Finite Group Databasebeta

← C7×#24731 #13201947

Subgroups — C7×#24731 #13201947

Order 1792.

1086 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#463C212yesmaximal, complemented
2C2×C7×#459C211yesmaximal, characteristic
3C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
4C7×#2548C211yesmaximal, characteristic, complemented
5C7×D4×C4⋊C4C211yesmaximal, characteristic, complemented
6C7×#2065C212yesmaximal, complemented
7C7×#2042C212yesmaximal, complemented
8C7×#1957C211yesmaximal, characteristic, complemented
9C7×#1939C211yesmaximal, characteristic, complemented
10C7×#1879C212yesmaximal, complemented
11C7×#1803C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2211yescharacteristic
13C22×C7×D8C2212yes—
14C2×C7×#127C2211yescharacteristic
15C2×C7×#124C2212yes—
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×D4C2212yes—
18C2×C4×C7×D4C2211yescharacteristic
19C22×C7×C4⋊C4—21no—
20C7×#463—28no—
21C7×#459—24no—
22C4×C7×SD16—24no—
23C2×C7×#106C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×C4⋊C8C2212yes—
26C2×C7×#103C2211yescharacteristic
27C2×C7×#103C2212yes—
28C2×C7×#102C2211yescharacteristic
29C2×C7×#102C2212yes—
30C2×C7×#102C2212yes—
31C2×C7×#102C2212yes—
32C2×C4×C7×C8C2211yescharacteristic
33C7×#394—22no—
34C7×#390—21no—
35C7×#390—21no—
36C7×#390C2212yes—
37C7×#389C2212yes—
38C7×#385—22no—
39C7×#384C2211yescharacteristic
40C7×#380—21no—
41C4×C7×C4⋊C4C2211yescharacteristic
42C7×#340C2212yes—
43C7×#336C2212yes—
44#24731C711yesmaximal, characteristic, direct product factor, Sylow
45C22×C7×Q8C2311yescharacteristic
46C22×C7×D4—21no—
47C22×C7×D4—21no—
48C22×C7×D4C2312yes—
49C22×C7×D4C2311yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7—22no—
52C2×C7×SD16—22no—
53C2×C7×SD16—22no—
54C2×C7×SD16D414yescomplemented
55C2×C7×D8—24no—
56C2×C7×D8—24no—
57C2×C7×D8—24no—
58C22×C7×C8C2311yescharacteristic
59C22×C7×C8C2312yes—
60C22×C7×C8C2311yescharacteristic
61C7×#127—22no—
62C7×#127—22no—
63C7×#124—24no—
64C7×#124—42no—
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×D4—24no—
68C4×C7×D4—22no—
69C4×C7×D4—42no—
70C4×C7×D4D414yes—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—44no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—22no—
83C2×C7×C22⋊C4—22no—
84C2×C7×C22⋊C4—22no—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—22no—
87C2×C42×C7—21no—
88C2×C42×C7C2312yes—
89C2×C42×C7C2311yescharacteristic
90C7×#106—22no—
91C7×C4⋊C8—24no—
92C7×C4⋊C8—22no—
93C7×#103—24no—
94C7×#103—22no—
95C7×#102—22no—
96C7×#102—24no—
97C7×#102—24no—
98C7×#102—42no—
99C4×C7×C8—22no—
100C7×#95—22no—
101C7×#95D412yes—
102C2×#463C2×C712yescomplemented
103C2×#459C2×C711yescharacteristic
104C2×C4×SD16C2×C711yescharacteristic, complemented
105#2548C2×C711yescharacteristic, complemented
106D4×C4⋊C4C2×C711yescharacteristic, complemented
107#2065C2×C712yescomplemented
108#2042C2×C712yescomplemented
109#1957C2×C711yescharacteristic, complemented
110#1939C2×C711yescharacteristic, complemented
111#1879C2×C712yescomplemented
112#1803C2×C711yescharacteristic, complemented
113C24×C7—21no—
114C24×C7—22no—
115C2×C7×Q8—21no—
116C2×C7×Q8—21no—
117C2×C7×Q8—21no—
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8C2×D412yes—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—21no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—21no—
127C2×C7×D4—22no—
128C2×C7×D4—21no—
129C2×C7×D4—22no—
130C2×C7×D4—42no—
131C2×C7×D4—42no—
132C2×C7×D4—42no—
133C2×C7×D4—42no—
134C2×C7×D4—81no—
135C2×C7×D4—81no—
136C2×C7×D4C2×D412yes—
137C2×C7×D4C2×D412yes—
138C2×C7×D4#4112yes—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—24no—
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—44no—
145C22×C4×C7—44no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7—82no—
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2411yescharacteristic
