Finite Group Databasebeta

← C2×#14531522 #17193152

Subgroups — C2×#14531522 #17193152

Order 4352.

2398 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587575C211yesmaximal, characteristic
2#14531522C2116yesmaximal, direct product factor
3C2×#11762706C211yesmaximal, characteristic
4C2×#11762705C211yesmaximal, characteristic
5C2×#11762704C211yesmaximal, characteristic
6C2×#11762704C211yesmaximal, characteristic
7C2×#11762704C211yesmaximal, characteristic
8C2×#11762695C211yesmaximal, characteristic
9C2×#11762693C211yesmaximal, characteristic
10C2×#11762692C211yesmaximal, characteristic
11C2×#11762690C211yesmaximal, characteristic
12C2×#11762678C211yesmaximal, characteristic
13C2×C4×#63225C211yesmaximal, characteristic
14C2×#11762660C211yesmaximal, characteristic
15C2×#11762654C211yesmaximal, characteristic
16C2×C17×#389C211yesmaximal, characteristic
17C2×#10587575C4116yescomplemented
18C2×#10587575C2218yes—
19C22×C4⋊Dic17C2211yescharacteristic
20C22×C4⋊Dic17C2211yescharacteristic
21C22×#63225C2211yescharacteristic
22C22×#63225C2211yescharacteristic
23C22×#63225C2211yescharacteristic
24C22×#63225C2211yescharacteristic
25C22×#63225C2211yescharacteristic
26C22×#63225C2211yescharacteristic
27C22×#63225C2211yescharacteristic
28C22×#63225C2211yescharacteristic
29C22×#63225C2211yescharacteristic
30C22×#63225C2211yescharacteristic
31C22×C4×Dic17C2211yescharacteristic
32C22×C4×Dic17C2211yescharacteristic
33C22×C4×Dic17C2211yescharacteristic
34C22×C4×Dic17C2211yescharacteristic
35C22×C4×Dic17C2211yescharacteristic
36C22×C4×Dic17C2211yescharacteristic
37#11762706C2218yes—
38#11762705C2218yes—
39#11762704C2218yes—
40#11762704C2218yes—
41#11762704C2218yes—
42#11762695C2218yes—
43#11762693C2218yes—
44#11762692C2218yes—
45#11762690C2218yes—
46#11762678C2218yes—
47C4×#63225C2218yes—
48#11762660C2218yes—
49#11762654C2218yes—
50C2×#10587502C2211yescharacteristic
51C2×#10587502C2211yescharacteristic
52C2×#10587502C2211yescharacteristic
53C2×#10587502C2211yescharacteristic
54C2×#10587502C2211yescharacteristic
55C2×#10587502C2211yescharacteristic
56C2×#10587502C2211yescharacteristic
57C2×#10587502C2211yescharacteristic
58C2×#10587502C2211yescharacteristic
59C2×#10587502C2211yescharacteristic
60C22×C4⋊C4×C17C2211yescharacteristic
61C22×C4⋊C4×C17C2211yescharacteristic
62C22×C4⋊C4×C17C2211yescharacteristic
63C22×C4⋊C4×C17C2211yescharacteristic
64C22×C42×C17C2211yescharacteristic
65C17×#389C2218yes—
66C2×C17×#95C2211yescharacteristic
67C2×C17×#95C2211yescharacteristic
68C23×C4×C17C2311yescharacteristic
69C23×C4×C17C2311yescharacteristic
70C23×C4×C17C2311yescharacteristic
71C23×C4×C17C2311yescharacteristic
72C23×C4×C17C2311yescharacteristic
73C23×C4×C17C2311yescharacteristic
74C23×C4×C17C2311yescharacteristic
75C23×Dic17C2311yescharacteristic
76C23×Dic17C2311yescharacteristic
