Finite Group Databasebeta

← D7×#1894 #13355474

Subgroups — D7×#1894 #13355474

Order 1792.

1966 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D7×M16C211yesmaximal, characteristic
2C2×D7×C22⋊Q8C211yesmaximal, characteristic
3C2×D7×#103C212yesmaximal, complemented
4D7×#380C211yesmaximal, characteristic
5C7×#1894C211yesmaximal, characteristic, complemented
6#11729743C211yesmaximal, characteristic, complemented
7#11729451C211yesmaximal, characteristic, complemented
8#11728025C212yesmaximal, complemented
9#11727270C211yesmaximal, characteristic, complemented
10#11726638C212yesmaximal, complemented
11D7×#340C212yesmaximal, complemented
12C22×Q8×D7C2211yescharacteristic
13C23×C4×D7C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C2×C7×C22⋊Q8C2211yescharacteristic
16C2×#6509C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×D7×M16—24no—
20C2×D7×M16—28no—
21C22×#1131C2212yes—
22C22×C8×D7C2212yes—
23D7×C22⋊Q8—24no—
24D7×C22⋊Q8C418yescomplemented
25C2×#6460C2211yescharacteristic
26C2×#6459—21no—
27C2×D7×C4⋊C4—21no—
28C2×D7×C4⋊C4C2212yes—
29C2×D7×C4⋊C4C2211yescharacteristic
30C2×D7×C22⋊C4—21no—
31C2×D7×C22⋊C4—21no—
32C2×#6441—21no—
33C2×C7×#103C2212yescomplemented
34C7×#380C2211yescharacteristic
35C2×#6409C2212yescomplemented
36#89984C2211yescharacteristic
37C2×#6390C2212yescomplemented
38#89759C2212yes—
39C2×#6384C2212yescomplemented
40#89714—21no—
41D7×#103—28no—
42#89439—21no—
43D7×#95—21no—
44C7×#340C2212yescomplemented
45#89346C2212yescomplemented
46#89324C2212yescomplemented
47#89324C2212yescomplemented
48C2×#1894—71nomaximal, Sylow
49C24×D7D411yescharacteristic
50C22×C7×Q8C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C23×Dic7D411yescharacteristic
53C2×Q8×D7—22no—
54C2×Q8×D7—24no—
55C2×Q8×D7—24no—
56C2×Q8×D7C2×C414yes—
57C22×C4×D7—24no—
58C22×C4×D7—24no—
59C22×C4×D7—21no—
60C22×C4×D7—21no—
61C22×C4×D7—22no—
62C22×C4×D7C2×C414yes—
63C22×C4×D7D411yescharacteristic
64C22×C4×D7D411yescharacteristic
65C22×C4×D7D414yes—
66C22×C4×D7C2311yescharacteristic
67C22×C4×D7C2311yescharacteristic
68C22×Dic14—21no—
69C22×Dic14C2311yescharacteristic
70C2×C7×M16—22no—
71C2×C7×M16—24no—
72C22×C7×C8C2312yes—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8C2×C414yescomplemented
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82#6509—22no—
83#6509C2×C414yescomplemented
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—22no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7C2312yes—
89C2×C4⋊Dic7C2311yescharacteristic
90#6487—22no—
91#6487C2×C414yescomplemented
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—22no—
96C2×#1136—22no—
97C2×#1136—24no—
98C22×C7⋊C8C2312yes—
99D7×M16—232no—
100C2×#1131—24no—
101C2×#1131—24no—
102C2×#1131—24no—