154C7×SD16—28no—
155C7×D8—48no—
156C2×C7×C8—21no—
157C2×C7×C8—21no—
158C2×C7×C8—21no—
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C2×C7×C8—22no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8C2×D412yes—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—44no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—81no—
176C7×C4⋊C4C2×D412yes—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—44no—
179C7×C22⋊C4—44no—
180C7×C22⋊C4—44no—
181C7×C22⋊C4—82no—
182C7×C22⋊C4—82no—
183C42×C7—22no—
184C42×C7—22no—
185C42×C7—42no—
186C42×C7C2×D412yes—
187C42×C7#4112yes—
188C22×SD16C22×C711yescharacteristic
189C22×D8C22×C712yes—
190C2×#127C22×C711yescharacteristic
191C2×#124C22×C712yes—
192C2×C4×Q8C22×C711yescharacteristic
193C2×C4×D4C22×C712yes—
194C2×C4×D4C22×C711yescharacteristic
195C22×C4⋊C4—21no—
196#463—28no—
197#459—24no—
198C4×SD16—24no—
199C2×#106C22×C711yescharacteristic
200C2×C4⋊C8C22×C711yescharacteristic
201C2×C4⋊C8C22×C712yes—
202C2×#103C22×C712yes—
203C2×#103C22×C711yescharacteristic
204C2×#102C22×C712yes—
205C2×#102C22×C712yes—
206C2×#102C22×C711yescharacteristic
207C2×#102C22×C712yes—
208C2×C4×C8C22×C711yescharacteristic
209#394—22no—
210#390—21no—
211#390—21no—
212#390C22×C712yes—
213#389C22×C712yes—
214#385—22no—
215#384C22×C711yescharacteristic
216#380—21no—
217C4×C4⋊C4C22×C711yescharacteristic
218#340C22×C712yes—
219#336C22×C712yes—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7C22×D411yescharacteristic
235C7×Q8—22no—
236C7×Q8—22no—
237C7×Q8—41no—
238C7×Q8—41no—
239C7×D4—22no—
240C7×D4—22no—
241C7×D4—22no—
242C7×D4—42no—
243C7×D4—42no—
244C7×D4—42no—
245C7×D4—42no—
246C7×D4—84no—
247C7×D4—84no—
248C7×D4#12114yes—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—48no—
264C2×C4×C7—41no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7—42no—
269C2×C4×C7—41no—
270C2×C4×C7—88no—
271C2×C4×C7#12112yes—
272C2×C4×C7#12112yes—
273C2×C4×C7#13512yes—
274C2×C4×C7AGL(1,ℤ/8)12yes—
275C2×C4×C7C22×D411yescharacteristic
276C2×C4×C7C22×D411yescharacteristic
277C2×C4×C7C2×#4111yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C2×C4×C7H(5,𝔽2)11yescharacteristic
280C2×C4×C72−1+411yescharacteristic
281C7×C8—22no—
282C7×C8—42no—
283C7×C8—41no—
284C22×Q8C23×C711yescharacteristic
285C22×D4—21no—
286C22×D4—21no—
287C22×D4C23×C712yes—
288C22×D4C23×C711yescharacteristic
289C23×C4—21no—
290C23×C4—22no—
291C2×SD16—22no—
292C2×SD16—22no—
293C2×SD16C7×D414yescomplemented
294C2×D8—24no—
295C2×D8—24no—
296C2×D8—24no—
297C22×C8C23×C712yes—
298C22×C8C23×C711yescharacteristic
299C22×C8C23×C711yescharacteristic
300#127—22no—
301#127—22no—
302#124—24no—
303#124—42no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×D4—24no—
307C4×D4—22no—
308C4×D4—42no—
309C4×D4C7×D414yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C7×D411yescharacteristic
316C2×C4⋊C4C7×D411yescharacteristic
317C2×C4⋊C4C23×C712yes—
318C2×C4⋊C4C23×C711yescharacteristic
319C2×C4⋊C4C23×C711yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C42—21no—
327C2×C42C23×C712yes—
328C2×C42C23×C711yescharacteristic
329#106—22no—
330C4⋊C8—24no—
331C4⋊C8—22no—
332#103—24no—
333#103—22no—
334#102—22no—
335#102—24no—
336#102—24no—
337#102—42no—
338C4×C8—22no—
339#95—22no—
340#95C7×D412yes—
341C22×C7—24no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—82no—
349C22×C7—82no—
350C22×C7—82no—
351C22×C7—82no—
352C22×C7—82no—
353C22×C7—82no—
354C22×C7—82no—
355C22×C7C2×#12111yescharacteristic
356C22×C7D4211yescharacteristic
357C22×C7#54811yescharacteristic
358C22×C7C2×#13511yescharacteristic
359C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C7#57611yescharacteristic
361C22×C7#57711yescharacteristic
362C4×C7—22no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—42no—
366C4×C7—42no—
