Finite Group Databasebeta

← C2×#14542864 #17942526

Subgroups — C2×#14542864 #17942526

Order 4352.

7758 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542864C2132yesmaximal, direct product factor
2C2×#11763545C213yesmaximal, complemented
3C2×Q8×Dic34C211yesmaximal, characteristic, complemented
4C2×#11763516C213yesmaximal, complemented
5C2×#11763466C212yesmaximal, complemented
6C2×#11763459C216yesmaximal, complemented
7C2×#11763426C216yesmaximal, complemented
8C2×#11763357C213yesmaximal, complemented
9C2×#11763341C213yesmaximal, complemented
10C2×#11763292C212yesmaximal, complemented
11C2×C4×Q8×D17C211yesmaximal, characteristic, complemented
12C2×C17×#550C211yesmaximal, characteristic, complemented
13C22×#63254C2216yes—
14C22×Q8×D17C2211yescharacteristic
15#11763545C22148yescomplemented
16Q8×Dic34C22116yescomplemented
17#11763516C22148yescomplemented
18#11763466C22132yescomplemented
19#11763459C22196yescomplemented
20#11763426C22196yescomplemented
21#11763357C22148yescomplemented
22C2×#10587555C2213yes—
23C2×#10587574C2216yes—
24#11763341C22148yescomplemented
25C2×#10587565C2216yes—
26C2×#10587576C2213yes—
27C2×#10587622C2213yes—
28C2×#10587622C2216yes—
29C2×#10587622C2216yes—
30C2×#10587569C2216yes—
31C2×#10587621C22112yes—
32C2×#10587583C2211yescharacteristic
33C2×#10587610C2216yes—
34C2×#10587564C22112yes—
35C2×#10587624C2216yes—
36C2×#10587624C2213yes—
37#11763292C22132yescomplemented
38C2×Q8×Dic17C2212yes—
39C2×Q8×Dic17C2211yescharacteristic
40C2×C4×Dic34C2213yes—
41C2×C4×Dic34C2213yes—
42C2×#10587573C2213yes—
43C4×Q8×D17C22116yescomplemented
44C2×D4×Dic17C2216yes—
45C2×D4×Dic17C2216yes—
46C2×D4×Dic17C2216yes—
47C2×C4×C22⋊D17C2216yes—
48C2×#10587612C2216yes—
49C22×C4⋊Dic17C2216yes—
50C2×C4⋊C4×D17C2213yes—
51C2×C42×D17C2213yes—
52C22×#41×C17C2212yes—
53C17×#550C22116yescomplemented
54C2×C17×#127C2213yes—
55C2×C17×#121C2216yes—
56C2×C4×Q8×C17C2211yescharacteristic
57C2×C4×D4×C17C2213yes—
58C2×#41×C17—216no—
59C2×#41×C17C23116yes—
60C22×Q8×C17C2311yescharacteristic
61C22×D4×C17C2313yes—
62C22×D4×C17C2316yes—
63C22×D4×C17C2316yes—
64C23×C4×C17C2316yes—
65C22×C22⋊D17C2316yes—
66C23×Dic17C23112yes—
67C2×Q8×D17—28no—
68C2×Q8×D17C2318yes—
69C2×#63254—424no—
70C2×#63254C23148yes—
71C22×C4×D17C2311yescharacteristic
72C22×C4×D17C2313yes—
73C22×C4×D17C2313yes—
74C22×Dic34C2313yes—
75C22×Dic34C2311yescharacteristic
76C22×Dic34C2316yes—
77C17×#127C23124yes—
78C17×#121C23148yes—
79C4×Q8×C17C2318yes—
80C4×D4×C17C23124yes—
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2313yes—
