Finite Group Databasebeta

← C2×#14531499 #17193116

Subgroups — C2×#14531499 #17193116

Order 4352.

2958 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587574C211yesmaximal, characteristic
2C22×C4×Dic34C211yesmaximal, characteristic
3C22×#10587573C211yesmaximal, characteristic
4#14531499C2116yesmaximal, direct product factor
5C2×#11762713C211yesmaximal, characteristic
6C2×#11762713C211yesmaximal, characteristic
7C2×#11762696C211yesmaximal, characteristic
8C2×#11762695C211yesmaximal, characteristic
9C2×#11762694C211yesmaximal, characteristic
10C2×#11762693C211yesmaximal, characteristic
11C2×#11762692C211yesmaximal, characteristic
12C2×#11762691C211yesmaximal, characteristic
13C2×#11762690C211yesmaximal, characteristic
14C2×#11762673C211yesmaximal, characteristic
15C2×#11762654C211yesmaximal, characteristic
16C2×C17×#387C211yesmaximal, characteristic
17C23×Dic34C2211yescharacteristic
18C23×Dic34C2211yescharacteristic
19C2×#10587574C4116yescomplemented
20C2×#10587574C2218yes—
21C2×C4×Dic34—28no—
22C2×C4×Dic34C2218yes—
23C2×#10587573—28no—
24C2×#10587573C2218yes—
25C22×C4⋊Dic17C2211yescharacteristic
26C22×C4⋊Dic17C2211yescharacteristic
27C22×#63225C2211yescharacteristic
28C22×#63225C2211yescharacteristic
29C22×#63225C2211yescharacteristic
30C22×#63225C2211yescharacteristic
31C22×#63225C2211yescharacteristic
32C22×#63225C2211yescharacteristic
33C22×C4×Dic17C2211yescharacteristic
34C22×C4×Dic17C2211yescharacteristic
35C22×C4×Dic17C2211yescharacteristic
36C22×C4×Dic17C2211yescharacteristic
37C22×C4×Dic17C2211yescharacteristic
38C22×C4×Dic17C2211yescharacteristic
39#11762713C2218yes—
40#11762713C2218yes—
41#11762696C2218yes—
42#11762695C2218yes—
43#11762694C2218yes—
44#11762693C2218yes—
45#11762692C2218yes—
46#11762691C2218yes—
47#11762690C2218yes—
48#11762673C2218yes—
49#11762654C2218yes—
50C2×#10587502C2211yescharacteristic
51C2×#10587502C2211yescharacteristic
52C2×#10587502C2211yescharacteristic
53C2×#10587502C2211yescharacteristic
54C2×#10587502C2211yescharacteristic
55C2×#10587502C2211yescharacteristic
56C2×#10587502C2211yescharacteristic
57C2×#10587502C2211yescharacteristic
58C2×#10587502C2211yescharacteristic
59C2×#10587502C2211yescharacteristic
60C2×#10587502C2211yescharacteristic
61C2×#10587502C2211yescharacteristic
62C22×C4⋊C4×C17C2211yescharacteristic
63C22×C4⋊C4×C17C2211yescharacteristic
64C22×C42×C17C2211yescharacteristic
65C17×#387C2218yes—
66C2×C17×#95C2211yescharacteristic
67C2×C17×#95C2211yescharacteristic
68C2×C17×#95C2211yescharacteristic
69C2×C17×#95C2211yescharacteristic
70C23×C4×C17C2311yescharacteristic
71C23×C4×C17C2311yescharacteristic
72C23×C4×C17C2311yescharacteristic
73C23×C4×C17C2311yescharacteristic
74C23×C4×C17C2311yescharacteristic
75C23×C4×C17C2311yescharacteristic
76C23×C4×C17C2311yescharacteristic
77C23×Dic17C2311yescharacteristic
78C23×Dic17C2311yescharacteristic
79C23×Dic17C2311yescharacteristic
80C23×Dic17C2311yescharacteristic
81C23×Dic17C2311yescharacteristic
82C23×Dic17C2311yescharacteristic
