Finite Group Databasebeta

← C7×#18782 #13195998

Subgroups — C7×#18782 #13195998

Order 1792.

1224 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic, complemented
2C7×#3227C211yesmaximal, characteristic, complemented
3C7×#3100C211yesmaximal, characteristic, complemented
4C7×#3036C211yesmaximal, characteristic, complemented
5C7×#1654C214yesmaximal, complemented
6C7×#1625C214yesmaximal, complemented
7C7×#1549C211yesmaximal, characteristic, complemented
8C7×#1486C211yesmaximal, characteristic, complemented
9C7×#1478C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C2×C7×AGL(1,ℤ/8)—21no—
12C7×#547—22no—
13C7×#547—22no—
14C7×#547—22no—
15C7×#547C2214yes—
16C7×#532C2211yescharacteristic
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#123—22no—
19C2×C7×C22⋊Q8—21no—
20C2×C7×#121—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C22×C7×C4⋊C4—21no—
24C7×#498C2212yescomplemented
25C7×#486C2212yescomplemented
26C7×#481—22no—
27C7×#480—21no—
28C7×#476—22no—
29C7×#474C2212yescomplemented
30C7×#463—22no—
31C7×#461C2212yescomplemented
32C7×#459C2212yescomplemented
33C7×#459C2212yescomplemented
34C7×#447—21no—
35C4×C7×D8—22no—
36C7×#431C2211yescharacteristic
37C7×#431C2211yescharacteristic
38C7×#423C2211yescharacteristic
39C7×#417—21no—
40C7×#332C2214yescomplemented
41C7×D4⋊C8C2214yescomplemented
42C7×D4⋊C8C2214yescomplemented
43#18782C711yesmaximal, characteristic, direct product factor, Sylow
44C2×C7×#41—24no—
45C2×C7×#41—24no—
46C2×C7×#41—21no—
47C2×C7×#41—48no—
48C22×C7×Q8—21no—
49C22×C7×D4—22no—
50C22×C7×D4D411yescharacteristic
51C22×C7×D4D411yescharacteristic
52C23×C4×C7—22no—
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C7×AGL(1,ℤ/8)—44no—
56C2×C7×SD16—22no—
57C2×C7×D8—22no—
58C2×C7×D8—22no—
59C2×C7×M16—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8D412yes—
63C7×C4⋊Q8C2311yescharacteristic
64C7×C4⋊Q8C2312yes—
65C7×#127C2311yescharacteristic
66C7×#124—21no—
67C7×#123—28no—
68C7×#123—28no—
69C7×C22⋊Q8—24no—
70C7×C22⋊Q8—24no—
71C7×#121—24no—
72C7×#121—24no—
73C7×#121—21no—
74C7×#121—21no—
75C4×C7×D4—21no—
76C4×C7×D4—21no—
77C4×C7×D4—21no—
78C4×C7×D4—21no—
79C4×C7×D4—21no—
80C4×C7×D4—21no—
81C4×C7×D4D412yes—
82C4×C7×D4D412yes—
83C4×C7×D4C2312yes—
84C4×C7×D4C2312yes—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4—24no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—24no—
89C2×C7×C4⋊C4D411yescharacteristic
90C2×C7×C4⋊C4D412yes—
91C2×C7×C4⋊C4D411yescharacteristic
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—21no—
94C2×C7×C22⋊C4—21no—
95C2×C42×C7C2311yescharacteristic
96C7×#107—22no—
97C7×#106—24no—
98C7×C4⋊C8—21no—
99C7×C4⋊C8—21no—
100C7×C4⋊C8C2312yes—
101C7×C4⋊C8C2312yes—
102C7×#103—22no—
103C7×#102—22no—
104C7×#102—24no—
105C7×#102—22no—
106C7×#102—22no—
107C7×C22⋊C8—21no—
108C7×C22⋊C8—21no—
109C4×C7×C8—21no—
110C4×C7×C8C2312yes—
111C2×#547C2×C711yescharacteristic, complemented
112#3227C2×C711yescharacteristic, complemented
113#3100C2×C711yescharacteristic, complemented
114#3036C2×C711yescharacteristic, complemented
115#1654C2×C714yescomplemented
116#1625C2×C714yescomplemented
117#1549C2×C711yescharacteristic, complemented
118#1486C2×C711yescharacteristic, complemented
119#1478C2×C711yescharacteristic, complemented
120C24×C7—21no—
121C24×C7—21no—
122C7×#41—416no—
123C7×#41—48no—
124C7×#41—48no—
125C7×#41—42no—
126C7×#41—42no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—21no—
130C2×C7×Q8—44no—
131C2×C7×D4—24no—
132C2×C7×D4—28no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—48no—
141C2×C7×D4—41no—
142C2×C7×D4C2×D412yes—
143C2×C7×D4C2×D412yes—
144C22×C4×C7—28no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—44no—