77C23×Dic17C2311yescharacteristic
78C23×Dic17C2311yescharacteristic
79C23×Dic17C2311yescharacteristic
80C23×Dic17C2311yescharacteristic
81C23×Dic17C2311yescharacteristic
82C23×Dic17C2311yescharacteristic
83C2×C4⋊C4×C17—42no—
84C2×C4⋊C4×C17—42no—
85C2×C4⋊C4×C17—42no—
86C2×C4⋊C4×C17C2×C418yes—
87C2×C4⋊C4×C17C2314yes—
88C2×C4⋊C4×C17C2314yes—
89C2×C4⋊C4×C17C2314yes—
90C2×C4⋊C4×C17C2314yes—
91C2×C42×C17—42no—
92C2×C42×C17C2314yes—
93C2×C4⋊Dic17—42no—
94C2×C4⋊Dic17C2×C418yes—
95C2×C4⋊Dic17C2314yes—
96C2×C4⋊Dic17C2314yes—
97C2×#63225—24no—
98C2×#63225—24no—
99C2×#63225—42no—
100C2×#63225—42no—
101C2×#63225—42no—
102C2×#63225—42no—
103C2×#63225C2×C418yes—
104C2×#63225C2×C418yes—
105C2×#63225C2×C418yes—
106C2×#63225C2×C418yes—
107C2×#63225C2314yes—
108C2×#63225C2314yes—
109C2×#63225C2314yes—
110C2×#63225C2314yes—
111C2×#63225C2314yes—
112C2×#63225C2314yes—
113C2×#63225C2314yes—
114C2×#63225C2314yes—
115C2×#63225C2314yes—
116C2×#63225C2314yes—
117C2×C4×Dic17—24no—
118C2×C4×Dic17—42no—
119C2×C4×Dic17—42no—
120C2×C4×Dic17—42no—
121C2×C4×Dic17C2×C418yes—
122C2×C4×Dic17Q818yes—
123C2×C4×Dic17C2314yes—
124C2×C4×Dic17C2314yes—
125C2×C4×Dic17C2314yes—
126C2×C4×Dic17C2314yes—
127C2×C4×Dic17C2314yes—
128C2×C4×Dic17C2314yes—
129#10587575C2×C4164yescomplemented
130C17×#95C2314yes—
131C17×#95C2314yes—
132#10587502C2314yes—
133#10587502C2314yes—
134#10587502C2314yes—
135#10587502C2314yes—
136#10587502C2314yes—
137#10587502C2314yes—
138#10587502C2314yes—
139#10587502C2314yes—
140#10587502C2314yes—
141#10587502C2314yes—
142C24×C17C2411yescharacteristic
143C22×C4×C17—22no—
144C22×C4×C17—22no—
145C22×C4×C17—22no—
146C22×C4×C17—22no—
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×C17C22×C414yes—
161C22×C4×C17C22×C414yes—
162C22×C4×C17C22×C414yes—
163C22×C4×C17C2×Q814yes—
164C22×C4×C17C2412yes—
165C22×C4×C17C2412yes—
166C22×C4×C17C2412yes—
167C22×C4×C17C2412yes—
168C22×C4×C17C2412yes—
169C22×C4×C17C2412yes—
170C22×C4×C17C2412yes—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17—22no—
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×Dic17C22×C414yes—
189C22×Dic17C22×C414yes—
190C22×Dic17C22×C414yes—
191C22×Dic17C22×C414yes—
192C22×Dic17C2×Q814yes—
193C22×Dic17C2×Q814yes—
194C22×Dic17#4114yes—
195C22×Dic17C2412yes—
196C22×Dic17C2412yes—
197C22×Dic17C2412yes—
198C22×Dic17C2412yes—
199C22×Dic17C2412yes—
200C22×Dic17C2412yes—
201C22×Dic17C2412yes—
202C22×Dic17C2412yes—
203C4⋊C4×C17—44no—
204C4⋊C4×C17—44no—
205C4⋊C4×C17—44no—
206C4⋊C4×C17C22×C4116yes—
207C42×C17—44no—
208C4⋊Dic17—44no—
209C4⋊Dic17C22×C4116yes—
210#63225—28no—
211#63225—28no—
212#63225—44no—
213#63225—44no—
214#63225—44no—
215#63225—44no—
216#63225C22×C4116yes—
217#63225C22×C4116yes—
218#63225C22×C4116yes—
219#63225C22×C4116yes—
220C4×Dic17—28no—
221C4×Dic17—44no—
222C4×Dic17—44no—
223C4×Dic17—44no—
