Finite Group Databasebeta

← C7×#18752 #13195968

Subgroups — C7×#18752 #13195968

Order 1792.

1020 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic, complemented
2C7×#3206C211yesmaximal, characteristic, complemented
3C7×#3100C211yesmaximal, characteristic
4C7×#3037C211yesmaximal, characteristic, complemented
5C7×#1644C214yesmaximal, complemented
6C7×#1622C214yesmaximal, complemented
7C7×#1548C211yesmaximal, characteristic, complemented
8C7×#1486C211yesmaximal, characteristic, complemented
9C7×#1479C211yesmaximal, characteristic, complemented
10C2×C7×#137—21no—
11C7×#554—22no—
12C7×#554—22no—
13C7×#554—22no—
14C7×#554C2214yes—
15C7×#532C2211yescharacteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#125—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8—22no—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4—21no—
22C2×C4×C7×D4C2211yescharacteristic
23C22×C7×C4⋊C4—21no—
24C7×#492C2212yescomplemented
25C7×#487C2212yescomplemented
26C7×#480—21no—
27C7×#474C2212yes—
28C7×#464—22no—
29C7×#462C2212yescomplemented
30C7×#461—22no—
31C7×#460C2212yescomplemented
32C7×#459C2212yes—
33C7×#448—22no—
34C7×#447—21no—
35C4×C7×D8—22no—
36C7×#431C2211yescharacteristic
37C7×#431C2211yescharacteristic
38C7×#423C2211yescharacteristic
39C7×#417—21no—
40C7×#333C2214yescomplemented
41C7×Q8⋊C8C2214yescomplemented
42C7×D4⋊C8C2214yes—
43#18752C711yesmaximal, characteristic, direct product factor, Sylow
44C2×C7×#41—21no—
45C22×C7×Q8D411yescharacteristic
46C22×C7×Q8D412yes—
47C22×C7×D4D411yescharacteristic
48C23×C4×C7—22no—
49C23×C4×C7—21no—
50C7×#137—44no—
51C2×C7×Q16—22no—
52C2×C7×SD16—22no—
53C2×C7×SD16—24no—
54C2×C7×D8—22no—
55C2×C7×M16—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8D412yes—
59C7×C4⋊Q8C2311yescharacteristic
60C7×C4⋊Q8C2312yes—
61C7×#127C2311yescharacteristic
62C7×#125—24no—
63C7×#125—24no—
64C7×#124—21no—
65C7×C22⋊Q8—24no—
66C7×C22⋊Q8—28no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—28no—
69C7×#121—21no—
70C7×#121—21no—
71C4×C7×Q8—21no—
72C4×C7×Q8—21no—
73C4×C7×Q8D412yes—
74C4×C7×Q8C2312yes—
75C4×C7×D4—24no—
76C4×C7×D4—24no—
77C4×C7×D4—21no—
78C4×C7×D4—21no—
79C4×C7×D4—21no—
80C4×C7×D4—21no—
81C4×C7×D4D412yes—
82C4×C7×D4C2312yes—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—24no—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—24no—
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4D411yescharacteristic
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—21no—
94C2×C42×C7—21no—
95C2×C42×C7C2311yescharacteristic
96C7×#107—22no—
97C7×C4⋊C8—21no—
98C7×C4⋊C8—21no—
99C7×C4⋊C8C2312yes—
100C7×C4⋊C8C2312yes—
101C7×#103—24no—
102C7×#103—22no—
103C7×#102—22no—
104C7×#102—22no—
105C7×#102—22no—
106C7×C22⋊C8—21no—
107C7×C22⋊C8—21no—
108C4×C7×C8—21no—
109C4×C7×C8C2312yes—
110C2×#554C2×C711yescharacteristic, complemented
111#3206C2×C711yescharacteristic, complemented
112#3100C2×C711yescharacteristic
113#3037C2×C711yescharacteristic, complemented
114#1644C2×C714yescomplemented
115#1622C2×C714yescomplemented
116#1548C2×C711yescharacteristic, complemented
117#1486C2×C711yescharacteristic, complemented
118#1479C2×C711yescharacteristic, complemented
119C24×C7—21no—
120C7×#41—42no—
121C7×#41—42no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—21no—
127C2×C7×Q8—82no—
128C2×C7×Q8C2×D412yes—
129C2×C7×Q8C2×D414yes—
130C2×C7×D4—24no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—41no—
136C2×C7×D4C2×D412yes—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—22no—
141C22×C4×C7—28no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7—41no—