83C2×C22⋊C4×C17C2316yes—
84C2×C42×C17C2313yes—
85C2×C22⋊Dic17C2316yes—
86C2×C22⋊Dic17C2316yes—
87C2×C22⋊Dic17C2316yes—
88C2×C22⋊Dic17C23112yes—
89Q8×Dic17C23116yes—
90Q8×Dic17C2318yes—
91#10587624C23148yes—
92#10587624C23124yes—
93#10587622C23148yes—
94#10587622C23124yes—
95#10587622C23148yes—
96#10587621C23196yes—
97D4×Dic17C23148yes—
98D4×Dic17C23148yes—
99D4×Dic17C23148yes—
100C4×C22⋊D17C23148yes—
101C2×#63227C2312yes—
102#10587612C23148yes—
103C2×C4⋊Dic17C23148yes—
104C2×C4⋊Dic17C2313yes—
105C2×C4⋊Dic17C2316yes—
106C2×C4⋊Dic17C2316yes—
107C2×C4⋊Dic17C2312yes—
108C2×C4⋊Dic17C2313yes—
109#10587610C23148yes—
110C2×#63225C2316yes—
111C2×#63225C2316yes—
112C2×#63225C2316yes—
113C2×C4×Dic17C2313yes—
114C2×C4×Dic17C2313yes—
115C2×C4×Dic17C2313yes—
116C2×C4×Dic17C2316yes—
117#10587583C2318yes—
118C4⋊C4×D17C23124yes—
119#10587576C23124yes—
120#10587574C23148yes—
121#10587573C23124yes—
122#10587569C23148yes—
123#10587565C23148yes—
124#10587564C23196yes—
125C42×D17C23124yes—
126#10587555C23124yes—
127C4×Dic34C23124yes—
128C4×Dic34C23124yes—
129C24×C17C2416yes—
130C23×D17C2411yescharacteristic
131#41×C17—264no—
132C2×Q8×C17#4118yescomplemented
133C2×Q8×C17C2414yes—
134C2×D4×C17—224no—
135C2×D4×C17—212no—
136C2×D4×C17—412no—
137C2×D4×C17C24112yes—
138C2×D4×C17C24124yes—
139C2×D4×C17C24124yes—
140C22×C4×C17—224no—
141C22×C4×C17C2413yes—
142C22×C4×C17C2412yes—
143C22×C4×C17C24124yes—
144C22×C4×C17C2413yes—
145C22×C4×C17C2411yescharacteristic
146C2×C22⋊D17—224no—
147C2×C22⋊D17C24124yes—
148C22×Dic17—424no—
149C22×Dic17C2413yes—
150C22×Dic17C2416yes—
151C22×Dic17C2412yes—
152C22×Dic17C2413yes—
153C22×Dic17C2411yescharacteristic
154C22×Dic17C24148yes—
155Q8×D17—232no—
156#63254—496no—
157C2×C4×D17—212no—
158C2×C4×D17—212no—
159C2×C4×D17—24no—
160C2×C4×D17C24112yes—
161C2×C4×D17C24112yes—
162C2×C4×D17C2414yes—
163C2×Dic34—212no—
164C2×Dic34—24no—
165C2×Dic34—412no—
166C2×Dic34C24112yes—
167C2×Dic34C2414yes—
168C2×Dic34C24124yes—
169C4⋊C4×C17C24112yes—
170C4⋊C4×C17C2414yes—
171C22⋊C4×C17C24124yes—
172C42×C17C24112yes—
173C22⋊Dic17C24148yes—
174C22⋊Dic17C24124yes—
175C22⋊Dic17C24124yes—
176C22⋊Dic17C24124yes—
177#63227C2418yes—
178C4⋊Dic17C24112yes—
179C4⋊Dic17C24112yes—
180C4⋊Dic17C24124yes—
181C4⋊Dic17C24124yes—
182C4⋊Dic17C2418yes—
183#63225C24124yes—
184#63225C24124yes—
185#63225C24124yes—
186C4×Dic17C24112yes—
187C4×Dic17C24124yes—
188C4×Dic17C24112yes—
189C4×Dic17C24112yes—
190C2×#3522—171nomaximal, Sylow
191C23×C17—212no—
192C23×C17—212no—
193C23×C17—212no—
194C23×C17C25112yes—
