Finite Group Databasebeta

← D7×#1506 #13320943

Subgroups — D7×#1506 #13320943

Order 1792.

1130 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#533C211yesmaximal, characteristic
2D7×#432C211yesmaximal, characteristic
3D7×#376C211yesmaximal, characteristic
4C7×#1506C211yesmaximal, characteristic, complemented
5#11729590C211yesmaximal, characteristic, complemented
6#11728590C211yesmaximal, characteristic, complemented
7#11727187C211yesmaximal, characteristic, complemented
8D7×#328C214yesmaximal
9#11726064C214yesmaximal, complemented
10C7×#533C2211yescharacteristic
11#90419C411yescharacteristic, complemented
12#90419C411yescharacteristic, complemented
13D7×C4⋊Q8C411yescharacteristic, complemented
14D7×C4⋊Q8C411yescharacteristic, complemented
15#90379—21no—
16D7×#125—21no—
17D7×C22⋊Q8—21no—
18D7×C22⋊Q8C2212yes—
19#90313—21no—
20#90313C2212yes—
21#90269—21no—
22#90269C412yescomplemented
23C4×Q8×D7—21no—
24C4×Q8×D7C412yescomplemented
25D7×#117—21no—
26#90208—21no—
27C2×C42×D7—21no—
28C2×C42×D7C2211yescharacteristic
29#90161C2211yescharacteristic
30C7×#432C2211yescharacteristic
31C7×#376C2211yescharacteristic
32#89704—21no—
33#89704C2211yescharacteristic
34#89698C2211yescharacteristic
35#89607—21no—
36D7×C4⋊C8—21no—
37D7×C22⋊C8C2212yes—
38#89491C2212yes—
39#89477—21no—
40D7×#97—21no—
41D7×#95—21no—
42D7×#95C2212yes—
43#89420—21no—
44#89420C2212yes—
45C7×#328C2214yes—
46#89333C2214yes—
47#89262C2214yes—
48#89247C2214yes—
49C2×#1506—71nomaximal, Sylow
50C2×Q8×D7—21no—
51C2×Q8×D7C2×C412yes—
52C22×C4×D7—22no—
53C22×C4×D7D411yescharacteristic
54C22×C4×D7D411yescharacteristic
55C22×C4×D7C2311yescharacteristic
56C7×C4⋊Q8C2×C411yescharacteristic, complemented
57C7×C4⋊Q8C2×C411yescharacteristic, complemented
58C7×#125—21no—
59C7×C22⋊Q8—21no—
60C7×C22⋊Q8C2312yes—
61C4×C7×Q8—21no—
62C4×C7×Q8C2×C412yescomplemented
63C7×#117—21no—
64C2×C42×C7—21no—
65C2×C42×C7C2311yescharacteristic
66#6509—21no—
67#6509C2×C412yes—
68Q8×Dic7—21no—
69Q8×Dic7C2×C412yescomplemented
70#6507—21no—
71#6507C2×C412yescomplemented
72#6489—21no—
73#6487—21no—
74#6487C2312yes—
75C2×C4×Dic7—21no—
76C2×C4×Dic7—22no—
77C2×C4×Dic7D411yescharacteristic
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7C2311yescharacteristic
80C2×C8×D7—22no—
81#6460—21no—
82#6460C2×C412yes—
83#6459—22no—
84#6459—22no—
85#6455—21no—
86#6455—21no—
87D7×C4⋊C4—21no—
88D7×C4⋊C4—21no—
89D7×C4⋊C4—22no—
90D7×C4⋊C4—21no—
91D7×C4⋊C4—22no—
92D7×C4⋊C4C2×C412yes—
93#6453—21no—
94#6453—21no—
95#6452—22no—
96#6452—22no—
97#6451—21no—
98#6451—21no—
99#6451—21no—
100#6451C2×C412yescomplemented
101#6450—22no—
102#6450—21no—
103#6450—22no—
104#6450C2×C412yescomplemented
105D7×C22⋊C4—22no—
106#6441—22no—
107#6441—22no—
108#6440—22no—
109#6435—22no—
