Finite Group Databasebeta

← Dic7×#540 #13512906

Subgroups — Dic7×#540 #13512906

Order 1792.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#540C211yesmaximal, characteristic
2#11740617C211yesmaximal, characteristic, complemented
3Dic7×#127C211yesmaximal, characteristic, complemented
4#11740468C211yesmaximal, characteristic, complemented
5Dic7×#125C211yesmaximal, characteristic, complemented
6#11740282C214yesmaximal, complemented
7Dic7×#123C214yesmaximal, complemented
8#11740080C214yesmaximal, complemented
9#11740078C212yesmaximal, complemented
10Dic7×#121C212yesmaximal, complemented
11Dic7×#121C214yesmaximal, complemented
12#11739770C212yesmaximal, complemented
13C4×D4×Dic7C212yesmaximal, complemented
14#11739676C211yesmaximal, characteristic, complemented
15Dic7×#117C211yesmaximal, characteristic, complemented
16C7×#540C4116yescomplemented
17C2×C7×#127C2211yescharacteristic
18C2×C7×#125C2211yescharacteristic
19C2×C7×#123C2214yes—
20C2×C7×#121C2212yes—
21C2×C7×#121C2214yes—
22C2×C4×C7×D4C2212yes—
23C2×C7×#117C2211yescharacteristic
24C2×D4×Dic7C2212yes—
25C2×D4×Dic7C2212yes—
26C2×D4×Dic7C2214yes—
27C2×D4×Dic7C2214yes—
28C2×#6489C2211yescharacteristic
29C22×C4×Dic7C2211yescharacteristic
30C22×C4×Dic7C2214yes—
31#89993C2212yes—
32#89993C2212yes—
33#89993C2214yes—
34#89993C2214yes—
35#89992C2214yes—
36#89992C2214yes—
37#89992C2214yes—
38#89992C2214yes—
39#89984C2212yes—
40#89984C2214yes—
41#89983C2214yes—
42C4×C22⋊Dic7C2214yes—
43#89757C2212yes—
44#89757C2212yes—
45#89757C2214yes—
46#89753C2214yes—
47#89753C2214yes—
48#89753C2214yes—
49C4⋊C4×Dic7C2212yes—
50C4⋊C4×Dic7C2212yes—
51C4⋊C4×Dic7C2212yes—
52C4⋊C4×Dic7C2214yes—
53C4⋊C4×Dic7C2212yes—
54#89723C2214yes—
55#89723C2214yes—
56#89723C2214yes—
57#89723C2214yes—
58#89723C2214yes—
59#89722C2214yes—
60C22⋊C4×Dic7C2212yes—
61C22⋊C4×Dic7C2212yes—
62C22⋊C4×Dic7C2214yes—
63C22⋊C4×Dic7C2214yes—
64C22⋊C4×Dic7C2214yes—
65#89708C2211yescharacteristic
66#89707C2211yescharacteristic
67C4×C4⋊Dic7C2212yes—
68#89704C2211yescharacteristic
69#89704C2211yescharacteristic
70C42×Dic7C2211yescharacteristic
71C42×Dic7C2211yescharacteristic
72C4×#540—71nomaximal, Sylow
73C22×C7×D4C2312yes—
74C22×C7×D4C2314yes—
75C22×C7×D4C2314yes—
76C23×C4×C7C2311yescharacteristic
77C23×C4×C7C2314yes—
78C23×Dic7C2311yescharacteristic
79C23×Dic7C2314yes—
80C23×Dic7C2311yescharacteristic
81C23×Dic7C2314yes—
82C7×#127C2×C418yescomplemented
83C7×#125C2×C418yescomplemented
84C7×#123C2×C4132yescomplemented
85C7×#121C2×C4132yescomplemented
86C7×#121C2×C4116yescomplemented
87C4×C7×D4C2×C4116yescomplemented
88C7×#117C2×C418yescomplemented
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C4⋊C4C2312yes—
91C2×C7×C4⋊C4C2314yes—
92C2×C7×C22⋊C4C2312yes—
93C2×C7×C22⋊C4C2314yes—
94C2×C7×C22⋊C4C2314yes—
95C2×C42×C7C2311yescharacteristic
