Finite Group Databasebeta

← C2×#11743170 #14261567

Subgroups — C2×#11743170 #14261567

Order 1792.

6566 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#125C211yesmaximal, characteristic, complemented
2#11743170C2132yesmaximal, direct product factor
3C2×#90390C212yesmaximal, complemented
4C2×#90391C211yesmaximal, characteristic, complemented
5C2×#90384C214yesmaximal, complemented
6C2×#90383C214yesmaximal, complemented
7C2×C7⋊#564C211yesmaximal, characteristic, complemented
8C2×#90387C214yesmaximal, complemented
9C2×#90205C214yesmaximal, complemented
10C2×#90206C214yesmaximal, complemented
11C2×#90202C214yesmaximal, complemented
12C2×#90170C212yesmaximal, complemented
13C2×C7×#125C2214yes—
14C2×C7×#125C2211yescharacteristic
15C2×C7×#125C2212yes—
16C2×C7×#125C2211yescharacteristic
17C2×C7×#125C22116yescomplemented
18C22×C7×C4⋊C4C2214yes—
19C22×C7×C4⋊C4C2212yes—
20C22×C42×C7C2211yescharacteristic
21C2×#6493C2212yes—
22C2×#6493C2214yes—
23C2×#6492C2214yes—
24C2×C4×C22⋊D7C2214yes—
25C2×C4×C22⋊D7C2214yes—
26C2×C4×C22⋊D7C2214yes—
27C2×#6487C2212yes—
28C2×#6487C2214yes—
29#90391C22116yescomplemented
30#90390C22132yescomplemented
31#90387C22164yescomplemented
32#90384C22164yescomplemented
33#90383C22164yescomplemented
34C7⋊#564C22116yescomplemented
35#90206C22164yescomplemented
36#90205C22164yescomplemented
37C2×#6461C2214yes—
38C2×#6461C2214yes—
39C2×#6461C2214yes—
40C2×#6461C2214yes—
41#90202C22164yescomplemented
42C2×#6460C2214yes—
43C2×#6459C2214yes—
44C2×#6459C2214yes—
45C2×#6459C2214yes—
46C2×#6459C2214yes—
47C2×#6458C2214yes—
48C2×#6457C2214yes—
49C2×#6457C2214yes—
50C2×#6457C2214yes—
51C2×#6457C2214yes—
52C2×#6456C2214yes—
53C2×#6455C2214yes—
54C2×D7×C4⋊C4C2214yes—
55C2×#6453C2214yes—
56C2×#6452C2214yes—
57C2×#6452C2214yes—
58C2×#6452C2214yes—
59C2×#6452C2214yes—
60C2×#6451C2214yes—
61C2×#6450C2214yes—
62#90170C22132yescomplemented
63C2×#6439C2214yes—
64C2×#6438C2212yes—
65C2×C4×D28C2212yes—
66C2×#6435C2214yes—
67C2×#6433C2212yes—
68C2×C4×Dic14C2212yes—
69C2×#3553—71nomaximal, Sylow
70C23×C4×C7C2311yescharacteristic
71C23×C4×C7C2312yes—
72C23×C4×C7C2314yes—
73C22×C22⋊D7C2314yes—
74C22×D28C2312yes—
75C22×D28C2314yes—
76C22×C4×D7C2314yes—
77C22×C4×D7C2314yes—
78C22×C4×D7C2314yes—
79C22×Dic14C2312yes—
80C22×Dic14C2314yes—
81C7×#125C2318yes—
82C7×#125C23132yes—
83C7×#125C23116yes—
84C7×#125C2318yes—
85C2×C7×C4⋊C4C2314yes—
86C2×C7×C4⋊C4C2314yes—
87C2×C7×C4⋊C4C2312yes—
88C2×C7×C4⋊C4C2314yes—
89C2×C7×C4⋊C4C2314yes—
90C2×C7×C4⋊C4C2314yes—
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C4⋊C4C23132yes—
93C2×C7×C4⋊C4C23116yes—