110C42×D7—28no—
111C42×D7—22no—
112C42×D7C2×C411yescharacteristic
113C42×D7C2×C411yescharacteristic
114C42×D7C2×C411yescharacteristic
115C42×D7C2×C411yescharacteristic
116C42×D7D411yescharacteristic
117C42×D7D411yescharacteristic
118C42×D7D411yescharacteristic
119C42×D7D411yescharacteristic
120#6433—21no—
121#6432C2×C411yescharacteristic, complemented
122#6432C2×C411yescharacteristic, complemented
123C4×Dic14—21no—
124C4×Dic14—22no—
125C4×Dic14—22no—
126C4×Dic14C2×C412yescomplemented
127C7×C4⋊C8—21no—
128C7×C22⋊C8C2312yes—
129C7×#97—21no—
130C7×#95—21no—
131C7×#95C2312yes—
132C7⋊#95—21no—
133C7⋊#95—22no—
134C7⋊#95—22no—
135C7⋊#95C2312yes—
136C22⋊(C7⋊C8)C2312yes—
137(C2×D7)⋊C8—22no—
138#6389—22no—
139Dic7⋊C8—22no—
140C4⋊(C7⋊C8)—21no—
141#6377—21no—
142C2×#533—71no—
143C2×#432—71no—
144C2×#376—71no—
145#1506—71no—
146#1506—71no—
147#1506—74no—
148#1506—71no—
149#1506D711yescharacteristic, direct product factor
150C2×#328—74no—
151C23×D7C2×D411yescharacteristic
152C2×C7×Q8—21no—
153C2×C7×Q8C22×C412yes—
154C22×C4×C7—22no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—22no—
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162Q8×D7—44no—
163Q8×D7—44no—
164C2×C4×D7—24no—
165C2×C4×D7—24no—
166C2×C4×D7—28no—
167C2×C4×D7—22no—
168C2×C4×D7—22no—
169C2×C4×D7—42no—
170C2×C4×D7—44no—
171C2×C4×D7—44no—
172C2×C4×D7C22⋊C411yescharacteristic
173C2×C4×D7C22⋊C411yescharacteristic
174C2×C4×D7C22⋊C411yescharacteristic
175C2×C4×D7C22⋊C411yescharacteristic
176C2×C4×D7C22⋊C411yescharacteristic
177C2×C4×D7C22⋊C411yescharacteristic
178C2×C4×D7C22⋊C411yescharacteristic
179C2×C4×D7C22⋊C411yescharacteristic
180C2×C4×D7C22×C411yescharacteristic
181C2×C4×D7C22×C411yescharacteristic
182C2×C4×D7C2×D411yescharacteristic
183C2×C4×D7C2×D411yescharacteristic
184C2×Dic14—21no—
185C2×Dic14—22no—
186C2×Dic14—22no—
187C2×Dic14C22×C412yes—
188C2×C7×C8—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4C22×C412yes—
195C7×C22⋊C4—22no—
196C42×C7—24no—
197C42×C7—22no—
198C42×C7C22×C411yescharacteristic
199C42×C7C22×C411yescharacteristic
200C42×C7C2×D411yescharacteristic
201C42×C7C2×D411yescharacteristic
202C22⋊Dic7—22no—
203(C2×D7)⋊C4—22no—
204(C2×D7)⋊C4—22no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—22no—
209C4⋊Dic7—22no—
210C4⋊Dic7C22×C412yes—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—22no—
219C4×Dic7—28no—
220C4×Dic7—24no—
221C4×Dic7—22no—
222C4×Dic7—22no—
223C4×Dic7—22no—
224C4×Dic7C22⋊C411yescharacteristic
225C4×Dic7C22⋊C411yescharacteristic
226C4×Dic7C22⋊C411yescharacteristic
227C4×Dic7C22⋊C411yescharacteristic
228C4×Dic7C22⋊C411yescharacteristic
229C4×Dic7C22⋊C411yescharacteristic
230C4×Dic7C22⋊C411yescharacteristic
231C4×Dic7C22⋊C411yescharacteristic
232C4×Dic7C22×C411yescharacteristic