83C23×Dic17C2311yescharacteristic
84C23×Dic17C2311yescharacteristic
85C22×Dic34—24no—
86C22×Dic34—24no—
87C22×Dic34—24no—
88C22×Dic34—24no—
89C22×Dic34C2×C418yes—
90C22×Dic34C2×C418yes—
91C22×Dic34C2314yes—
92C22×Dic34C2314yes—
93C2×C4⋊C4×C17—24no—
94C2×C4⋊C4×C17C2×C418yes—
95C2×C4⋊C4×C17C2314yes—
96C2×C4⋊C4×C17C2314yes—
97C2×C42×C17—24no—
98C2×C42×C17C2314yes—
99C2×C4⋊Dic17—24no—
100C2×C4⋊Dic17C2×C418yes—
101C2×C4⋊Dic17C2314yes—
102C2×C4⋊Dic17C2314yes—
103C2×#63225—24no—
104C2×#63225—24no—
105C2×#63225—24no—
106C2×#63225—24no—
107C2×#63225C2×C418yes—
108C2×#63225C2×C418yes—
109C2×#63225C2314yes—
110C2×#63225C2314yes—
111C2×#63225C2314yes—
112C2×#63225C2314yes—
113C2×#63225C2314yes—
114C2×#63225C2314yes—
115C2×C4×Dic17—24no—
116C2×C4×Dic17—24no—
117C2×C4×Dic17—24no—
118C2×C4×Dic17—24no—
119C2×C4×Dic17C2×C418yes—
120C2×C4×Dic17D418yes—
121C2×C4×Dic17C2314yes—
122C2×C4×Dic17C2314yes—
123C2×C4×Dic17C2314yes—
124C2×C4×Dic17C2314yes—
125C2×C4×Dic17C2314yes—
126C2×C4×Dic17C2314yes—
127#10587574C2×C4164yescomplemented
128#10587573—232no—
129C4×Dic34—232no—
130C17×#95C2314yes—
131C17×#95C2314yes—
132C17×#95C2314yes—
133C17×#95C2314yes—
134#10587502C2314yes—
135#10587502C2314yes—
136#10587502C2314yes—
137#10587502C2314yes—
138#10587502C2314yes—
139#10587502C2314yes—
140#10587502C2314yes—
141#10587502C2314yes—
142#10587502C2314yes—
143#10587502C2314yes—
144#10587502C2314yes—
145#10587502C2314yes—
146C24×C17C2411yescharacteristic
147C22×C4×C17—22no—
148C22×C4×C17—22no—
149C22×C4×C17—22no—
150C22×C4×C17—22no—
151C22×C4×C17—22no—
152C22×C4×C17—22no—
153C22×C4×C17—22no—
154C22×C4×C17—22no—
155C22×C4×C17—22no—
156C22×C4×C17—22no—
157C22×C4×C17—22no—
158C22×C4×C17—22no—
159C22×C4×C17—22no—
160C22×C4×C17—22no—
161C22×C4×C17—22no—
162C22×C4×C17—22no—
163C22×C4×C17—22no—
164C22×C4×C17C22×C414yes—
165C22×C4×C17C22×C414yes—
166C22×C4×C17C22×C414yes—
167C22×C4×C17C2×D414yes—
168C22×C4×C17C2412yes—
169C22×C4×C17C2412yes—
170C22×C4×C17C2412yes—
171C22×C4×C17C2412yes—
172C22×C4×C17C2412yes—
173C22×C4×C17C2412yes—
174C22×C4×C17C2412yes—
175C22×Dic17—22no—
176C22×Dic17—22no—
177C22×Dic17—22no—
178C22×Dic17—22no—
179C22×Dic17—22no—
180C22×Dic17—22no—
181C22×Dic17—22no—
182C22×Dic17—22no—
183C22×Dic17—22no—
184C22×Dic17—22no—
185C22×Dic17—22no—
186C22×Dic17—22no—
187C22×Dic17—22no—
188C22×Dic17—22no—
189C22×Dic17—22no—
190C22×Dic17—22no—
191C22×Dic17—22no—
192C22×Dic17C22×C414yes—
193C22×Dic17C22×C414yes—
194C22×Dic17C22×C414yes—
195C22×Dic17C22×C414yes—
196C22×Dic17C2×D414yes—
197C22×Dic17C2×D414yes—
198C22×Dic17#4114yes—
199C22×Dic17C2412yes—
200C22×Dic17C2412yes—
201C22×Dic17C2412yes—
202C22×Dic17C2412yes—
203C22×Dic17C2412yes—
204C22×Dic17C2412yes—
205C22×Dic17C2412yes—
206C22×Dic17C2412yes—
207C2×Dic34—232no—
208C2×Dic34—232no—
209C2×Dic34—28no—
210C2×Dic34—28no—
211C2×Dic34—28no—
212C2×Dic34—28no—
213C2×Dic34C22×C4116yes—
214C2×Dic34C22×C4116yes—
215C4⋊C4×C17—28no—
216C4⋊C4×C17C22×C4116yes—
217C42×C17—28no—
218C4⋊Dic17—28no—
219C4⋊Dic17C22×C4116yes—
