Finite Group Databasebeta

← C2×#14543119 #17942005

Subgroups — C2×#14543119 #17942005

Order 4352.

7086 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14543119C2132yesmaximal, direct product factor
2C2×#11763559C214yesmaximal, complemented
3C2×#11763539C211yesmaximal, characteristic, complemented
4C2×#11763516C212yesmaximal, complemented
5C2×#11763498C214yesmaximal, complemented
6C2×#11763459C214yesmaximal, complemented
7C2×#11763437C214yesmaximal, complemented
8C2×#11763426C214yesmaximal, complemented
9C2×#11763379C214yesmaximal, complemented
10C2×C4⋊Q8×D17C211yesmaximal, characteristic, complemented
11C2×#11763341C212yesmaximal, complemented
12C2×C17×#555C211yesmaximal, characteristic, complemented
13C22×#63254C2214yes—
14C22×Q8×D17C2212yes—
15#11763559C22164yescomplemented
16#11763539C22116yescomplemented
17#11763516C22132yescomplemented
18#11763498C22164yescomplemented
19#11763459C22164yescomplemented
20#11763437C22164yescomplemented
21#11763426C22164yescomplemented
22#11763379C22164yescomplemented
23C2×#10587555C2214yes—
24C2×#10587555C2211yescharacteristic
25C2×#10587574C2214yes—
26C2×#10587574C2214yes—
27C2×#10587630C2214yes—
28C2×#10587630C2212yes—
29C4⋊Q8×D17C22116yescomplemented
30#11763341C22132yescomplemented
31C2×#10587565C2214yes—
32C2×#10587565C2218yes—
33C2×#10587556C2218yes—
34C2×#10587576C2212yes—
35C2×#10587576C2214yes—
36C2×#10587622C2214yes—
37C2×#10587622C2212yes—
38C2×#10587569C2214yes—
39C2×#10587569C2218yes—
40C2×#10587621C2218yes—
41C2×#10587583C2212yes—
42C2×#10587610C2214yes—
43C2×#10587564C2218yes—
44C2×#10587564C2218yes—
45C2×#10587564C2218yes—
46C2×#10587624C2212yes—
47C2×C4×Dic34C2212yes—
48C2×#10587573C2214yes—
49C2×D4×Dic17C2214yes—
50C2×C4×C22⋊D17C2214yes—
51C2×#10587612C2214yes—
52C2×#10587558C2214yes—
53C22×C4⋊Dic17C2214yes—
54C2×C4⋊C4×D17C2214yes—
55C2×C42×D17C2211yescharacteristic
56C17×#555C22116yescomplemented
57C2×C17×C4⋊Q8C2211yescharacteristic
58C2×C17×#124C2214yes—
59C2×C17×#121C2214yes—
60C2×C4×D4×C17C2212yes—
61C2×C17×#117C2214yes—
62C22×Q8×C17C2312yes—
63C22×D4×C17C2312yes—
64C22×D4×C17C2314yes—
65C23×C4×C17C2314yes—
66C22×C22⋊D17C2314yes—
67C23×Dic17C2318yes—
68C2×Q8×D17—48no—
69C2×Q8×D17C23116yes—
70C2×#63254—416no—
71C2×#63254C23132yes—
72C22×C4×D17C2312yes—
73C22×C4×D17C2311yescharacteristic
74C22×C4×D17C2314yes—
75C22×Dic34C2312yes—
76C22×Dic34C2312yes—
77C22×Dic34C2314yes—
78C22×Dic34C2314yes—
79C17×C4⋊Q8C2318yes—
80C17×#124C23132yes—
81C17×#121C23132yes—
82C4×D4×C17C23116yes—
83C17×#117C23132yes—
84C2×C4⋊C4×C17C2312yes—
