Finite Group Databasebeta

← C7×#24018 #13201234

Subgroups — C7×#24018 #13201234

Order 1792.

1578 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3484C211yesmaximal, characteristic
2C7×#3092C212yesmaximal, complemented
3C2×C7×#390C211yesmaximal, characteristic, complemented
4C7×#2037C214yesmaximal, complemented
5C7×#1896C212yesmaximal, complemented
6C7×#1856C212yesmaximal, complemented
7C7×#1792C212yesmaximal, complemented
8C7×#1783C211yesmaximal, characteristic, complemented
9C2×C7×H(5,𝔽2)C2211yescharacteristic
10C23×C7×D4C2211yescharacteristic
11C7×#546—22no—
12C7×#546—22no—
13C7×D42—22no—
14C7×D42—22no—
15C7×#535C2212yes—
16C7×#534—22no—
17C7×#525C2212yes—
18C2×C7×#120—22no—
19C7×#518C2211yescharacteristic
20C22×C7×C4⋊C4C2211yescharacteristic, complemented
21C22×C7×C22⋊C4—22no—
22C22×C42×C7C2211yescharacteristic, complemented
23C7×#484—22no—
24C7×#483—22no—
25C7×#481—22no—
26C7×#480—22no—
27C7×#428C2214yescomplemented
28C2×C7×#104C2214yescomplemented
29C7×#417C2214yescomplemented
30C2×C7×#100C2212yes—
31C2×C7×#100C2212yescomplemented
32C7×#407C2212yescomplemented
33C7×#390—22no—
34C7×#390C2212yes—
35C7×#390C2212yescomplemented
36C7×#339C2214yescomplemented
37#24018C711yesmaximal, characteristic, direct product factor, Sylow
38C25×C7D412yes—
39C7×H(5,𝔽2)—24no—
40C7×H(5,𝔽2)—42no—
41C2×C7×#41—22no—
42C2×C7×#41C2312yes—
43C22×C7×D4—22no—
44C22×C7×D4—22no—
45C22×C7×D4—22no—
46C22×C7×D4—22no—
47C22×C7×D4—24no—
48C22×C7×D4—22no—
49C22×C7×D4—22no—
50C22×C7×D4—42no—
51C22×C7×D4D412yes—
52C22×C7×D4C2311yescharacteristic
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7—21no—
56C23×C4×C7C2311yescharacteristic
57C2×C7×M16C2314yes—
58C7×#127—22no—
59C7×#124—22no—
60C7×#123—24no—
61C7×#123—22no—
62C7×C22⋊Q8D412yes—
63C7×C22⋊Q8D412yes—
64C7×#121—24no—
65C7×#121—24no—
66C7×#121—22no—
67C7×#121—22no—
68C7×#121D412yes—
69C7×#121D412yes—
70C7×#120—24no—
71C7×#120—24no—
72C7×#120—24no—
73C7×#120—22no—
74C7×#120—24no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C4×C7×D4—22no—
79C7×#117C2312yes—
80C2×C7×C4⋊C4—22no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C22⋊C4—22no—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—22no—
87C2×C7×C22⋊C4—24no—
88C2×C7×C22⋊C4—24no—
89C2×C7×C22⋊C4—48no—
90C2×C42×C7—21no—
91C2×C42×C7—42no—
92C2×C42×C7C2312yes—
93C7×#107—24no—
94C7×#106—24no—
95C7×#104—44no—
96C7×#103—24no—
97C7×#102—24no—
98C7×#100—24no—
99C7×#100—42no—
100C7×C22⋊C8—22no—
101C7×C22⋊C8—22no—
102#3484C2×C711yescharacteristic
103#3092C2×C712yescomplemented
104C2×#390C2×C711yescharacteristic, complemented
105#2037C2×C714yescomplemented
106#1896C2×C712yescomplemented
107#1856C2×C712yescomplemented
108#1792C2×C712yescomplemented
109#1783C2×C711yescharacteristic, complemented
110C24×C7—22no—
111C24×C7—24no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—42no—
115C24×C7—44no—
116C24×C7—44no—
117C24×C7C2×D411yescharacteristic
118C24×C7C2×D412yes—
119C24×C7C2×D412yes—
120C7×#41—24no—
121C7×#41—24no—
122C7×#41—48no—
123C2×C7×Q8C2×D412yes—
124C2×C7×D4—24no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—24no—
128C2×C7×D4—22no—
129C2×C7×D4—24no—
130C2×C7×D4—24no—
131C2×C7×D4—24no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—48no—
138C2×C7×D4—48no—
139C2×C7×D4—42no—
140C2×C7×D4—48no—
141C2×C7×D4—44no—
142C2×C7×D4—44no—
143C2×C7×D4—84no—
144C2×C7×D4—84no—
145C2×C7×D4C2×D412yes—
146C2×C7×D4#4114yes—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—24no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—44no—
159C22×C4×C7—41no—
