Finite Group Databasebeta

← C2×#11726032 #13320939

Subgroups — C2×#11726032 #13320939

Order 1792.

1410 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90287C211yesmaximal, characteristic
2C2×#89491C211yesmaximal, characteristic, complemented
3C2×#89420C211yesmaximal, characteristic, complemented
4C2×C7×#327C211yesmaximal, characteristic, complemented
5C2×#89332C211yesmaximal, characteristic, complemented
6C2×#89265C211yesmaximal, characteristic, complemented
7C2×#89246C211yesmaximal, characteristic, complemented
8#11726032C218yesmaximal, direct product factor
9C22×#1159—21no—
10C2×C7×#121C2211yescharacteristic
11C2×#6503C411yescharacteristic, complemented
12C2×#6502—21no—
13C2×D4×Dic7C411yescharacteristic, complemented
14C2×#6493C2211yescharacteristic
15C22×C4×Dic7—21no—
16C22×C4×Dic7C2211yescharacteristic
17#90287—22no—
18#90287—22no—
19#90287C414yescomplemented
20#90287C2214yes—
21C2×#6456C411yescharacteristic, complemented
22C2×#6451C411yescharacteristic, complemented
23C2×#6448—21no—
24C2×(C22⋊D7)⋊C4—21no—
25C2×C7×C22⋊C8C2211yescharacteristic
26C2×C7×#95C2211yescharacteristic
27C2×C7⋊#95—21no—
28C2×C7⋊#95C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×#6389—21no—
31C2×Dic7⋊C8—21no—
32#89491—22no—
33#89491C2214yescomplemented
34#89420—22no—
35#89420C2214yescomplemented
36C7×#327C2214yescomplemented
37#89332C2214yescomplemented
38#89265C2214yescomplemented
39#89246C2214yescomplemented
40C2×#1505—71nomaximal, Sylow
41C22×C7×D4—21no—
42C22×C7×D4C2×C411yescharacteristic
43C23×C4×C7—21no—
44C23×C4×C7C2311yescharacteristic
45C22×C22⋊D7—21no—
46C22×C22⋊D7—21no—
47C22×C22⋊D7—21no—
48C23×Dic7—21no—
49C23×Dic7—21no—
50C23×Dic7D411yescharacteristic
51C23×Dic7D411yescharacteristic
52C2×#1159—24no—
53C2×#1159—24no—
54C2×#1159—44no—
55C2×#1159—44no—
56C22×D28C2×C411yescharacteristic
57C22×C4×D7—21no—
58C22×Dic14—21no—
59C22×C7×C8—21no—
60C7×#121—21no—
61C7×#121—21no—
62C7×#121C2×C412yes—
63C7×#121C2312yes—
64C2×C7×C4⋊C4C2×C411yescharacteristic
65C2×C7×C22⋊C4—21no—
66C2×C22⋊Dic7—21no—
67#6503—22no—
68#6503C2×C414yescomplemented
69#6502—24no—
70#6502—24no—
71D4×Dic7—22no—
72D4×Dic7C2×C414yescomplemented
73#6493—21no—
74#6493—21no—
75#6493C2×C412yes—
76#6493C2312yes—
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—21no—
79C2×C4⋊Dic7C2×C411yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×C4×Dic7—28no—
82C2×C4×Dic7—24no—
83C2×C4×Dic7C2×C411yescharacteristic
84C2×C4×Dic7C2×C411yescharacteristic
85C2×C4×Dic7C2×C412yes—
86C2×C4×Dic7D412yes—
87C2×C4×Dic7D411yescharacteristic
88C2×C4×Dic7D411yescharacteristic
89C2×C4×Dic7D412yes—
90C2×C4×Dic7C2312yes—
91C22×C7⋊C8—21no—
92#6456—22no—
93#6456C2×C414yescomplemented
94#6451—22no—
95#6451C2×C414yescomplemented
96#6448—24no—
97#6448—24no—
98(C22⋊D7)⋊C4—24no—
99(C22⋊D7)⋊C4—24no—
100C7×C22⋊C8—21no—
101C7×C22⋊C8C2312yes—
102C7×#95—21no—
103C7×#95C2312yes—
104C7⋊#95—21no—
105C7⋊#95—24no—
106C7⋊#95C2312yes—
107C22⋊(C7⋊C8)—21no—
108C22⋊(C7⋊C8)C2312yes—
109#6389—24no—
110Dic7⋊C8—24no—
111C2×#532—71no—
112C2×#432—71no—
113C2×#376—71no—
114#1505—78no—
115C2×#327—71no—
116C2×#327—71no—
117C2×#327—71no—
118C2×#327D711yescharacteristic, complemented
119C24×C7—21no—
