Finite Group Databasebeta

← C7×#18316 #13195532

Subgroups — C7×#18316 #13195532

Order 1792.

1516 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#534C211yesmaximal, characteristic, complemented
2C7×#3192C212yesmaximal, complemented
3C7×#2539C211yesmaximal, characteristic, complemented
4C7×#1613C214yesmaximal, complemented
5C7×#1595C214yesmaximal, complemented
6C7×#1507C212yesmaximal, complemented
7C7×#1462C211yesmaximal, characteristic, complemented
8C23×C7×D4—21no—
9C7×#534—24no—
10C7×#534—22no—
11C7×#534C2214yes—
12C2×C7×#123—21no—
13C2×C7×#123—22no—
14C2×C7×#121—21no—
15C2×C7×#121C2212yes—
16C2×C7×#121C2211yescharacteristic, complemented
17C2×C7×#120—22no—
18C2×C7×#120—21no—
19C22×C7×C22⋊C4C2211yescharacteristic
20C7×#481—22no—
21C7×#481C2214yescomplemented
22C7×#469—24no—
23C7×#447—22no—
24C7×#447C2214yescomplemented
25C7×#407C2212yescomplemented
26C7×#399C2212yescomplemented
27C7×#392C2212yescomplemented
28C7×#390—22no—
29C4×C7×C22⋊C4C2211yescharacteristic, complemented
30C7×#327C2218yescomplemented
31C7×#323C2214yescomplemented
32#18316C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7—21no—
34C25×C7D411yescharacteristic
35C22×C7×D4—22no—
36C22×C7×D4—24no—
37C22×C7×D4—22no—
38C22×C7×D4—24no—
39C22×C7×D4—21no—
40C22×C7×D4—21no—
41C22×C7×D4—22no—
42C22×C7×D4—48no—
43C22×C7×D4D412yescomplemented
44C22×C7×D4C2311yescharacteristic, complemented
45C23×C4×C7—21no—
46C23×C4×C7D411yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C2×C7×D8—24no—
49C2×C7×M16—22no—
50C7×#123—28no—
51C7×#123—28no—
52C7×#123—44no—
53C7×#121—24no—
54C7×#121—24no—
55C7×#121—22no—
56C7×#121—22no—
57C7×#121—22no—
58C7×#121—22no—
59C7×#121D414yescomplemented
60C7×#121C2314yes—
61C7×#120—28no—
62C7×#120—28no—
63C7×#120—44no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4D412yescomplemented
67C2×C7×C4⋊C4C2311yescharacteristic, complemented
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4D412yes—
77C2×C7×C22⋊C4C2312yes—
78C2×C42×C7—21no—
79C7×#106—24no—
80C7×#102—24no—
81C7×#102—24no—
82C7×C22⋊C8—22no—
83C7×C22⋊C8C2314yescomplemented
84C7×#95C2312yescomplemented
85C2×#534C2×C711yescharacteristic, complemented
86#3192C2×C712yescomplemented
87#2539C2×C711yescharacteristic, complemented
88#1613C2×C714yescomplemented
89#1595C2×C714yescomplemented
90#1507C2×C712yescomplemented
91#1462C2×C711yescharacteristic, complemented
92C24×C7—21no—
93C24×C7—21no—
94C24×C7—24no—
95C24×C7—24no—
96C24×C7—22no—
97C24×C7—21no—
98C24×C7—44no—
99C24×C7—42no—
100C24×C7—81no—
101C24×C7—82no—
102C24×C7—82no—
103C24×C7C2×D411yescharacteristic
104C24×C7C2×D412yes—
105C2×C7×D4—28no—
106C2×C7×D4—24no—
107C2×C7×D4—24no—
108C2×C7×D4—24no—
109C2×C7×D4—22no—
110C2×C7×D4—22no—
111C2×C7×D4—44no—
112C2×C7×D4—48no—
113C2×C7×D4—416no—
114C2×C7×D4—48no—
115C2×C7×D4—48no—
116C2×C7×D4—48no—
117C2×C7×D4—42no—
118C2×C7×D4—42no—
119C2×C7×D4—41no—
120C2×C7×D4—41no—
121C2×C7×D4—44no—
122C2×C7×D4—81no—
123C2×C7×D4—82no—
124C2×C7×D4—82no—
125C2×C7×D4—82no—
126C2×C7×D4C2×D414yes—
127C22×C4×C7—24no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—44no—
134C22×C4×C7—42no—
135C22×C4×C7—81no—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2411yescharacteristic
141C7×D8—84no—
142C7×M16—82no—
143C2×C7×C8—24no—
144C7×C4⋊C4—24no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—44no—
148C7×C4⋊C4C2×D414yes—
149C7×C22⋊C4—28no—
150C7×C22⋊C4—24no—
151C7×C22⋊C4—24no—
152C7×C22⋊C4—24no—
153C7×C22⋊C4—24no—
154C7×C22⋊C4—24no—
155C7×C22⋊C4—44no—
156C7×C22⋊C4—44no—