153C22×C4×C7—48no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—41no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C7×SD16—44no—
161C7×D8—44no—
162C7×D8—44no—
163C7×M16—42no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C7×C4⋊C4—28no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4C2×D412yes—
176C7×C4⋊C4C2×D414yes—
177C7×C4⋊C4C2×D412yes—
178C7×C22⋊C4—28no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—41no—
184C7×C22⋊C4—41no—
185C42×C7C2×D411yescharacteristic
186C42×C7C2×D411yescharacteristic
187C42×C7C2×D411yescharacteristic
188C42×C7C2411yescharacteristic
189C22×#41—21no—
190C2×AGL(1,ℤ/8)—21no—
191#547—22no—
192#547—22no—
193#547—22no—
194#547C22×C714yes—
195#532C22×C711yescharacteristic
196C2×C4⋊Q8C22×C711yescharacteristic
197C2×#123—22no—
198C2×C22⋊Q8—21no—
199C2×#121—21no—
200C2×C4×D4C22×C711yescharacteristic
201C2×C4×D4C22×C711yescharacteristic
202C22×C4⋊C4—21no—
203#498C22×C712yescomplemented
204#486C22×C712yescomplemented
205#481—22no—
206#480—21no—
207#476—22no—
208#474C22×C712yescomplemented
209#463—22no—
210#461C22×C712yescomplemented
211#459C22×C712yescomplemented
212#459C22×C712yescomplemented
213#447—21no—
214C4×D8—22no—
215#431C22×C711yescharacteristic
216#431C22×C711yescharacteristic
217#423C22×C711yescharacteristic
218#417—21no—
219#332C22×C714yescomplemented
220D4⋊C8C22×C714yescomplemented
221D4⋊C8C22×C714yescomplemented
222C23×C7—24no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—41no—
231C23×C7#12011yescharacteristic
232C7×Q8—44no—
233C7×Q8—44no—
234C7×Q8—41no—
235C7×Q8—41no—
236C7×D4—24no—
237C7×D4—22no—
238C7×D4—22no—
239C7×D4—22no—
240C7×D4—42no—
241C7×D4—416no—
242C7×D4—41no—
243C7×D4—41no—
244C7×D4—42no—
245C7×D4—42no—
246C7×D4—82no—
247C7×D4AGL(1,ℤ/8)14yescomplemented
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—41no—
253C2×C4×C7—44no—
254C2×C4×C7—42no—
255C2×C4×C7—44no—
256C2×C4×C7—416no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—44no—
261C2×C4×C7—44no—
262C2×C4×C7—41no—
263C2×C4×C7—82no—
264C2×C4×C7#12011yescharacteristic
265C2×C4×C7#12011yescharacteristic
266C2×C4×C7#12011yescharacteristic
267C2×C4×C7#13512yes—
268C2×C4×C7AGL(1,ℤ/8)12yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C7×C8—42no—
273C7×C8—42no—
274C7×C8—42no—
275C2×#41—24no—
276C2×#41—24no—
277C2×#41—21no—
278C2×#41—48no—
279C22×Q8—21no—
280C22×D4—22no—
281C22×D4C7×D411yescharacteristic
282C22×D4C7×D411yescharacteristic
283C23×C4—22no—
284C23×C4—21no—
285C23×C4—21no—
286AGL(1,ℤ/8)—44no—
287C2×SD16—22no—
288C2×D8—22no—
289C2×D8—22no—
290C2×M16—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8C7×D412yes—
294C4⋊Q8C23×C711yescharacteristic
295C4⋊Q8C23×C712yes—
296#127C23×C711yescharacteristic
297#124—21no—
298#123—28no—
299#123—28no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302#121—24no—
303#121—24no—
304#121—21no—
305#121—21no—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4C7×D412yes—
313C4×D4C7×D412yes—
314C4×D4C23×C712yes—
315C4×D4C23×C712yes—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4C7×D411yescharacteristic
321C2×C4⋊C4C7×D411yescharacteristic
322C2×C4⋊C4C7×D412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C42C23×C711yescharacteristic
327#107—22no—
328#106—24no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8C23×C712yes—
332C4⋊C8C23×C712yes—
333#103—22no—
334#102—22no—
335#102—22no—
336#102—22no—
337#102—24no—
338C22⋊C8—21no—
339C22⋊C8—21no—
340C4×C8—21no—
341C4×C8C23×C712yes—
342C22×C7—21no—
343C22×C7—21no—
344C22×C7—24no—
345C22×C7—48no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—84no—
352C22×C7—81no—
353C22×C7—81no—
354C22×C7—81no—
355C22×C7#44912yes—
356C22×C7C2×#12011yescharacteristic
357C4×C7—21no—
358C4×C7—21no—
359C4×C7—21no—
360C4×C7—42no—
361C4×C7—42no—
362C4×C7—44no—
