Finite Group Databasebeta

← C7×#18317 #13195533

Subgroups — C7×#18317 #13195533

Order 1792.

1402 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#534C211yesmaximal, characteristic
2C7×#3193C212yesmaximal, complemented
3C7×#2561C211yesmaximal, characteristic, complemented
4C7×#1607C214yesmaximal, complemented
5C7×#1601C214yesmaximal, complemented
6C7×#1507C212yesmaximal, complemented
7C7×#1462C211yesmaximal, characteristic, complemented
8C23×C7×D4—21no—
9C7×#534—24no—
10C7×#534—22no—
11C7×#534C2214yes—
12C2×C7×#123—22no—
13C2×C7×C22⋊Q8C2211yescharacteristic, complemented
14C2×C7×#121—21no—
15C2×C7×#121C2212yes—
16C2×C7×#120—22no—
17C22×C7×C22⋊C4C2211yescharacteristic
18C7×#484—22no—
19C7×#480C2214yescomplemented
20C7×#468—24no—
21C7×#450—22no—
22C7×#448C2214yescomplemented
23C7×#407C2212yescomplemented
24C7×#397C2212yescomplemented
25C7×#393C2212yescomplemented
26C7×#388—22no—
27C7×#381—21no—
28C7×#379—21no—
29C4×C7×C22⋊C4C2211yescharacteristic, complemented
30C7×#327C2218yescomplemented
31C7×#323C2214yescomplemented
32#18317C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7—21no—
34C25×C7D411yescharacteristic
35C22×C7×Q8C2311yescharacteristic, complemented
36C22×C7×D4—22no—
37C22×C7×D4—24no—
38C22×C7×D4—22no—
39C22×C7×D4—24no—
40C22×C7×D4—21no—
41C22×C7×D4—48no—
42C22×C7×D4D412yes—
43C23×C4×C7—21no—
44C23×C4×C7D411yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C2×C7×SD16—24no—
47C2×C7×M16—22no—
48C7×#123—28no—
49C7×#123—28no—
50C7×C22⋊Q8—22no—
51C7×C22⋊Q8—22no—
52C7×#121—24no—
53C7×#121—24no—
54C7×#121—22no—
55C7×#121—22no—
56C7×#121D414yes—
57C7×#121C2314yes—
58C7×#120—28no—
59C7×#120—28no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4D412yes—
63C2×C7×C4⋊C4C2311yescharacteristic, complemented
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4D412yes—
73C2×C7×C22⋊C4C2312yes—
74C2×C42×C7—21no—
75C7×#107—24no—
76C7×#103—24no—
77C7×#102—24no—
78C7×C22⋊C8—22no—
79C7×C22⋊C8C2314yescomplemented
80C7×#95—22no—
81C7×#95—22no—
82C7×#95C2312yescomplemented
83C2×#534C2×C711yescharacteristic
84#3193C2×C712yescomplemented
85#2561C2×C711yescharacteristic, complemented
86#1607C2×C714yescomplemented
87#1601C2×C714yescomplemented
88#1507C2×C712yescomplemented
89#1462C2×C711yescharacteristic, complemented
90C24×C7—21no—
91C24×C7—21no—
92C24×C7—24no—
93C24×C7—24no—
94C24×C7—22no—
95C24×C7—44no—
96C24×C7—42no—
97C24×C7—81no—
98C24×C7—82no—
99C24×C7—82no—
100C24×C7C2×D411yescharacteristic
101C24×C7C2×D412yes—
102C2×C7×Q8—22no—
103C2×C7×Q8—81no—
104C2×C7×D4—28no—
105C2×C7×D4—24no—
106C2×C7×D4—24no—
107C2×C7×D4—24no—
108C2×C7×D4—22no—
109C2×C7×D4—48no—
110C2×C7×D4—416no—
111C2×C7×D4—48no—
112C2×C7×D4—48no—
113C2×C7×D4—48no—
114C2×C7×D4—82no—
115C2×C7×D4—82no—
116C2×C7×D4C2×D414yes—
117C22×C4×C7—24no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—44no—
125C22×C4×C7—42no—
126C22×C4×C7—81no—
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2×D412yes—
131C22×C4×C7C2411yescharacteristic
132C7×SD16—84no—
133C7×M16—82no—
134C2×C7×C8—24no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—42no—
139C7×C4⋊C4C2×D414yes—
140C7×C22⋊C4—28no—
141C7×C22⋊C4—24no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C7×C22⋊C4—24no—
