Finite Group Databasebeta

← C7×#18329 #13195545

Subgroups — C7×#18329 #13195545

Order 1792.

1078 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#538C211yesmaximal, characteristic
2C7×#3195C211yesmaximal, characteristic, complemented
3C7×#3193C211yesmaximal, characteristic, complemented
4C7×#2561C211yesmaximal, characteristic, complemented
5C7×#1612C214yesmaximal, complemented
6C7×#1606C214yesmaximal, complemented
7C7×#1508C211yesmaximal, characteristic, complemented
8C7×#1507C211yesmaximal, characteristic, complemented
9C7×#1462C211yesmaximal, characteristic, complemented
10C7×#538—22no—
11C7×#538—22no—
12C7×#538—22no—
13C7×#538C2214yes—
14C2×C7×#126—21no—
15C2×C7×#124—21no—
16C2×C7×#123—21no—
17C2×C7×C22⋊Q8C2211yescharacteristic
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×#121—21no—
20C2×C7×#121C2211yescharacteristic
21C2×C7×#120—21no—
22C2×C4×C7×D4—21no—
23C22×C7×C22⋊C4C2211yescharacteristic
24C7×#484—21no—
25C7×#483—21no—
26C7×#483C2212yescomplemented
27C7×#480C2212yescomplemented
28C7×#470—22no—
29C7×#468—22no—
30C7×#451—21no—
31C7×#451C2212yescomplemented
32C7×#450—21no—
33C7×#448C2212yescomplemented
34C7×#407C2211yescharacteristic
35C7×#407C2211yescharacteristic
36C7×#397C2212yescomplemented
37C7×#393C2212yescomplemented
38C7×#388—22no—
39C7×#381—21no—
40C7×#379—21no—
41C4×C7×C22⋊C4C2211yescharacteristic
42C7×#328C2214yescomplemented
43C7×#327C2214yescomplemented
44C7×#323C2214yescomplemented
45#18329C711yesmaximal, characteristic, direct product factor, Sylow
46C25×C7D411yescharacteristic
47C22×C7×Q8D411yescharacteristic
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4—21no—
52C22×C7×D4D411yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7D411yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C2×C7×Q16—22no—
57C2×C7×SD16—22no—
58C2×C7×M16—21no—
59C2×C7×M16—21no—
60C7×#126—24no—
61C7×#126—24no—
62C7×#124—24no—
63C7×#124—24no—
64C7×#123—24no—
65C7×#123—24no—
66C7×C22⋊Q8—21no—
67C7×C22⋊Q8—21no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8—21no—
71C7×C22⋊Q8—21no—
72C7×C22⋊Q8D412yes—
73C7×C22⋊Q8C2312yes—
74C7×#121—24no—
75C7×#121—24no—
76C7×#121—21no—
77C7×#121—21no—
78C7×#121D412yes—
79C7×#121C2312yes—
80C7×#120—24no—
81C7×#120—24no—
82C4×C7×D4—24no—
83C4×C7×D4—24no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4D411yescharacteristic
88C2×C7×C4⋊C4D411yescharacteristic
89C2×C7×C4⋊C4C2311yescharacteristic
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4—21no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—22no—
102C2×C7×C22⋊C4—22no—
103C2×C7×C22⋊C4D412yes—
104C2×C7×C22⋊C4C2312yes—
105C2×C42×C7—21no—
106C2×C42×C7—21no—
107C7×#107—22no—
108C7×#106—22no—
109C7×#103—22no—
110C7×#103—22no—
111C7×#103—22no—
112C7×#102—22no—
113C7×C22⋊C8—21no—
114C7×C22⋊C8—21no—
115C7×C22⋊C8C2312yes—
116C7×C22⋊C8C2312yes—
117C7×#95—22no—
118C7×#95—22no—
119C7×#95C2312yes—
120C2×#538C2×C711yescharacteristic
121#3195C2×C711yescharacteristic, complemented
122#3193C2×C711yescharacteristic, complemented
123#2561C2×C711yescharacteristic, complemented
124#1612C2×C714yescomplemented
125#1606C2×C714yescomplemented
126#1508C2×C711yescharacteristic, complemented
127#1507C2×C711yescharacteristic, complemented
128#1462C2×C711yescharacteristic, complemented
129C24×C7—21no—
130C24×C7—21no—
131C24×C7—21no—
132C24×C7—42no—
133C24×C7—42no—
134C24×C7—81no—
135C24×C7C2×D411yescharacteristic
136C24×C7C2×D412yes—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—81no—
141C2×C7×Q8—81no—
142C2×C7×Q8C2×D412yes—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C2×C7×D4—24no—
146C2×C7×D4—24no—
147C2×C7×D4—21no—
148C2×C7×D4—44no—