220#63225—28no—
221#63225—28no—
222#63225—28no—
223#63225—28no—
224#63225C22×C4116yes—
225#63225C22×C4116yes—
226C4×Dic17—28no—
227C4×Dic17—28no—
228C4×Dic17—28no—
229C4×Dic17—28no—
230C4×Dic17C22×C4116yes—
231C4×Dic17C2×D4116yes—
232C2×#2327—171nomaximal, Sylow
233C23×C17C23×C412yes—
234C23×C17C22×D412yes—
235C23×C17C2×#4112yes—
236C23×C172−1+412yes—
237C23×C172−1+412yes—
238C23×C172−1+412yes—
239C23×C172−1+412yes—
240C23×C17C2511yescharacteristic
241C2×C4×C17—28no—
242C2×C4×C17—28no—
243C2×C4×C17—28no—
244C2×C4×C17—28no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—22no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—22no—
258C2×C4×C17—22no—
259C2×C4×C17—22no—
260C2×C4×C17—22no—
261C2×C4×C17—22no—
262C2×C4×C17—44no—
263C2×C4×C17—44no—
264C2×C4×C17—44no—
265C2×C4×C17C23×C414yes—
266C2×C4×C17C23×C414yes—
267C2×C4×C17C23×C414yes—
268C2×C4×C17C22×D414yes—
269C2×Dic17—28no—
270C2×Dic17—28no—
271C2×Dic17—28no—
272C2×Dic17—28no—
273C2×Dic17—22no—
274C2×Dic17—22no—
275C2×Dic17—22no—
276C2×Dic17—22no—
277C2×Dic17—22no—
278C2×Dic17—22no—
279C2×Dic17—22no—
280C2×Dic17—22no—
281C2×Dic17—22no—
282C2×Dic17—22no—
283C2×Dic17—22no—
284C2×Dic17—22no—
285C2×Dic17—22no—
286C2×Dic17—22no—
287C2×Dic17—22no—
288C2×Dic17—22no—
289C2×Dic17—22no—
290C2×Dic17—44no—
291C2×Dic17—44no—
292C2×Dic17—44no—
293C2×Dic17C4×D4116yes—
294C2×Dic17C23×C414yes—
295C2×Dic17C23×C414yes—
296C2×Dic17C23×C414yes—
297C2×Dic17C23×C414yes—
298C2×Dic17C22×D414yes—
299C2×Dic17C22×D414yes—
300C2×Dic17C2×#4114yes—
301Dic34—232no—
302Dic34—232no—
303C22×C4⋊Q8—171no—
304C22×C4×Q8—171no—
305C22×C4×Q8—171no—
306#2327—1716no—
307C2×#391—171no—
308C2×#391—171no—
309C2×#387—171no—
310C2×#387—171no—
311C2×#387—171no—
312C2×#387—171no—
313C2×#387—171no—
314C2×#387—171no—
315C2×#387—171no—
316C2×#387D1711yescharacteristic
317C2×#382—171no—
318C2×#376—171no—
319C22×C17C2×C4×D414yes—
320C22×C17#51914yes—
321C22×C17#51914yes—
322C22×C17#53614yes—
323C22×C17#53614yes—
324C22×C17#54114yes—
325C22×C17#54114yes—
326C22×C17C24×C411yescharacteristic
327C22×C17C23×D411yescharacteristic
328C22×C17C22×#4111yescharacteristic
329C22×C17C2×2−1+411yescharacteristic
330C22×C17C2×2−1+411yescharacteristic
331C22×C17C2×2−1+411yescharacteristic
332C22×C17C2×2−1+411yescharacteristic
333C4×C17—24no—
334C4×C17—24no—
335C4×C17—24no—
336C4×C17—24no—
337C4×C17—42no—
338C4×C17—42no—
339C4×C17—42no—
340Dic17—24no—
341Dic17—24no—
342Dic17—24no—
343Dic17—24no—
344Dic17—42no—
345Dic17—42no—
346Dic17—42no—
347Dic17C2×C4×D418yes—
348C23×Q8—171no—
349C23×Q8—171no—
350C2×C4⋊Q8—1716no—
351C2×C4⋊Q8—178no—
352C2×C4×Q8—178no—
353C2×C4×Q8—178no—
354C2×C4×Q8—348no—
355C2×C4×Q8—348no—
356C22×C4⋊C4—171no—
357C22×C4⋊C4—171no—
358C22×C4⋊C4—171no—
359C22×C4⋊C4—171no—
360C22×C4⋊C4—171no—
361C22×C4⋊C4—171no—
362C22×C4⋊C4—171no—
363C22×C4⋊C4—171no—
364C22×C4⋊C4C2×D1711yescharacteristic
365C22×C4⋊C4C2×D1711yescharacteristic
366C22×C42—171no—
367C22×C42—171no—
368C22×C42—171no—
369C22×C42—171no—
370C22×C42—171no—
371C22×C42—171no—
