Finite Group Databasebeta

← C7×#18993 #13196209

Subgroups — C7×#18993 #13196209

Order 1792.

1012 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×#562C211yesmaximal, characteristic, complemented
2C7×#3234C211yesmaximal, characteristic, complemented
3C7×#3213C211yesmaximal, characteristic, complemented
4C7×#3153C211yesmaximal, characteristic, complemented
5C7×#1661C214yesmaximal, complemented
6C7×#1656C214yesmaximal, complemented
7C7×#1533C211yesmaximal, characteristic, complemented
8C7×#1528C211yesmaximal, characteristic, complemented
9C7×#1527C211yesmaximal, characteristic, complemented
10C2×C7×#137—21no—
11C2×C7×AGL(1,ℤ/8)—21no—
12C7×#562—22no—
13C7×#562—22no—
14C7×#562—22no—
15C7×#562C2214yes—
16C7×#532C2211yescharacteristic
17C2×C7×#125C2211yescharacteristic
18C2×C7×#124C2211yescharacteristic
19C2×C7×#124C2211yescharacteristic
20C2×C7×#123—21no—
21C2×C7×#123—21no—
22C2×C7×C22⋊Q8—21no—
23C2×C7×C22⋊Q8—21no—
24C2×C7×#121—21no—
25C2×C7×#121—21no—
26C7×#501C2212yescomplemented
27C7×#497C2212yescomplemented
28C7×#496C2212yescomplemented
29C7×#496C2212yescomplemented
30C7×#495—22no—
31C7×#490C2212yescomplemented
32C7×#489C2212yescomplemented
33C7×#481—21no—
34C7×#480—21no—
35C7×#478—22no—
36C7×#463—22no—
37C7×#448—21no—
38C7×#447—21no—
39C7×#432C2211yescharacteristic
40C7×#432C2211yescharacteristic
41C4×C7×M16C2211yescharacteristic
42C7×#330C2214yescomplemented
43C7×#329C2214yescomplemented
44C7×#329C2214yescomplemented
45#18993C711yesmaximal, characteristic, direct product factor, Sylow
46C2×C7×#41—21no—
47C22×C7×Q8D411yescharacteristic
48C22×C7×Q8D411yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4D411yescharacteristic
52C22×C7×D4D411yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C7×#137—44no—
56C7×AGL(1,ℤ/8)—44no—
57C2×C7×Q16—22no—
58C2×C7×SD16—22no—
59C2×C7×SD16—22no—
60C2×C7×SD16—22no—
61C2×C7×D8—22no—
62C2×C7×D8—22no—
63C2×C7×M16—21no—
64C7×C4⋊Q8C2311yescharacteristic
65C7×#127C2311yescharacteristic
66C7×#125—21no—
67C7×#125—21no—
68C7×#125D412yes—
69C7×#125C2312yes—
70C7×#124—21no—
71C7×#124—21no—
72C7×#124—21no—
73C7×#124—21no—
74C7×#124—21no—
75C7×#124D412yes—
76C7×#124D412yes—
77C7×#124C2312yes—
78C7×#124C2312yes—
79C7×#123—24no—
80C7×#123—24no—
81C7×#123—24no—
82C7×#123—24no—
83C7×C22⋊Q8—24no—
84C7×C22⋊Q8—24no—
85C7×C22⋊Q8—24no—
86C7×C22⋊Q8—24no—
87C7×#121—24no—
88C7×#121—24no—
89C7×#121—24no—
90C7×#121—24no—
91C7×#121—21no—
92C7×#121—21no—
93C4×C7×D4—21no—
94C4×C7×D4—21no—
95C2×C7×C4⋊C4—21no—
96C2×C7×C4⋊C4—21no—
97C2×C7×C4⋊C4—21no—
98C2×C7×C4⋊C4—21no—
99C2×C7×C4⋊C4D411yescharacteristic
100C2×C7×C4⋊C4D411yescharacteristic
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4—21no—
105C2×C7×C22⋊C4—21no—
106C2×C7×C22⋊C4—21no—
107C2×C42×C7C2311yescharacteristic
108C7×#107—22no—
109C7×#106—22no—
110C7×C4⋊C8—21no—
111C7×C4⋊C8—21no—
112C7×#103—22no—
113C7×#102—22no—
114C7×#102—22no—
115C7×#102—22no—
116C7×C22⋊C8—21no—
117C7×C22⋊C8—21no—
118C7×#97C2312yes—
119C7×#97C2312yes—
120C7×#97C2312yes—
121C4×C7×C8—21no—
122C2×#562C2×C711yescharacteristic, complemented
123#3234C2×C711yescharacteristic, complemented
124#3213C2×C711yescharacteristic, complemented
125#3153C2×C711yescharacteristic, complemented
126#1661C2×C714yescomplemented
127#1656C2×C714yescomplemented
128#1533C2×C711yescharacteristic, complemented
129#1528C2×C711yescharacteristic, complemented
130#1527C2×C711yescharacteristic, complemented
131C24×C7—21no—
132C24×C7—21no—
133C7×#41—42no—
134C7×#41—42no—
135C2×C7×Q8—21no—
136C2×C7×Q8—21no—
137C2×C7×Q8—21no—
138C2×C7×Q8—42no—
139C2×C7×Q8—81no—
140C2×C7×Q8C2×D412yes—
141C2×C7×Q8C2×D412yes—
142C2×C7×D4—24no—