224C4×Dic17C22×C4116yes—
225C4×Dic17C2×Q8116yes—
226C2×#2330—171nomaximal, Sylow
227C23×C17C23×C412yes—
228C23×C17C22×Q812yes—
229C23×C17C2×#4112yes—
230C23×C17H(5,𝔽2)12yes—
231C23×C17H(5,𝔽2)12yes—
232C23×C172−1+412yes—
233C23×C172−1+412yes—
234C23×C17C2511yescharacteristic
235C2×C4×C17—28no—
236C2×C4×C17—28no—
237C2×C4×C17—28no—
238C2×C4×C17—28no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
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—44no—
257C2×C4×C17—44no—
258C2×C4×C17—44no—
259C2×C4×C17C23×C414yes—
260C2×C4×C17C23×C414yes—
261C2×C4×C17C23×C414yes—
262C2×C4×C17C22×Q814yes—
263C2×Dic17—28no—
264C2×Dic17—28no—
265C2×Dic17—28no—
266C2×Dic17—28no—
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—22no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—22no—
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—44no—
285C2×Dic17—44no—
286C2×Dic17—44no—
287C2×Dic17C4×Q8116yes—
288C2×Dic17C23×C414yes—
289C2×Dic17C23×C414yes—
290C2×Dic17C23×C414yes—
291C2×Dic17C23×C414yes—
292C2×Dic17C22×Q814yes—
293C2×Dic17C22×Q814yes—
294C2×Dic17C2×#4114yes—
295C22×#125—171no—
296#2330—1716no—
297C2×#389—171no—
298C2×#389—171no—
299C2×#389—171no—
300C2×#389—171no—
301C2×#389—171no—
302C2×#389D1711yescharacteristic
303C2×#387—171no—
304C2×#387—171no—
305C2×#387—171no—
306C2×#387—171no—
307C2×#384—171no—
308C2×C4×C4⋊C4—171no—
309C2×C4×C4⋊C4—171no—
310C2×#376—171no—
311C22×C17C2×C4×Q814yes—
312C22×C17#52014yes—
313C22×C17#52014yes—
314C22×C17#54014yes—
315C22×C17#54114yes—
316C22×C17#54414yes—
317C22×C17#54414yes—
318C22×C17C24×C411yescharacteristic
319C22×C17C23×Q811yescharacteristic
320C22×C17C22×#4111yescharacteristic
321C22×C17C2×H(5,𝔽2)11yescharacteristic
322C22×C17C2×H(5,𝔽2)11yescharacteristic
323C22×C17C2×2−1+411yescharacteristic
324C22×C17C2×2−1+411yescharacteristic
325C4×C17—24no—
326C4×C17—24no—
327C4×C17—24no—
328C4×C17—24no—
329C4×C17—42no—
330C4×C17—42no—
331C4×C17—42no—
332Dic17—24no—
333Dic17—24no—
334Dic17—24no—
335Dic17—24no—
336Dic17—42no—
337Dic17—42no—
338Dic17—42no—
339Dic17C2×C4×Q818yes—
340C2×#125—1716no—
341C2×#125—178no—
342C22×C4⋊C4—171no—
343C22×C4⋊C4—171no—
344C22×C4⋊C4—171no—
345C22×C4⋊C4—171no—
346C22×C4⋊C4—171no—
347C22×C4⋊C4—171no—
348C22×C4⋊C4—171no—
349C22×C4⋊C4—171no—
350C22×C4⋊C4—171no—
351C22×C4⋊C4—171no—
352C22×C4⋊C4—171no—
353C22×C4⋊C4—171no—
354C22×C4⋊C4C2×D1711yescharacteristic
355C22×C4⋊C4C2×D1711yescharacteristic
356C22×C4⋊C4C2×D1711yescharacteristic
357C22×C4⋊C4C2×D1711yescharacteristic
358C22×C42—171no—
359C22×C42—171no—
360C22×C42—171no—
361C22×C42—171no—
362C22×C42—171no—
363C22×C42—171no—
364C22×C42C2×D1711yescharacteristic
365#389—178no—
366#389—178no—
367#389—178no—
368#389—178no—
369#389—178no—
370#389C2×D1718yes—
371#387—178no—
372#387—178no—
373#387—178no—
374#387—178no—
375#384—178no—
376C4×C4⋊C4—178no—
