Finite Group Databasebeta

← C7×#18132 #13195348

Subgroups — C7×#18132 #13195348

Order 1792.

1654 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#534C211yesmaximal, characteristic, complemented
2C7×#3006C212yesmaximal, complemented
3C7×#2404C211yesmaximal, characteristic, complemented
4C7×#1601C214yesmaximal, complemented
5C7×#1595C214yesmaximal, complemented
6C7×#1510C212yesmaximal, complemented
7C7×#1462C211yesmaximal, characteristic, complemented
8C22×C7×#41C2211yescharacteristic, complemented
9C23×C7×D4—21no—
10C2×C7×AGL(1,ℤ/8)—22no—
11C7×#534—24no—
12C7×#534—22no—
13C7×#534C2214yes—
14C2×C7×#123—22no—
15C2×C7×#121—21no—
16C2×C7×#121C2212yes—
17C2×C7×#120—22no—
18C2×C7×#120—21no—
19C22×C7×C22⋊C4C2211yescharacteristic
20C7×#449—24no—
21C7×#448C2214yescomplemented
22C7×#447C2214yescomplemented
23C7×#417—22no—
24C7×#407C2212yescomplemented
25C7×#393C2212yescomplemented
26C7×#392C2212yescomplemented
27C7×#386—21no—
28C7×#386—22no—
29C7×#381C2211yescharacteristic, complemented
30C7×#327C2218yescomplemented
31C7×#323C2214yescomplemented
32#18132C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7—21no—
34C25×C7D411yescharacteristic
35C2×C7×#41—22no—
36C2×C7×#41—22no—
37C2×C7×#41—44no—
38C22×C7×Q8C2311yescharacteristic, complemented
39C22×C7×D4—22no—
40C22×C7×D4—24no—
41C22×C7×D4—22no—
42C22×C7×D4—24no—
43C22×C7×D4—21no—
44C22×C7×D4—21no—
45C22×C7×D4—22no—
46C22×C7×D4—48no—
47C22×C7×D4D412yescomplemented
48C22×C7×D4C2311yescharacteristic, complemented
49C23×C4×C7—21no—
50C23×C4×C7—21no—
51C23×C4×C7D411yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C7×AGL(1,ℤ/8)—48no—
54C2×C7×SD16—24no—
55C2×C7×D8—24no—
56C2×C7×M16—22no—
57C7×#123—28no—
58C7×#123—28no—
59C7×#121—24no—
60C7×#121—24no—
61C7×#121—22no—
62C7×#121—22no—
63C7×#121D414yescomplemented
64C7×#121C2314yes—
65C7×#120—28no—
66C7×#120—28no—
67C7×#120—44no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4D412yescomplemented
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⋊C4—22no—
77C2×C7×C22⋊C4D412yes—
78C2×C7×C22⋊C4C2312yes—
79C7×#103—24no—
80C7×#102—24no—
81C7×C22⋊C8—22no—
82C7×C22⋊C8C2314yescomplemented
83C7×#95—21no—
84C7×#95—22no—
85C7×#95C2312yescomplemented
86C2×#534C2×C711yescharacteristic, complemented
87#3006C2×C712yescomplemented
88#2404C2×C711yescharacteristic, complemented
89#1601C2×C714yescomplemented
90#1595C2×C714yescomplemented
91#1510C2×C712yescomplemented
92#1462C2×C711yescharacteristic, complemented
93C24×C7—21no—
94C24×C7—21no—
95C24×C7—24no—
96C24×C7—24no—
97C24×C7—22no—
98C24×C7—21no—
99C24×C7—44no—
100C24×C7—42no—
101C24×C7—81no—
102C24×C7—82no—
103C24×C7—82no—
104C24×C7C2×D411yescharacteristic
105C24×C7C2×D412yes—
106C7×#41—28no—
107C7×#41—44no—
108C7×#41—44no—
109C7×#41—44no—
110C2×C7×Q8—22no—
111C2×C7×Q8—42no—
112C2×C7×D4—28no—
113C2×C7×D4—24no—
114C2×C7×D4—24no—
115C2×C7×D4—24no—
116C2×C7×D4—24no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4—44no—
120C2×C7×D4—42no—
121C2×C7×D4—48no—
122C2×C7×D4—416no—
123C2×C7×D4—48no—
124C2×C7×D4—48no—
125C2×C7×D4—42no—
126C2×C7×D4—41no—
127C2×C7×D4—41no—
128C2×C7×D4—48no—
129C2×C7×D4—44no—
130C2×C7×D4—44no—
131C2×C7×D4—82no—
132C2×C7×D4—82no—
133C2×C7×D4C2×D414yes—
134C22×C4×C7—24no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—42no—
141C22×C4×C7—42no—
142C22×C4×C7—44no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—81no—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2411yescharacteristic
151C7×SD16—48no—
152C7×D8—48no—
153C7×M16—44no—
154C2×C7×C8—24no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4C2×D414yes—
158C7×C22⋊C4—28no—
159C7×C22⋊C4—24no—
160C7×C22⋊C4—24no—
161C7×C22⋊C4—24no—
162C7×C22⋊C4—24no—