120C24×C7C2×D411yescharacteristic
121C23×D7—21no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—21no—
125C2×C7×D4—42no—
126C2×C7×D4—42no—
127C2×C7×D4—42no—
128C2×C7×D4C22×C412yes—
129C22×C4×C7—24no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7—42no—
135C22×C4×C7C22×C411yescharacteristic
136C22×C4×C7C22×C411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2411yescharacteristic
141C2×C22⋊D7—22no—
142C2×C22⋊D7—22no—
143C2×C22⋊D7—24no—
144C2×C22⋊D7—24no—
145C2×C22⋊D7—44no—
146C2×C22⋊D7—42no—
147C2×C22⋊D7—42no—
148C2×C22⋊D7—44no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—24no—
152C22×Dic7—24no—
153C22×Dic7—21no—
154C22×Dic7—22no—
155C22×Dic7—24no—
156C22×Dic7—44no—
157C22×Dic7—42no—
158C22×Dic7—42no—
159C22×Dic7C22⋊C411yescharacteristic
160C22×Dic7C22⋊C412yes—
161C22×Dic7C22⋊C411yescharacteristic
162C22×Dic7C22⋊C411yescharacteristic
163C22×Dic7C22⋊C412yes—
164C22×Dic7C22⋊C411yescharacteristic
165C22×Dic7C2×D412yes—
166C22×Dic7C2×D412yes—
167#1159—48no—
168#1159—48no—
169#1159—416no—
170C2×D28—21no—
171C2×D28—42no—
172C2×D28C22×C412yes—
173C2×C4×D7—24no—
174C2×C4×D7—44no—
175C2×Dic14—24no—
176C2×Dic14—44no—
177C2×C7×C8—22no—
178C2×C7×C8—42no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4C22×C412yes—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C22⋊Dic7—24no—
184(C2×D7)⋊C4—22no—
185(C2×D7)⋊C4—22no—
186(C2×D7)⋊C4—24no—
187C4⋊Dic7—21no—
188C4⋊Dic7C22×C412yes—
189Dic7⋊C4—24no—
190C4×Dic7—216no—
191C4×Dic7C22⋊C412yes—
192C4×Dic7C22⋊C412yes—
193C4×Dic7C22⋊C412yes—
194C4×Dic7C22⋊C412yes—
195C4×Dic7C22×C412yes—
196C4×Dic7C22×C412yes—
197C4×Dic7C2×D412yes—
198C4×Dic7C2×D412yes—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—42no—
201C22×#41—141no—
202#532—74no—
203#532—74no—
204#532—142no—
205#532—142no—
206C2×C4⋊Q8—71no—
207C2×#127—71no—
208C2×#124—141no—
209C2×#121—71no—
210C2×#121—141no—
211C2×#121C2×D711yescharacteristic
212C2×C4×D4—71no—
213C2×C4×D4—71no—
214C2×C4×D4—141no—
215C22×C42—71no—
216C22×C42—141no—
217#432—74no—
218#432—142no—
219C2×C4⋊C8—141no—
220C2×C22⋊C8—71no—
221C2×C22⋊C8C2×D711yescharacteristic
222C2×#97—141no—
223#376—74no—
224#376—142no—
225C2×#95—71no—
226C2×#95—141no—
227C2×#95C2×D711yescharacteristic
228#327—74no—
229#327—74no—
230#327—74no—
231#327C2×D714yescomplemented
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7C2×C22⋊C411yescharacteristic
239C23×C7C2×C22⋊C411yescharacteristic
240C23×C7AGL(1,ℤ/8)12yes—
241C23×C7AGL(1,ℤ/8)12yes—
242C23×C7C22×D411yescharacteristic
243C22×D7—22no—
244C22×D7—42no—
245C22×D7—42no—
246C22×D7—42no—
247C7×D4—42no—
248C7×D4—42no—
249C7×D4—42no—
250C7×D4—42no—
251C7×D4—44no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—24no—
255C2×C4×C7—22no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
261C2×C4×C7—48no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7C2×C22⋊C411yescharacteristic
265C2×C4×C7C2×C22⋊C411yescharacteristic
266C2×C4×C7C23×C411yescharacteristic
267C2×C4×C7C22×D411yescharacteristic
268C22⋊D7—48no—
269C22⋊D7—42no—
270C22⋊D7—42no—
271C22⋊D7—44no—
272C22⋊D7—48no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—24no—
278C2×Dic7—22no—
279C2×Dic7—24no—
280C2×Dic7—24no—
281C2×Dic7—24no—