195C23×C17C2511yescharacteristic
196C22×D17—22no—
197C22×D17—22no—
198C22×D17H(5,𝔽2)14yes—
199C22×D17C2512yes—
200Q8×C17C2×#41116yescomplemented
201D4×C17—248no—
202D4×C17—224no—
203D4×C17—424no—
204C2×C4×C17—24no—
205C2×C4×C17—26no—
206C2×C4×C17—248no—
207C2×C4×C17C2×#41112yes—
208C2×C4×C172−1+414yes—
209C2×C4×C17C2516yes—
210C2×C4×C17C2514yes—
211C2×C4×C17C2516yes—
212C2×C4×C17C2512yes—
213C22⋊D17—248no—
214C2×Dic17—26no—
215C2×Dic17—212no—
216C2×Dic17—26no—
217C2×Dic17—24no—
218C2×Dic17—22no—
219C2×Dic17—448no—
220C2×Dic17C2516yes—
221C2×Dic17C2514yes—
222C2×Dic17C2512yes—
223C2×Dic17C25112yes—
224C2×Dic17C2516yes—
225C4×D17—224no—
226C4×D17—224no—
227C4×D17—28no—
228Dic34—224no—
229Dic34—28no—
230Dic34—424no—
231#3522—1732no—
232C2×#559—173no—
233C2×Q82—171no—
234C2×#554—173no—
235C2×#550D1711yescharacteristic, complemented
236C2×#548—172no—
237C2×#547—176no—
238C2×#542—176no—
239C2×#532—173no—
240C2×#529—173no—
241C2×#520—172no—
242C22×C4×Q8—171no—
243C22×C17—212no—
244C22×C17—212no—
245C22×C17—212no—
246C22×C17—424no—
247C22×C17C22×#4112yes—
248C22×C17C2×H(5,𝔽2)12yes—
249C22×C17C2×2−1+412yes—
250C22×C17C2611yescharacteristic
251C2×D17—22no—
252C2×D17—22no—
253C2×D17—28no—
254C2×D17C2×H(5,𝔽2)14yes—
255C4×C17—24no—
256C4×C17—26no—
257C4×C17C22×#41112yes—
258C4×C17C2×2−1+414yes—
259Dic17—26no—
260Dic17—212no—
261Dic17—26no—
262Dic17—24no—
263Dic17—22no—
264C22×#41—176no—
265C22×#41C2×D1712yes—
266C23×Q8—171no—
267#559—1748no—
268Q82—1716no—
269#554—1748no—
270#550C2×D17116yescomplemented
271#548—1732no—
272#547—1796no—
273#542—1796no—
274#532—1748no—
275C2×C4⋊Q8—176no—
276C2×C4⋊Q8—173no—
277C2×#127C2×D1713yes—
278#529—1748no—
279C2×#126—176no—
280C2×#125—173no—
281C2×#124—173no—
282C2×#124—176no—
283C2×#124—176no—
284C2×#123—1712no—
285C2×#123—176no—
286C2×C22⋊Q8—176no—
287C2×C22⋊Q8—171no—
288C2×C22⋊Q8—1712no—
289C2×#121—173no—
290C2×#121—176no—
291C2×#121C2×D1716yes—
292#520—1732no—
293C2×C4×Q8—171no—
294C2×C4×Q8—172no—
295C2×C4×Q8—1716no—
296C2×C4×Q8—173no—
297C2×C4×Q8—173no—
298C2×C4×Q8—173no—
299C2×C4×Q8C2×D1711yescharacteristic
300C2×C4×D4—176no—
301C2×C4×D4—176no—
302C2×C4×D4—176no—
303C2×C4×D4—176no—
304C2×C4×D4C2×D1713yes—
305C2×#117—176no—
306C22×C4⋊C4—176no—
307C22×C4⋊C4—173no—
308C22×C42—173no—
309C2×C17—412no—
310C2×C17#352214yescomplemented
311C2×C17C23×#4111yescharacteristic
312C2×C17C22×H(5,𝔽2)11yescharacteristic
313C2×C17C22×2−1+411yescharacteristic
314D17—24no—
315C2×#41—216no—
316C2×#41—1748no—
317C2×#41—6824no—
318C2×#41C22×D17116yes—