163C7×C22⋊C4—44no—
164C7×C22⋊C4—42no—
165C7×C22⋊C4—82no—
166C22×#41C22×C711yescharacteristic, complemented
167C23×D4—21no—
168C2×AGL(1,ℤ/8)—22no—
169#534—24no—
170#534—22no—
171#534C22×C714yes—
172C2×#123—22no—
173C2×#121—21no—
174C2×#121C22×C712yes—
175C2×#120—22no—
176C2×#120—21no—
177C22×C22⋊C4C22×C711yescharacteristic
178#449—24no—
179#448C22×C714yescomplemented
180#447C22×C714yescomplemented
181#417—22no—
182#407C22×C712yescomplemented
183#393C22×C712yescomplemented
184#392C22×C712yescomplemented
185#386—21no—
186#386—22no—
187#381C22×C711yescharacteristic, complemented
188#327C22×C718yescomplemented
189#323C22×C714yescomplemented
190C23×C7—24no—
191C23×C7—24no—
192C23×C7—22no—
193C23×C7—22no—
194C23×C7—22no—
195C23×C7—22no—
196C23×C7—22no—
197C23×C7—24no—
198C23×C7—41no—
199C23×C7—42no—
200C23×C7—42no—
201C23×C7—42no—
202C23×C7—42no—
203C23×C7—41no—
204C23×C7—44no—
205C23×C7—48no—
206C23×C7—44no—
207C23×C7—42no—
208C23×C7—41no—
209C23×C7—42no—
210C23×C7—44no—
211C23×C7—81no—
212C23×C7—81no—
213C23×C7—88no—
214C23×C7—84no—
215C23×C7—84no—
216C23×C7—816no—
217C23×C7#12011yescharacteristic
218C23×C7#12011yescharacteristic
219C23×C7AGL(1,ℤ/8)14yescomplemented
220C23×C7C22×D411yescharacteristic
221C7×Q8—24no—
222C7×Q8—42no—
223C7×D4—24no—
224C7×D4—44no—
225C7×D4—42no—
226C7×D4—416no—
227C7×D4—48no—
228C7×D4—44no—
229C7×D4—48no—
230C7×D4—44no—
231C7×D4—82no—
232C7×D4—84no—
233C7×D4—88no—
234C2×C4×C7—24no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—24no—
238C2×C4×C7—41no—
239C2×C4×C7—42no—
240C2×C4×C7—41no—
241C2×C4×C7—48no—
242C2×C4×C7—44no—
243C2×C4×C7—48no—
244C2×C4×C7—41no—
245C2×C4×C7—42no—
246C2×C4×C7—41no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—81no—
252C2×C4×C7#12012yes—
253C2×C4×C7C22×D412yes—
254C7×C8—44no—
255C25—21no—
256C25C7×D411yescharacteristic
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—44no—
260C22×Q8C23×C711yescharacteristic, complemented
261C22×D4—22no—
262C22×D4—24no—
263C22×D4—21no—
264C22×D4—22no—
265C22×D4—24no—
266C22×D4—21no—
267C22×D4—22no—
268C22×D4—48no—
269C22×D4C7×D412yescomplemented
270C22×D4C23×C711yescharacteristic, complemented
271C23×C4—21no—
272C23×C4—21no—
273C23×C4C7×D411yescharacteristic
274C23×C4C23×C711yescharacteristic
275AGL(1,ℤ/8)—48no—
276C2×SD16—24no—
277C2×D8—24no—
278C2×M16—22no—
279#123—28no—
280#123—28no—
281#121—24no—
282#121—24no—
283#121—22no—
284#121—22no—
285#121C7×D414yescomplemented
286#121C23×C714yes—
287#120—28no—
288#120—28no—
289#120—44no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C7×D412yescomplemented
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4C7×D412yes—
300C2×C22⋊C4C23×C712yes—
301#103—24no—
302#102—24no—
303C22⋊C8—22no—
304C22⋊C8C23×C714yescomplemented
305#95—21no—
306#95—22no—
307#95C23×C712yescomplemented
308C22×C7—21no—
309C22×C7—21no—
310C22×C7—21no—
311C22×C7—21no—
312C22×C7—24no—
313C22×C7—21no—
314C22×C7—21no—
315C22×C7—21no—
316C22×C7—44no—
317C22×C7—44no—
318C22×C7—48no—
319C22×C7—44no—
320C22×C7—42no—
321C22×C7—42no—
322C22×C7—44no—
323C22×C7—44no—
324C22×C7—42no—
325C22×C7—44no—
326C22×C7—82no—
327C22×C7—82no—
328C22×C7—88no—
329C22×C7—84no—
330C22×C7—816no—
331C22×C7—81no—
332C22×C7—81no—
333C22×C7B(4,𝔽2)12yes—
334C22×C7C2×#12011yescharacteristic
335C22×C7C2×AGL(1,ℤ/8)12yes—
336C4×C7—22no—
337C4×C7—22no—
338C4×C7—44no—
339C4×C7—42no—
340C4×C7—42no—
341C4×C7—82no—
342C24—21no—
343C24—21no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—21no—
348C24—44no—
349C24—42no—
350C24—81no—
351C24—82no—
352C24—82no—
353C24C2×C7×D411yescharacteristic
354C24C2×C7×D412yes—
355#41—28no—
356#41—44no—
357#41—44no—
358#41—44no—
359C2×Q8—22no—
