Finite Group Databasebeta

← C2×#14543129 #17942009

Subgroups — C2×#14543129 #17942009

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#14543129C2132yesmaximal, direct product factor
2C2×#11763558C214yesmaximal, complemented
3C2×#11763525C211yesmaximal, characteristic, complemented
4C2×#11763519C212yesmaximal, complemented
5C2×#11763500C214yesmaximal, complemented
6C2×#11763462C214yesmaximal, complemented
7C2×#11763436C214yesmaximal, complemented
8C2×#11763431C214yesmaximal, complemented
9C2×#11763384C214yesmaximal, complemented
10C2×C4⋊Q8×D17C211yesmaximal, characteristic, complemented
11C2×#11763346C212yesmaximal, complemented
12C2×Q82×C17C211yesmaximal, characteristic, complemented
13C22×#63256C2214yes—
14C22×Q8×D17C2212yes—
15#11763558C22164yescomplemented
16#11763525C22116yescomplemented
17#11763519C22132yescomplemented
18#11763500C22164yescomplemented
19#11763462C22164yescomplemented
20#11763436C22164yescomplemented
21#11763431C22164yescomplemented
22#11763384C22164yescomplemented
23C2×#10587555C2211yescharacteristic
24C2×#10587574C2214yes—
25C2×#10587630C2214yes—
26C2×#10587630C2212yes—
27C4⋊Q8×D17C22116yescomplemented
28#11763346C22132yescomplemented
29C2×#10587584C2214yes—
30C2×#10587562C2218yes—
31C2×#10587556C2214yes—
32C2×#10587576C2212yes—
33C2×#10587575C2218yes—
34C2×#10587633C2214yes—
35C2×#10587633C2212yes—
36C2×#10587561C2214yes—
37C2×#10587580C2214yes—
38C2×#10587580C2218yes—
39C2×#10587580C2218yes—
40C2×#10587582C2218yes—
41C2×#10587582C2218yes—
42C2×#10587632C2218yes—
43C2×#10587632C2214yes—
44C2×#10587632C2212yes—
45C2×#10587581C2214yes—
46C2×#10587581C2212yes—
47C2×C4×D68C2214yes—
48C2×C4×D68C2214yes—
49C2×C4×D68C2212yes—
50C2×#10587578C2214yes—
51C2×#10587558C2214yes—
52C2×#10587558C2214yes—
53C2×C4⋊C4×D17C2214yes—
54C2×C4⋊C4×D17C2214yes—
55C2×C42×D17C2211yescharacteristic
56Q82×C17C22116yescomplemented
57C2×C17×C4⋊Q8C2214yes—
58C2×C17×C4⋊Q8C2214yes—
59C2×C17×C4⋊Q8C2211yescharacteristic
60C2×C4×Q8×C17C2214yes—
61C2×C4×Q8×C17C2212yes—
62C22×Q8×C17C2312yes—
63C22×Q8×C17C2314yes—
64C22×Q8×C17C2312yes—
65C2×#63256—416no—
66C2×#63256C23132yes—
67C2×Q8×D17—48no—
68C2×Q8×D17C23116yes—
69C22×D68C2312yes—
70C22×D68C2314yes—
71C22×D68C2314yes—
72C22×C4×D17C2312yes—
73C22×C4×D17C2311yescharacteristic
74C22×C4×D17C2314yes—
75C22×C4×D17C2314yes—
76C22×C4×D17C2318yes—
77C22×Dic34C2312yes—
78C22×Dic34C2314yes—
79C17×C4⋊Q8C23132yes—
80C17×C4⋊Q8C2318yes—
81C17×C4⋊Q8C23132yes—
82C4×Q8×C17C23132yes—
83C4×Q8×C17C23116yes—