96C2×C42×C7C2311yescharacteristic
97C2×C22⋊Dic7C2312yes—
98C2×C22⋊Dic7C2314yes—
99C2×C22⋊Dic7C2314yes—
100C2×C22⋊Dic7C2314yes—
101C2×C22⋊Dic7C2314yes—
102C2×C22⋊Dic7C2314yes—
103C2×C22⋊Dic7C2314yes—
104C2×C22⋊Dic7C2314yes—
105D4×Dic7—28no—
106D4×Dic7—28no—
107D4×Dic7—48no—
108D4×Dic7—48no—
109#6489—24no—
110C2×C4⋊Dic7C2312yes—
111C2×C4⋊Dic7C2312yes—
112C2×C4⋊Dic7C2312yes—
113C2×C4⋊Dic7C2314yes—
114C2×C4⋊Dic7C2314yes—
115C2×C4⋊Dic7C2314yes—
116C2×C4×Dic7—24no—
117C2×C4×Dic7—48no—
118C2×C4×Dic7C2311yescharacteristic
119C2×C4×Dic7C2312yes—
120C2×C4×Dic7C2312yes—
121C2×C4×Dic7C2311yescharacteristic
122C2×C4×Dic7C2311yescharacteristic
123C2×C4×Dic7C2314yes—
124C2×C4×Dic7C2311yescharacteristic
125C2×C4×Dic7C2312yes—
126C2×C4×Dic7C2312yes—
127C2×C4×Dic7C2314yes—
128C2×C4×Dic7C2314yes—
129C2×C4×Dic7C2311yescharacteristic
130C2×C4×Dic7C2311yescharacteristic
131C7⋊#95C2312yes—
132C7⋊#95C2312yes—
133C7⋊#95C2312yes—
134C7⋊#95C2312yes—
135C7⋊#95C2314yes—
136C7⋊#95C2314yes—
137C7⋊#95C2314yes—
138C7⋊#95C2314yes—
139C7⋊#95C2314yes—
140C7⋊#95C2314yes—
141C7⋊#95C2314yes—
142C2×#540D711yescharacteristic
143#2335—74no—
144#2333—74no—
145#2330—71no—
146#2328—71no—
147#2324—72no—
148#2319—72no—
149#2314—71no—
150C4×#127—71no—
151C4×#125—71no—
152C4×#123—74no—
153C4×#121—72no—
154C4×#121—74no—
155C42×D4—72no—
156C4×#117—71no—
157C24×C7C2411yescharacteristic
158C24×C7C2414yes—
159C2×C7×D4—24no—
160C2×C7×D4—24no—
161C2×C7×D4—44no—
162C2×C7×D4—44no—
163C2×C7×D4—44no—
164C2×C7×D4—44no—
165C2×C7×D4C22×C418yes—
166C2×C7×D4C22×C4116yes—
167C2×C7×D4C22×C4116yes—
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7—44no—
171C22×C4×C7—44no—
172C22×C4×C7C22×C414yes—
173C22×C4×C7C22×C4116yes—
174C22×C4×C7C2412yes—
175C22×C4×C7C2412yes—
176C22×C4×C7C2411yescharacteristic
177C22×C4×C7C2414yes—
178C22×C4×C7C2411yescharacteristic
179C22×Dic7—22no—
180C22×Dic7—22no—
181C22×Dic7—28no—
182C22×Dic7—28no—
183C22×Dic7—28no—
184C22×Dic7—24no—
185C22×Dic7—28no—
186C22×Dic7—44no—
187C22×Dic7—42no—
188C22×Dic7—44no—
189C22×Dic7C2411yescharacteristic
190C22×Dic7C2414yes—
191C22×Dic7C2412yes—
192C22×Dic7C2411yescharacteristic
193C22×Dic7C2414yes—
194C22×Dic7C2411yescharacteristic
195C22×Dic7C2412yes—
196C22×Dic7C2411yescharacteristic
197C7×C4⋊C4C22×C418yes—
198C7×C4⋊C4C22×C418yes—
199C7×C4⋊C4C22×C4116yes—
200C7×C22⋊C4C22×C418yes—
201C7×C22⋊C4C22×C4116yes—
202C7×C22⋊C4C22×C4116yes—
203C42×C7C22×C414yes—
204C42×C7C22×C414yes—
205C22⋊Dic7—24no—
206C22⋊Dic7—28no—
207C22⋊Dic7—28no—
208C22⋊Dic7—44no—
209C22⋊Dic7—44no—
210C22⋊Dic7—44no—
211C22⋊Dic7—44no—
212C22⋊Dic7—44no—
213C4⋊Dic7—24no—
214C4⋊Dic7—24no—
215C4⋊Dic7—24no—
216C4⋊Dic7—44no—
217C4⋊Dic7—44no—
218C4⋊Dic7—44no—
219C4×Dic7—22no—
220C4×Dic7—22no—
221C4×Dic7—22no—
222C4×Dic7—22no—
223C4×Dic7—24no—
