Finite Group Databasebeta

← C7×#24110 #13201326

Subgroups — C7×#24110 #13201326

Order 1792.

1088 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#552C211yesmaximal, characteristic
2C2×C7×#484C211yesmaximal, characteristic, complemented
3C2×C7×#428C211yesmaximal, characteristic, complemented
4C7×#2486C211yesmaximal, characteristic
5C7×#2059C212yesmaximal, complemented
6C7×#2035C212yesmaximal, complemented
7C7×#1894C211yesmaximal, characteristic, complemented
8C7×#1870C211yesmaximal, characteristic
9C7×#1863C212yesmaximal, complemented
10C7×#1852C212yesmaximal, complemented
11C7×#1825C211yesmaximal, characteristic, complemented
12C22×C7×M16C2211yescharacteristic
13C7×#552—24no—
14C7×#552—24no—
15C2×C7×#126—21no—
16C2×C7×#125—21no—
17C2×C7×#125C2211yescharacteristic
18C2×C7×#123—21no—
19C2×C7×#123—21no—
20C2×C7×C22⋊Q8C2211yescharacteristic
21C2×C7×C22⋊Q8C2211yescharacteristic
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C2×C7×#117—21no—
25C2×C7×#117C2211yescharacteristic
26C22×C7×C4⋊C4C2211yescharacteristic
27C22×C7×C4⋊C4C2211yescharacteristic
28C7×#484—24no—
29C7×#428—24no—
30C2×C7×#107C2212yes—
31C2×C7×#107C2212yes—
32C2×C7×#106C2212yes—
33C2×C7×#103C2212yes—
34C2×C7×#103C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C7×C22⋊C8C2211yescharacteristic
37C7×#398—21no—
38C7×#397—21no—
39C7×#389C2212yes—
40C7×#381—21no—
41C7×#380C2212yes—
42C7×#380C2211yescharacteristic
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#340C2212yes—
46C7×#336C2212yes—
47#24110C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4D411yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7—21no—
52C23×C4×C7C2311yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C2×C7×M16—22no—
55C2×C7×M16—24no—
56C22×C7×C8C2312yes—
57C22×C7×C8C2312yes—
58C7×#126—28no—
59C7×#125—28no—
60C7×#125—22no—
61C7×#125—22no—
62C7×#123—28no—
63C7×#123—44no—
64C7×C22⋊Q8—22no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8D414yes—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C7×#117—28no—
73C7×#117—22no—
74C7×#117D414yes—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—22no—
87C2×C7×C4⋊C4D411yescharacteristic
88C2×C7×C4⋊C4C2312yes—
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C4⋊C4C2311yescharacteristic
91C2×C7×C4⋊C4C2311yescharacteristic
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C22⋊C4—21no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4D411yescharacteristic
99C2×C7×C22⋊C4D411yescharacteristic
100C2×C42×C7—21no—
101C2×C42×C7—21no—
102C2×C42×C7C2311yescharacteristic
103C7×#107—24no—
104C7×#107—24no—
105C7×#106—24no—
106C7×#103—24no—
107C7×#103—24no—
108C7×C22⋊C8—22no—
109C7×C22⋊C8—24no—
110C7×#95—22no—
111C7×#95—21no—
112C7×#95—22no—
113C2×#552C2×C711yescharacteristic
114C2×#484C2×C711yescharacteristic, complemented
115C2×#428C2×C711yescharacteristic, complemented
116#2486C2×C711yescharacteristic
117#2059C2×C712yescomplemented
118#2035C2×C712yescomplemented
119#1894C2×C711yescharacteristic, complemented
120#1870C2×C711yescharacteristic
121#1863C2×C712yescomplemented
122#1852C2×C712yescomplemented
123#1825C2×C711yescharacteristic, complemented
124C24×C7C2×D411yescharacteristic
125C24×C7C2×D411yescharacteristic
126C2×C7×Q8—21no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8C2×D412yes—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—41no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—24no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2411yescharacteristic
158C7×M16—28no—
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8#4114yes—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—44no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4C2×D412yes—
182C7×C4⋊C4C2×D412yes—
183C7×C4⋊C4C2×D412yes—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—24no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—42no—
190C7×C22⋊C4—44no—
191C42×C7—24no—
192C42×C7—24no—
193C42×C7—21no—
194C42×C7—21no—
195C22×M16C22×C711yescharacteristic
196#552—24no—
197#552—24no—
198C2×#126—21no—
199C2×#125—21no—
200C2×#125C22×C711yescharacteristic
201C2×#123—21no—
202C2×#123—21no—
203C2×C22⋊Q8C22×C711yescharacteristic
204C2×C22⋊Q8C22×C711yescharacteristic
205C2×C4×Q8C22×C711yescharacteristic
206C2×C4×D4C22×C711yescharacteristic
207C2×#117—21no—
208C2×#117C22×C711yescharacteristic
209C22×C4⋊C4C22×C711yescharacteristic
210C22×C4⋊C4C22×C711yescharacteristic
211#484—24no—
212#428—24no—
213C2×#107C22×C712yes—
214C2×#107C22×C712yes—
215C2×#106C22×C712yes—
216C2×#103C22×C712yes—
217C2×#103C22×C712yes—
218C2×C22⋊C8C22×C711yescharacteristic
219C2×C22⋊C8C22×C712yes—
220#398—21no—
221#397—21no—
222#389C22×C712yes—
223#381—21no—
224#380C22×C711yescharacteristic
225#380C22×C712yes—
226#340C22×C712yes—
227#340C22×C712yes—
228#340C22×C712yes—
229#336C22×C712yes—