233C4×Dic7C22×C411yescharacteristic
234C4×Dic7C2×D411yescharacteristic
235C4×Dic7C2×D411yescharacteristic
236C2×C7⋊C8—22no—
237C8×D7—44no—
238#533—71no—
239#533—71no—
240#533—72no—
241#533—72no—
242#533—71no—
243#533—141no—
244#533—141no—
245#533—141no—
246#533—141no—
247#533C2×D711yescharacteristic
248C2×C4⋊Q8—71no—
249C2×C4⋊Q8—71no—
250C2×#125—141no—
251C2×C22⋊Q8—72no—
252C2×C22⋊Q8—141no—
253C2×C4×Q8—72no—
254C2×C4×Q8—141no—
255C2×#117—141no—
256C22×C42—71no—
257C22×C42—141no—
258#432—71no—
259#432—72no—
260#432—141no—
261#432—141no—
262#432C2×D711yescharacteristic
263C2×C4⋊C8—141no—
264C2×C22⋊C8—72no—
265C2×#97—141no—
266#376—71no—
267#376—72no—
268#376—141no—
269#376—141no—
270#376C2×D711yescharacteristic
271C2×#95—72no—
272C2×#95—141no—
273#328—74no—
274#328—74no—
275#328—74no—
276#328C2×D714yes—
277C23×C7C22×D411yescharacteristic
278C22×D7—22no—
279C22×D7—22no—
280C22×D7C2×C22⋊C411yescharacteristic
281C22×D7C2×C22⋊C411yescharacteristic
282C22×D7#13714yes—
283C7×Q8—42no—
284C7×Q8—42no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—24no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7C2×C22⋊C411yescharacteristic
294C2×C4×C7C2×C22⋊C411yescharacteristic
295C2×C4×C7C2×C22⋊C411yescharacteristic
296C2×C4×C7C2×C22⋊C411yescharacteristic
297C2×C4×C7C23×C411yescharacteristic
298C2×C4×C7C22×D411yescharacteristic
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—24no—
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7—41no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7C2×C22⋊C411yescharacteristic
308C2×Dic7C2×C22⋊C411yescharacteristic
309C2×Dic7C2×C22⋊C411yescharacteristic
310C2×Dic7C2×C22⋊C411yescharacteristic
311C2×Dic7C2×C22⋊C411yescharacteristic
312C2×Dic7C2×C22⋊C411yescharacteristic
313C4×D7—22no—
314C4×D7—22no—
315C4×D7—22no—
316C4×D7—22no—
317C4×D7—22no—
318C4×D7—22no—
319C4×D7—22no—
320C4×D7—22no—
321C4×D7—22no—
322C4×D7—41no—
323C4×D7—41no—
324C4×D7—416no—
325C4×D7—44no—
326C4×D7—44no—
327C4×D7#10414yes—
328Dic14—44no—
329Dic14—42no—
330Dic14—44no—
331Dic14—42no—
332C7×C8—42no—
333C7⋊C8—42no—
334C22×Q8—72no—
335C22×Q8—141no—
336C23×C4—71no—
337C23×C4—71no—
338C23×C4—71no—
339C23×C4—142no—
340C22×C8—142no—
341C4⋊Q8—72no—
342C4⋊Q8—72no—
343C4⋊Q8—71no—
344C4⋊Q8—71no—
345C4⋊Q8—141no—
346C4⋊Q8—141no—
347C4⋊Q8—141no—
348C4⋊Q8—141no—
349C4⋊Q8C4×D711yescharacteristic, complemented
350C4⋊Q8C4×D711yescharacteristic, complemented
351#125—21no—
352#125—142no—
353#125—142no—
354#125—141no—
355#125—141no—
356#125—141no—
357C22⋊Q8—21no—
358C22⋊Q8—72no—
359C22⋊Q8—72no—
360C22⋊Q8—72no—
361C22⋊Q8—142no—
362C22⋊Q8—142no—
363C22⋊Q8—142no—
364C22⋊Q8—142no—
365C22⋊Q8—141no—
366C22⋊Q8—141no—
367C22⋊Q8—141no—
368C22⋊Q8C22×D712yes—
369C4×Q8—21no—
370C4×Q8—72no—