372C22×C42C2×D1711yescharacteristic
373#391—178no—
374#391—178no—
375#387—178no—
376#387—178no—
377#387—178no—
378#387—178no—
379#387—178no—
380#387—178no—
381#387—178no—
382#387C2×D1718yes—
383#382—178no—
384#376—178no—
385C2×#95—171no—
386C2×#95—171no—
387C2×#95—171no—
388C2×#95—171no—
389C2×#95—171no—
390C2×#95—171no—
391C2×#95—171no—
392C2×#95—171no—
393C2×#95—171no—
394C2×#95—171no—
395C2×#95—171no—
396C2×#95—171no—
397C2×#95C2×D1711yescharacteristic
398C2×#95C2×D1711yescharacteristic
399C2×#95C2×D1711yescharacteristic
400C2×#95C2×D1711yescharacteristic
401C2×C17#232718yescomplemented
402C2×C17C22×C4×D411yescharacteristic
403C2×C17C2×#51911yescharacteristic
404C2×C17C2×#51911yescharacteristic
405C2×C17C2×#53611yescharacteristic
406C2×C17C2×#53611yescharacteristic
407C2×C17C2×#54111yescharacteristic
408C2×C17C2×#54111yescharacteristic
409C22×Q8—178no—
410C22×Q8—178no—
411C22×Q8—174no—
412C22×Q8—174no—
413C22×Q8—344no—
414C22×Q8—344no—
415C22×Q8—344no—
416C22×Q8—344no—
417C23×C4—171no—
418C23×C4—171no—
419C23×C4—171no—
420C23×C4—171no—
421C23×C4—171no—
422C23×C4—171no—
423C23×C4—171no—
424C23×C4—171no—
425C23×C4C22×D1711yescharacteristic
426C23×C4C22×D1711yescharacteristic
427C23×C4C22×D1711yescharacteristic
428C23×C4C22×D1711yescharacteristic
429C23×C4C22×D1711yescharacteristic
430C23×C4C22×D1711yescharacteristic
431C23×C4C22×D1711yescharacteristic
432C4⋊Q8—1764no—
433C4×Q8—3432no—
434C4×Q8—3432no—
435C2×C4⋊C4—24no—
436C2×C4⋊C4—178no—
437C2×C4⋊C4—178no—
438C2×C4⋊C4—178no—
439C2×C4⋊C4—174no—
440C2×C4⋊C4—174no—
441C2×C4⋊C4—174no—
442C2×C4⋊C4—174no—
443C2×C4⋊C4—174no—
444C2×C4⋊C4—174no—
445C2×C4⋊C4—174no—
446C2×C4⋊C4—174no—
447C2×C4⋊C4—344no—
448C2×C4⋊C4—344no—
449C2×C4⋊C4—344no—
450C2×C4⋊C4—344no—
451C2×C4⋊C4—344no—
452C2×C4⋊C4C4×D1718yes—
453C2×C4⋊C4C22×D1714yes—
454C2×C4⋊C4C22×D1714yes—
455C2×C42—24no—
456C2×C42—178no—
457C2×C42—178no—
458C2×C42—174no—
459C2×C42—174no—
460C2×C42—174no—
461C2×C42—174no—
462C2×C42—174no—
463C2×C42—174no—
464C2×C42—344no—
465C2×C42—344no—
466C2×C42—344no—
467C2×C42—344no—
468C2×C42C22×D1714yes—
469#95—174no—
470#95—174no—
471#95—174no—
472#95—174no—
473#95—174no—
474#95—174no—
475#95—174no—
476#95—174no—
477#95—174no—
478#95—174no—
479#95—174no—
480#95—174no—
481#95C22×D1714yes—
482#95C22×D1714yes—
483#95C22×D1714yes—
484#95C22×D1714yes—
485C17C2×#232711yescharacteristic, complemented, Sylow
486C24C23×D1711yescharacteristic
487C2×Q8—1716no—
488C2×Q8—1716no—
489C2×Q8—348no—
490C2×Q8—348no—
491C2×Q8—3432no—
492C2×Q8—3432no—
493C2×Q8—348no—
494C2×Q8—348no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—174no—
513C22×C4—174no—
514C22×C4—174no—
515C22×C4—172no—
516C22×C4—174no—
517C22×C4—174no—
518C22×C4—174no—
519C22×C4—174no—
520C22×C4—172no—
521C22×C4—172no—
522C22×C4—172no—
523C22×C4—172no—
524C22×C4—172no—
525C22×C4—172no—
526C22×C4—172no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—342no—
531C22×C4—342no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—342no—
542C22×C4—342no—