230C23×C7—21no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7#12012yes—
239C23×C7AGL(1,ℤ/8)12yes—
240C23×C7C22×D411yescharacteristic
241C7×Q8—24no—
242C7×Q8—41no—
243C7×Q8—41no—
244C7×D4—44no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—21no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—22no—
260C2×C4×C7—48no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
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—84no—
277C2×C4×C7#12012yes—
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C2×#4112yes—
281C2×C4×C72−1+412yes—
282C7×C8—24no—
283C7×C8—42no—
284C22×Q8C23×C711yescharacteristic
285C22×D4C7×D411yescharacteristic
286C23×C4—21no—
287C23×C4—21no—
288C23×C4C23×C711yescharacteristic
289C23×C4C23×C711yescharacteristic
290C2×M16—22no—
291C2×M16—24no—
292C22×C8C23×C712yes—
293C22×C8C23×C712yes—
294#126—28no—
295#125—28no—
296#125—22no—
297#125—22no—
298#123—28no—
299#123—44no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8C7×D414yes—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×D4—22no—
307C4×D4—22no—
308#117—28no—
309#117—22no—
310#117C7×D414yes—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4C7×D411yescharacteristic
324C2×C4⋊C4C23×C712yes—
325C2×C4⋊C4C23×C712yes—
326C2×C4⋊C4C23×C711yescharacteristic
327C2×C4⋊C4C23×C711yescharacteristic
328C2×C4⋊C4C23×C711yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C7×D411yescharacteristic
335C2×C22⋊C4C7×D411yescharacteristic
336C2×C42—21no—
337C2×C42—21no—
338C2×C42C23×C711yescharacteristic
339#107—24no—
340#107—24no—
341#106—24no—
342#103—24no—
343#103—24no—
344C22⋊C8—22no—
345C22⋊C8—24no—
346#95—22no—
347#95—21no—
348#95—22no—
349C22×C7—24no—
350C22×C7—22no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—41no—
358C22×C7—41no—
359C22×C7C2×#12011yescharacteristic
360C22×C7#54712yes—
361C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
362C22×C7#57511yescharacteristic
363C22×C7#57812yes—
364C4×C7—22no—
365C4×C7—41no—
366C4×C7—42no—
367C4×C7—41no—
368C4×C7—42no—
369C4×C7—42no—
370C4×C7—41no—
371C4×C7—42no—
372C4×C7—82no—
373C4×C7—82no—
374C4×C7#52511yescharacteristic
375C4×C7#53611yescharacteristic
376C4×C7#54712yes—
377C24C2×C7×D411yescharacteristic
378C24C2×C7×D411yescharacteristic
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8C2×C7×D412yes—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—41no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4C2×C7×D411yescharacteristic
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C2×C7×D412yes—
409C22×C4C2×C7×D412yes—
410C22×C4C24×C711yescharacteristic
411M16—28no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8C7×#4114yes—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4C2×C7×D412yes—
435C4⋊C4C2×C7×D412yes—
436C4⋊C4C2×C7×D412yes—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—42no—
443C22⋊C4—44no—
444C42—24no—
445C42—24no—
446C42—21no—
447C42—21no—
448C2×C7—42no—
449C2×C7—42no—
450C2×C7#240611yescharacteristic
451C2×C7#300611yescharacteristic
452C2×C7#301111yescharacteristic
453C2×C7#334312yes—
454C2×C7#335212yes—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23C7×#12012yes—
464C23C7×AGL(1,ℤ/8)12yes—
465C23C22×C7×D411yescharacteristic
466Q8—24no—
467Q8—41no—
468Q8—41no—
469D4—44no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4C7×#12012yes—
503C2×C4C22×C7×D411yescharacteristic
504C2×C4C22×C7×D411yescharacteristic
505C2×C4C2×C7×#4112yes—
506C2×C4C7×2−1+412yes—
507C8—24no—
508C8—42no—
509C7#2411011yescharacteristic, direct product factor, Sylow
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—41no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22C2×C7×#12011yescharacteristic
521C22C7×#54712yes—
522C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
523C22C7×#57511yescharacteristic
524C22C7×#57812yes—
525C4—22no—
526C4—41no—
527C4—42no—
528C4—41no—
529C4—42no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—82no—
534C4—82no—
535C4C7×#52511yescharacteristic
536C4C7×#53611yescharacteristic
537C4C7×#54712yes—
538C2—42no—
539C2—42no—
540C2C7×#240611yescharacteristic
541C2C7×#300611yescharacteristic
542C2C7×#301111yescharacteristic
543C2C7×#334312yes—
544C2C7×#335212yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#484C2semidirect
3. C2×C7×#428C2semidirect
5. C7×#2059C2semidirect
6. C7×#2035C2semidirect
7. C7×#1894C2semidirect
9. C7×#1863C2semidirect
10. C7×#1852C2semidirect
11. C7×#1825C2semidirect
47. #24110C7direct (N × H)
114. C2×#484C2×C7semidirect
115. C2×#428C2×C7semidirect
117. #2059C2×C7semidirect
118. #2035C2×C7semidirect
119. #1894C2×C7semidirect
121. #1863C2×C7semidirect
122. #1852C2×C7semidirect
123. #1825C2×C7semidirect
509. C7#24110direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5522
C2×C7×#4842
C2×C7×#4282
C7×#24862
C7×#20592
C7×#20352
C7×#18942
C7×#18702
C7×#18632
C7×#18522
C7×#18252
#241107