157C7×C22⋊C4—42no—
158C42×C7—81no—
159C23×D4—21no—
160#534—24no—
161#534—22no—
162#534C22×C714yes—
163C2×#123—21no—
164C2×#123—22no—
165C2×#121—21no—
166C2×#121C22×C712yes—
167C2×#121C22×C711yescharacteristic, complemented
168C2×#120—22no—
169C2×#120—21no—
170C22×C22⋊C4C22×C711yescharacteristic
171#481—22no—
172#481C22×C714yescomplemented
173#469—24no—
174#447—22no—
175#447C22×C714yescomplemented
176#407C22×C712yescomplemented
177#399C22×C712yescomplemented
178#392C22×C712yescomplemented
179#390—22no—
180C4×C22⋊C4C22×C711yescharacteristic, complemented
181#327C22×C718yescomplemented
182#323C22×C714yescomplemented
183C23×C7—24no—
184C23×C7—24no—
185C23×C7—22no—
186C23×C7—22no—
187C23×C7—22no—
188C23×C7—22no—
189C23×C7—24no—
190C23×C7—41no—
191C23×C7—42no—
192C23×C7—42no—
193C23×C7—42no—
194C23×C7—44no—
195C23×C7—48no—
196C23×C7—44no—
197C23×C7—41no—
198C23×C7—41no—
199C23×C7—42no—
200C23×C7—42no—
201C23×C7—44no—
202C23×C7—41no—
203C23×C7—42no—
204C23×C7—81no—
205C23×C7—81no—
206C23×C7—88no—
207C23×C7—84no—
208C23×C7—84no—
209C23×C7—816no—
210C23×C7#12011yescharacteristic
211C23×C7#12011yescharacteristic
212C23×C7AGL(1,ℤ/8)14yes—
213C23×C7C22×D411yescharacteristic
214C7×D4—44no—
215C7×D4—44no—
216C7×D4—416no—
217C7×D4—48no—
218C7×D4—48no—
219C7×D4—82no—
220C7×D4—84no—
221C7×D4—88no—
222C7×D4—82no—
223C7×D4—82no—
224C2×C4×C7—24no—
225C2×C4×C7—22no—
226C2×C4×C7—24no—
227C2×C4×C7—41no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—44no—
233C2×C4×C7—48no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—81no—
241C2×C4×C7#12012yes—
242C2×C4×C7C22×D412yes—
243C7×C8—44no—
244C25—21no—
245C25C7×D411yescharacteristic
246C22×D4—22no—
247C22×D4—24no—
248C22×D4—22no—
249C22×D4—24no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—48no—
254C22×D4C7×D412yescomplemented
255C22×D4C23×C711yescharacteristic, complemented
256C23×C4—21no—
257C23×C4C7×D411yescharacteristic
258C23×C4C23×C711yescharacteristic
259C2×D8—24no—
260C2×M16—22no—
261#123—28no—
262#123—28no—
263#123—44no—
264#121—24no—
265#121—24no—
266#121—22no—
267#121—22no—
268#121—22no—
269#121—22no—
270#121C7×D414yescomplemented
271#121C23×C714yes—
272#120—28no—
273#120—28no—
274#120—44no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4C7×D412yescomplemented
278C2×C4⋊C4C23×C711yescharacteristic, complemented
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4C7×D412yes—
288C2×C22⋊C4C23×C712yes—
289C2×C42—21no—
290#106—24no—
291#102—24no—
292#102—24no—
293C22⋊C8—22no—
294C22⋊C8C23×C714yescomplemented
295#95C23×C712yescomplemented
296C22×C7—21no—
297C22×C7—21no—
298C22×C7—21no—
299C22×C7—21no—
300C22×C7—24no—
301C22×C7—21no—
302C22×C7—21no—
303C22×C7—21no—
304C22×C7—44no—
305C22×C7—44no—
306C22×C7—48no—
307C22×C7—42no—
308C22×C7—42no—
309C22×C7—44no—
310C22×C7—44no—
311C22×C7—44no—
312C22×C7—82no—
313C22×C7—88no—
314C22×C7—82no—
315C22×C7—82no—
316C22×C7—84no—
317C22×C7—816no—
318C22×C7—81no—
319C22×C7—81no—
320C22×C7—81no—
321C22×C7#45312yes—
322C22×C7C2×#12011yescharacteristic
323C22×C7C2×AGL(1,ℤ/8)12yes—
324C4×C7—22no—
325C4×C7—22no—
326C4×C7—44no—
327C4×C7—44no—
328C4×C7—42no—
329C4×C7—81no—
330C24—21no—
331C24—21no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—21no—
336C24—44no—
337C24—42no—
338C24—81no—
339C24—82no—
340C24—82no—
341C24C2×C7×D411yescharacteristic
342C24C2×C7×D412yes—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—416no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—44no—
360C2×D4—81no—
361C2×D4—82no—
362C2×D4—82no—
