Finite Group Databasebeta

← C7×#18321 #13195537

Subgroups — C7×#18321 #13195537

Order 1792.

982 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#539C211yesmaximal, characteristic
2C7×#3195C211yesmaximal, characteristic, complemented
3C7×#3193C211yesmaximal, characteristic
4C7×#2583C211yesmaximal, characteristic, complemented
5C7×#1609C214yesmaximal, complemented
6C7×#1603C214yesmaximal, complemented
7C7×#1508C211yesmaximal, characteristic, complemented
8C7×#1507C211yesmaximal, 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×C22⋊Q8C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C2×C4×C7×D4—21no—
23C22×C7×C4⋊C4C2211yescharacteristic
24C7×#484—21no—
25C7×#483—21no—
26C7×#483C2212yescomplemented
27C7×#480C2212yes—
28C7×#470—22no—
29C7×#468—22no—
30C7×#451—21no—
31C7×#451C2212yescomplemented
32C7×#450—21no—
33C7×#448C2212yes—
34C7×#407C2211yescharacteristic
35C7×#407C2211yescharacteristic
36C7×#398C2212yescomplemented
37C7×#395C2212yescomplemented
38C7×#389—22no—
39C7×#385—21no—
40C7×#385—21no—
41C4×C7×C22⋊C4C2211yescharacteristic
42C7×#328C2214yescomplemented
43C7×#327C2214yes—
44C7×#324C2214yescomplemented
45#18321C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4—21no—
49C22×C7×D4D411yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7D411yescharacteristic
52C23×C4×C7D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C2×C7×Q16—22no—
55C2×C7×SD16—22no—
56C2×C7×M16—21no—
57C2×C7×M16—21no—
58C7×#126—24no—
59C7×#126—24no—
60C7×#125—24no—
61C7×#125—24no—
62C7×#123—24no—
63C7×#123—24no—
64C7×#123—24no—
65C7×#123—24no—
66C7×C22⋊Q8—24no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8—21no—
71C7×C22⋊Q8—21no—
72C7×C22⋊Q8—21no—
73C7×C22⋊Q8—21no—
74C7×C22⋊Q8D412yes—
75C7×C22⋊Q8C2312yes—
76C7×#121—21no—
77C7×#121—21no—
78C7×#121D412yes—
79C7×#121C2312yes—
80C4×C7×D4—24no—
81C4×C7×D4—24no—
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—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4D412yes—
94C2×C7×C4⋊C4D411yescharacteristic
95C2×C7×C4⋊C4D411yescharacteristic
96C2×C7×C4⋊C4C2311yescharacteristic
97C2×C7×C4⋊C4C2312yes—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4—21no—
105C2×C42×C7—21no—
106C2×C42×C7—21no—
107C7×#107—22no—
108C7×#106—22no—
109C7×#103—22no—
110C7×#103—22no—
111C7×#103—22no—
112C7×#102—22no—
113C7×C22⋊C8—21no—
114C7×C22⋊C8—21no—
115C7×C22⋊C8C2312yes—
116C7×C22⋊C8C2312yes—
117C7×#95—22no—
118C7×#95—22no—
119C7×#95C2312yes—
120C2×#539C2×C711yescharacteristic
121#3195C2×C711yescharacteristic, complemented
122#3193C2×C711yescharacteristic
123#2583C2×C711yescharacteristic, complemented
124#1609C2×C714yescomplemented
125#1603C2×C714yescomplemented
126#1508C2×C711yescharacteristic, complemented
127#1507C2×C711yescharacteristic
128#1463C2×C711yescharacteristic, complemented
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8—21no—
134C2×C7×Q8—81no—
135C2×C7×Q8—81no—
136C2×C7×Q8C2×D412yes—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—21no—
140C2×C7×D4—44no—
141C2×C7×D4—81no—
142C2×C7×D4C2×D412yes—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—24no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—42no—
155C22×C4×C7—81no—
156C22×C4×C7—81no—
157C22×C4×C7—82no—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2411yescharacteristic
166C7×Q16—82no—
167C7×SD16—82no—
168C7×M16—81no—
169C7×M16—81no—
170C2×C7×C8—22no—
171C2×C7×C8—22no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—44no—
186C7×C4⋊C4—41no—
187C7×C4⋊C4—41no—
188C7×C4⋊C4—82no—
189C7×C4⋊C4C2×D412yes—
190C7×C4⋊C4C2×D412yes—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—24no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—41no—
200C7×C22⋊C4—41no—
201C42×C7—24no—
202C42×C7—81no—
203#539—22no—
204#539—22no—
205#539—22no—
206#539C22×C714yes—
207C2×#126—21no—
208C2×#125—21no—
209C2×#123—21no—
210C2×#123—21no—
211C2×C22⋊Q8—21no—
212C2×C22⋊Q8C22×C711yescharacteristic
213C2×C22⋊Q8C22×C711yescharacteristic
214C2×#121C22×C711yescharacteristic
215C2×C4×D4—21no—
216C22×C4⋊C4C22×C711yescharacteristic
217#484—21no—
218#483—21no—
219#483C22×C712yescomplemented
220#480C22×C712yes—
221#470—22no—
222#468—22no—
223#451—21no—
224#451C22×C712yescomplemented
225#450—21no—
226#448C22×C712yes—
227#407C22×C711yescharacteristic
228#407C22×C711yescharacteristic
229#398C22×C712yescomplemented
230#395C22×C712yescomplemented
231#389—22no—
232#385—21no—
233#385—21no—
234C4×C22⋊C4C22×C711yescharacteristic
235#328C22×C714yescomplemented
236#327C22×C714yes—
237#324C22×C714yescomplemented
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—41no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7#12011yescharacteristic
246C23×C7#12011yescharacteristic
247C23×C7C22×D411yescharacteristic