371C4×Q8—72no—
372C4×Q8—72no—
373C4×Q8—142no—
374C4×Q8—142no—
375C4×Q8—142no—
376C4×Q8—141no—
377C4×Q8—142no—
378C4×Q8—141no—
379C4×Q8—141no—
380C4×Q8C4×D712yescomplemented
381#117—21no—
382#117—142no—
383#117—142no—
384#117—141no—
385#117—141no—
386#117—141no—
387C2×C4⋊C4—72no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4—142no—
391C2×C4⋊C4—142no—
392C2×C4⋊C4—141no—
393C2×C22⋊C4—142no—
394C2×C42—21no—
395C2×C42—71no—
396C2×C42—71no—
397C2×C42—71no—
398C2×C42—71no—
399C2×C42—71no—
400C2×C42—71no—
401C2×C42—71no—
402C2×C42—71no—
403C2×C42—71no—
404C2×C42—71no—
405C2×C42—71no—
406C2×C42—142no—
407C2×C42—148no—
408C2×C42—142no—
409C2×C42—141no—
410C2×C42C22×D711yescharacteristic
411C4⋊C8—21no—
412C4⋊C8—142no—
413C4⋊C8—141no—
414C22⋊C8—72no—
415C22⋊C8—142no—
416C22⋊C8C22×D712yes—
417#97—21no—
418#97—142no—
419#97—141no—
420#95—21no—
421#95—72no—
422#95—142no—
423#95—142no—
424#95—141no—
425#95C22×D712yes—
426C22×C7—21no—
427C22×C7—21no—
428C22×C7C22×C22⋊C411yescharacteristic
429C22×C7C2×#13712yes—
430C2×D7—21no—
431C2×D7—21no—
432C2×D7—28no—
433C2×D7#41012yes—
434C2×D7#41912yes—
435C2×D7#41912yes—
436C2×D7C2×#10412yes—
437C4×C7—21no—
438C4×C7—21no—
439C4×C7—21no—
440C4×C7—21no—
441C4×C7—21no—
442C4×C7—44no—
443C4×C7—42no—
444C4×C7—42no—
445C4×C7C2×#10412yes—
446Dic7—21no—
447Dic7—21no—
448Dic7—21no—
449Dic7—21no—
450Dic7—21no—
451Dic7—44no—
452Dic7—42no—
453Dic7—42no—
454Dic7C2×#10412yes—
455C24—71no—
456C2×Q8—21no—
457C2×Q8—72no—
458C2×Q8—142no—
459C2×Q8—142no—
460C2×Q8—141no—
461C2×Q8—284no—
462C2×Q8—284no—
463C2×Q8C2×C4×D712yes—
464C22×C4—22no—
465C22×C4—71no—
466C22×C4—71no—
467C22×C4—71no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—71no—
474C22×C4—71no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—71no—
478C22×C4—71no—
479C22×C4—71no—
480C22×C4—144no—
481C22×C4—144no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—148no—
485C22×C4—142no—
486C22×C4—282no—
487C22×C4—284no—
488C22×C4—284no—
489C22×C4D4×D711yescharacteristic
490C22×C4D4×D711yescharacteristic
491C22×C4C23×D711yescharacteristic
492C2×C8—22no—
493C2×C8—142no—
494C2×C8—284no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—72no—
501C4⋊C4—142no—
502C4⋊C4—142no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—142no—
506C4⋊C4—142no—
507C4⋊C4—142no—
508C4⋊C4—142no—
509C4⋊C4—142no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—142no—
514C4⋊C4C2×C4×D712yes—
515C22⋊C4—22no—
516C22⋊C4—142no—
517C22⋊C4—142no—
518C22⋊C4—142no—
519C42—24no—
520C42—22no—
521C42—71no—
522C42—71no—
523C42—71no—
524C42—71no—
525C42—71no—
526C42—71no—
527C42—71no—
528C42—71no—
529C42—71no—
530C42—71no—
531C42—71no—
532C42—71no—
533C42—142no—
534C42—142no—
535C42—144no—
536C42—148no—
537C42—142no—