363C4×C7—81no—
364C4×C7#44912yes—
365C4×C7C2×#13511yescharacteristic
366C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
367C4×C7C2×AGL(1,ℤ/8)12yes—
368C24—21no—
369C24—21no—
370#41—416no—
371#41—48no—
372#41—48no—
373#41—42no—
374#41—42no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—44no—
379C2×D4—24no—
380C2×D4—28no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—48no—
389C2×D4—41no—
390C2×D4C2×C7×D412yes—
391C2×D4C2×C7×D412yes—
392C22×C4—28no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408SD16—44no—
409D8—44no—
410D8—44no—
411M16—42no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C4⋊C4—28no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4C2×C7×D412yes—
424C4⋊C4C2×C7×D414yes—
425C4⋊C4C2×C7×D412yes—
426C22⋊C4—28no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42C2×C7×D411yescharacteristic
434C42C2×C7×D411yescharacteristic
435C42C2×C7×D411yescharacteristic
436C42C24×C711yescharacteristic
437C2×C7—22no—
438C2×C7—44no—
439C2×C7—82no—
440C2×C7—161no—
441C2×C7C2×#44911yescharacteristic
442C2×C7#300311yescharacteristic
443C2×C7#302011yescharacteristic
444C23—24no—
445C23—22no—
446C23—22no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23C7×#12011yescharacteristic
454Q8—44no—
455Q8—44no—
456Q8—41no—
457Q8—41no—
458D4—24no—
459D4—22no—
460D4—22no—
461D4—22no—
462D4—42no—
463D4—416no—
464D4—41no—
465D4—41no—
466D4—42no—
467D4—42no—
468D4—82no—
469D4C7×AGL(1,ℤ/8)14yescomplemented
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—416no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4C7×#12011yescharacteristic
487C2×C4C7×#12011yescharacteristic
488C2×C4C7×#12011yescharacteristic
489C2×C4C7×#13512yes—
490C2×C4C7×AGL(1,ℤ/8)12yes—
491C2×C4C22×C7×D411yescharacteristic
492C2×C4C22×C7×D411yescharacteristic
493C2×C4C22×C7×D411yescharacteristic
494C8—42no—
495C8—42no—
496C8—42no—
497C7#1878211yescharacteristic, direct product factor, Sylow
498C22—21no—
499C22—21no—
500C22—24no—
501C22—48no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—84no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22C7×#44912yes—
512C22C2×C7×#12011yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—44no—
519C4—81no—
520C4C7×#44912yes—
521C4C2×C7×#13511yescharacteristic
522C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
523C4C2×C7×AGL(1,ℤ/8)12yes—
524C2—22no—
525C2—44no—
526C2—82no—
527C2—161no—
528C2C2×C7×#44911yescharacteristic
529C2C7×#300311yescharacteristic
530C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#547C2semidirect
2. C7×#3227C2semidirect
3. C7×#3100C2semidirect
4. C7×#3036C2semidirect
5. C7×#1654C2semidirect
6. C7×#1625C2semidirect
7. C7×#1549C2semidirect
8. C7×#1486C2semidirect
9. C7×#1478C2semidirect
24. C7×#498C22semidirect
25. C7×#486C22semidirect
29. C7×#474C22semidirect
31. C7×#461C22semidirect
32. C7×#459C22semidirect
33. C7×#459C22semidirect
40. C7×#332C22semidirect
41. C7×D4⋊C8C22semidirect
42. C7×D4⋊C8C22semidirect
43. #18782C7direct (N × H)
111. C2×#547C2×C7semidirect
112. #3227C2×C7semidirect
113. #3100C2×C7semidirect
114. #3036C2×C7semidirect
115. #1654C2×C7semidirect
116. #1625C2×C7semidirect
117. #1549C2×C7semidirect
118. #1486C2×C7semidirect
119. #1478C2×C7semidirect
203. #498C22×C7semidirect
204. #486C22×C7semidirect
208. #474C22×C7semidirect
210. #461C22×C7semidirect
211. #459C22×C7semidirect
212. #459C22×C7semidirect
219. #332C22×C7semidirect
220. D4⋊C8C22×C7semidirect
221. D4⋊C8C22×C7semidirect
247. C7×D4AGL(1,ℤ/8)semidirect
469. D4C7×AGL(1,ℤ/8)semidirect
497. C7#18782direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C7×#32272
C7×#31002
C7×#30362
C7×#16542
C7×#16252
C7×#15492
C7×#14862
C7×#14782
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