143C2×C7×D4—24no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—42no—
150C2×C7×D4—41no—
151C2×C7×D4—81no—
152C2×C7×D4—82no—
153C2×C7×D4—82no—
154C2×C7×D4C2×D412yes—
155C2×C7×D4C2×D412yes—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—24no—
161C22×C4×C7—24no—
162C22×C4×C7—41no—
163C22×C4×C7—82no—
164C22×C4×C7—82no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C7×Q16—44no—
169C7×SD16—44no—
170C7×SD16—44no—
171C7×SD16—82no—
172C7×D8—44no—
173C7×D8—82no—
174C7×M16—24no—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4C2×D412yes—
189C7×C4⋊C4C2×D412yes—
190C7×C22⋊C4—24no—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—22no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—41no—
201C7×C22⋊C4—41no—
202C42×C7C2×D411yescharacteristic
203C42×C7C2×D411yescharacteristic
204C42×C7C2×D411yescharacteristic
205C42×C7C2411yescharacteristic
206C2×#137—21no—
207C2×AGL(1,ℤ/8)—21no—
208#562—22no—
209#562—22no—
210#562—22no—
211#562C22×C714yes—
212#532C22×C711yescharacteristic
213C2×#125C22×C711yescharacteristic
214C2×#124C22×C711yescharacteristic
215C2×#124C22×C711yescharacteristic
216C2×#123—21no—
217C2×#123—21no—
218C2×C22⋊Q8—21no—
219C2×C22⋊Q8—21no—
220C2×#121—21no—
221C2×#121—21no—
222#501C22×C712yescomplemented
223#497C22×C712yescomplemented
224#496C22×C712yescomplemented
225#496C22×C712yescomplemented
226#495—22no—
227#490C22×C712yescomplemented
228#489C22×C712yescomplemented
229#481—21no—
230#480—21no—
231#478—22no—
232#463—22no—
233#448—21no—
234#447—21no—
235#432C22×C711yescharacteristic
236#432C22×C711yescharacteristic
237C4×M16C22×C711yescharacteristic
238#330C22×C714yescomplemented
239#329C22×C714yescomplemented
240#329C22×C714yescomplemented
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7—41no—
250C23×C7#12011yescharacteristic
251C7×Q8—42no—
252C7×Q8—42no—
253C7×Q8—41no—
254C7×Q8—41no—
255C7×Q8—81no—
256C7×Q8—81no—
257C7×D4—42no—
258C7×D4—42no—
259C7×D4—41no—
260C7×D4—42no—
261C7×D4—41no—
262C7×D4—42no—
263C7×D4—81no—
264C7×D4—81no—
265C7×D4—84no—
266C7×D4—84no—
267C7×D4—82no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—41no—
280C2×C4×C7—82no—
281C2×C4×C7—84no—
282C2×C4×C7—84no—
283C2×C4×C7#12011yescharacteristic
284C2×C4×C7#12011yescharacteristic
285C2×C4×C7#12011yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C22×D411yescharacteristic
289C7×C8—24no—
290C7×C8—42no—
291C7×C8—42no—
292C2×#41—21no—
293C22×Q8C7×D411yescharacteristic
294C22×Q8C7×D411yescharacteristic
295C22×D4—21no—
296C22×D4—21no—
297C22×D4C7×D411yescharacteristic
298C22×D4C7×D411yescharacteristic
299C23×C4—21no—
300C23×C4—21no—
301#137—44no—
302AGL(1,ℤ/8)—44no—
303C2×Q16—22no—
304C2×SD16—22no—
305C2×SD16—22no—
306C2×SD16—22no—
307C2×D8—22no—
308C2×D8—22no—
309C2×M16—21no—
310C4⋊Q8C23×C711yescharacteristic
311#127C23×C711yescharacteristic
312#125—21no—
313#125—21no—
314#125C7×D412yes—
315#125C23×C712yes—
316#124—21no—
317#124—21no—
318#124—21no—
319#124—21no—
320#124—21no—
321#124C7×D412yes—
322#124C7×D412yes—
323#124C23×C712yes—
324#124C23×C712yes—
325#123—24no—
326#123—24no—
327#123—24no—
328#123—24no—
329C22⋊Q8—24no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332C22⋊Q8—24no—
333#121—24no—
334#121—24no—
335#121—24no—
336#121—24no—
337#121—21no—
338#121—21no—
339C4×D4—21no—
340C4×D4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C7×D411yescharacteristic
346C2×C4⋊C4C7×D411yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C42C23×C711yescharacteristic
354#107—22no—
355#106—22no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358#103—22no—
359#102—22no—
360#102—22no—
361#102—22no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364#97C23×C712yes—