282C2×Dic7—22no—
283C2×Dic7—48no—
284C2×Dic7—42no—
285C2×Dic7—48no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7#10414yes—
289C2×Dic7C2×C22⋊C412yes—
290C2×Dic7C2×C22⋊C412yes—
291C2×Dic7C2×C22⋊C412yes—
292C2×Dic7C2×C22⋊C412yes—
293D28—42no—
294D28—42no—
295C4×D7—48no—
296Dic14—48no—
297C7×C8—42no—
298C7⋊C8—42no—
299C2×#41—144no—
300C2×#41—144no—
301C2×#41—284no—
302C2×#41—284no—
303C22×Q8—141no—
304C22×D4—21no—
305C22×D4—71no—
306C22×D4—141no—
307C22×D4—141no—
308C22×D4—141no—
309C22×D4C4×D711yescharacteristic
310C23×C4—21no—
311C23×C4—71no—
312C23×C4—71no—
313C23×C4—141no—
314C23×C4—141no—
315C23×C4—141no—
316C23×C4C22×D711yescharacteristic
317C22×C8—21no—
318C22×C8—141no—
319C4⋊Q8—74no—
320C4⋊Q8—142no—
321#127—74no—
322#127—142no—
323#124—144no—
324#124—144no—
325#121—21no—
326#121—21no—
327#121—72no—
328#121—72no—
329#121—144no—
330#121—144no—
331#121—141no—
332#121—141no—
333#121C4×D712yes—
334#121C22×D712yes—
335C4×D4—74no—
336C4×D4—74no—
337C4×D4—144no—
338C4×D4—144no—
339C4×D4—142no—
340C4×D4—142no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4C4×D711yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—141no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C42—71no—
349C2×C42—71no—
350C2×C42—72no—
351C2×C42—71no—
352C2×C42—71no—
353C2×C42—72no—
354C2×C42—72no—
355C2×C42—72no—
356C2×C42—144no—
357C2×C42—148no—
358C4⋊C8—144no—
359C22⋊C8—21no—
360C22⋊C8—72no—
361C22⋊C8—141no—
362C22⋊C8C22×D712yes—
363#97—144no—
364#95—21no—
365#95—72no—
366#95—144no—
367#95—141no—
368#95C22×D712yes—
369C22×C7—21no—
370C22×C7—21no—
371C22×C7—21no—
372C22×C7—21no—
373C22×C7—24no—
374C22×C7—24no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—84no—
379C22×C7#41012yes—
380C22×C7#41812yes—
381C22×C7#41812yes—
382C22×C7C2×#10412yes—
383C22×C7C22×C22⋊C411yescharacteristic
384C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
386C2×D7—42no—
387C2×D7—42no—
388C2×D7—42no—
389C2×D7—84no—
390C4×C7—21no—
391C4×C7—21no—
392C4×C7—21no—
393C4×C7—21no—
394C4×C7—44no—
395C4×C7—42no—
396Dic7—22no—
397Dic7—22no—
398Dic7—22no—
399Dic7—44no—
400Dic7—42no—
401Dic7C2×#10414yes—
402C24—21no—
403C24—141no—
404C24D4×D711yescharacteristic
405#41—288no—
406#41—288no—
407#41—2816no—
408C2×Q8—144no—
409C2×Q8—284no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—72no—
417C2×D4—144no—
418C2×D4—144no—
419C2×D4—142no—
420C2×D4—142no—
421C2×D4—141no—
422C2×D4—282no—
423C2×D4—284no—
424C2×D4—284no—
425C2×D4—282no—
426C2×D4—282no—
427C2×D4C2×C4×D712yes—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—71no—
435C22×C4—72no—
436C22×C4—72no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—72no—
441C22×C4—72no—
442C22×C4—144no—
443C22×C4—144no—
444C22×C4—144no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—144no—
448C22×C4—141no—
449C22×C4—142no—
450C22×C4—284no—
451C22×C4—284no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4C2×C4×D711yescharacteristic
455C22×C4C2×C4×D711yescharacteristic
456C22×C4D4×D711yescharacteristic
457C22×C4D4×D711yescharacteristic
458C22×C4D4×D711yescharacteristic
459C22×C4C23×D711yescharacteristic
460C2×C8—22no—
461C2×C8—42no—
462C2×C8—142no—
463C2×C8—282no—
464C4⋊C4—21no—
465C4⋊C4—72no—