94C2×C42×C7C2312yes—
95C2×C42×C7C2311yescharacteristic
96C2×C42×C7C2311yescharacteristic
97C2×C42×C7C2318yes—
98C2×C22⋊Dic7C2314yes—
99#6493C23116yes—
100#6493C23132yes—
101#6492C23132yes—
102C4×C22⋊D7C23132yes—
103C4×C22⋊D7C23132yes—
104C4×C22⋊D7C23132yes—
105C2×(C2×D7)⋊C4C2314yes—
106C2×(C2×D7)⋊C4C2314yes—
107C2×(C2×D7)⋊C4C2314yes—
108C2×(C2×D7)⋊C4C2314yes—
109C2×(C2×D7)⋊C4C2314yes—
110C2×(C2×D7)⋊C4C2314yes—
111C2×(C2×D7)⋊C4C2314yes—
112C2×(C2×D7)⋊C4C2314yes—
113C2×C4⋊Dic7C2312yes—
114C2×C4⋊Dic7C2312yes—
115C2×C4⋊Dic7C2314yes—
116C2×C4⋊Dic7C2314yes—
117#6487C23116yes—
118#6487C23132yes—
119C2×Dic7⋊C4C2314yes—
120C2×Dic7⋊C4C2314yes—
121C2×Dic7⋊C4C2314yes—
122C2×Dic7⋊C4C2314yes—
123C2×Dic7⋊C4C2314yes—
124C2×Dic7⋊C4C2314yes—
125C2×Dic7⋊C4C2314yes—
126C2×Dic7⋊C4C2314yes—
127C2×C4×Dic7C2314yes—
128C2×C4×Dic7C2314yes—
129C2×C4×Dic7C2314yes—
130#6461C23132yes—
131#6461C23132yes—
132#6461C23132yes—
133#6461C23132yes—
134#6460C23132yes—
135#6459C23132yes—
136#6459C23132yes—
137#6459C23132yes—
138#6459C23132yes—
139#6458C23132yes—
140#6457C23132yes—
141#6457C23132yes—
142#6457C23132yes—
143#6457C23132yes—
144#6456C23132yes—
145#6455C23132yes—
146D7×C4⋊C4C23132yes—
147#6453C23132yes—
148#6452C23132yes—
149#6452C23132yes—
150#6452C23132yes—
151#6452C23132yes—
152#6451C23132yes—
153#6450C23132yes—
154#6439C23132yes—
155#6438C23116yes—
156C4×D28C23116yes—
157#6435C23132yes—
158#6433C23116yes—
159C4×Dic14C23116yes—
160#3553—732no—
161C2×#564—71no—
162C2×#563—72no—
163C2×#562—71no—
164C2×#554—74no—
165C2×#553—74no—
166C2×#552—74no—
167C2×#541—74no—
168C2×#539—74no—
169C2×#539—74no—
170C2×#529—72no—
171C22×#125D711yescharacteristic, complemented
172C24×C7C2411yescharacteristic
173C23×D7C2414yes—
174C22×C4×C7—24no—
175C22×C4×C7—48no—
176C22×C4×C7—44no—
177C22×C4×C7C2412yes—
178C22×C4×C7C2414yes—
179C22×C4×C7C2414yes—
180C22×C4×C7C2411yescharacteristic
181C22×C4×C7C2411yescharacteristic
182C22×C4×C7C2412yes—
183C22×C4×C7C2414yes—
184C22×C4×C7C2418yes—
185C22×C4×C7C24116yes—
186C2×C22⋊D7—216no—
187C2×C22⋊D7C24116yes—
188C22×Dic7C2414yes—
189C22×Dic7C2414yes—
190C22×Dic7C2414yes—
191C2×D28—44no—
192C2×D28—48no—
193C2×D28C2418yes—
194C2×D28C24116yes—
195C2×C4×D7—48no—
196C2×C4×D7—48no—
197C2×C4×D7—48no—
198C2×C4×D7C24116yes—
199C2×C4×D7C24116yes—
200C2×C4×D7C24116yes—
201C2×Dic14—44no—
202C2×Dic14—48no—
203C2×Dic14C2418yes—
204C2×Dic14C24116yes—
205C7×C4⋊C4C24116yes—
206C7×C4⋊C4C24116yes—
207C7×C4⋊C4C2418yes—
208C7×C4⋊C4C24116yes—
209C7×C4⋊C4C24116yes—
210C7×C4⋊C4C24116yes—
211C7×C4⋊C4C2418yes—
212C42×C7C2418yes—
213C42×C7C2414yes—