377C4×C4⋊C4—178no—
378#376—178no—
379C2×#95—171no—
380C2×#95—171no—
381C2×#95—171no—
382C2×#95—171no—
383C2×#95—171no—
384C2×#95—171no—
385C2×#95—171no—
386C2×#95—171no—
387C2×#95—171no—
388C2×#95—171no—
389C2×#95C2×D1711yescharacteristic
390C2×#95C2×D1711yescharacteristic
391C2×C17#233018yescomplemented
392C2×C17C22×C4×Q811yescharacteristic
393C2×C17C2×#52011yescharacteristic
394C2×C17C2×#52011yescharacteristic
395C2×C17C2×#54011yescharacteristic
396C2×C17C2×#54111yescharacteristic
397C2×C17C2×#54411yescharacteristic
398C2×C17C2×#54411yescharacteristic
399C23×C4—171no—
400C23×C4—171no—
401C23×C4—171no—
402C23×C4—171no—
403C23×C4—171no—
404C23×C4—171no—
405C23×C4—171no—
406C23×C4—171no—
407C23×C4C22×D1711yescharacteristic
408C23×C4C22×D1711yescharacteristic
409C23×C4C22×D1711yescharacteristic
410C23×C4C22×D1711yescharacteristic
411C23×C4C22×D1711yescharacteristic
412C23×C4C22×D1711yescharacteristic
413C23×C4C22×D1711yescharacteristic
414#125—1764no—
415C2×C4⋊C4—42no—
416C2×C4⋊C4—42no—
417C2×C4⋊C4—42no—
418C2×C4⋊C4—178no—
419C2×C4⋊C4—178no—
420C2×C4⋊C4—178no—
421C2×C4⋊C4—178no—
422C2×C4⋊C4—178no—
423C2×C4⋊C4—174no—
424C2×C4⋊C4—174no—
425C2×C4⋊C4—174no—
426C2×C4⋊C4—174no—
427C2×C4⋊C4—174no—
428C2×C4⋊C4—174no—
429C2×C4⋊C4—174no—
430C2×C4⋊C4—174no—
431C2×C4⋊C4—174no—
432C2×C4⋊C4—174no—
433C2×C4⋊C4—174no—
434C2×C4⋊C4—174no—
435C2×C4⋊C4—344no—
436C2×C4⋊C4—344no—
437C2×C4⋊C4—682no—
438C2×C4⋊C4—682no—
439C2×C4⋊C4—682no—
440C2×C4⋊C4—682no—
441C2×C4⋊C4—682no—
442C2×C4⋊C4C4×D1718yes—
443C2×C4⋊C4C22×D1714yes—
444C2×C4⋊C4C22×D1714yes—
445C2×C4⋊C4C22×D1714yes—
446C2×C4⋊C4C22×D1714yes—
447C2×C42—42no—
448C2×C42—178no—
449C2×C42—178no—
450C2×C42—174no—
451C2×C42—174no—
452C2×C42—174no—
453C2×C42—174no—
454C2×C42—174no—
455C2×C42—174no—
456C2×C42—344no—
457C2×C42—682no—
458C2×C42—682no—
459C2×C42—682no—
460C2×C42C22×D1714yes—
461#95—174no—
462#95—174no—
463#95—174no—
464#95—174no—
465#95—174no—
466#95—174no—
467#95—174no—
468#95—174no—
469#95—174no—
470#95—174no—
471#95C22×D1714yes—
472#95C22×D1714yes—
473C17C2×#233011yescharacteristic, complemented, Sylow
474C24C23×D1711yescharacteristic
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—174no—
493C22×C4—174no—
494C22×C4—174no—
495C22×C4—172no—
496C22×C4—174no—
497C22×C4—174no—
498C22×C4—174no—
499C22×C4—174no—
500C22×C4—172no—
501C22×C4—172no—
502C22×C4—172no—
503C22×C4—172no—
504C22×C4—172no—
505C22×C4—172no—
506C22×C4—172no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—342no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—342no—
519C22×C4—342no—
520C22×C4—342no—
521C22×C4—342no—
522C22×C4—342no—
523C22×C4—342no—
524C22×C4C2×C4×D1714yes—
525C22×C4C2×C4×D1714yes—
526C22×C4C2×C4×D1714yes—
527C22×C4Q8×D1714yes—
528C22×C4C23×D1712yes—
529C22×C4C23×D1712yes—
530C22×C4C23×D1712yes—
531C22×C4C23×D1712yes—