149C2×C7×D4—81no—
150C2×C7×D4—82no—
151C2×C7×D4—82no—
152C2×C7×D4C2×D412yes—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—24no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—42no—
163C22×C4×C7—81no—
164C22×C4×C7—82no—
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C7×Q16—82no—
173C7×SD16—82no—
174C7×M16—81no—
175C7×M16—81no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—41no—
186C7×C4⋊C4—41no—
187C7×C4⋊C4C2×D412yes—
188C7×C4⋊C4C2×D412yes—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—24no—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—22no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—42no—
201C7×C22⋊C4—44no—
202C7×C22⋊C4—41no—
203C7×C22⋊C4—41no—
204C7×C22⋊C4—82no—
205C42×C7—24no—
206C42×C7—81no—
207#538—22no—
208#538—22no—
209#538—22no—
210#538C22×C714yes—
211C2×#126—21no—
212C2×#124—21no—
213C2×#123—21no—
214C2×C22⋊Q8C22×C711yescharacteristic
215C2×C22⋊Q8C22×C711yescharacteristic
216C2×#121—21no—
217C2×#121C22×C711yescharacteristic
218C2×#120—21no—
219C2×C4×D4—21no—
220C22×C22⋊C4C22×C711yescharacteristic
221#484—21no—
222#483—21no—
223#483C22×C712yescomplemented
224#480C22×C712yescomplemented
225#470—22no—
226#468—22no—
227#451—21no—
228#451C22×C712yescomplemented
229#450—21no—
230#448C22×C712yescomplemented
231#407C22×C711yescharacteristic
232#407C22×C711yescharacteristic
233#397C22×C712yescomplemented
234#393C22×C712yescomplemented
235#388—22no—
236#381—21no—
237#379—21no—
238C4×C22⋊C4C22×C711yescharacteristic
239#328C22×C714yescomplemented
240#327C22×C714yescomplemented
241#323C22×C714yescomplemented
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7—41no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—42no—
256C23×C7—42no—
257C23×C7—42no—
258C23×C7—81no—
259C23×C7—81no—
260C23×C7—88no—
261C23×C7#12011yescharacteristic
262C23×C7#12011yescharacteristic
263C23×C7C22×D411yescharacteristic
264C7×Q8—42no—
265C7×Q8—42no—
266C7×Q8—42no—
267C7×Q8—81no—
268C7×D4—42no—
269C7×D4—44no—
270C7×D4—44no—
271C7×D4—81no—
272C7×D4—84no—
273C7×D4—84no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—81no—
297C2×C4×C7—84no—
298C2×C4×C7#12011yescharacteristic
299C2×C4×C7#12011yescharacteristic
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C7×C8—42no—
303C7×C8—42no—
304C25C7×D411yescharacteristic
305C22×Q8C7×D411yescharacteristic
306C22×Q8C23×C711yescharacteristic
307C22×D4—21no—
308C22×D4—21no—
309C22×D4—21no—
310C22×D4C7×D411yescharacteristic
311C23×C4—21no—
312C23×C4C7×D411yescharacteristic
313C23×C4C23×C711yescharacteristic
314C2×Q16—22no—
315C2×SD16—22no—
316C2×M16—21no—
317C2×M16—21no—
318#126—24no—
319#126—24no—
320#124—24no—
321#124—24no—
322#123—24no—
323#123—24no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C22⋊Q8C7×D412yes—
331C22⋊Q8C23×C712yes—
332#121—24no—
333#121—24no—
334#121—21no—
335#121—21no—
336#121C7×D412yes—
337#121C23×C712yes—
338#120—24no—
339#120—24no—
340C4×D4—24no—
341C4×D4—24no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C7×D411yescharacteristic
346C2×C4⋊C4C7×D411yescharacteristic
347C2×C4⋊C4C23×C711yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4C7×D412yes—
362C2×C22⋊C4C23×C712yes—
363C2×C42—21no—
364C2×C42—21no—
365#107—22no—
366#106—22no—
367#103—22no—
368#103—22no—
369#103—22no—
370#102—22no—
371C22⋊C8—21no—
372C22⋊C8—21no—
373C22⋊C8C23×C712yes—
374C22⋊C8C23×C712yes—
375#95—22no—
376#95—22no—
377#95C23×C712yes—
378C22×C7—21no—
379C22×C7—21no—
380C22×C7—21no—
381C22×C7—21no—
382C22×C7—21no—
383C22×C7—21no—
384C22×C7—21no—
385C22×C7—44no—
386C22×C7—44no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7—42no—
392C22×C7—42no—
393C22×C7—82no—
394C22×C7—84no—