224C4×Dic7—28no—
225C4×Dic7—24no—
226C4×Dic7—44no—
227C4×Dic7—44no—
228C4×Dic7—42no—
229C4×Dic7—42no—
230C4×Dic7#4114yes—
231C4×Dic7#4114yes—
232#540Dic7116yesdirect product factor
233C2×#127C2×D711yescharacteristic
234C2×#125C2×D711yescharacteristic
235C2×#123C2×D714yes—
236C2×#121C2×D712yes—
237C2×#121C2×D714yes—
238C2×C4×D4—72no—
239C2×C4×D4—72no—
240C2×C4×D4—74no—
241C2×C4×D4—74no—
242C2×C4×D4C2×D712yes—
243C2×#117—71no—
244C2×#117C2×D711yescharacteristic
245C22×C42—71no—
246C22×C42—74no—
247#390—72no—
248#390—74no—
249#390—72no—
250#390—74no—
251#389—72no—
252#389—72no—
253#389—74no—
254#388—74no—
255#388—74no—
256#388—74no—
257#388—74no—
258#388—74no—
259#387—74no—
260#387—74no—
261#387—74no—
262#386—74no—
263#386—74no—
264#386—74no—
265#386—74no—
266#385—74no—
267#384—71no—
268#382—71no—
269#381—74no—
270#380—72no—
271#380—74no—
272C4×C4⋊C4—72no—
273C4×C4⋊C4—74no—
274C4×C4⋊C4—72no—
275C4×C4⋊C4—72no—
276C4×C4⋊C4—72no—
277C4×C4⋊C4—72no—
278C4×C22⋊C4—74no—
279C4×C22⋊C4—72no—
280C4×C22⋊C4—74no—
281C4×C22⋊C4—72no—
282C4×C22⋊C4—74no—
283C4×C22⋊C4—74no—
284#376—71no—
285#376—71no—
286C43—71no—
287C43—71no—
288C23×C7—21no—
289C23×C7—24no—
290C23×C7—24no—
291C23×C7—21no—
292C23×C7—24no—
293C23×C7—22no—
294C23×C7—24no—
295C23×C7—24no—
296C23×C7—22no—
297C23×C7—24no—
298C23×C7C23×C412yes—
299C23×C7C23×C418yes—
300C23×C7C2511yescharacteristic
301C7×D4—216no—
302C7×D4—416no—
303C7×D4—416no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—22no—
307C2×C4×C7—24no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—24no—
312C2×C4×C7—28no—
313C2×C4×C7—416no—
314C2×C4×C7C23×C414yes—
315C2×C4×C7C23×C414yes—
316C2×C4×C7C23×C412yes—
317C2×C4×C7C23×C418yes—
318C2×C4×C7C23×C412yes—
319C2×C4×C7C2×#4112yes—
320C2×C4×C7C2×#4112yes—
321C2×Dic7—22no—
322C2×Dic7—22no—
323C2×Dic7—22no—
324C2×Dic7—21no—
325C2×Dic7—21no—
326C2×Dic7—22no—
327C2×Dic7—24no—
328C2×Dic7—24no—
329C2×Dic7—24no—
330C2×Dic7—22no—
331C2×Dic7—24no—
332C2×Dic7—22no—
333C2×Dic7—24no—
334C2×Dic7—22no—
335C2×Dic7—24no—
336C2×Dic7—22no—
337C2×Dic7—44no—
338C2×Dic7—416no—
339C2×Dic7—44no—
340C2×Dic7—416no—
341C2×Dic7C2×#4112yes—
342C2×Dic7C2×#4114yes—
343C2×Dic7C2×#4112yes—
344C2×Dic7H(5,𝔽2)14yes—
345C22×D4C22×D712yes—
346C22×D4C22×D714yes—
347C22×D4C22×D714yes—
348C23×C4—71no—
349C23×C4—74no—
350C23×C4—74no—
351C23×C4—71no—
352C23×C4C22×D711yescharacteristic
353C23×C4C22×D714yes—
354#127C2×Dic718yescomplemented
355#125C2×Dic718yescomplemented
356#123C2×Dic7132yescomplemented
357#121C2×Dic7132yescomplemented
358#121C2×Dic7116yescomplemented
359C4×D4—148no—
360C4×D4—148no—
361C4×D4—288no—
362C4×D4—288no—
363C4×D4C2×Dic7116yescomplemented
364#117—144no—
365#117C2×Dic718yescomplemented
366C2×C4⋊C4—72no—
367C2×C4⋊C4—72no—
368C2×C4⋊C4—72no—
369C2×C4⋊C4—74no—
370C2×C4⋊C4—74no—
