Finite Group Databasebeta

← C7×#24058 #13201274

Subgroups — C7×#24058 #13201274

Order 1792.

1556 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3484C211yesmaximal, characteristic
2C7×#2622C212yesmaximal, complemented
3C7×#2308C211yesmaximal, characteristic, complemented
4C7×#2044C214yesmaximal, complemented
5C7×#1898C212yesmaximal, complemented
6C7×#1867C212yesmaximal, complemented
7C7×#1790C212yesmaximal, complemented
8C7×#1781C211yesmaximal, characteristic, complemented
9C2×C7×H(5,𝔽2)C2211yescharacteristic
10C23×C7×D4C2211yescharacteristic
11C7×#546—24no—
12C7×D42—24no—
13C7×#535—21no—
14C7×#534—21no—
15C7×#534C2212yes—
16C7×#525—21no—
17C2×C7×#120—21no—
18C2×C7×#120C2212yes—
19C7×#518C2211yescharacteristic
20C22×C7×C22⋊C4C2211yescharacteristic, complemented
21C22×C7×C22⋊C4C2211yescharacteristic, complemented
22C2×C7×#99C2212yes—
23C2×C7×#99C2212yescomplemented
24C2×C7×#99C2214yescomplemented
25C7×#397—24no—
26C7×#394—24no—
27C7×#386—21no—
28C7×#386—21no—
29C7×#386C2212yes—
30C7×#386C2212yescomplemented
31C7×#386C2214yescomplemented
32C7×#385C2214yescomplemented
33C7×#381—21no—
34C7×#379—21no—
35C7×#379C2212yescomplemented
36C7×#342C2214yescomplemented
37#24058C711yesmaximal, characteristic, direct product factor, Sylow
38C25×C7D411yescharacteristic
39C25×C7C2311yescharacteristic
40C7×H(5,𝔽2)—24no—
41C7×H(5,𝔽2)—42no—
42C2×C7×#41—21no—
43C2×C7×#41—22no—
44C22×C7×D4—24no—
45C22×C7×D4—21no—
46C22×C7×D4—22no—
47C22×C7×D4—22no—
48C22×C7×D4—24no—
49C22×C7×D4—42no—
50C22×C7×D4D411yescharacteristic
51C22×C7×D4D411yescharacteristic
52C22×C7×D4D414yes—
53C22×C7×D4C2311yescharacteristic
54C22×C7×D4C2312yes—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7D411yescharacteristic
58C23×C4×C7C2312yes—
59C7×#127—22no—
60C7×#124—22no—
61C7×#123—22no—
62C7×#123—22no—
63C7×#123—22no—
64C7×C22⋊Q8—22no—
65C7×#121—22no—
66C7×#121—24no—
67C7×#121—24no—
68C7×#121—24no—
69C7×#120—24no—
70C7×#120—22no—
71C7×#120—22no—
72C7×#120—22no—
73C7×#120—22no—
74C7×#120—24no—
75C7×#120—22no—
76C4×C7×D4—24no—
77C4×C7×D4—24no—
78C7×#117—21no—
79C2×C7×C4⋊C4—24no—
80C2×C7×C22⋊C4—22no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—24no—
86C2×C7×C22⋊C4—22no—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—24no—
89C2×C7×C22⋊C4—24no—
90C2×C7×C22⋊C4—42no—
91C2×C7×C22⋊C4D414yes—
92C2×C7×C22⋊C4C2312yes—
93C2×C7×C22⋊C4C2314yes—
94C2×C7×C22⋊C4C2312yes—
95C7×#99—24no—
96C7×#99—42no—
97C7×#99—44no—
98C7×#95—22no—
99C7×#95—22no—
100C7×#95—24no—
101C7×#95—24no—
102#3484C2×C711yescharacteristic
103#2622C2×C712yescomplemented
104#2308C2×C711yescharacteristic, complemented
105#2044C2×C714yescomplemented
106#1898C2×C712yescomplemented
107#1867C2×C712yescomplemented
108#1790C2×C712yescomplemented
109#1781C2×C711yescharacteristic, complemented
110C24×C7—22no—
111C24×C7—22no—
112C24×C7—21no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—41no—
116C24×C7—82no—
117C24×C7—82no—
118C24×C7C2×D411yescharacteristic
119C24×C7C2×D412yes—
120C24×C7C2×D414yes—
121C24×C7C2×D411yescharacteristic
122C24×C7C2×D411yescharacteristic
123C24×C7C2×D411yescharacteristic
124C24×C7C2411yescharacteristic
125C7×#41—48no—
126C7×#41—42no—
127C7×#41—42no—
128C2×C7×Q8—21no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—21no—
133C2×C7×D4—24no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—24no—
142C2×C7×D4—22no—
143C2×C7×D4—24no—
144C2×C7×D4—24no—
145C2×C7×D4—24no—
146C2×C7×D4—22no—
147C2×C7×D4—41no—
148C2×C7×D4—41no—
149C2×C7×D4—416no—
150C2×C7×D4—48no—
151C2×C7×D4—42no—
152C2×C7×D4—42no—
153C2×C7×D4—84no—
154C2×C7×D4—84no—
155C2×C7×D4#4114yes—
156C22×C4×C7—21no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—24no—
163C22×C4×C7—22no—
164C22×C4×C7—24no—
165C22×C4×C7—41no—