103C2×C8×D7—24no—
104C2×C8×D7—24no—
105C2×C8×D7D418yes—
106#6460—22no—
107#6460C2×C414yescomplemented
108#6459—28no—
109D7×C4⋊C4—28no—
110D7×C4⋊C4—22no—
111D7×C4⋊C4—24no—
112D7×C4⋊C4—24no—
113D7×C4⋊C4C2×C414yes—
114D7×C22⋊C4—28no—
115D7×C22⋊C4—44no—
116#6441—28no—
117C7×#103—24no—
118C7×#95—21no—
119#6409—24no—
120C7⋊#95—21no—
121C7⋊#95—22no—
122#6390—24no—
123#6384—24no—
124C23×M16—71no—
125C22×C22⋊Q8—71no—
126C22×#103—72no—
127C2×#380—71no—
128#1894—72no—
129#1894—71no—
130#1894—72no—
131#1894—71no—
132#1894—71no—
133#1894D711yescharacteristic, direct product factor
134C2×#340—72no—
135C24×C7C2×D411yescharacteristic
136C23×D7—24no—
137C23×D7—22no—
138C23×D7—22no—
139C23×D7—24no—
140C23×D7C22⋊C414yes—
141C23×D7C2×D411yescharacteristic
142C23×D7C2×D411yescharacteristic
143C2×C7×Q8—21no—
144C2×C7×Q8—22no—
145C2×C7×Q8—22no—
146C2×C7×Q8C22×C412yes—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—22no—
152C22×C4×C7C22×C412yes—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—24no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—22no—
160C22×Dic7C22⋊C414yes—
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163Q8×D7—216no—
164Q8×D7—44no—
165Q8×D7—44no—
166C2×C4×D7—24no—
167C2×C4×D7—22no—
168C2×C4×D7—22no—
169C2×C4×D7—24no—
170C2×C4×D7—216no—
171C2×C4×D7—28no—
172C2×C4×D7—28no—
173C2×C4×D7—24no—
174C2×C4×D7—24no—
175C2×C4×D7—44no—
176C2×C4×D7—42no—
177C2×C4×D7—42no—
178C2×C4×D7—44no—
179C2×C4×D7—44no—
180C2×C4×D7—44no—
181C2×C4×D7C22⋊C414yes—
182C2×C4×D7C22⋊C414yes—
183C2×C4×D7C22×C412yes—
184C2×C4×D7C22×C412yes—
185C2×C4×D7C2×D412yes—
186C2×C4×D7C2×D414yes—
187C2×C4×D7C2×D412yes—
188C2×C4×D7#4114yes—
189C2×Dic14—22no—
190C2×Dic14—22no—
191C2×Dic14—21no—
192C2×Dic14—24no—
193C2×Dic14—44no—
194C2×Dic14C22×C412yes—
195C7×M16—28no—
196C2×C7×C8—22no—
197C2×C7×C8—22no—
198C2×C7×C8C2×D414yes—
199C7×C4⋊C4—24no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4C22×C412yes—
204C7×C22⋊C4—24no—
205C7×C22⋊C4—42no—
206C22⋊Dic7—24no—
207C22⋊Dic7—42no—
208(C2×D7)⋊C4—24no—
209(C2×D7)⋊C4—24no—
210(C2×D7)⋊C4—44no—
211C4⋊Dic7—21no—
212C4⋊Dic7—24no—
213C4⋊Dic7—22no—
214C4⋊Dic7—22no—
215C4⋊Dic7C22×C412yes—
216Dic7⋊C4—24no—
217Dic7⋊C4—24no—
218Dic7⋊C4—24no—
219Dic7⋊C4—44no—
220#1136—28no—
221C2×C7⋊C8—22no—
222C2×C7⋊C8—22no—
223C2×C7⋊C8C2×D414yes—
224#1131—216no—
225C8×D7—216no—
226C23×Q8—71no—
227C24×C4—71no—
228C22×M16—71no—
229C22×M16—72no—
230C22×M16—148no—
231C22×M16—144no—
232C22×M16C2×D711yescharacteristic
233C23×C8—72no—
234C2×C22⋊Q8—71no—
235C2×C22⋊Q8—71no—
236C2×C22⋊Q8—78no—
237C2×C22⋊Q8—71no—
238C2×C22⋊Q8—141no—
239C2×C22⋊Q8—141no—
240C2×C22⋊Q8—144no—
241C2×C22⋊Q8C2×D711yescharacteristic
242C22×C4⋊C4—71no—
243C22×C4⋊C4—72no—
244C22×C4⋊C4—141no—
245C22×C22⋊C4—141no—