543C22×C4—342no—
544C22×C4C2×C4×D1714yes—
545C22×C4C2×C4×D1714yes—
546C22×C4C2×C4×D1714yes—
547C22×C4D4×D1714yes—
548C22×C4C23×D1712yes—
549C22×C4C23×D1712yes—
550C22×C4C23×D1712yes—
551C22×C4C23×D1712yes—
552C22×C4C23×D1712yes—
553C22×C4C23×D1712yes—
554C22×C4C23×D1712yes—
555C4⋊C4—28no—
556C4⋊C4—1716no—
557C4⋊C4—1716no—
558C4⋊C4—1716no—
559C4⋊C4—348no—
560C4⋊C4—348no—
561C4⋊C4—348no—
562C4⋊C4—348no—
563C4⋊C4—348no—
564C4⋊C4C2×C4×D17116yes—
565C42—28no—
566C42—1716no—
567C42—1716no—
568C42—348no—
569C42—348no—
570C42—348no—
571C42—348no—
572C23C22×C4×D1712yes—
573C23C2×D4×D1712yes—
574C23#1058769412yes—
575C23#41×D1712yes—
576C23#1058769712yes—
577C23#1058769712yes—
578C23#1058769712yes—
579C23C24×D1711yescharacteristic
580Q8—3432no—
581Q8—3432no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—1716no—
607C2×C4—174no—
608C2×C4—174no—
609C2×C4—174no—
610C2×C4—174no—
611C2×C4—174no—
612C2×C4—174no—
613C2×C4—174no—
614C2×C4—348no—
615C2×C4—348no—
616C2×C4—348no—
617C2×C4—342no—
618C2×C4—342no—
619C2×C4—342no—
620C2×C4—342no—
621C2×C4—348no—
622C2×C4—342no—
623C2×C4—342no—
624C2×C4—342no—
625C2×C4—342no—
626C2×C4—342no—
627C2×C4—342no—
628C2×C4—342no—
629C2×C4—342no—
630C2×C4—342no—
631C2×C4—342no—
632C2×C4—342no—
633C2×C4—342no—
634C2×C4—342no—
635C2×C4—684no—
636C2×C4—684no—
637C2×C4—684no—
638C2×C4C22×C4×D1714yes—
639C2×C4C22×C4×D1714yes—
640C2×C4C22×C4×D1714yes—
641C2×C4C2×D4×D1714yes—
642C22C4×D4×D1714yes—
643C22#1176328514yes—
644C22#1176328614yes—
645C22#1176338114yes—
646C22#1176338214yes—
647C22#1176341614yes—
648C22#1176341814yes—
649C22C23×C4×D1711yescharacteristic
650C22C22×D4×D1711yescharacteristic
651C22C2×#41×D1711yescharacteristic
652C22C2×#1058769711yescharacteristic
653C22C2×#1058769711yescharacteristic
654C22C2×#1058769711yescharacteristic
655C22C2×#1058769411yescharacteristic
656C4—24no—
657C4—24no—
658C4—24no—
659C4—24no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—178no—
664C4—344no—
665C4—344no—
666C4—344no—
667C4—344no—
668C4—682no—
669C4—682no—
670C4—682no—
671C2#1453149918yesdirect product factor
672C2C2×C4×D4×D1711yescharacteristic
673C2C2×#1176328511yescharacteristic
674C2C2×#1176328611yescharacteristic
675C2C2×#1176338111yescharacteristic
676C2C2×#1176338211yescharacteristic
677C2C2×#1176341611yescharacteristic
678C2C2×#1176341811yescharacteristic
subgroupG/H
centerC24C23×D17
commutatorC22×C17C24×C4
FrattiniC23C24×D17
FittingC2×C17×#387C2
socle, prime radicalC24×C17C24
cosocleC23×C17C25
hypercenterC2×#387D17
hypocenterC17C2×#2327
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
4. #14531499C2direct (N × H)
19. C2×#10587574C4semidirect
127. #10587574C2×C4semidirect
401. C2×C17#2327semidirect
485. C17C2×#2327semidirect
671. C2#14531499direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875742
C22×C4×Dic342
C22×#105875732
#145314992
C2×#117627132
C2×#117626962
C2×#117626952
C2×#117626942
C2×#117626932
C2×#117626922
C2×#117626912
C2×#117626902
C2×#117626732
C2×#117626542
C2×C17×#3872
C2×#232717