360C2×Q8—42no—
361C2×D4—28no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—44no—
369C2×D4—42no—
370C2×D4—48no—
371C2×D4—416no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4C2×C7×D414yes—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—81no—
395C22×C4C2×C7×D412yes—
396C22×C4C2×C7×D412yes—
397C22×C4C2×C7×D411yescharacteristic
398C22×C4C2×C7×D412yes—
399C22×C4C24×C711yescharacteristic
400SD16—48no—
401D8—48no—
402M16—44no—
403C2×C8—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4C2×C7×D414yes—
407C22⋊C4—28no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—82no—
415C2×C7—22no—
416C2×C7—22no—
417C2×C7—41no—
418C2×C7—44no—
419C2×C7—82no—
420C2×C7—84no—
421C2×C7#300311yescharacteristic
422C2×C7#301911yescharacteristic
423C2×C7C2×B(4,𝔽2)11yescharacteristic
424C23—24no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—44no—
439C23—48no—
440C23—44no—
441C23—42no—
442C23—41no—
443C23—42no—
444C23—44no—
445C23—81no—
446C23—81no—
447C23—88no—
448C23—84no—
449C23—84no—
450C23—816no—
451C23C7×#12011yescharacteristic
452C23C7×#12011yescharacteristic
453C23C7×AGL(1,ℤ/8)14yescomplemented
454C23C22×C7×D411yescharacteristic
455Q8—24no—
456Q8—42no—
457D4—24no—
458D4—44no—
459D4—42no—
460D4—416no—
461D4—48no—
462D4—44no—
463D4—48no—
464D4—44no—
465D4—82no—
466D4—84no—
467D4—88no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—48no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—81no—
486C2×C4C7×#12012yes—
487C2×C4C22×C7×D412yes—
488C8—44no—
489C7#1813211yescharacteristic, direct product factor, Sylow
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—24no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—44no—
499C22—44no—
500C22—48no—
501C22—44no—
502C22—42no—
503C22—42no—
504C22—44no—
505C22—44no—
506C22—42no—
507C22—44no—
508C22—82no—
509C22—88no—
510C22—82no—
511C22—84no—
512C22—816no—
513C22—81no—
514C22—81no—
515C22C7×B(4,𝔽2)12yes—
516C22C2×C7×#12011yescharacteristic
517C22C2×C7×AGL(1,ℤ/8)12yes—
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—82no—
524C2—22no—
525C2—22no—
526C2—41no—
527C2—44no—
528C2—84no—
529C2—82no—
530C2C7×#300311yescharacteristic
531C2C7×#301911yescharacteristic
532C2C2×C7×B(4,𝔽2)11yescharacteristic
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×#3006C2semidirect
3. C7×#2404C2semidirect
4. C7×#1601C2semidirect
5. C7×#1595C2semidirect
6. C7×#1510C2semidirect
7. C7×#1462C2semidirect
8. C22×C7×#41C22semidirect
21. C7×#448C22semidirect
22. C7×#447C22semidirect
24. C7×#407C22semidirect
25. C7×#393C22semidirect
26. C7×#392C22semidirect
29. C7×#381C22semidirect
30. C7×#327C22semidirect
31. C7×#323C22semidirect
32. #18132C7direct (N × H)
38. C22×C7×Q8C23semidirect
47. C22×C7×D4D4semidirect
48. C22×C7×D4C23semidirect
63. C7×#121D4semidirect
69. C2×C7×C4⋊C4D4semidirect
82. C7×C22⋊C8C23semidirect
85. C7×#95C23semidirect
86. C2×#534C2×C7semidirect
87. #3006C2×C7semidirect
88. #2404C2×C7semidirect
89. #1601C2×C7semidirect
90. #1595C2×C7semidirect
91. #1510C2×C7semidirect
92. #1462C2×C7semidirect
166. C22×#41C22×C7semidirect
179. #448C22×C7semidirect
180. #447C22×C7semidirect
182. #407C22×C7semidirect
183. #393C22×C7semidirect
184. #392C22×C7semidirect
187. #381C22×C7semidirect
188. #327C22×C7semidirect
189. #323C22×C7semidirect
219. C23×C7AGL(1,ℤ/8)semidirect
260. C22×Q8C23×C7semidirect
269. C22×D4C7×D4semidirect
270. C22×D4C23×C7semidirect
285. #121C7×D4semidirect
291. C2×C4⋊C4C7×D4semidirect
304. C22⋊C8C23×C7semidirect
307. #95C23×C7semidirect
453. C23C7×AGL(1,ℤ/8)semidirect
489. C7#18132direct (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×#30062
C7×#24042
C7×#16012
C7×#15952
C7×#15102
C7×#14622
#181327