145C7×C22⋊C4—42no—
146C7×C22⋊C4—41no—
147C7×C22⋊C4—41no—
148C7×C22⋊C4—44no—
149C7×C22⋊C4—82no—
150C42×C7—81no—
151C23×D4—21no—
152#534—24no—
153#534—22no—
154#534C22×C714yes—
155C2×#123—22no—
156C2×C22⋊Q8C22×C711yescharacteristic, complemented
157C2×#121—21no—
158C2×#121C22×C712yes—
159C2×#120—22no—
160C22×C22⋊C4C22×C711yescharacteristic
161#484—22no—
162#480C22×C714yescomplemented
163#468—24no—
164#450—22no—
165#448C22×C714yescomplemented
166#407C22×C712yescomplemented
167#397C22×C712yescomplemented
168#393C22×C712yescomplemented
169#388—22no—
170#381—21no—
171#379—21no—
172C4×C22⋊C4C22×C711yescharacteristic, complemented
173#327C22×C718yescomplemented
174#323C22×C714yescomplemented
175C23×C7—24no—
176C23×C7—24no—
177C23×C7—22no—
178C23×C7—22no—
179C23×C7—22no—
180C23×C7—22no—
181C23×C7—24no—
182C23×C7—41no—
183C23×C7—42no—
184C23×C7—42no—
185C23×C7—42no—
186C23×C7—44no—
187C23×C7—48no—
188C23×C7—44no—
189C23×C7—42no—
190C23×C7—44no—
191C23×C7—81no—
192C23×C7—81no—
193C23×C7—88no—
194C23×C7—84no—
195C23×C7—84no—
196C23×C7—816no—
197C23×C7#12011yescharacteristic
198C23×C7#12011yescharacteristic
199C23×C7AGL(1,ℤ/8)14yes—
200C23×C7C22×D411yescharacteristic
201C7×Q8—44no—
202C7×D4—44no—
203C7×D4—416no—
204C7×D4—48no—
205C7×D4—48no—
206C7×D4—82no—
207C7×D4—84no—
208C2×C4×C7—22no—
209C2×C4×C7—24no—
210C2×C4×C7—41no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7—41no—
214C2×C4×C7—44no—
215C2×C4×C7—48no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—81no—
226C2×C4×C7#12012yes—
227C2×C4×C7C22×D412yes—
228C7×C8—44no—
229C25—21no—
230C25C7×D411yescharacteristic
231C22×Q8C23×C711yescharacteristic, complemented
232C22×D4—22no—
233C22×D4—24no—
234C22×D4—22no—
235C22×D4—24no—
236C22×D4—21no—
237C22×D4—48no—
238C22×D4C7×D412yes—
239C23×C4—21no—
240C23×C4C7×D411yescharacteristic
241C23×C4C23×C711yescharacteristic
242C2×SD16—24no—
243C2×M16—22no—
244#123—28no—
245#123—28no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248#121—24no—
249#121—24no—
250#121—22no—
251#121—22no—
252#121C7×D414yes—
253#121C23×C714yes—
254#120—28no—
255#120—28no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C7×D412yes—
259C2×C4⋊C4C23×C711yescharacteristic, complemented
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4C7×D412yes—
269C2×C22⋊C4C23×C712yes—
270C2×C42—21no—
271#107—24no—
272#103—24no—
273#102—24no—
274C22⋊C8—22no—
275C22⋊C8C23×C714yescomplemented
276#95—22no—
277#95—22no—
278#95C23×C712yescomplemented
279C22×C7—21no—
280C22×C7—21no—
281C22×C7—21no—
282C22×C7—21no—
283C22×C7—24no—
284C22×C7—21no—
285C22×C7—21no—
286C22×C7—21no—
287C22×C7—44no—
288C22×C7—44no—
289C22×C7—48no—
290C22×C7—42no—
291C22×C7—42no—
292C22×C7—44no—
293C22×C7—44no—
294C22×C7—44no—
295C22×C7—82no—
296C22×C7—88no—
297C22×C7—84no—
298C22×C7—816no—
299C22×C7#45512yes—
300C22×C7C2×#12011yescharacteristic
301C22×C7C2×AGL(1,ℤ/8)12yes—
302C4×C7—22no—
303C4×C7—22no—
304C4×C7—44no—
305C4×C7—42no—
306C4×C7—82no—
307C4×C7—81no—
308C4×C7—81no—
309C24—21no—
310C24—21no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—44no—
315C24—42no—
316C24—81no—
317C24—82no—
318C24—82no—
319C24C2×C7×D411yescharacteristic
320C24C2×C7×D412yes—
321C2×Q8—22no—
322C2×Q8—81no—