363C2×D4—82no—
364C2×D4C2×C7×D414yes—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—81no—
374C22×C4C2×C7×D412yes—
375C22×C4C2×C7×D411yescharacteristic
376C22×C4C2×C7×D412yes—
377C22×C4C2×C7×D412yes—
378C22×C4C24×C711yescharacteristic
379D8—84no—
380M16—82no—
381C2×C8—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—44no—
386C4⋊C4C2×C7×D414yes—
387C22⋊C4—28no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—42no—
396C42—81no—
397C2×C7—22no—
398C2×C7—22no—
399C2×C7—41no—
400C2×C7—44no—
401C2×C7—84no—
402C2×C7—161no—
403C2×C7#300311yescharacteristic
404C2×C7C2×#45311yescharacteristic
405C2×C7#302611yescharacteristic
406C23—24no—
407C23—24no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—41no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—48no—
419C23—44no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—41no—
426C23—42no—
427C23—81no—
428C23—81no—
429C23—88no—
430C23—84no—
431C23—84no—
432C23—816no—
433C23C7×#12011yescharacteristic
434C23C7×#12011yescharacteristic
435C23C7×AGL(1,ℤ/8)14yes—
436C23C22×C7×D411yescharacteristic
437D4—44no—
438D4—44no—
439D4—416no—
440D4—48no—
441D4—48no—
442D4—82no—
443D4—84no—
444D4—88no—
445D4—82no—
446D4—82no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—81no—
464C2×C4C7×#12012yes—
465C2×C4C22×C7×D412yes—
466C8—44no—
467C7#1831611yescharacteristic, direct product factor, Sylow
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—24no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—44no—
477C22—44no—
478C22—48no—
479C22—42no—
480C22—42no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—82no—
485C22—88no—
486C22—82no—
487C22—82no—
488C22—84no—
489C22—816no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22C7×#45312yes—
494C22C2×C7×#12011yescharacteristic
495C22C2×C7×AGL(1,ℤ/8)12yes—
496C4—22no—
497C4—22no—
498C4—44no—
499C4—44no—
500C4—42no—
501C4—81no—
502C2—22no—
503C2—22no—
504C2—41no—
505C2—44no—
506C2—84no—
507C2—161no—
508C2C7×#300311yescharacteristic
509C2C2×C7×#45311yescharacteristic
510C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
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
1. C2×C7×#534C2semidirect
2. C7×#3192C2semidirect
3. C7×#2539C2semidirect
4. C7×#1613C2semidirect
5. C7×#1595C2semidirect
6. C7×#1507C2semidirect
7. C7×#1462C2semidirect
16. C2×C7×#121C22semidirect
21. C7×#481C22semidirect
24. C7×#447C22semidirect
25. C7×#407C22semidirect
26. C7×#399C22semidirect
27. C7×#392C22semidirect
29. C4×C7×C22⋊C4C22semidirect
30. C7×#327C22semidirect
31. C7×#323C22semidirect
32. #18316C7direct (N × H)
43. C22×C7×D4D4semidirect
44. C22×C7×D4C23semidirect
59. C7×#121D4semidirect
66. C2×C7×C4⋊C4D4semidirect
67. C2×C7×C4⋊C4C23semidirect
83. C7×C22⋊C8C23semidirect
84. C7×#95C23semidirect
85. C2×#534C2×C7semidirect
86. #3192C2×C7semidirect
87. #2539C2×C7semidirect
88. #1613C2×C7semidirect
89. #1595C2×C7semidirect
90. #1507C2×C7semidirect
91. #1462C2×C7semidirect
167. C2×#121C22×C7semidirect
172. #481C22×C7semidirect
175. #447C22×C7semidirect
176. #407C22×C7semidirect
177. #399C22×C7semidirect
178. #392C22×C7semidirect
180. C4×C22⋊C4C22×C7semidirect
181. #327C22×C7semidirect
182. #323C22×C7semidirect
254. C22×D4C7×D4semidirect
255. C22×D4C23×C7semidirect
270. #121C7×D4semidirect
277. C2×C4⋊C4C7×D4semidirect
278. C2×C4⋊C4C23×C7semidirect
294. C22⋊C8C23×C7semidirect
295. #95C23×C7semidirect
467. C7#18316direct (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×#31922
C7×#25392
C7×#16132
C7×#15952
C7×#15072
C7×#14622
#183167