319C22×Q8—178no—
320C22×Q8—171no—
321C22×Q8—173no—
322C22×Q8—176no—
323C22×Q8—348no—
324C22×Q8C22×D1711yescharacteristic
325C22×D4—176no—
326C22×D4C22×D1713yes—
327C22×D4C22×D1716yes—
328C22×D4C22×D1716yes—
329C23×C4—1712no—
330C23×C4—173no—
331C23×C4—173no—
332C23×C4—171no—
333C23×C4C22×D1716yes—
334C4⋊Q8—1748no—
335C4⋊Q8—1724no—
336#127C22×D17124yes—
337#126—1748no—
338#125—1724no—
339#124—1724no—
340#124—1748no—
341#124—1748no—
342#123—1796no—
343#123—1748no—
344C22⋊Q8—1748no—
345C22⋊Q8—178no—
346C22⋊Q8—1796no—
347#121—1724no—
348#121—1748no—
349#121C22×D17148yes—
350C4×Q8—178no—
351C4×Q8—1716no—
352C4×Q8—1724no—
353C4×Q8—1724no—
354C4×Q8—1724no—
355C4×Q8C22×D1718yes—
356C4×D4—1748no—
357C4×D4—1748no—
358C4×D4—1748no—
359C4×D4—1748no—
360C4×D4C22×D17124yes—
361#117—1748no—
362C2×C4⋊C4—173no—
363C2×C4⋊C4—176no—
364C2×C4⋊C4—176no—
365C2×C4⋊C4—176no—
366C2×C4⋊C4—1748no—
367C2×C4⋊C4—172no—
368C2×C4⋊C4—173no—
369C2×C4⋊C4—176no—
370C2×C4⋊C4—176no—
371C2×C4⋊C4—1724no—
372C2×C4⋊C4C22×D1713yes—
373C2×C4⋊C4C22×D1711yescharacteristic
374C2×C22⋊C4—176no—
375C2×C22⋊C4—176no—
376C2×C22⋊C4—1712no—
377C2×C22⋊C4—176no—
378C2×C22⋊C4—172no—
379C2×C22⋊C4C22×D1716yes—
380C2×C42—173no—
381C2×C42—176no—
382C2×C42—173no—
383C2×C42—173no—
384C2×C42—1724no—
385C2×C42C22×D1713yes—
386C17C2×#352211yescharacteristic, complemented, Sylow
387C24—171no—
388C24C23×D1716yes—
389#41—264no—
390#41—6896no—
391C2×Q8—174no—
392C2×Q8—1712no—
393C2×Q8—1724no—
394C2×Q8—3432no—
395C2×Q8—3412no—
396C2×Q8—344no—
397C2×Q8—6812no—
398C2×Q8#6325418yescomplemented
399C2×Q8C23×D1714yes—
400C2×D4—224no—
401C2×D4—212no—
402C2×D4—412no—
403C2×D4—1724no—
404C2×D4—3424no—
405C2×D4C23×D17112yes—
406C2×D4C23×D17124yes—
407C2×D4C23×D17124yes—
408C22×C4—224no—
409C22×C4—171no—
410C22×C4—172no—
411C22×C4—173no—
412C22×C4—173no—
413C22×C4—176no—
414C22×C4—1748no—
415C22×C4—174no—
416C22×C4—1712no—
417C22×C4—1712no—
418C22×C4—3412no—
419C22×C4—3412no—
420C22×C4—344no—
421C22×C4—6824no—
422C22×C4C23×D17124yes—
423C22×C4C23×D1713yes—
424C22×C4C23×D1712yes—
425C22×C4C23×D1713yes—
426C22×C4C23×D1711yescharacteristic
427C4⋊C4—1712no—
428C4⋊C4—1724no—
429C4⋊C4—1724no—
430C4⋊C4—1724no—
431C4⋊C4—178no—
432C4⋊C4—1712no—
433C4⋊C4—1724no—
434C4⋊C4—1724no—
435C4⋊C4C23×D17112yes—
436C4⋊C4C23×D1714yes—
437C22⋊C4—1724no—
438C22⋊C4—1724no—
439C22⋊C4—1748no—
440C22⋊C4—1724no—
441C22⋊C4—178no—
442C22⋊C4C23×D17124yes—
443C42—1712no—
444C42—1712no—
445C42—1712no—
446C42—1724no—
447C42C23×D17112yes—
448C23—212no—
449C23—212no—
450C23—212no—
451C23—174no—