160C22×C4×C7—41no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2411yescharacteristic
165C7×M16—24no—
166C7×M16—24no—
167C2×C7×C8—24no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4C2×D412yes—
173C7×C4⋊C4C2×D412yes—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—24no—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—48no—
183C7×C22⋊C4—48no—
184C42×C7—22no—
185C42×C7—24no—
186C42×C7—42no—
187C2×H(5,𝔽2)C22×C711yescharacteristic
188C23×D4C22×C711yescharacteristic
189#546—22no—
190#546—22no—
191D42—22no—
192D42—22no—
193#535C22×C712yes—
194#534—22no—
195#525C22×C712yes—
196C2×#120—22no—
197#518C22×C711yescharacteristic
198C22×C4⋊C4C22×C711yescharacteristic, complemented
199C22×C22⋊C4—22no—
200C22×C42C22×C711yescharacteristic, complemented
201#484—22no—
202#483—22no—
203#481—22no—
204#480—22no—
205#428C22×C714yescomplemented
206C2×#104C22×C714yescomplemented
207#417C22×C714yescomplemented
208C2×#100C22×C712yes—
209C2×#100C22×C712yescomplemented
210#407C22×C712yescomplemented
211#390—22no—
212#390C22×C712yes—
213#390C22×C712yescomplemented
214#339C22×C714yescomplemented
215C23×C7—21no—
216C23×C7—21no—
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—22no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—44no—
226C23×C7—44no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—416no—
232C23×C7—44no—
233C23×C7—44no—
234C23×C7—44no—
235C23×C7—82no—
236C23×C7—82no—
237C23×C7—88no—
238C23×C7—82no—
239C23×C7—82no—
240C23×C7#12011yescharacteristic
241C23×C7#12011yescharacteristic
242C23×C7#12314yes—
243C23×C7C22×D411yescharacteristic
244C7×Q8—24no—
245C7×D4—28no—
246C7×D4—24no—
247C7×D4—48no—
248C7×D4—48no—
249C7×D4—42no—
250C7×D4—42no—
251C7×D4—44no—
252C7×D4—44no—
253C7×D4—44no—
254C7×D4—84no—
255C7×D4—84no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—48no—
265C2×C4×C7—42no—
266C2×C4×C7—44no—
267C2×C4×C7—42no—
268C2×C4×C7—44no—
269C2×C4×C7—44no—
270C2×C4×C7—42no—
271C2×C4×C7—82no—
272C2×C4×C7—82no—
273C2×C4×C7—82no—
274C2×C4×C7—82no—
275C2×C4×C7#12012yes—
276C2×C4×C7C22×D412yes—
277C2×C4×C7C2×#4112yes—
278C2×C4×C7H(5,𝔽2)12yes—
279C7×C8—44no—
280C25C7×D412yes—
281H(5,𝔽2)—24no—
282H(5,𝔽2)—42no—
283C2×#41—22no—
284C2×#41C23×C712yes—
285C22×D4—22no—
286C22×D4—22no—
287C22×D4—22no—
288C22×D4—22no—
289C22×D4—24no—
290C22×D4—22no—
291C22×D4—22no—
292C22×D4—42no—
293C22×D4C7×D412yes—
294C22×D4C23×C711yescharacteristic
295C22×D4C23×C711yescharacteristic
296C23×C4—21no—
297C23×C4—21no—
298C23×C4C23×C711yescharacteristic
299C2×M16C23×C714yes—
300#127—22no—
301#124—22no—
302#123—24no—
303#123—22no—
304C22⋊Q8C7×D412yes—
305C22⋊Q8C7×D412yes—
306#121—24no—
307#121—24no—
308#121—22no—
309#121—22no—
310#121C7×D412yes—
311#121C7×D412yes—
312#120—24no—
313#120—24no—
314#120—24no—
315#120—22no—
316#120—24no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321#117C23×C712yes—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C23×C712yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—48no—
332C2×C42—21no—
333C2×C42—42no—
334C2×C42C23×C712yes—
335#107—24no—
336#106—24no—
337#104—44no—
338#103—24no—
339#102—24no—
340#100—24no—
341#100—42no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344C22×C7—24no—
345C22×C7—24no—
346C22×C7—21no—
347C22×C7—21no—
348C22×C7—21no—
349C22×C7—21no—
350C22×C7—21no—
351C22×C7—21no—
352C22×C7—24no—
353C22×C7—44no—
354C22×C7—44no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—82no—
364C22×C7—88no—
365C22×C7—88no—
366C22×C7—82no—
367C22×C7—164no—
368C22×C7#45312yes—
369C22×C7#45512yes—
370C22×C7C2×#12011yescharacteristic
371C22×C7C2×#12311yescharacteristic