214C42×C7C2414yes—
215C22⋊Dic7C24116yes—
216(C2×D7)⋊C4C24116yes—
217(C2×D7)⋊C4C24116yes—
218(C2×D7)⋊C4C24116yes—
219(C2×D7)⋊C4C24116yes—
220(C2×D7)⋊C4C24116yes—
221(C2×D7)⋊C4C24116yes—
222(C2×D7)⋊C4C24116yes—
223(C2×D7)⋊C4C24116yes—
224C4⋊Dic7C2418yes—
225C4⋊Dic7C2418yes—
226C4⋊Dic7C24116yes—
227C4⋊Dic7C24116yes—
228Dic7⋊C4C24116yes—
229Dic7⋊C4C24116yes—
230Dic7⋊C4C24116yes—
231Dic7⋊C4C24116yes—
232Dic7⋊C4C24116yes—
233Dic7⋊C4C24116yes—
234Dic7⋊C4C24116yes—
235Dic7⋊C4C24116yes—
236C4×Dic7C24116yes—
237C4×Dic7C24116yes—
238C4×Dic7C24116yes—
239#564—716no—
240#563—732no—
241#562—716no—
242#554—764no—
243#553—764no—
244#552—764no—
245#541—764no—
246#539—764no—
247#539—764no—
248C2×C4⋊Q8—74no—
249#529—732no—
250C2×#126—74no—
251C2×#126—74no—
252C2×#126—74no—
253C2×#126—74no—
254C2×#126—74no—
255C2×#125—74no—
256C2×#125—74no—
257C2×#125—74no—
258C2×#125—74no—
259C2×#125—74no—
260C2×#125—72no—
261C2×#125C2×D714yes—
262C2×#125C2×D712yes—
263C2×#125C2×D711yescharacteristic
264C2×#125C2×D711yescharacteristic
265C2×#125C2×D7116yescomplemented
266C2×#124—72no—
267C2×#123—74no—
268C2×#123—74no—
269C2×#123—74no—
270C2×#123—74no—
271C2×#123—74no—
272C2×C22⋊Q8—72no—
273C2×C22⋊Q8—74no—
274C2×C22⋊Q8—74no—
275C2×C22⋊Q8—74no—
276C2×C22⋊Q8—74no—
277C2×C22⋊Q8—74no—
278C2×C22⋊Q8—74no—
279C2×#121—72no—
280C2×#121—74no—
281C2×#121—74no—
282C2×C4×Q8—74no—
283C2×C4×Q8—72no—
284C2×C4×D4—74no—
285C2×C4×D4—74no—
286C2×C4×D4—74no—
287C2×C4×D4—74no—
288C2×C4×D4—72no—
289C2×#117—74no—
290C2×#117—74no—
291C22×C4⋊C4—74no—
292C22×C4⋊C4C2×D712yes—
293C22×C4⋊C4C2×D714yes—
294C22×C42C2×D711yescharacteristic
295C23×C7—24no—
296C23×C72−1+414yes—
297C23×C7C2512yes—
298C23×C7C2511yescharacteristic
299C22×D7—28no—
300C22×D7—28no—
301C22×D7—28no—
302C22×D7C2518yes—
303C2×C4×C7—28no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—28no—
307C2×C4×C7—24no—
308C2×C4×C7—24no—
309C2×C4×C7—28no—
310C2×C4×C7—416no—
311C2×C4×C7—48no—
312C2×C4×C7C2514yes—
313C2×C4×C7C2518yes—
314C2×C4×C7C2518yes—
315C2×C4×C7C2512yes—
316C2×C4×C7C2512yes—
317C2×C4×C7C2514yes—
318C22⋊D7—232no—
319C2×Dic7—28no—
320C2×Dic7—28no—
321C2×Dic7—28no—
322C2×Dic7C2518yes—
323C2×Dic7C2518yes—
324C2×Dic7C2518yes—
325D28—48no—
326D28—416no—
327C4×D7—416no—
328C4×D7—416no—
329C4×D7—416no—
330Dic14—48no—
331Dic14—416no—
332C22×Q8—74no—
333C22×Q8—72no—
334C22×D4—74no—
335C22×D4—74no—
336C22×D4—72no—
337C23×C4—74no—
338C23×C4—74no—
339C23×C4—74no—
340C23×C4C22×D711yescharacteristic
341C23×C4C22×D714yes—
342C23×C4C22×D712yes—
343C4⋊Q8—732no—
344#126—732no—
345#126—732no—
346#126—732no—
347#126—732no—