323C2×D4—28no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—48no—
329C2×D4—416no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—82no—
334C2×D4—82no—
335C2×D4C2×C7×D414yes—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—81no—
346C22×C4C2×C7×D412yes—
347C22×C4C2×C7×D411yescharacteristic
348C22×C4C2×C7×D412yes—
349C22×C4C2×C7×D412yes—
350C22×C4C24×C711yescharacteristic
351SD16—84no—
352M16—82no—
353C2×C8—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—42no—
358C4⋊C4C2×C7×D414yes—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—42no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—44no—
368C22⋊C4—82no—
369C42—81no—
370C2×C7—22no—
371C2×C7—22no—
372C2×C7—41no—
373C2×C7—44no—
374C2×C7—84no—
375C2×C7#300311yescharacteristic
376C2×C7C2×#45511yescharacteristic
377C2×C7#302611yescharacteristic
378C23—24no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—48no—
391C23—44no—
392C23—42no—
393C23—44no—
394C23—81no—
395C23—81no—
396C23—88no—
397C23—84no—
398C23—84no—
399C23—816no—
400C23C7×#12011yescharacteristic
401C23C7×#12011yescharacteristic
402C23C7×AGL(1,ℤ/8)14yes—
403C23C22×C7×D411yescharacteristic
404Q8—44no—
405D4—44no—
406D4—416no—
407D4—48no—
408D4—48no—
409D4—82no—
410D4—84no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—48no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—81no—
429C2×C4C7×#12012yes—
430C2×C4C22×C7×D412yes—
431C8—44no—
432C7#1831711yescharacteristic, direct product factor, Sylow
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—44no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—82no—
450C22—88no—
451C22—84no—
452C22—816no—
453C22C7×#45512yes—
454C22C2×C7×#12011yescharacteristic
455C22C2×C7×AGL(1,ℤ/8)12yes—
456C4—22no—
457C4—22no—
458C4—44no—
459C4—42no—
460C4—82no—
461C4—81no—
462C4—81no—
463C2—22no—
464C2—22no—
465C2—41no—
466C2—44no—
467C2—84no—
468C2C7×#300311yescharacteristic
469C2C2×C7×#45511yescharacteristic
470C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#534C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3193C2semidirect
3. C7×#2561C2semidirect
4. C7×#1607C2semidirect
5. C7×#1601C2semidirect
6. C7×#1507C2semidirect
7. C7×#1462C2semidirect
13. C2×C7×C22⋊Q8C22semidirect
19. C7×#480C22semidirect
22. C7×#448C22semidirect
23. C7×#407C22semidirect
24. C7×#397C22semidirect
25. C7×#393C22semidirect
29. C4×C7×C22⋊C4C22semidirect
30. C7×#327C22semidirect
31. C7×#323C22semidirect
32. #18317C7direct (N × H)
35. C22×C7×Q8C23semidirect
63. C2×C7×C4⋊C4C23semidirect
79. C7×C22⋊C8C23semidirect
82. C7×#95C23semidirect
84. #3193C2×C7semidirect
85. #2561C2×C7semidirect
86. #1607C2×C7semidirect
87. #1601C2×C7semidirect
88. #1507C2×C7semidirect
89. #1462C2×C7semidirect
156. C2×C22⋊Q8C22×C7semidirect
162. #480C22×C7semidirect
165. #448C22×C7semidirect
166. #407C22×C7semidirect
167. #397C22×C7semidirect
168. #393C22×C7semidirect
172. C4×C22⋊C4C22×C7semidirect
173. #327C22×C7semidirect
174. #323C22×C7semidirect
231. C22×Q8C23×C7semidirect
259. C2×C4⋊C4C23×C7semidirect
275. C22⋊C8C23×C7semidirect
278. #95C23×C7semidirect
432. C7#18317direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5342
C7×#31932
C7×#25612
C7×#16072
C7×#16012
C7×#15072
C7×#14622
#183177