147C22×C4×C7—84no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C7×Q16—44no—
152C7×SD16—44no—
153C7×SD16—84no—
154C7×D8—44no—
155C7×M16—42no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C7×C4⋊C4—28no—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4C2×D412yes—
172C7×C4⋊C4C2×D412yes—
173C7×C22⋊C4—28no—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—22no—
176C7×C22⋊C4—41no—
177C7×C22⋊C4—41no—
178C42×C7—22no—
179C42×C7—22no—
180C42×C7C2×D411yescharacteristic
181C42×C7C2×D411yescharacteristic
182C42×C7C2×D411yescharacteristic
183C42×C7C2411yescharacteristic
184C2×#137—21no—
185#554—22no—
186#554—22no—
187#554—22no—
188#554C22×C714yes—
189#532C22×C711yescharacteristic
190C2×C4⋊Q8C22×C711yescharacteristic
191C2×#125—21no—
192C2×C22⋊Q8—21no—
193C2×C22⋊Q8—22no—
194C2×C4×Q8C22×C711yescharacteristic
195C2×C4×D4—21no—
196C2×C4×D4C22×C711yescharacteristic
197C22×C4⋊C4—21no—
198#492C22×C712yescomplemented
199#487C22×C712yescomplemented
200#480—21no—
201#474C22×C712yes—
202#464—22no—
203#462C22×C712yescomplemented
204#461—22no—
205#460C22×C712yescomplemented
206#459C22×C712yes—
207#448—22no—
208#447—21no—
209C4×D8—22no—
210#431C22×C711yescharacteristic
211#431C22×C711yescharacteristic
212#423C22×C711yescharacteristic
213#417—21no—
214#333C22×C714yescomplemented
215Q8⋊C8C22×C714yescomplemented
216D4⋊C8C22×C714yes—
217C23×C7—24no—
218C23×C7—22no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—41no—
222C23×C7#12011yescharacteristic
223C7×Q8—22no—
224C7×Q8—22no—
225C7×Q8—44no—
226C7×Q8—42no—
227C7×Q8—41no—
228C7×Q8—41no—
229C7×Q8—82no—
230C7×D4—24no—
231C7×D4—22no—
232C7×D4—42no—
233C7×D4—41no—
234C7×D4—41no—
235C7×D4—42no—
236C7×D4—82no—
237C7×D4#13714yescomplemented
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—24no—
243C2×C4×C7—41no—
244C2×C4×C7—42no—
245C2×C4×C7—44no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—44no—
251C2×C4×C7—41no—
252C2×C4×C7—88no—
253C2×C4×C7—82no—
254C2×C4×C7#12011yescharacteristic
255C2×C4×C7#12011yescharacteristic
256C2×C4×C7#12011yescharacteristic
257C2×C4×C7#13512yes—
258C2×C4×C7AGL(1,ℤ/8)12yes—
259C2×C4×C7C22×D411yescharacteristic
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C22×D411yescharacteristic
262C7×C8—42no—
263C7×C8—42no—
264C7×C8—42no—
265C2×#41—21no—
266C22×Q8C7×D411yescharacteristic
267C22×Q8C7×D412yes—
268C22×D4C7×D411yescharacteristic
269C23×C4—22no—
270C23×C4—21no—
271#137—44no—
272C2×Q16—22no—
273C2×SD16—22no—
274C2×SD16—24no—
275C2×D8—22no—
276C2×M16—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8C7×D412yes—
280C4⋊Q8C23×C711yescharacteristic
281C4⋊Q8C23×C712yes—
282#127C23×C711yescharacteristic
283#125—24no—
284#125—24no—
285#124—21no—
286C22⋊Q8—24no—
287C22⋊Q8—28no—
288C22⋊Q8—24no—
289C22⋊Q8—28no—
290#121—21no—
291#121—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8C7×D412yes—
295C4×Q8C23×C712yes—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4C7×D412yes—
303C4×D4C23×C712yes—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4C7×D411yescharacteristic
312C2×C4⋊C4C7×D411yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C42—21no—
316C2×C42C23×C711yescharacteristic
317#107—22no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8C23×C712yes—
321C4⋊C8C23×C712yes—
322#103—24no—
323#103—22no—
324#102—22no—
325#102—22no—
326#102—22no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C4×C8—21no—
330C4×C8C23×C712yes—
331C22×C7—21no—
332C22×C7—21no—
333C22×C7—24no—
334C22×C7—48no—
335C22×C7—42no—
336C22×C7—42no—
337C22×C7—81no—
338C22×C7—81no—