85C2×C4⋊C4×C17C2314yes—
86C2×C22⋊C4×C17C2318yes—
87C2×C22⋊C4×C17C2314yes—
88C2×C42×C17C2314yes—
89C2×C42×C17C2311yescharacteristic
90C2×C22⋊Dic17C2314yes—
91C2×C22⋊Dic17C2314yes—
92C2×C22⋊Dic17C2314yes—
93C2×C22⋊Dic17C2318yes—
94#10587630C23132yes—
95#10587630C23116yes—
96#10587624C23116yes—
97#10587622C23116yes—
98#10587622C23132yes—
99#10587621C23164yes—
100D4×Dic17C23132yes—
101C4×C22⋊D17C23132yes—
102C2×#63227C2314yes—
103#10587612C23132yes—
104C2×C4⋊Dic17C23132yes—
105C2×C4⋊Dic17C2314yes—
106C2×C4⋊Dic17C2312yes—
107C2×C4⋊Dic17C2318yes—
108C2×C4⋊Dic17C2314yes—
109C2×C4⋊Dic17C2314yes—
110#10587610C23132yes—
111C2×#63225C2318yes—
112C2×#63225C2318yes—
113C2×#63225C2314yes—
114C2×#63225C2318yes—
115C2×#63225C2314yes—
116C2×C4×Dic17C2312yes—
117C2×C4×Dic17C2311yescharacteristic
118C2×C4×Dic17C2314yes—
119C2×C4×Dic17C2314yes—
120#10587583C23116yes—
121C4⋊C4×D17C23132yes—
122#10587576C23132yes—
123#10587576C23116yes—
124#10587574C23132yes—
125#10587574C23132yes—
126#10587573C23132yes—
127#10587569C23164yes—
128#10587569C23132yes—
129#10587565C23164yes—
130#10587565C23132yes—
131#10587564C23164yes—
132#10587564C23164yes—
133#10587564C23164yes—
134#10587558C23132yes—
135C42×D17C2318yes—
136#10587556C23164yes—
137#10587555C2318yes—
138#10587555C23132yes—
139C4×Dic34C23116yes—
140C24×C17C2414yes—
141C23×D17C2411yescharacteristic
142C2×Q8×C17—44no—
143C2×Q8×C17C2418yes—
144C2×D4×C17—28no—
145C2×D4×C17—48no—
146C2×D4×C17C2418yes—
147C2×D4×C17C24116yes—
148C22×C4×C17—216no—
149C22×C4×C17C2414yes—
150C22×C4×C17C2411yescharacteristic
151C22×C4×C17C2414yes—
152C22×C4×C17C24116yes—
153C22×C4×C17C2412yes—
154C2×C22⋊D17—216no—
155C2×C22⋊D17C24116yes—
156C22×Dic17—416no—
157C22×Dic17C2411yescharacteristic
158C22×Dic17C2414yes—
159C22×Dic17C2414yes—
160C22×Dic17C2414yes—
161C22×Dic17C2412yes—
162C22×Dic17C24132yes—
163Q8×D17—432no—
164#63254—464no—
165C2×C4×D17—216no—
166C2×C4×D17—24no—
167C2×C4×D17—28no—
168C2×C4×D17C24116yes—
169C2×C4×D17C2414yes—
170C2×C4×D17C2418yes—
171C2×Dic34—28no—
172C2×Dic34—48no—
173C2×Dic34—48no—
174C2×Dic34—44no—
175C2×Dic34C2418yes—
176C2×Dic34C24116yes—
177C2×Dic34C2418yes—
178C2×Dic34C24116yes—
179C4⋊C4×C17C24116yes—
180C4⋊C4×C17C2418yes—
181C22⋊C4×C17C24116yes—
182C22⋊C4×C17C24132yes—
183C42×C17C24116yes—
184C42×C17C2414yes—
185C22⋊Dic17C24116yes—
186C22⋊Dic17C24116yes—
187C22⋊Dic17C24132yes—
188C22⋊Dic17C24116yes—
189#63227C24116yes—
190C4⋊Dic17C2418yes—
191C4⋊Dic17C24116yes—
192C4⋊Dic17C24132yes—
193C4⋊Dic17C24116yes—
194C4⋊Dic17C24116yes—
195#63225C24116yes—
196#63225C24116yes—