246C22×C22⋊C4—141no—
247C2×#103—72no—
248C2×#103—72no—
249C2×#103—72no—
250C2×#103—148no—
251C2×#103C2×D712yescomplemented
252#380—71no—
253#380—72no—
254#380—141no—
255#380—141no—
256#380C2×D711yescharacteristic
257C2×#95—141no—
258#340—72no—
259#340—72no—
260#340—72no—
261#340C2×D712yescomplemented
262C23×C7—22no—
263C23×C7—21no—
264C23×C7—22no—
265C23×C7—21no—
266C23×C7C2×C22⋊C412yes—
267C23×C7C22×D411yescharacteristic
268C22×D7—22no—
269C22×D7—21no—
270C22×D7—22no—
271C22×D7—21no—
272C22×D7—28no—
273C22×D7—28no—
274C22×D7—28no—
275C22×D7—48no—
276C22×D7—48no—
277C22×D7C2×C22⋊C412yes—
278C22×D7C2×C22⋊C412yes—
279C22×D7#12114yes—
280C22×D7AGL(1,ℤ/8)12yes—
281C22×D7#13712yes—
282C22×D7#13714yes—
283C7×Q8—24no—
284C7×Q8—41no—
285C7×Q8—41no—
286C2×C4×C7—22no—
287C2×C4×C7—24no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—42no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7C2×C22⋊C412yes—
299C2×C4×C7C23×C411yescharacteristic
300C2×C4×C7C22×D412yes—
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C2×#4112yes—
303C2×Dic7—22no—
304C2×Dic7—28no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—42no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312C2×Dic7—42no—
313C2×Dic7C2×C22⋊C412yes—
314C2×Dic7C2×C22⋊C412yes—
315C2×Dic7#12114yes—
316C4×D7—28no—
317C4×D7—24no—
318C4×D7—22no—
319C4×D7—22no—
320C4×D7—28no—
321C4×D7—48no—
322C4×D7—48no—
323C4×D7—88no—
324C4×D7C4×D418yes—
325C4×D7#12014yes—
326C4×D7#12314yes—
327Dic14—24no—
328Dic14—41no—
329Dic14—41no—
330Dic14—48no—
331C7×C8—24no—
332C7⋊C8—24no—
333C25—71no—
334C22×Q8—74no—
335C22×Q8—71no—
336C22×Q8—141no—
337C22×Q8—142no—
338C22×Q8—144no—
339C22×Q8—144no—
340C22×Q8C22×D711yescharacteristic
341C23×C4—71no—
342C23×C4—71no—
343C23×C4—74no—
344C23×C4—71no—
345C23×C4—74no—
346C23×C4—71no—
347C23×C4—71no—
348C23×C4—144no—
349C23×C4—144no—
350C23×C4—141no—
351C23×C4—141no—
352C23×C4—142no—
353C23×C4C22×D711yescharacteristic
354C2×M16—22no—
355C2×M16—24no—
356C2×M16—144no—
357C2×M16—144no—
358C2×M16—144no—
359C2×M16—1432no—
360C2×M16—142no—
361C2×M16—144no—
362C22×C8—72no—
363C22×C8—78no—
364C22×C8—144no—
365C22×C8—144no—
366C22×C8C22×D712yes—
367C22⋊Q8—22no—
368C22⋊Q8—74no—
369C22⋊Q8—74no—
370C22⋊Q8—74no—
371C22⋊Q8—148no—
372C22⋊Q8—148no—
373C22⋊Q8—142no—
374C22⋊Q8—142no—
375C22⋊Q8—142no—
376C22⋊Q8C4×D714yescomplemented
377C2×C4⋊C4—21no—
378C2×C4⋊C4—71no—
379C2×C4⋊C4—74no—
380C2×C4⋊C4—72no—
381C2×C4⋊C4—142no—
382C2×C4⋊C4—148no—
383C2×C4⋊C4—144no—
384C2×C4⋊C4—144no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—142no—
390C2×C4⋊C4C22×D712yes—
391C2×C4⋊C4C22×D711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—148no—
395C2×C22⋊C4—141no—
396C2×C22⋊C4—141no—
397C2×C22⋊C4—141no—
398C2×C22⋊C4—142no—