348#126—732no—
349#125—732no—
350#125—732no—
351#125—732no—
352#125—732no—
353#125—732no—
354#125—716no—
355#125C22×D7132yes—
356#125C22×D7116yes—
357#125C22×D718yes—
358#125C22×D718yes—
359#124—716no—
360#123—732no—
361#123—732no—
362#123—732no—
363#123—732no—
364#123—732no—
365C22⋊Q8—716no—
366C22⋊Q8—732no—
367C22⋊Q8—732no—
368C22⋊Q8—732no—
369C22⋊Q8—732no—
370C22⋊Q8—732no—
371C22⋊Q8—732no—
372#121—716no—
373#121—732no—
374#121—732no—
375C4×Q8—732no—
376C4×Q8—716no—
377C4×D4—732no—
378C4×D4—732no—
379C4×D4—732no—
380C4×D4—732no—
381C4×D4—716no—
382#117—732no—
383#117—732no—
384C2×C4⋊C4—74no—
385C2×C4⋊C4—74no—
386C2×C4⋊C4—74no—
387C2×C4⋊C4—74no—
388C2×C4⋊C4—74no—
389C2×C4⋊C4—72no—
390C2×C4⋊C4—74no—
391C2×C4⋊C4—74no—
392C2×C4⋊C4—74no—
393C2×C4⋊C4—72no—
394C2×C4⋊C4—74no—
395C2×C4⋊C4—74no—
396C2×C4⋊C4—732no—
397C2×C4⋊C4C22×D714yes—
398C2×C4⋊C4C22×D714yes—
399C2×C4⋊C4C22×D714yes—
400C2×C4⋊C4C22×D712yes—
401C2×C4⋊C4C22×D714yes—
402C2×C4⋊C4C22×D714yes—
403C2×C4⋊C4C22×D712yes—
404C2×C4⋊C4C22×D7116yes—
405C2×C4⋊C4C22×D7132yes—
406C2×C22⋊C4—74no—
407C2×C22⋊C4—74no—
408C2×C22⋊C4—74no—
409C2×C22⋊C4—74no—
410C2×C22⋊C4—74no—
411C2×C22⋊C4—74no—
412C2×C22⋊C4—74no—
413C2×C22⋊C4—74no—
414C2×C22⋊C4—74no—
415C2×C42—74no—
416C2×C42—74no—
417C2×C42—74no—
418C2×C42C22×D711yescharacteristic
419C2×C42C22×D712yes—
420C2×C42C22×D711yescharacteristic
421C2×C42C22×D718yes—
422C22×C7—24no—
423C22×C7—28no—
424C22×C7C2×2−1+414yes—
425C22×C7C2×2−1+412yes—
426C22×C7#58514yes—
427C22×C7C2611yescharacteristic
428C2×D7—28no—
429C2×D7—28no—
430C2×D7—28no—
431C2×D7—416no—
432C4×C7—22no—
433C4×C7—22no—
434C4×C7—28no—
435C4×C7—24no—
436C4×C7—24no—
437C4×C7—28no—
438Dic7—28no—
439Dic7—28no—
440Dic7—28no—
441C24—74no—
442C24C23×D711yescharacteristic
443C2×Q8—716no—
444C2×Q8—78no—
445C2×Q8—288no—
446C2×Q8—284no—
447C2×D4—716no—
448C2×D4—716no—
449C2×D4—78no—
450C2×D4—1416no—
451C2×D4—288no—
452C2×D4—284no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—74no—
457C22×C4—74no—
458C22×C4—74no—
459C22×C4—716no—
460C22×C4—716no—
461C22×C4—716no—
462C22×C4—288no—
463C22×C4—288no—
464C22×C4—288no—
465C22×C4C23×D714yes—
466C22×C4C23×D712yes—
467C22×C4C23×D714yes—
468C22×C4C23×D714yes—
469C22×C4C23×D711yescharacteristic
470C22×C4C23×D711yescharacteristic
471C22×C4C23×D712yes—
472C22×C4C23×D718yes—
473C22×C4C23×D7116yes—
474C4⋊C4—716no—
475C4⋊C4—716no—
476C4⋊C4—716no—
477C4⋊C4—716no—
478C4⋊C4—716no—
479C4⋊C4—716no—
480C4⋊C4—716no—
481C4⋊C4—716no—
482C4⋊C4—716no—
483C4⋊C4—716no—
484C4⋊C4—78no—
485C4⋊C4—78no—
486C4⋊C4C23×D7116yes—
487C4⋊C4C23×D7116yes—
488C4⋊C4C23×D7116yes—