197#63225C24132yes—
198#63225C24132yes—
199#63225C24132yes—
200C4×Dic17C24116yes—
201C4×Dic17C2414yes—
202C4×Dic17C2418yes—
203C4×Dic17C24116yes—
204C2×#3544—171nomaximal, Sylow
205C23×C17—28no—
206C23×C17—28no—
207C23×C17—28no—
208C23×C17C2518yes—
209C23×C17C2511yescharacteristic
210C22×D17—24no—
211C22×D17H(5,𝔽2)14yes—
212C22×D17C2512yes—
213Q8×C17—48no—
214D4×C17—216no—
215D4×C17—416no—
216C2×C4×C17—28no—
217C2×C4×C17—22no—
218C2×C4×C17—28no—
219C2×C4×C17—232no—
220C2×C4×C172−1+418yes—
221C2×C4×C17C2518yes—
222C2×C4×C17C2512yes—
223C2×C4×C17C2518yes—
224C2×C4×C17C2514yes—
225C22⋊D17—232no—
226C2×Dic17—22no—
227C2×Dic17—28no—
228C2×Dic17—28no—
229C2×Dic17—28no—
230C2×Dic17—24no—
231C2×Dic17—432no—
232C2×Dic17C2512yes—
233C2×Dic17C2518yes—
234C2×Dic17C2514yes—
235C2×Dic17C2518yes—
236C2×Dic17C2518yes—
237C4×D17—232no—
238C4×D17—28no—
239C4×D17—216no—
240Dic34—216no—
241Dic34—416no—
242Dic34—416no—
243Dic34—48no—
244#3544—1732no—
245C2×#563—174no—
246C2×Q82—171no—
247C2×#555D1711yescharacteristic, complemented
248C2×#554—172no—
249C2×#552—174no—
250C2×#547—174no—
251C2×#544—174no—
252C2×#542—174no—
253C2×#536—174no—
254C22×C4⋊Q8—171no—
255C2×#529—172no—
256C22×C17—28no—
257C22×C17—28no—
258C22×C17—28no—
259C22×C17—416no—
260C22×C17C2×H(5,𝔽2)12yes—
261C22×C17C2×2−1+414yes—
262C22×C17C2611yescharacteristic
263C2×D17—24no—
264C2×D17—28no—
265C2×D17C2×H(5,𝔽2)14yes—
266C4×C17—28no—
267C4×C17—22no—
268C4×C17—28no—
269C4×C17C2×2−1+418yes—
270Dic17—22no—
271Dic17—28no—
272Dic17—28no—
273Dic17—28no—
274Dic17—24no—
275C22×#41—174no—
276C23×Q8—172no—
277#563—1764no—
278Q82—1716no—
279#555C2×D17116yescomplemented
280#554—1732no—
281#552—1764no—
282#547—1764no—
283#544—1764no—
284#542—1764no—
285#536—1764no—
286C2×C4⋊Q8—172no—
287C2×C4⋊Q8—174no—
288C2×C4⋊Q8—1716no—
289C2×C4⋊Q8—174no—
290C2×C4⋊Q8—174no—
291C2×C4⋊Q8—171no—
292C2×C4⋊Q8—174no—
293C2×C4⋊Q8C2×D1711yescharacteristic
294#529—1732no—
295C2×#126—174no—
296C2×#126—178no—
297C2×#125—172no—
298C2×#125—174no—
299C2×#125—178no—
300C2×#124—172no—
301C2×#124—174no—
302C2×#124C2×D1714yes—
303C2×#123—178no—
304C2×#123—174no—
305C2×#123—178no—
306C2×C22⋊Q8—174no—
307C2×C22⋊Q8—172no—
308C2×C22⋊Q8—178no—
309C2×C22⋊Q8—178no—
310C2×C22⋊Q8—178no—
311C2×#121—172no—
312C2×#121C2×D1714yes—
313C2×C4×Q8—174no—
314C2×C4×Q8—172no—
315C2×C4×D4—174no—
316C2×C4×D4—174no—
317C2×C4×D4C2×D1712yes—
318C2×#117—174no—
319C2×#117—174no—
320C2×#117C2×D1714yes—
321C22×C4⋊C4—174no—
322C22×C4⋊C4—174no—
323C22×C42—171no—
324C2×C17—48no—
325C2×C17#354414yescomplemented