399C2×C22⋊C4—141no—
400C2×C22⋊C4—284no—
401#103—24no—
402#103—144no—
403#103—144no—
404#103—144no—
405#95—21no—
406#95—142no—
407#95—141no—
408C22×C7—22no—
409C22×C7—22no—
410C22×C7—22no—
411C22×C7—42no—
412C22×C7—42no—
413C22×C7C22×C22⋊C411yescharacteristic
414C22×C7C2×#12112yes—
415C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
416C22×C7C2×#13712yes—
417C22×C7C2×#13711yescharacteristic
418C2×D7—22no—
419C2×D7—22no—
420C2×D7—22no—
421C2×D7—42no—
422C2×D7—42no—
423C2×D7—416no—
424C2×D7#38614yes—
425C2×D7#41714yes—
426C2×D7#41914yes—
427C2×D7#46818yes—
428C2×D7#47018yes—
429C4×C7—22no—
430C4×C7—42no—
431C4×C7—42no—
432C4×C7—82no—
433C4×C7C2×C4×D412yes—
434C4×C7C2×#12011yescharacteristic
435C4×C7C2×#12311yescharacteristic
436Dic7—22no—
437Dic7—42no—
438Dic7—42no—
439Dic7—82no—
440Dic7#38614yes—
441C24—71no—
442C24—74no—
443C24—71no—
444C24—142no—
445C24—144no—
446C24—142no—
447C24—144no—
448C24D4×D711yescharacteristic
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—72no—
453C2×Q8—144no—
454C2×Q8—1416no—
455C2×Q8—141no—
456C2×Q8—142no—
457C2×Q8—142no—
458C2×Q8—284no—
459C2×Q8—284no—
460C2×Q8—284no—
461C2×Q8C2×C4×D712yes—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—71no—
468C22×C4—72no—
469C22×C4—74no—
470C22×C4—72no—
471C22×C4—74no—
472C22×C4—74no—
473C22×C4—71no—
474C22×C4—74no—
475C22×C4—74no—
476C22×C4—72no—
477C22×C4—72no—
478C22×C4—148no—
479C22×C4—144no—
480C22×C4—1416no—
481C22×C4—148no—
482C22×C4—144no—
483C22×C4—144no—
484C22×C4—144no—
485C22×C4—144no—
486C22×C4—142no—
487C22×C4—142no—
488C22×C4—141no—
489C22×C4—141no—
490C22×C4—142no—
491C22×C4—282no—
492C22×C4—282no—
493C22×C4—284no—
494C22×C4—284no—
495C22×C4—284no—
496C22×C4—284no—
497C22×C4C2×C4×D712yes—
498C22×C4D4×D711yescharacteristic
499C22×C4D4×D712yes—
500C22×C4C23×D711yescharacteristic
501M16—28no—
502M16—1416no—
503M16—148no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—74no—
507C2×C8—1416no—
508C2×C8—142no—
509C2×C8—142no—
510C2×C8D4×D714yes—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—72no—
516C4⋊C4—144no—
517C4⋊C4—144no—
518C4⋊C4—144no—
519C4⋊C4—144no—
520C4⋊C4—141no—
521C4⋊C4—142no—
522C4⋊C4—142no—
523C4⋊C4—284no—
524C4⋊C4C2×C4×D712yes—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—144no—
528C22⋊C4—144no—
529C22⋊C4—144no—
530C22⋊C4—284no—
531C22⋊C4—282no—
532C2×C7—42no—
533C2×C7C2×#38611yescharacteristic
534C2×C7C2×#41911yescharacteristic
535C2×C7C2×#41711yescharacteristic
536C2×C7C2×#46812yes—
537C2×C7C2×#47012yes—
538D7—42no—
539D7#189418yesdirect product factor
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—72no—
545C23—72no—
546C23—72no—
547C23—74no—
548C23—74no—
549C23—72no—
550C23—148no—
551C23—148no—
552C23—148no—
553C23—141no—
