23—681no—
543C23—681no—
544C23—682no—
545C23C22×C4×D1711yescharacteristic
546C23C2×#6325211yescharacteristic
547C23C2×#6325212yes—
548C23C2×#6325212yes—
549C23C2×D4×D1712yes—
550C23C2×Q8×D1711yescharacteristic
551C23C2×#6325611yescharacteristic
552C23#41×D1712yes—
553C23#41×D1711yescharacteristic
554C23#41×D1711yescharacteristic
555C23C22×C22⋊D1711yescharacteristic
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
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—178no—
580C2×C4—344no—
581C2×C4—344no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—344no—
585C2×C4—344no—
586C2×C4—344no—
587C2×C4—342no—
588C2×C4—342no—
589C2×C4—344no—
590C2×C4—344no—
591C2×C4—342no—
592C2×C4—344no—
593C2×C4—344no—
594C2×C4—344no—
595C2×C4—681no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—681no—
599C2×C4—682no—
600C2×C4—682no—
601C2×C4—682no—
602C2×C4—682no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—682no—
608C2×C4—6816no—
609C2×C4#1058758018yes—
610C2×C4#1058758218yes—
611C2×C4C4×C22⋊D1714yes—
612C2×C4#1058761514yes—
613C2×C4#1058762614yes—
614C2×C4#1058763214yes—
615C22—344no—
616C22—348no—
617C22—682no—
618C22—682no—
619C22—682no—
620C22—684no—
621C22—682no—
622C22—684no—
623C22—682no—
624C22—682no—
625C22—684no—
626C22C4×D4×D1712yes—
627C22C2×C4×C22⋊D1711yescharacteristic
628C22C4×Q8×D1711yescharacteristic
629C22C4×#6325212yes—
630C22C4×#6325611yescharacteristic
631C22C2×#1058763211yescharacteristic
632C22C2×#1058758212yes—
633C22C2×#1058761511yescharacteristic
634C22C2×#1058758012yes—
635C22#1176331611yescharacteristic
636C22#1176331812yes—
637C22#1176331812yes—
638C22#1176332011yescharacteristic
639C22#1176333712yes—
640C22#1176333712yes—
641C22#1176334312yes—
642C22#1176334611yescharacteristic
643C22#1176334612yes—
644C22C2×#1058762611yescharacteristic
645C22#1176335511yescharacteristic
646C22#1176335612yes—
647C22#1176336312yes—
648C22#1176336311yescharacteristic
649C4—24no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—348no—
654C4—682no—
655C4—682no—
656C4—684no—
657C4—682no—
658C4#1176274418yesdirect product factor
659C2—684no—
660C2C4×#1058761511yescharacteristic
661C2C4×#1058758012yes—
662C2C4×#1058763211yescharacteristic
663C2C4×#1058758212yes—
664C2C4×#1058762611yescharacteristic
665C2C2×#1176274411yescharacteristic
666C2#1453204111yescharacteristic
667C2#1453204712yes—
668C2#1453205411yescharacteristic
669C2#1453206012yes—
670C2#1453206311yescharacteristic
subgroupG/H
centerC23×C4C22×D17
commutatorC23×C17C23×C4
FrattiniC24C23×D17
FittingC2×C4×C4⋊C4×C17C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC2×C4×C4⋊C4D17
hypocenterC17C4×#399
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23×C4 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#63225C2semidirect
5. #14527052C2semidirect
6. #14527050C2semidirect
7. #14527047C2semidirect
10. C4×#10587502C2semidirect
11. C2×C4×C4⋊C4×C17C2semidirect
21. #11762744C4direct (N × H)
60. C2×C4⋊C4×C17C2×C4semidirect
83. C2×#63225C2×C4semidirect
98. #10587502C2×C4semidirect
282. C2×C4×C4⋊C4D17semidirect
329. C4×C17#399semidirect
401. C2×C4⋊C4C4×D17semidirect
434. C17C4×#399semidirect
658. C4#11762744direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117627442
C2×C4×#632252
C2×C4×#632272
#145270522
#145270502
#145270472
#145270282
#145270262
C4×#105875022
C2×C4×C4⋊C4×C172
C4×#39917