84C2×C4⋊C4×C17C2312yes—
85C2×C4⋊C4×C17C2314yes—
86C2×C4⋊C4×C17C2314yes—
87C2×C4⋊C4×C17C2318yes—
88C2×C42×C17C2311yescharacteristic
89C2×C42×C17C2314yes—
90C2×C42×C17C2314yes—
91#10587633C23132yes—
92#10587633C23116yes—
93#10587632C23164yes—
94#10587632C23116yes—
95#10587632C23132yes—
96#10587630C23116yes—
97#10587630C23132yes—
98C2×#63227C2314yes—
99C2×#63227C2314yes—
100C2×#63227C2314yes—
101C2×#63227C2318yes—
102C2×#63227C2318yes—
103C2×#63227C2314yes—
104C2×#63227C2314yes—
105C2×C4⋊Dic17C2314yes—
106C2×C4⋊Dic17C2312yes—
107C2×C4⋊Dic17C2314yes—
108C2×#63225C2318yes—
109C2×#63225C2318yes—
110C2×#63225C2318yes—
111C2×#63225C2314yes—
112C2×#63225C2314yes—
113C2×C4×Dic17C2312yes—
114C2×C4×Dic17C2311yescharacteristic
115C2×C4×Dic17C2314yes—
116#10587584C23132yes—
117#10587582C23164yes—
118#10587582C23164yes—
119#10587581C23116yes—
120#10587581C23132yes—
121#10587580C23164yes—
122#10587580C23164yes—
123#10587580C23132yes—
124#10587578C23132yes—
125C4⋊C4×D17C23132yes—
126C4⋊C4×D17C23132yes—
127#10587576C23116yes—
128#10587575C23164yes—
129#10587574C23132yes—
130#10587562C23164yes—
131#10587561C23132yes—
132C4×D68C23132yes—
133C4×D68C23132yes—
134C4×D68C23116yes—
135#10587558C23132yes—
136#10587558C23132yes—
137C42×D17C2318yes—
138#10587556C23132yes—
139#10587555C2318yes—
140C23×D17C2414yes—
141C23×D17C2411yescharacteristic
142C2×Q8×C17—28no—
143C2×Q8×C17—44no—
144C2×Q8×C17—48no—
145C2×Q8×C17C2418yes—
146C2×Q8×C17C2418yes—
147C2×Q8×C17C24116yes—
148C22×C4×C17C2411yescharacteristic
149C22×C4×C17C2414yes—
150C22×C4×C17C2414yes—
151C22×C4×C17C2414yes—
152C22×C4×C17C2412yes—
153C22×Dic17C2411yescharacteristic
154C22×Dic17C2414yes—
155C22×Dic17C2412yes—
156C22×Dic17C2414yes—
157#63256—464no—
158Q8×D17—432no—
159C2×D68—216no—
160C2×D68—28no—
161C2×D68—48no—
162C2×D68C24116yes—
163C2×D68C2418yes—
164C2×D68C24116yes—
165C2×C4×D17—24no—
166C2×C4×D17—216no—
167C2×C4×D17—28no—
168C2×C4×D17—216no—
169C2×C4×D17—416no—
170C2×C4×D17C2414yes—
171C2×C4×D17C24116yes—
172C2×C4×D17C2418yes—
173C2×C4×D17C24116yes—
174C2×C4×D17C24132yes—
175C2×Dic34—48no—
176C2×Dic34—44no—
177C2×Dic34C2418yes—
178C2×Dic34C24116yes—
179C4⋊C4×C17C24116yes—
180C4⋊C4×C17C24132yes—
181C4⋊C4×C17C2418yes—
182C4⋊C4×C17C24116yes—
183C42×C17C2414yes—
184C42×C17C24116yes—
185C42×C17C24116yes—
186#63227C24116yes—
187#63227C24116yes—
188#63227C24116yes—
189#63227C24132yes—
190#63227C24116yes—
191#63227C24116yes—
192#63227C24132yes—
193C4⋊Dic17C24116yes—
194C4⋊Dic17C2418yes—