554C23—141no—
555C23—142no—
556C23—142no—
557C23—288no—
558C23—288no—
559C23D7×C22⋊C412yes—
560C23C2×D4×D711yescharacteristic
561Q8—24no—
562Q8—41no—
563Q8—41no—
564Q8—144no—
565Q8—288no—
566Q8—281no—
567Q8—281no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—72no—
581C2×C4—74no—
582C2×C4—78no—
583C2×C4—74no—
584C2×C4—74no—
585C2×C4—72no—
586C2×C4—148no—
587C2×C4—148no—
588C2×C4—148no—
589C2×C4—142no—
590C2×C4—144no—
591C2×C4—142no—
592C2×C4—142no—
593C2×C4—142no—
594C2×C4—142no—
595C2×C4—288no—
596C2×C4—288no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—282no—
600C2×C4—282no—
601C2×C4—282no—
602C2×C4—282no—
603C2×C4—568no—
604C2×C4D7×C22⋊C412yes—
605C2×C4C22×C4×D711yescharacteristic
606C2×C4C2×D4×D712yes—
607C2×C4C2×D4×D711yescharacteristic
608C2×C4D7×#4112yes—
609C8—24no—
610C8—144no—
611C7C2×#189411yescharacteristic, complemented, Sylow
612C22—22no—
613C22—22no—
614C22—22no—
615C22—42no—
616C22—42no—
617C22—74no—
618C22—74no—
619C22—74no—
620C22—78no—
621C22—78no—
622C22—142no—
623C22—142no—
624C22—142no—
625C22—2816no—
626C22—282no—
627C22—282no—
628C22C2×D7×C22⋊C411yescharacteristic
629C22D7×#12112yes—
630C22D7×AGL(1,ℤ/8)11yescharacteristic
631C22D7×#13711yescharacteristic
632C22D7×#13712yes—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—74no—
637C4—82no—
638C4—142no—
639C4—282no—
640C4—282no—
641C4—562no—
642C4C4×D4×D712yes—
643C4D7×#12011yescharacteristic
644C4D7×#12311yescharacteristic
645C2—42no—
646C2—78no—
647C2—282no—
648C2D7×#38611yescharacteristic
649C2D7×#41711yescharacteristic
650C2D7×#41911yescharacteristic
651C2D7×#46812yes—
652C2D7×#47012yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1894C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenter#1894D7
hypocenterC7C2×#1894
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
3. C2×D7×#103C2semidirect
5. C7×#1894C2semidirect
6. #11729743C2semidirect
7. #11729451C2semidirect
8. #11728025C2semidirect
9. #11727270C2semidirect
10. #11726638C2semidirect
11. D7×#340C2semidirect
24. D7×C22⋊Q8C4semidirect
33. C2×C7×#103C22semidirect
35. C2×#6409C22semidirect
37. C2×#6390C22semidirect
39. C2×#6384C22semidirect
44. C7×#340C22semidirect
45. #89346C22semidirect
46. #89324C22semidirect
47. #89324C22semidirect
74. C7×C22⋊Q8C2×C4semidirect
83. #6509C2×C4semidirect
91. #6487C2×C4semidirect
107. #6460C2×C4semidirect
133. #1894D7direct (N × H)
251. C2×#103C2×D7semidirect
261. #340C2×D7semidirect
376. C22⋊Q8C4×D7semidirect
539. D7#1894direct (N × H)
611. C7C2×#1894semidirect

Maximal subgroups

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

subgroupindex
C22×D7×M162
C2×D7×C22⋊Q82
C2×D7×#1032
D7×#3802
C7×#18942
#117297432
#117294512
#117280252
#117272702
#117266382
D7×#3402
C2×#18947