466C4⋊C4—144no—
467C4⋊C4—141no—
468C4⋊C4C2×C4×D712yes—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—144no—
472C22⋊C4—144no—
473C22⋊C4—142no—
474C22⋊C4—142no—
475C42—72no—
476C42—72no—
477C42—72no—
478C42—72no—
479C42—72no—
480C42—72no—
481C42—72no—
482C42—72no—
483C42—1416no—
484C2×C7—21no—
485C2×C7—21no—
486C2×C7—22no—
487C2×C7—82no—
488C2×C7#150514yescomplemented
489C2×C7C2×#41011yescharacteristic
490C2×C7C2×#41811yescharacteristic
491C2×C7C22×#10411yescharacteristic
492D7—82no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—142no—
500C23—282no—
501C23—282no—
502C23—282no—
503C23D7×C22⋊C411yescharacteristic
504C23D7×C22⋊C411yescharacteristic
505C23#647412yes—
506C23#647712yes—
507C23C2×D4×D711yescharacteristic
508Q8—288no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—44no—
514D4—282no—
515D4—282no—
516D4—288no—
517D4—288no—
518D4—282no—
519D4—282no—
520D4—284no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—72no—
534C2×C4—74no—
535C2×C4—72no—
536C2×C4—72no—
537C2×C4—72no—
538C2×C4—144no—
539C2×C4—144no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—144no—
545C2×C4—142no—
546C2×C4—144no—
547C2×C4—142no—
548C2×C4—288no—
549C2×C4—288no—
550C2×C4—282no—
551C2×C4—288no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4D7×C22⋊C411yescharacteristic
555C2×C4D7×C22⋊C411yescharacteristic
556C2×C4C22×C4×D711yescharacteristic
557C2×C4C2×D4×D711yescharacteristic
558C8—42no—
559C8—282no—
560C7C2×#150511yescharacteristic, complemented, Sylow
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—84no—
571C22—282no—
572C22—282no—
573C22—282no—
574C22—564no—
575C22#8951212yes—
576C22#8954212yes—
577C22#8954212yes—
578C22D7×#10412yes—
579C22C2×D7×C22⋊C411yescharacteristic
580C22C2×#647411yescharacteristic
581C22C2×#647711yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—44no—
587C4—42no—
588C4—74no—
589C4—142no—
590C4—142no—
591C4—142no—
592C4—284no—
593C4—282no—
594C2—21no—
595C2—21no—
596C2—22no—
597C2—82no—
598C2—562no—
599C2#1172603214yesdirect product factor
600C2C2×#8951211yescharacteristic
601C2C2×#8954211yescharacteristic
602C2C2×D7×#10411yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#327C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#90287C2
cosocleC22×C4×C7C24
hypercenterC2×#327D7
hypocenterC7C2×#1505
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#89491C2semidirect
3. C2×#89420C2semidirect
4. C2×C7×#327C2semidirect
5. C2×#89332C2semidirect
6. C2×#89265C2semidirect
7. C2×#89246C2semidirect
8. #11726032C2direct (N × H)
11. C2×#6503C4semidirect
13. C2×D4×Dic7C4semidirect
19. #90287C4semidirect
21. C2×#6456C4semidirect
22. C2×#6451C4semidirect
33. #89491C22semidirect
35. #89420C22semidirect
36. C7×#327C22semidirect
37. #89332C22semidirect
38. #89265C22semidirect
39. #89246C22semidirect
68. #6503C2×C4semidirect
72. D4×Dic7C2×C4semidirect
93. #6456C2×C4semidirect
95. #6451C2×C4semidirect
118. C2×#327D7semidirect
231. #327C2×D7semidirect
488. C2×C7#1505semidirect
560. C7C2×#1505semidirect
599. C2#11726032direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902872
C2×#894912
C2×#894202
C2×C7×#3272
C2×#893322
C2×#892652
C2×#892462
#117260322
C2×#15057