248C7×Q8—42no—
249C7×Q8—42no—
250C7×Q8—42no—
251C7×Q8—81no—
252C7×D4—42no—
253C7×D4—44no—
254C7×D4—44no—
255C7×D4—81no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—81no—
284C2×C4×C7—84no—
285C2×C4×C7#12011yescharacteristic
286C2×C4×C7#12011yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C22×D411yescharacteristic
289C7×C8—42no—
290C7×C8—42no—
291C22×Q8C7×D411yescharacteristic
292C22×Q8C23×C711yescharacteristic
293C22×D4—21no—
294C22×D4C7×D411yescharacteristic
295C23×C4—21no—
296C23×C4C7×D411yescharacteristic
297C23×C4C7×D411yescharacteristic
298C23×C4C23×C711yescharacteristic
299C2×Q16—22no—
300C2×SD16—22no—
301C2×M16—21no—
302C2×M16—21no—
303#126—24no—
304#126—24no—
305#125—24no—
306#125—24no—
307#123—24no—
308#123—24no—
309#123—24no—
310#123—24no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—21no—
319C22⋊Q8C7×D412yes—
320C22⋊Q8C23×C712yes—
321#121—21no—
322#121—21no—
323#121C7×D412yes—
324#121C23×C712yes—
325C4×D4—24no—
326C4×D4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4C7×D412yes—
339C2×C4⋊C4C7×D411yescharacteristic
340C2×C4⋊C4C7×D411yescharacteristic
341C2×C4⋊C4C23×C711yescharacteristic
342C2×C4⋊C4C23×C712yes—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C42—21no—
351C2×C42—21no—
352#107—22no—
353#106—22no—
354#103—22no—
355#103—22no—
356#103—22no—
357#102—22no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8C23×C712yes—
361C22⋊C8C23×C712yes—
362#95—22no—
363#95—22no—
364#95C23×C712yes—
365C22×C7—21no—
366C22×C7—21no—
367C22×C7—21no—
368C22×C7—21no—
369C22×C7—21no—
370C22×C7—21no—
371C22×C7—21no—
372C22×C7—44no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—84no—
377C22×C7#45612yes—
378C22×C7C2×#12011yescharacteristic
379C22×C7#57512yes—
380C4×C7—21no—
381C4×C7—21no—
382C4×C7—21no—
383C4×C7—21no—
384C4×C7—44no—
385C4×C7—42no—
386C4×C7—42no—
387C4×C7—42no—
388C4×C7—42no—
389C4×C7—42no—
390C4×C7—81no—
391C4×C7—81no—
392C24—21no—
393C24C2×C7×D411yescharacteristic
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×Q8C2×C7×D412yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—44no—
404C2×D4—81no—
405C2×D4C2×C7×D412yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—82no—
421C22×C4C2×C7×D412yes—
422C22×C4C2×C7×D412yes—
423C22×C4C2×C7×D411yescharacteristic
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D411yescharacteristic
426C22×C4C2×C7×D411yescharacteristic
427C22×C4C2×C7×D411yescharacteristic
428C22×C4C24×C711yescharacteristic
429Q16—82no—
430SD16—82no—
431M16—81no—
432M16—81no—
433C2×C8—22no—
434C2×C8—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—44no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—82no—
452C4⋊C4C2×C7×D412yes—
453C4⋊C4C2×C7×D412yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C42—24no—
465C42—81no—
466C2×C7—22no—
467C2×C7—22no—
468C2×C7—41no—
469C2×C7—82no—
470C2×C7#300511yescharacteristic
471C2×C7C2×#45611yescharacteristic
472C2×C7#302611yescharacteristic
473C23—22no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23C7×#12011yescharacteristic
481C23C7×#12011yescharacteristic
482C23C22×C7×D411yescharacteristic
483Q8—42no—
484Q8—42no—
485Q8—42no—
486Q8—81no—
487D4—42no—
488D4—44no—
489D4—44no—
490D4—81no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—81no—
519C2×C4—84no—
520C2×C4C7×#12011yescharacteristic
521C2×C4C7×#12011yescharacteristic
522C2×C4C22×C7×D411yescharacteristic
523C2×C4C22×C7×D411yescharacteristic
524C8—42no—
525C8—42no—
526C7#1832111yescharacteristic, direct product factor, Sylow
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—84no—
539C22C7×#45612yes—
540C22C2×C7×#12011yescharacteristic
541C22C7×#57512yes—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—81no—
553C4—81no—
554C2—22no—
555C2—22no—
556C2—41no—
557C2—82no—
558C2C7×#300511yescharacteristic
559C2C2×C7×#45611yescharacteristic
560C2C7×#302611yescharacteristic
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×#3195C2semidirect
4. C7×#2583C2semidirect
5. C7×#1609C2semidirect
6. C7×#1603C2semidirect
7. C7×#1508C2semidirect
9. C7×#1463C2semidirect
26. C7×#483C22semidirect
31. C7×#451C22semidirect
36. C7×#398C22semidirect
37. C7×#395C22semidirect
42. C7×#328C22semidirect
44. C7×#324C22semidirect
45. #18321C7direct (N × H)
121. #3195C2×C7semidirect
123. #2583C2×C7semidirect
124. #1609C2×C7semidirect
125. #1603C2×C7semidirect
126. #1508C2×C7semidirect
128. #1463C2×C7semidirect
219. #483C22×C7semidirect
224. #451C22×C7semidirect
229. #398C22×C7semidirect
230. #395C22×C7semidirect
235. #328C22×C7semidirect
237. #324C22×C7semidirect
526. C7#18321direct (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×#31952
C7×#31932
C7×#25832
C7×#16092
C7×#16032
C7×#15082
C7×#15072
C7×#14632
#183217