166C22×C4×C7—42no—
167C22×C4×C7—82no—
168C22×C4×C7—82no—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D414yes—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—84no—
176C7×C22⋊C4—22no—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—24no—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—42no—
187C7×C22⋊C4—42no—
188C7×C22⋊C4—42no—
189C7×C22⋊C4—84no—
190C7×C22⋊C4—84no—
191C7×C22⋊C4—82no—
192C7×C22⋊C4—82no—
193C42×C7—22no—
194C2×H(5,𝔽2)C22×C711yescharacteristic
195C23×D4C22×C711yescharacteristic
196#546—24no—
197D42—24no—
198#535—21no—
199#534—21no—
200#534C22×C712yes—
201#525—21no—
202C2×#120—21no—
203C2×#120C22×C712yes—
204#518C22×C711yescharacteristic
205C22×C22⋊C4C22×C711yescharacteristic, complemented
206C22×C22⋊C4C22×C711yescharacteristic, complemented
207C2×#99C22×C712yes—
208C2×#99C22×C714yescomplemented
209C2×#99C22×C712yescomplemented
210#397—24no—
211#394—24no—
212#386—21no—
213#386—21no—
214#386C22×C712yes—
215#386C22×C712yescomplemented
216#386C22×C714yescomplemented
217#385C22×C714yescomplemented
218#381—21no—
219#379—21no—
220#379C22×C712yescomplemented
221#342C22×C714yescomplemented
222C23×C7—21no—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—21no—
226C23×C7—21no—
227C23×C7—21no—
228C23×C7—28no—
229C23×C7—21no—
230C23×C7—22no—
231C23×C7—21no—
232C23×C7—21no—
233C23×C7—24no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—44no—
238C23×C7—44no—
239C23×C7—44no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—44no—
243C23×C7—42no—
244C23×C7—44no—
245C23×C7—82no—
246C23×C7—84no—
247C23×C7—84no—
248C23×C7—84no—
249C23×C7—82no—
250C23×C7—84no—
251C23×C7#12012yes—
252C23×C7#12012yes—
253C23×C7#12312yes—
254C23×C7C22×D411yescharacteristic
255C23×C7C22×D412yes—
256C23×C7C2×#4112yes—
257C23×C7H(5,𝔽2)12yes—
258C7×Q8—42no—
259C7×D4—28no—
260C7×D4—44no—
261C7×D4—42no—
262C7×D4—44no—
263C7×D4—44no—
264C7×D4—48no—
265C7×D4—42no—
266C7×D4—42no—
267C7×D4—44no—
268C7×D4—44no—
269C7×D4—84no—
270C7×D4—84no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—24no—
274C2×C4×C7—22no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—24no—
279C2×C4×C7—44no—
280C2×C4×C7—44no—
281C2×C4×C7—42no—
282C2×C4×C7—44no—
283C2×C4×C7—44no—
284C2×C4×C7—44no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—44no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—44no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—44no—
294C2×C4×C7—82no—
295C2×C4×C7—84no—
296C2×C4×C7#12312yes—
297C25C7×D411yescharacteristic
298C25C23×C711yescharacteristic
299H(5,𝔽2)—24no—
300H(5,𝔽2)—42no—
301C2×#41—21no—
302C2×#41—22no—
303C22×D4—21no—
304C22×D4—22no—
305C22×D4—24no—
306C22×D4—22no—
307C22×D4—24no—
308C22×D4—42no—
309C22×D4C7×D411yescharacteristic
310C22×D4C7×D411yescharacteristic
311C22×D4C7×D414yes—
312C22×D4C23×C712yes—
313C22×D4C23×C711yescharacteristic
314C22×D4C23×C711yescharacteristic
315C23×C4—21no—
316C23×C4C7×D411yescharacteristic
317C23×C4C23×C712yes—
318#127—22no—
319#124—22no—
320#123—22no—
321#123—22no—
322#123—22no—
323C22⋊Q8—22no—
324#121—22no—
325#121—24no—
326#121—24no—
327#121—24no—
328#120—24no—
329#120—22no—
330#120—22no—
331#120—22no—
332#120—22no—
333#120—24no—
334#120—22no—
335C4×D4—24no—
336C4×D4—24no—
337#117—21no—
338C2×C4⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4C7×D414yes—
351C2×C22⋊C4C23×C712yes—
352C2×C22⋊C4C23×C714yes—
353C2×C22⋊C4C23×C712yes—
354#99—24no—
355#99—42no—
356#99—44no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361C22×C7—24no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—22no—
365C22×C7—22no—
366C22×C7—22no—
367C22×C7—24no—
368C22×C7—22no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—41no—
380C22×C7—44no—
381C22×C7—42no—
382C22×C7—81no—
383C22×C7—88no—
384C22×C7—84no—