489C4⋊C4C23×D718yes—
490C4⋊C4C23×D7116yes—
491C4⋊C4C23×D7116yes—
492C4⋊C4C23×D718yes—
493C22⋊C4—716no—
494C22⋊C4—716no—
495C22⋊C4—716no—
496C22⋊C4—716no—
497C22⋊C4—716no—
498C22⋊C4—716no—
499C22⋊C4—716no—
500C22⋊C4—716no—
501C22⋊C4—716no—
502C42—716no—
503C42—716no—
504C42—716no—
505C42C23×D714yes—
506C42C23×D718yes—
507C42C23×D714yes—
508C2×C7—24no—
509C2×C7#355314yescomplemented
510C2×C7C22×2−1+411yescharacteristic
511C2×C7C2×#58512yes—
512D7—48no—
513C23—24no—
514C23—78no—
515C23—148no—
516C23—148no—
517C23—148no—
518C23#655114yes—
519C23C24×D712yes—
520C23C24×D711yescharacteristic
521Q8—2816no—
522Q8—288no—
523D4—1432no—
524D4—2816no—
525D4—288no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—48no—
534C2×C4—416no—
535C2×C4—78no—
536C2×C4—78no—
537C2×C4—78no—
538C2×C4—148no—
539C2×C4—148no—
540C2×C4—148no—
541C2×C4—2816no—
542C2×C4—2816no—
543C2×C4—2816no—
544C2×C4C24×D714yes—
545C2×C4C24×D718yes—
546C2×C4C24×D718yes—
547C2×C4C24×D712yes—
548C2×C4C24×D712yes—
549C2×C4C24×D714yes—
550C7C2×#355311yescharacteristic, complemented, Sylow
551C22—24no—
552C22—28no—
553C22—148no—
554C22—148no—
555C22—148no—
556C22—2816no—
557C22C2×#655114yes—
558C22C2×#655112yes—
559C22#9061614yes—
560C22C25×D711yescharacteristic
561C4—22no—
562C4—22no—
563C4—24no—
564C4—28no—
565C4—28no—
566C4—24no—
567C4—148no—
568C4—148no—
569C4—148no—
570C2—24no—
571C2—288no—
572C2#1174317014yesdirect product factor
573C2C22×#655111yescharacteristic
574C2C2×#9061612yes—
subgroupG/H
centerC23C24×D7
commutator, cosocleC22×C7C26
FrattiniC22C25×D7
FittingC22×C7×#125C2
socleC23×C7C25
hypercenterC22×#125D7
hypocenterC7C2×#3553
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
1. C22×C7×#125C2semidirect
2. #11743170C2direct (N × H)
3. C2×#90390C2semidirect
4. C2×#90391C2semidirect
5. C2×#90384C2semidirect
6. C2×#90383C2semidirect
7. C2×C7⋊#564C2semidirect
8. C2×#90387C2semidirect
9. C2×#90205C2semidirect
10. C2×#90206C2semidirect
11. C2×#90202C2semidirect
12. C2×#90170C2semidirect
17. C2×C7×#125C22semidirect
29. #90391C22semidirect
30. #90390C22semidirect
31. #90387C22semidirect
32. #90384C22semidirect
33. #90383C22semidirect
34. C7⋊#564C22semidirect
35. #90206C22semidirect
36. #90205C22semidirect
41. #90202C22semidirect
62. #90170C22semidirect
171. C22×#125D7semidirect
265. C2×#125C2×D7semidirect
509. C2×C7#3553semidirect
550. C7C2×#3553semidirect
572. C2#11743170direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1252
#117431702
C2×#903902
C2×#903912
C2×#903842
C2×#903832
C2×C7⋊#5642
C2×#903872
C2×#902052
C2×#902062
C2×#902022
C2×#901702
C2×#35537