Finite Group Databasebeta

← C7×#18757 #13195973

Subgroups — C7×#18757 #13195973

Order 1792.

1170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2C7×#3229C211yesmaximal, characteristic, complemented
3C7×#3101C211yesmaximal, characteristic, complemented
4C7×#3077C211yesmaximal, characteristic, complemented
5C7×#1654C214yesmaximal, complemented
6C7×#1635C214yesmaximal, complemented
7C7×#1546C211yesmaximal, characteristic, complemented
8C7×#1486C211yesmaximal, characteristic, complemented
9C7×#1478C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C7×#547—22no—
12C7×#547—22no—
13C7×#547—22no—
14C7×#547C2214yes—
15C7×#533C2211yescharacteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#123—22no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×#121—21no—
20C2×C4×C7×D4C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C22×C7×C4⋊C4—21no—
23C7×#498C2212yescomplemented
24C7×#487C2212yescomplemented
25C7×#483—22no—
26C7×#483—21no—
27C7×#478—22no—
28C7×#476C2212yescomplemented
29C7×#476C2212yescomplemented
30C7×#475—22no—
31C7×#474C2212yescomplemented
32C7×#461C2212yescomplemented
33C7×#450—21no—
34C4×C7×SD16—22no—
35C7×#431C2211yescharacteristic
36C7×#431C2211yescharacteristic
37C7×#428—21no—
38C7×#423C2211yescharacteristic
39C7×#418—21no—
40C7×#332C2214yescomplemented
41C7×D4⋊C8C2214yescomplemented
42C7×D4⋊C8C2214yescomplemented
43#18757C711yesmaximal, characteristic, direct product factor, Sylow
44C2×C7×#41—24no—
45C2×C7×#41—24no—
46C2×C7×#41—44no—
47C2×C7×#41—44no—
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—
55C2×C7×SD16—22no—
56C2×C7×M16—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8D412yes—
60C7×C4⋊Q8C2311yescharacteristic
61C7×C4⋊Q8C2311yescharacteristic
62C7×C4⋊Q8C2312yes—
63C7×#125—21no—
64C7×#123—28no—
65C7×#123—28no—
66C7×C22⋊Q8—24no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—21no—
70C7×#121—24no—
71C7×#121—24no—
72C4×C7×Q8—21no—
73C4×C7×Q8—21no—
74C4×C7×D4—21no—
75C4×C7×D4—21no—
76C4×C7×D4—21no—
77C4×C7×D4—21no—
78C4×C7×D4D412yes—
79C4×C7×D4D412yes—
80C4×C7×D4C2312yes—
81C4×C7×D4C2312yes—
82C7×#117—21no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—24no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—24no—
87C2×C7×C4⋊C4D411yescharacteristic
88C2×C7×C4⋊C4D412yes—
89C2×C7×C4⋊C4D411yescharacteristic
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—21no—
93C2×C42×C7C2311yescharacteristic
94C7×#107—22no—
95C7×#106—24no—
96C7×#106—22no—
97C7×#106—22no—
98C7×C4⋊C8—21no—
99C7×C4⋊C8—21no—
100C7×C4⋊C8C2312yes—
101C7×C4⋊C8C2312yes—
102C7×#103—24no—
103C7×#103—22no—
104C7×#102—22no—
105C7×#102—22no—
106C7×#102—22no—
107C7×C22⋊C8—21no—
108C7×C22⋊C8—21no—
109C4×C7×C8—21no—
110C4×C7×C8C2312yes—
111C2×#547C2×C711yescharacteristic
112#3229C2×C711yescharacteristic, complemented
113#3101C2×C711yescharacteristic, complemented
114#3077C2×C711yescharacteristic, complemented
115#1654C2×C714yescomplemented
116#1635C2×C714yescomplemented
117#1546C2×C711yescharacteristic, complemented
118#1486C2×C711yescharacteristic, complemented
119#1478C2×C711yescharacteristic, complemented
120C24×C7—21no—
121C24×C7—21no—
122C7×#41—48no—
123C7×#41—416no—
124C7×#41—48no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—42no—
130C2×C7×Q8—42no—
131C2×C7×D4—28no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—22no—
135C2×C7×D4—42no—
136C2×C7×D4—48no—
137C2×C7×D4SD1614yescomplemented
138C2×C7×D4C2×D412yes—
139C2×C7×D4C2×D412yes—
140C22×C4×C7—28no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—44no—
149C22×C4×C7—48no—
150C22×C4×C7—42no—
151C22×C4×C7—42no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C7×SD16—44no—