452C23—172no—
453C23—342no—
454C23—342no—
455C23C24×D17112yes—
456C23C24×D1711yescharacteristic
457Q8—3424no—
458Q8—348no—
459Q8—6824no—
460Q8C2×#63254116yescomplemented
461D4—248no—
462D4—224no—
463D4—424no—
464D4—3448no—
465C2×C4—24no—
466C2×C4—26no—
467C2×C4—248no—
468C2×C4—172no—
469C2×C4—174no—
470C2×C4—176no—
471C2×C4—176no—
472C2×C4—1712no—
473C2×C4—3424no—
474C2×C4—346no—
475C2×C4—346no—
476C2×C4—3412no—
477C2×C4—342no—
478C2×C4—344no—
479C2×C4—3424no—
480C2×C4—348no—
481C2×C4—6848no—
482C2×C4C2×#63254112yes—
483C2×C4#1058769714yes—
484C2×C4C24×D1716yes—
485C2×C4C24×D1714yes—
486C2×C4C24×D1716yes—
487C2×C4C24×D1712yes—
488C22—212no—
489C22—212no—
490C22—212no—
491C22—424no—
492C22—174no—
493C22—348no—
494C22—342no—
495C22—342no—
496C22C22×#6325412yes—
497C22H(5,𝔽2)×D1712yes—
498C22C2×#1058769712yes—
499C22C25×D1711yescharacteristic
500C4—24no—
501C4—26no—
502C4—346no—
503C4—346no—
504C4—3412no—
505C4—342no—
506C4—344no—
507C4C22×#63254112yes—
508C4C2×#1058769714yes—
509C2—412no—
510C2—344no—
511C2#1454286414yesdirect product factor
512C2C23×#6325411yescharacteristic
513C2C2×H(5,𝔽2)×D1711yescharacteristic
514C2C22×#1058769711yescharacteristic
subgroupG/H
centerC23C24×D17
commutator, cosocleC22×C17C26
FrattiniC22C25×D17
FittingC2×C17×#550C2
socleC23×C17C25
hypercenterC2×#550D17
hypocenterC17C2×#3522
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #14542864C2direct (N × H)
2. C2×#11763545C2semidirect
3. C2×Q8×Dic34C2semidirect
4. C2×#11763516C2semidirect
5. C2×#11763466C2semidirect
6. C2×#11763459C2semidirect
7. C2×#11763426C2semidirect
8. C2×#11763357C2semidirect
9. C2×#11763341C2semidirect
10. C2×#11763292C2semidirect
11. C2×C4×Q8×D17C2semidirect
12. C2×C17×#550C2semidirect
15. #11763545C22semidirect
16. Q8×Dic34C22semidirect
17. #11763516C22semidirect
18. #11763466C22semidirect
19. #11763459C22semidirect
20. #11763426C22semidirect
21. #11763357C22semidirect
24. #11763341C22semidirect
37. #11763292C22semidirect
43. C4×Q8×D17C22semidirect
53. C17×#550C22semidirect
132. C2×Q8×C17#41semidirect
200. Q8×C17C2×#41semidirect
235. C2×#550D17semidirect
270. #550C2×D17semidirect
310. C2×C17#3522semidirect
386. C17C2×#3522semidirect
398. C2×Q8#63254semidirect
460. Q8C2×#63254semidirect
511. C2#14542864direct (N × H)

Maximal subgroups

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

subgroupindex
#145428642
C2×#117635452
C2×Q8×Dic342
C2×#117635162
C2×#117634662
C2×#117634592
C2×#117634262
C2×#117633572
C2×#117633412
C2×#117632922
C2×C4×Q8×D172
C2×C17×#5502
C2×#352217