195C4⋊Dic17C24116yes—
196#63225C24132yes—
197#63225C24132yes—
198#63225C24116yes—
199#63225C24116yes—
200#63225C24132yes—
201C4×Dic17C2418yes—
202C4×Dic17C2414yes—
203C4×Dic17C24116yes—
204C2×#3544—171nomaximal, Sylow
205C23×C17C2511yescharacteristic
206C22×D17—24no—
207C22×D17—28no—
208C22×D17—28no—
209C22×D17—28no—
210C22×D17H(5,𝔽2)14yes—
211C22×D17C2518yes—
212C22×D17C2512yes—
213Q8×C17—216no—
214Q8×C17—48no—
215Q8×C17—416no—
216C2×C4×C17—22no—
217C2×C4×C17—28no—
218C2×C4×C17—28no—
219C2×C4×C17—28no—
220C2×C4×C172−1+418yes—
221C2×C4×C17C2518yes—
222C2×C4×C17C2512yes—
223C2×C4×C17C2518yes—
224C2×C4×C17C2518yes—
225C2×C4×C17C2514yes—
226C2×Dic17—22no—
227C2×Dic17—28no—
228C2×Dic17—24no—
229C2×Dic17—28no—
230C2×Dic17C2512yes—
231C2×Dic17C2518yes—
232C2×Dic17C2514yes—
233C2×Dic17C2518yes—
234D68—232no—
235D68—216no—
236D68—416no—
237C4×D17—28no—
238C4×D17—232no—
239C4×D17—216no—
240C4×D17—232no—
241C4×D17—432no—
242Dic34—416no—
243Dic34—48no—
244#3544—1732no—
245C2×#563—174no—
246C2×Q82D1711yescharacteristic, complemented
247C2×#555—171no—
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×C17C2×H(5,𝔽2)12yes—
257C22×C17C2×2−1+414yes—
258C22×C17C2611yescharacteristic
259C2×D17—24no—
260C2×D17—28no—
261C2×D17—28no—
262C2×D17—28no—
263C2×D17—28no—
264C2×D17—416no—
265C2×D17C2×H(5,𝔽2)14yes—
266C4×C17—22no—
267C4×C17—28no—
268C4×C17—28no—
269C4×C17—28no—
270C4×C17C2×2−1+418yes—
271Dic17—22no—
272Dic17—28no—
273Dic17—24no—
274Dic17—28no—
275C22×#41—174no—
276C23×Q8—172no—
277#563—1764no—
278Q82C2×D17116yescomplemented
279#555—1716no—
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—171no—
291C2×C4⋊Q8C2×D1711yescharacteristic
292C2×C4⋊Q8C2×D1714yes—
293C2×C4⋊Q8C2×D1714yes—
294#529—1732no—
295C2×#126—174no—
296C2×#126—178no—
297C2×#125—172no—
298C2×#125—178no—
299C2×#125—174no—
300C2×#124—172no—
301C2×#124—174no—
302C2×#124—174no—
303C2×#123—174no—
304C2×#123—178no—
305C2×#123—178no—
306C2×C22⋊Q8—172no—
307C2×C22⋊Q8—174no—
308C2×C22⋊Q8—178no—
309C2×C22⋊Q8—178no—
310C2×C22⋊Q8—178no—
311C2×#121—172no—
312C2×#121—174no—
313C2×C4×Q8C2×D1712yes—
314C2×C4×Q8C2×D1714yes—
315C2×C4×D4—174no—
316C2×C4×D4—172no—
317C2×C4×D4—174no—
318C2×#117—174no—
319C2×#117—174no—
320C2×#117—174no—
321C22×C4⋊C4—174no—
322C22×C4⋊C4—174no—
323C22×C42—171no—
324C2×C17#354414yescomplemented
325C2×C17C22×H(5,𝔽2)11yescharacteristic
326C2×C17C22×2−1+412yes—
327D17—24no—
328D17—48no—
329C2×#41—1732no—
330C2×#41—6816no—
331C22×Q8—1716no—
332C22×Q8—172no—