156C7×M16—42no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—41no—
171C7×C4⋊C4—41no—
172C7×C4⋊C4C2×D412yes—
173C7×C4⋊C4C2×D414yes—
174C7×C4⋊C4C2×D412yes—
175C7×C22⋊C4—28no—
176C7×C22⋊C4—22no—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—41no—
181C42×C7—41no—
182C42×C7C2×D411yescharacteristic
183C42×C7C2×D411yescharacteristic
184C42×C7C2×D411yescharacteristic
185C42×C7C2411yescharacteristic
186C22×#41—21no—
187#547—22no—
188#547—22no—
189#547—22no—
190#547C22×C714yes—
191#533C22×C711yescharacteristic
192C2×C4⋊Q8C22×C711yescharacteristic
193C2×#123—22no—
194C2×C22⋊Q8—21no—
195C2×#121—21no—
196C2×C4×D4C22×C711yescharacteristic
197C2×C4×D4C22×C711yescharacteristic
198C22×C4⋊C4—21no—
199#498C22×C712yescomplemented
200#487C22×C712yescomplemented
201#483—21no—
202#483—22no—
203#478—22no—
204#476C22×C712yescomplemented
205#476C22×C712yescomplemented
206#475—22no—
207#474C22×C712yescomplemented
208#461C22×C712yescomplemented
209#450—21no—
210C4×SD16—22no—
211#431C22×C711yescharacteristic
212#431C22×C711yescharacteristic
213#428—21no—
214#423C22×C711yescharacteristic
215#418—21no—
216#332C22×C714yescomplemented
217D4⋊C8C22×C714yescomplemented
218D4⋊C8C22×C714yescomplemented
219C23×C7—22no—
220C23×C7—24no—
221C23×C7—22no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7#12011yescharacteristic
228C7×Q8—42no—
229C7×Q8—42no—
230C7×Q8—42no—
231C7×Q8—44no—
232C7×Q8—42no—
233C7×D4—22no—
234C7×D4—22no—
235C7×D4—22no—
236C7×D4—24no—
237C7×D4—42no—
238C7×D4—416no—
239C7×D4C2×SD1614yes—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—24no—
244C2×C4×C7—41no—
245C2×C4×C7—44no—
246C2×C4×C7—42no—
247C2×C4×C7—416no—
248C2×C4×C7—42no—
249C2×C4×C7—44no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—44no—
253C2×C4×C7—44no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7#12011yescharacteristic
257C2×C4×C7#12011yescharacteristic
258C2×C4×C7#12011yescharacteristic
259C2×C4×C7C2×SD1612yes—
260C2×C4×C7AGL(1,ℤ/8)12yes—
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C22×D411yescharacteristic
264C7×C8—42no—
265C7×C8—42no—
266C7×C8—42no—
267C2×#41—24no—
268C2×#41—24no—
269C2×#41—44no—
270C2×#41—44no—
271C22×Q8—21no—
272C22×D4—22no—
273C22×D4C7×D411yescharacteristic
274C22×D4C7×D411yescharacteristic
275C23×C4—22no—
276C23×C4—21no—
277C23×C4—21no—
278C2×SD16—22no—
279C2×M16—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8C7×D412yes—
283C4⋊Q8C23×C711yescharacteristic
284C4⋊Q8C23×C711yescharacteristic
285C4⋊Q8C23×C712yes—
286#125—21no—
287#123—28no—
288#123—28no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293#121—24no—
294#121—24no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4C7×D412yes—
302C4×D4C7×D412yes—
303C4×D4C23×C712yes—
304C4×D4C23×C712yes—
305#117—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4C7×D411yescharacteristic
311C2×C4⋊C4C7×D411yescharacteristic
312C2×C4⋊C4C7×D412yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C42C23×C711yescharacteristic
317#107—22no—
318#106—22no—
319#106—22no—
320#106—24no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8C23×C712yes—
324C4⋊C8C23×C712yes—
325#103—22no—
326#103—24no—
327#102—22no—
328#102—22no—
329#102—22no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C4×C8—21no—
333C4×C8C23×C712yes—
334C22×C7—21no—
335C22×C7—21no—
336C22×C7—24no—
337C22×C7—48no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—84no—
344C22×C7#45012yes—
345C22×C7C2×#12011yescharacteristic
346C4×C7—21no—
347C4×C7—21no—
348C4×C7—21no—
349C4×C7—42no—
350C4×C7—44no—
351C4×C7—42no—
352C4×C7—81no—
353C4×C7—81no—
