Finite Group Databasebeta

← C7×#24727 #13201943

Subgroups — C7×#24727 #13201943

Order 1792.

1050 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#463C212yesmaximal
2C2×C7×#461C211yesmaximal, characteristic, complemented
3C2×C4×C7×D8C211yesmaximal, characteristic
4C7×#2547C211yesmaximal, characteristic, complemented
5C7×D4×C4⋊C4C211yesmaximal, characteristic, complemented
6C7×#2068C212yesmaximal, complemented
7C7×#2042C212yesmaximal, complemented
8C7×#1956C211yesmaximal, characteristic, complemented
9C7×#1939C211yesmaximal, characteristic, complemented
10C7×#1877C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2212yes—
13C22×C7×D8C2211yescharacteristic
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#124C2212yes—
16C2×C4×C7×D4C2211yescharacteristic
17C2×C4×C7×D4C2212yes—
18C2×C4×C7×D4C2211yescharacteristic
19C22×C7×C4⋊C4—21no—
20C7×#463—28no—
21C7×#461—24no—
22C4×C7×D8—24no—
23C2×C7×#107C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×C4⋊C8C2212yes—
26C2×C7×#103C2212yes—
27C2×C7×#103C2212yes—
28C2×C7×#103C2212yes—
29C2×C7×#102C2211yescharacteristic
30C2×C7×#102C2211yescharacteristic
31C2×C7×#102C2212yes—
32C2×C4×C7×C8C2211yescharacteristic
33C7×#397—22no—
34C7×#391C2212yes—
35C7×#389C2212yes—
36C7×#385—22no—
37C7×#384C2211yescharacteristic
38C7×#380—21no—
39C7×#380—21no—
40C7×#380—21no—
41C4×C7×C4⋊C4C2211yescharacteristic
42C7×#340C2212yes—
43C7×#336C2212yes—
44#24727C711yesmaximal, characteristic, direct product factor, Sylow
45C22×C7×Q8C2312yes—
46C22×C7×D4C2311yescharacteristic
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C23×C4×C7—21no—
50C23×C4×C7—22no—
51C2×C7×SD16—24no—
52C2×C7×SD16—24no—
53C2×C7×SD16—24no—
54C2×C7×D8—22no—
55C2×C7×D8—22no—
56C2×C7×D8D414yes—
57C22×C7×C8C2311yescharacteristic
58C22×C7×C8C2312yes—
59C22×C7×C8C2311yescharacteristic
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8—22no—
62C7×#124—24no—
63C7×#124—42no—
64C4×C7×D4—24no—
65C4×C7×D4—22no—
66C4×C7×D4—22no—
67C4×C7×D4—22no—
68C4×C7×D4—42no—
69C4×C7×D4D414yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—21no—
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—21no—
84C2×C7×C22⋊C4—22no—
85C2×C7×C22⋊C4—22no—
86C2×C42×C7C2312yes—
87C2×C42×C7C2311yescharacteristic
88C7×#107—22no—
89C7×C4⋊C8—24no—
90C7×C4⋊C8—22no—
91C7×#103—24no—
92C7×#103—24no—
93C7×#103—42no—
94C7×#102—22no—
95C7×#102—24no—
96C7×#102—22no—
97C4×C7×C8—22no—
98C7×#95—22no—
99C7×#95—22no—
100C7×#95—22no—
101C7×#95D412yes—
102C2×#463C2×C712yes—
103C2×#461C2×C711yescharacteristic, complemented
104C2×C4×D8C2×C711yescharacteristic
105#2547C2×C711yescharacteristic, complemented
106D4×C4⋊C4C2×C711yescharacteristic, complemented
107#2068C2×C712yescomplemented
108#2042C2×C712yescomplemented
109#1956C2×C711yescharacteristic, complemented
110#1939C2×C711yescharacteristic, complemented
111#1877C2×C712yescomplemented
112#1802C2×C711yescharacteristic, complemented
113C24×C7—21no—
114C24×C7—21no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4C2×D412yes—
130C2×C7×D4C2×D412yes—
131C2×C7×D4C2×D412yes—
132C2×C7×D4#4112yes—
133C22×C4×C7—21no—
134C22×C4×C7—24no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—44no—
139C22×C4×C7—44no—
140C22×C4×C7—42no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—82no—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2411yescharacteristic
151C7×SD16—48no—
152C7×D8—28no—
153C2×C7×C8—21no—
154C2×C7×C8—21no—
