Finite Group Databasebeta

← C7×#18139 #13195355

Subgroups — C7×#18139 #13195355

Order 1792.

962 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#539C211yesmaximal, characteristic
2C7×#3095C211yesmaximal, characteristic, complemented
3C7×#3094C211yesmaximal, characteristic
4C7×#2484C211yesmaximal, characteristic, complemented
5C7×#1615C214yesmaximal, complemented
6C7×#1609C214yesmaximal, complemented
7C7×#1511C211yesmaximal, characteristic, complemented
8C7×#1510C211yesmaximal, characteristic
9C7×#1463C211yesmaximal, characteristic, complemented
10C7×#539—22no—
11C7×#539—22no—
12C7×#539—22no—
13C7×#539C2214yes—
14C2×C7×#126—21no—
15C2×C7×#125—21no—
16C2×C7×#123—21no—
17C2×C7×#123—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C7×#121C2211yescharacteristic
21C2×C4×C7×D4—21no—
22C2×C7×#117C2211yescharacteristic
23C22×C7×C4⋊C4C2211yescharacteristic
24C7×#485—22no—
25C7×#484C2212yescomplemented
26C7×#483C2212yescomplemented
27C7×#482—22no—
28C7×#481C2212yes—
29C7×#480C2212yes—
30C7×#428—21no—
31C7×#428—21no—
32C7×#419—21no—
33C7×#418—21no—
34C7×#407C2211yescharacteristic
35C7×#407C2211yescharacteristic
36C7×#400C2212yescomplemented
37C7×#398C2212yescomplemented
38C7×#387—22no—
39C7×#385—21no—
40C7×#381—21no—
41C7×#381C2211yescharacteristic
42C7×#328C2214yescomplemented
43C7×#327C2214yes—
44C7×#324C2214yescomplemented
45#18139C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×D4—21no—
48C22×C7×D4D411yescharacteristic
49C23×C4×C7—21no—
50C23×C4×C7D411yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C2×C7×M16—21no—
54C2×C7×M16—21no—
55C7×#126—24no—
56C7×#126—24no—
57C7×#125—24no—
58C7×#125—24no—
59C7×#123—24no—
60C7×#123—24no—
61C7×#123—24no—
62C7×#123—24no—
63C7×C22⋊Q8—24no—
64C7×C22⋊Q8—24no—
65C7×C22⋊Q8—21no—
66C7×C22⋊Q8—21no—
67C7×C22⋊Q8D412yes—
68C7×C22⋊Q8C2312yes—
69C7×#121—21no—
70C7×#121—21no—
71C7×#121D412yes—
72C7×#121C2312yes—
73C4×C7×D4—24no—
74C4×C7×D4—24no—
75C7×#117—21no—
76C7×#117—21no—
77C7×#117—21no—
78C7×#117—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—22no—
88C2×C7×C4⋊C4D412yes—
89C2×C7×C4⋊C4D411yescharacteristic
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4C2311yescharacteristic
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C42×C7—21no—
102C2×C42×C7—21no—
103C7×#107—22no—
104C7×#107—22no—
105C7×#106—22no—
106C7×#106—22no—
107C7×#103—22no—
108C7×#103—22no—
109C7×#102—22no—
110C7×#102—22no—
111C7×C22⋊C8—21no—
112C7×C22⋊C8—21no—
113C7×C22⋊C8C2312yes—
114C7×C22⋊C8C2312yes—
115C7×#95—22no—
116C7×#95—22no—
117C7×#95—21no—
118C7×#95—22no—
119C7×#95C2312yes—
120C2×#539C2×C711yescharacteristic
121#3095C2×C711yescharacteristic, complemented
122#3094C2×C711yescharacteristic
123#2484C2×C711yescharacteristic, complemented
124#1615C2×C714yescomplemented
125#1609C2×C714yescomplemented
126#1511C2×C711yescharacteristic, complemented
127#1510C2×C711yescharacteristic
128#1463C2×C711yescharacteristic, complemented
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C2×C7×Q8—21no—
132C2×C7×Q8—42no—
133C2×C7×Q8C2×D412yes—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—21no—
137C2×C7×D4—42no—
138C2×C7×D4—44no—