354C4×C7#45012yes—
355C4×C7C22×SD1611yescharacteristic
356C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
357C4×C7#57512yes—
358C24—21no—
359C24—21no—
360#41—48no—
361#41—416no—
362#41—48no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×D4—28no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—48no—
375C2×D4C7×SD1614yescomplemented
376C2×D4C2×C7×D412yes—
377C2×D4C2×C7×D412yes—
378C22×C4—28no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4C2×C7×D411yescharacteristic
391C22×C4C2×C7×D411yescharacteristic
392C22×C4C2×C7×D411yescharacteristic
393SD16—44no—
394M16—42no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C4⋊C4—28no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4C2×C7×D412yes—
411C4⋊C4C2×C7×D414yes—
412C4⋊C4C2×C7×D412yes—
413C22⋊C4—28no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—41no—
419C42—41no—
420C42C2×C7×D411yescharacteristic
421C42C2×C7×D411yescharacteristic
422C42C2×C7×D411yescharacteristic
423C42C24×C711yescharacteristic
424C2×C7—22no—
425C2×C7—44no—
426C2×C7—82no—
427C2×C7C2×#45011yescharacteristic
428C2×C7#300511yescharacteristic
429C2×C7#302011yescharacteristic
430C23—22no—
431C23—24no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23C7×#12011yescharacteristic
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—44no—
443Q8—42no—
444D4—22no—
445D4—22no—
446D4—22no—
447D4—24no—
448D4—42no—
449D4—416no—
450D4C2×C7×SD1614yes—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—416no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4C7×#12011yescharacteristic
468C2×C4C7×#12011yescharacteristic
469C2×C4C7×#12011yescharacteristic
470C2×C4C2×C7×SD1612yes—
471C2×C4C7×AGL(1,ℤ/8)12yes—
472C2×C4C22×C7×D411yescharacteristic
473C2×C4C22×C7×D411yescharacteristic
474C2×C4C22×C7×D411yescharacteristic
475C8—42no—
476C8—42no—
477C8—42no—
478C7#1875711yescharacteristic, direct product factor, Sylow
479C22—21no—
480C22—21no—
481C22—24no—
482C22—48no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22C7×#45012yes—
490C22C2×C7×#12011yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—44no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4C7×#45012yes—
500C4C22×C7×SD1611yescharacteristic
501C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
502C4C7×#57512yes—
503C2—22no—
504C2—44no—
505C2—82no—
506C2C2×C7×#45011yescharacteristic
507C2C7×#300511yescharacteristic
508C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×#547C2
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
2. C7×#3229C2semidirect
3. C7×#3101C2semidirect
4. C7×#3077C2semidirect
5. C7×#1654C2semidirect
6. C7×#1635C2semidirect
7. C7×#1546C2semidirect
8. C7×#1486C2semidirect
9. C7×#1478C2semidirect
23. C7×#498C22semidirect
24. C7×#487C22semidirect
28. C7×#476C22semidirect
29. C7×#476C22semidirect
31. C7×#474C22semidirect
32. C7×#461C22semidirect
40. C7×#332C22semidirect
41. C7×D4⋊C8C22semidirect
42. C7×D4⋊C8C22semidirect
43. #18757C7direct (N × H)
112. #3229C2×C7semidirect
113. #3101C2×C7semidirect
114. #3077C2×C7semidirect
115. #1654C2×C7semidirect
116. #1635C2×C7semidirect
117. #1546C2×C7semidirect
118. #1486C2×C7semidirect
119. #1478C2×C7semidirect
137. C2×C7×D4SD16semidirect
199. #498C22×C7semidirect
200. #487C22×C7semidirect
204. #476C22×C7semidirect
205. #476C22×C7semidirect
207. #474C22×C7semidirect
208. #461C22×C7semidirect
216. #332C22×C7semidirect
217. D4⋊C8C22×C7semidirect
218. D4⋊C8C22×C7semidirect
375. C2×D4C7×SD16semidirect
478. C7#18757direct (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×#32292
C7×#31012
C7×#30772
C7×#16542
C7×#16352
C7×#15462
C7×#14862
C7×#14782
#187577