371C2×C4⋊C4—74no—
372C2×C4⋊C4C22×D712yes—
373C2×C4⋊C4C22×D712yes—
374C2×C4⋊C4C22×D714yes—
375C2×C22⋊C4—72no—
376C2×C22⋊C4—74no—
377C2×C22⋊C4—74no—
378C2×C22⋊C4—74no—
379C2×C22⋊C4—74no—
380C2×C22⋊C4—74no—
381C2×C22⋊C4—74no—
382C2×C22⋊C4—74no—
383C2×C22⋊C4C22×D712yes—
384C2×C22⋊C4C22×D714yes—
385C2×C22⋊C4C22×D714yes—
386C2×C42—71no—
387C2×C42—71no—
388C2×C42—71no—
389C2×C42—72no—
390C2×C42—74no—
391C2×C42—72no—
392C2×C42—71no—
393C2×C42—71no—
394C2×C42—71no—
395C2×C42—72no—
396C2×C42—72no—
397C2×C42—74no—
398C2×C42—74no—
399C2×C42—144no—
400C2×C42—288no—
401C2×C42C22×D711yescharacteristic
402C2×C42C22×D711yescharacteristic
403#95—72no—
404#95—72no—
405#95—72no—
406#95—72no—
407#95—74no—
408#95—74no—
409#95—74no—
410#95—74no—
411#95—74no—
412#95—74no—
413#95—74no—
414C22×C7—22no—
415C22×C7—28no—
416C22×C7—28no—
417C22×C7—24no—
418C22×C7—28no—
419C22×C7—44no—
420C22×C7—41no—
421C22×C7—41no—
422C22×C7—44no—
423C22×C7—44no—
424C22×C7—42no—
425C22×C7—44no—
426C22×C7C24×C411yescharacteristic
427C22×C7C22×#4111yescharacteristic
428C22×C7C22×#4111yescharacteristic
429C22×C7C2×H(5,𝔽2)12yes—
430C22×C7#58512yes—
431C4×C7—22no—
432C4×C7—24no—
433C4×C7—28no—
434C4×C7—24no—
435C4×C7C4×#4114yes—
436Dic7—22no—
437Dic7—24no—
438Dic7—41no—
439Dic7—44no—
440Dic7—41no—
441Dic7—42no—
442Dic7—44no—
443Dic7#54014yesdirect product factor
444C24C23×D711yescharacteristic
445C24C23×D714yes—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4C22×Dic718yes—
453C2×D4C22×Dic7116yes—
454C2×D4C22×Dic7116yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—71no—
460C22×C4—71no—
461C22×C4—72no—
462C22×C4—71no—
463C22×C4—74no—
464C22×C4—72no—
465C22×C4—71no—
466C22×C4—74no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—148no—
470C22×C4—148no—
471C22×C4—148no—
472C22×C4—144no—
473C22×C4—148no—
474C22×C4—284no—
475C22×C4—282no—
476C22×C4—284no—
477C22×C4C22×Dic714yes—
478C22×C4C22×Dic7116yes—
479C22×C4C23×D712yes—
480C22×C4C23×D712yes—
481C22×C4C23×D711yescharacteristic
482C22×C4C23×D714yes—
483C22×C4C23×D711yescharacteristic
484C4⋊C4—144no—
485C4⋊C4—144no—
486C4⋊C4—144no—
487C4⋊C4—284no—
488C4⋊C4—284no—
489C4⋊C4—284no—
490C4⋊C4C22×Dic718yes—
491C4⋊C4C22×Dic718yes—
492C4⋊C4C22×Dic7116yes—
493C22⋊C4—144no—
494C22⋊C4—148no—
495C22⋊C4—148no—
496C22⋊C4—284no—
497C22⋊C4—284no—
498C22⋊C4—284no—
499C22⋊C4—284no—
500C22⋊C4—284no—
501C22⋊C4C22×Dic718yes—
502C22⋊C4C22×Dic7116yes—
503C22⋊C4C22×Dic7116yes—
504C42—74no—
505C42—74no—
506C42—142no—
507C42—142no—
508C42—144no—
509C42—148no—
510C42—144no—
511C42—142no—
512C42—142no—
513C42—282no—
514C42—282no—
515C42—284no—
516C42—284no—
517C42C22×Dic714yes—
518C42C22×Dic714yes—
519C2×C7—42no—
520C2×C7—48no—
521C2×C7C2×C4×#4111yescharacteristic
522C2×C7C4×H(5,𝔽2)12yes—
523C2×C7C2×#54011yescharacteristic
524C2×C7#345511yescharacteristic