367C4×C7—82no—
368C4×C7—81no—
369C4×C7#46314yescomplemented
370C4×C7C2×#12111yescharacteristic
371C4×C7#53711yescharacteristic
372C4×C7#54711yescharacteristic
373C4×C7#55311yescharacteristic
374C24—21no—
375C24—22no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8C2×C7×D412yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D4C2×C7×D412yes—
398C2×D4C2×C7×D412yes—
399C2×D4C7×#4112yes—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—82no—
410C22×C4C2×C7×D412yes—
411C22×C4C2×C7×D412yes—
412C22×C4C2×C7×D411yescharacteristic
413C22×C4C2×C7×D411yescharacteristic
414C22×C4C24×C711yescharacteristic
415SD16—28no—
416D8—48no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8C2×C7×D412yes—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—81no—
437C4⋊C4C2×C7×D412yes—
438C22⋊C4—24no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C42—22no—
445C42—22no—
446C42—42no—
447C42C2×C7×D412yes—
448C42C7×#4112yes—
449C2×C7—44no—
450C2×C7—162no—
451C2×C7#241611yescharacteristic
452C2×C7C2×#46311yescharacteristic
453C2×C7#304511yescharacteristic
454C2×C7#327411yescharacteristic
455C2×C7#328411yescharacteristic
456C2×C7#336111yescharacteristic
457C2×C7#337311yescharacteristic
458C23—22no—
459C23—22no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23C22×C7×D411yescharacteristic
473Q8—22no—
474Q8—22no—
475Q8—41no—
476Q8—41no—
477D4—22no—
478D4—22no—
479D4—22no—
480D4—42no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—84no—
485D4—84no—
486D4C7×#12114yes—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—48no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—88no—
509C2×C4C7×#12112yes—
510C2×C4C7×#12112yes—
511C2×C4C7×#13512yes—
512C2×C4C7×AGL(1,ℤ/8)12yes—
513C2×C4C22×C7×D411yescharacteristic
514C2×C4C22×C7×D411yescharacteristic
515C2×C4C2×C7×#4111yescharacteristic
516C2×C4C2×C7×#4111yescharacteristic
517C2×C4C7×H(5,𝔽2)11yescharacteristic
518C2×C4C7×2−1+411yescharacteristic
519C8—22no—
520C8—42no—
521C8—41no—
522C7#2473111yescharacteristic, direct product factor, Sylow
523C22—24no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22C2×C7×#12111yescharacteristic
538C22C7×D4211yescharacteristic
539C22C7×#54811yescharacteristic
540C22C2×C7×#13511yescharacteristic
541C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
542C22C7×#57611yescharacteristic
543C22C7×#57711yescharacteristic
544C4—22no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—82no—
550C4—81no—
551C4C7×#46314yescomplemented
552C4C2×C7×#12111yescharacteristic
553C4C7×#53711yescharacteristic
554C4C7×#54711yescharacteristic
555C4C7×#55311yescharacteristic
556C2—44no—
557C2—162no—
558C2C7×#241611yescharacteristic
559C2C2×C7×#46311yescharacteristic
560C2C7×#304511yescharacteristic
561C2C7×#327411yescharacteristic
562C2C7×#328411yescharacteristic
563C2C7×#336111yescharacteristic
564C2C7×#337311yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C4×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#463C2semidirect
3. C2×C4×C7×SD16C2semidirect
4. C7×#2548C2semidirect
5. C7×D4×C4⋊C4C2semidirect
6. C7×#2065C2semidirect
7. C7×#2042C2semidirect
8. C7×#1957C2semidirect
9. C7×#1939C2semidirect
10. C7×#1879C2semidirect
11. C7×#1803C2semidirect
44. #24731C7direct (N × H)
54. C2×C7×SD16D4semidirect
102. C2×#463C2×C7semidirect
104. C2×C4×SD16C2×C7semidirect
105. #2548C2×C7semidirect
106. D4×C4⋊C4C2×C7semidirect
107. #2065C2×C7semidirect
108. #2042C2×C7semidirect
109. #1957C2×C7semidirect
110. #1939C2×C7semidirect
111. #1879C2×C7semidirect
112. #1803C2×C7semidirect
293. C2×SD16C7×D4semidirect
369. C4×C7#463semidirect
522. C7#24731direct (N × H)
551. C4C7×#463semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4632
C2×C7×#4592
C2×C4×C7×SD162
C7×#25482
C7×D4×C4⋊C42
C7×#20652
C7×#20422
C7×#19572
C7×#19392
C7×#18792
C7×#18032
#247317