155C2×C7×C8—21no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8C2×D412yes—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—44no—
170C7×C4⋊C4—42no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—42no—
173C7×C4⋊C4—81no—
174C7×C4⋊C4C2×D412yes—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—44no—
177C7×C22⋊C4—44no—
178C7×C22⋊C4—44no—
179C7×C22⋊C4—42no—
180C42×C7—22no—
181C42×C7—22no—
182C42×C7C2×D412yes—
183C42×C7#4112yes—
184C22×SD16C22×C712yes—
185C22×D8C22×C711yescharacteristic
186C2×C4⋊Q8C22×C711yescharacteristic
187C2×#124C22×C712yes—
188C2×C4×D4C22×C711yescharacteristic
189C2×C4×D4C22×C712yes—
190C2×C4×D4C22×C711yescharacteristic
191C22×C4⋊C4—21no—
192#463—28no—
193#461—24no—
194C4×D8—24no—
195C2×#107C22×C711yescharacteristic
196C2×C4⋊C8C22×C711yescharacteristic
197C2×C4⋊C8C22×C712yes—
198C2×#103C22×C712yes—
199C2×#103C22×C712yes—
200C2×#103C22×C712yes—
201C2×#102C22×C712yes—
202C2×#102C22×C711yescharacteristic
203C2×#102C22×C711yescharacteristic
204C2×C4×C8C22×C711yescharacteristic
205#397—22no—
206#391C22×C712yes—
207#389C22×C712yes—
208#385—22no—
209#384C22×C711yescharacteristic
210#380—21no—
211#380—21no—
212#380—21no—
213C4×C4⋊C4C22×C711yescharacteristic
214#340C22×C712yes—
215#336C22×C712yes—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—22no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7C22×D411yescharacteristic
231C7×Q8—42no—
232C7×Q8—42no—
233C7×Q8—42no—
234C7×Q8—42no—
235C7×D4—22no—
236C7×D4—22no—
237C7×D4—22no—
238C7×D4—22no—
239C7×D4—22no—
240C7×D4—41no—
241C7×D4—41no—
242C7×D4#12114yes—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—48no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—88no—
265C2×C4×C7—84no—
266C2×C4×C7#12112yes—
267C2×C4×C7#12112yes—
268C2×C4×C7#13512yes—
269C2×C4×C7AGL(1,ℤ/8)12yes—
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7H(5,𝔽2)11yescharacteristic
275C2×C4×C72−1+411yescharacteristic
276C7×C8—22no—
277C7×C8—42no—
278C7×C8—41no—
279C22×Q8C23×C712yes—
280C22×D4C23×C711yescharacteristic
281C22×D4C23×C711yescharacteristic
282C23×C4—21no—
283C23×C4—21no—
284C23×C4—22no—
285C2×SD16—24no—
286C2×SD16—24no—
287C2×SD16—24no—
288C2×D8—22no—
289C2×D8—22no—
290C2×D8C7×D414yes—
291C22×C8C23×C712yes—
292C22×C8C23×C711yescharacteristic
293C22×C8C23×C711yescharacteristic
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296#124—24no—
297#124—42no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—42no—
303C4×D4C7×D414yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C7×D411yescharacteristic
311C2×C4⋊C4C7×D411yescharacteristic
312C2×C4⋊C4C23×C711yescharacteristic
313C2×C4⋊C4C23×C712yes—
314C2×C4⋊C4C23×C711yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C42C23×C712yes—
321C2×C42C23×C711yescharacteristic
322#107—22no—
323C4⋊C8—24no—
324C4⋊C8—22no—
325#103—24no—
326#103—24no—
327#103—42no—
328#102—22no—
329#102—24no—
330#102—22no—
331C4×C8—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95C7×D412yes—
336C22×C7—24no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—81no—
347C22×C7—81no—
348C22×C7—81no—
349C22×C7—81no—
350C22×C7C2×#12111yescharacteristic
351C22×C7D4211yescharacteristic
352C22×C7#54811yescharacteristic
353C22×C7C2×#13511yescharacteristic
354C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
355C22×C7#57711yescharacteristic
356C22×C7#57811yescharacteristic
357C4×C7—22no—
358C4×C7—42no—
359C4×C7—42no—
360C4×C7—42no—
361C4×C7—82no—
362C4×C7—82no—
363C4×C7—81no—
364C4×C7#46314yescomplemented
365C4×C7C2×#12111yescharacteristic