532C22×C4C23×D1712yes—
533C22×C4C23×D1712yes—
534C22×C4C23×D1712yes—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—1716no—
539C4⋊C4—1716no—
540C4⋊C4—1716no—
541C4⋊C4—1716no—
542C4⋊C4—1716no—
543C4⋊C4—348no—
544C4⋊C4—348no—
545C4⋊C4—684no—
546C4⋊C4—684no—
547C4⋊C4—684no—
548C4⋊C4—684no—
549C4⋊C4—684no—
550C4⋊C4C2×C4×D17116yes—
551C42—44no—
552C42—1716no—
553C42—1716no—
554C42—348no—
555C42—684no—
556C42—684no—
557C42—684no—
558C23C22×C4×D1712yes—
559C23#1058769112yes—
560C23#1058769112yes—
561C23C2×Q8×D1712yes—
562C23#1058769412yes—
563C23#41×D1712yes—
564C23#1058769712yes—
565C23C24×D1711yescharacteristic
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—1716no—
591C2×C4—174no—
592C2×C4—174no—
593C2×C4—174no—
594C2×C4—174no—
595C2×C4—174no—
596C2×C4—174no—
597C2×C4—174no—
598C2×C4—348no—
599C2×C4—348no—
600C2×C4—348no—
601C2×C4—342no—
602C2×C4—342no—
603C2×C4—342no—
604C2×C4—342no—
605C2×C4—342no—
606C2×C4—342no—
607C2×C4—342no—
608C2×C4—342no—
609C2×C4—348no—
610C2×C4—342no—
611C2×C4—342no—
612C2×C4—342no—
613C2×C4—342no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—342no—
618C2×C4—342no—
619C2×C4—684no—
620C2×C4—684no—
621C2×C4—684no—
622C2×C4C22×C4×D1714yes—
623C2×C4C22×C4×D1714yes—
624C2×C4C22×C4×D1714yes—
625C2×C4C2×Q8×D1714yes—
626C22C4×Q8×D1714yes—
627C22#1176328814yes—
628C22#1176329014yes—
629C22#1176340914yes—
630C22#1176341814yes—
631C22#1176343614yes—
632C22#1176343814yes—
633C22C23×C4×D1711yescharacteristic
634C22C22×Q8×D1711yescharacteristic
635C22C2×#41×D1711yescharacteristic
636C22C2×#1058769111yescharacteristic
637C22C2×#1058769111yescharacteristic
638C22C2×#1058769711yescharacteristic
639C22C2×#1058769411yescharacteristic
640C4—24no—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—178no—
648C4—344no—
649C4—344no—
650C4—344no—
651C4—344no—
652C4—682no—
653C4—682no—
654C4—682no—
655C2#1453152218yesdirect product factor
656C2C2×C4×Q8×D1711yescharacteristic
657C2C2×#1176328811yescharacteristic
658C2C2×#1176329011yescharacteristic
659C2C2×#1176340911yescharacteristic
660C2C2×#1176341811yescharacteristic
661C2C2×#1176343611yescharacteristic
662C2C2×#1176343811yescharacteristic
subgroupG/H
centerC24C23×D17
commutatorC22×C17C24×C4
FrattiniC23C24×D17
FittingC2×C17×#389C2
socle, prime radicalC24×C17C24
cosocleC23×C17C25
hypercenterC2×#389D17
hypocenterC17C2×#2330
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. #14531522C2direct (N × H)
17. C2×#10587575C4semidirect
129. #10587575C2×C4semidirect
391. C2×C17#2330semidirect
473. C17C2×#2330semidirect
655. C2#14531522direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875752
#145315222
C2×#117627062
C2×#117627052
C2×#117627042
C2×#117626952
C2×#117626932
C2×#117626922
C2×#117626902
C2×#117626782
C2×C4×#632252
C2×#117626602
C2×#117626542
C2×C17×#3892
C2×#233017