372C22×C7#53411yescharacteristic
373C4×C7—22no—
374C4×C7—44no—
375C4×C7—42no—
376C4×C7—42no—
377C4×C7—42no—
378C4×C7—82no—
379C4×C7#54614yes—
380C24—22no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—42no—
385C24—44no—
386C24—44no—
387C24C2×C7×D411yescharacteristic
388C24C2×C7×D412yes—
389C24C2×C7×D412yes—
390#41—24no—
391#41—24no—
392#41—48no—
393C2×Q8C2×C7×D412yes—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—48no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4C2×C7×D412yes—
416C2×D4C7×#4114yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4C2×C7×D411yescharacteristic
432C22×C4C2×C7×D412yes—
433C22×C4C2×C7×D412yes—
434C22×C4C24×C711yescharacteristic
435M16—24no—
436M16—24no—
437C2×C8—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—42no—
442C4⋊C4C2×C7×D412yes—
443C4⋊C4C2×C7×D412yes—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C42—22no—
455C42—24no—
456C42—42no—
457C2×C7—21no—
458C2×C7—21no—
459C2×C7—22no—
460C2×C7—22no—
461C2×C7—44no—
462C2×C7—82no—
463C2×C7—82no—
464C2×C7#240511yescharacteristic
465C2×C7C2×#45311yescharacteristic
466C2×C7C2×#45511yescharacteristic
467C23—21no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—416no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—82no—
488C23—82no—
489C23—88no—
490C23—82no—
491C23—82no—
492C23C7×#12011yescharacteristic
493C23C7×#12011yescharacteristic
494C23C7×#12314yes—
495C23C22×C7×D411yescharacteristic
496Q8—24no—
497D4—28no—
498D4—24no—
499D4—48no—
500D4—48no—
501D4—42no—
502D4—42no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—84no—
507D4—84no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4C7×#12012yes—
528C2×C4C22×C7×D412yes—
529C2×C4C2×C7×#4112yes—
530C2×C4C7×H(5,𝔽2)12yes—
531C8—44no—
532C7#2401811yescharacteristic, direct product factor, Sylow
533C22—24no—
534C22—24no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—24no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—88no—
554C22—88no—
555C22—82no—
556C22—164no—
557C22C7×#45312yes—
558C22C7×#45512yes—
559C22C2×C7×#12011yescharacteristic
560C22C2×C7×#12311yescharacteristic
561C22C7×#53411yescharacteristic
562C4—22no—
563C4—44no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—82no—
568C4C7×#54614yes—
569C2—21no—
570C2—21no—
571C2—22no—
572C2—22no—
573C2—44no—
574C2—82no—
575C2—82no—
576C2C7×#240511yescharacteristic
577C2C2×C7×#45311yescharacteristic
578C2C2×C7×#45511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3484C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3092C2semidirect
3. C2×C7×#390C2semidirect
4. C7×#2037C2semidirect
5. C7×#1896C2semidirect
6. C7×#1856C2semidirect
7. C7×#1792C2semidirect
8. C7×#1783C2semidirect
20. C22×C7×C4⋊C4C22semidirect
22. C22×C42×C7C22semidirect
27. C7×#428C22semidirect
28. C2×C7×#104C22semidirect
29. C7×#417C22semidirect
31. C2×C7×#100C22semidirect
32. C7×#407C22semidirect
35. C7×#390C22semidirect
36. C7×#339C22semidirect
37. #24018C7direct (N × H)
103. #3092C2×C7semidirect
104. C2×#390C2×C7semidirect
105. #2037C2×C7semidirect
106. #1896C2×C7semidirect
107. #1856C2×C7semidirect
108. #1792C2×C7semidirect
109. #1783C2×C7semidirect
198. C22×C4⋊C4C22×C7semidirect
200. C22×C42C22×C7semidirect
205. #428C22×C7semidirect
206. C2×#104C22×C7semidirect
207. #417C22×C7semidirect
209. C2×#100C22×C7semidirect
210. #407C22×C7semidirect
213. #390C22×C7semidirect
214. #339C22×C7semidirect
532. C7#24018direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34842
C7×#30922
C2×C7×#3902
C7×#20372
C7×#18962
C7×#18562
C7×#17922
C7×#17832
#240187