333C22×Q8—174no—
334C22×Q8—688no—
335C22×Q8C22×D1712yes—
336C22×Q8C22×D1714yes—
337C22×Q8C22×D1712yes—
338C22×D4—172no—
339C22×D4—174no—
340C22×D4—174no—
341C23×C4—174no—
342C23×C4—172no—
343C23×C4—174no—
344C23×C4—171no—
345C23×C4—178no—
346C4⋊Q8—1716no—
347C4⋊Q8—1732no—
348C4⋊Q8—1732no—
349C4⋊Q8—178no—
350C4⋊Q8C22×D1718yes—
351C4⋊Q8C22×D17132yes—
352C4⋊Q8C22×D17132yes—
353#126—1732no—
354#126—1764no—
355#125—1716no—
356#125—1764no—
357#125—1732no—
358#124—1716no—
359#124—1732no—
360#124—1732no—
361#123—1732no—
362#123—1764no—
363#123—1764no—
364C22⋊Q8—1716no—
365C22⋊Q8—1732no—
366C22⋊Q8—1764no—
367C22⋊Q8—1764no—
368C22⋊Q8—1764no—
369#121—1716no—
370#121—1732no—
371C4×Q8C22×D17116yes—
372C4×Q8C22×D17132yes—
373C4×D4—1732no—
374C4×D4—1716no—
375C4×D4—1732no—
376#117—1732no—
377#117—1732no—
378#117—1732no—
379C2×C4⋊C4—174no—
380C2×C4⋊C4—178no—
381C2×C4⋊C4—178no—
382C2×C4⋊C4—172no—
383C2×C4⋊C4—178no—
384C2×C4⋊C4—174no—
385C2×C4⋊C4—174no—
386C2×C4⋊C4—174no—
387C2×C4⋊C4—1732no—
388C2×C4⋊C4—1732no—
389C2×C4⋊C4C22×D1712yes—
390C2×C4⋊C4C22×D1714yes—
391C2×C4⋊C4C22×D1714yes—
392C2×C4⋊C4C22×D1718yes—
393C2×C22⋊C4—174no—
394C2×C22⋊C4—174no—
395C2×C22⋊C4—178no—
396C2×C22⋊C4—174no—
397C2×C22⋊C4—178no—
398C2×C22⋊C4—174no—
399C2×C22⋊C4—174no—
400C2×C42—174no—
401C2×C42—172no—
402C2×C42—171no—
403C2×C42—178no—
404C2×C42C22×D1714yes—
405C2×C42C22×D1714yes—
406C2×C42C22×D1711yescharacteristic
407C17C2×#354411yescharacteristic, complemented, Sylow
408C24—171no—
409C24—174no—
410#41—6864no—
411C2×Q8—28no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—178no—
415C2×Q8—1716no—
416C2×Q8—688no—
417C2×Q8—684no—
418C2×Q8—6832no—
419C2×Q8C23×D1718yes—
420C2×Q8C23×D17116yes—
421C2×Q8C23×D1718yes—
422C2×D4—178no—
423C2×D4—1716no—
424C2×D4—1716no—
425C2×D4—3416no—
426C2×D4—348no—
427C2×D4—688no—
428C22×C4—174no—
429C22×C4—172no—
430C22×C4—171no—
431C22×C4—174no—
432C22×C4—178no—
433C22×C4—174no—
434C22×C4—1716no—
435C22×C4—1716no—
436C22×C4—1732no—
437C22×C4—344no—
438C22×C4—3416no—
439C22×C4—348no—
440C22×C4—3416no—
441C22×C4—6816no—
442C22×C4C23×D1711yescharacteristic
443C22×C4C23×D1714yes—
444C22×C4C23×D1714yes—
445C22×C4C23×D1714yes—
446C22×C4C23×D1712yes—
447C4⋊C4—1716no—
448C4⋊C4—1732no—
449C4⋊C4—1732no—
450C4⋊C4—178no—
451C4⋊C4—1732no—
452C4⋊C4—1716no—
453C4⋊C4—1716no—
454C4⋊C4—1716no—
455C4⋊C4C23×D1718yes—
456C4⋊C4C23×D17116yes—
457C4⋊C4C23×D17116yes—
458C4⋊C4C23×D17132yes—
459C22⋊C4—1716no—
460C22⋊C4—1716no—
461C22⋊C4—1732no—
462C22⋊C4—1716no—