365#97C23×C712yes—
366#97C23×C712yes—
367C4×C8—21no—
368C22×C7—21no—
369C22×C7—21no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—84no—
377C22×C7—84no—
378C22×C7—81no—
379C22×C7—81no—
380C22×C7—81no—
381C22×C7#45312yes—
382C22×C7C2×#12011yescharacteristic
383C4×C7—21no—
384C4×C7—21no—
385C4×C7—21no—
386C4×C7—21no—
387C4×C7—21no—
388C4×C7—21no—
389C4×C7—42no—
390C4×C7—42no—
391C4×C7—42no—
392C4×C7—42no—
393C4×C7—81no—
394C24—21no—
395C24—21no—
396#41—42no—
397#41—42no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—42no—
402C2×Q8—81no—
403C2×Q8C2×C7×D412yes—
404C2×Q8C2×C7×D412yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—41no—
414C2×D4—81no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4C2×C7×D412yes—
418C2×D4C2×C7×D412yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—41no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C2×C7×D411yescharacteristic
430C22×C4C2×C7×D411yescharacteristic
431Q16—44no—
432SD16—44no—
433SD16—44no—
434SD16—82no—
435D8—44no—
436D8—82no—
437M16—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4C2×C7×D412yes—
452C4⋊C4C2×C7×D412yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C42C2×C7×D411yescharacteristic
466C42C2×C7×D411yescharacteristic
467C42C2×C7×D411yescharacteristic
468C42C24×C711yescharacteristic
469C2×C7—22no—
470C2×C7—82no—
471C2×C7—82no—
472C2×C7—161no—
473C2×C7C2×#45311yescharacteristic
474C2×C7#301911yescharacteristic
475C2×C7#302011yescharacteristic
476C23—22no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23C7×#12011yescharacteristic
486Q8—42no—
487Q8—42no—
488Q8—41no—
489Q8—41no—
490Q8—81no—
491Q8—81no—
492D4—42no—
493D4—42no—
494D4—41no—
495D4—42no—
496D4—41no—
497D4—42no—
498D4—81no—
499D4—81no—
500D4—84no—
501D4—84no—
502D4—82no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4C7×#12011yescharacteristic
519C2×C4C7×#12011yescharacteristic
520C2×C4C7×#12011yescharacteristic
521C2×C4C22×C7×D411yescharacteristic
522C2×C4C22×C7×D411yescharacteristic
523C2×C4C22×C7×D411yescharacteristic
524C8—24no—
525C8—42no—
526C8—42no—
527C7#1899311yescharacteristic, direct product factor, Sylow
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—84no—
537C22—84no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C7×#45312yes—
542C22C2×C7×#12011yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—81no—
554C2—22no—
555C2—82no—
556C2—82no—
557C2—161no—
558C2C2×C7×#45311yescharacteristic
559C2C7×#301911yescharacteristic
560C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#562C2semidirect
2. C7×#3234C2semidirect
3. C7×#3213C2semidirect
4. C7×#3153C2semidirect
5. C7×#1661C2semidirect
6. C7×#1656C2semidirect
7. C7×#1533C2semidirect
8. C7×#1528C2semidirect
9. C7×#1527C2semidirect
26. C7×#501C22semidirect
27. C7×#497C22semidirect
28. C7×#496C22semidirect
29. C7×#496C22semidirect
31. C7×#490C22semidirect
32. C7×#489C22semidirect
42. C7×#330C22semidirect
43. C7×#329C22semidirect
44. C7×#329C22semidirect
45. #18993C7direct (N × H)
122. C2×#562C2×C7semidirect
123. #3234C2×C7semidirect
124. #3213C2×C7semidirect
125. #3153C2×C7semidirect
126. #1661C2×C7semidirect
127. #1656C2×C7semidirect
128. #1533C2×C7semidirect
129. #1528C2×C7semidirect
130. #1527C2×C7semidirect
222. #501C22×C7semidirect
223. #497C22×C7semidirect
224. #496C22×C7semidirect
225. #496C22×C7semidirect
227. #490C22×C7semidirect
228. #489C22×C7semidirect
238. #330C22×C7semidirect
239. #329C22×C7semidirect
240. #329C22×C7semidirect
527. C7#18993direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5622
C7×#32342
C7×#32132
C7×#31532
C7×#16612
C7×#16562
C7×#15332
C7×#15282
C7×#15272
#189937