339C22×C7—81no—
340C22×C7#44912yes—
341C22×C7C2×#12011yescharacteristic
342C4×C7—21no—
343C4×C7—21no—
344C4×C7—21no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7—42no—
348C4×C7—44no—
349C4×C7—81no—
350C4×C7#44912yes—
351C4×C7C2×#13511yescharacteristic
352C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
353C4×C7C2×#13712yes—
354C24—21no—
355#41—42no—
356#41—42no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—82no—
363C2×Q8C2×C7×D412yes—
364C2×Q8C2×C7×D414yes—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—41no—
371C2×D4C2×C7×D412yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—84no—
383C22×C4C2×C7×D411yescharacteristic
384C22×C4C2×C7×D411yescharacteristic
385C22×C4C2×C7×D411yescharacteristic
386Q16—44no—
387SD16—44no—
388SD16—84no—
389D8—44no—
390M16—42no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4C2×C7×D412yes—
407C4⋊C4C2×C7×D412yes—
408C22⋊C4—28no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C42—22no—
414C42—22no—
415C42C2×C7×D411yescharacteristic
416C42C2×C7×D411yescharacteristic
417C42C2×C7×D411yescharacteristic
418C42C24×C711yescharacteristic
419C2×C7—22no—
420C2×C7—44no—
421C2×C7—161no—
422C2×C7C2×#44911yescharacteristic
423C2×C7#300411yescharacteristic
424C2×C7#301911yescharacteristic
425C23—24no—
426C23—22no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23C7×#12011yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—44no—
434Q8—42no—
435Q8—41no—
436Q8—41no—
437Q8—82no—
438D4—24no—
439D4—22no—
440D4—42no—
441D4—41no—
442D4—41no—
443D4—42no—
444D4—82no—
445D4C7×#13714yescomplemented
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—41no—
460C2×C4—88no—
461C2×C4—82no—
462C2×C4C7×#12011yescharacteristic
463C2×C4C7×#12011yescharacteristic
464C2×C4C7×#12011yescharacteristic
465C2×C4C7×#13512yes—
466C2×C4C7×AGL(1,ℤ/8)12yes—
467C2×C4C22×C7×D411yescharacteristic
468C2×C4C22×C7×D411yescharacteristic
469C2×C4C22×C7×D411yescharacteristic
470C8—42no—
471C8—42no—
472C8—42no—
473C7#1875211yescharacteristic, direct product factor, Sylow
474C22—21no—
475C22—21no—
476C22—24no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22C7×#44912yes—
484C22C2×C7×#12011yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—44no—
492C4—81no—
493C4C7×#44912yes—
494C4C2×C7×#13511yescharacteristic
495C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
496C4C2×C7×#13712yes—
497C2—22no—
498C2—44no—
499C2—161no—
500C2C2×C7×#44911yescharacteristic
501C2C7×#300411yescharacteristic
502C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3100C2
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×#554C2semidirect
2. C7×#3206C2semidirect
4. C7×#3037C2semidirect
5. C7×#1644C2semidirect
6. C7×#1622C2semidirect
7. C7×#1548C2semidirect
8. C7×#1486C2semidirect
9. C7×#1479C2semidirect
24. C7×#492C22semidirect
25. C7×#487C22semidirect
29. C7×#462C22semidirect
31. C7×#460C22semidirect
40. C7×#333C22semidirect
41. C7×Q8⋊C8C22semidirect
43. #18752C7direct (N × H)
110. C2×#554C2×C7semidirect
111. #3206C2×C7semidirect
113. #3037C2×C7semidirect
114. #1644C2×C7semidirect
115. #1622C2×C7semidirect
116. #1548C2×C7semidirect
117. #1486C2×C7semidirect
118. #1479C2×C7semidirect
198. #492C22×C7semidirect
199. #487C22×C7semidirect
203. #462C22×C7semidirect
205. #460C22×C7semidirect
214. #333C22×C7semidirect
215. Q8⋊C8C22×C7semidirect
237. C7×D4#137semidirect
445. D4C7×#137semidirect
473. C7#18752direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C7×#32062
C7×#31002
C7×#30372
C7×#16442
C7×#16222
C7×#15482
C7×#14862
C7×#14792
#187527