525C2×C7#346012yes—
526C23—21no—
527C23—24no—
528C23—24no—
529C23—21no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23C23×Dic712yes—
537C23C23×Dic718yes—
538C23C24×D711yescharacteristic
539D4—216no—
540D4—416no—
541D4—416no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—416no—
552C2×C4—72no—
553C2×C4—74no—
554C2×C4—72no—
555C2×C4—74no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—141no—
560C2×C4—141no—
561C2×C4—142no—
562C2×C4—144no—
563C2×C4—144no—
564C2×C4—144no—
565C2×C4—142no—
566C2×C4—144no—
567C2×C4—142no—
568C2×C4—144no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—144no—
572C2×C4—284no—
573C2×C4—2816no—
574C2×C4—284no—
575C2×C4—2816no—
576C2×C4D7×#4112yes—
577C2×C4D7×#4112yes—
578C2×C4C23×Dic714yes—
579C2×C4C23×Dic714yes—
580C2×C4C23×Dic712yes—
581C2×C4C23×Dic718yes—
582C2×C4C23×Dic712yes—
583C7C4×#54011yescharacteristic, complemented, Sylow
584C22—22no—
585C22—28no—
586C22—28no—
587C22—24no—
588C22—28no—
589C22—41no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—41no—
596C22C2×D7×#4111yescharacteristic
597C22C2×D7×#4111yescharacteristic
598C22D7×H(5,𝔽2)12yes—
599C22#9061812yes—
600C22C24×Dic711yescharacteristic
601C4—22no—
602C4—24no—
603C4—28no—
604C4—24no—
605C4—74no—
606C4—142no—
607C4—144no—
608C4—281no—
609C4—284no—
610C4—282no—
611C4—281no—
612C4—284no—
613C4#41×Dic714yes—
614C2—42no—
615C2—48no—
616C2#1174353511yescharacteristic
617C2#1174353612yes—
618C2D7×#54011yescharacteristic
619C2C2×#41×Dic711yescharacteristic
620C2Dic7×H(5,𝔽2)12yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
Fitting, prime radicalC2×C7×#540C2
socle, cosocleC23×C7C25
hypercenterC2×#540D7
hypocenterC7C4×#540
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
2. #11740617C2semidirect
3. Dic7×#127C2semidirect
4. #11740468C2semidirect
5. Dic7×#125C2semidirect
6. #11740282C2semidirect
7. Dic7×#123C2semidirect
8. #11740080C2semidirect
9. #11740078C2semidirect
10. Dic7×#121C2semidirect
11. Dic7×#121C2semidirect
12. #11739770C2semidirect
13. C4×D4×Dic7C2semidirect
14. #11739676C2semidirect
15. Dic7×#117C2semidirect
16. C7×#540C4semidirect
82. C7×#127C2×C4semidirect
83. C7×#125C2×C4semidirect
84. C7×#123C2×C4semidirect
85. C7×#121C2×C4semidirect
86. C7×#121C2×C4semidirect
87. C4×C7×D4C2×C4semidirect
88. C7×#117C2×C4semidirect
232. #540Dic7direct (N × H)
354. #127C2×Dic7semidirect
355. #125C2×Dic7semidirect
356. #123C2×Dic7semidirect
357. #121C2×Dic7semidirect
358. #121C2×Dic7semidirect
363. C4×D4C2×Dic7semidirect
365. #117C2×Dic7semidirect
443. Dic7#540direct (N × H)
583. C7C4×#540semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5402
#117406172
Dic7×#1272
#117404682
Dic7×#1252
#117402822
Dic7×#1232
#117400802
#117400782
Dic7×#1212
#117397702
C4×D4×Dic72
#117396762
Dic7×#1172
C4×#5407