139C2×C7×D4C2×D412yes—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—24no—
147C22×C4×C7—21no—
148C22×C4×C7—22no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—82no—
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2411yescharacteristic
163C7×M16—42no—
164C7×M16—42no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—44no—
183C7×C4⋊C4C2×D412yes—
184C7×C4⋊C4C2×D412yes—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—24no—
187C7×C22⋊C4—24no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—22no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—41no—
194C7×C22⋊C4—41no—
195C42×C7—24no—
196C42×C7—41no—
197C42×C7—41no—
198#539—22no—
199#539—22no—
200#539—22no—
201#539C22×C714yes—
202C2×#126—21no—
203C2×#125—21no—
204C2×#123—21no—
205C2×#123—21no—
206C2×C22⋊Q8—21no—
207C2×C22⋊Q8C22×C711yescharacteristic
208C2×#121C22×C711yescharacteristic
209C2×C4×D4—21no—
210C2×#117C22×C711yescharacteristic
211C22×C4⋊C4C22×C711yescharacteristic
212#485—22no—
213#484C22×C712yescomplemented
214#483C22×C712yescomplemented
215#482—22no—
216#481C22×C712yes—
217#480C22×C712yes—
218#428—21no—
219#428—21no—
220#419—21no—
221#418—21no—
222#407C22×C711yescharacteristic
223#407C22×C711yescharacteristic
224#400C22×C712yescomplemented
225#398C22×C712yescomplemented
226#387—22no—
227#385—21no—
228#381—21no—
229#381C22×C711yescharacteristic
230#328C22×C714yescomplemented
231#327C22×C714yes—
232#324C22×C714yescomplemented
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7—41no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7#12011yescharacteristic
241C23×C7#12011yescharacteristic
242C23×C7C22×D411yescharacteristic
243C7×Q8—42no—
244C7×Q8—81no—
245C7×D4—42no—
246C7×D4—44no—
247C7×D4—44no—
248C7×D4—81no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—81no—
277C2×C4×C7—84no—
278C2×C4×C7#12011yescharacteristic
279C2×C4×C7#12011yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C7×C8—42no—
283C7×C8—42no—
284C22×Q8C7×D411yescharacteristic
285C22×D4—21no—
286C22×D4C7×D411yescharacteristic
287C23×C4—21no—
288C23×C4C7×D411yescharacteristic
289C23×C4C7×D411yescharacteristic
290C23×C4C23×C711yescharacteristic
291C2×M16—21no—
292C2×M16—21no—
293#126—24no—
294#126—24no—
295#125—24no—
296#125—24no—
297#123—24no—
298#123—24no—
299#123—24no—
300#123—24no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8C7×D412yes—
306C22⋊Q8C23×C712yes—
307#121—21no—
308#121—21no—
309#121C7×D412yes—
310#121C23×C712yes—
311C4×D4—24no—
312C4×D4—24no—
313#117—21no—
314#117—21no—
315#117—21no—
316#117—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4C7×D412yes—
327C2×C4⋊C4C7×D411yescharacteristic
328C2×C4⋊C4C7×D411yescharacteristic
329C2×C4⋊C4C23×C711yescharacteristic
330C2×C4⋊C4C23×C711yescharacteristic
331C2×C4⋊C4C23×C712yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C42—21no—
340C2×C42—21no—
341#107—22no—
342#107—22no—
343#106—22no—
344#106—22no—
345#103—22no—
346#103—22no—
347#102—22no—
348#102—22no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8C23×C712yes—
352C22⋊C8C23×C712yes—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—22no—
357#95C23×C712yes—
358C22×C7—21no—
359C22×C7—21no—
360C22×C7—21no—
361C22×C7—21no—