385C22×C7—81no—
386C22×C7—88no—
387C22×C7—162no—
388C22×C7C2×#12011yescharacteristic
389C22×C7C2×#12311yescharacteristic
390C22×C7#53411yescharacteristic
391C22×C7#54614yes—
392C4×C7—22no—
393C4×C7—41no—
394C4×C7—42no—
395C4×C7—44no—
396C4×C7—84no—
397C4×C7—82no—
398C4×C7—82no—
399C24—22no—
400C24—22no—
401C24—21no—
402C24—22no—
403C24—22no—
404C24—41no—
405C24—82no—
406C24—82no—
407C24C2×C7×D411yescharacteristic
408C24C2×C7×D412yes—
409C24C2×C7×D414yes—
410C24C2×C7×D411yescharacteristic
411C24C2×C7×D411yescharacteristic
412C24C2×C7×D411yescharacteristic
413C24C24×C711yescharacteristic
414#41—48no—
415#41—42no—
416#41—42no—
417C2×Q8—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—416no—
439C2×D4—48no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4C7×#4114yes—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4C2×C7×D412yes—
459C22×C4C2×C7×D411yescharacteristic
460C22×C4C2×C7×D411yescharacteristic
461C22×C4C2×C7×D414yes—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—84no—
465C22⋊C4—22no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C42—22no—
483C2×C7—21no—
484C2×C7—24no—
485C2×C7—21no—
486C2×C7—42no—
487C2×C7—42no—
488C2×C7—42no—
489C2×C7—81no—
490C2×C7—82no—
491C2×C7#240511yescharacteristic
492C2×C7#302612yes—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—28no—
500C23—21no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—44no—
516C23—82no—
517C23—84no—
518C23—84no—
519C23—84no—
520C23—82no—
521C23—84no—
522C23C7×#12012yes—
523C23C7×#12012yes—
524C23C7×#12312yes—
525C23C22×C7×D411yescharacteristic
526C23C22×C7×D412yes—
527C23C2×C7×#4112yes—
528C23C7×H(5,𝔽2)12yes—
529Q8—42no—
530D4—28no—
531D4—44no—
532D4—42no—
533D4—44no—
534D4—44no—
535D4—48no—
536D4—42no—
537D4—42no—
538D4—44no—
539D4—44no—
540D4—84no—
541D4—84no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—82no—
566C2×C4—84no—
567C2×C4C7×#12312yes—
568C7#2405811yescharacteristic, direct product factor, Sylow
569C22—24no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—24no—
576C22—22no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—44no—
589C22—42no—
590C22—81no—
591C22—88no—
592C22—81no—
593C22—84no—
594C22—88no—
595C22—162no—
596C22C2×C7×#12011yescharacteristic
597C22C2×C7×#12311yescharacteristic
598C22C7×#53411yescharacteristic
599C22C7×#54614yes—
600C4—22no—
601C4—41no—
602C4—42no—
603C4—44no—
604C4—82no—
605C4—84no—
606C4—82no—
607C2—21no—
608C2—24no—
609C2—21no—
610C2—42no—
611C2—42no—
612C2—42no—
613C2—81no—
614C2—82no—
615C2C7×#240511yescharacteristic
616C2C7×#302612yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC7×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C25×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2622C2semidirect
3. C7×#2308C2semidirect
4. C7×#2044C2semidirect
5. C7×#1898C2semidirect
6. C7×#1867C2semidirect
7. C7×#1790C2semidirect
8. C7×#1781C2semidirect
20. C22×C7×C22⋊C4C22semidirect
21. C22×C7×C22⋊C4C22semidirect
23. C2×C7×#99C22semidirect
24. C2×C7×#99C22semidirect
30. C7×#386C22semidirect
31. C7×#386C22semidirect
32. C7×#385C22semidirect
35. C7×#379C22semidirect
36. C7×#342C22semidirect
37. #24058C7direct (N × H)
103. #2622C2×C7semidirect
104. #2308C2×C7semidirect
105. #2044C2×C7semidirect
106. #1898C2×C7semidirect
107. #1867C2×C7semidirect
108. #1790C2×C7semidirect
109. #1781C2×C7semidirect
205. C22×C22⋊C4C22×C7semidirect
206. C22×C22⋊C4C22×C7semidirect
208. C2×#99C22×C7semidirect
209. C2×#99C22×C7semidirect
215. #386C22×C7semidirect
216. #386C22×C7semidirect
217. #385C22×C7semidirect
220. #379C22×C7semidirect
221. #342C22×C7semidirect
568. C7#24058direct (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×#26222
C7×#23082
C7×#20442
C7×#18982
C7×#18672
C7×#17902
C7×#17812
#240587