538C42C2×C4×D711yescharacteristic
539C42C2×C4×D711yescharacteristic
540C42D4×D711yescharacteristic
541C42D4×D711yescharacteristic
542C2×C7—22no—
543C2×C7C2×#41011yescharacteristic
544C2×C7C2×#41911yescharacteristic
545C2×C7C22×#10411yescharacteristic
546D7—22no—
547D7#150614yesdirect product factor
548C23—71no—
549C23—71no—
550C23—74no—
551C23—142no—
552C23—142no—
553C23C2×D4×D711yescharacteristic
554Q8—42no—
555Q8—42no—
556Q8—282no—
557Q8—284no—
558Q8—284no—
559Q8—282no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—71no—
569C2×C4—74no—
570C2×C4—71no—
571C2×C4—71no—
572C2×C4—71no—
573C2×C4—71no—
574C2×C4—71no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—142no—
582C2×C4—142no—
583C2×C4—142no—
584C2×C4—142no—
585C2×C4—142no—
586C2×C4—144no—
587C2×C4—142no—
588C2×C4—142no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4—284no—
592C2×C4—284no—
593C2×C4—2816no—
594C2×C4—281no—
595C2×C4—282no—
596C2×C4—282no—
597C2×C4D7×C22⋊C411yescharacteristic
598C2×C4D7×C22⋊C411yescharacteristic
599C2×C4D7×C22⋊C411yescharacteristic
600C2×C4D7×C22⋊C411yescharacteristic
601C2×C4C22×C4×D711yescharacteristic
602C2×C4C2×D4×D711yescharacteristic
603C8—42no—
604C8—282no—
605C7C2×#150611yescharacteristic, complemented, Sylow
606C22—21no—
607C22—21no—
608C22—72no—
609C22—72no—
610C22—72no—
611C22—72no—
612C22—148no—
613C22—141no—
614C22—141no—
615C22C2×D7×C22⋊C411yescharacteristic
616C22D7×#13712yes—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—72no—
626C4—141no—
627C4—141no—
628C4—141no—
629C4—141no—
630C4—141no—
631C4—284no—
632C4—282no—
633C4—282no—
634C4D7×#10412yes—
635C2—22no—
636C2—74no—
637C2—142no—
638C2D7×#41011yescharacteristic
639C2D7×#41911yescharacteristic
640C2C2×D7×#10411yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1506C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1506D7
hypocenterC7C2×#1506
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
4. C7×#1506C2semidirect
5. #11729590C2semidirect
6. #11728590C2semidirect
7. #11727187C2semidirect
9. #11726064C2semidirect
11. #90419C4semidirect
12. #90419C4semidirect
13. D7×C4⋊Q8C4semidirect
14. D7×C4⋊Q8C4semidirect
22. #90269C4semidirect
24. C4×Q8×D7C4semidirect
56. C7×C4⋊Q8C2×C4semidirect
57. C7×C4⋊Q8C2×C4semidirect
62. C4×C7×Q8C2×C4semidirect
69. Q8×Dic7C2×C4semidirect
71. #6507C2×C4semidirect
100. #6451C2×C4semidirect
104. #6450C2×C4semidirect
121. #6432C2×C4semidirect
122. #6432C2×C4semidirect
126. C4×Dic14C2×C4semidirect
149. #1506D7direct (N × H)
349. C4⋊Q8C4×D7semidirect
350. C4⋊Q8C4×D7semidirect
380. C4×Q8C4×D7semidirect
547. D7#1506direct (N × H)
605. C7C2×#1506semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
D7×#4322
D7×#3762
C7×#15062
#117295902
#117285902
#117271872
D7×#3282
#117260642
C2×#15067