362C22×C7—21no—
363C22×C7—21no—
364C22×C7—21no—
365C22×C7—44no—
366C22×C7—42no—
367C22×C7—42no—
368C22×C7—42no—
369C22×C7—84no—
370C22×C7#45812yes—
371C22×C7C2×#12011yescharacteristic
372C22×C7#57512yes—
373C4×C7—21no—
374C4×C7—21no—
375C4×C7—21no—
376C4×C7—21no—
377C4×C7—42no—
378C4×C7—42no—
379C4×C7—42no—
380C4×C7—42no—
381C4×C7—42no—
382C4×C7—82no—
383C4×C7—81no—
384C4×C7—81no—
385C24—21no—
386C24C2×C7×D411yescharacteristic
387C2×Q8—21no—
388C2×Q8—42no—
389C2×Q8C2×C7×D412yes—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4C2×C7×D412yes—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—82no—
411C22×C4C2×C7×D412yes—
412C22×C4C2×C7×D411yescharacteristic
413C22×C4C2×C7×D411yescharacteristic
414C22×C4C2×C7×D412yes—
415C22×C4C2×C7×D411yescharacteristic
416C22×C4C2×C7×D411yescharacteristic
417C22×C4C2×C7×D411yescharacteristic
418C22×C4C24×C711yescharacteristic
419M16—42no—
420M16—42no—
421C2×C8—22no—
422C2×C8—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—44no—
439C4⋊C4C2×C7×D412yes—
440C4⋊C4C2×C7×D412yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—42no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C42—24no—
452C42—41no—
453C42—41no—
454C2×C7—22no—
455C2×C7—22no—
456C2×C7—41no—
457C2×C7—82no—
458C2×C7#300511yescharacteristic
459C2×C7#302011yescharacteristic
460C2×C7C2×#45811yescharacteristic
461C23—22no—
462C23—22no—
463C23—22no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23C7×#12011yescharacteristic
469C23C7×#12011yescharacteristic
470C23C22×C7×D411yescharacteristic
471Q8—42no—
472Q8—81no—
473D4—42no—
474D4—44no—
475D4—44no—
476D4—81no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—81no—
505C2×C4—84no—
506C2×C4C7×#12011yescharacteristic
507C2×C4C7×#12011yescharacteristic
508C2×C4C22×C7×D411yescharacteristic
509C2×C4C22×C7×D411yescharacteristic
510C8—42no—
511C8—42no—
512C7#1813911yescharacteristic, direct product factor, Sylow
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—84no—
525C22C7×#45812yes—
526C22C2×C7×#12011yescharacteristic
527C22C7×#57512yes—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—82no—
538C4—81no—
539C4—81no—
540C2—22no—
541C2—22no—
542C2—41no—
543C2—82no—
544C2C7×#300511yescharacteristic
545C2C7×#302011yescharacteristic
546C2C2×C7×#45811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
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×#3095C2semidirect
4. C7×#2484C2semidirect
5. C7×#1615C2semidirect
6. C7×#1609C2semidirect
7. C7×#1511C2semidirect
9. C7×#1463C2semidirect
25. C7×#484C22semidirect
26. C7×#483C22semidirect
36. C7×#400C22semidirect
37. C7×#398C22semidirect
42. C7×#328C22semidirect
44. C7×#324C22semidirect
45. #18139C7direct (N × H)
121. #3095C2×C7semidirect
123. #2484C2×C7semidirect
124. #1615C2×C7semidirect
125. #1609C2×C7semidirect
126. #1511C2×C7semidirect
128. #1463C2×C7semidirect
213. #484C22×C7semidirect
214. #483C22×C7semidirect
224. #400C22×C7semidirect
225. #398C22×C7semidirect
230. #328C22×C7semidirect
232. #324C22×C7semidirect
512. C7#18139direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5392
C7×#30952
C7×#30942
C7×#24842
C7×#16152
C7×#16092
C7×#15112
C7×#15102
C7×#14632
#181397