326C2×C17C22×H(5,𝔽2)11yescharacteristic
327C2×C17C22×2−1+412yes—
328D17—24no—
329C2×#41—1732no—
330C2×#41—6816no—
331C22×Q8—1716no—
332C22×Q8—172no—
333C22×Q8—172no—
334C22×Q8—174no—
335C22×Q8—174no—
336C22×Q8—688no—
337C22×Q8C22×D1712yes—
338C22×D4—174no—
339C22×D4C22×D1712yes—
340C22×D4C22×D1714yes—
341C23×C4—178no—
342C23×C4—174no—
343C23×C4—172no—
344C23×C4—171no—
345C23×C4C22×D1714yes—
346C4⋊Q8—1716no—
347C4⋊Q8—1732no—
348C4⋊Q8—1732no—
349C4⋊Q8—1732no—
350C4⋊Q8—178no—
351C4⋊Q8—1732no—
352C4⋊Q8C22×D1718yes—
353#126—1732no—
354#126—1764no—
355#125—1716no—
356#125—1732no—
357#125—1764no—
358#124—1716no—
359#124—1732no—
360#124C22×D17132yes—
361#123—1764no—
362#123—1732no—
363#123—1764no—
364C22⋊Q8—1732no—
365C22⋊Q8—1716no—
366C22⋊Q8—1764no—
367C22⋊Q8—1764no—
368C22⋊Q8—1764no—
369#121—1716no—
370#121C22×D17132yes—
371C4×Q8—1732no—
372C4×Q8—1716no—
373C4×D4—1732no—
374C4×D4—1732no—
375C4×D4C22×D17116yes—
376#117—1732no—
377#117—1732no—
378#117C22×D17132yes—
379C2×C4⋊C4—174no—
380C2×C4⋊C4—172no—
381C2×C4⋊C4—178no—
382C2×C4⋊C4—178no—
383C2×C4⋊C4—1732no—
384C2×C4⋊C4—178no—
385C2×C4⋊C4—174no—
386C2×C4⋊C4—174no—
387C2×C4⋊C4—174no—
388C2×C4⋊C4—174no—
389C2×C4⋊C4—178no—
390C2×C4⋊C4—1732no—
391C2×C4⋊C4C22×D1714yes—
392C2×C4⋊C4C22×D1712yes—
393C2×C22⋊C4—174no—
394C2×C22⋊C4—174no—
395C2×C22⋊C4—178no—
396C2×C22⋊C4—174no—
397C2×C22⋊C4—174no—
398C2×C22⋊C4C22×D1718yes—
399C2×C22⋊C4C22×D1714yes—
400C2×C42—172no—
401C2×C42—174no—
402C2×C42—174no—
403C2×C42—171no—
404C2×C42—178no—
405C2×C42C22×D1714yes—
406C2×C42C22×D1711yescharacteristic
407C17C2×#354411yescharacteristic, complemented, Sylow
408C24—171no—
409C24C23×D1714yes—
410#41—6864no—
411C2×Q8—44no—
412C2×Q8—178no—
413C2×Q8—178no—
414C2×Q8—1716no—
415C2×Q8—1716no—
416C2×Q8—348no—
417C2×Q8—688no—
418C2×Q8—684no—
419C2×Q8—688no—
420C2×Q8—6832no—
421C2×Q8C23×D1718yes—
422C2×D4—28no—
423C2×D4—48no—
424C2×D4—1716no—
425C2×D4—3416no—
426C2×D4C23×D1718yes—
427C2×D4C23×D17116yes—
428C22×C4—216no—
429C22×C4—174no—
430C22×C4—172no—
431C22×C4—171no—
432C22×C4—174no—
433C22×C4—174no—
434C22×C4—1732no—
435C22×C4—178no—
436C22×C4—174no—
437C22×C4—1716no—
438C22×C4—344no—
439C22×C4—3416no—
440C22×C4—348no—
441C22×C4—6816no—
442C22×C4C23×D17116yes—
443C22×C4C23×D1711yescharacteristic
444C22×C4C23×D1714yes—
445C22×C4C23×D1714yes—
446C22×C4C23×D1712yes—
447C4⋊C4—1716no—
448C4⋊C4—178no—
449C4⋊C4—1732no—
450C4⋊C4—1732no—
451C4⋊C4—1732no—
452C4⋊C4—1716no—
453C4⋊C4—1716no—
454C4⋊C4—1716no—
455C4⋊C4—1716no—
456C4⋊C4—1732no—
457C4⋊C4C23×D17116yes—
458C4⋊C4C23×D1718yes—
459C22⋊C4—1716no—