463C22⋊C4—1732no—
464C22⋊C4—1716no—
465C22⋊C4—1716no—
466C42—178no—
467C42—174no—
468C42—1716no—
469C42C23×D1714yes—
470C42C23×D17116yes—
471C42C23×D17116yes—
472C23—174no—
473C23—172no—
474C23—178no—
475C23—344no—
476C23—348no—
477C23—348no—
478C23—348no—
479C23C24×D1711yescharacteristic
480Q8—216no—
481Q8—48no—
482Q8—416no—
483Q8—6816no—
484Q8—688no—
485D4—3432no—
486D4—3416no—
487D4—6816no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—174no—
493C2×C4—178no—
494C2×C4—172no—
495C2×C4—178no—
496C2×C4—348no—
497C2×C4—3432no—
498C2×C4—348no—
499C2×C4—342no—
500C2×C4—348no—
501C2×C4—344no—
502C2×C4—3416no—
503C2×C4—3432no—
504C2×C4—6832no—
505C2×C4#1058769418yes—
506C2×C4C24×D1712yes—
507C2×C4C24×D1718yes—
508C2×C4C24×D1718yes—
509C2×C4C24×D1718yes—
510C2×C4C24×D1714yes—
511C22—174no—
512C22—348no—
513C22—344no—
514C22—348no—
515C22—348no—
516C22—348no—
517C22—6816no—
518C22H(5,𝔽2)×D1712yes—
519C22C2×#1058769414yes—
520C22C25×D1711yescharacteristic
521C4—28no—
522C4—22no—
523C4—28no—
524C4—28no—
525C4—348no—
526C4—342no—
527C4—344no—
528C4—348no—
529C4C2×#1058769418yes—
530C2—344no—
531C2—688no—
532C2#1454312914yesdirect product factor
533C2C2×H(5,𝔽2)×D1711yescharacteristic
534C2C22×#1058769412yes—
subgroupG/H
centerC23C24×D17
commutator, cosocleC22×C17C26
FrattiniC22C25×D17
FittingC2×Q82×C17C2
socleC23×C17C25
hypercenterC2×Q82D17
hypocenterC17C2×#3544
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. #14543129C2direct (N × H)
2. C2×#11763558C2semidirect
3. C2×#11763525C2semidirect
4. C2×#11763519C2semidirect
5. C2×#11763500C2semidirect
6. C2×#11763462C2semidirect
7. C2×#11763436C2semidirect
8. C2×#11763431C2semidirect
9. C2×#11763384C2semidirect
10. C2×C4⋊Q8×D17C2semidirect
11. C2×#11763346C2semidirect
12. C2×Q82×C17C2semidirect
15. #11763558C22semidirect
16. #11763525C22semidirect
17. #11763519C22semidirect
18. #11763500C22semidirect
19. #11763462C22semidirect
20. #11763436C22semidirect
21. #11763431C22semidirect
22. #11763384C22semidirect
27. C4⋊Q8×D17C22semidirect
28. #11763346C22semidirect
56. Q82×C17C22semidirect
246. C2×Q82D17semidirect
278. Q82C2×D17semidirect
324. C2×C17#3544semidirect
407. C17C2×#3544semidirect
532. C2#14543129direct (N × H)

Maximal subgroups

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

subgroupindex
#145431292
C2×#117635582
C2×#117635252
C2×#117635192
C2×#117635002
C2×#117634622
C2×#117634362
C2×#117634312
C2×#117633842
C2×C4⋊Q8×D172
C2×#117633462
C2×Q82×C172
C2×#354417