395C22×C7—84no—
396C22×C7—84no—
397C22×C7#45512yes—
398C22×C7C2×#12011yescharacteristic
399C22×C7#57512yes—
400C4×C7—21no—
401C4×C7—21no—
402C4×C7—21no—
403C4×C7—21no—
404C4×C7—42no—
405C4×C7—42no—
406C4×C7—42no—
407C4×C7—42no—
408C4×C7—82no—
409C4×C7—81no—
410C4×C7—81no—
411C24—21no—
412C24—21no—
413C24—21no—
414C24—42no—
415C24—42no—
416C24—81no—
417C24C2×C7×D411yescharacteristic
418C24C2×C7×D412yes—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—81no—
423C2×Q8—81no—
424C2×Q8C2×C7×D412yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—44no—
431C2×D4—81no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4C2×C7×D412yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—81no—
446C22×C4—82no—
447C22×C4C2×C7×D412yes—
448C22×C4C2×C7×D411yescharacteristic
449C22×C4C2×C7×D411yescharacteristic
450C22×C4C2×C7×D411yescharacteristic
451C22×C4C2×C7×D411yescharacteristic
452C22×C4C2×C7×D411yescharacteristic
453C22×C4C24×C711yescharacteristic
454Q16—82no—
455SD16—82no—
456M16—81no—
457M16—81no—
458C2×C8—22no—
459C2×C8—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4C2×C7×D412yes—
470C4⋊C4C2×C7×D412yes—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—82no—
487C42—24no—
488C42—81no—
489C2×C7—22no—
490C2×C7—22no—
491C2×C7—41no—
492C2×C7—82no—
493C2×C7—82no—
494C2×C7#300511yescharacteristic
495C2×C7C2×#45511yescharacteristic
496C2×C7#302611yescharacteristic
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—81no—
514C23—81no—
515C23—88no—
516C23C7×#12011yescharacteristic
517C23C7×#12011yescharacteristic
518C23C22×C7×D411yescharacteristic
519Q8—42no—
520Q8—42no—
521Q8—42no—
522Q8—81no—
523D4—42no—
524D4—44no—
525D4—44no—
526D4—81no—
527D4—84no—
528D4—84no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—81no—
552C2×C4—84no—
553C2×C4C7×#12011yescharacteristic
554C2×C4C7×#12011yescharacteristic
555C2×C4C22×C7×D411yescharacteristic
556C2×C4C22×C7×D411yescharacteristic
557C8—42no—
558C8—42no—
559C7#1832911yescharacteristic, direct product factor, Sylow
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—44no—
568C22—44no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—82no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22C7×#45512yes—
580C22C2×C7×#12011yescharacteristic
581C22C7×#57512yes—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—82no—
591C4—81no—
592C4—81no—
593C2—22no—
594C2—22no—
595C2—41no—
596C2—82no—
597C2—82no—
598C2C7×#300511yescharacteristic
599C2C2×C7×#45511yescharacteristic
600C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#538C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3195C2semidirect
3. C7×#3193C2semidirect
4. C7×#2561C2semidirect
5. C7×#1612C2semidirect
6. C7×#1606C2semidirect
7. C7×#1508C2semidirect
8. C7×#1507C2semidirect
9. C7×#1462C2semidirect
26. C7×#483C22semidirect
27. C7×#480C22semidirect
31. C7×#451C22semidirect
33. C7×#448C22semidirect
36. C7×#397C22semidirect
37. C7×#393C22semidirect
42. C7×#328C22semidirect
43. C7×#327C22semidirect
44. C7×#323C22semidirect
45. #18329C7direct (N × H)
121. #3195C2×C7semidirect
122. #3193C2×C7semidirect
123. #2561C2×C7semidirect
124. #1612C2×C7semidirect
125. #1606C2×C7semidirect
126. #1508C2×C7semidirect
127. #1507C2×C7semidirect
128. #1462C2×C7semidirect
223. #483C22×C7semidirect
224. #480C22×C7semidirect
228. #451C22×C7semidirect
230. #448C22×C7semidirect
233. #397C22×C7semidirect
234. #393C22×C7semidirect
239. #328C22×C7semidirect
240. #327C22×C7semidirect
241. #323C22×C7semidirect
559. C7#18329direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5382
C7×#31952
C7×#31932
C7×#25612
C7×#16122
C7×#16062
C7×#15082
C7×#15072
C7×#14622
#183297