460C22⋊C4—1716no—
461C22⋊C4—1732no—
462C22⋊C4—1716no—
463C22⋊C4—1716no—
464C22⋊C4C23×D17132yes—
465C22⋊C4C23×D17116yes—
466C42—178no—
467C42—174no—
468C42—1716no—
469C42—1716no—
470C42C23×D1714yes—
471C42C23×D17116yes—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—174no—
476C23—172no—
477C23—344no—
478C23C24×D1718yes—
479C23C24×D1711yescharacteristic
480Q8—48no—
481Q8—3416no—
482Q8—6816no—
483Q8—688no—
484Q8—6816no—
485D4—216no—
486D4—416no—
487D4—3432no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—232no—
492C2×C4—178no—
493C2×C4—174no—
494C2×C4—172no—
495C2×C4—178no—
496C2×C4—178no—
497C2×C4—348no—
498C2×C4—3432no—
499C2×C4—348no—
500C2×C4—342no—
501C2×C4—348no—
502C2×C4—344no—
503C2×C4—348no—
504C2×C4—3416no—
505C2×C4—6832no—
506C2×C4#1058769718yes—
507C2×C4C24×D1712yes—
508C2×C4C24×D1718yes—
509C2×C4C24×D1718yes—
510C2×C4C24×D1714yes—
511C22—28no—
512C22—28no—
513C22—28no—
514C22—416no—
515C22—174no—
516C22—348no—
517C22—344no—
518C22H(5,𝔽2)×D1712yes—
519C22C2×#1058769714yes—
520C22C25×D1711yescharacteristic
521C4—28no—
522C4—22no—
523C4—28no—
524C4—348no—
525C4—342no—
526C4—348no—
527C4—344no—
528C4—348no—
529C4C2×#1058769718yes—
530C2—48no—
531C2—344no—
532C2#1454311914yesdirect product factor
533C2C2×H(5,𝔽2)×D1711yescharacteristic
534C2C22×#1058769712yes—
subgroupG/H
centerC23C24×D17
commutator, cosocleC22×C17C26
FrattiniC22C25×D17
FittingC2×C17×#555C2
socleC23×C17C25
hypercenterC2×#555D17
hypocenterC17C2×#3544
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #14543119C2direct (N × H)
2. C2×#11763559C2semidirect
3. C2×#11763539C2semidirect
4. C2×#11763516C2semidirect
5. C2×#11763498C2semidirect
6. C2×#11763459C2semidirect
7. C2×#11763437C2semidirect
8. C2×#11763426C2semidirect
9. C2×#11763379C2semidirect
10. C2×C4⋊Q8×D17C2semidirect
11. C2×#11763341C2semidirect
12. C2×C17×#555C2semidirect
15. #11763559C22semidirect
16. #11763539C22semidirect
17. #11763516C22semidirect
18. #11763498C22semidirect
19. #11763459C22semidirect
20. #11763437C22semidirect
21. #11763426C22semidirect
22. #11763379C22semidirect
29. C4⋊Q8×D17C22semidirect
30. #11763341C22semidirect
56. C17×#555C22semidirect
247. C2×#555D17semidirect
279. #555C2×D17semidirect
325. C2×C17#3544semidirect
407. C17C2×#3544semidirect
532. C2#14543119direct (N × H)

Maximal subgroups

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

subgroupindex
#145431192
C2×#117635592
C2×#117635392
C2×#117635162
C2×#117634982
C2×#117634592
C2×#117634372
C2×#117634262
C2×#117633792
C2×C4⋊Q8×D172
C2×#117633412
C2×C17×#5552
C2×#354417