366C4×C7#53711yescharacteristic
367C4×C7#54711yescharacteristic
368C4×C7#55311yescharacteristic
369C24—21no—
370C24—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4C2×C7×D412yes—
386C2×D4C2×C7×D412yes—
387C2×D4C2×C7×D412yes—
388C2×D4C7×#4112yes—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4C2×C7×D412yes—
403C22×C4C2×C7×D412yes—
404C22×C4C2×C7×D411yescharacteristic
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C24×C711yescharacteristic
407SD16—48no—
408D8—28no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8C2×C7×D412yes—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—81no—
430C4⋊C4C2×C7×D412yes—
431C22⋊C4—24no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—42no—
436C42—22no—
437C42—22no—
438C42C2×C7×D412yes—
439C42C7×#4112yes—
440C2×C7—44no—
441C2×C7—82no—
442C2×C7#241611yescharacteristic
443C2×C7C2×#46311yescharacteristic
444C2×C7#304611yescharacteristic
445C2×C7#327211yescharacteristic
446C2×C7#328311yescharacteristic
447C2×C7#336011yescharacteristic
448C2×C7#336711yescharacteristic
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23C22×C7×D411yescharacteristic
464Q8—42no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468D4—22no—
469D4—22no—
470D4—22no—
471D4—22no—
472D4—22no—
473D4—41no—
474D4—41no—
475D4C7×#12114yes—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—48no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—88no—
498C2×C4—84no—
499C2×C4C7×#12112yes—
500C2×C4C7×#12112yes—
501C2×C4C7×#13512yes—
502C2×C4C7×AGL(1,ℤ/8)12yes—
503C2×C4C22×C7×D411yescharacteristic
504C2×C4C22×C7×D411yescharacteristic
505C2×C4C2×C7×#4111yescharacteristic
506C2×C4C2×C7×#4111yescharacteristic
507C2×C4C7×H(5,𝔽2)11yescharacteristic
508C2×C4C7×2−1+411yescharacteristic
509C8—22no—
510C8—42no—
511C8—41no—
512C7#2472711yescharacteristic, direct product factor, Sylow
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22C2×C7×#12111yescharacteristic
528C22C7×D4211yescharacteristic
529C22C7×#54811yescharacteristic
530C22C2×C7×#13511yescharacteristic
531C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
532C22C7×#57711yescharacteristic
533C22C7×#57811yescharacteristic
534C4—22no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—82no—
539C4—82no—
540C4—81no—
541C4C7×#46314yescomplemented
542C4C2×C7×#12111yescharacteristic
543C4C7×#53711yescharacteristic
544C4C7×#54711yescharacteristic
545C4C7×#55311yescharacteristic
546C2—44no—
547C2—82no—
548C2C7×#241611yescharacteristic
549C2C2×C7×#46311yescharacteristic
550C2C7×#304611yescharacteristic
551C2C7×#327211yescharacteristic
552C2C7×#328311yescharacteristic
553C2C7×#336011yescharacteristic
554C2C7×#336711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×D8C22
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
2. C2×C7×#461C2semidirect
4. C7×#2547C2semidirect
5. C7×D4×C4⋊C4C2semidirect
6. C7×#2068C2semidirect
7. C7×#2042C2semidirect
8. C7×#1956C2semidirect
9. C7×#1939C2semidirect
10. C7×#1877C2semidirect
11. C7×#1802C2semidirect
44. #24727C7direct (N × H)
103. C2×#461C2×C7semidirect
105. #2547C2×C7semidirect
106. D4×C4⋊C4C2×C7semidirect
107. #2068C2×C7semidirect
108. #2042C2×C7semidirect
109. #1956C2×C7semidirect
110. #1939C2×C7semidirect
111. #1877C2×C7semidirect
112. #1802C2×C7semidirect
364. C4×C7#463semidirect
512. C7#24727direct (N × H)
541. 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×#4612
C2×C4×C7×D82
C7×#25472
C7×D4×C4⋊C42
C7×#20682
C7×#20422
C7×#19562
C7×#19392
C7×#18772
C7×#18022
#247277